# A Cooperative Coevolutionary Cuckoo Search Algorithm for Optimization Problem

Authors: Hongqing Zheng; Yongquan Zhou
Journal: Journal of Applied Mathematics
 (2013)

---

## Abstract

Taking inspiration from an organizational evolutionary algorithm for numerical optimization, this paper designs a kind of dynamic population and combining evolutionary operators to form a novel algorithm, a cooperative coevolutionary cuckoo search algorithm (CCCS), for solving both unconstrained, constrained optimization and engineering problems. A population of this algorithm consists of organizations, and an organization consists of dynamic individuals. In experiments, fifteen unconstrained functions, eleven constrained functions, and two engineering design problems are used to validate the performance of CCCS, and thorough comparisons are made between the CCCS and the existing approaches. The results show that the CCCS obtains good performance in the solution quality. Moreover, for the constrained problems, the good performance is obtained by only incorporating a simple constraint handling technique into the CCCS. The results show that the CCCS is quite robust and easy to use.


---

## Body

## 1. Introduction

High dimension numerical optimization problems tend to be complex, and general basic intelligent algorithms are difficult to obtain the global optimal solution. In order to solve this problem, many improved methods are put forward, such as evolutionary programming made faster [1], orthogonal genetic algorithm [2], and good point set based genetic algorithm [3], and these methods have achieved good effect.Recently, a novel heuristic search algorithm, called Cuckoo Search (CS) in  [4],   has been proposed by Yang and Deb in 2009. The CS is a search algorithm based on the interesting breeding behavior such as brood parasitism of certain species of cuckoos. Each nest within the swarm is represented by a vector in multidimensional search space; the CS algorithm also determines how to update the position of cuckoo laid egg. Each cuckoo updates it position of lay egg based on current step size via Lévy flights. It has been shown that this simple model has been applied successfully to continuous nonlinear function, engineering optimization problem in [5], and so forth. Intelligent algorithm is based on the selection of the fittest in biological systems which have evolved by natural selection over millions of years, between organisms in nature not only competition but also cooperation. Potter et al. earlier proposed a cooperative coevolutionary genetic algorithm to function optimization (CCGA) [6], and Bergh et al. apply this idea to the standard particle group Algorithm to construct a new collaborative model (CPSO-SK) [7]. Liu et al. proposed an organizational evolutionary algorithm for numerical optimization [8], Mu et al. put forward M-elite coevolutionary algorithm for numerical optimization [9], and  Fister et al. proposed memetic artificial bee colony algorithm for large-scale global optimization [10]. They have obtained the good effect in the numerical optimization problem. Based on this idea and combining evolutionary operators, this paper proposes  a new algorithm of solving high-dimensional unconstrained, constrained, and engineering optimization problem, namely, a cooperative coevolutionary cuckoo search algorithm (CCCS) algorithm. Population of the algorithm is divided into M groups, each group has a leader, by annexation and collaborative operation between different organizations, and uses the cuboids crossover operator, discrete crossover operator, flip crossover operator, and mutation operator to achieve the exchange of information between individuals, to promote the evolution of the population. Simulation experiments show that CCCS optimization ability is very strong, can well solve the unconstrained optimization, constrained optimization, and engineering optimization problems and so on.The remainder of this paper is organized as follows: Section2 briefly introduces the original cuckoo search algorithm. This is followed in Section 3 by new cooperative coevolutionary implements of the CS algorithm. Section 4 describes the definition of a constrained optimization and a unconstrained optimization problem in the penalty function approach. The results can be found and discussed in Section 5. Finally, some directions for future research are discussed in Section 6.


## 2. Original CS

CS is a heuristic search algorithm which has been proposed recently by Yang and Deb. The algorithm is inspired by the reproduction strategy of cuckoos. At the most basic level, cuckoos lay their eggs in the nests of other host birds, which may be of different species. The host bird may discover that the eggs are not it’s own and either destroy the egg or abandon the nest all together. This has resulted in the evolution of cuckoo eggs which mimic the eggs of local host birds. To apply this as an optimization tool, Yang and Deb [4] used three ideal rules.(1)
Each cuckoo lays one egg, which represents a set of solution coordinates, at a time, and dumps it in a random nest.(2)
A fraction of the nests containing the best eggs, or solutions, will carry over to the next generation.(3)
The number of nests is fixed, and there is a probability that a host can discover an alien   egg. If this happens, the host can either discard the egg or the nest, and this results in building a new nest in a new location.This algorithm uses a balanced combination of a local random walk and the global explorative random walk, controlled by a switching parameterpa. The local random walk can be written as
(1)xit+1=xit+∂s⊗H(pa-ε)⊗(xjt-xkt),
where xjt and xkt are two different solutions selected randomly by random permutation, H(u) is a Heaviside function, ε is a random number drawn from a uniform distribution, and s is the step size. On the other hand, the global random walk is carried out by using Lévy flights
(2)xit+1=xit+∂L(s,λ),
where
(3)L(s,λ)=λΓ(λ)sin(πλ/2)π1s1+λ,(s≫s0>0).
Here ∂>0 is the step-size-scaling factor, which should be related to the scales of the problem of interests. In most cases, we can use ∂=O(L/10), and ∂=O(L/100) can be more effective and can avoid flying too far.


## 3. A Cooperative Coevolutionary Cuckoo Search Algorithm

In section, taking inspiration from an organizational evolutionary algorithm, we present a cooperative coevolutionary cuckoo search algorithm (CCCS) which integers annexing operator and cooperating operator, in the core the cuckoo search algorithm. This proposed model will focus on enhancing diversity and the performance of the cuckoo search algorithm.


### 3.1. Splitting Operator

When a size is too large usually it is split into several small organizations; letMaxor be the parameter controlling the maximum size of organization.


### 3.2. Annexing Operator

Two organizations,orgp1={x1,x2,…,xM} and orgp2={y1,y2,…,yN}, are randomly selected from the current generation. Choose their leaders using CS algorithm. If the orgp1 is the winner; thus orgp1 will annex orgp2 to generate a new organization,  orgc={z1,z2,…,zM,zM+1,zM+2,…zM+N}, where zi=xi, i=1,2,…,M. Let AS be a predefined parameter. Then, if rand<AS, zj, j=M+1,M+2,…,M+N are generated by (4). Otherwise they are generated by (5). xk is the leader of an organization and rjk  are   new member
(4)rjk={xk_βk<xk_,xk¯βk>xk¯,βk=xk+∂k×(xk-yjk),k=1,2,…,n,βkotherwise,(5)rjk={xk_+∂×(xk¯-xk_)βk(0,1)<1n,xkotherwise.
After rj, j=1,2,…N are generated, zj+M, j=1,2,…,M are determined in (6):
(6)zj+M={rjrj<yj,rj(yj>rj),{Uj(0,1)}<exp(f(rj)-f(y)),yjotherwise.


### 3.3. Cooperating Operator

Two organizations,orgp1={x1,x2,…,xM} and orgp2={y1,y2,…,yN}, are randomly selected from the current generation. Let CS∈(0,1) be a predefined parameter; if  rand<CS, the child organization is generated in (7). xk, yk  are the leader   of organization respectively. Otherwise use (8); i is random integer and uses flip operator;
(7)qk=∂k×xk+(1-∂k)×yk,rk=(1-∂k)×xk+∂k×yk,k=1,2,…,n,(8)


### 3.4. The Pseudo Code of the Proposed Algorithm is Shown in Algorithm1

Algorithm 1:A cooperative coevolutionary cuckoo search algorithm (minimum).
Begin
Initializing populationp0 with n0 organizations, and each organization has one member;
t←0;
While (the termination criteria are not reached) do
Begin
For each organization inpt, if the number of it more than 20, performing the splitting
operator on it, deleting it frompt, and adding the child organizations into pt+1;
While (the number of organizations in is pt greater than 1) do
Begin
Randomly selecting two parent organizationsorgp1 and orgp2 from pt;
Performing the CS and selecting their leaders;
If rand < 0.5
Annexing operator;
Else
Cooperating operator;
Iff(w)<f(x) and f(z)<f(y)
x=w;
y=z;
End
adding the child organizations intopt+1;
End
Deletingorgp1and orgp2form pt;
End
Deleting theu organizations form pt+1;
%u is the child number of join organizations
pt←p(t+1);
t←t+1;
End
output the best solution inpt
EndIn the initialization, each organization has only one member, and the population has totaln0 organizations. During the evolutionary process, the number of the organizations changes; this is just to maintain the diversity of the population. The main difference between the CCCS and the OEA is that populations changes during the optimization process, in OEA, the number of populations in the optimization process is the same. In contrast, the number of the population in CCCS is changing. In addition, cooperating operators of CCCS and OEA are also different.


## 3.1. Splitting Operator

When a size is too large usually it is split into several small organizations; letMaxor be the parameter controlling the maximum size of organization.


## 3.2. Annexing Operator

Two organizations,orgp1={x1,x2,…,xM} and orgp2={y1,y2,…,yN}, are randomly selected from the current generation. Choose their leaders using CS algorithm. If the orgp1 is the winner; thus orgp1 will annex orgp2 to generate a new organization,  orgc={z1,z2,…,zM,zM+1,zM+2,…zM+N}, where zi=xi, i=1,2,…,M. Let AS be a predefined parameter. Then, if rand<AS, zj, j=M+1,M+2,…,M+N are generated by (4). Otherwise they are generated by (5). xk is the leader of an organization and rjk  are   new member
(4)rjk={xk_βk<xk_,xk¯βk>xk¯,βk=xk+∂k×(xk-yjk),k=1,2,…,n,βkotherwise,(5)rjk={xk_+∂×(xk¯-xk_)βk(0,1)<1n,xkotherwise.
After rj, j=1,2,…N are generated, zj+M, j=1,2,…,M are determined in (6):
(6)zj+M={rjrj<yj,rj(yj>rj),{Uj(0,1)}<exp(f(rj)-f(y)),yjotherwise.


## 3.3. Cooperating Operator

Two organizations,orgp1={x1,x2,…,xM} and orgp2={y1,y2,…,yN}, are randomly selected from the current generation. Let CS∈(0,1) be a predefined parameter; if  rand<CS, the child organization is generated in (7). xk, yk  are the leader   of organization respectively. Otherwise use (8); i is random integer and uses flip operator;
(7)qk=∂k×xk+(1-∂k)×yk,rk=(1-∂k)×xk+∂k×yk,k=1,2,…,n,(8)


## 3.4. The Pseudo Code of the Proposed Algorithm is Shown in Algorithm1

Algorithm 1:A cooperative coevolutionary cuckoo search algorithm (minimum).
Begin
Initializing populationp0 with n0 organizations, and each organization has one member;
t←0;
While (the termination criteria are not reached) do
Begin
For each organization inpt, if the number of it more than 20, performing the splitting
operator on it, deleting it frompt, and adding the child organizations into pt+1;
While (the number of organizations in is pt greater than 1) do
Begin
Randomly selecting two parent organizationsorgp1 and orgp2 from pt;
Performing the CS and selecting their leaders;
If rand < 0.5
Annexing operator;
Else
Cooperating operator;
Iff(w)<f(x) and f(z)<f(y)
x=w;
y=z;
End
adding the child organizations intopt+1;
End
Deletingorgp1and orgp2form pt;
End
Deleting theu organizations form pt+1;
%u is the child number of join organizations
pt←p(t+1);
t←t+1;
End
output the best solution inpt
EndIn the initialization, each organization has only one member, and the population has totaln0 organizations. During the evolutionary process, the number of the organizations changes; this is just to maintain the diversity of the population. The main difference between the CCCS and the OEA is that populations changes during the optimization process, in OEA, the number of populations in the optimization process is the same. In contrast, the number of the population in CCCS is changing. In addition, cooperating operators of CCCS and OEA are also different.


## 4. Problem Definition

A unconstrained optimization problems (UCOPs) are formulated as solving the objective function(9)minimizef(x),x=(x1,x2,…,xn)∈s,
where s⊆ℜn defines the search space which is an n-dimensional space bounded by the parametric constraints xi_≤xi≤xi¯, i=1,2,…,n. Thus, s=[x_,x¯], where x_=(x1_,x2_,…,xn_) and   x¯=(x1¯,x2¯,…,xn¯).A constrained optimization problems (COPs) can be formulated as solving objective function(10)minimizef(x),x=(x1,x2,…,xn)∈s∩χ,
where s is the same with that of (9), and the feasible region χ is
(11)χ={x∈ℜn∣gj(x)≤0,j=1,2,…,m},
where gj(x), j=1,2,…,m are constraints.


### 4.1. Constraint Handling

In the penalty function approach, nonlinear constraints can be collapsed with the cost function into a response functional. By doing this, the constrained optimization problem is transformed into an unconstrained optimization problem simpler to solve [35]. For example, if there are some nonlinear equality constraints ϕi and some inequality constraints φj, the response functional PI can be defined as follows:
(12)∏(x,μi,vj)=f(x)+∑i=1Mμiϕi2(x)+∑j=1Nvjφj2(x),
where 1≤μi and 0≤vi. The coefficients of penalty terms should be large enough; their values may depend on the specific optimization problem. The contribution of any equality constraints function to the response functional ∏ is null but increases significantly as soon as the constraint is violated. The same applies to inequality constraints when they become critical.


## 4.1. Constraint Handling

In the penalty function approach, nonlinear constraints can be collapsed with the cost function into a response functional. By doing this, the constrained optimization problem is transformed into an unconstrained optimization problem simpler to solve [35]. For example, if there are some nonlinear equality constraints ϕi and some inequality constraints φj, the response functional PI can be defined as follows:
(12)∏(x,μi,vj)=f(x)+∑i=1Mμiϕi2(x)+∑j=1Nvjφj2(x),
where 1≤μi and 0≤vi. The coefficients of penalty terms should be large enough; their values may depend on the specific optimization problem. The contribution of any equality constraints function to the response functional ∏ is null but increases significantly as soon as the constraint is violated. The same applies to inequality constraints when they become critical.


## 5. Implementation in Optimization Problems

All computational experiments are conducted with Matlab R2010a and run on Celeron(R) Dual-core CPU T3100, 1.90 GHZ with 2 GB memory capacity under windows7.


### 5.1. Experimental Studies on Unconstrained Optimization Problems

In this section, 15 benchmark functions  (F01–F15)   are used to test the performance of CCCS in solving UCOPS. F01–F13 are f1–f13 in [1] and F14, F15 are f7, f9 in [36]. The problem dimension is set to 30 for F01–F13 and 100 for F14 and F15. In this manner, these functions have so many local minima that they are challenging enough for performance evaluation. The parameters of CCCS are set as follows: the number of organization is 10, others refer to [8], and the number of iterations is 2500.


#### 5.1.1. Experimental Results of CCCS

Table1 summarizes the experimental results of CCCS, which include the best, the mean, the standard deviation, and the worst function values found. As can be seen, besides F05, F14, other function values reached the theoretical value or are very close to the theoretical value. What should be noted is that the global optimum for F11 is 0 every time, but both MECA (M-elite co-evolutionary algorithm for numerical optimization) and OEA (An organizational evolutionary algorithm for numerical optimization) cannot find the global optimal solution. Convergence curve of F08 and F15 as shown in Figures 1 and 2 respectively, other convergence curve figure has been omitted.Table 1
Experimental result of CCCS on 15 unconstrained benchmark functions over 50 trails.
f
f
min
Best function value
Mean function value
Standard deviation
Worst function value
F
01
0
8.804
e
-
242
2.011
e
-
217
0
2.011
e
-
216
F
02
0
4.271
e
-
143
1.628
e
-
130
4.142
e
-
130
1.323
e
-
129
F
03
0
6.280
e
-
192
5.464
e
-
154
1.729
e
-
153
5.464
e
-
153
F
04
0
1.085
e
-
121
1.333
e
-
098
4.080
e
-
098
1.294
e
-
097
F
05
0
25.0366
25.4182
0.3013
25.8868
F
06
0
0
0
0
0
F
07
0
5.309
e
-
005
3.080
e
-
004
2.391
e
-
004
8.047
e
-
004
F
08
-
12569.5
-
1.25694870
e
+
004
-
1.25694867
e
+
004
4.830
e
-
004
-
1.25694860
e
+
004
F
09
0
0
0
0
0
F
10
0
1.655
e
-
7
3.663
e
-
5
2.654
e
-
9
3.188
e
-
4
F
11
0
0
0
0
0
F
12
0
6.845
e
-
011
9.518
e
-
007
2.780
e
-
006
8.635
e
-
006
F
13
0
2.073
e
-
013
1.723
e
-
007
2.657
e
-
007
6.775
e
-
007
F
14
-
99.60
-
96.6008
-
83.7328
6.1910
-
75.6993
F
15
-
78.33236
-
78.33233
-
78.33232
8.654
e
-
006
-
78.33230Figure 1
Convergence curve ofF08.Figure 2
Convergence curve ofF15.


#### 5.1.2. Comparison between CCCS, MECA, and OEA

Table2 shows statistical results of the CCCS optimization. As can be seen, as for functions F01, F02, F03, F04, F07, F08, and F11,  the mean function values of CCCS is better than MECA; as for functions F06 and F09, both CCCS and MECA can find the global optimal solution. It is a pity that the results of these functions (F05, F10, F12, F13, and F14) is worse than MECA.Table 2
Comparison between MECA, OEA, and CCCS over 50 trials.
f
f
min
Mean function value
Standard deviation
CCCS
MECA
OEA
CCCS
MECA
OEA
F
01
0
2.011e – 217
4.228
e
-
183
2.481
e
-
30
0
0
1.128
e
-
29
F
02
0
1.628e – 130
1.845
e
-
110
2.068
e
-
13
4.142e − 130
3.113
e
-
110
1.440
e
-
12
F
03
0
5.464e – 154
3.274
e
-
95
1.883
e
-
9
1.729e − 153
2.313
e
-
94
3.726
e
-
9
F
04
0
1.333e – 098
5.124
e
-
2
8.821
e
-
2
4.080e − 098
9.732
e
-
2
2.356
e
-
2
F
05
0
25.4182
7.973
e
-
2
0.227
0.3013
5.638
e
-
1
0.941
F
06
0
0
0
0
0
0
0
F
07
0
3.080e – 004
4.084
e
-
4
3.297
e
-
3
2.391e − 004
3.800
e
-
4
1.096
e
-
3
F
08
-
12569.5
−12569.4867
-
12569.4866
-
12
569.4866
4.830e − 004
7.350
e
-
12
5.555
e
-
12
F
09
0
0
0
5.430
e
-
17
0
0
1.683
e
-
16
F
10
0
3.663e− 5
0
5.336
e
-
14
2.654e − 9
0
2.945
e
-
13
F
11
0
0
3.844
e
-
3
1.317
e
-
2
0
7.130
e
-
3
1.561
e
-
2
F
12
0
9.518e – 007
1.571
e
-
32
9.207
e
-
30
2.780e − 006
5.529
e
-
48
6.436
e
-
31
F
13
0
1.723e – 007
1.350
e
-
32
4.323
e
-
19
2.657e − 007
1.106
e
-
47
2.219
e
-
18
F
14
-
99.60
−83.7328
-
98.7094891
-
99.5024042
6.1910
1.450
e
-
1
2.526
e
-
2
F
15
-
78.33236
−78.33232
-
78.3323314
-
78.3323314
8.654e − 6
1.005
e
-
13
2.804
e
-
11


#### 5.1.3. Comparison between MECA, CCGA, CPSO, and CCCS

Table3 summarizes the experimental results of comparison between MECA, CCGA, CPSO, and CCCS. Data of CCGA and CPSO algorithm results from the literature [9], MECA, CCGA, and CPSO requite 200000 function evaluations. But CCCS requires 2500 function evaluations to complete the optimization process. As a whole, the results of CCCS are better than those of MECA, OEA.Table 3
Comparison between MECA, CCGA, CPSO, and CCCS over 50 trials.
f
CCCS
MECA
CCGA
CPSO
F
05 (f0)
25.0366
6.43
e
-
1
3.80
1.94
e
-
1
F
03 (f1)
5.464e − 154
1.23
e
-
64
1.38
e
+
2
2.55
e
-
128
F
10 (f2)
1.655e − 7
0
9.51
e
-
2
2.78
e
-
14
F
09 (f3)
0
0
1.22
0
F
11 (f4)
0
3.94
e
-
3
2.20
e
-
1
1.86
e
-
2


### 5.2. Experimental Studies on Constrained Optimization Problems

In this section, 11 benchmark functions (G01–G11) and 2 engineering design problems (welded beam design, pressure vessel design) are used to validate the performance of CCCS in solving constrained optimization problems. These functions are described in [37]. The equality constraints have been converted into inequality constraints, |g(x)-δ|≤0, using the degree of violation δ=0.00001, the same with that of [37] (see Figures 3 and 5).Figure 3
Schematic of the welded beam design problem.The parameters of CCCS are set as follows: the number of iterations is 2500. However, the others of OEA are 24000. The experimental results of OEA are obtained over 50 independent trials. The running environment is the same as the previously. Table4 shows the comparison results between OEA, SMES, and CCCS. As can be seen, besides G02, other function values reached the theoretical value or are very close to the theoretical value. On the whole, the result of CCCS is better than those of SMES and CCCS.Table 4
The comparison between OEA, SMES, and CCCS.
f
f
min
Method
Best FV
Mean FV
St. dev
Worst FV
G
01
-
15
CCCS
−15
−15
0
−15
OEA CHp
-
15
-
15
0
-
15
OEA CHc
-
15
-
14.78
7.637
e
-
1
-
12
SMES
-
15.000
-
15.000
0
-
15.000
G
02
-
0.803619
CCCS
−0.7947
−0.7826
0.0093
−0.7694
OEA CHp
-
0.803605
-
0.782518
1.483
e
-
2
-
0.746283
OEA CHc
-
0.803589
-
0.782676
1.512
e
-
2
-
0.743149
SMES
-
0.803601
-
0.795238
1.67
e
-
2
-
0.751322
G
03
-
1
CCCS
−1.000
−1.000
0
−1.000
OEA CHp
-
1.000
-
1.000
8.470
e
-
6
-
1.000
OEA CHc
-
1.000
-
1.000
1.131
e
-
5
-
1.000
SMES
-
1.000
-
1.000
2.09
e
-
4
-
1.000
G
04
-
30665.539
CCCS
−3.0665539e + 004
−3.0665539e + 004
3.8348e − 012
−3.0665539e + 004
OEA CHp
-
30665.539
-
30665.539
1.735
e
-
11
-
30665.539
OEA CHc
-
30665.539
-
30665.539
1.837
e
-
11
-
30665.539
SMES
-
30665.539
-
30665.539
0
-
30665.539
G
05
5126.498
CCCS
5.1264981e + 003
5.1264981e + 003
9.5869e − 013
5.1264981e + 003
OEA CHp
5126.497
5127.048
7.071
e
-
1
5130.051
OEA CHc
5126.532
5315.975
145.473
5900.26
SMES
5126.599
5174.492
50.06
5304.167
G
06
-
6961.814
CCCS
−6.9618139e + 003
−6.9618139e + 003
9.5869e − 013
−6.9618139e + 003
OEA CHp
-
6961.814
-
6961.814
5.512
e
-
12
-
6961.814
OEA CHc
-
6961.814
-
6961.814
5.512
e
-
12
-
6961.814
SMES
-
6961.814
-
6861.284
1.85
-
6952.482
G
07
24.306
CCCS
2.4306209e + 001
2.4306209e + 001
4.2164e − 007
2.4306209e + 001
OEA CHp
24.308
24.373
7.615
e
-
2
24.655
OEA CHc
24.307
24.392
1.178
e
-
1
24.973
SMES
24.327
24.475
1.32
e
-
1
24.843
G
08
-
0.095825
CCCS
−9.5825041e − 002
−9.5825041e − 002
0
−9.5825041e − 002
OEA CHp
-
0.095825
-
0.095825
0
-
0.095825
OEA CHc
-
0.095825
-
0.095825
0
-
0.095825
SMES
-
0.095825
-
0.095825
0
-
0.095825
G
09
680.630
CCCS
6.8063006e + 002
6.8063006e + 002
1.1984e − 013
6.8063006e + 002
OEA CHp
680.630
680.632
1.718
e
-
3
680.638
OEA CHc
680.630
680.632
2.163
e
-
3
680.641
SMES
680.632
680.643
1.55
e
-
2
680.719
G
10
7049.331
CCCS
7.0492e + 003
7.0492e + 003
4.7796e − 004
7.0492e + 003
OEA CHp
7052.236
7219.011
60.737
7326.032
OEA CHc
7100.030
7231.357
86.409
7469.047
SMES
7051.903
7253.603
136.02
7638.366
G
11
0.750
CCCS
0.7500
0.7500
0
0.7500
OEA CHp
0.750
0.750
5.993
e
-
5
0.750
OEA CHc
0.750
0.750
1.881
e
-
7
0.750
SMES
0.75
0.75
1.52
e
-
4
0.75


### 5.3. Implementation in Structural Optimization Problems

Case I (welded beam design).
The objective function and parameters of   Case I are refers to in   [32]. With 25 nests, CCCS found the global optimum requiring 1000 iterations per optimization run. Table 5 compares the optimization results found by CCCS with similar data reported in the literature. CCCS obtained the best design overall of 2.1730. Mahdavi et al. [25] and Fesanghary et al. [26] found a better desgin. “But for the continuous optimization problem equal to 1.7248’’ is deleted.  In addition, CCCS requires only 25,000 function evaluations to complete the optimization process, hence much less than the literature. Convergence curve of Case I as shown in Figure 4.Table 5
Welded beam problem: comparison of CS results with the literature.
Researcher(s)
Method
h
l
t
b
Cost
No. of evaluation
Coello [11]
GA
0.2088
3.4205
8.9975
0.2100
1.7483
N.A.
Leite and Topping [12]
GA
0.2489
6.1097
8.2484
0.2485
2.4000
6273
Deb [13]
GA
0.2489
6.1730
8.1789
0.2533
2.4331
320,080
Lemonge and Barbosa [14]
GA
0.2443
6.2117
8.3015
0.2443
2.3816
320,000
Bernardino et al. [15]
AISa-GA
0.2444
6.2183
8.2912
0.2444
2.3812
320,000
Atiqullah and Rao [16]
SAb
0.2471
6.1451
8.2721
0.2495
2.4148
N.A.
Hedar and Fukushima [17]
SA
0.2444
6.2175
8.2915
0.2444
2.3810
N.A.
Liu [18]
SA-GA
0.2231
1.5815
12.8468
0.2245
2.2500
26,466
Hwang and He [19]
SA-DSc
0.2444
6.2158
8.2939
0.2444
2.3811
56,243
Parsopoulos and Vrahatis [20]
PSO
N.A.
N.A.
N.A.
N.A.
1.9220
100,000
He et al. [21]
PSO
0.2444
6.2175
8.2915
0.2444
2.3810
30,000
Zhang et al. [22]
EAd
0.2443
6.2201
8.2940
0.2444
2.3816
28,897
Coello [23]
EA
N.A.
N.A.
N.A.
N.A.
1.8245
N.A.
Lee and Geem [24]
HSe
0.2442
6.2231
8.2915
0.2443
2.381
110,000
Mahdavi et al. [25]
HS
0.2057
3.4705
9.0366
0.2057
1.7248
200,000
Fesanghary et al. [26]
HS-SQP
0.2057
3.4706
9.0368
0.2057
1.7248
90,000
Siddall [27]
RAg
0.2444
6.2819
8.2915
0.2444
2.3815
N.A.
Akhtar et al. [28]
SBMh
0.2407
6.4851
8.2399
0.2497
2.4426
19,259
Ray and Liew [29]
SCAi
0.2444
6.2380
8.2886
0.2446
2.3854
33,095
Montes and Oca [30]
BFOj
0.2536
7.1410
7.1044
0.2536
2.3398
N.A.
Zhang et al. [31]
DEk
0.2444
6.2175
8.2915
0.2444
2.3810
24,000
Gandomi et al. [32]
FA
0.2015
3.562
9.0414
0.2057
1.7312
50,000
Present study
CCCS
0.3312
10.0000
2.4095
0.3456
2.1730
25,000Figure 4
Convergence curve of Case I.Figure 5
Schematic of the pressure vessel design problem.Case II (pressure vessel design).
The objective function and parameters of the Case II are refers to in [33]. With 25 nests, CCCS found the global optimum requiring 1000 iterations per optimization run. Table 6 compares the optimization results found by CCCS with similar data reported in the literature. CCCS obtained the best design overall of 5885.3. In addition, CCCS requires only 25,000 function evaluations to complete the optimization process, hence much less than the literature. Here, N/A represents no records in the literature. Convergence curve of Case II is shown in Figure 6.Table 6
Pressure vessel design: comparison of CCCS results with the literature.
Reference method
R
L
T
s
T
h
f
min
ite
r
max
CCCS
40.3196
200.0000
0.7782
0.3846
5885.3
1000
ALPSO [33]
41.35
200
0.798
0.395
6234
7590
PSOA [34]
N/A
N/A
N/A
N/A
6292
6506
PSOSTR [34]
N/A
N/A
N/A
N/A
6272
3723
He et al.  [21]
N/A
N/A
N/A
N/A
6290
30000
Akhtar et al.  [28]
N/A
N/A
N/A
N/A
6335
12630Figure 6
Convergence curve of Case II.


## 5.1. Experimental Studies on Unconstrained Optimization Problems

In this section, 15 benchmark functions  (F01–F15)   are used to test the performance of CCCS in solving UCOPS. F01–F13 are f1–f13 in [1] and F14, F15 are f7, f9 in [36]. The problem dimension is set to 30 for F01–F13 and 100 for F14 and F15. In this manner, these functions have so many local minima that they are challenging enough for performance evaluation. The parameters of CCCS are set as follows: the number of organization is 10, others refer to [8], and the number of iterations is 2500.


### 5.1.1. Experimental Results of CCCS

Table1 summarizes the experimental results of CCCS, which include the best, the mean, the standard deviation, and the worst function values found. As can be seen, besides F05, F14, other function values reached the theoretical value or are very close to the theoretical value. What should be noted is that the global optimum for F11 is 0 every time, but both MECA (M-elite co-evolutionary algorithm for numerical optimization) and OEA (An organizational evolutionary algorithm for numerical optimization) cannot find the global optimal solution. Convergence curve of F08 and F15 as shown in Figures 1 and 2 respectively, other convergence curve figure has been omitted.Table 1
Experimental result of CCCS on 15 unconstrained benchmark functions over 50 trails.
f
f
min
Best function value
Mean function value
Standard deviation
Worst function value
F
01
0
8.804
e
-
242
2.011
e
-
217
0
2.011
e
-
216
F
02
0
4.271
e
-
143
1.628
e
-
130
4.142
e
-
130
1.323
e
-
129
F
03
0
6.280
e
-
192
5.464
e
-
154
1.729
e
-
153
5.464
e
-
153
F
04
0
1.085
e
-
121
1.333
e
-
098
4.080
e
-
098
1.294
e
-
097
F
05
0
25.0366
25.4182
0.3013
25.8868
F
06
0
0
0
0
0
F
07
0
5.309
e
-
005
3.080
e
-
004
2.391
e
-
004
8.047
e
-
004
F
08
-
12569.5
-
1.25694870
e
+
004
-
1.25694867
e
+
004
4.830
e
-
004
-
1.25694860
e
+
004
F
09
0
0
0
0
0
F
10
0
1.655
e
-
7
3.663
e
-
5
2.654
e
-
9
3.188
e
-
4
F
11
0
0
0
0
0
F
12
0
6.845
e
-
011
9.518
e
-
007
2.780
e
-
006
8.635
e
-
006
F
13
0
2.073
e
-
013
1.723
e
-
007
2.657
e
-
007
6.775
e
-
007
F
14
-
99.60
-
96.6008
-
83.7328
6.1910
-
75.6993
F
15
-
78.33236
-
78.33233
-
78.33232
8.654
e
-
006
-
78.33230Figure 1
Convergence curve ofF08.Figure 2
Convergence curve ofF15.


### 5.1.2. Comparison between CCCS, MECA, and OEA

Table2 shows statistical results of the CCCS optimization. As can be seen, as for functions F01, F02, F03, F04, F07, F08, and F11,  the mean function values of CCCS is better than MECA; as for functions F06 and F09, both CCCS and MECA can find the global optimal solution. It is a pity that the results of these functions (F05, F10, F12, F13, and F14) is worse than MECA.Table 2
Comparison between MECA, OEA, and CCCS over 50 trials.
f
f
min
Mean function value
Standard deviation
CCCS
MECA
OEA
CCCS
MECA
OEA
F
01
0
2.011e – 217
4.228
e
-
183
2.481
e
-
30
0
0
1.128
e
-
29
F
02
0
1.628e – 130
1.845
e
-
110
2.068
e
-
13
4.142e − 130
3.113
e
-
110
1.440
e
-
12
F
03
0
5.464e – 154
3.274
e
-
95
1.883
e
-
9
1.729e − 153
2.313
e
-
94
3.726
e
-
9
F
04
0
1.333e – 098
5.124
e
-
2
8.821
e
-
2
4.080e − 098
9.732
e
-
2
2.356
e
-
2
F
05
0
25.4182
7.973
e
-
2
0.227
0.3013
5.638
e
-
1
0.941
F
06
0
0
0
0
0
0
0
F
07
0
3.080e – 004
4.084
e
-
4
3.297
e
-
3
2.391e − 004
3.800
e
-
4
1.096
e
-
3
F
08
-
12569.5
−12569.4867
-
12569.4866
-
12
569.4866
4.830e − 004
7.350
e
-
12
5.555
e
-
12
F
09
0
0
0
5.430
e
-
17
0
0
1.683
e
-
16
F
10
0
3.663e− 5
0
5.336
e
-
14
2.654e − 9
0
2.945
e
-
13
F
11
0
0
3.844
e
-
3
1.317
e
-
2
0
7.130
e
-
3
1.561
e
-
2
F
12
0
9.518e – 007
1.571
e
-
32
9.207
e
-
30
2.780e − 006
5.529
e
-
48
6.436
e
-
31
F
13
0
1.723e – 007
1.350
e
-
32
4.323
e
-
19
2.657e − 007
1.106
e
-
47
2.219
e
-
18
F
14
-
99.60
−83.7328
-
98.7094891
-
99.5024042
6.1910
1.450
e
-
1
2.526
e
-
2
F
15
-
78.33236
−78.33232
-
78.3323314
-
78.3323314
8.654e − 6
1.005
e
-
13
2.804
e
-
11


### 5.1.3. Comparison between MECA, CCGA, CPSO, and CCCS

Table3 summarizes the experimental results of comparison between MECA, CCGA, CPSO, and CCCS. Data of CCGA and CPSO algorithm results from the literature [9], MECA, CCGA, and CPSO requite 200000 function evaluations. But CCCS requires 2500 function evaluations to complete the optimization process. As a whole, the results of CCCS are better than those of MECA, OEA.Table 3
Comparison between MECA, CCGA, CPSO, and CCCS over 50 trials.
f
CCCS
MECA
CCGA
CPSO
F
05 (f0)
25.0366
6.43
e
-
1
3.80
1.94
e
-
1
F
03 (f1)
5.464e − 154
1.23
e
-
64
1.38
e
+
2
2.55
e
-
128
F
10 (f2)
1.655e − 7
0
9.51
e
-
2
2.78
e
-
14
F
09 (f3)
0
0
1.22
0
F
11 (f4)
0
3.94
e
-
3
2.20
e
-
1
1.86
e
-
2


## 5.1.1. Experimental Results of CCCS

Table1 summarizes the experimental results of CCCS, which include the best, the mean, the standard deviation, and the worst function values found. As can be seen, besides F05, F14, other function values reached the theoretical value or are very close to the theoretical value. What should be noted is that the global optimum for F11 is 0 every time, but both MECA (M-elite co-evolutionary algorithm for numerical optimization) and OEA (An organizational evolutionary algorithm for numerical optimization) cannot find the global optimal solution. Convergence curve of F08 and F15 as shown in Figures 1 and 2 respectively, other convergence curve figure has been omitted.Table 1
Experimental result of CCCS on 15 unconstrained benchmark functions over 50 trails.
f
f
min
Best function value
Mean function value
Standard deviation
Worst function value
F
01
0
8.804
e
-
242
2.011
e
-
217
0
2.011
e
-
216
F
02
0
4.271
e
-
143
1.628
e
-
130
4.142
e
-
130
1.323
e
-
129
F
03
0
6.280
e
-
192
5.464
e
-
154
1.729
e
-
153
5.464
e
-
153
F
04
0
1.085
e
-
121
1.333
e
-
098
4.080
e
-
098
1.294
e
-
097
F
05
0
25.0366
25.4182
0.3013
25.8868
F
06
0
0
0
0
0
F
07
0
5.309
e
-
005
3.080
e
-
004
2.391
e
-
004
8.047
e
-
004
F
08
-
12569.5
-
1.25694870
e
+
004
-
1.25694867
e
+
004
4.830
e
-
004
-
1.25694860
e
+
004
F
09
0
0
0
0
0
F
10
0
1.655
e
-
7
3.663
e
-
5
2.654
e
-
9
3.188
e
-
4
F
11
0
0
0
0
0
F
12
0
6.845
e
-
011
9.518
e
-
007
2.780
e
-
006
8.635
e
-
006
F
13
0
2.073
e
-
013
1.723
e
-
007
2.657
e
-
007
6.775
e
-
007
F
14
-
99.60
-
96.6008
-
83.7328
6.1910
-
75.6993
F
15
-
78.33236
-
78.33233
-
78.33232
8.654
e
-
006
-
78.33230Figure 1
Convergence curve ofF08.Figure 2
Convergence curve ofF15.


## 5.1.2. Comparison between CCCS, MECA, and OEA

Table2 shows statistical results of the CCCS optimization. As can be seen, as for functions F01, F02, F03, F04, F07, F08, and F11,  the mean function values of CCCS is better than MECA; as for functions F06 and F09, both CCCS and MECA can find the global optimal solution. It is a pity that the results of these functions (F05, F10, F12, F13, and F14) is worse than MECA.Table 2
Comparison between MECA, OEA, and CCCS over 50 trials.
f
f
min
Mean function value
Standard deviation
CCCS
MECA
OEA
CCCS
MECA
OEA
F
01
0
2.011e – 217
4.228
e
-
183
2.481
e
-
30
0
0
1.128
e
-
29
F
02
0
1.628e – 130
1.845
e
-
110
2.068
e
-
13
4.142e − 130
3.113
e
-
110
1.440
e
-
12
F
03
0
5.464e – 154
3.274
e
-
95
1.883
e
-
9
1.729e − 153
2.313
e
-
94
3.726
e
-
9
F
04
0
1.333e – 098
5.124
e
-
2
8.821
e
-
2
4.080e − 098
9.732
e
-
2
2.356
e
-
2
F
05
0
25.4182
7.973
e
-
2
0.227
0.3013
5.638
e
-
1
0.941
F
06
0
0
0
0
0
0
0
F
07
0
3.080e – 004
4.084
e
-
4
3.297
e
-
3
2.391e − 004
3.800
e
-
4
1.096
e
-
3
F
08
-
12569.5
−12569.4867
-
12569.4866
-
12
569.4866
4.830e − 004
7.350
e
-
12
5.555
e
-
12
F
09
0
0
0
5.430
e
-
17
0
0
1.683
e
-
16
F
10
0
3.663e− 5
0
5.336
e
-
14
2.654e − 9
0
2.945
e
-
13
F
11
0
0
3.844
e
-
3
1.317
e
-
2
0
7.130
e
-
3
1.561
e
-
2
F
12
0
9.518e – 007
1.571
e
-
32
9.207
e
-
30
2.780e − 006
5.529
e
-
48
6.436
e
-
31
F
13
0
1.723e – 007
1.350
e
-
32
4.323
e
-
19
2.657e − 007
1.106
e
-
47
2.219
e
-
18
F
14
-
99.60
−83.7328
-
98.7094891
-
99.5024042
6.1910
1.450
e
-
1
2.526
e
-
2
F
15
-
78.33236
−78.33232
-
78.3323314
-
78.3323314
8.654e − 6
1.005
e
-
13
2.804
e
-
11


## 5.1.3. Comparison between MECA, CCGA, CPSO, and CCCS

Table3 summarizes the experimental results of comparison between MECA, CCGA, CPSO, and CCCS. Data of CCGA and CPSO algorithm results from the literature [9], MECA, CCGA, and CPSO requite 200000 function evaluations. But CCCS requires 2500 function evaluations to complete the optimization process. As a whole, the results of CCCS are better than those of MECA, OEA.Table 3
Comparison between MECA, CCGA, CPSO, and CCCS over 50 trials.
f
CCCS
MECA
CCGA
CPSO
F
05 (f0)
25.0366
6.43
e
-
1
3.80
1.94
e
-
1
F
03 (f1)
5.464e − 154
1.23
e
-
64
1.38
e
+
2
2.55
e
-
128
F
10 (f2)
1.655e − 7
0
9.51
e
-
2
2.78
e
-
14
F
09 (f3)
0
0
1.22
0
F
11 (f4)
0
3.94
e
-
3
2.20
e
-
1
1.86
e
-
2


## 5.2. Experimental Studies on Constrained Optimization Problems

In this section, 11 benchmark functions (G01–G11) and 2 engineering design problems (welded beam design, pressure vessel design) are used to validate the performance of CCCS in solving constrained optimization problems. These functions are described in [37]. The equality constraints have been converted into inequality constraints, |g(x)-δ|≤0, using the degree of violation δ=0.00001, the same with that of [37] (see Figures 3 and 5).Figure 3
Schematic of the welded beam design problem.The parameters of CCCS are set as follows: the number of iterations is 2500. However, the others of OEA are 24000. The experimental results of OEA are obtained over 50 independent trials. The running environment is the same as the previously. Table4 shows the comparison results between OEA, SMES, and CCCS. As can be seen, besides G02, other function values reached the theoretical value or are very close to the theoretical value. On the whole, the result of CCCS is better than those of SMES and CCCS.Table 4
The comparison between OEA, SMES, and CCCS.
f
f
min
Method
Best FV
Mean FV
St. dev
Worst FV
G
01
-
15
CCCS
−15
−15
0
−15
OEA CHp
-
15
-
15
0
-
15
OEA CHc
-
15
-
14.78
7.637
e
-
1
-
12
SMES
-
15.000
-
15.000
0
-
15.000
G
02
-
0.803619
CCCS
−0.7947
−0.7826
0.0093
−0.7694
OEA CHp
-
0.803605
-
0.782518
1.483
e
-
2
-
0.746283
OEA CHc
-
0.803589
-
0.782676
1.512
e
-
2
-
0.743149
SMES
-
0.803601
-
0.795238
1.67
e
-
2
-
0.751322
G
03
-
1
CCCS
−1.000
−1.000
0
−1.000
OEA CHp
-
1.000
-
1.000
8.470
e
-
6
-
1.000
OEA CHc
-
1.000
-
1.000
1.131
e
-
5
-
1.000
SMES
-
1.000
-
1.000
2.09
e
-
4
-
1.000
G
04
-
30665.539
CCCS
−3.0665539e + 004
−3.0665539e + 004
3.8348e − 012
−3.0665539e + 004
OEA CHp
-
30665.539
-
30665.539
1.735
e
-
11
-
30665.539
OEA CHc
-
30665.539
-
30665.539
1.837
e
-
11
-
30665.539
SMES
-
30665.539
-
30665.539
0
-
30665.539
G
05
5126.498
CCCS
5.1264981e + 003
5.1264981e + 003
9.5869e − 013
5.1264981e + 003
OEA CHp
5126.497
5127.048
7.071
e
-
1
5130.051
OEA CHc
5126.532
5315.975
145.473
5900.26
SMES
5126.599
5174.492
50.06
5304.167
G
06
-
6961.814
CCCS
−6.9618139e + 003
−6.9618139e + 003
9.5869e − 013
−6.9618139e + 003
OEA CHp
-
6961.814
-
6961.814
5.512
e
-
12
-
6961.814
OEA CHc
-
6961.814
-
6961.814
5.512
e
-
12
-
6961.814
SMES
-
6961.814
-
6861.284
1.85
-
6952.482
G
07
24.306
CCCS
2.4306209e + 001
2.4306209e + 001
4.2164e − 007
2.4306209e + 001
OEA CHp
24.308
24.373
7.615
e
-
2
24.655
OEA CHc
24.307
24.392
1.178
e
-
1
24.973
SMES
24.327
24.475
1.32
e
-
1
24.843
G
08
-
0.095825
CCCS
−9.5825041e − 002
−9.5825041e − 002
0
−9.5825041e − 002
OEA CHp
-
0.095825
-
0.095825
0
-
0.095825
OEA CHc
-
0.095825
-
0.095825
0
-
0.095825
SMES
-
0.095825
-
0.095825
0
-
0.095825
G
09
680.630
CCCS
6.8063006e + 002
6.8063006e + 002
1.1984e − 013
6.8063006e + 002
OEA CHp
680.630
680.632
1.718
e
-
3
680.638
OEA CHc
680.630
680.632
2.163
e
-
3
680.641
SMES
680.632
680.643
1.55
e
-
2
680.719
G
10
7049.331
CCCS
7.0492e + 003
7.0492e + 003
4.7796e − 004
7.0492e + 003
OEA CHp
7052.236
7219.011
60.737
7326.032
OEA CHc
7100.030
7231.357
86.409
7469.047
SMES
7051.903
7253.603
136.02
7638.366
G
11
0.750
CCCS
0.7500
0.7500
0
0.7500
OEA CHp
0.750
0.750
5.993
e
-
5
0.750
OEA CHc
0.750
0.750
1.881
e
-
7
0.750
SMES
0.75
0.75
1.52
e
-
4
0.75


## 5.3. Implementation in Structural Optimization Problems

Case I (welded beam design).
The objective function and parameters of   Case I are refers to in   [32]. With 25 nests, CCCS found the global optimum requiring 1000 iterations per optimization run. Table 5 compares the optimization results found by CCCS with similar data reported in the literature. CCCS obtained the best design overall of 2.1730. Mahdavi et al. [25] and Fesanghary et al. [26] found a better desgin. “But for the continuous optimization problem equal to 1.7248’’ is deleted.  In addition, CCCS requires only 25,000 function evaluations to complete the optimization process, hence much less than the literature. Convergence curve of Case I as shown in Figure 4.Table 5
Welded beam problem: comparison of CS results with the literature.
Researcher(s)
Method
h
l
t
b
Cost
No. of evaluation
Coello [11]
GA
0.2088
3.4205
8.9975
0.2100
1.7483
N.A.
Leite and Topping [12]
GA
0.2489
6.1097
8.2484
0.2485
2.4000
6273
Deb [13]
GA
0.2489
6.1730
8.1789
0.2533
2.4331
320,080
Lemonge and Barbosa [14]
GA
0.2443
6.2117
8.3015
0.2443
2.3816
320,000
Bernardino et al. [15]
AISa-GA
0.2444
6.2183
8.2912
0.2444
2.3812
320,000
Atiqullah and Rao [16]
SAb
0.2471
6.1451
8.2721
0.2495
2.4148
N.A.
Hedar and Fukushima [17]
SA
0.2444
6.2175
8.2915
0.2444
2.3810
N.A.
Liu [18]
SA-GA
0.2231
1.5815
12.8468
0.2245
2.2500
26,466
Hwang and He [19]
SA-DSc
0.2444
6.2158
8.2939
0.2444
2.3811
56,243
Parsopoulos and Vrahatis [20]
PSO
N.A.
N.A.
N.A.
N.A.
1.9220
100,000
He et al. [21]
PSO
0.2444
6.2175
8.2915
0.2444
2.3810
30,000
Zhang et al. [22]
EAd
0.2443
6.2201
8.2940
0.2444
2.3816
28,897
Coello [23]
EA
N.A.
N.A.
N.A.
N.A.
1.8245
N.A.
Lee and Geem [24]
HSe
0.2442
6.2231
8.2915
0.2443
2.381
110,000
Mahdavi et al. [25]
HS
0.2057
3.4705
9.0366
0.2057
1.7248
200,000
Fesanghary et al. [26]
HS-SQP
0.2057
3.4706
9.0368
0.2057
1.7248
90,000
Siddall [27]
RAg
0.2444
6.2819
8.2915
0.2444
2.3815
N.A.
Akhtar et al. [28]
SBMh
0.2407
6.4851
8.2399
0.2497
2.4426
19,259
Ray and Liew [29]
SCAi
0.2444
6.2380
8.2886
0.2446
2.3854
33,095
Montes and Oca [30]
BFOj
0.2536
7.1410
7.1044
0.2536
2.3398
N.A.
Zhang et al. [31]
DEk
0.2444
6.2175
8.2915
0.2444
2.3810
24,000
Gandomi et al. [32]
FA
0.2015
3.562
9.0414
0.2057
1.7312
50,000
Present study
CCCS
0.3312
10.0000
2.4095
0.3456
2.1730
25,000Figure 4
Convergence curve of Case I.Figure 5
Schematic of the pressure vessel design problem.Case II (pressure vessel design).
The objective function and parameters of the Case II are refers to in [33]. With 25 nests, CCCS found the global optimum requiring 1000 iterations per optimization run. Table 6 compares the optimization results found by CCCS with similar data reported in the literature. CCCS obtained the best design overall of 5885.3. In addition, CCCS requires only 25,000 function evaluations to complete the optimization process, hence much less than the literature. Here, N/A represents no records in the literature. Convergence curve of Case II is shown in Figure 6.Table 6
Pressure vessel design: comparison of CCCS results with the literature.
Reference method
R
L
T
s
T
h
f
min
ite
r
max
CCCS
40.3196
200.0000
0.7782
0.3846
5885.3
1000
ALPSO [33]
41.35
200
0.798
0.395
6234
7590
PSOA [34]
N/A
N/A
N/A
N/A
6292
6506
PSOSTR [34]
N/A
N/A
N/A
N/A
6272
3723
He et al.  [21]
N/A
N/A
N/A
N/A
6290
30000
Akhtar et al.  [28]
N/A
N/A
N/A
N/A
6335
12630Figure 6
Convergence curve of Case II.


## 6. Conclusions

Taking inspiration from the OEA, a new numerical optimization algorithm, CCCS, has been proposed in this paper. The experimental results in Tables1–6 indicate that CCCS outperforms the MECA and OEA. On the whole, CCCS obtains a good performance for the unconstrained functions, constrained functions, and 2 engineering design problems. These benefits mainly from the following three aspects. One is the dynamics population, and the other is three evolutionary operators. The third aspect is a combination of CS algorithm. 28 experiments illustrate that CCCS has an effective searching mechanism. However, the number of dynamic populations, is difficult to control, which spends lots of computational cost. How to control the number of dynamic population is the future research work.



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*Source: 912056-2013-08-27.xml*