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.
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 [
Recently, a novel heuristic search algorithm, called Cuckoo Search (CS) in [
The remainder of this paper is organized as follows: Section
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 [ Each cuckoo lays one egg, which represents a set of solution coordinates, at a time, and dumps it in a random nest. A fraction of the nests containing the best eggs, or solutions, will carry over to the next generation. 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 parameter
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.
When a size is too large usually it is split into several small organizations; let
Two organizations,
Two organizations,
Initializing population For each organization in operator on it, deleting it from Randomly selecting two parent organizations Performing the CS and selecting their leaders; Annexing operator; Cooperating operator; adding the child organizations into Deleting Deleting the % output the best solution in
In the initialization, each organization has only one member, and the population has total
A unconstrained optimization problems (UCOPs) are formulated as solving the objective function
A constrained optimization problems (COPs) can be formulated as solving objective function
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 [
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.
In this section, 15 benchmark functions (
Table
Experimental result of CCCS on 15 unconstrained benchmark functions over 50 trails.
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Best function value | Mean function value | Standard deviation | Worst function value |
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0 | 25.0366 | 25.4182 | 0.3013 | 25.8868 |
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0 | 0 | 0 | 0 | 0 |
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0 | 0 | 0 | 0 | 0 |
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6.1910 |
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Convergence curve of
Convergence curve of
Table
Comparison between MECA, OEA, and CCCS over 50 trials.
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Mean function value | Standard deviation | ||||
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CCCS | MECA | OEA | CCCS | MECA | OEA | ||
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0.227 |
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0.941 |
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Table
Comparison between MECA, CCGA, CPSO, and CCCS over 50 trials.
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CCCS | MECA | CCGA | CPSO |
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3.80 |
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0 | 1.22 | 0 |
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In this section, 11 benchmark functions (
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. Table
The comparison between OEA, SMES, and CCCS.
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Method | Best FV | Mean FV | St. dev | Worst FV |
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OEA CHp |
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OEA CHc |
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SMES |
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OEA CHp |
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OEA CHc |
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SMES |
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OEA CHp |
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OEA CHc |
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SMES |
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OEA CHp |
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OEA CHc |
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SMES |
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5126.498 |
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OEA CHp | 5126.497 | 5127.048 |
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5130.051 | ||
OEA CHc | 5126.532 | 5315.975 | 145.473 | 5900.26 | ||
SMES | 5126.599 | 5174.492 | 50.06 | 5304.167 | ||
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OEA CHp |
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OEA CHc |
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SMES |
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24.306 |
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OEA CHp | 24.308 | 24.373 |
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24.655 | ||
OEA CHc | 24.307 | 24.392 |
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24.973 | ||
SMES | 24.327 | 24.475 |
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24.843 | ||
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OEA CHp |
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OEA CHc |
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SMES |
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680.630 |
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OEA CHp | 680.630 | 680.632 |
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680.638 | ||
OEA CHc | 680.630 | 680.632 |
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680.641 | ||
SMES | 680.632 | 680.643 |
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680.719 | ||
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7049.331 |
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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 | ||
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0.750 |
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OEA CHp | 0.750 | 0.750 |
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0.750 | ||
OEA CHc | 0.750 | 0.750 |
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0.750 | ||
SMES | 0.75 | 0.75 |
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0.75 |
The objective function and parameters of Case I are refers to in [
Welded beam problem: comparison of CS results with the literature.
Researcher(s) | Method |
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Cost | No. of evaluation |
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Coello [ |
GA | 0.2088 | 3.4205 | 8.9975 | 0.2100 | 1.7483 | N.A. |
Leite and Topping [ |
GA | 0.2489 | 6.1097 | 8.2484 | 0.2485 | 2.4000 | 6273 |
Deb [ |
GA | 0.2489 | 6.1730 | 8.1789 | 0.2533 | 2.4331 | 320,080 |
Lemonge and Barbosa [ |
GA | 0.2443 | 6.2117 | 8.3015 | 0.2443 | 2.3816 | 320,000 |
Bernardino et al. [ |
AISa-GA | 0.2444 | 6.2183 | 8.2912 | 0.2444 | 2.3812 | 320,000 |
Atiqullah and Rao [ |
SAb | 0.2471 | 6.1451 | 8.2721 | 0.2495 | 2.4148 | N.A. |
Hedar and Fukushima [ |
SA | 0.2444 | 6.2175 | 8.2915 | 0.2444 | 2.3810 | N.A. |
Liu [ |
SA-GA | 0.2231 | 1.5815 | 12.8468 | 0.2245 | 2.2500 | 26,466 |
Hwang and He [ |
SA-DSc | 0.2444 | 6.2158 | 8.2939 | 0.2444 | 2.3811 | 56,243 |
Parsopoulos and Vrahatis [ |
PSO | N.A. | N.A. | N.A. | N.A. | 1.9220 | 100,000 |
He et al. [ |
PSO | 0.2444 | 6.2175 | 8.2915 | 0.2444 | 2.3810 | 30,000 |
Zhang et al. [ |
EAd | 0.2443 | 6.2201 | 8.2940 | 0.2444 | 2.3816 | 28,897 |
Coello [ |
EA | N.A. | N.A. | N.A. | N.A. | 1.8245 | N.A. |
Lee and Geem [ |
HSe | 0.2442 | 6.2231 | 8.2915 | 0.2443 | 2.381 | 110,000 |
Mahdavi et al. [ |
HS | 0.2057 | 3.4705 | 9.0366 | 0.2057 | 1.7248 | 200,000 |
Fesanghary et al. [ |
HS-SQP | 0.2057 | 3.4706 | 9.0368 | 0.2057 | 1.7248 | 90,000 |
Siddall [ |
RAg | 0.2444 | 6.2819 | 8.2915 | 0.2444 | 2.3815 | N.A. |
Akhtar et al. [ |
SBMh | 0.2407 | 6.4851 | 8.2399 | 0.2497 | 2.4426 | 19,259 |
Ray and Liew [ |
SCAi | 0.2444 | 6.2380 | 8.2886 | 0.2446 | 2.3854 | 33,095 |
Montes and Oca [ |
BFOj | 0.2536 | 7.1410 | 7.1044 | 0.2536 | 2.3398 | N.A. |
Zhang et al. [ |
DEk | 0.2444 | 6.2175 | 8.2915 | 0.2444 | 2.3810 | 24,000 |
Gandomi et al. [ |
FA | 0.2015 | 3.562 | 9.0414 | 0.2057 | 1.7312 | 50,000 |
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Convergence curve of Case I.
Schematic of the pressure vessel design problem.
The objective function and parameters of the Case II are refers to in [
Pressure vessel design: comparison of CCCS results with the literature.
Reference method |
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ALPSO [ |
41.35 | 200 | 0.798 | 0.395 | 6234 | 7590 |
PSOA [ |
N/A | N/A | N/A | N/A | 6292 | 6506 |
PSOSTR [ |
N/A | N/A | N/A | N/A | 6272 | 3723 |
He et al. [ |
N/A | N/A | N/A | N/A | 6290 | 30000 |
Akhtar et al. [ |
N/A | N/A | N/A | N/A | 6335 | 12630 |
Convergence curve of Case II.
Taking inspiration from the OEA, a new numerical optimization algorithm, CCCS, has been proposed in this paper. The experimental results in Tables
This work is supported by the National Science Foundation of China under Grant no. 61165015, Key Project of Guangxi Science Foundation under Grant no. 2012GXNSFDA053028, Key Project of Guangxi High School Science Foundation under Grant no. 20121ZD008, Open Research Fund Program of Key Lab of Intelligent Perception and Image Understanding of Ministry of Education of China under Grant no. IPIU01201100 and the Innovation Project of Guangxi Graduate Education under Grant no. YCSZ2012063.