Cuckoo search algorithm is a novel nature-inspired optimization technique based on the obligate brood parasitic behavior of some cuckoo species. It iteratively employs Lévy flights random walk with a scaling factor and biased/selective random walk with a fraction probability. Unfortunately, these two parameters are used in constant value schema, resulting in a problem sensitive to solution quality and convergence speed. In this paper, we proposed a variable value schema cuckoo search algorithm with chaotic maps, called CCS. In CCS, chaotic maps are utilized to, respectively, define the scaling factor and the fraction probability to enhance the solution quality and convergence speed. Extensive experiments with different chaotic maps demonstrate the improvement in efficiency and effectiveness.
Cuckoo search algorithm (CS) is a novel nature-inspired approach based on the obligate brood parasitic behavior of some cuckoo species in combination with the Lévy flights behavior of some birds and fruit flies [
Due to its promising performance, CS has received much attention. Some studies have focused on improving LFRW [
One of the mathematical approaches for the variable value schema is chaos. Chaos theory is related to the study of chaotic dynamical systems that are highly sensitive to the initial conditions [
The main contribution of this paper is to define the variable value for the scaling factor and the fraction probability using chaotic maps. This leads to the major advantages of our approach as follows: (i) since the scaling factor and the fraction probability are used in constant value way, the variable value schema of two parameters is generally more suitable for the optimization problems, resulting in better performance; (ii) due to the simpleness of chaotic maps, our approach does not increase the overall complexity of CS; (iii) our approach does not destroy the structure of CS; thus, it is still very simple.
The remainder of this paper is organized as follows. Section
CS, developed recently by Yang and Deb [
In the initialization phase, CS initializes solutions that are randomly sampled from solution space by
After initialization, CS goes into an iterative phase where two random walks: Lévy flights random walk and biased/selective random walk, are employed to search for new solutions. After each random walk, CS selects a better solution according to the new generated and current solutions fitness using the greedy strategy. At the end of each iteration process, the best solution is updated.
Broadly speaking, LFRW is a random walk whose step-size is drawn from Lévy distribution. At generation
The product
In implementation, Lévy(
Obviously, (
BSRW is used to discover new solutions far enough away from the current best solution by far field randomization [
In this section, we first present different chaotic maps. Then, we apply them to define the scaling factor and the fraction probability. We last propose the framework of cuckoo search algorithm with chaotic maps, called CCS.
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, engineering, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect. One of ways to make quantitative statements about the behavior of chaotic systems is chaotic map like Circle map [
Chaotic maps.
Number | Name | Chaotic map | Range |
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1 | Circle |
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2 | Gauss |
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3 | Logistic |
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4 | Piecewise |
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5 | Sine |
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6 | Singer |
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(0, 1) |
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7 | Sinusoidal |
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8 | Tent |
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(0, 1) |
Visualization of different chaotic maps.
Circle map
Gauss map
Logistic map
Piecewise map
Sine map
Singer map
Sinusoidal map
Tent map
As seen from (
In (
According to the above descriptions, we give the framework of CCS in Algorithm
Nest0 = ( Fitness
WHILE ( FOR ( ENDFOR FOR ( ENDFOR ENDWHILE
In this section, a suit of 20 benchmark functions used in [
It can observed from Figure
Average Error values obtained by CCS with different chaotic maps for 20 benchmark functions at
cCCS | gCCS | lgCCS | pCCS | seCCS | srCCS | slCCS | tCCS | |
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Average ranking of eight algorithms by the Friedman test for 20 functions at
Algorithm | cCCS | gCCS | lgCCS | pCCS | seCCS | srCCS | slCCS | tCCS |
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Ranking |
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6.10 | 4.92 | 4.22 | 4.33 | 3.95 | 4.47 | 4.30 |
As observed from Table
Note that the random value can also be regarded as the variable value schema. To show the advantage of CS with chaotic maps, CS with random value, called rCS, is tested on 20 benchmark functions at
Error obtained by rCS and CCS at
rCS | CCS | |||
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−/=/+ |
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Results of the multiple-problem Wilcoxon’s test for CCS and rCS for 20 functions at
Algorithm |
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CCS versus rCS | 150.125 | 59.875 | 0.092059 | = | + |
We can find from Table
Additionally, it can be seen from Table
To show how chaotic maps can improve the performance of CS, we carry out experiments on the 20 benchmark functions at
Error obtained by CS and CCS for 20 functions at
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CS | CCS | CS | CCS | CS | CCS | ||||
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Table
In the case of
In the case of
When
Furthermore, to show the convergence speed of CCS reaching the accuracy level
Average Evaluation obtained by CS and CCS at
CS | CCS | |
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88868 ± 1709 (25) | 52414 ± 1420 (25) |
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292080 ± 0 (1) | 268680 ± 10353 (2) |
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166055 ± 17655 (24) | 79652 ± 2091 (25) |
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141264 ± 31378 (25) | 67294 ± 26143 (25) |
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153843 ± 24442 (25) | 59170 ± 5439 (25) |
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109678 ± 12077 (25) | 57044 ± 2408 (25) |
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93476 ± 1709 (25) | 53530 ± 1538 (25) |
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— | 240891 ± 11343 (25) |
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— | 259483 ± 49358 (14) |
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165087 ± 27396 (25) | 101414 ± 45112 (18) |
Additionally, convergence graphs of CS and CCS for some functions at
Convergence graphs of CS and CCS.
According to
It is worthy pointing out that the chaotic sequences are highly sensitive to initial condition. To show the performance of CCS affected by the initial value, we perform the experiments on chaotic maps with different initial values. The results are listed in Table
Error obtained by CCS25, CCS5, and CCS for 20 functions at
CCS25 | CCS | CCS5 | |||||
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Seen from Table
To show the competitiveness of CCS with the other improved CS algorithms, we compare it at
Error obtained by ICS, CSPSO, OLCS, and CCS for 20 functions at
ICS | CSPSO | OLCS | CCS | |
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Results of the multiple-problem Wilcoxon’s test for ICS, CSPSO, OLCS, and CCS for 20 functions at
Algorithm |
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CCS versus ICS | 156.375 | 53.625 | 0.055115 | = | + |
CCS versus CSPSO | 206 | 4 | 0.000163 | + | + |
CCS versus OLCS | 171.5 | 38.5 | 0.013042 | + | + |
Average ranking of eight algorithms by the Friedman test for 20 functions at
Algorithm | ICS | CSPSO | OLCS | CCS |
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Ranking | 2.23 | 3.45 | 2.70 |
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As observed from Table
CCS shows its promising performance by using two chaotic maps simultaneously to define the scaling factor and the fraction probability. In this case, two chaotic maps make cooperative contribution to the performance of CCS. In this section, therefore, we discuss the contribution of each chaotic map to the performance of CCS. To analyze the contribution of each chaotic map, we consider two derived algorithms: CCS1 and CCS2. The former uses chaotic map to define the scaling factor and keeps the original BSRW, while the later utilizes chaotic map to define the fraction probability and keeps the original LFRW. CCS1 and CCS2 are performed on 20 benchmark functions at
Error obtained by CS, CCS1, CCS2, and CCS for 20 functions at
CS | CCS1 | CCS | CCS2 | |||||
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−/=/+ |
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It can be observed from Table
In CS, the scaling factor and the fraction probability parameters are used in constant value way, resulting in a problem sensitive to solution quality and convergence speed. In this paper, we employed chaotic maps to define the scaling factor and the fraction probability in variable value schema and proposed chaotic cuckoo search algorithm, called CCS. Comprehensive experiments were carried out on 20 benchmark functions to test the performances of CCS. The results show that chaotic maps can improve the performance of CS effectively and efficiently. The scalability study reveals that the advantage of CCS over CS is overall stable when increasing the dimensionality of problems. The results in comparison with another study on the scaling factor and the fraction probability verify that chaotic maps are a better selection to define the variable value schema.
There are several interesting directions for future work. First, it is interesting to test the different combinations of chaotic maps to find the optimal one. Second, we plan to integrate chaotic maps into improved CS algorithms to further verify their efficiency and effectiveness. Last but not least, we also plan to apply CCS to some real-world optimization problems for further examinations.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are very grateful to the editor and the anonymous reviewers for their constructive comments and suggestions to this paper. This work was supported by the Natural Science Foundation of Fujian Province of China under Grant no. 2013J01216.