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Mathematical Problems in Engineering

Volume 2015, Article ID 715635, 14 pages

http://dx.doi.org/10.1155/2015/715635

## Cuckoo Search Algorithm with Chaotic Maps

College of Computer and & Information Science, Fujian Agriculture and Forestry University, Fuzhou 350002, China

Received 5 March 2015; Revised 25 June 2015; Accepted 28 June 2015

Academic Editor: Evangelos J. Sapountzakis

Copyright © 2015 Lijin Wang and Yiwen Zhong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2]. Subsequent investigations [2, 3] have demonstrated that CS is a simple yet very promising population-based stochastic search technique by using Lévy flights random walk (LFRW) and biased/selective random walk (BSRW). LFRW with a scaling factor parameter uses a mutation operator to generate new solutions based on a best solution obtained so far, while BSRW with a fraction probability parameter employs a complex crossover operator to search new solutions. After each random walk, a greedy strategy is utilized to select a better solution from the current and new generated solutions according to their fitness.

Due to its promising performance, CS has received much attention. Some studies have focused on improving LFRW [4–10] and BSRW [11–15]. Some attempts have been made to combine CS with other optimization techniques like particle swarm optimization [16, 17], Tabu search [18], differential evolution [19], ant colony optimization [20], and cooperative coevolutionary framework [21, 22]. The above studies have shown their contribution to the research on CS. Except for the literatures [9, 10], however, these studies used the definition of the scaling factor and the fraction probability in the constant value way, resulting in making CS sensitive to the optimization problems. This motivates us to study the scaling factor and the fraction probability using the variable value schema.

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 [23]. Recently, chaos theory has been integrated into genetic algorithm [24], differential evolution [25], firefly algorithm [26], krill herd [27, 28], and biogeography-based optimization [23, 29], and these have shown the effectiveness and efficiency of chaos theory. In light of the above investigations, we propose chaotic cuckoo search algorithm, called CCS, which utilizes chaotic maps to define the scaling factor and the fraction probability. The comprehensive experiments are carried out on 20 benchmark functions, and the results show that chaotic maps can improve the solution quality and convergence speed of CS effectively and efficiently.

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 2 describes the standard cuckoo search algorithm. Section 3 presents the cuckoo search algorithm with chaos. Section 4 reports the experimental results. Section 5 draws conclusion on this paper.

#### 2. Cuckoo Search Algorithm

CS, developed recently by Yang and Deb [1, 2], is a simple yet very promising population-based stochastic search technique. In general, when CS is used to solve an objective function with the solution space [], , a nest represents a candidate solution .

In the initialization phase, CS initializes solutions that are randomly sampled from solution space bywhere represents a uniformly distributed random variable on the range and is the population size.

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.

##### 2.1. Lévy Flights Random Walk

Broadly speaking, LFRW is a random walk whose step-size is drawn from Lévy distribution. At generation , LFRW can be formulated as follows:where *α* is a step-size which is related to the scales of the problem. In CS, LFRW is employed to search for new solutions around the best solution obtained so far. Therefore, the step-size can be obtained by the following equation [2]:where is a scaling factor (generally, ) and represents the best solution obtained so far.

The product means entry-wise multiplications. Lévy(*β*) is a random number, which is drawn from a Lévy distribution for large steps:

In implementation, Lévy(*β*) can be calculated as follows [2]:where is a constant and set to 1.5 in the standard software implementation of CS [2], and are random numbers drawn from a normal distribution with mean of 0 and standard deviation of 1, and is a* gamma function*.

Obviously, (2) can be reformulated as

##### 2.2. Biased/Selective Random Walk

BSRW is used to discover new solutions far enough away from the current best solution by far field randomization [1]. First, a trial solution is built with a mutation of the current solution as base vector and two randomly selected solutions as perturbed vectors. Second, a new solution is generated by a crossover operator from the current and the trial solutions. BSRW can be formulated as follows:where the random indexes and are the th and th solutions in the population, respectively, is the th dimension of the solution, and are random numbers on the range , and is a fraction probability.

#### 3. Chaotic Cuckoo Search Algorithm

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.

##### 3.1. Chaotic Maps

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 [30], Gauss map [30], Logistic map [31], Piecewise map [32], Sine map [33], Singer map [34], Sinusoidal map [31], and Tent map [35], shown in Table 1. Additionally, the visualization of these chaotic maps with the initial point at 0.7 is plotted in Figure 1. The other chaotic maps can be found in [26, 28].