Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 4839763, 11 pages

http://dx.doi.org/10.1155/2016/4839763

## Cuckoo Search Algorithm with Hybrid Factor Using Dimensional Distance

^{1}College of Computer and Information Science, Fujian Agriculture and Forestry University, Fuzhou 350002, China^{2}College of Management, Fujian University of Traditional Chinese Medicine, Fuzhou 350002, China

Received 15 May 2016; Accepted 6 November 2016

Academic Editor: Salvatore Alfonzetti

Copyright © 2016 Yaohua Lin et al. 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

This paper proposes a hybrid factor strategy for cuckoo search algorithm by combining constant factor and varied factor. The constant factor is used to the dimensions of each solution which are closer to the corresponding dimensions of the best solution, while the varied factor using a random or a chaotic sequence is utilized to farer dimensions. For each solution, the dimension whose distance to the corresponding one of the best solution is shorter than mean distance of all dimensional distances will be regarded as the closer one, otherwise as the farer one. A suit of 20 benchmark functions are employed to verify the performance of the proposed strategy, and the results show the improvement in effectiveness and efficiency of the hybridization.

#### 1. Introduction

Cuckoo search algorithm (CS), proposed by Yang and Deb in 2009, is a new nature-inspired method for solving real-valued numerical optimization problems [1, 2]. The method utilizes Lévy flights random walk (LFRW) and biased random walk to search for new solutions and achieves the promising performance for many tough problems. This has attracted a lot of researchers, and many studies have been proposed. Some studies have focused on the combination CS with other optimization methods [3–13]. Some attempts have been made to improve search ability of LFRW and BSW [14–32]. Other attentions have been played on CS for the combinational and multiobject problems [33–41].

The above studies have made great contributions to CS. Nevertheless, according to the implementation in the literature [2], Lévy flights random walk (LFRW), one of search components, is used iteratively to search for new solutions. LFRW uses a mutation operator to generate new solutions based on the best solution obtained so far. A factor in LFRW is utilized to control Lévy flights not to be too aggressive; thus, it is suggested to a constant value, typically 0.01 [2]. In this case, it is beneficial to the solutions which are close to the best one, but it is a disadvantage to those far away from the best one. To avoid the above, Wang et al. [22] proposed a varied factor strategy for CS, named as VCS, where the random sequence factor obeying uniformly distribution is used to replace the constant one. Wang and Zhong [23] used a chaotic sequence factor instead of the constant one, called CCS. However, the above researches make the scale of step size of all dimensions of one solution not different due to the same factor. This may cause a part of dimensions to be too aggressive when the large factor is sampled or too inefficient in the case of the small factor.

In this paper, we aim at avoiding the above problem by using the different factor for the dimensions of each solution and then propose a hybrid factor based cuckoo search algorithm, termed as HFCS. The hybrid factor strategy (HF) combines the constant factor and the varied factor. The constant factor, typically 0.01, is used to benefit the dimensions which are closer to the corresponding ones of the best solution. The varied factor using a random sequence or a chaotic sequence is employed to drive the farer dimensions to be near the corresponding ones of the best one. HFCS selects the dimension of each solution by using the dimensional distance that can be defined as the distance between one dimension and the corresponding one of the best solution. If the dimensional distance of one dimension is shorter than the average of all dimensional distances, then this dimension is selected as the closer one, else as the farer one. The experiments are carried out on 20 benchmark functions to test HFCS, and the results show the improvement in effectiveness and efficiency of hybrid factor strategy.

The remainder of this paper is organized as follows. Section 2 describes the cuckoo search algorithm and the variants. Section 3 presents the proposed algorithm. Section 4 reports the experimental results. Section 5 concludes this paper.

#### 2. Cuckoo Search Algorithm

##### 2.1. CS

CS, a new nature-inspired algorithm 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] is a simple yet very promising population-based stochastic search technique. Generally, a nest represents a candidate solution , when solving an objective function with the solution space , , . Like evolutionary algorithms, the iteration process of CS includes the initial phase and evolutional phase.

In the initial phase, the whole population called solution is randomly sampled from solution space bywhere represents a uniformly distributed random variable on the range [] and is the population size.

According to the implementation of CS shown in the literature [2], CS iteratively uses two random walks: Lévy flights random walk (LFRW) and biased random walk (BRW) to search for new solutions.

LFRW is a random walk whose step size is drawn from Lévy distribution. At generation (), LFRW can be formulated as follows:where is a step size related to the scales of the problem. The ⊕ means entry-wise multiplications. Lévy(* β*) is drawn from a Lévy distribution for large steps:In CS, LFRW is employed to search for new solutions around the best solution obtained so far and implemented according to the following equation [2]:where is a factor (generally, ) and represents the best solution obtained so far:where is a constant and suggested to be 1.5, and are random numbers drawn from a normal distribution with mean of 0 and standard deviation of 1, and is a

*gamma function*.

BRW 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.

After each random walk, CS selects a better solution according to the new generated and the current solutions fitness using the greedy strategy. At the end of each iteration process, the best solution is updated.

##### 2.2. Variants of CS

CS is developed recently, but this algorithm has been researched a lot.

Some studies are an attempt to combine CS with other optimization techniques. Wang et al. [3] and Ghodrati and Lotfi [4], respectively, proposed a hybrid CS with particle swarm optimization. Wang et al. [5] applied differential evolution to optimize the process of selecting cuckoo of the CS model during the process of cuckoo in nest updating. Babukartik and Dhavachelvan [6] proposed the hybrid algorithm combining ant colony optimization and CS. Srivastava et al. [7] combined the CS algorithm's strength of converging to the solution in minimal time along with the tabu mechanism of backtracking from local optimal by Lévy flight. Liu and Fu [8] applied the local search mechanism of the frog leaping algorithm to enhance the local search ability of cuckoo search. Other techniques, such as the orthogonal learning strategy [9], cooperative coevolutionary (CC) framework [10–12], and the teaching-learning-based optimization [13], are also hybridized to enhance the search ability of cuckoo search.

Some variants have paid attention to improve search ability of LFRW and BSW. Walton et al. [14] made a modification to the step size of Lévy flights decreasing as the number of generations increases in order to increase the convergence rate. Ljouad et al. [15] modified Lévy flights model with an adaptive step size based on the number of generations. Valian et al. [16], Wang and Zhou [17], and Mohapatra et al. [18] proposed adaptive step sizes of Lévy flights according to different equations with maximal and minimal step sizes, respectively. Wang et al. [19] and Huang et al. [20] used the chaotic sequence to change the step size of Lévy flights, respectively. Jia et al. [21] proposed the variable step length of Lévy flights and a method of discovering probability. Wang et al. [22, 23], respectively, used random sequence and chaotic sequence as the factor instead of the constant 0.01 in Lévy flights. Coelho et al. [24] integrated the differential operator into Lévy flights to search for new solutions. Mlakar et al. [25] proposed the hybrid algorithm using explicit control of exploration search strategies with the CS algorithm. Ding et al. [26] proposed heterogeneous search strategies based on the quantum mechanism. Wang et al. [27] employed a probabilistic mutation to enhance the Lévy flights. Wang and Zhong [28] added a crossover-like operator in search schema of Lévy flights using one-position inheritance mechanism. Inspired by the social learning and cognitive learning, Li and Yin [29] added these two learning parts into Lévy flights and into BSW. Wang et al. [30, 31] utilized orthogonal crossover and dimension by dimension improvement to enhance the search ability of BSW, respectively. Li and Yin [32] used two new mutation operators based on the rand and best individuals among the entire population to enhance the search ability of BSW.

Other versions have focused on the combinational and multiobject problems. Yang and Deb [33], Hanoun et al. [34], and Chandrasekaran and Simon [35] modified the cuckoo search to solve multiobjective optimization problems. Ouyang et al. [36] and Quaarab et al. [37] proposed the improved CS to solve the travelling salesman problem. Zhou et al. [38] applied an improved CS for solving planar graph coloring problem. Marichelvam et al. [39] and Dasgupta and Das [40] presented the discrete versions for the flow shop scheduling problem. Teymourian et al. [41] applied CS for solving the capacitated vehicle routing problem.

#### 3. HFCS

According to the implementation of CS [2], the factor, for example, 0.01, is used to control Lévy flights not to be too aggressive and to try to make the solutions not jump outside of the search space. In this case, a small step size will be got due to this small factor. Obviously, this makes a contribution to the solutions nearby the best one, but it is not more helpful to the solutions far away from the best one, resulting in the slow convergence. Wang et al. [22, 23] utilized the random sequence and the chaotic sequence instead of the constant one and proved the improvement on the convergence and solution quality. However, the factor with constant or with random sequence or chaotic sequence makes the scale of each dimension of each solution be the same. This perhaps results in a problem that some dimensions near the corresponding dimensions of the best one have larger scale when using random or chaotic sequence, while some dimensions far away from the corresponding dimensions of the best one get the smaller scale in the case of the constant factor. To remedy the above problem, a hybrid factor based cuckoo search is proposed, called HFCS, and presented in Algorithm 1.