Computational Intelligence and Neuroscience

Volume 2016, Article ID 6097484, 17 pages

http://dx.doi.org/10.1155/2016/6097484

## A Novel Quantum-Behaved Bat Algorithm with Mean Best Position Directed for Numerical Optimization

^{1}Communications Engineering, Chongqing University, Chongqing 400030, China^{2}Jiuquan Satellite Launch Center, Jiuquan 732750, China

Received 26 January 2016; Revised 11 April 2016; Accepted 26 April 2016

Academic Editor: Christian W. Dawson

Copyright © 2016 Binglian Zhu 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 novel quantum-behaved bat algorithm with the direction of mean best position (QMBA). In QMBA, the position of each bat is mainly updated by the current optimal solution in the early stage of searching and in the late search it also depends on the mean best position which can enhance the convergence speed of the algorithm. During the process of searching, quantum behavior of bats is introduced which is beneficial to jump out of local optimal solution and make the quantum-behaved bats not easily fall into local optimal solution, and it has better ability to adapt complex environment. Meanwhile, QMBA makes good use of statistical information of best position which bats had experienced to generate better quality solutions. This approach not only inherits the characteristic of quick convergence, simplicity, and easy implementation of original bat algorithm, but also increases the diversity of population and improves the accuracy of solution. Twenty-four benchmark test functions are tested and compared with other variant bat algorithms for numerical optimization the simulation results show that this approach is simple and efficient and can achieve a more accurate solution.

#### 1. Introduction

In recent years, with the need of optimization problems in reality, all kinds of bioinspired optimization algorithms or swarm intelligence optimization algorithms have been proposed, such as the genetic algorithm (GA) [1], differential evolution (DE) [2], ant colony optimization (ACO) [3], firefly algorithm (FA) [4, 5], cuckoo search (CS) algorithm [6, 7], particle swarm optimization (PSO) [8], artificial bee colony (ABC) optimization [9], and bat algorithm (BA) [10, 11]. These bionic intelligent algorithms are random search methods which mimic natural biological systems [12]; entirely instinct depends on the organism itself, by the unconscious optimization behavior to optimize its survival to adapt to the environment. Compared with the traditional optimal algorithms, they do not depend on the characteristic of strict mathematical optimization problem itself and have the characteristic of strong parallelism. Each individual has self-organization and evolution; as a result, some high dimensional optimization problems are superior to the traditional methods.

The bat algorithm is a new metaheuristic method which was proposed by Yang in 2010 [10, 11]. The capability of echolocation of microbats is fascinating as these bats can find their prey and discriminate different types of insects even in complete darkness [13]. Inspired by the echolocative behavior of bats, this algorithm carries the search process using artificial bats as search agents mimicking the natural pulse loudness and emission rate of real bats. When these bats are chasing prey, they tend to decrease the loudness and increase the rate of pulse emission. Bat algorithm has a good optimization performance in low dimensional case [14–16] and is widely used in engineering optimization [17, 18] and multiobjective optimization [19]; however, due to low population diversity, it severely suffers from premature convergence problem in the high dimensional case [20]. So many variant bat algorithms are proposed to enhance the population diversity and to avoid being trapped into local optimum. In [21], Lin et al. put forward a chaotic Levy flight bat algorithm (CLBA) for parameter estimation in nonlinear dynamic biological system; in [13], Xie et al. introduced the difference operator and Levy flight trajectory into the bat algorithm (DLBA) to solve function optimization problems; in [22], Yılmaz and Kucuksille, inspired by standard particle swarm optimization (PSO) [8] algorithm and artificial bee colony (ABC) algorithm [9], proposed improved bat algorithm (IBA). In IBA, the velocity of each bat is updated with linear decreasing inertia weight factor and the frequency is self-adaptive to improve the exploration and exploitation. In [23], Wang and Guo put the harmony search method into the bat algorithm and developed a hybrid metaheuristic HSBA method for optimization problem, speeding up convergence of bat algorithm; in [24], Yılmaz et al. have studied the mechanism of updating loudness and pulse emission rate of BA and they found that loudness and pulse emission rate provide a balance between exploitation and exploration. So they modify the equations to improve the exploration capability of BA. Afrabandpey et al. [25] introduced the chaotic sequences into bat algorithm in different ways to avoid premature convergence. In [26], four parameters of BA are replaced by the chaotic maps separately; meanwhile, in [20], loudness and pulse emission rate are tuned via multiplying a linear decreasing or increasing function by a chaotic map function; in [25], chaotic map takes place of random number for parameter initialization. These approaches, to some extent, can avoid getting trapped into local minimum. However, in QMBA, bats adopt different search strategies in different times and have the mechanism of jumping out of local optimal solution; these strategies enhance the convergence speed of the algorithm and improve the accuracy of solutions.

The rest of paper is organized as follows. Section 2 describes the standard BA and Section 3 presents the quantum-behaved bat algorithm with the direction of mean best position. The simulation and comparison of this proposed algorithm are presented in Section 4. Finally, general conclusions are drawn in Section 5.

#### 2. The Bat Algorithm

Echolocation is a very important character of bats; Yang proposed bat algorithm by mimics of bats’ foraging behavior. Bats fly randomly in the air or in the process of searching for prey by using echolocation to catch food and to avoid obstacles. In order to transform these behaviors of bats to algorithm, there are some approximations and idealized rules [10].(i)All bats use echolocation to sense distance, and they also “know” the difference between food/prey and background barriers in some magical way.(ii)Bats fly randomly with velocity at position with a fixed frequency , varying wavelength , and loudness to search for prey. They can automatically adjust the wavelength (or frequency) of their emitted pulses and adjust the rate of pulse emission , depending on the proximity of their target.(iii)Although the loudness can vary in many ways, we assume that the loudness varies from a large (positive) to a minimum constant value .

In the BA, for the th bats of swarm having position (solution), velocity , and frequency , each bat will move toward the current best position (solution), and its position, velocity, and frequency are updated during the course of iteration as follows:where is a random number of a uniform distribution in and represents the current global best solution (position) after comparing all the solutions (positions) among all the bats. These equations can guarantee the exploration ability of BA.

For the local search, when a solution is selected among the current best solutions, a new candidate solution can be generated as where is a random number in and directs new solution apart from or close to the current best solution. Here, is mean value of all bats of loudness.

When finding prey, bat will gradually decrease the loudness and increase the rate of pulse emission in order to track its prey to capture it. The loudness and pulse emission rate update accordingly as the iterations proceed as shown in where and are constants. In fact, the parameter controls the convergence of bat algorithm and therefore plays a similar role as the cooling factor in the simulated annealing algorithm [27]. For simplicity, we set in our simulations.

The basic steps of BA can be summarized as the pseudocode shown in Algorithm 1.