Computational Intelligence and Neuroscience

Volume 2016, Article ID 9820294, 13 pages

http://dx.doi.org/10.1155/2016/9820294

## A New Modified Artificial Bee Colony Algorithm with Exponential Function Adaptive Steps

^{1}Department of Mathematics, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, China^{2}Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things, Zigong, Sichuan 643000, China^{3}School of Automation and Electronic Information, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, China

Received 26 July 2015; Revised 10 October 2015; Accepted 19 October 2015

Academic Editor: Yufeng Zheng

Copyright © 2016 Wei Mao 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

As one of the most recent popular swarm intelligence techniques, artificial bee colony algorithm is poor at exploitation and has some defects such as slow search speed, poor population diversity, the stagnation in the working process, and being trapped into the local optimal solution. The purpose of this paper is to develop a new modified artificial bee colony algorithm in view of the initial population structure, subpopulation groups, step updating, and population elimination. Further, depending on opposition-based learning theory and the new modified algorithms, an improved -type grouping method is proposed and the original way of roulette wheel selection is substituted through sensitivity-pheromone way. Then, an adaptive step with exponential functions is designed for replacing the original random step. Finally, based on the new test function versions CEC13, six benchmark functions with the dimensions and are chosen and applied in the experiments for analyzing and comparing the iteration speed and accuracy of the new modified algorithms. The experimental results show that the new modified algorithm has faster and more stable searching and can quickly increase poor population diversity and bring out the global optimal solutions.

#### 1. Introduction

It is well known that algorithms for solving various characteristics optimization problems can be classified into different groups, such as population-based algorithms, stochastic algorithms, deterministic algorithms, and iterative algorithms. An algorithm is called population-based [1] if one works with a group of solutions and tries to improve them. Two important classes of population-based optimization algorithms are exactly evolutionary algorithms and swarm intelligence-based algorithms [2]. Swarm intelligence is an innovative artificial intelligence technique with collective behavior of self-organized systems [3]. Since many swarm intelligence algorithms, such as genetic algorithm (GA) [4], particle swarm optimization (PSO) [5], ant colony optimization (ACO) [6], and biogeography-based optimization [7], have simplicity, ease of implementation, outstanding performance, and other advantages [8], they have shown great success in solving some nonconvex, discontinuous, or nondifferentiable optimization problems. However, these intelligence algorithms are sensitive to value and precision. Thus, inspired by the behavior of honey bees, Karaboga [9] introduced basic artificial bee colony algorithm (ABC) in 2005 and constituted one of the most prominent approaches in the field of bee-inspired algorithms. Further, in consideration of the solution reaching speed, the success rate, and the performance rate, El-Abd [10] provided a complete performance assessment of ABC and compared it with the widely known differential evolution (DE), GA, heuristic algorithms, PSO, and other foraging algorithms (e.g., bacterial algorithm, ACO, and bacterial foraging optimization) by using the well-known benchmark functions in [11].

It was claimed that ABC is the most successful algorithm for multimodal and hybrid functions [12]. This is because ABC has no demand to the objective function, constraint, and external information and is only based on fitness function in the search process [13]. Further, ABC has the following advantages. (i) The mechanism of multiple roles: using different methods, bees adjust quality of the solutions spontaneously, so as to adapt to the next search process [14]. (ii) The cooperative working mechanism: according to the information from other bees, bees decide whether to find the optimal solution with larger probability [15]. (iii) The strong robustness: the search rules are not certain but are probabilistic and have excellent robustness and a wide range of applicability [16]. (iv) The stability: even if the individual fails, the entire swarm can still complete the task [17]. (v) Less control parameters, simple operation, and ease of implementation [18]: indeed, ABC has been shown to be very competitive with respect to other state-of-the-art foraging and evolutionary algorithms.

However, there exists still an insufficiency to ABC because ABC does well in exploration but is poor at exploitation. As for the improvement and development of ABC, Karaboga and Gorkemli [19] proposed a new update rule for onlooker bees in the hive to improve the local search and convergence characteristics of ABC. Inspired by PSO, Imanian et al. [20] changed the update rule of basic ABC to increase the convergence speed for solving high dimensional and continuous optimization problems. Wang et al. [21] proposed multistrategy ensemble artificial bee colony algorithm (MEABC) to improve the local and global search capability of basic ABC and tested the performance of MEABC by using basic, shifted, and rotated benchmark functions. Gao et al. [22] developed new search equations to adjust exploration and exploitation capability of ABC. In a different approach for ABC, Das et al. [23] proposed a learning routine based on fitness and proximity stimuli and tested the method with standard benchmark functions. Zang et al. [24] designed a logarithmic function adaptive step instead of the original random step.

Moreover, in dealing with some complex problems by applying ABC, there are some defects such as slow search speed, poor population diversity, stagnation in the working process, and trapping into the local optimal solution [13]. Recently, ABC has been extended and improved by many researches. But since ABC is relatively new, the researches in the literatures lack systematicness and are scattered. See, for example, [1–3, 12–30] and the references therein. In 2012, Li et al. [31] pointed out that “ABC has no mechanism to use the global information in the search space, so it easily results in a waste of computing power and gets trapped in local optima”. Further, the authors proposed a novel algorithm (named as DEABC, i.e., differential evolution artificial bee colony algorithm), which synthesizes DE and ABC and enhances individuals by sharing information between DE population and bee colony. For related works, one can see [1, 30] and the references therein.

Motivated and inspired by the above works, a new modified artificial bee colony (MABC) algorithm shall be constructed based on adaptive step with exponential functions. By using opposition-based learning theory and an improved -type grouping method, the initial population for MABC will be given, and the original way of roulette wheel selection shall be substituted by sensitivity-pheromone way. Specifically, an adaptive step with exponential functions will be designed to replace the original random step. In order to verify the validity of the improved algorithm, MABC, it is compared with DEABC [30], novel artificial bee colony algorithm (NABC) [8], and ABC; the experiment results tested with six well-known benchmark optimization functions, which are chosen from new test function versions CEC13, show that MABC is superior to ABC, NABC, and DEABC.

The rest of this paper is organized as follows. Section 2 gives a brief introduction to ABC. The new MABC is presented and analyzed in Section 3. Section 4 presents and discusses the experimental results of six benchmark functions with the dimensions and , respectively. Finally, the conclusion is drawn in Section 5.

#### 2. Brief Review to ABC

In this section, a brief review on ABC is going to be given.

##### 2.1. Thoughts of the Algorithm

A honey bee colony can successfully discover the highest quality food sources in nature. Hence, the idea of ABC comes from intelligent foraging behavior of honey bees to finding good solutions for solving optimization problems. In a general way, according to the ways of searching food, the colony of bees is divided into three kinds: employed bees, onlooker bees, and scout bees. The employed bees are responsible for exploiting the nectar sources. They explore the beforehand food source position and give the quality information of the food to the onlooker bees. The onlooker bees wait in the hive and decide to exploit a food source based on the information shared by the employed bees. In order to find a new nectar source, the scout bees randomly search environment either depending on an internal motivation or based on possible external clues [32]. The position of a nectar source implies a possible solution of the optimization problems, and the profitability of a nectar source corresponds to the quality (fitness) of the possible solution. Each nectar source is exploited by only one employed bee. In other words, the number of nectar sources equals the number of employed bees or onlooker bees [25]. In this process, the employed bees maintain good solution, the onlooker bees improve convergence speed, and the scout bees enhance the ability to remove local optimum [26, 31].

##### 2.2. ABC Iteration Steps

From [24, 33], it follows that main iteration steps of ABC can be listed as follows.

*Step 1 (initialization). *Randomly generate solutions (i.e., food sources) as initial population in a dimension searching space, where is the number of food sources, which equals the half of the colony size, is a -dimensional solution vector and the th food source in the initial population for , and also denotes the number of optimization parameters.

*Step 2 (renewing population). *In the stage of collecting honey, each employed bee produces a new nectar source within the neighborhood of the food source. After comparing the new nectar source with the old ones, the high probability will be memorized. Next, in the stage of follow, every onlooker bee evaluates the profitability of nectar sources taken from all employed bees and then chooses a food source at a certain probability. As in the case of the employed bees, she produces a modification on the source position in her memory and keeps the better nectar source. The regeneration of nectar sources in these two stages is based on the following formula:where , , , and is a random number, which controls the generation range of neighborhood of . With the search being close to optimal solution, the range of neighborhood will become smaller and smaller.

*Step 3 (nectar source selection). *In the stage of follow, the onlooker bees choose food source through comparing the probability which is computed by the fitness value. The nectar sources of high probability are selected in large probability. And the probability of being selected for the food sources is calculated as follows:where is the fitness (profitability) value of the th solution, which is obtained by the following equation:where is the objective function value for , which is specific for the optimization problem. If the new food source position has a quality equal to or better than the old one, then the old food source position is replaced by the new one. Otherwise, the old one is retained, which is the same as the employed bees stage.

*Step 4 (population elimination). *If a solution has not been improved significantly through a predetermined number of trials, called “max iteration,” then the solution is regarded as falling into a local optimal solution and their original position will be abandoned. Thus, the corresponding employed bees will become scout bees and a new solution instead of the eliminated solution is randomly generated, which can be expressed as follows:where , are the same as in (1) and and denote the th individual maximum and the th individual minimum values, respectively.

Based on the above iteration steps, the process of ABC [12] can be shown in Algorithm 1.