Computational Intelligence and Neuroscience

Volume 2016, Article ID 8085953, 13 pages

http://dx.doi.org/10.1155/2016/8085953

## An Enhanced Artificial Bee Colony Algorithm with Solution Acceptance Rule and Probabilistic Multisearch

Department of Industrial Engineering, Uludag University, Görükle Campus, 16059 Bursa, Turkey

Received 3 July 2015; Accepted 9 September 2015

Academic Editor: Ezequiel López-Rubio

Copyright © 2016 Alkın Yurtkuran and Erdal Emel. 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

The artificial bee colony (ABC) algorithm is a popular swarm based technique, which is inspired from the intelligent foraging behavior of honeybee swarms. This paper proposes a new variant of ABC algorithm, namely, enhanced ABC with solution acceptance rule and probabilistic multisearch (ABC-SA) to address global optimization problems. A new solution acceptance rule is proposed where, instead of greedy selection between old solution and new candidate solution, worse candidate solutions have a probability to be accepted. Additionally, the acceptance probability of worse candidates is nonlinearly decreased throughout the search process adaptively. Moreover, in order to improve the performance of the ABC and balance the intensification and diversification, a probabilistic multisearch strategy is presented. Three different search equations with distinctive characters are employed using predetermined search probabilities. By implementing a new solution acceptance rule and a probabilistic multisearch approach, the intensification and diversification performance of the ABC algorithm is improved. The proposed algorithm has been tested on well-known benchmark functions of varying dimensions by comparing against novel ABC variants, as well as several recent state-of-the-art algorithms. Computational results show that the proposed ABC-SA outperforms other ABC variants and is superior to state-of-the-art algorithms proposed in the literature.

#### 1. Introduction

Optimization techniques play an important role in the field of science and engineering. Over the last five decades, numerous algorithms have been developed to solve complex optimization algorithms. Since more and more present-day problems turn out to be nonlinear, multimodal, discontinuous, or dynamic in nature, derivative-free, nonexact solution methods attract ever-increasing attention. Evolutionary biology or swarm behaviors inspired most of these methods. There have been several classes of algorithms proposed in this evolutionary or swarm intelligence framework including genetic algorithms [1, 2], memetic algorithms [3], differential evolution (DE) [4], ant colony optimization (ACO) [5], particle swarm optimization (PSO) [6], artificial bee colony algorithm (ABC) [7], cuckoo search [8], and firefly algorithm [9].

The ABC is a biologically inspired population-based metaheuristic algorithm that mimics the foraging behavior of honeybee swarms [7]. Due to its simplicity and ease of application, the ABC has been widely used to solve both continuous and discrete optimization problems since its introduction [10]. It has been shown that ABC tends to suffer poor intensification performance on complex problems [11–13]. To improve the intensification performance of ABC, many researchers have focused on the search rules as they control the tradeoff between diversification and intensification. Diversification means the ability of an algorithm to search for unvisited points in the search region, whereas intensification is the process of refining those points within the neighborhood of previously visited locations to improve solution quality. Various new search strategies, mostly inspired from PSO and DE, have been proposed in the literature. Zhu and Kwong [14] proposed a global best guided ABC, which utilizes the global best individual’s information within the search equation similar to PSO. Gao et al. [15] introduced another variant of global best ABC. Inspired by DE, Gao and Liu [13] introduced a modified version of the ABC in which ABC/Best/1 and ABC/Rand/1 were employed as local search equations. Kang et al. [16] described the Rosenbrock ABC, which combines Rosenbrock’s rotational method with the original ABC. To improve diversification, Alatas [11] employed chaotic maps for initialization and chaotic searches within a search strategy. Akay and Karaboga [17] introduced a modified version of the ABC in which frequency of perturbation is controlled adaptively and the ratio of variance operator was introduced. Liao et al. [18] proposed a detailed experimental analysis and comparison of an ABC variant with different search equations. Gao et al. [19] introduced two new search equations for onlooker and employed bee phases and a new robust comparison technique for candidate solutions. ABC. Qiu et al. [20] were inspired from DE/current-to-best/1 strategy in DE algorithm and proposed a modified ABC. Banitalebi et al. [21] proposed an enhanced compact ABC, which did not store the actual population of candidate solution; instead their approach employed probabilistic representation. Wang et al. [22] presented multistrategy ABC, in which a pool of different search strategies was constructed and various search strategies were used during the search process. Gao et al. [23] introduced a bare bones ABC with parameter adaptation and fitness-based neighborhood to improve the intensification performance of standard ABC. Ma et al. [24] reduced the redundant search moves and maintained the diversity of the swarm by introducing hybrid ABC with life cycle and social learning. Furthermore, ABC has been successfully applied to solve various types of optimization problems, such as production scheduling [25, 26], vehicle routing [27], location-allocation problem [28], image segmentation [29], wireless sensor network routing [30], leaf-constrained minimum spanning tree problem [31], clustering problem [32], fuel management optimization [33], and many others [34–36]. Readers can refer to Karaboga et al. [10] for an extensive literature review of the ABC and its applications.

This study presents an enhanced ABC with solution acceptance rule and probabilistic multisearch (ABC-SA) in order to solve global optimization problems efficiently. In ABC-SA, three search mechanisms with different diversification and intensification characteristics are employed. Moreover, search mechanism selection probabilities , and are introduced to control the balance between diversification and intensification. In our proposed approach, a search mechanism is established using the selection probabilities to generate a new neighbor solution from the current one. Additionally, a solution acceptance rule is implemented, in which not only better solutions but also worse solutions may be accepted by using a probability function. A nonlinearly decreasing acceptance probability function is employed, thus allowing worse solutions to be more likely accepted in the early phases of the search. Therefore, ABC-SA algorithm explores the search space more widespread, especially in the early phases of the search process. By using solution acceptance rule and implementing different search mechanisms of contrasting nature, ABC-SA balances the trade-off between diversification and intensification efficiently. The proposed approach is tested on six different benchmark functions with varying dimensions and compared to novel ABC, PSO, and DE variants. Computational results reveal that ABC-SA outperforms competitor algorithms in terms of solution quality.

The main contributions of the proposed study are as follows:(i)Three different search mechanisms are employed with varying diversification and intensification abilities. Probabilistic multisearch with predetermined probability values are employed to determine the search mechanism to be used to generate candidate solutions. Therefore, ABC-SA explores and exploits the search space efficiently.(ii)Instead of a greedy selection, a new candidate solution acceptance rule is integrated, where a worse solution may have a chance to be accepted as new solution. By the help of this new acceptance rule, ABC-SA achieves better diversification performance, specifically in the early phases of the search.

The remainder of this paper is structured as follows: Section 2 presents the traditional ABC; Section 3 introduces the proposed framework; the instances, parameter settings, and computational results are presented in Section 4 and finally Section 5 concludes the paper.

#### 2. Artificial Bee Colony Algorithm

The ABC has inspired from the organizational nature and foraging behavior of honeybee swarms. In the ABC algorithm, the bee colony comprises three kinds of bees: employed bees, onlooker bees, and scout bees. Each bee has a specialized task in the colony to maximize the nectar amount that is stored in the hive. In ABC, each food source is placed in the -dimensional search space and represents a potential solution to the optimization problem. The amount of nectar in the food source is assumed to be the fitness value of a food source. Generally, the number of employed and onlooker bees is the same and equal to the number of food sources.

Each employed bee belongs to a food source and is responsible for mining the corresponding food source. Then, employed bees pass the nectar information to onlooker bees in the “dance area.” Onlooker bees wait in the hive and select a food source to mine based on the information coming from the employed bees. Here, more beneficial food sources will have higher selection probabilities to be selected by onlooker bees. In ABC, in order to decide if a food source is abandoned or not, trial counters and a predetermined limit parameter are used. If a solution represented by a food source does not improve during a number of trials (limit), the food source is abandoned. When the food source is abandoned, the corresponding employed bee will become a scout bee and randomly generate a new food source and replace it with the abandoned one.

The ABC algorithm consists of four main steps: initialization, employed bee phase, onlooker bee phase, and scout bee phase. After the initialization step, the other three main steps of the algorithm are carried out repeatedly in a loop until the termination condition is met. The main steps of the ABC algorithm are as follows.

*Step 1 (initialization). *In the initialization step, the ABC generates a randomly distributed population of* SN* solutions (food sources), where* SN* also denotes the number of employed or onlooker bees. Let represent the th food source, where is the problem size. Each food source is generated within the limited range of th index bywhere , , is a uniformly distributed random real number in , and and are the lower and upper bounds for the dimension , respectively. Moreover, a trial counter for each food source is initialized.

*Step 2 (employed bee phase). *In the employed bee phase, each employed bee visits a food source and generates a neighboring food source in the vicinity of the selected food source. Employed bees search a new solution, , by performing a local search around each food source as follows:where is a randomly selected index and is a randomly chosen food source that is not equal to ; that is, . is a random number within the range generated specifically for each and combination. A greedy selection is applied between and by selecting the better one.

*Step 3 (onlooker bee phase). *Unlike the employed bees, onlooker bees select a food source depending on the probability value , which is determined by nectar amount associated with that food source. The value of is calculated for th food source as follows:where is the fitness value of solution and calculated as in (4) for minimization problems. Different fitness functions are employed for maximization problems. By using this type of roulette wheel based probabilistic selection, better food sources will more likely be visited by onlooker bees. Therefore, onlooker bees try to find new candidate food sources around good solutions. Once the onlooker bee chooses the food source, it generates a new solution using (2). Similar to the employed bee phase, a greedy selection is carried out between and .

*Step 4 (scout bee phase). *A trial counter is associated with each food source, which depicts the number of tries that the food source cannot be improved. If a food source cannot be improved for a predetermined number of tries (limit) during the onlooker and employed bee phases, then the employed bee associated with that food source becomes a scout bee. Then, the scout bee finds a new food source using (1). By implementing the scout bee phase, the ABC algorithm easily escapes from minimums and improves its diversification performance.

It should be noted that, in the employed bee phase, a local search is applied to each food source, whereas in the onlooker bee phase better food sources will more likely be updated. Therefore, in ABC algorithm, the employed bee phase is responsible for diversification whereas the onlooker bee phase is responsible of intensification. The flow chart of the ABC is given in Figure 1.