1. INTRODUCTION

Evolutionary algorithms (EAs), generally known as general-purpose optimization algorithms, are often used to find, within a reasonable compilation time, near-optimal solutions to numerical, real-valued test problems. Differential evolution algorithms (DEs) are one type of recently introduced EA (Price, etc. 2005). DEs have been proposed to overcome the poor local search ability of genetic algorithms (GAs) (Holland, 1975). Selection operations used represent an important difference between GAs and DEs. For GAs, the chance of being selected as a parent solution depends on the relevant solution’s fitness value (Krink, etc., 2004). In DEs, all solutions have an equal chance of being selected as parents, i.e., the chance does not depend on fitness values. After a new solution is produced using self-adjusting mutation and crossover operations, the new solution competes with its parent for the next generation, with the better one winning the competition. In other words, a greedy scheme is applied to select one of the two for the next generation. Using a mutation operation, which is able to self adapt, perform crossover operations and make selections via a greedy process, makes DEs fast-converging evolutionary algorithms (Krink, etc., 2004). This has made them the subject of significant interest by researchers from a diverse range of fields, who have applied DEs to a variety of real world problems (Price, etc., 2005; Krink, etc., 2004).

Swarm intelligence (SI) has been of increasing interest to research scientists in recent years. Swarm intelligence was defined by Bonabeau et al. as any attempt to design algorithms or distributed problem-solving devices based on the collective behavior of social insect colonies or other animals (Bonabeau, etc., 2004). Bonabeau et al. focused primarily on the social behavior of ants (Dorigo, 1992), fish (Li, 2003), birds (Kennedy, etc., 1995) and bees (Pham, etc., 2006) etc. However, the term “swarm” can be applied more generally to refer to any restrained collection of interacting agents or individuals. Although bees swarming around a hive is the classical example of “swarm”, swarms can easily be extended to other systems with similar architectures.

A few models have been developed to model the intelligent behaviors of honeybee swarms and applied to solve combinatorial type problems. Yang (2006) presented a virtual bee algorithm (VBA) that is effective when applied to function optimization problems. However, while the proposed algorithm was similar to GA, it was much more efficient due to the parallelism of multiple independent bees. VBA was tested on two functions with two parameters, single-peaked and multi-peaked, respectively. Results show the VBA as significantly more efficient than GA. Karaboga et al. (2009) presented an artificial bee colony (ABC) algorithm and expanded its experimental results (Basturk, etc., 2006). It has been pointed out that the ABC algorithm outperforms GA for functions exhibiting multi-modality or uni-modality. Pham et al. (2006) presented an original bee algorithm (BA) and applied to two standard functional optimization problems with two and six dimensions. Results demonstrated the BA able to find solutions very close to the optimum, showing that BA generally outperformed GA. Ozbakir et al. (2010) developed a modified BA (Pham, etc., 2006) to solve generalized assignment problems (GAP) that presented an ejection chain neighborhood mechanism. This study found that the proposed BA offers the potential to solve GAP. However, while BA (Pham, etc., 2006) offers the potential to conduct global searches and uses a simpler mechanism in comparison with GA, it is weak in local searching and does not records past searching experiences during the optimization search process.

For instance, a flock of birds may be thought of as a swarm whose individual agents are birds. Particle swarm optimization (PSO), which has become quite popular for many researchers recently (Tsai, 2010; Parsopoulos, etc. 2007), models the social behavior of birds (Kennedy, 1995). PSO is a population-based stochastic optimization technique that is well adapted to the optimization of nonlinear functions in multi-dimensional space. PSO consists of a swarm of particles moving in a search space of potential problem solutions. Every particle has a position vector representing a candidate solution to the problem and a velocity vector. Moreover, each particle contains a small memory that stores its own best position so far and a global best position obtained through communication with neighbor particles. PSO potentially used in local searching, and records past searching experiences during optimization search process. However, it converges early in highly discrete problems (Korenaga, etc., 2006).

Hence, in order to improve BA and PSO, Cheng (2012) and Lien (2012, 2014) proposed an optimization hybrid swarm algorithm, named the particle bee algorithm (PBA), based on intelligent behavior traits of bird and honeybee swarms. PBA has been successful applied to many case studies (Cheng and Lien, 2012; Lien and Cheng, 2012, 2014). PBA integrates their advantages and a self-parameter-updating technique to prevent becoming trapped in a local optimum in high dimensional problems. This study compares the performance of the PBA algorithm with that of DE, EA, PSO (Krink, etc., 2004) and BA (Pham, etc., 2006) for a set of well-known test functions (Krink, etc., 2004). Also, the performance of PBA is analyzed under conditions in which control parameter values change. In Section 2 and 3, bee algorithm (BA) and particle swarm optimization (PSO) are described and then the particle bee algorithm (PBA) is introduced in Section 4. In Section 5, the experimental study is described. Obtained simulation results are presented and discussed in Section 6.

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