IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 8, AUGUST 2013 4811 An Improved Artificial Bee Colony Algorithm for Optimal Design of Electromagnetic Devices Xin Zhang , Xiu Zhang , Shiu Yin Yuen , S. L. Ho , and W. N. Fu Department of Electronic Engineering, City University of Hong Kong, Hong Kong Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong Optimal design problems of electromagnetic devices are generally multimodal, nondifferentiable, and constrained. This makes meta- heuristic algorithm a good choice for solving such problems. In this paper, a newly developed metaheuristic algorithm is presented to address the aforementioned issues. The proposed algorithm is based on the paradigm of artificial bee colony (ABC). A drawback of the original ABC algorithm is because its solution variation is only 1-D, as this decreases its convergence speed. In this paper, a one-position inheritance scheme is proposed to alleviate this drawback. An opposite directional (OD) search is also proposed to accelerate the conver- gence of the ABC algorithm. The novel algorithm is applied to both TEAM Workshop problem 22 and a loudspeaker design problem. Both discrete and continuous cases of problem 22 are tested. The effectiveness and efficiency of the proposed algorithm are demonstrated by comparing its performance with those of the original ABC, an improved ABC known as Gaussian ABC, and differential evolution algorithms. Index Terms—Artificial bee colony (ABC), global optimization, loudspeaker design problem, metaheuristic algorithm, TEAM Work- shop problem. I. INTRODUCTION T HE designs of electromagnetic devices can be formulated as optimization problems. As these problems are gener- ally multimodal, nondifferentiable, and constrained, traditional optimization methods are not particularly applicable [1]. In- stead, metaheuristic algorithm is a good choice, because it needs few or even no assumptions about the optimization problems. Recently, metaheuristic algorithms have been widely em- ployed to resolve inverse electromagnetic problems. These algorithms are efficient, robust, and easy to use. Typical paradigms of metaheuristic algorithms include particle swarm optimization (PSO) [2], bacterial foraging strategies (BFSs) [3], cross-entropy method (CE) [4], and differential evolution (DE) [5]. Although a number of inverse electromagnetic problems have been effectively addressed by PSO, BFS, CE, and DE, the effectiveness and efficiency of the paradigm of artificial bee colony (ABC) [6] have hardly been reported. Thus, this paper will concentrate on utilizing the paradigm of ABC to resolve optimization problems. The original ABC algorithm is first reviewed. Based on the original ABC algorithm, two modifications, which are, namely, the one-position inheritance (OPI) mechanism and the opposite directional (OD) search, are proposed to speed up the conver- gence of the ABC algorithm. Essentially, both methods can bal- ance the exploration and exploitation of the search process in a self-adaptive manner. Simulation experiments are conducted on mathematical functions and inverse electromagnetic prob- lems (TEAM Workshop problem 22 [7], [8] and a loudspeaker Manuscript received October 03, 2012; revised January 10, 2013; accepted January 14, 2013. Date of publication January 18, 2013; date of current version July 23, 2013. Corresponding author: X. Zhang (e-mail: eex.zhang@connect. polyu.hk). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2013.2241447 design problem [9]). The findings are then analyzed and com- pared with those obtained using the original ABC algorithm and DE and Gaussian ABC [10] algorithms. It is found that the pro- posed algorithm outperforms these algorithms. II. ARTIFICIAL BEE COLONY At the initialization phase, a population of solutions is randomly created, where denotes population size. For the ABC algorithm, a solution is likened to a food source, which could attract bees to gather nectar and make honey. Correspond- ingly, the function value of a solution is equivalent to the nectar amount of a food source. After initialization, a cycle of the ABC algorithm includes the employed bee phase, the onlooker bee phase, and the scout bee phase, as described below. At the employed bee phase, employed bees are sent out and the ratio of employed bees and food sources is one to one. In ABC, an employed bee searching around the associated food source is implemented by if otherwise (1) where , , and denote the th element of , , and , respectively; is a random number; is a random integer, and is the problem dimension; and are different solutions in the current population; and is a newly generated candidate solution. After evaluating ,a greedy selection between and is executed and the winner is stored as the new . At the onlooker bee phase, onlooker bees are sent out; however, unlike the employed bee phase, an onlooker bee se- lects the food source according to its nectar amount. In ABC, this behavior is implemented by first calculating a probability 0018-9464/$31.00 © 2013 IEEE