An Entropy-Based Multiobjective Evolutionary Algorithm with an Enhanced Elite Mechanism

Hindawi Publishing Corporation Applied Computational Intelligence and Soft Computing Volume 2012, Article ID 682372, 11 pages doi:10.1155/2012/682372 Research Article An Entropy-Based Multiobjective Evolutionary Algorithm with an Enhanced Elite Mechanism Yufang Qin, Junzhong Ji, and Chunnian Liu Beijing Municipal Key Laboratory of Multimedia and Intelligent Software Technology, College of Computer Science and Technology, Beijing University of Technology, Beijing 100124, China Correspondence should be addressed to Junzhong Ji, Received 26 December 2011; Accepted 11 June 2012 Academic Editor: Christian W. Dawson Copyright © 2012 Yufang Qin 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. Multiobjective optimization problem (MOP) is an important and challenging topic in the fields of industrial design and scientific research. Multi-objective evolutionary algorithm (MOEA) has proved to be one of the most efficient algorithms solving the multiobjective optimization. In this paper, we propose an entropy-based multi-objective evolutionary algorithm with an enhanced elite mechanism (E-MOEA), which improves the convergence and diversity of solution set in MOPs effectively. In this algorithm, an enhanced elite mechanism is applied to guide the direction of the evolution of the population. Specifically, it accelerates the population to approach the true Pareto front at the early stage of the evolution process. A strategy based on entropy is used to maintain the diversity of population when the population is near to the Pareto front. The proposed algorithm is executed on widely used test problems, and the simulated results show that the algorithm has better or comparative performances in convergence and diversity of solutions compared with two state-of-the-art evolutionary algorithms: NSGA-II, SPEA2 and the MOSADE. 1. Introduction Optimization problems exist in all kinds of engineering and scientific areas. When there is more than one objective in an optimization problem, it is called a multiobjective optimization problem (MOP). Since these objectives are usually in conflict with each other, the goal of solving a MOP is to find a set of compromise solutions regarding all objectives rather than a best one as in single-objective optimization problems. The solutions of MOP, also called as the Pareto-optimal solutions, are optimal in the sense that there exist no other feasible solutions which would decrease some criteria without causing the increase of at least one other criterion. Evolutionary algorithm (EA) is an optimization algorithm based on the evolution of a population. As it can search for multiple solutions in parallel, it has gained great attention from researchers. In recent years, many excellent EAs [1–4] have been proposed to solve the MOPs efficiently and MOEA has been recognized as one of the best methods to solve the MOPs. Generally, there are two performance measures in evaluating the Pareto-optimal solutions obtained by MOEA. One is the convergence measurement, which evaluates the adjacent degree between the Pareto solutions and the true optimal front. Another one is the diversity measurement, which evaluates the distribution of solutions in the objective space. In order to achieve good performance, many excellent strategies and methods have been presented in MOEA [1, 2, 5–9]. For the convergence, the elite mechanism has proved to be very helpful to accelerate the evolution of population [6]. The basic idea of the elite mechanism is that the information of good solutions, which have occurred in the progress of the evolution, is used to ensure the solution set converge to the optimal front as soon as possible. Its usual practice is that a certain number of best solutions are selected from the population as the parents to produce the good offspring [1]. However, in the early stage of the algorithm applying this strategy, because there are many dominated solutions existing in the population selected as the parents, the population cannot converge at a fast speed. In order to maintain the diversity of nondominated solutions, two main methods are applied. The first is using the grid to maintain the diversity [7]. It draws grids in the objective space and controls the number of solutions in a grid. Although this way can find the better solutions quickly, sometimes it cannot accurately reflect the global distribution of solutions because 2 Applied Computational Intelligence and Soft Computing the grid position is fixed. The second one is the way based on the density [2, 8, 9]. Every solution obtains a value of the density and an outstanding density calculation can help to form a good distribution of solutions. Since 1948, Shannon [10] introduced the information theoretic entropy to measure information content of a stochastic process, which led to the establishment of the field of information theory. Then, many different applications for entropy are given in various fields. In solving the multiobjective optimization problems, Farhang-Mehr and Azarm [11] and Gunawan et al. [12] have applied the entropy to maintain the diversity of the solution set well in multiobjective problems and multilevel multiobjective problems. Wang et al. proposed the MOSADE algorithm [13], which combines the self-adaptive differential evolution and the crowding entropy-based diversity measure to obtain the nondominated solution set. In this algorithm, every solution can calculate its crowding degree through the improved the information entropy formula according to solutions’ distribution. In essence, this method is similar to the crowding distance in NSGA-II. Thus, for some three objective problems, this algorithm cannot obtain the very ideal solution set. In this paper, we propose a new MOEA to solve the MOP more effectively, in which an enhanced elitism makes the nondominated solutions play the better guide role and an entropy-based strategy is applied to preserve the diversity of the population. We call it an entropy-based multiobjective evolution algorithm with an enhanced elitism, namely, E-MOEA in brief. Specifically, we employ the enhanced elitism in which only the nondominated solutions in the union population are selected as the parents to ensure that the solution set converges to the optimal front more quickly. With the algorithm going on, the number of the nondominated solutions in union population will increase gradually. In order to keep the size of the elitist population (the maximum number of the elitist population in our algorithm is set as N) and maintain the diversity of solutions, the strategy based on entropy is applied. In this strategy, a region is determined by taking a solution as its center and the most crowded regions with the most uneven distribution of solutions are found through applying the entropy; t (...truncated)