Complexity

Volume 2018, Article ID 1753071, 20 pages

https://doi.org/10.1155/2018/1753071

## A Decomposition-Based Multiobjective Evolutionary Algorithm with Adaptive Weight Adjustment

College of Computer Science, Shaanxi Normal University, Xi’an 710119, China

Correspondence should be addressed to Xiujuan Lei; nc.ude.unns@ieljx

Received 24 April 2018; Revised 25 July 2018; Accepted 1 August 2018; Published 12 September 2018

Academic Editor: Michele Scarpiniti

Copyright © 2018 Cai Dai and Xiujuan Lei. 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

Recently, decomposition-based multiobjective evolutionary algorithms have good performances in the field of multiobjective optimization problems (MOPs) and have been paid attention by many scholars. Generally, a MOP is decomposed into a number of subproblems through a set of weight vectors with good uniformly and aggregate functions. The main role of weight vectors is to ensure the diversity and convergence of obtained solutions. However, these algorithms with uniformity of weight vectors cannot obtain a set of solutions with good diversity on some MOPs with complex Pareto optimal fronts (PFs) (i.e., PFs with a sharp peak or low tail or discontinuous PFs). To deal with this problem, an improved decomposition-based multiobjective evolutionary algorithm with adaptive weight adjustment (IMOEA/DA) is proposed. Firstly, a new method based on uniform design and crowding distance is used to generate a set of weight vectors with good uniformly. Secondly, according to the distances of obtained nondominated solutions, an adaptive weight vector adjustment strategy is proposed to redistribute the weight vectors of subobjective spaces. Thirdly, a selection strategy is used to help each subobjective space to obtain a nondominated solution (if have). Comparing with six efficient state-of-the-art algorithms, for example, NSGAII, MOEA/D, MOEA/D-AWA, EMOSA, RVEA, and KnEA on some benchmark functions, the proposed algorithm is able to find a set of solutions with better diversity and convergence.

#### 1. Introduction

In real-world applications, there are many problems needed to simultaneously optimize multiple objectives which are typically characterized by conflicting objectives. These problems are called as multiobjective optimization problems (MOPs). A continuous optimization problem can be formulated as follows [1]: where is a -dimensional decision variable bounded in the decision space , and is the number of objective functions. is the th objective function to be minimized, defines the th inequality constraint, and defines the th equality constraint. Moreover, all the inequality and equality constraints determine a set of feasible solutions which is denoted by , and is denoted as the objective space. Because the objectives often contradict each other, the improvement of one objective may cause to the deterioration of other objectives. So, MOPs have many optimal solutions which can be called nondominated solutions [2]. Some important definitions are introduced as follows. Let , is said to be better than , if and for . If there is no other such that is better than , is called Pareto optimal solution. The set of all the Pareto optimal solutions is defined as the Pareto set (PS). The image of the PS is called the Pareto optimal front (PF) [2].

Because the number of PF may be infinite, it is impractical to obtain all the Pareto optimal solutions. Thus, the principal goal of solving MOPs is to find out a set of solutions with good diversity and convergence. Currently, multiobjective evolutionary algorithms (MOEAs) use the strategy of the population evolution to simultaneously optimize the solutions of the population in a run. MOEAs can well deal with some complex problems which are characterized with discontinuity, multimodality, and nonlinearity [3]. Nowadays, many MOEAs [4–30] with good performance have been proposed, such as multiobjective genetic algorithms [3], multiobjective particle swarm optimization algorithms [7–10], multiobjective differential evolution algorithms [10, 11], multiobjective immune clone algorithms [12], group search optimizer [13], evolutionary algorithms based on decomposition [14–17], and hybrid algorithms [8, 22]. Moreover, many MOEAs are used to solve numerous applications [31–34].

Recently, Zhang and Li [16, 21] introduce the decomposition approaches into MOEA and developed an outstanding MOEA, MOEA/D, which has a superior performance for many problems. MOEA/D decomposes the MOP into a number of subproblems and uses the EA to optimize these subproblems simultaneously. The two main advantages of MOEA/D are that it uses the neighbor strategy to improve the search efficiency and well maintain the diversity of obtained solutions by the given weight vectors. In the last decade, MOEA/D has attracted many research interests and many related articles [17–22] have been published.

In this work, we mainly study the refinement of weight vectors in MOEA/D to enhance the diversity of obtained solutions. Zhang and Li [16] claim that the weight vectors should be selected properly to obtain the nondominated solutions evenly distributed over the true PF. The basic assumption of MOEA/D is that the set of weight vectors with good uniformity can help obtained nondominated solutions to maintain the diversity. However, recent studies have suggested that MOEA/D which uses the fixed weight vectors might not well solve MOPs with complex PFs [35].

In this paper, we develop an improved decomposition-based multiobjective evolutionary algorithm with adaptive weight vector adjustment (IMOEA/DA) to solve MOPs. The main contributions of this paper are as follows: firstly, a new method [36] based on uniform design and crowding distance [5] is used to generate a uniformity of weight vectors; secondly, some weight vectors are adaptively deleted or added according to the distances of obtained nondominated solutions to solve the problems with complex PF; thirdly, a selection strategy is used to help each subobjective space to obtain a nondominated solution (if have). The frame of decomposition-based multiobjective evolutionary algorithm with adaptive weight vector adjustment and the initialization method of weight vectors has been studied. Moreover, the research result has been presented in the conference “2017 13th International Conference on Computational Intelligence and Security (CIS)”. In this conference paper, the adaptive weight adjustment [37] is used. In this new paper, a new adaptive weight adjustment is proposed.

The rest of this paper is organized as follows: Section 2 summarizes the related works of refinements of the weight vectors. Section 3 presents the proposed algorithm IMOEA/DA in detail, while the experiment results of the proposed algorithm and the related analysis are given in Section 5; finally, Section 5 provides the conclusions and proposes the future work.

#### 2. Related Works

MOEA/D uses the predetermined uniformly distributed weight vectors. Recent studies have shown that the fixed weight vector used in MOEA/D might not be able to cover the whole PF very well [35]. Therefore, some researches have refined the weight vectors in MOEA/D. Gu and Liu [38] periodically create the new weight vectors according to the distribution of the current set of weight vectors. Li and Landa-Silva [35] suggest that according to the strategy, the solution of each subproblem should be a long way from the corresponding nearest neighbor to adjust each weight vector. Qi et al. [37] propose an adaptive weight adjustment which utilizes the obtained nondominated solutions to reinstall the weight vectors. In the adaptive weight adjustment, the intersection angle of the target vector of each nondominated solution and the corresponding weight vector of this nondominated solution is zero. Jiang et al. [39] develop an adaptive weight adjustment by sampling the regression curve of objective vectors of the solution in an external population.

Other MOEAs use reference points to solve MOPs. These algorithms guide solutions to converge to the reference points. The principle of algorithms based on reference points or weight vectors is the same. Jain and Deb [40] adjust the reference points in terms of the distribution of candidate solutions in the current population at each generation. Jain and Deb [40] delete reference points with an empty niche and randomly add new reference points inside each crowded reference point with a high niche count. Cheng et al. [41] design two sets of reference vectors, where one maintains uniformly distributed and the other one is adaptively adjusted. Asafuddoula et al. [42] also adopt two sets of reference vectors, where one is called active set which is adaptively adjusted and the other one is called inactive set which stores the discarded reference vectors. In this algorithm [42], the two sets of reference vectors are tuned dynamically over the course of evolution.

#### 3. The Proposed Algorithm

In this paper, an improvement decomposition-based multiobjective evolutionary algorithm with adaptive weight vector adjustment (IMOEA/DA) is proposed to address the MOPs with complex PF. The proposed algorithm mainly consists of two parts: a new weight vector initialization method based on uniform design and crowding distance and adaptive weight vector adjustment strategy, which will be introduced in this section.

##### 3.1. Motivation

The main goal of this paper is to use decomposition-based multiobjective evolutionary algorithm to obtain a set of nondominated solutions which evenly distribute on the true PF and have a good convergence. In this paper, we adaptively add or delete some weight vectors to achieve this goal. In decomposition-based multiobjective evolutionary algorithms, the main role of weight vectors is to improve the convergence of obtained solutions by guiding the search of subproblems. Thus, weight vectors should maintain relative stability to improve the convergence of obtained solutions. We adaptively delete or add some weight vectors by the distances of obtained nondominated solutions to maintain relative stability of weight vectors and solve the problems with complex PF. In addition, we consider that the current optimal solutions of some subproblems are dominated solution, but their optimal solutions are nondominated solution. Some of corresponding weight vectors of these subproblems are retained, and a selection strategy is used to make each subproblem obtain a nondominated solution (if have).

##### 3.2. A New Weight Vector Initialization Method-Based Uniform Design

In this subsection, the new weight vector initialization-based [36] uniform design and crowding distance is presented. Firstly, a uniform design method is briefly shown. For a given bounded and closed set (where is the dimension of the set ), the uniform design was developed to sample some points which have a small number and are uniformly scattered on . In this paper, we only consider a specific case of and introduce the main features of uniform design. More details can be obtained by referring the literature [34].

For a given set , in general, a set of exactly uniformly scattered points on is very difficult to be found. However, there are some efficient methods that can find a set of well approximately uniformly scattered points on . The good lattice point method (GLP) [43] is one of the simple and efficient methods and can generate a set of uniformly scattered points on . The details of GLP are as follows. For given integers , , and , a integer matrix called uniform array is denoted by
where , and is the remainder of . Thus, there are different integer matrices be generated by these all *μ*. So, for given and , they can determine a number (Table 1 lists the vales of for different values of and ) which determines an integer matrix with the smallest discrepancy among these different integer matrices. In this paper, the discrepancy is denoted as , where is the fraction of the points falling in the hyperrectangle . In practice, the greatest common divisor of and should be 1 to reduce the amount of calculation, which is because that the integer matrix with the smallest discrepancy must be determined by these [43]. Each row of matrix determiners a point of by