A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) COMBINATION OF THE ORIGINAL DECOMPOSITION METHODS <s> This paper proposes an alternate method for finding several Pareto optimal points for a general nonlinear multicriteria optimization problem. Such points collectively capture the trade-off among the various conflicting objectives. It is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the Pareto set given an evenly distributed set of parameters, a property which the popular method of minimizing weighted combinations of objective functions lacks. Further, this method can handle more than two objectives while retaining the computational efficiency of continuation-type algorithms. This is an improvement over continuation techniques for tracing the trade-off curve since continuation strategies cannot easily be extended to handle more than two objectives. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) COMBINATION OF THE ORIGINAL DECOMPOSITION METHODS <s> The use of Pareto dominance for fitness evaluation has been the mainstream in evolutionary multiobjective optimization for the last two decades. Recently, it has been pointed out in some studies that Pareto dominance-based algorithms do not always work well on multiobjective problems with many objectives. Scalarizing function-based fitness evaluation is a promising alternative to Pareto dominance especially for the case of many objectives. A representative scalarizing function-based algorithm is MOEA/D (multiobjective evolutionary algorithm based on decomposition) of Zhang & Li (2007). Its high search ability has already been shown for various problems. One important implementation issue of MOEA/D is a choice of a scalarizing function because its search ability strongly depends on this choice. It is, however, not easy to choose an appropriate scalarizing function for each multiobjective problem. In this paper, we propose an idea of using different types of scalarizing functions simultaneously. For example, both the weighted Tchebycheff (Chebyshev) and the weighted sum are used for fitness evaluation. We examine two methods for implementing our idea. One is to use multiple grids of weight vectors and the other is to assign a different scalarizing function alternately to each weight vector in a single grid. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) COMBINATION OF THE ORIGINAL DECOMPOSITION METHODS <s> MOEA/D is a generic multiobjective evolutionary optimization algorithm. MOEA/D needs a approach to decompose a multiobjective optimization problem into a number of single objective optimization problems. The commonly-used weighted sum approach and the Tchebycheff approach may not be able to handle disparately scaled objectives. This paper suggests a new decomposition approach, called NBI-style Tchebycheff approach, for MOEA/D to deal with such objectives. A portfolio management MOP has been used as an example to test the effectiveness of MOEA/D with NBI-style Tchebycheff approach. <s> BIB003 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> 2) COMBINATION OF THE ORIGINAL DECOMPOSITION METHODS <s> MOEA/D is one of the most popular multi-objective evolutionary algorithms. To extend the effective application scope of MOEA/D for high-dimensional objectives, an improved MOEA/D design for many-objective optimization problems, named I-MOEA/D, is proposed in this paper. Comparing with the original MOEA/D, we redesigned the weight vectors used in the subproblems, making the distribution of the weight vectors broader and more effective to ensure the diversity and convergence of solutions in the objective space. Moreover, a new decomposition approach, called the weighted mixture-style method, which combines the advantages of the weighted sum decomposition and the Tchebycheff decomposition approaches, is adopted in I-MOEA/D to improve the effectiveness of the algorithm. A three-part experimental comparison using DTLZ1-DTLZ4, with the number of objectives ranging from three to fifteen, is performed. Experimental results verify the effectiveness of each strategy and reveal that the proposed I-MOEA/D method achieves better performance than the other related state-of-the-art algorithms in solving this type of many-objective optimization problems. <s> BIB004
Each decomposition method used widely has its own advantages and disadvantages. Therefore, it is a simple and effective strategy to combine the complementary advantages of different decomposition methods, which makes the algorithm obtain more solutions. TCH usually performs well on MOPs with convex and nonconvex PFs, while WS is only suitable for the optimization problems with convex PFs. Further, WS converges faster than TCH and gets a smoother PF. Ishibuchi et al. BIB002 used both WS and TCH for fitness assessment, which is mainly realized by two mechanisms based on a grid that consists of weight vectors. One mechanism assigns separate decomposition methods to different grids, i.e., all weight vectors within the same grid use the same decomposition method. The other mechanism alternately uses one of these two different decomposition methods to each of all weight vectors in a single grid. Wei et al. BIB004 presented a weighted mixture-style (WM) decomposition method that combines the advantages of WS and TCH, which allows the algorithm to obtain more solutions. WM decomposition method was adopted in MOEA/D to transform an MaOP into a set of single-objective subproblems and to solve them in parallel. The experimental results showed that the WM decomposition method outperforms the three typical methods, i.e., WS, TCH, and PBI under the framework of MOEA/D. WS and TCH are unsuitable for handling some MOPs with very different objective sizes. A direction-based decomposition method, termed normal boundary intersection (NBI) approach BIB001 , is relatively insensitive to the scales of objective functions. Therefore, Zhang et al. BIB003 suggested a new decomposition method, called NBI-style Tchebycheff, which fully combines the characteristics of TCH and NBI. The algorithm that integrates this decomposition method into MOEA/D was presented, which finds two extreme points with minimum function values and then lets the solution mapping vectors approximate PF by minimizing the maximum distance between the points on the line composed of two extreme points and the point (solution) on the weight vector. Finally, the experimental study on portfolio optimization of real-world problems demonstrated the effectiveness of the proposed algorithm.