A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> F. COMPUTATIONAL RESOURCES ALLOCATION <s> Recently, MOEA/D multi-objective evolutionary algorithm based on decomposition has achieved great success in the field of evolutionary multi-objective optimization and has attracted a lot of attention. It decomposes a multi-objective optimization problem MOP into a set of scalar subproblems using uniformly distributed aggregation weight vectors and provides an excellent general algorithmic framework of evolutionary multi-objective optimization. Generally, the uniformity of weight vectors in MOEA/D can ensure the diversity of the Pareto optimal solutions, however, it cannot work as well when the target MOP has a complex Pareto front PF; i.e., discontinuous PF or PF with sharp peak or low tail. To remedy this, we propose an improved MOEA/D with adaptive weight vector adjustment MOEA/D-AWA. According to the analysis of the geometric relationship between the weight vectors and the optimal solutions under the Chebyshev decomposition scheme, a new weight vector initialization method and an adaptive weight vector adjustment strategy are introduced in MOEA/D-AWA. The weights are adjusted periodically so that the weights of subproblems can be redistributed adaptively to obtain better uniformity of solutions. Meanwhile, computing efforts devoted to subproblems with duplicate optimal solution can be saved. Moreover, an external elite population is introduced to help adding new subproblems into real sparse regions rather than pseudo sparse regions of the complex PF, that is, discontinuous regions of the PF. MOEA/D-AWA has been compared with four state of the art MOEAs, namely the original MOEA/D, Adaptive-MOEA/D, -MOEA/D, and NSGA-II on 10 widely used test problems, two newly constructed complex problems, and two many-objective problems. Experimental results indicate that MOEA/D-AWA outperforms the benchmark algorithms in terms of the IGD metric, particularly when the PF of the MOP is complex. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> F. COMPUTATIONAL RESOURCES ALLOCATION <s> This letter suggests an approach for decomposing a multiobjective optimization problem (MOP) into a set of simple multiobjective optimization subproblems. Using this approach, it proposes MOEA/D-M2M, a new version of multiobjective optimization evolutionary algorithm-based decomposition. This proposed algorithm solves these subproblems in a collaborative way. Each subproblem has its own population and receives computational effort at each generation. In such a way, population diversity can be maintained, which is critical for solving some MOPs. Experimental studies have been conducted to compare MOEA/D-M2M with classic MOEA/D and NSGA-II. This letter argues that population diversity is more important than convergence in multiobjective evolutionary algorithms for dealing with some MOPs. It also explains why MOEA/D-M2M performs better. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> F. COMPUTATIONAL RESOURCES ALLOCATION <s> The resulting set of solutions obtained by MOEA/D depends on the weights used in the decomposition. In this work, we use this feature to incorporate user preferences into the search. We use co-evolutionary approach to change the weights adaptively during the run of the algorithm. After the user specifies their preferences by assigning binary preference values to the individuals, the co-evolutionary step improves the distribution of weights by creating new (offspring) weights and selecting those that better match the user preferences. The algorithm is tested on a set of benchmark functions with a set of different user preferences. <s> BIB003
The computational cost increases sharply as the number of objectives increases and becomes extremely high in solving MaOPs. Therefore, it is significant to allocate computational resources reasonably. Many algorithms have been proposed to assign weight vectors that are used to allocate different amounts of computational resources for subproblems according to some criteria, such as computational difficulties of solving subproblems  , geometrical characteristics of a PF BIB001 , and user preferences BIB003 . To realize the adaptive decomposition for a region BIB002 and overcome the defect of wasting resources due to fixed and uniform weight vectors, another potential method is to assign subpopulations with different sizes to their corresponding subproblems.