A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> C. UNIFORM DESIGN <s> The 0/1 knapsack problem is a well known problem occurring in many real world problems. The problem is NP-complete. The multiobjective 0/1 knapsack problem is a generalization of the 0/1 knapsack problem in which multiple knapsacks are considered. A new version of MOEA/D for solving multiobjective 0/1 knapsack problem is proposed in this paper. This algorithm adopts the uniform design method to generate the aggregation coefficient vectors so that the decomposed scalar optimization sub problems are uniformly scattered, and therefore the algorithm could explore uniformly the region of interest from the initial iteration. Experimental results demonstrate that the proposed algorithm outperforms NSGA-II, SPEA2 and PESA significantly for the 2-objective, 3-objective and 4-objective benchmark knapsack problems. <s> BIB001 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> C. UNIFORM DESIGN <s> In order to exploit the enhancement of the multi-objective evolutionary algorithm based on decomposition (MOEA/D), we propose an improved algorithm with uniform design (UD), i.e. MOEA/D-UD. Three mechanisms in MOEA/D-UD are modified by introducing an experimental design method called UD. To fully employ the information contained in the domain of the multi-objective problem, we apply UD to initialize a uniformly scattered population. Then, motivated by the analysis of the relationship between weight vectors and optimal solutions of scalar subproblems in the study of MOEA/D with adaptive weight adjustment (MOEA/D-AWA), a new weight vector design method based on UD is introduced. To distinguish real sparse regions from pseudo sparse regions, i.e. discontinuous regions, of the complex Pareto front, the weight vector adjustment strategy in MOEA/D-UD adequately utilizes the information from neighbors of individuals. In the experimental study, we compare MOEA/D-UD with three outstanding algorithms, namely MOEA/D with the differential evolution operator (MOEA/D-DE), MOEA/D-AWA and the nondominated sorting genetic algorithm II (NSGA-II) on nineteen test instances. The experimental results show that MOEA/D-UD is capable of obtaining a well-converged and well diversified set of solutions within an acceptable execution time. <s> BIB002 </s> A Survey of Multiobjective Evolutionary Algorithms Based on Decomposition: Variants, Challenges and Future Directions <s> C. UNIFORM DESIGN <s> Any numbers of weight vectors with good diversity can be generated by a uniform design method.A sub-population strategy is used to enhance the local search ability of the algorithm.An update strategy based on decomposition is proposed to maintain the diversity. For many-objective optimization problems, how to get a set of solutions with good convergence and diversity is a difficult and challenging work. In this paper, a new decomposition based evolutionary algorithm with uniform designs is proposed to achieve the goal. The proposed algorithm adopts the uniform design method to set the weight vectors which are uniformly distributed over the design space, and the size of the weight vectors neither increases nonlinearly with the number of objectives nor considers a formulaic setting. A crossover operator based on the uniform design method is constructed to enhance the search capacity of the proposed algorithm. Moreover, in order to improve the convergence performance of the algorithm, a sub-population strategy is used to optimize each sub-problem. Comparing with some efficient state-of-the-art algorithms, e.g., NSGAII-CE, MOEA/D and HypE, on six benchmark functions, the proposed algorithm is able to find a set of solutions with better diversity and convergence. <s> BIB003
The distribution of POSs obtained depends highly on the weight vector generation method. Uniform design methods like the good grid point can be used to initialize population BIB002 , construct crossover operators BIB003 , and generate weight vectors BIB001 , making the solution mapping vectors be distributed more evenly in the objective space.