Evolutionary Method for Weight Vector Generation in Multi-Objective Evolutionary Algorithms based on Decomposition and Aggregation

The generation of weight vectors is the primary step in MOEA based on decomposition and aggregation methods, affecting the diversity of the Pareto approximation and overall performance of the algorithm. The basic methods, following the method proposed by Scheffe, have some limitations mainly when the number of objectives increases, because the number of weight vectors and hence the population size becomes very large. In this paper, we present a new method for weight vector generation that can create an arbitrary number of weight vectors, almost equally spaced, located in a surface in the first orthant of the objective space, with free choice of norm. The proposed evolutionary algorithm is able to prevent the creation of weight vectors along the border of the orthant, which is a region that contains solutions of little interest to the decision maker. With a small modification in the proposed method it is also possible to create cones of weight vectors, useful to explore specific regions of the decision space defined by preference directions. In our experiments, different sets of weight vectors were generated, varying the number of vectors and the dimension of the space. The validation of the results was given by the mean distance of each vector to its nearest neighbor, as well as the standard deviation and the Pearson coefficient of variation for this mean value. The results indicate that the proposed method is able to produce a distribution of vectors close to a uniform distribution, with no clustering of points, being useful for guiding decomposition-based MOEA.