Relation between Weight Vectors and Solutions in MOEA/D

An important implementation issue in MOEA/D (multiobjective evolutionary algorithm based on decomposition) is the specification of a scalarizing function. For its appropriate specification, it is necessary to understand the search behavior of MOEA/D for various settings of a scalarizing function. Especially, it is important to understand the relation between weight vectors and obtained solutions. The understanding of this relation is also very important for the incorporation of preference information into MOEA/D through weight vector specification. In this paper, we examine the mapping from weight vectors to solutions by monitoring which solution is obtained from each weight vector. MOEA/D with a number of different settings of a scalarizing function is applied to knapsack problems and DTLZ2 with 2-6 objectives. As a scalarizing function, we use the weighted sum, the weighted Tchebycheff and the PBI (penalty-based boundary intersection). We report some interesting observations obtained from computational experiments. Among them are the existence of many duplicated solutions, their positive and negative effects, and a dominant effect of the penalty parameter value in the PBI.