Relation Between Objective Space Normalization and Weight Vector Scaling in Decomposition-Based Multiobjective Evolutionary Algorithms

Real-world multiobjective optimization problems (MOPs) usually have conflicting and differently scaled objectives. To deal with such problems, objective space normalization is widely used in the multiobjective evolutionary algorithm (MOEA) design, especially, in the design of decomposition-based MOEAs. It has been demonstrated that uniformly distributed solutions can be obtained for badly scaled MOPs by decomposition-based MOEAs with objective space normalization. Recently, weight vector scaling has also been used for badly scaled MOPs. In some studies, it was argued that weight vector scaling and objective space normalization are essentially the same when applied to decomposition-based MOEAs. In this article, we theoretically and empirically show the relation between objective space normalization and weight vector scaling. Our results demonstrate that similarities and differences between the two methods depend on the choice of a scalarizing function. How the choice between normalization and weight vector scaling affects decomposition-based MOEAs with solution assignment mechanisms is also analyzed.