Multi-objectivization Via Decomposition: An analysis of helper-objectives and complete decomposition

Multi-objectivization has been used to solve several single objective problems with improved results over traditional genetically inspired optimization methods. Multi-objectivization reformulates the single objective problem into a multiple objective problem. The reformulated problem is then solved with a multiple objective method to obtain a resulting solution to the original problem. Multi-objectivization Via Decomposition (MVD) and the addition of novel objectives are the two major approaches used in multi-objectivization. This paper focuses on analysis of two major MVD methods: helper-objectives and complete decomposition. Helper-objectives decomposition methods identify one or more decomposed objectives that are used simultaneously with the main objective to focus attention on components of the decomposed objectives. Complete decomposition, unlike helper-objectives does not explicitly use the main objective and instead uses decomposed objectives that exhaustively cover all portions of the main objective. This work examines the relationship between helper-objective decompositions and complete decomposition using both an analytic and experimental methodology. Pareto dominance relationships are examined analytically to clarify the relationship between dominant solutions in both types of decompositions. These results more clearly characterize how solutions from the two approaches rank in Pareto-frontier based fitness algorithms such as NSGA-II. An empirical study on job shop scheduling problems shows how fitness signal and fitness noise are affected by the balance of decomposition size. Additionally we provide evidence that, for the settings and instances studied, complete decompositions have a better on-average performance when compared to analogous helper-objective decompositions. Lastly we examine the underlying forces that determine effective decomposition size. We argue that it is advantageous to use less balanced decompositions as within-decomposition conflict increases and as heuristic strength increases.