An algorithm for calculating the hypervolume contribution of a set

The reliability model can be optimized with a multi-objective optimization algorithm, while hypervolume-based multi-objective evolutionary algorithms (MOEAs) have been shown to produce better results for multi-objective problem in practice. When hypervolume is used in some MOEAs as archiving strategy, diversity mechanism or selection criterion to guide the search, it is necessary to determine which subset contributes the least hypervolume contribution. Few algorithms have been designed for this purpose. In this paper a new algorithm based on HSO (hypervolume by slicing objective) is proposed for calculating the exclusive hypervolume contributions of each subset to the whole nondominated set directly for small dimension. The new algorithm is composed of two parts: the algorithm SHSO (set hypervolume contribution by slicing objective) and the algorithm SHSO*. SHSO is used to calculate the exclusive hypervolume contribution of a subset to the whole nondominated set. SHSO* is applied to select the subset which contributes the least hypervolume contribution by repeated application of SHSO. Compared with HSO adopted for calculating the exclusive hypervolume contribution, SHSO* outperforms HSO for all of the test fronts with small dimension.