A new performance metric for multiobjective optimization: the integrated sphere counting

A large number of evolutionary algorithms for solving multiobjective optimization problems has been already developed. Several merit factors for comparing the outcomes of these algorithms have also been proposed. However, evaluating Pareto-surface sample sets is still considered an open problem, since the result of a multiobjective evolutionary algorithm is a collection of vectors forming a nondominated set, that can be viewed under rather different merit criteria. In this paper, we present a new performance metric: the Integrated Sphere Counting. This metric is motivated on two reasoning principles: (i) the Pareto-surface is an object that is to be described via sample sets, in a sense that is similar to the sampled function description in signal processing; and (ii) the resolution that is to be employed in the Pareto-surface sample set depends on the decision-making procedure resolution, instead of the surface structure itself. We test this metric with two benchmark problems: the 0/1 Knapsack Problem and ZDT number 6 test suite.