Modified distance calculation in generational distance and inverted generational distance

In this paper, we propose the use of modified distance calculation in generational distance (GD) and inverted generational distance (IGD). These performance indicators evaluate the quality of an obtained solution set in comparison with a pre-specified reference point set. Both indicators are based on the distance between a solution and a reference point. The Euclidean distance in an objective space is usually used for distance calculation. Our idea is to take into account the dominance relation between a solution and a reference point when we calculate their distance. If a solution is dominated by a reference point, the Euclidean distance is used for their distance calculation with no modification. However, if they are non-dominated with each other, we calculate the minimum distance from the reference point to the dominated region by the solution. This distance can be viewed as an amount of the inferiority of the solution (i.e., the insufficiency of its objective values) in comparison with the reference point. We demonstrate using simple examples that some Pareto non-compliant results of GD and IGD are resolved by the modified distance calculation. We also show that IGD with the modified distance calculation is weakly Pareto compliant whereas the original IGD is Pareto non-compliant.