3.5. Performance Metrics:
In multi-objective optimization algorithms, the solution set generated approximates the actual Pareto front. The approximated fronts are compared based on convergence, diversity, and uniformity. Various performance matrices are available that account for one, two, or all of the three aspects of the Pareto fronts. These metrics can be unary or binary. In this work, six metrics are used, namely, epsilon, spread, generalized spread, generational distance, inverted generational distance, and hypervolume (Table 4). The matrices require a reference point or reference Pareto front. The non-dominated set from the union of the points obtained from the compared Pareto front is used as the reference Pareto front and a value lower than that of the anti-utopia point of this reference front is taken as a reference point (1e4, 5e4). A detailed comparison of various performance metrics can be found in Zitzler et. al. (2003).
The unary e-indicator metric represents the smallest distance that an approximate Pareto front must be translated to completely dominate the reference Pareto set (Kollat et. al., 2005). The average Euclidean distance between the reference Pareto set and the approximate Pareto solutions is called the Generational distance (GD). Both, epsilon and generational distance measure the convergence. The lower the value of epsilon and generational distance, the better is the convergence. The diversity of the Pareto front can be compared based on the distribution of the solution set (uniformity) and its extent. Spread and generalized spread quantifies the non-uniformity of approximate Pareto front. Small the value of these matrices indicates, a better and diverse set of approximate Pareto front. The inverted GD is the average distance between each member of the reference Pareto front and the approximate Pareto front. It is highly influenced by the distribution of the approximate Pareto front. Hypervolume measures the size of the space enclosed by the approximate Pareto front. Both, inverted generational distance and hypervolume are a unary metric that considers convergence and diversity. In addition, hypervolume is the most widely used performance metric (Riquelme et. al. 2015). Auger A. et. al. (2009) have provided a method to choose the value of reference point for the calculation of hypervolume. Unlike inverted generational distance, the higher the hypervolume better is the combined effect of convergence and diversity of the approximate Pareto front.