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.