8 An Introduction to Many-Objective Evolutionary Optimization
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8.3.1.3 Progressive Methods
In the two sections above, we have mentioned that the decision makers can
input their preferences before or after the optimization loop. The other possibility
is to input their preferences during the optimization. The decision making and
optimization are intertwined, i.e., within the optimization loop, the decision makers
need to give preference information [41].
One way to do this is by generating a set of solutions and requiring the decision
makers to pick their most favorite. These favorite solutions are taken to update the
preference information, and then new solutions are generated. The process could be
repeated until a stopping criterion is reached.
8.3.2 Solution Quality Assessment
Comparing solutions in multi- and many-objective problems is not a trivial task
because what we have is a set of non-dominated solutions instead of only one
solution. This would imply that we need to define what makes a non-dominated set
better than another non-dominated set. These measurements are called performance
metrics or performance indices.
Performance metrics are usually based on three criteria: cardinality, accuracy,
and diversity [35, 36]. Cardinality simply means the number of points in the
non-dominated set; accuracy measures convergence to the real Pareto front; and
diversity measures how well spread the solutions are in the objective space. A
performance metric can measure more than one criteria simultaneously.
The number of performance metrics currently available is vast. This section will
only introduce the top two most used metrics between 2005 and 2013: hypervolume
and generational distance. Some other metrics are also described in Sects. 8.3.3
and 8.4.2; the metrics in the section are used to rank solutions within the population;
e.g., the crowding distance, non-dominated ranking, etc. Other popular metrics are
the -indicator [48] and R-metric [24] which can compare performances of a pair of
solution sets in all three aforementioned criteria simultaneously [36].
8.3.2.1 Hypervolume
The hypervolume is the most widely used performance metric [36]. Hypervolume
is a generalization of the area (2D), or volume (3D) in higher dimensions. In a
biobjective problem, the hypervolume is measured as the area covered by the nondominated set with respect to a reference point (see Fig. 8.8).
The maximum hypervolume can only be achieved by the real (possibly continuous) Pareto front, thus maximizing hypervolume is a straightforward and general
goal to approach the real Pareto front [43]. The hypervolume can also measure
279
8.3.1.3 Progressive Methods
In the two sections above, we have mentioned that the decision makers can
input their preferences before or after the optimization loop. The other possibility
is to input their preferences during the optimization. The decision making and
optimization are intertwined, i.e., within the optimization loop, the decision makers
need to give preference information [41].
One way to do this is by generating a set of solutions and requiring the decision
makers to pick their most favorite. These favorite solutions are taken to update the
preference information, and then new solutions are generated. The process could be
repeated until a stopping criterion is reached.
8.3.2 Solution Quality Assessment
Comparing solutions in multi- and many-objective problems is not a trivial task
because what we have is a set of non-dominated solutions instead of only one
solution. This would imply that we need to define what makes a non-dominated set
better than another non-dominated set. These measurements are called performance
metrics or performance indices.
Performance metrics are usually based on three criteria: cardinality, accuracy,
and diversity [35, 36]. Cardinality simply means the number of points in the
non-dominated set; accuracy measures convergence to the real Pareto front; and
diversity measures how well spread the solutions are in the objective space. A
performance metric can measure more than one criteria simultaneously.
The number of performance metrics currently available is vast. This section will
only introduce the top two most used metrics between 2005 and 2013: hypervolume
and generational distance. Some other metrics are also described in Sects. 8.3.3
and 8.4.2; the metrics in the section are used to rank solutions within the population;
e.g., the crowding distance, non-dominated ranking, etc. Other popular metrics are
the -indicator [48] and R-metric [24] which can compare performances of a pair of
solution sets in all three aforementioned criteria simultaneously [36].
8.3.2.1 Hypervolume
The hypervolume is the most widely used performance metric [36]. Hypervolume
is a generalization of the area (2D), or volume (3D) in higher dimensions. In a
biobjective problem, the hypervolume is measured as the area covered by the nondominated set with respect to a reference point (see Fig. 8.8).
The maximum hypervolume can only be achieved by the real (possibly continuous) Pareto front, thus maximizing hypervolume is a straightforward and general
goal to approach the real Pareto front [43]. The hypervolume can also measure