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Fig. 2.
Power profiles considered in the study
-
The number of (different) non-dominated solutions (ND).
-
Generational Distance (GD): the sum of the distances between the non-
dominated solutions in the computed Pareto front and a set of uniformly
distributed points in the true Pareto front (Eq. 4). Smaller values of GD
mean a better approximation to the Pareto front.
-
Spread (
s
): evaluates the dispersion of non-dominated solutions in the com-
puted Pareto front, including the distance from the extreme points of the
true Pareto front (Eq. 5);
d
i
is the distance between solution
i
in the com-
puted Pareto front and its nearest neighbor,
d
is the average of all
d
i
and
d
h
is the distance between the extreme of the
h
-thobjectiveinthetrue
Pareto front and the closest point in the computed Pareto front [5]. Smaller
values of spread mean a better distribution of non-dominated solutions.
d − d
i
h
=1
i
=1
k
ND
v∈P
∗
d
h
+
d
(
v,P
)
s
=
(5)
GD
=
(4)
h
=1
k
|P
∗
|
d
d
h
+
ND×
-
Relative hypervolume (RHV): the ratio of the volume (in the objective func-
tions space) covered by the computed Pareto front and the one covered by
the true Pareto front. The ideal RHV value is 1.
The true Pareto front—unknown for the problem instances studied—was ap-
proximated by gathering the non-dominated solutions found for each instance,
in each execution performed for the three variants of the proposed MOEA.
5.3 Results and Discussion
All the reported results were computed in 15 independent executions of the
proposed MOEA for each problem instance dimension solved.
Parameter setting
. The proposed MOEA was executed using the best parame-
ter values from a previous analysis that studied: population size (#P, candidate
values
{
50,75,100
}
), and the probabilities for crossover (
p
C
, candidate values
{
). The best
results were obtained using the parameter setting #P = 75,
p
C
=0.9,and
p
M
=
0.01. The stopping criterion is 500 generations, providing a reasonable trade-off
between results quality and execution time.
0.6,0.75,0.9
}
) and mutation (
p
M
, candidate values
{
0.01,0.05,0.1
}
