Page 343 - Decision Making Applications in Modern Power Systems

Heuristic methods for the evaluation of environmental impacts Chapter | 12 305

The NSGA-II, the first version based on GAs, is classified as an elitist type,
since it incorporates a preservation mechanism of the dominant solutions
through several generations of a GA.
The process starts from a set of size N solutions (couple) obtained ran-
domly or methodically. Later generations are determined using modified
mechanisms of selection, crossover, and mutation defined by classic GA.

12.2.3.1 Selection process, crossover, and mutation
On the current population (pair), randomly selected N pairs of solutions are
selected. Each pair competes in a tournament in winning alternative that
belongs to the category of best quality. If the dominance of alternatives
belongs to the same front, then winning it introduces a greater degree of
diversity to all that are under construction. The winners of each tournament
are allowed only for seed; the crossover and mutation are handled in the
same way as shown by the classic GA.
Thus what is expected is that the genetic information of the dominant
alternative be present in the following generations and attract the rest of the
population to their respective neighborhoods.

12.2.3.2 Stacking operator
The multiobjective algorithms seek to find a big number of solutions that
belong to the Pareto front. Therefore it is necessary that the population be
kept as much diverse as possible. The stacking operator quantifies the space
around an alternative that is not occupied by any other solution. This is due
to calculating the perimeter of the cuboid formed by neighboring solutions
that have the same category of the alternative dominance i, which is
described by the following equation:

m
M I ð i11 Þ I ð i21 Þ
m
f m 2 f m
X
d i 5 ð12:1Þ
f 2 f
max min
m51 m m
m
where I is a vector indicating the nearby alternative solution alternative i,
f max and f min are the maximum and minimum values in the function of the
m m
solution space object m, respectively, and M is the number of optimized
objective functions.
12.2.3.3 Selection by tournament second stacking operator
This procedure replaces the selection used in traditional GA. They consist of
comparing two solutions; each one of them has two attributes:
A range of nondomination r i , according to the Pareto front.
A local stacking distance, d i .