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x
10
4
14
GA with traditional crossover
GA with multiple crossover
Standard PSO
Hybrid algorithm
12
10
8
6
4
2
0
0
2000
4000
6000
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10000
Ite
r
ation
1000
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0
2000
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9000
10000
Fig. 7 The evolutionary trajectory of the single-objective optimization algorithm on the Step test
function
measures is used (Mahmoodabadi et al.
2013
). To this end, a neighborhood radius
R
neighborhood
is de
ned for the whole non-dominated solutions. Two non-dominated
solutions are regarded neighbors in case the Euclidean distance of them is less than
R
neighborhood
. Based upon this radius, the number of neighbors of each non-domi-
nated solution is computed in the objective function domain and the particle having
fewer neighbors is chosen as leaders. Furthermore, for particle i, the nearest
member of the archive is devoted to
!
pbest
i
. At this stage, a multi-objective opti-
mization algorithm using the hybridization of genetic operators and PSO formula
can be presented (Mahmoodabadi et al.
2013
). In elaboration, the population is
randomly generated. Once the
rst
archive can be produced. The inertia weight, the learning factors and operator
fitness values of all members are computed, the
s
probabilities are computed at each iteration. The genetic operators, that is, mutation
operators, traditional crossover and multiple-crossover are utilized to change some
chromosomes selected randomly. Each chromosome corresponds to a particle in it
and the group of chromosome can be regarded as a swarm. On the other hand, the
'
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