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operators introduce new individuals into a population by creating a
variation of a current individual, thus adding variability into the population
and preventing stagnation of the search in local optima. Several modified
PSO algorithms with mutation operators have been proposed
17,18,42-44
to
improve the global search capability of PSO. They use a mutation operator
to change a particle dimension value using a random number drawn from
a probability distribution, such as Gaussian or Cauchy distribution. A
particle is selected for mutation using a mutation rate that is decreased
during a run, i.e., as the number of iterations increases, the effect of the
mutation operator decreases.
4.5.2.
Adaptive evolutionary particle swarm optimization
(AEPSO) algorithm
To enhance the global exploratory capability of PSO while maintaining
a fast rate of convergence, especially in the context of multi-objective
optimization, we incorporate non-dominated sorting, adaptive inertia
weight and a special mutation operator into the particle swarm optimization
algorithm.
17
With this strategy, the particle's velocity in Equation (4.6) is
modified as follows:
v
i
(
t
+1)=
wv
i
(
t
)+[
r
1
(
p
i
−
x
i
(
t
)) +
r
2
(
p
g
−
x
i
(
t
))] +
v
m
(
t
)
.
(4.8)
The second term in Equation (4.8) can be viewed as an acceleration
term, which depends on the distances between the current position
x
i
,the
personal best
p
i
, and the global best
p
g
. The acceleration factor is defined
as follows:
t
N
t
,
α
=
α
0
+
t
=1
,
2
,...,N
t
,
(4.9)
where
N
t
denotes the number of iterations,
t
represents the current
generation, and the suggested range for
α
0
is [0.5, 1].
As can be seen from Equation (4.9), the acceleration term will increase
as the number of iterations increases, which will enhance the global search
ability as the search proceeds and help the algorithm to jump out of local
optima, especially in the case of multimodal problems.
Furthermore, instead of using a linearly-decreasing inertia weight, we
use a random number, which has been shown by Zhang
et al.
16
to improve
the performance of the PSO in some benchmark functions. Hence, in this
