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In-Depth Information
PSO has been enormously successful in several and various industrial domains and
engineering
fields (Bouall
è
gue et al.
2012a
;Dr
é
o et al.
2006
; Rao and Savsani
2012
;
Siarry and Michalewicz
2008
).
The basic PSO algorithm uses a swarm consisting of N particles N
k
, randomly
distributed in the considered initial search space, to
find an optimal solution x
¼
D
of a generic optimization problem. Each particle, that represents
a potential solution, is characterised by its position and its velocity x
i
;
d
k
argminfx
ðÞ2
R
and v
i
;
d
k
,
respectively.
At each iteration of the algorithm, and in the dth direction, the ith particle
position evolves based on the following update rules:
x
i
;
d
x
i
;
k
þ
v
i
;
d
k
þ
1
k
þ
1
¼
ð
27
Þ
c
2
r
i
2
;
k
p
g
;
d
v
i
;
d
k
w
k
þ
1
v
i
;
d
c
1
r
i
1
;
k
p
i
;
d
x
i
;
d
k
x
i
;
d
k
¼
þ
þ
ð
28
Þ
þ
1
k
k
k
where w
k
þ
1
the inertia factor, c
1
and c
2
the cognitive and the social scaling factors
respectively, r
i
1
;
k
and r
i
2
;
k
the random numbers uniformly distributed in the interval
the best previously obtained position of the ith particle and p
g
;
d
k
[0,1], p
i
;
d
k
the best
obtained position in the entire swarm at the current iteration k.
In order to improve the exploration and exploitation capacities of the proposed
PSO algorithm, we choose for the inertia factor a linear evolution with respect to the
algorithm iteration (Bouall
è
gue et al.
2011
,
2012a
,
b
; Madiouni et al.
2013
):
k
w
max
w
min
k
max
w
k
þ
1
¼
w
max
ð
29
Þ
where w
max
¼
0
:
9 and w
min
¼
0
:
4 represent the maximum and minimum inertia
factor values, respectively.
Finally, the steps of the original version of PSO algorithm, as described in
Eberhart and Kennedy (
1995
), Kennedy and Eberhart (
1995
), can be summarized as
follows:
1. De
ne all PSO algorithm parameters such as swarm size N, maximum and
minimum inertia factor values, cognitive and social coef
…
2. Initialize the particles with random positions and velocities. Evaluate the initial
population and determine p
i
;
d
0
and p
g
;
0
.
3. For each particle apply the update Eqs. (
27
)
cients,
(
29
).
-
4. Evaluate the corresponding
fitness values and select the best solutions.
5. Repeat the steps 3
4 until the termination criterion is reached.
-
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