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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
8000
10000
Iteration
10
5
0
2000
3000
4000
5000
6000
7000
8000
9000
1000
Fig. 4 The evolutionary trajectory of the single-objective optimization algorithm on the Sphere
test function
solutions which are non-dominated with respect to each other and the designer can
use each of them based upon the design criteria.
Findx ¼½
x 1 ;
x 2 ; ...;
x n 2
R n
To minimize f
R m
ð~
x
Þ ¼½
f 1 ð~
x
Þ;
f 2 ð~
x
Þ; ...;
f m ð~
x
Þ 2
By regarding p equality constraints g i ð~
x
Þ ¼
0
;
i
¼
1
;
2
; ...;
p and q inequality
constraints h j ð~
x represents the vector of decision
variables and~f f ð x Þ denotes the vector of objective functions.
As it is mentioned earlier, there is not one unique optimal solution for multi-
objective problems. There exists a set of optimal solutions called Pareto-optimal
solutions. The following de
x
Þ
0
;
j
¼
1
;
2
; ...;
q, where
~
nitions are needed to describe the concept of opti-
mality (Deb et al. 2002 ).
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