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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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