A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 70

Fig. 13 The non-dominated
solutions of the hybrid
method for the SCH test
function

4

Pareto optimal front
Hybrid algorithm

3.5

3

2.5

2

1.5

1

0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4

f 1 (x)

Fig. 14 The non-dominated
solutions of the hybrid
method for the FON test
function

1

Pareto optimal front
Hybrid algorithm

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0 0.2 0.4 0.6 0.8 1

f 1 (x)

(1) A proper indication of the gap between the non-dominated solution members
and the Pareto optimal front is gained by means of the metric of distance ( ! )

(Deb et al. 2002 ) as follows:

X

n

d i

! ¼ ð 13 Þ

i ¼ 1

where n is the number of members in the set of non-dominated solutions and di i is
the least Euclidean distance between the member i in the set of non-dominated