A Hybrid Global Optimization Algorithm: Particle Swarm Optimization in Association with a Genetic Algorithm - Complex System Modelling and Control Through Intelligent Soft Computations - page 60

Table 10 The comparison results among single-objective optimization algorithms for the

Rastringin function
GA (traditional
crossover)

GA (multiple-
crossover)

PSO
(standard)

The hybrid
algorithm

7.54 × 10 +1 1.52 × 10 +2 4.11 × 10 +1 2.15 × 10 − 3

Mean

10 +1 10 +1 10 0 10 − 2 Standard

deviation

1.90 × 4.37 × 9.19 × 1.18 ×

Table 11 The comparison results among single-objective optimization algorithms for the

Griewank function

GA (traditional
crossover)

GA (multiple-
crossover)

PSO
(standard)

The hybrid
algorithm

7.16 × 10 − 1 4.86 × 10 − 1 2.15 × 10 − 2 2.03 × 10 − 2

Mean

10 0 10 0 10 − 2 10 − 2

Standard
deviation

2.87 × 1.4478 × 2.31 × 2.21 ×

Table 12 The comparison results among single-objective optimization algorithms for the Ackley
function

GA (traditional
crossover)

GA (multiple-
crossover)

PSO
(standard)

The hybrid
algorithm

1.57 × 10 +1 1.80 × 10 +1 2.81 × 10 − 1 4.30 × 10 − 8

Mean

10 0 10 − 1 10 − 1 10 − 8 Standard

deviation

1.13 × 6.23 × 5.92 × 9.90 ×

but the result is very close to the results of the hybrid method. Figures 4 , 5 , 6 , 7 , 8 , 9
and 10 illustrate the evolutionary traces of some test functions of Table 2 . In these

figures, the mean best values are gained for thirty runs. The maximum iteration,
population size, and dimension are set at 1,000, 10 and 50, respectively. In these

figures, the vertical axis is the value of the best function found after each iteration of
the algorithms and the horizontal axis is the iteration. By comparing these figures, it

is obtained that the combination of the traditional crossover, multiple-crossover and
mutation operator can enhance the performance of particle swarm optimization.

5.2 Multi-objective Optimization

5.2.1 De nition of Multi-objective Optimization Problem

In most of real problems, there is more than one objective function required to be
optimized. Furthermore, most of these functions are in con

ict with each other.
Hence, there is not just one solution for the problem and there are some optimal

fl