Engineering Optimization: Theory and Practice, Fourth Edition

714 Modern Methods of Optimization

6.Find the objective function values at the currentxj(i):

f[x 1 ( 2 )]= 4. 4480 , f[x 2 ( 2 )]= 10. 1721 , f[x 3 ( 2 )]= 11. 2138 ,

f[x 4 ( 2 )]= 11. 9644

Check the convergence of the process. Since the values ofxj(i) id not con-d

verge, we increment the iteration number asi=3 and go to step 4. Repeat

step 4 until the convergence of the process is achieved.

13.5 Ant Colony Optimization

13.5.1 Basic Concept

Ant colony optimization (ACO) is based on the cooperative behavior of real ant

colonies, which are able to find the shortest path from their nest to a food source.

The method was developed by Dorigo and his associates in the early 1990s [13.31,

13.32]. The ant colony optimization process can be explained by representing the opti-

mization problem as a multilayered graph as shown in Fig. 13.3, where the number of

Home

Destination

(Food)

Layer 1 (x 1 ) x 11 x^12 x 13 x 14 x 15 x 16 x 17 x 18

x 21 x 22 x^23 x 24 x 25 x 26 x 27 x 28

x 31

x 32 x 33 x 34 x 35 x 36 x 37 x 38

x 41 x 42 x 43 x 44 x^45 x 46 x 47 x 48

x 51 x 52 x 53 x 54 x^55 x^56 x 57 x 58

x 61 x^62 x^63 x^64 x^65 x^66

x (^67) x 68
Layer 2 (x 2 )
Layer 3 (x 3 )
Layer 4 (x 4 )
Layer 5 (x 5 )
Layer 6 (x 6 )
Figure 13.3 Graphical representation of the ACO process in the form of a multi-layered
network.