Engineering Optimization: Theory and Practice, Fourth Edition

13.4 Particle Swarm Optimization 713

so that

v 1 ( 1 )= 0 + 0. 3294 (− 1. 5 + 1. 5 )+ 0. 9542 ( 1. 25 + 1. 5 )= 2. 6241

v 2 ( 1 )= 0 + 0. 3294 ( 0. 0 − 0. 0 )+ 0. 9542 ( 1. 25 − 0. 0 )= 1. 1927

v 3 ( 1 )= 0 + 0. 3294 ( 0. 5 − 0. 5 )+ 0. 9542 ( 1. 25 − 0. 5 )= 0. 7156

v 4 ( 1 )= 0 + 0. 3294 ( 1. 25 − 1. 25 )+ 0. 9542 ( 1. 25 − 1. 25 )= 0. 0

(c) Find the new values ofxj( 1 ), j= 1 , 2 , 3 , 4 , asxj(i)=xj(i− 1 )+vj(i):

x 1 ( 1 )=− 1. 5 + 2. 6241 = 1. 1241

x 2 ( 1 )= 0. 0 + 1. 1927 = 1. 1927

x 3 ( 1 )= 0. 5 + 0. 7156 = 1. 2156

x 4 ( 1 )= 1. 25 + 0. 0 = 1. 25

5.Evaluate the objective function values at the currentxj(i):

f[x 1 ( 1 )]= 11. 9846 , f[x 2 ( 1 )]= 11. 9629 , f[x 3 ( 1 )]= 11. 9535 ,

f[x 4 ( 1 )]= 11. 9375

Check the convergence of the current solution. Since the values ofxj(i) did

not converge, we increment the iteration number asi=2 and go to step 4.

4.(a) FindPbest, 1 = 1. 1241 , Pbest, 2 = 1. 1927 , Pbest, 3 = 1. 2156 , Pbest, 4 = 1. 2 5,
andGbest= 1. 1 241.
(b) Compute the new velocities of particles (by assumingc 1 =c 2 = and using 1
the random numbers in the range (0, 1) asr 1 = 0. 1 482 andr 2 = 0. 4 867):

vj(i)=vj(i− 1 )+r 1 (Pbest ,j−xj(i))+r 2 (Gbest−xj(i));j= 1 , 2 , 3 , 4

so that

v 1 ( 2 )= 2. 6240 + 0. 1482 ( 1. 1241 − 1. 1241 )+ 0. 4867 ( 1. 1241 − 1. 1241 )= 2. 6240

v 2 ( 2 )= 1. 1927 + 0. 1482 ( 1. 1927 − 1. 1927 )+ 0. 4867 ( 1. 1241 − 1. 1927 )= 1. 1593

v 3 ( 2 )= 0. 7156 + 0. 1482 ( 1. 2156 − 1. 2156 )+ 0. 4867 ( 1. 1241 − 1. 2156 )= 0. 6711

v 4 ( 2 )= 0. 0 + 0. 1482 ( 1. 25 − 1. 25 )+ 0. 4867 ( 1. 1241 − 1. 25 )= − 0. 0613

(c) Compute the current values ofxj(i) sa xj(i)=xj(i− 1 )+vj(i), j= 1 , 2 , 3 , 4 :

x 1 ( 2 )= 1. 1241 + 2. 6240 = 3. 7481

x 2 ( 2 )= 1. 1927 + 1. 1593 = 2. 3520

x 3 ( 2 )= 1. 2156 + 0. 6711 = 1. 8867

x 4 ( 2 )= 1. 25 − 0. 0613 = 1. 1887