Mathematical Problems in Engineering

Research Article

Solving Constrained Global Optimization Problems by Using Hybrid Evolutionary Computing and Artificial Life Approaches

Table 4

The best solutions obtained using the AIA-PSO algorithm from TPs 1–13.
TP number 𝑓 ( x ∗ A I A - P S O ) x ∗ A I A - P S O
1 24.358 x ∗ A I A - P S O = ( 2.18024296, 2.35746157, 8.75670935, 5.11326109, 1.03976363 1.54784227, 1.32994030, 9.83127443, 8.27618717, 8.32717779)
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( − 0 . 0 0 0 0 7 1 ≤ 0 , − 0 . 0 0 3 6 9 9 ≤ 0 , − 0 . 0 0 0 4 4 0 ≤ 0 , − 0 . 6 8 4 0 2 5 ≤ 0 , − 0 . 0 0 0 0 8 6 ≤ 0 , − 0 . 0 0 1 0 2 4 ≤ 0 , − 5 . 9 7 3 8 6 6 ≤ 0 , − 4 7 . 3 7 1 9 5 8 ≤ 0 )
2 −30665.539 𝐱 ∗ A I A - P S O = ( 78, 33, 29.99525527, 45, 36.77581286)
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( − 9 2 . 0 0 0 0 0 0 ≤ 0 , 5 . 5 7 𝐸 − 0 8 ≤ 0 , − 8 . 8 4 0 5 0 0 ≤ 0 , − 1 1 . 1 5 9 5 0 0 ≤ 0 , 2 . 7 6 𝐸 − 0 7 ≤ 0 , − 5 . 0 0 0 0 0 0 ≤ 0 )
3 680.633 𝐱 ∗ A I A - P S O = ( 2.32925164, 1.95481519, −0.47307614, 4.35691576, −0.62313420, 1.05236194, 1.59750978)
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( − 4 . 2 6 𝐸 − 0 6 ≤ 0 , − 2 5 2 . 6 1 2 7 3 3 ≤ 0 , − 1 4 4 . 7 4 1 1 9 4 ≤ 0 , − 1 . 0 0 𝐸 − 0 5 ≤ 0 )
4 −15 𝐱 ∗ A I A - P S O = ( 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1)
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( 0 ≤ 0 , 0 ≤ 0 , 0 ≤ 0 , − 5 ≤ 0 , − 5 ≤ 0 , − 5 ≤ 0 , 0 ≤ 0 , 0 ≤ 0 , 0 ≤ 0 )
5 1227.1598 𝐱 ∗ A I A - P S O = ( 1 6 9 7 . 9 1 6 4 5 0 4 4 , 5 3 . 7 8 1 3 2 8 2 5 , 3 0 3 1 . 0 7 9 8 9 0 5 9 , 9 0 . 1 2 6 5 1 3 0 1 ,
9 4 . 9 9 9 9 9 9 9 5 , 1 0 . 4 8 1 0 3 7 0 8 , 1 5 3 . 5 3 5 9 4 8 6 1 )
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( 1 . 0 0 0 0 0 0 ≤ 1 , 0 . 9 8 0 1 0 0 ≤ 1 , 1 . 0 0 0 0 0 2 ≤ 1 , 0 . 9 8 0 0 9 9 ≤ 1 , 0 . 9 9 0 5 6 2 ≤ 1 , 1 . 0 0 0 0 0 1 ≤ 1 , 1 . 0 0 0 0 0 2 ≤ 1 , 0 . 9 7 6 7 1 5 ≤ 1 , 1 . 0 0 0 0 0 1 ≤ 1 , 0 . 4 5 9 4 2 5 ≤ 1 , 0 . 3 8 7 5 1 5 ≤ 1 , 0 . 9 8 1 8 5 6 ≤ 1 , 0 . 9 8 7 2 4 5 ≤ 1 , − 8 . 3 0 2 5 3 3 ≤ 1 )
6 3.9516 𝐱 ∗ A I A - P S O = ( 6 . 4 4 3 7 3 6 4 7 , 2 . 2 3 3 3 5 1 1 1 , 0 . 6 8 2 3 3 3 0 3 , 0 . 6 0 0 2 6 6 1 7 ,
5 . 9 3 7 4 5 1 1 9 , 5 . 5 3 1 4 6 1 8 6 , 1 . 0 1 8 6 2 9 5 8 , 0 . 4 0 6 7 3 6 6 1 )
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( 0 . 9 9 9 9 9 9 ≤ 1 , 0 . 9 9 9 9 9 9 ≤ 1 , 0 . 9 9 9 9 9 4 ≤ 1 , 0 . 9 9 9 9 8 5 ≤ 1 )
7 −5.7398 𝐱 ∗ A I A - P S O = ( 8 . 1 3 0 4 2 9 8 5 , 0 . 6 1 7 5 8 1 3 6 , 0 . 5 6 3 9 3 6 0 3 , 5 . 6 3 6 1 8 1 9 1 )
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( 0 . 9 9 9 9 9 9 ≤ 1 , 0 . 9 9 9 9 9 8 ≤ 1 )
8 −83.2497 𝐱 ∗ A I A - P S O = ( 88.35635930, 7.67202113, 1.31765768)
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( 3 . 0 0 𝐸 − 1 8 ≤ 1 )
9 −6.0467 𝐱 ∗ A I A - P S O = ( 6 . 4 6 3 9 3 1 7 3 , 0.67575021, 1.01188804, 5.94072071, 2.24639462, 0.60683404, 0.39677469, 5.52596342)
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( 0 . 9 9 9 9 8 0 ≤ 1 , 0 . 9 9 9 9 9 9 ≤ 1 , 0 . 9 9 8 7 3 9 ≤ 1 , 0 . 9 9 9 9 2 8 ≤ 1 )
10 6299.8395 𝐱 ∗ A I A - P S O = ( 108.97708780, 85.02248757, 204.45439729)
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( 1 . 0 0 0 0 0 3 ≤ 1 )
11 10122.4852 𝐱 ∗ A I A - P S O = ( 78, 33, 29.99571251, 45, 36.77534593)
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( − 0 . 3 0 9 9 9 1 ≤ 1 , 1 . 0 0 0 0 0 1 ≤ 1 , − 0 . 0 2 1 3 8 0 ≤ 1 , 0 . 6 2 1 4 0 3 ≤ 1 , 1 . 0 0 0 0 0 1 ≤ 1 , 0 . 6 8 1 5 1 7 ≤ 1 )
12 0.012667 𝐱 ∗ A I A - P S O = ( 0.05164232, 0.35558085, 11.35742676)
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( − 8 . 8 3 𝐸 − 0 5 ≤ 0 , − 3 . 0 4 𝐸 − 0 5 ≤ 0 , − 4 . 0 5 0 9 2 4 ≤ 0 , − 0 . 7 2 8 5 1 8 ≤ 0 )
13 5885.3310 𝐱 ∗ A I A - P S O = ( 0.77816843, 0.38464909, 40.31961929, 199.99999330)
𝑔 𝑚 ( 𝐱 ∗ A I A - P S O ) = ( 2 . 2 2 𝐸 − 0 7 ≤ 0 , 7 . 8 0 𝐸 − 0 8 ≤ 0 , − 0 . 0 0 6 0 1 5 ≤ 0 , − 4 0 . 0 0 0 0 0 7 ≤ 0 )