Mathematical Problems in Engineering

Research Article

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

Table 3

The best solutions obtained using the RGA-PSO algorithm from TPs 1–13.
TP number 𝑓 ( x ∗ R G A - P S O ) x ∗ R G A - P S O
1 24.323 𝐱 ∗ R G A - P S O = ( 2 . 1 6 7 2 7 7 6 0 , 2.37789634, 8.78162804, 5.12372885, 0.97991270, 1.39940993, 1.31127142, 9.81945011, 8.30004549, 8.45891329)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( − 0 . 0 0 0 1 7 1 ≤ 0 , − 0 . 0 0 3 1 0 9 ≤ 0 , − 0 . 0 0 0 0 2 7 ≤ 0 , − 0 . 0 0 0 1 2 3 ≤ 0 , − 0 . 0 0 1 3 7 1 ≤ 0 , − 0 . 0 0 2 1 0 1 ≤ 0 , − 6 . 2 4 5 9 5 7 ≤ 0 , − 4 7 . 8 4 6 3 0 2 ≤ 0 )
2 −30665.539 𝐱 ∗ R G A - P S O = (78, 33, 29.99525450, 45, 36.77581373)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( − 9 2 . 0 0 0 0 0 0 ≤ 0 , 2 . 2 4 𝐸 − 0 7 ≤ 0 , − 8 . 8 4 0 5 0 0 ≤ 0 , − 1 1 . 1 5 9 5 0 0 ≤ 0 , 4 . 2 8 𝐸 − 0 7 ≤ 0 , − 5 . 0 0 0 0 0 0 ≤ 0 )
3 680.632 𝐱 ∗ R G A - P S O = ( 2.33860239, 1.95126191, −0.45483579, 4.36300325, −0.62317747, 1.02938443, 1.59588410)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( − 7 . 7 6 𝐸 − 0 7 ≤ 0 , − 2 5 2 . 7 2 1 1 ≤ 0 , − 1 4 4 . 8 1 4 0 ≤ 0 , − 6 . 1 5 𝐸 − 0 5 ≤ 0 )
4 −15 𝐱 ∗ R G A - P S O = ( 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( 0 ≤ 0 , 0 ≤ 0 , 0 ≤ 0 , − 5 ≤ 0 , − 5 ≤ 0 , − 5 ≤ 0 , 0 ≤ 0 , 0 ≤ 0 , 0 ≤ 0 )
5 1227.1139 𝐱 ∗ R G A - P S O = ( 1697.13793410, 54.17332240, 3030.44600072, 90.18199040, 94.99999913, 10.42097385, 153.53771370)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( 0 . 9 9 9 9 7 8 ≤ 1 , 0 . 9 8 0 1 1 4 ≤ 1 , 1 . 0 0 0 0 0 5 ≤ 1 , 0 . 9 8 0 0 9 7 ≤ 1 , 0 . 9 9 0 5 6 5 ≤ 1 , 1 . 0 0 0 0 0 5 ≤ 1 , 1 . 0 0 0 0 0 ≤ 1 , 0 . 9 7 6 7 0 1 ≤ 1 , 1 . 0 0 0 0 0 3 ≤ 1 , 0 . 4 5 9 3 1 1 ≤ 1 , 0 . 3 8 7 4 3 2 ≤ 1 , 0 . 9 8 1 9 9 7 ≤ 1 , 0 . 9 8 0 3 6 4 ≤ 1 , − 8 . 2 4 1 9 2 6 ≤ 1 )
6 3.9521 𝐱 ∗ R G A - P S O = ( 6.444100620, 2.243029250, 0.642672939, 0.582321363, 5.940008650, 5.531235784, 1.018087316, 0.403665649)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( 1 . 0 0 0 0 0 0 ≤ 1 , 1 . 0 0 0 0 0 0 ≤ 1 , 0 . 9 9 9 9 9 6 ≤ 1 , 0 . 9 9 9 8 6 ≤ 1 )
7 −5.7398 𝐱 ∗ R G A - P S O = ( 8.12997229, 0.61463971, 0.56407162, 5.63623069)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( 1 . 0 0 0 0 0 0 ≤ 1 , 1 . 0 0 0 0 0 0 ≤ 1 )
8 −83.2497 𝐱 ∗ R G A - P S O = ( 88.35595404, 7.67259607, 1.31787691)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( 1 . 0 0 0 0 0 0 ≤ 1 )
9 −6.0441 𝐱 ∗ R G A - P S O = ( 6.40497368, 0.64284563, 1.02766984, 5.94729224, 2.21044814, 0.59816471, 0.42450835, 5.54339987)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( 0 . 9 9 9 9 9 7 ≤ 1 , 0 . 9 9 9 9 5 9 ≤ 1 , 0 . 9 9 9 9 3 0 ≤ 1 , 0 . 9 9 9 9 7 5 ≤ 1 )
10 6299.8374 𝐱 ∗ R G A - P S O = ( 108.66882633, 85.10837983, 204.35894362)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( 1 . 0 0 0 0 0 2 ≤ 1 )
11 10122.4732 𝐱 ∗ R G A - P S O = ( 78, 33, 29.99564864, 45, 36.77547104)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( − 0 . 3 0 9 9 9 1 ≤ 1 , 1 . 0 0 0 0 0 4 ≤ 1 , − 0 . 0 2 1 3 7 9 ≤ 1 , 0 . 6 2 1 4 0 3 ≤ 1 , 1 . 0 0 0 0 0 2 ≤ 1 , 0 . 6 8 1 5 1 6 ≤ 1 )
12 0.012692 𝐱 ∗ R G A - P S O = ( 0.050849843, 0.336663305, 12.579603478)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( − 0 . 0 0 0 1 4 3 ≤ 0 , − 0 . 0 0 0 4 6 9 ≤ 0 , − 4 . 0 0 9 0 2 1 ≤ 0 , − 0 . 7 4 1 6 5 8 ≤ 0 )
13 5885.3018 𝐱 ∗ R G A - P S O = ( 0.77816852, 0.38464913, 40.31961883, 199.99999988)
𝑔 𝑚 ( 𝐱 ∗ R G A - P S O ) = ( 1 . 2 3 𝐸 − 0 7 ≤ 0 , 3 . 3 6 𝐸 − 0 8 ≤ 0 , − 0 . 0 0 6 9 1 6 ≤ 0 , − 4 0 . 0 0 0 0 0 0 1 2 ≤ 0 )