Research Article
Synthesis of Sparse Concentric Ring Arrays Based on Fitness-Associated Differential Evolution Algorithm
Table 3
Test results for ten benchmark functions at D = 50 and D = 100.
| | | D = 50 | D = 100 | | Mean | SD | PSR (%) | Mean | SD | PSR (%) |
| | f1 | DE | 3.52 × 10−64 | 6.76 × 10−128 | 100 | 2.00 × 10−14 | 5.29 × 10−29 | 100 | | CR1-DE | 5.63 × 10−76 | 1.46 × 10−150 | 100 | 1.16 × 10−40 | 2.09 × 10−80 | 100 | | CR2-DE | 1.04 × 10−56 | 2.74 × 10−112 | 100 | 7.59 × 10−26 | 4.05 × 10−51 | 100 | | CR3-DE | 2.79 × 10−50 | 6.18 × 10−100 | 100 | 1.68 × 10−16 | 1.67 × 10−32 | 100 | | DELF | 4.90 × 10−44 | 1.05 × 10−87 | 100 | 8.90 × 10−21 | 1.97 × 10−41 | 100 | | PSO | 6.72 × 10−58 | 6.49 × 10−96 | 100 | 3.03 × 10−27 | 8.16 × 10−52 | 100 | | IWO | 2.20 × 10−54 | 3.92 × 10−90 | 100 | 1.07 × 10−32 | 5.50 × 10−45 | 100 | | HS | 3.71 × 10−67 | 6.09 × 10−102 | 100 | 6.23 × 10−30 | 3.61 × 10−36 | 100 | | CS | 2.05 × 10−62 | 2.46 × 10−118 | 100 | 1.36 × 10−23 | 9.82 × 10−34 | 100 |
| | f2 | DE | 4.60 × 10−37 | 6.36 × 10−74 | 100 | 8.42 × 10−09 | 2.00 × 10−18 | 0 | | CR1-DE | 5.49 × 10−38 | 5.17 × 10−75 | 100 | 3.80 × 10−21 | 1.98 × 10−41 | 100 | | CR2-DE | 3.38 × 10−29 | 1.39 × 10−57 | 100 | 1.73 × 10−13 | 1.73 × 10−26 | 100 | | CR3-DE | 1.36 × 10−26 | 1.74 × 10−52 | 100 | 4.38 × 10−09 | 3.73 × 10−18 | 0 | | DELF | 3.00 × 10−24 | 1.24 × 10−48 | 100 | 7.09 × 10−11 | 2.68 × 10−22 | 0 | | PSO | 6.47 × 10−20 | 2.56 × 10−51 | 100 | 3.06 × 10−13 | 8.57 × 10−26 | 100 | | IWO | 5.03 × 10−31 | 6.61 × 10−62 | 100 | 8.42 × 10−14 | 1.38 × 10−30 | 100 | | HS | 1.83 × 10−32 | 5.77 × 10−61 | 100 | 2.44 × 10−15 | 3.65 × 10−12 | 92 | | CS | 2.41 × 10−37 | 4.01 × 10−53 | 100 | 3.74 × 10−12 | 1.90 × 10−08 | 83 |
| | f3 | DE | 7.46 × 10+04 | 4.06 × 10+07 | 0 | 3.40 × 10+05 | 5.80 × 10+08 | 0 | | CR1-DE | 1.21 × 10−02 | 1.37 × 10−04 | 0 | 1.16 × 10+04 | 1.78 × 10+07 | 0 | | CR2-DE | 3.37 × 10+03 | 2.53 × 10+06 | 0 | 2.22 × 10+05 | 9.18 × 10+08 | 0 | | CR3-DE | 4.36 × 10+04 | 7.83 × 10+07 | 0 | 3.00 × 10+05 | 6.54 × 10+08 | 0 | | DELF | 2.66 × 10+01 | 3.31 × 10+01 | 0 | 1.60 × 10+02 | 8.12 × 10+02 | 0 | | PSO | 1.02 × 10+04 | 4.38 × 10+07 | 0 | 4.55 × 10+04 | 2.68 × 10+08 | 0 | | IWO | 1.56 × 10−02 | 5.59 × 10−08 | 0 | 3.24 × 10+00 | 1.01 × 10+00 | 0 | | HS | 2.02 × 10+04 | 1.90 × 10+07 | 0 | 2.45 × 10+05 | 5.61 × 10+08 | 0 | | CS | 2.27 × 10−02 | 5.22 × 10−04 | 0 | 5.72 × 10+01 | 9.53 × 10+01 | 0 |
| | f4 | DE | 7.14 × 10−03 | 3.70 × 10−06 | 0 | 8.65 × 10+01 | 2.16 × 10+01 | 0 | | CR1-DE | 7.65 × 10+00 | 1.64 × 10+01 | 0 | 9.32 × 10+01 | 1.09 × 10+02 | 0 | | CR2-DE | 9.68 × 10−01 | 2.23 × 10+00 | 0 | 9.21 × 10+01 | 1.71 × 10+02 | 0 | | CR3-DE | 1.75 × 10−03 | 7.96 × 10−05 | 0 | 9.09 × 10+01 | 2.45 × 10+02 | 0 | | DELF | 6.34 × 10−05 | 1.95 × 10−10 | 0 | 1.38 × 10−01 | 4.50 × 10−03 | 0 | | PSO | 1.12 × 10+01 | 3.94 × 10+00 | 0 | 1.56 × 10+01 | 4.59 × 10+00 | 0 | | IWO | 5.30 × 10−03 | 1.38 × 10−07 | 0 | 1.22 × 10−01 | 1.28 × 10−01 | 0 | | HS | 9.19 × 10+00 | 4.99 × 10−01 | 0 | 4.19 × 10+01 | 7.51 × 10−01 | 0 | | CS | 1.64 × 10−01 | 5.71 × 10−04 | 0 | 1.67 × 10+00 | 5.24 × 10−02 | 0 |
| | f5 | DE | 2.09 × 10+01 | 5.66 × 10−01 | 0 | 8.94 × 10+01 | 1.27 × 10−01 | 0 | | CR1-DE | 1.59 × 10−01 | 6.23 × 10−01 | 0 | 1.01 × 10+02 | 1.25 × 10+03 | 0 | | CR2-DE | 8.34 × 10−04 | 1.13 × 10−06 | 0 | 7.86 × 10+01 | 2.13 × 10+02 | 0 | | CR3-DE | 1.05 × 10+00 | 3.33 × 10−01 | 0 | 8.31 × 10+01 | 8.24 × 10+01 | 0 | | DELF | 3.43 × 10+01 | 2.07 × 10+00 | 0 | 8.79 × 10+01 | 3.16 × 10−01 | 0 | | PSO | 2.93 × 10+01 | 3.36 × 10+01 | 0 | 2.17 × 10+05 | 1.74 × 10+10 | 0 | | IWO | 5.93 × 10+01 | 2.93 × 10+03 | 0 | 1.32 × 10+02 | 7.34 × 10+03 | 0 | | HS | 1.36 × 10+03 | 1.56 × 10+05 | 0 | 3.77 × 10+06 | 2.67 × 10+11 | 0 | | CS | 4.71 × 10+01 | 1.45 × 10+00 | 0 | 2.13 × 10+02 | 5.33 × 10+02 | 0 |
| | f6 | DE | 0 | 0 | 100 | 1.99 × 10−14 | 6.35 × 10−29 | 100 | | CR1-DE | 0 | 0 | 100 | 4.73 × 10−32 | 1.84 × 10−63 | 100 | | CR2-DE | 0 | 0 | 100 | 1.07 × 10−25 | 1.62 × 10−50 | 100 | | CR3-DE | 0 | 0 | 100 | 2.19 × 10−16 | 3.94 × 10−32 | 100 | | DELF | 0 | 0 | 100 | 8.56 × 10−21 | 1.27 × 10−41 | 100 | | PSO | 0 | 0 | 100 | 3.10 × 10−17 | 9.39 × 10−35 | 100 | | IWO | 0 | 0 | 100 | 1.09 × 10−20 | 4.10 × 10−25 | 100 | | HS | 0 | 0 | 100 | 6.14 × 10−24 | 2.85 × 10−27 | 100 | | CS | 0 | 0 | 100 | 1.48 × 10−21 | 1.37 × 10−22 | 100 |
| | f7 | DE | 8.03 × 10−03 | 1.16 × 10−06 | 0 | 7.22 × 10−02 | 5.75 × 10−05 | 0 | | CR1-DE | 3.42 × 10−03 | 7.39 × 10−07 | 0 | 1.65 × 10−02 | 1.77 × 10−05 | 0 | | CR2-DE | 5.53 × 10−03 | 1.55 × 10−06 | 0 | 1.71 × 10−02 | 1.26 × 10−05 | 0 | | CR3-DE | 7.85 × 10−03 | 2.84 × 10−06 | 0 | 3.58 × 10−02 | 4.64 × 10−05 | 0 | | DELF | 7.01 × 10−03 | 2.14 × 10−06 | 0 | 1.89 × 10−02 | 1.06 × 10−05 | 0 | | PSO | 2.06 × 10−01 | 2.79 × 10−02 | 0 | 1.23 × 10−01 | 3.95 × 10−01 | 0 | | IWO | 5.11 × 10−03 | 2.50 × 10−06 | 0 | 2.20 × 10−02 | 2.27 × 10−05 | 0 | | HS | 6.99 × 10−02 | 1.81 × 10−04 | 0 | 5.20 × 10+00 | 4.85 × 10−01 | 0 | | CS | 1.10 × 10−02 | 9.37 × 10−06 | 0 | 8.05 × 10−02 | 4.31 × 10−04 | 0 |
| | f8 | DE | 2.40 × 10+02 | 1.24 × 10+02 | 0 | 7.66 × 10+02 | 4.52 × 10+02 | 0 | | CR1-DE | 8.57 × 10+01 | 2.38 × 10+03 | 0 | 2.54 × 10+02 | 3.08 × 10+04 | 0 | | CR2-DE | 2.56 × 10+02 | 1.05 × 10+03 | 0 | 7.17 × 10+02 | 2.07 × 10+03 | 0 | | CR3-DE | 2.99 × 10+02 | 3.33 × 10+02 | 0 | 7.96 × 10+02 | 8.25 × 10+02 | 0 | | DELF | 2.65 × 10+02 | 1.51 × 10+01 | 0 | 1.43 × 10+03 | 1.30 × 10+02 | 0 | | PSO | 1.68 × 10+02 | 9.18 × 10−02 | 0 | 4.99 × 10+02 | 2.31 × 10+03 | 0 | | IWO | 3.55 × 10+02 | 1.46 × 10+03 | 0 | 7.52 × 10+02 | 3.83 × 10+03 | 0 | | HS | 5.72 × 10+02 | 1.60 × 10+00 | 0 | 1.13 × 10+03 | 5.46 × 10+01 | 0 | | CS | 2.14 × 10+02 | 4.05 × 10+02 | 0 | 6.22 × 10+02 | 1.13 × 10+03 | 0 |
| | f9 | DE | 7.99 × 10−15 | 0 | 100 | 2.81 × 10−08 | 1.49 × 10−17 | 0 | | CR1-DE | 7.57 × 10−15 | 1.36 × 10−30 | 100 | 2.36 × 10+00 | 4.18 × 10+01 | 88 | | CR2-DE | 7.43 × 10−15 | 1.73 × 10−30 | 100 | 7.98 × 10−01 | 1.56 × 10+01 | 96 | | CR3-DE | 7.78 × 10−15 | 7.26 × 10−31 | 100 | 5.28 × 10−10 | 5.31 × 10−20 | 0 | | DELF | 7.64 × 10−15 | 1.16 × 10−30 | 100 | 2.20 × 10−11 | 3.32 × 10−23 | 0 | | PSO | 7.82 × 10−15 | 1.85 × 10−30 | 100 | 9.24 × 10+00 | 9.53 × 10−01 | 83 | | IWO | 1.79 × 10−14 | 2.09 × 10−25 | 100 | 1.93 × 10−01 | 2.34 × 10−02 | 0 | | HS | 2.11 × 10−14 | 5.54 × 10−30 | 100 | 9.35 × 10+00 | 8.35 × 10−02 | 0 | | CS | 2.37 × 10−14 | 9.89 × 10−25 | 100 | 4.37 × 10−01 | 6.17 × 10−03 | 0 |
| | f10 | DE | 0 | 0 | 100 | 1.12 × 10−14 | 1.26 × 10−29 | 100 | | CR1-DE | 0 | 0 | 100 | 1.15 × 10−16 | 4.83 × 10−34 | 100 | | CR2-DE | 0 | 0 | 100 | 1.11 × 10−16 | 0 | 100 | | CR3-DE | 0 | 0 | 100 | 1.67 × 10−16 | 3.14 × 10−33 | 100 | | DELF | 0 | 0 | 100 | 1.12 × 10−16 | 0 | 100 | | PSO | 0 | 0 | 100 | 2.55 × 10−16 | 5.54 × 10−33 | 100 | | IWO | 0 | 0 | 100 | 2.88 × 10−13 | 1.90 × 10−34 | 100 | | HS | 0 | 0 | 100 | 5.43 × 10−15 | 1.87 × 10−28 | 100 | | CS | 0 | 0 | 100 | 5.32 × 10−13 | 1.37 × 10−16 | 100 |
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