Table 5.
Simulation results for 23 benchmark functions.
| Function | ESMA | SMA | ROA | AOA | AO | SSA | WOA | SCA | |
|---|---|---|---|---|---|---|---|---|---|
| F1 | Mean | 0.00 × 10+00 | 3.83 × 10−320 | 5.93× 10−323 | 2.05× 10−13 | 1.19 × 10−104 | 1.31 × 10−07 | 2.30 × 10−68 | 2.25 × 10+01 |
| Std | 0.00 × 10+00 | 0.00 × 1000 | 0.00 × 1000 | 1.12 × 10−12 | 6.49 × 10−104 | 1.15 × 10−07 | 1.26 × 10−67 | 6.73 × 10+01 | |
| F2 | Mean | 1.12 × 10−188 | 1.68 × 10−148 | 6.68 × 10−162 | 0.00 × 10+00 | 2.45 × 10−53 | 1.96 × 10+00 | 3.57 × 10−52 | 1.84 × 10−02 |
| Std | 0.00 × 10+00 | 9.20 × 10−148 | 3.61 × 10−161 | 0.00 × 10+00 | 1.34 × 10−52 | 1.49 × 10+00 | 8.24 × 10−52 | 3.52 × 10−02 | |
| F3 | Mean | 0.00 × 10+00 | 3.03 × 10−285 | 5.68 × 10−286 | 3.47 × 10−03 | 3.16 × 10−97 | 1.66 × 10+03 | 4.50 × 10+04 | 1.04 × 10+04 |
| Std | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 8.24 × 10−03 | 1.73 × 10−96 | 1.32 × 10+03 | 1.64 × 10+04 | 5.62 × 10+03 | |
| F4 | Mean | 5.48 × 10−222 | 9.79 × 10−161 | 2.33 × 10−153 | 2.62 × 10−02 | 3.78 × 10−53 | 1.13 × 10+01 | 5.27 × 10+01 | 3.50 × 10+01 |
| Std | 0.00 × 10+00 | 5.08 × 10−160 | 1.27 × 10−152 | 2.02 × 10−02 | 2.07 × 10−52 | 2.92 × 10+00 | 2.75 × 10+01 | 1.48 × 10+01 | |
| F5 | Mean | 3.79 × 10−03 | 6.04 × 10+00 | 2.71 × 10+01 | 2.83 × 10+01 | 4.02 × 10−03 | 1.78 × 10+02 | 2.79 × 10+01 | 9.83 × 10+04 |
| Std | 2.33 × 10−03 | 1.01 × 10+01 | 4.41 × 10−01 | 4.22 × 10−01 | 7.30 × 10−03 | 3.08 × 10+02 | 4.92 × 10−01 | 1.99 × 10+05 | |
| F6 | Mean | 5.80 × 10−07 | 6.08 × 10−03 | 9.77 × 10−02 | 3.08 × 10+00 | 9.27 × 10−05 | 1.71 × 10−07 | 3.71 × 10−01 | 1.26 × 10+01 |
| Std | 1.76 × 10−07 | 3.84 × 10−03 | 1.04 × 10−01 | 3.20 × 10−01 | 1.26 × 10−04 | 1.50 × 10−07 | 2.29 × 10−01 | 1.02 × 10+01 | |
| F7 | Mean | 5.24 × 10−05 | 1.84 × 10−04 | 1.48 × 10−04 | 5.37 × 10−05 | 7.57 × 10−05 | 1.61 × 10−01 | 4.74 × 10−03 | 9.19 × 10−02 |
| Std | 4.96 × 10−05 | 1.50 × 10−04 | 1.27 × 10−04 | 4.21 × 10−05 | 7.75 × 10−05 | 7.12 × 10−02 | 6.51 × 10−03 | 1.01 × 10−01 | |
| F8 | Mean | −1.26 × 10+04 | −1.26 × 10+04 | −1.24 × 10+04 | −5.20 × 10+03 | −8.88 × 10+03 | −7.34 × 10+03 | −1.03 × 10+04 | −3.72 × 10+03 |
| Std | 4.07 × 10−03 | 3.91 × 10−01 | 4.39 × 10+02 | 4.69 × 10+02 | 3.74 × 10+03 | 6.61 × 10+02 | 2.01 × 10+03 | 2.65 × 10+02 | |
| F9 | Mean | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 5.79 × 10+01 | 4.11 × 10+00 | 4.28 × 10+01 |
| Std | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 1.87 × 10+01 | 2.25 × 10+01 | 3.24 × 10+01 | |
| F10 | Mean | 8.88 × 10−16 | 8.88 × 10−16 | 8.88 × 10−16 | 8.88 × 10−16 | 8.88 × 10−16 | 2.77 × 10+00 | 4.80 × 10−15 | 1.26 × 10+01 |
| Std | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 8.52 × 10−01 | 2.35 × 10−15 | 8.96 × 10+00 | |
| F11 | Mean | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 1.78 × 10−02 | 0.00 × 10+00 | 9.69 × 10−01 |
| Std | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 0.00 × 10+00 | 1.23 × 10−02 | 0.00 × 10+00 | 3.69 × 10−01 | |
| F12 | Mean | 2.18 × 10−05 | 4.44 × 10−03 | 1.04 × 10−02 | 4.99 × 10−01 | 2.64 × 10−06 | 6.84 × 10+00 | 2.53 × 10−02 | 2.92 × 10+05 |
| Std | 7.96 × 10−05 | 7.53 × 10−03 | 5.91 × 10−03 | 4.80 × 10−02 | 5.61 × 10−06 | 3.30 × 10+00 | 1.62 × 10−02 | 1.19 × 10+06 | |
| F13 | Mean | 3.62 × 10−07 | 5.78 × 10−03 | 2.25 × 10−01 | 2.83 × 10+00 | 1.99 × 10−05 | 1.56 × 10+01 | 5.31 × 10−01 | 4.50 × 10+04 |
| Std | 1.69 × 10−07 | 5.70 × 10−03 | 1.51 × 10−01 | 1.08 × 10−01 | 3.79 × 10−05 | 1.47 × 10+01 | 2.84 × 10−01 | 1.76 × 10+05 | |
| F14 | Mean | 9.98 × 10−01 | 9.98 × 10−01 | 4.45 × 10+00 | 9.54 × 10+00 | 2.50 × 10+00 | 1.10 × 10+00 | 2.12 × 10+00 | 2.25 × 10+00 |
| Std | 5.17 × 10−16 | 6.55 × 10−13 | 4.85 × 10+00 | 4.22 × 10+00 | 3.33 × 10+00 | 4.00 × 10−01 | 2.12 × 10+00 | 2.49 × 10+00 | |
| F15 | Mean | 6.07 × 10−04 | 5.57 × 10−04 | 4.23 × 10−04 | 1.80 × 10−02 | 4.89 × 10−04 | 2.92 × 10−03 | 5.83 × 10−04 | 8.49 × 10−04 |
| Std | 2.67 × 10−04 | 2.83 × 10−04 | 2.92 × 10−04 | 2.86 × 10−02 | 3.29 × 10−04 | 5.93 × 10−03 | 3.84 × 10−04 | 2.32 × 10−04 | |
| F16 | Mean | −1.03 × 10+00 | −1.03 × 10+00 | −1.03 × 10+00 | −1.03 × 10+00 | −1.03 × 10+00 | −1.03 × 10+00 | −1.03 × 10+00 | −1.03 × 10+00 |
| Std | 7.70 × 10−15 | 3.95 × 10−10 | 5.90 × 10−08 | 1.65 × 10−07 | 3.69 × 10−04 | 4.13 × 10−14 | 1.32 × 10−09 | 4.90 × 10−05 | |
| F17 | Mean | 3.98 × 10−01 | 3.98 × 10−01 | 3.98 × 10−01 | 3.98 × 10−01 | 3.98 × 10−01 | 3.98 × 10−01 | 3.98 × 10−01 | 4.00 × 10−01 |
| Std | 2.82 × 10−13 | 2.77 × 10−08 | 4.26 × 10−06 | 8.49 × 10−08 | 2.67 × 10−04 | 9.08 × 10−15 | 5.79 × 10−06 | 2.15 × 10−03 | |
| F18 | Mean | 1.02 × 10+01 | 3.00 × 10+00 | 3.00 × 10+00 | 1.02 × 10+01 | 3.03 × 10+00 | 3.00 × 10+00 | 3.00 × 10+01 | 3.00 × 10+00 |
| Std | 1.21 × 10+01 | 7.33 × 10−11 | 6.72 × 10−05 | 1.21 × 10+01 | 2.65 × 10−02 | 1.90 × 10−13 | 4.08 × 10−05 | 2.37 × 10−04 | |
| F19 | Mean | −3.86 × 10+00 | −3.86 × 10+00 | −3.86 × 10+00 | −3.85 × 10+00 | −3.85 × 10+00 | −3.86 × 10+00 | −3.83 × 10+00 | −3.85 × 10+00 |
| Std | 1.85 × 10−11 | 5.00 × 10−07 | 2.07 × 10−03 | 6.68 × 10−03 | 9.15 × 10−03 | 6.05 × 10−10 | 1.40 × 10−01 | 1.17 × 10−02 | |
| F20 | Mean | −3.26 × 10+00 | −3.25 × 10+00 | −3.28 × 10+00 | −3.06 × 10+00 | −3.17 × 10+00 | −3.23 × 10+00 | −3.18 × 10+00 | −2.86 × 10+00 |
| Std | 3.05 × 10−02 | 5.96 × 10−02 | 6.88 × 10−02 | 9.11 × 10−02 | 7.18 × 10−02 | 5.77 × 10−02 | 1.88 × 10−01 | 4.10 × 10−01 | |
| F21 | Mean | −1.02 × 10+01 | −1.02 × 10+01 | −1.01 × 10+01 | −3.47 × 10+00 | −1.01 × 10+01 | −7.73 × 10+00 | −8.03 × 10+00 | −2.73 × 10+00 |
| Std | 5.52 × 10−08 | 3.30 × 10−04 | 1.25 × 10−02 | 1.24 × 10+00 | 3.68 × 10−02 | 3.32 × 10+00 | 2.89 × 10+00 | 2.28 × 10+00 | |
| F22 | Mean | −1.04 × 10+01 | −1.04 × 10+01 | −1.04 × 10+01 | −4.00 × 10+00 | −1.04 × 10+01 | −8.42 × 10+00 | −7.67 × 10+00 | −2.86 × 10+00 |
| Std | 5.77 × 10−08 | 3.07 × 10−04 | 1.58 × 10−02 | 1.51 × 10+00 | 9.40 × 10−03 | 3.14 × 10+00 | 3.54 × 10+00 | 1.77 × 10+00 | |
| F23 | Mean | −1.05 × 10+01 | −1.05 × 10+01 | −1.05 × 10+01 | −3.97 × 10+00 | −1.05 × 10+01 | −8.00 × 10+00 | −6.60 × 10+00 | −3.31 × 10+00 |
| Std | 3.17 × 10−08 | 3.92 × 10−04 | 1.94 × 10−02 | 1.63 × 10+00 | 2.59 × 10−02 | 3.47 × 10+00 | 3.32 × 10+00 | 1.98 × 10+00 | |