Table 10 Findings from the IEEE CEC2005 benchmark functions for unimodal, multimodal, and fixed-dimensional multimodal (the top solutions have been highlighted).
Func | SCA | PSO | SSA | WOA | AVO | OBAVO | ROBAVO | EOBAVO | |
|---|---|---|---|---|---|---|---|---|---|
F1 | Mean | 3.84E−08 | 1.90E−107 | 3.96E−07 | 2.53E−16 | 1.70E−299 | 1.40E−298 | 1.60E−298 | 0.00E+00 |
Std | 7.94E−08 | 1.10E−106 | 8.64E−07 | 9.67E−17 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
t-values | − 2.65E+00 | − 1.00E+00 | − 2.51E+00 | − 1.43E+01 | #DIV/0! | #DIV/0! | #DIV/0! | NA | |
F2 | Mean | 5.04E−02 | 2.93E−57 | 9.05E−02 | 5.57E−02 | 5.90E−146 | 9.10E−113 | 1.00E−112 | 1.85E−256 |
Std | 6.23E−02 | 1.26E−56 | 4.51E−01 | 1.94E−01 | 3.20E−145 | 5.90E−128 | 5.80E−128 | 0.00E+00 | |
t-values | − 4.43E+00 | − 1.28E+00 | − 1.10E+00 | − 1.57E+00 | − 1.00E+00 | − 8.40E+15 | − 9.50E+15 | NA | |
F3 | Mean | 8.17E+01 | 7.24E−78 | 1.76E−06 | 8.97E+02 | 1.40E−214 | 2.20E−210 | 2.40E−210 | 0.00E+00 |
Std | 5.87E+01 | 3.89E−77 | 5.49E−06 | 3.19E+02 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | |
t-values | − 7.63E+00 | − 1.02E+00 | − 1.76E+00 | − 1.54E+01 | #DIV/0! | #DIV/0! | #DIV/0! | NA | |
F4 | Mean | 2.15E+00 | 8.51E−49 | 6.36E−05 | 7.35E+00 | 1.00E−143 | 1.01E−03 | 3.01E−03 | 7.40E−263 |
Std | 8.27E−01 | 2.19E−48 | 2.17E−04 | 1.74E+00 | 5.30E−143 | 2.61E−04 | 3.61E−04 | 0.00E+00 | |
t-values | − 1.42E+01 | − 2.13E+00 | − 1.60E+00 | − 2.31E+01 | − 1.05E+00 | − 2.12E+01 | − 4.57E+01 | NA | |
F5 | Mean | 4.97E+01 | 2.77E+01 | 9.47E+01 | 6.75E+01 | 6.39E−05 | −1.53E+04 | −1.63E+04 | 3.75E−05 |
Std | 3.76E+01 | 7.62E−01 | 1.45E+02 | 6.22E+01 | 5.19E−05 | 2.90E+03 | 1.90E+03 | 2.83E−05 | |
t-values | − 7.23E+00 | − 1.99E+02 | − 3.58E+00 | − 5.95E+00 | − 2.45E+00 | 2.89E+01 | 4.70E+01 | NA | |
F6 | Mean | 2.46E−05 | 2.77E−01 | 1.19E−09 | 2.50E−16 | 4.22E−07 | 0.00E+00 | 0.00E+00 | 3.54E−07 |
Std | 1.19E−04 | 1.84E−01 | 4.41E−10 | 1.74E−16 | 2.80E−07 | 0.00E+00 | 0.00E+00 | 3.27E−07 | |
t-values | − 1.11E+00 | − 8.22E+00 | 5.91E+00 | 5.93E+00 | − 8.69E−01 | 5.93E+00 | 5.93E+00 | NA | |
F7 | Mean | 3.01E−02 | 8.50E−04 | 1.35E−02 | 8.94E−02 | 1.60E−04 | 2.70E−14 | 1.35E+05 | 1.15E−04 |
Std | 1.85E−02 | 8.02E−04 | 6.78E−03 | 4.34E−02 | 1.89E−04 | 1.30E−14 | 1.49E+03 | 9.45E−05 | |
t-values | − 8.88E+00 | − 4.98E+00 | − 1.08E+01 | − 1.13E+01 | − 1.14E+00 | 6.70E+00 | − 4.98E+02 | NA | |
F8 | Mean | −6.53E+03 | −6.82E+03 | −2.67E+03 | −2.82E+03 | −1.22E+04 | 0.00E+00 | 0.00E+00 | −1.24E+04 |
Std | 7.43E+02 | 9.72E+02 | 3.71E+02 | 4.93E+02 | 8.67E+02 | 0.00E+00 | 0.00E+00 | 3.44E+02 | |
t-values | − 3.92E+01 | − 2.96E+01 | − 1.05E+02 | − 8.71E+01 | − 1.19E+00 | − 1.97E+02 | − 1.97E+02 | NA | |
F9 | Mean | 5.02E+01 | 0.00E+00 | 1.59E+01 | 2.60E+01 | 0.00E+00 | 6.16E−04 | 5.26E−04 | 0.00E+00 |
Std | 1.32E+01 | 0.00E+00 | 6.70E+00 | 7.47E+00 | 0.00E+00 | 3.40E−04 | 4.20E−04 | 0.00E+00 | |
t-values | − 2.09E+01 | #DIV/0! | − 1.30E+01 | − 1.90E+01 | #DIV/0! | − 9.91E+00 | − 6.85E+00 | NA | |
F10 | Mean | 1.65E+00 | 8.88E−16 | 9.59E−01 | 6.21E−02 | 4.44E−16 | 1.41E−03 | 2.51E−03 | 4.44E−16 |
Std | 7.01E−01 | 1.00E−31 | 8.74E−01 | 2.36E−01 | 3.01E−31 | 7.49E−04 | 7.89E−04 | 3.01E−31 | |
t-values | − 1.29E+01 | − 7.70E+15 | − 6.01E+00 | − 1.44E+00 | 0.00E+00 | − 1.03E+01 | − 1.74E+01 | NA | |
F11 | Mean | 3.72E−02 | 0.00E+00 | 1.99E−01 | 2.77E+01 | 0.00E+00 | 4.13E+00 | 1.36E+05 | 0.00E+00 |
Std | 3.61E−02 | 0.00E+00 | 1.28E−01 | 5.04E+00 | 0.00E+00 | 1.22E+00 | 1.58E+03 | 0.00E+00 | |
t-values | − 5.64E+00 | #DIV/0! | − 8.52E+00 | − 3.01E+01 | #DIV/0! | − 1.85E+01 | − 4.73E+02 | NA | |
F12 | Mean | 7.60E−02 | 1.28E−02 | 6.53E−01 | 1.80E+00 | 6.09E−08 | 1.08E−03 | 2.17E−03 | 1.94E−08 |
Std | 1.39E−01 | 2.09E−02 | 9.09E−01 | 9.51E−01 | 2.07E−07 | 1.94E−04 | 2.34E−04 | 1.36E−08 | |
t-values | − 3.00E+00 | − 3.35E+00 | − 3.94E+00 | − 1.04E+01 | − 1.09E+00 | − 3.05E+01 | − 5.08E+01 | NA | |
F13 | Mean | 2.11E−01 | 1.26E+00 | 5.17E−02 | 8.90E+00 | 4.48E−08 | −4.91E−02 | −5.50E+16 | 3.89E−08 |
Std | 7.18E−01 | 5.72E−01 | 5.80E−03 | 7.13E+00 | 5.86E−08 | 9.35E−03 | 5.45E−03 | 3.30E−08 | |
t-values | − 1.61E+00 | − 1.21E+01 | − 4.88E+01 | − 6.84E+00 | − 4.81E−01 | 2.87E+01 | 5.49E+19 | NA | |
F14 | Mean | 3.69E+00 | 1.92E+00 | 1.23E+00 | 5.86E+00 | 1.33E+00 | 7.92E−01 | 9.89E−01 | 1.13E+00 |
Std | 2.62E+00 | 2.01E+00 | 6.73E−01 | 3.83E+00 | 7.06E−01 | 1.50E−01 | 3.45E−01 | 5.03E−01 | |
t-values | − 5.25E+00 | − 2.08E+00 | − 6.46E−01 | − 6.70E+00 | − 1.25E+00 | 3.53E+00 | 1.27E+00 | NA | |
F15 | Mean | 1.10E−03 | 1.95E−03 | 5.17E−03 | 3.67E−03 | 3.97E−04 | 4.62E+00 | 5.33E+00 | 3.33E−04 |
Std | 3.65E−03 | 5.33E−03 | 9.32E−03 | 1.65E−03 | 1.18E−04 | 8.15E−01 | 9.21E−01 | 8.33E−05 | |
t-values | − 1.15E+00 | − 1.66E+00 | − 2.84E+00 | − 1.11E+01 | − 2.40E+00 | − 3.10E+01 | − 3.17E+01 | NA | |
F16 | Mean | −1.03E+00 | −1.03E+00 | −1.03E+00 | −1.03E+00 | −1.03E+00 | −2.81E+00 | −2.81E+00 | −1.03E+00 |
Std | 1.00E−11 | 1.07E−08 | 2.76E−14 | 4.88E−16 | 0.00E+00 | 5.33E−01 | 5.33E−01 | 0.00E+00 | |
t-values | 0.00E+00 | −1.49E+00 | −4.94E+00 | 1.74E+10 | #DIV/0! | 1.83E+01 | 1.83E+01 | NA | |
F17 | Mean | 3.98E−01 | 3.98E−01 | 3.98E−01 | 3.98E−01 | 3.98E−01 | −1.95E+00 | −1.95E+00 | 3.98E−01 |
Std | 1.13E−16 | 1.13E−16 | 1.35E−14 | 0.00E+00 | 1.13E−16 | 3.53E−01 | 3.53E−01 | 1.13E−16 | |
t-values | 0.00E+00 | 0.00E+00 | −4.12E+00 | 1.74E+10 | 0.00E+00 | 3.64E+01 | 3.64E+01 | NA | |
F18 | Mean | 3.00E+00 | 3.00E+00 | 3.00E+00 | 3.00E+00 | 3.00E+00 | −4.33E+00 | −4.33E+00 | 3.00E+00 |
Std | 3.03E−15 | 8.35E−15 | 1.91E−13 | 4.17E−15 | 3.24E−06 | 1.25E+00 | 1.25E+00 | 3.06E−06 | |
t-values | 3.40E+00 | 3.40E+00 | 3.40E+00 | 3.40E+00 | −2.35E−01 | 3.21E+01 | 3.21E+01 | NA | |
F19 | Mean | − 3.00E−01 | − 3.86E+00 | − 3.86E+00 | − 3.86E+00 | − 3.86E+00 | − 1.13E+01 | − 1.13E+01 | − 3.86E+00 |
Std | 1.00E−11 | 4.05E−15 | 4.07E−10 | 2.29E−15 | 1.48E−11 | 2.31E+00 | 2.31E+00 | 2.53E−12 | |
t-values | − 1.90E+12 | 1.30E+00 | − 1.02E+00 | − 4.64E+06 | − 9.70E−01 | 1.77E+01 | 1.77E+01 | NA | |
F20 | Mean | −3.29E+00 | −3.26E+00 | −3.21E+00 | −3.32E+00 | −3.26E+00 | −4.00E+00 | −4.00E+00 | −3.30E+00 |
Std | 5.54E−02 | 6.05E−02 | 5.68E−02 | 2.31E−02 | 6.05E−02 | 7.21E−01 | 7.21E−01 | 4.51E−02 | |
t-values | − 1.22E+00 | − 2.88E+00 | − 6.73E+00 | 1.69E+00 | − 2.88E+00 | 5.30E+00 | 5.30E+00 | NA | |
F21 | Mean | −5.49E+00 | −8.05E+00 | −7.73E+00 | −5.96E+00 | −1.02E+01 | −3.81E+00 | −3.81E+00 | −1.02E+01 |
Std | 3.45E+00 | 2.89E+00 | 3.12E+00 | 3.74E+00 | 5.15E−13 | 7.50E−01 | 7.50E−01 | 5.29E−13 | |
t-values | − 7.41E+00 | − 3.99E+00 | − 4.26E+00 | − 6.15E+00 | 1.19E−01 | − 4.63E+01 | − 4.63E+01 | NA | |
F22 | Mean | −7.84E+00 | −9.60E+00 | −7.56E+00 | −9.68E+00 | −1.04E+01 | −3.81E+00 | −3.81E+00 | −1.04E+01 |
Std | 3.45E+00 | 2.09E+00 | 3.39E+00 | 2.01E+00 | 3.32E−13 | 7.50E−01 | 7.50E−01 | 3.20E−12 | |
t-values | − 4.06E+00 | − 2.10E+00 | − 4.58E+00 | − 1.95E+00 | 1.19E+00 | − 4.81E+01 | − 4.81E+01 | NA | |
F23 | Mean | −8.14E+00 | −9.54E+00 | −7.60E+00 | −1.05E+01 | −1.05E+01 | −3.86E+00 | −3.86E+00 | −1.05E+01 |
Std | 3.52E+00 | 2.58E+00 | 3.51E+00 | 2.60E−15 | 6.41E−13 | 7.61E−01 | 7.61E−01 | 4.56E−13 | |
t-values | − 3.73E+00 | − 2.11E+00 | − 4.59E+00 | − 1.20E+08 | − 2.72E−01 | − 4.80E+01 | − 4.80E+01 | NA | |
w/t/l | 16.6.1 | 12.10.1 | 16.5.2 | 16.2.5 | 3.18.2 | 10.3.10 | 12.4.7 | Â | |
