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. 2023 Dec 6;13:21472. doi: 10.1038/s41598-023-48462-1

Table 7.

Evaluation results of CEC 2011 test suite.

PEOA RSA MPA TSA WOA MVO GWO TLBO GSA PSO GA
C11-F1 Mean 5.92E+00 2.11E+01 8.08E+00 1.79E+01 1.32E+01 1.39E+01 1.11E+01 1.79E+01 2.08E+01 1.75E+01 2.24E+01
Best 2.00E−10 1.83E+01 4.85E−01 1.68E+01 7.56E+00 1.15E+01 1.16E+00 1.68E+01 1.78E+01 1.18E+01 2.08E+01
Worst 1.23E+01 2.32E+01 1.28E+01 1.91E+01 1.68E+01 1.68E+01 1.80E+01 2.03E+01 2.22E+01 2.33E+01 2.45E+01
Std 7.40E+00 2.30E+00 5.97E+00 1.04E+00 4.50E+00 2.93E+00 7.71E+00 1.82E+00 2.22E+00 5.61E+00 1.67E+00
Median 5.69E+00 2.14E+01 9.51E+00 1.78E+01 1.42E+01 1.36E+01 1.25E+01 1.73E+01 2.16E+01 1.74E+01 2.21E+01
Rank 1 10 2 7 4 5 3 8 9 6 11
C11-F2 Mean −2.63E+01 −1.26E+01 −2.47E+01 −1.24E+01 −1.89E+01 −1.02E+01 −2.25E+01 −1.20E+01 −1.62E+01 −2.26E+01 −1.39E+01
Best −2.71E+01 −1.30E+01 −2.55E+01 −1.60E+01 −2.23E+01 −1.21E+01 −2.47E+01 −1.31E+01 −2.07E+01 −2.39E+01 −1.62E+01
Worst −2.54E+01 −1.24E+01 −2.31E+01 −1.05E+01 −1.54E+01 −8.33E+00 −1.89E+01 −1.13E+01 −1.21E+01 −2.05E+01 −1.22E+01
Std 7.60E−01 2.81E−01 1.20E+00 2.78E+00 3.61E+00 1.71E+00 2.79E+00 9.15E−01 4.20E+00 1.64E+00 2.11E+00
Median −2.64E+01 −1.26E+01 −2.52E+01 −1.15E+01 −1.91E+01 −1.01E+01 −2.33E+01 −1.19E+01 −1.60E+01 −2.30E+01 −1.35E+01
Rank 1 8 2 9 5 11 4 10 6 3 7
C11-F4 Mean 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05
Best 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05
Worst 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05
Std 2.06E−19 4.53E−11 1.63E−15 2.12E−14 5.07E−16 9.04E−13 3.89E−15 7.15E−14 5.07E−16 5.06E−16 5.06E−16
Median 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05 1.15E−05
Rank 1 11 6 8 4 10 7 9 3 2 5
C11-F4 Mean 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
Best 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
Worst 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
Std 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
Median 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00
Rank 1 1 1 1 1 1 1 1 1 1 1
C11-F5 Mean −3.41E+01 −2.12E+01 −3.30E+01 −2.76E+01 −2.80E+01 −2.75E+01 −3.15E+01 −1.31E+01 −2.78E+01 −1.11E+01 −1.19E+01
Best −3.47E+01 −2.33E+01 −3.37E+01 −3.19E+01 −2.83E+01 −3.11E+01 −3.42E+01 −1.53E+01 −3.10E+01 −1.38E+01 −1.27E+01
Worst −3.34E+01 −1.86E+01 −3.13E+01 −2.29E+01 −2.76E+01 −2.55E+01 −2.74E+01 −1.17E+01 −2.53E+01 −9.68E+00 −1.05E+01
Std 6.07E−01 2.47E+00 1.24E+00 3.97E+00 3.22E−01 2.83E+00 3.12E+00 1.69E+00 2.69E+00 2.08E+00 1.11E+00
Median −3.42E+01 −2.15E+01 −3.35E+01 −2.78E+01 −2.81E+01 −2.66E+01 −3.23E+01 −1.26E+01 −2.74E+01 −1.06E+01 −1.22E+01
Rank 1 8 2 6 4 7 3 9 5 11 10
C11-F6 Mean −2.41E+01 −1.37E+01 −2.22E+01 −8.83E+00 −1.98E+01 −1.06E+01 −1.96E+01 −4.18E+00 −2.16E+01 −4.95E+00 −5.75E+00
Best −2.74E+01 −1.44E+01 −2.53E+01 −1.66E+01 −2.26E+01 −1.77E+01 −2.23E+01 −4.76E+00 −2.55E+01 −7.84E+00 −1.02E+01
Worst −2.30E+01 −1.26E+01 −2.09E+01 −5.73E+00 −1.34E+01 −3.89E+00 −1.79E+01 −3.89E+00 −1.82E+01 −3.89E+00 −3.89E+00
Std 2.39E+00 8.31E−01 2.27E+00 5.63E+00 4.67E+00 7.85E+00 2.29E+00 4.46E−01 3.48E+00 2.09E+00 3.22E+00
Median −2.30E+01 −1.39E+01 −2.13E+01 −6.49E+00 −2.17E+01 −1.04E+01 −1.90E+01 −4.03E+00 −2.12E+01 −4.03E+00 −4.47E+00
Rank 1 6 2 8 4 7 5 11 3 10 9
C11-F7 Mean 8.61E−01 1.82E+00 9.49E−01 1.28E+00 1.67E+00 9.06E−01 1.07E+00 1.64E+00 1.08E+00 1.12E+00 1.66E+00
Best 5.82E−01 1.61E+00 8.06E−01 1.13E+00 1.59E+00 8.25E−01 8.12E−01 1.47E+00 9.18E−01 8.59E−01 1.33E+00
Worst 1.03E+00 1.95E+00 1.04E+00 1.60E+00 1.78E+00 1.00E+00 1.30E+00 1.74E+00 1.29E+00 1.36E+00 1.84E+00
Std 2.17E−01 1.62E−01 1.13E−01 2.32E−01 8.99E−02 9.09E−02 2.18E−01 1.36E−01 1.87E−01 2.86E−01 2.47E−01
Median 9.18E−01 1.87E+00 9.76E−01 1.19E+00 1.64E+00 8.99E−01 1.09E+00 1.68E+00 1.06E+00 1.13E+00 1.74E+00
Rank 1 11 3 7 10 2 4 8 5 6 9
C11-F8 Mean 2.20E+02 3.14E+02 2.23E+02 2.54E+02 2.62E+02 2.25E+02 2.27E+02 2.25E+02 2.44E+02 4.42E+02 2.23E+02
Best 2.20E+02 2.79E+02 2.20E+02 2.20E+02 2.42E+02 2.20E+02 2.20E+02 2.20E+02 2.20E+02 2.47E+02 2.20E+02
Worst 2.20E+02 3.53E+02 2.26E+02 3.41E+02 3.04E+02 2.34E+02 2.35E+02 2.36E+02 2.87E+02 5.32E+02 2.31E+02
Std 0.00E+00 3.28E+01 3.92E+00 6.32E+01 3.04E+01 7.17E+00 9.32E+00 8.56E+00 3.44E+01 1.45E+02 5.56E+00
Median 2.20E+02 3.13E+02 2.23E+02 2.27E+02 2.51E+02 2.22E+02 2.27E+02 2.21E+02 2.35E+02 4.94E+02 2.21E+02
Rank 1 9 2 7 8 4 5 4 6 10 3
C11-F9 Mean 8.79E+03 9.47E+05 2.34E+04 6.42E+04 3.38E+05 1.24E+05 4.37E+04 3.68E+05 7.35E+05 9.65E+05 1.73E+06
Best 5.46E+03 6.20E+05 1.21E+04 5.20E+04 1.86E+05 7.13E+04 1.86E+04 3.04E+05 6.30E+05 7.72E+05 1.66E+06
Worst 1.40E+04 1.11E+06 3.53E+04 7.88E+04 5.67E+05 1.85E+05 7.65E+04 4.70E+05 7.88E+05 1.18E+06 1.83E+06
Std 4.00E+03 2.40E+05 1.08E+04 1.32E+04 1.87E+05 5.08E+04 2.64E+04 7.82E+04 7.69E+04 2.36E+05 8.95E+04
Median 7.83E+03 1.03E+06 2.32E+04 6.30E+04 2.99E+05 1.19E+05 3.98E+04 3.48E+05 7.61E+05 9.53E+05 1.71E+06
Rank 1 9 2 4 6 5 3 7 8 10 11
C11-F10 Mean −2.15E+01 −1.24E+01 −1.83E+01 −1.42E+01 −1.29E+01 −1.45E+01 −1.40E+01 −1.15E+01 −1.31E+01 −1.16E+01 −1.13E+01
Best −2.18E+01 −1.28E+01 −1.87E+01 −1.80E+01 −1.35E+01 −2.03E+01 −1.45E+01 −1.16E+01 −1.36E+01 −1.17E+01 −1.14E+01
Worst −2.08E+01 −1.22E+01 −1.79E+01 −1.22E+01 −1.25E+01 −1.15E+01 −1.28E+01 −1.13E+01 −1.23E+01 −1.14E+01 −1.12E+01
Std 5.13E−01 2.92E−01 4.55E−01 2.83E+00 4.77E−01 4.26E+00 8.62E−01 1.57E−01 7.05E−01 1.21E−01 1.15E−01
Median −2.17E+01 −1.22E+01 −1.83E+01 −1.33E+01 −1.28E+01 −1.31E+01 −1.43E+01 −1.15E+01 −1.33E+01 −1.16E+01 −1.13E+01
Rank 1 8 2 4 7 3 5 10 6 9 11
C11-F11 Mean 5.72E+05 8.34E+06 1.97E+06 5.76E+06 1.57E+06 1.66E+06 3.89E+06 5.11E+06 1.75E+06 5.12E+06 5.92E+06
Best 2.61E+05 8.13E+06 1.84E+06 4.85E+06 1.45E+06 1.03E+06 3.70E+06 5.08E+06 1.59E+06 5.08E+06 5.88E+06
Worst 8.29E+05 8.49E+06 2.07E+06 6.85E+06 1.70E+06 2.89E+06 4.26E+06 5.14E+06 1.89E+06 5.16E+06 5.96E+06
Std 2.68E+05 1.63E+05 1.06E+05 8.89E+05 1.09E+05 9.06E+05 2.73E+05 2.55E+04 1.36E+05 3.51E+04 3.57E+04
Median 5.99E+05 8.37E+06 1.98E+06 5.67E+06 1.57E+06 1.35E+06 3.80E+06 5.11E+06 1.75E+06 5.12E+06 5.91E+06
Rank 1 11 5 9 2 3 6 7 4 8 10
C11-F12 Mean 1.20E+06 1.19E+07 1.30E+06 4.62E+06 5.31E+06 1.34E+06 1.43E+06 1.28E+07 5.29E+06 2.22E+06 1.30E+07
Best 1.16E+06 1.10E+07 1.21E+06 4.40E+06 4.95E+06 1.21E+06 1.26E+06 1.21E+07 5.04E+06 2.08E+06 1.29E+07
Worst 1.25E+06 1.26E+07 1.38E+06 4.73E+06 5.48E+06 1.47E+06 1.57E+06 1.34E+07 5.45E+06 2.40E+06 1.31E+07
Std 4.85E+04 7.12E+05 8.16E+04 1.69E+05 2.70E+05 1.15E+05 1.37E+05 5.83E+05 1.95E+05 1.44E+05 1.05E+05
Median 1.20E+06 1.19E+07 1.30E+06 4.67E+06 5.41E+06 1.34E+06 1.44E+06 1.29E+07 5.34E+06 2.19E+06 1.30E+07
Rank 1 9 2 6 8 3 4 10 7 5 11
C11-F13 Mean 1.54E+04 1.62E+04 1.55E+04 1.55E+04 1.55E+04 1.55E+04 1.55E+04 1.59E+04 1.15E+05 1.55E+04 2.83E+04
Best 1.54E+04 1.58E+04 1.55E+04 1.55E+04 1.55E+04 1.55E+04 1.55E+04 1.56E+04 8.39E+04 1.55E+04 1.55E+04
Worst 1.54E+04 1.71E+04 1.55E+04 1.55E+04 1.56E+04 1.55E+04 1.55E+04 1.64E+04 1.58E+05 1.55E+04 6.67E+04
Std 9.35E−03 6.66E+02 3.87E+00 1.04E+01 4.51E+01 2.54E+01 9.22E+00 3.76E+02 3.61E+04 2.47E+01 2.76E+04
Median 1.54E+04 1.59E+04 1.55E+04 1.55E+04 1.55E+04 1.55E+04 1.55E+04 1.58E+04 1.10E+05 1.55E+04 1.56E+04
Rank 1 9 2 3 7 6 5 8 11 4 10
C11-F14 Mean 1.83E+04 2.05E+05 1.87E+04 1.95E+04 1.92E+04 1.94E+04 1.92E+04 2.77E+05 1.91E+04 1.92E+04 1.91E+04
Best 1.82E+04 1.52E+05 1.86E+04 1.93E+04 1.91E+04 1.93E+04 1.91E+04 2.90E+04 1.89E+04 1.90E+04 1.89E+04
Worst 1.84E+04 2.95E+05 1.88E+04 2.00E+04 1.94E+04 1.95E+04 1.94E+04 5.34E+05 1.93E+04 1.93E+04 1.94E+04
Std 7.36E+01 6.92E+04 8.23E+01 3.72E+02 1.40E+02 9.34E+01 1.61E+02 2.62E+05 2.06E+02 1.31E+02 2.45E+02
Median 1.83E+04 1.87E+05 1.87E+04 1.94E+04 1.93E+04 1.94E+04 1.92E+04 2.74E+05 1.92E+04 1.92E+04 1.91E+04
Rank 1 10 2 9 6 8 7 11 3 5 4
C11-F15 Mean 3.29E+04 1.70E+06 3.30E+04 5.21E+04 1.98E+05 3.31E+04 3.31E+04 1.37E+07 2.69E+05 3.33E+04 7.04E+06
Best 3.28E+04 7.13E+05 3.29E+04 3.31E+04 3.30E+04 3.30E+04 3.30E+04 2.87E+06 2.39E+05 3.33E+04 3.21E+06
Worst 3.30E+04 4.43E+06 3.30E+04 1.09E+05 2.81E+05 3.32E+04 3.31E+04 2.04E+07 2.90E+05 3.33E+04 1.21E+07
Std 7.91E+01 1.97E+06 6.26E+01 4.10E+04 1.21E+05 6.06E+01 5.04E+01 8.61E+06 2.59E+04 1.37E+01 4.39E+06
Median 3.29E+04 8.28E+05 3.30E+04 3.32E+04 2.38E+05 3.31E+04 3.31E+04 1.57E+07 2.74E+05 3.33E+04 6.44E+06
Rank 1 9 2 6 7 4 3 11 8 5 10
C11-F16 Mean 1.34E+05 1.74E+06 1.39E+05 1.45E+05 1.43E+05 1.43E+05 1.46E+05 7.88E+07 1.66E+07 7.05E+07 6.77E+07
Best 1.31E+05 4.36E+05 1.37E+05 1.43E+05 1.37E+05 1.35E+05 1.44E+05 7.68E+07 8.44E+06 5.83E+07 5.47E+07
Worst 1.36E+05 4.31E+06 1.43E+05 1.47E+05 1.48E+05 1.50E+05 1.52E+05 8.11E+07 3.00E+07 8.43E+07 8.66E+07
Std 2.46E+03 1.88E+06 2.92E+03 2.08E+03 4.91E+03 6.98E+03 4.08E+03 1.94E+06 1.01E+07 1.21E+07 1.46E+07
Median 1.33E+05 1.12E+06 1.38E+05 1.46E+05 1.43E+05 1.43E+05 1.45E+05 7.87E+07 1.40E+07 6.98E+07 6.48E+07
Rank 1 7 2 5 4 3 6 11 8 10 9
C11-F17 Mean 1.93E+06 1.38E+10 2.39E+06 1.14E+09 8.59E+09 3.12E+06 3.04E+06 1.98E+10 9.94E+09 1.85E+10 1.94E+10
Best 1.92E+06 9.88E+09 1.97E+06 9.37E+08 6.13E+09 2.37E+06 2.04E+06 1.90E+10 8.74E+09 1.63E+10 1.81E+10
Worst 1.94E+06 1.68E+10 3.19E+06 1.30E+09 1.14E+10 3.59E+06 4.94E+06 2.06E+10 1.05E+10 2.13E+10 2.19E+10
Std 1.23E+04 3.22E+09 5.92E+05 2.01E+08 2.41E+09 5.78E+05 1.41E+06 7.21E+08 8.75E+08 2.46E+09 1.85E+09
Median 1.92E+06 1.42E+10 2.21E+06 1.15E+09 8.41E+09 3.26E+06 2.59E+06 1.97E+10 1.02E+10 1.81E+10 1.88E+10
Rank 1 8 2 5 6 4 3 11 7 9 10
C11-F18 Mean 9.42E+05 1.05E+08 9.80E+05 1.94E+06 8.61E+06 9.95E+05 1.03E+06 2.76E+07 9.98E+06 1.20E+08 1.02E+08
Best 9.38E+05 7.26E+07 9.53E+05 1.70E+06 3.76E+06 9.90E+05 9.67E+05 2.19E+07 7.47E+06 1.00E+08 9.79E+07
Worst 9.45E+05 1.20E+08 1.05E+06 2.23E+06 1.50E+07 9.97E+05 1.21E+06 2.99E+07 1.26E+07 1.33E+08 1.05E+08
Std 2.85E+03 2.40E+07 5.35E+04 2.70E+05 5.15E+06 3.14E+03 1.25E+05 4.13E+06 2.47E+06 1.57E+07 3.30E+06
Median 9.43E+05 1.14E+08 9.57E+05 1.91E+06 7.81E+06 9.96E+05 9.79E+05 2.93E+07 9.94E+06 1.23E+08 1.02E+08
Rank 1 10 2 5 6 3 4 8 7 11 9
C11-F19 Mean 1.03E+06 1.03E+08 1.17E+06 2.34E+06 9.22E+06 1.47E+06 1.37E+06 3.17E+07 5.72E+06 1.53E+08 1.02E+08
Best 9.68E+05 8.89E+07 1.09E+06 2.12E+06 1.98E+06 1.16E+06 1.24E+06 2.22E+07 2.30E+06 1.39E+08 9.95E+07
Worst 1.17E+06 1.29E+08 1.33E+06 2.74E+06 1.66E+07 1.90E+06 1.55E+06 3.95E+07 7.45E+06 1.77E+08 1.05E+08
Std 1.02E+05 2.04E+07 1.16E+05 2.92E+05 7.43E+06 3.34E+05 1.42E+05 8.08E+06 2.53E+06 1.79E+07 2.49E+06
Median 9.83E+05 9.67E+07 1.13E+06 2.26E+06 9.16E+06 1.41E+06 1.34E+06 3.25E+07 6.56E+06 1.48E+08 1.02E+08
Rank 1 10 2 5 7 4 3 8 6 11 9
C11-F20 Mean 9.41E+05 1.11E+08 9.66E+05 1.73E+06 6.57E+06 9.77E+05 1.00E+06 3.08E+07 1.28E+07 1.41E+08 1.02E+08
Best 9.36E+05 9.72E+07 9.60E+05 1.57E+06 6.20E+06 9.68E+05 9.78E+05 3.01E+07 8.52E+06 1.29E+08 9.74E+07
Worst 9.47E+05 1.32E+08 9.70E+05 2.00E+06 7.07E+06 9.88E+05 1.02E+06 3.15E+07 1.97E+07 1.53E+08 1.06E+08
Std 5.16E+03 1.60E+07 4.57E+03 2.21E+05 4.03E+05 1.06E+04 1.78E+04 6.30E+05 5.28E+06 1.46E+07 3.96E+06
Median 9.41E+05 1.08E+08 9.67E+05 1.68E+06 6.50E+06 9.76E+05 1.00E+06 3.07E+07 1.14E+07 1.41E+08 1.03E+08
Rank 1 10 2 5 6 3 4 8 7 11 9
C11-F21 Mean 1.27E+01 7.05E+01 1.69E+01 2.92E+01 3.71E+01 2.71E+01 2.26E+01 9.20E+01 3.88E+01 9.65E+01 9.37E+01
Best 9.97E+00 5.34E+01 1.48E+01 2.60E+01 3.45E+01 2.44E+01 2.08E+01 4.56E+01 3.48E+01 8.38E+01 5.49E+01
Worst 1.50E+01 8.78E+01 1.92E+01 3.05E+01 4.07E+01 3.01E+01 2.49E+01 1.34E+02 4.14E+01 1.07E+02 1.14E+02
Std 2.48E+00 1.64E+01 2.19E+00 2.32E+00 2.97E+00 3.34E+00 1.98E+00 3.91E+01 3.18E+00 1.25E+01 2.96E+01
Median 1.30E+01 7.04E+01 1.67E+01 3.01E+01 3.67E+01 2.70E+01 2.23E+01 9.42E+01 3.95E+01 9.76E+01 1.03E+02
Rank 1 8 2 5 6 4 3 9 7 11 10
C11-F22 Mean 1.61E+01 5.93E+01 1.99E+01 3.15E+01 4.41E+01 3.16E+01 2.52E+01 9.40E+01 4.44E+01 9.76E+01 8.51E+01
Best 1.15E+01 4.37E+01 1.75E+01 2.81E+01 3.87E+01 2.51E+01 2.40E+01 6.20E+01 3.74E+01 8.24E+01 8.44E+01
Worst 1.96E+01 6.76E+01 2.17E+01 3.37E+01 4.81E+01 3.60E+01 2.60E+01 1.11E+02 5.25E+01 1.07E+02 8.64E+01
Std 4.32E+00 1.15E+01 2.14E+00 2.64E+00 4.61E+00 5.30E+00 1.02E+00 2.36E+01 6.70E+00 1.22E+01 1.02E+00
Median 1.67E+01 6.29E+01 2.02E+01 3.21E+01 4.48E+01 3.27E+01 2.53E+01 1.02E+02 4.39E+01 1.00E+02 8.48E+01
Rank 1 8 2 4 6 5 3 10 7 11 9
Sum rank 22 190 51 128 124 105 91 189 134 169 187
Mean rank 1.00e+00 8.64E+00 2.32E+00 5.82E+00 5.64E+00 4.77E+00 4.14E+00 8.59E+00 6.09E+00 7.68E+00 8.50E+00
Total rank 1 2 12 4 13 3 11 9 6 7 10
Wilcoxon: p-value 1.37E−15 1.54E−03 4.30E−15 4.62E−15 1.41E−11 1.69E−12 2.94E−15 7.07E−15 1.37E−15 2.01E−15