Research Article
A Chaotic Multi-Objective Runge–Kutta Optimization Algorithm for Optimized Circuit Design
Table 5
Statistical results of CMRUN and 11 other algorithms using 15 benchmark functions.
| | FUNCTION | CMRUN | MOEAD | MOPSO | NSGA-II | MOFA | MOBA | SPEA-2 | MOCS | MOFPA | MOMA | NSGA-II/MOPSO | NSMFO |
| | F1 | avg | 2.14E-03 | 1.75E + 01 | 1.18E-04 | 1.77E-09 | 0.00E + 00 | 6.39E + 00 | 9.82E-40 | 0.00E + 00 | 0.00E + 00 | 9.22E-10 | 2.05E-06 | 9.00E-14 | | std | 2.89E-03 | 5.23E + 01 | 1.38E-04 | 5.27E-09 | 0.00E + 00 | 6.42E + 00 | 2.94E-39 | 0.00E + 00 | 0.00E + 00 | 2.76E-09 | 6.14E-06 | 2.70E-13 | | F2 | avg | −4.00E + 07 | −3.61E + 07 | −4.00E + 07 | −2.86E + 08 | −4.00E + 07 | −2.38E + 07 | −3.97E + 07 | −4.00E + 07 | −4.00E + 07 | −4.00E + 07 | -−4.00E + 07 | −4.00E + 07 | | std | 5.48E + 01 | 4.41E + 06 | 9.50E + 04 | 8.48E + 07 | 0.00E + 00 | 5.94E + 06 | 4.07E + 05 | 6.69E-02 | 1.76E + 00 | 1.66E-01 | 4.26E-01 | 0.00E + 00 | | F3 | avg | −1.40E + 01 | −1.39E + 01 | −1.40E + 01 | −1.40E + 01 | −1.40E + 01 | −1.39E + 01 | −1.40E + 01 | −1.40E + 01 | −1.40E + 01 | −1.40E + 01 | −1.40E + 01 | −3.80E + 00 | | std | 2.90E-02 | 8.64E-02 | 4.37E-02 | 2.22E-03 | 0.00E + 00 | 7.09E-02 | 2.00E-03 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 1.75E-04 | 1.56E + 01 | | F4 | avg | 1.52E + 01 | 1.60E + 01 | 1.52E + 01 | 1.50E + 01 | 1.50E + 01 | 1.53E + 01 | 1.50E + 01 | 1.50E + 01 | 1.50E + 01 | 1.50E + 01 | 1.50E + 01 | 1.50E + 01 | | std | 2.19E-01 | 3.67E-01 | 2.56E-01 | 8.40E-03 | 0.00E + 00 | 1.92E-01 | 3.03E-02 | 5.00E-05 | 6.00E-05 | 2.52E-04 | 2.38E-04 | 1.85E-04 | | F5 | avg | 0.00E + 00 | 1.06E-03 | 2.56E-03 | −1.08E + 00 | 0.00E + 00 | 1.34E-01 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 1.00E-05 | 3.67E-18 | 6.74E-15 | | std | 0.00E + 00 | 3.19E-03 | 5.66E-03 | 1.39E-02 | 0.00E + 00 | 5.78E-02 | 0.00E + 00 | 0.00E + 00 | 0.00E + 00 | 3.00E-05 | 5.61E-18 | 6.74E-15 | | F6 | avg | −7.89E + 03 | −5.34E + 05 | −2.85E + 11 | −1.15E + 10 | −4.84E + 10 | −2.23E + 03 | −1.22E + 07 | −3.27E + 06 | −2.42E + 19 | −1.42E + 11 | −3.37E + 06 | −4.92E + 18 | | std | 6.35E + 03 | 1.52E + 06 | 6.46E + 11 | 1.82E + 10 | 1.37E + 11 | 2.14E + 03 | 2.86E + 07 | 5.95E + 06 | 2.09E + 19 | 2.00E + 11 | 9.88E + 06 | 1.34E + 19 | | F7 | avg | −1.55E + 02 | −1.44E + 02 | −6.06E + 02 | −6.90E + 02 | −4.30E + 02 | −6.25E-02 | −6.35E + 02 | −5.93E + 02 | −5.07E + 02 | −5.50E + 02 | −6.99E + 02 | −5.99E + 02 | | std | 1.15E + 02 | 6.55E + 01 | 9.45E + 01 | 2.44E + 02 | 1.45E + 01 | 4.97E + 00 | 4.64E + 01 | 2.63E + 00 | 3.58E + 01 | 2.49E + 01 | 1.14E-13 | 2.45E-02 | | F8 | avg | 1.85E + 02 | 8.83E + 01 | 1.68E + 01 | NA | 1.11E + 01 | 2.08E + 02 | 5.70E + 01 | 1.10E + 01 | 1.35E + 01 | 1.23E + 01 | 0.00E+00 | 1.22E + 01 | | std | 6.19E + 01 | 2.35E + 01 | 8.85E + 00 | NA | 1.53E + 00 | 1.93E + 01 | 1.32E + 01 | 2.34E + 00 | 3.00E + 00 | 1.86E + 00 | 0.00E+00 | 6.98E + 00 | | F9 | avg | 7.15E-01 | 1.87E + 00 | 6.58E + 00 | 2.39E + 00 | −7.17E-01 | 7.59E + 00 | 1.91E-01 | −6.32E + 00 | −4.73E + 00 | −2.45E + 00 | 1.00E+00 | −6.22E + 00 | | std | 6.74E-02 | 9.78E-01 | 2.58E + 00 | 1.01E + 00 | 8.59E-01 | 8.93E-01 | 8.75E-01 | 8.88E-16 | 1.56E + 00 | 4.77E-01 | 0.00E+00 | 1.05E-01 | | F10 | avg | 8.26E + 00 | 6.75E + 00 | 4.88E + 00 | −1.16E + 02 | 2.32E + 00 | 8.32E + 00 | 4.97E + 00 | 6.79E-01 | 5.28E + 00 | 2.06E + 00 | 2.81E-01 | 2.81E-01 | | std | 1.38E-01 | 3.59E-01 | 1.30E + 00 | 2.99E + 02 | 1.91E-01 | 1.91E-01 | 2.01E-01 | 5.50E-01 | 1.32E + 00 | 1.30E + 00 | 0.00E+00 | 5.55E-17 | | F11 | avg | −1.06E + 07 | −1.00E + 07 | −1.04E + 07 | −3.36E + 07 | −2.09E + 07 | −1.07E + 07 | −1.96E + 07 | −2.88E + 07 | −2.88E + 07 | −1.76E + 07 | −1.00E + 06 | −2.81E + 07 | | std | 8.85E + 05 | 1.13E + 06 | 3.47E + 06 | 4.45E + 06 | 7.61E + 05 | 7.93E + 05 | 8.58E + 05 | 5.34E + 04 | 2.40E + 05 | 7.99E + 05 | 4.89E + 03 | 7.07E + 05 | | F12 | avg | −5.03E + 05 | −2.38E + 05 | −5.24E + 05 | −1.04E + 06 | −4.73E + 05 | −1.81E + 05 | −4.87E + 05 | −5.80E + 05 | −5.77E + 05 | −4.93E + 05 | −1.99E + 04 | −5.69E + 05 | | std | 1.62E + 04 | 2.33E+04 | 2.92E + 04 | 7.51E + 04 | 1.30E + 04 | 6.01E + 04 | 1.36E + 04 | 1.19E + 03 | 4.77E + 03 | 2.40E + 04 | 1.38E + 02 | 1.05E + 04 | | F13 | avg | −4.96E + 05 | −2.38E + 05 | −5.59E + 05 | −1.17E + 06 | −5.01E + 05 | −2.30E + 05 | −5.12E + 05 | −5.80E + 05 | −5.79E + 05 | −5.42E + 05 | −2.02E + 04 | −5.77E + 05 | | std | 8.87E + 03 | 2.33E + 04 | 1.15E + 04 | 9.96E + 04 | 8.89E + 03 | 1.24E + 04 | 7.79E + 03 | 1.06E + 03 | 7.01E + 02 | 1.54E + 04 | 9.47E-03 | 9.17E + 03 | | F14 | avg | −3.70E + 01 | −2.87E + 01 | −3.00E + 01 | −4.76E + 01 | −4.77E + 01 | −3.51E + 01 | −4.57E + 01 | −4.77E + 01 | −4.76E + 01 | −4.74E + 01 | NA | −4.77E + 01 | | std | 1.97E + 00 | 1.38E + 00 | 1.86E + 00 | 2.68E-02 | 5.62E-02 | 1.98E + 00 | 3.78E-01 | 5.82E-03 | 1.70E-01 | 2.19E-01 | NA | 2.01E-02 | | F15 | avg | −6.10E-04 | 1.70E-02 | 8.80E-03 | −2.04E-03 | −2.80E-03 | 4.64E-01 | −2.05E-03 | −2.80E-03 | −2.80E-03 | −2.68E-03 | −2.63E-03 | −2.72E-03 | | std | 2.22E-03 | 2.68E-02 | 1.38E-02 | 1.70E-03 | 0.00E + 00 | 4.09E-01 | 2.28E-04 | 0.00E + 00 | 0.00E + 00 | 3.83E-05 | 1.47E-04 | 9.80E-05 |
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Bold values are the best results obtained for each function to show the algorithms that performs best.
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