Table 8 Experimental and statistical results of HGMRFO and seven competitive algorithms on IEEE CEC2017 benchmark functions with 10 dimensions, where HGMRFO is the main algorithm in statistical results (W/T/L).
| Â | HGMRFO | MRFO | FCMRFO | CFDBWDO | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | Std | Â | Mean | Std | Â | Mean | Std | Â | Mean | Std | Â | |
F1 | 8.6683E+02 | 1.2267E+03 | Â | 2.1391E+03 | 2.2607E+03 | \(+\) | 1.4390E+03 | 1.6856E+03 | \(+\) | 2.9078E+08 | 3.2182E+08 | \(+\) |
F3 | 1.6099E−07 | 3.5662E−07 |  | 0.0000E+00 | 0.0000E+00 | – | 0.0000E+00 | 0.0000E+00 | – | 9.6125E+01 | 3.4366E+01 | \(+\) |
F4 | 1.0595E+00 | 6.4060E−01 |  | 4.5960E−01 | 3.0959E−01 | – | 8.0051E−01 | 5.8132E−01 | – | 2.6648E+01 | − 8.6780E+01 | \(+\) |
F5 | 9.6075E+00 | 4.2259E+00 |  | 1.8975E+01 | 9.2856E+00 | \(+\) | 1.3435E+01 | 7.1257E+00 | \(+\) | 3.9411E+01 | − 8.6301E+01 | \(+\) |
F6 | 3.3411E−05 | 1.3797E−04 |  | 1.3743E−01 | 5.2944E−01 | \(\approx\) | 6.9261E−06 | 3.5272E−05 | \(\approx\) | 9.8519E+00 | − 9.6736E+01 | \(+\) |
F7 | 2.1261E+01 | 5.8644E+00 |  | 3.8274E+01 | 1.3229E+01 | \(+\) | 2.9335E+01 | 9.5837E+00 | \(+\) | 4.0592E+01 | − 8.6953E+01 | \(+\) |
F8 | 8.4670E+00 | 4.4534E+00 |  | 2.0916E+01 | 7.8338E+00 | \(+\) | 1.7061E+01 | 7.9085E+00 | \(+\) | 1.8946E+01 | − 9.2168E+01 | \(+\) |
F9 | 1.7817E−02 | 8.9065E−02 |  | 7.3773E−01 | 1.6240E+00 | \(+\) | 7.4777E−02 | 1.6626E−01 | \(+\) | 3.3138E+01 | − 6.6699E+01 | \(+\) |
F10 | 8.9371E+02 | 3.6271E+02 |  | 6.9056E+02 | 2.8553E+02 | – | 5.7381E+02 | 2.4583E+02 | – | 9.8173E+02 | 2.2739E+02 | \(\approx\) |
F11 | 3.9538E+00 | 2.7002E+00 |  | 9.0631E+00 | 7.2642E+00 | \(+\) | 5.6738E+00 | 4.3209E+00 | \(+\) | 2.6683E+01 | − 7.9023E+01 | \(+\) |
F12 | 1.0481E+04 | 9.3136E+03 | Â | 1.4007E+04 | 1.4340E+04 | \(\approx\) | 1.3380E+04 | 1.1107E+04 | \(\approx\) | 3.6156E+06 | 5.5755E+06 | \(+\) |
F13 | 8.1528E+02 | 6.5802E+02 |  | 5.5364E+02 | 3.9581E+02 | – | 7.4775E+02 | 6.5096E+02 | \(\approx\) | 1.3126E+04 | 1.3402E+04 | \(+\) |
F14 | 5.4780E+01 | 1.2855E+01 | Â | 5.3370E+01 | 1.4746E+01 | \(\approx\) | 5.5322E+01 | 1.3296E+01 | \(\approx\) | 1.5092E+02 | 2.9695E+01 | \(+\) |
F15 | 1.1020E+02 | 6.2245E+01 |  | 8.3502E+01 | 4.7708E+01 | – | 1.0481E+02 | 5.6166E+01 | \(\approx\) | 3.8884E+02 | 1.8017E+02 | \(+\) |
F16 | 4.7362E+00 | 1.8420E+01 |  | 4.4551E+01 | 7.2817E+01 | \(\approx\) | 3.4733E+01 | 6.7734E+01 | \(\approx\) | 9.6849E+01 | − 2.8928E+01 | \(+\) |
F17 | 2.1034E+01 | 1.4086E+01 |  | 3.6949E+01 | 2.6993E+01 | \(+\) | 2.4964E+01 | 1.7624E+01 | \(\approx\) | 5.2097E+01 | − 8.1448E+01 | \(+\) |
F18 | 3.0874E+03 | 2.8832E+03 | Â | 3.8506E+03 | 4.1892E+03 | \(\approx\) | 3.3755E+03 | 4.5487E+03 | \(\approx\) | 9.6431E+04 | 1.3356E+05 | \(+\) |
F19 | 5.0447E+01 | 3.6361E+01 | Â | 6.8366E+01 | 4.0008E+01 | \(+\) | 7.7273E+01 | 9.4133E+01 | \(+\) | 8.8857E+02 | 1.2132E+03 | \(+\) |
F20 | 5.6400E+00 | 7.4941E+00 |  | 1.0803E+01 | 1.0352E+01 | \(+\) | 7.8981E+00 | 9.8061E+00 | \(\approx\) | 7.0253E+01 | − 7.5543E+01 | \(+\) |
F21 | 1.0008E+02 | 4.3074E−01 |  | 1.1221E+02 | 3.4553E+01 | \(\approx\) | 9.6376E+01 | 1.9681E+01 | \(\approx\) | 1.0635E+02 | − 9.5544E+01 | \(+\) |
F22 | 8.5595E+01 | 3.3210E+01 |  | 9.5237E+01 | 2.0674E+01 | \(\approx\) | 9.6959E+01 | 1.9299E+01 | \(\approx\) | 1.0640E+02 | − 7.9196E+01 | \(+\) |
F23 | 3.1220E+02 | 5.4232E+00 | Â | 3.2371E+02 | 1.1675E+01 | \(+\) | 2.9770E+02 | 7.5336E+01 | \(+\) | 3.3531E+02 | 1.4766E+01 | \(+\) |
F24 | 1.6727E+02 | 1.1556E+02 |  | 2.4005E+02 | 1.2523E+02 | \(+\) | 2.1898E+02 | 1.2736E+02 | \(+\) | 1.1414E+02 | − 7.5293E+01 | \(+\) |
F25 | 4.2531E+02 | 2.2570E+01 |  | 4.2692E+02 | 2.2906E+01 | \(\approx\) | 4.3201E+02 | 2.0885E+01 | \(+\) | 4.2233E+02 | − 7.9955E+01 | \(\approx\) |
F26 | 2.9504E+02 | 4.2654E+01 |  | 3.2154E+02 | 7.0915E+01 | \(+\) | 2.9101E+02 | 4.9507E+01 | \(\approx\) | 4.1408E+02 | − 2.4751E+01 | \(+\) |
F27 | 3.9532E+02 | 2.3788E+00 |  | 4.0115E+02 | 1.3473E+01 | \(+\) | 3.9973E+02 | 6.4132E+00 | \(+\) | 4.4057E+02 | − 9.0851E+01 | \(+\) |
F28 | 3.1168E+02 | 5.8445E+01 |  | 4.0899E+02 | 1.3246E+02 | \(+\) | 3.6977E+02 | 1.2028E+02 | \(\approx\) | 4.0513E+02 | − 2.0984E+01 | \(+\) |
F29 | 2.6882E+02 | 1.6980E+01 |  | 2.8829E+02 | 3.4224E+01 | \(+\) | 2.6969E+02 | 2.4738E+01 | \(\approx\) | 3.3722E+02 | − 6.2970E+01 | \(+\) |
F30 | 2.0845E+04 | 8.4126E+04 |  | 2.6334E+05 | 4.8761E+05 | \(\approx\) | 1.2791E+05 | 3.4531E+05 | – | 7.8889E+05 | 8.6737E+05 | \(+\) |
W/T/L | –/–/– | 15/9/5 | 11/14/4 | 27/2/0 | ||||||||
| Â | SIS | RSA | FOFPA | SE | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | Std | Â | Mean | Std | Â | Mean | Std | Â | Mean | Std | Â | |
F1 | 2.8508E+03 | 3.2684E+03 | \(+\) | 1.0959E+10 | 3.5843E+09 | \(+\) | 3.3990E+07 | 3.1633E+07 | \(+\) | 9.4515E+02 | 1.0747E+03 | \(+\) |
F3 | 1.8415E+03 | 4.1076E+03 | \(+\) | 8.6206E+03 | 2.4497E+03 | \(+\) | 1.9081E+02 | 6.5715E+01 | \(+\) | 5.7662E+02 | 2.3915E+02 | \(+\) |
F4 | 3.3205E−01 | 1.9930E−01 | – | 7.5599E+02 | 4.8506E+02 | \(+\) | 2.8533E+00 | 1.8370E+00 | \(+\) | 5.6375E+00 | 4.8616E−01 | \(+\) |
F5 | 1.7357E+01 | 9.2791E+00 | \(+\) | 9.0559E+01 | 1.4777E+01 | \(+\) | 2.6142E+01 | 5.4053E+00 | \(+\) | 8.6108E+00 | 1.5538E+00 | \(\approx\) |
F6 | 6.3752E−02 | 2.3733E−01 | \(+\) | 4.8054E+01 | 8.0604E+00 | \(+\) | 1.8584E+01 | 3.6507E+00 | \(+\) | 1.5604E−14 | 3.9511E−14 | – |
F7 | 2.7258E+01 | 6.6245E+00 | \(+\) | 1.0736E+02 | 9.9270E+00 | \(+\) | 3.6236E+01 | 5.2421E+00 | \(+\) | 2.0692E+01 | 1.7287E+00 | \(\approx\) |
F8 | 1.8040E+01 | 7.6703E+00 | \(+\) | 5.2143E+01 | 6.7164E+00 | \(+\) | 1.9637E+01 | 4.0122E+00 | \(+\) | 9.0938E+00 | 1.6881E+00 | \(+\) |
F9 | 6.0602E−02 | 2.6142E−01 | \(\approx\) | 6.4497E+02 | 2.9999E+02 | \(+\) | 1.3895E+02 | 4.4131E+01 | \(+\) | 2.3718E−10 | 1.6133E−09 | \(+\) |
F10 | 9.0434E+02 | 4.2094E+02 | \(\approx\) | 1.5393E+03 | 2.3004E+02 | \(+\) | 6.7542E+02 | 9.6887E+01 | – | 4.0393E+02 | 1.1324E+02 | – |
F11 | 3.2850E+01 | 2.3526E+01 | \(+\) | 2.9215E+03 | 2.0410E+03 | \(+\) | 3.1319E+01 | 9.3211E+00 | \(+\) | 4.0058E+00 | 8.1692E−01 | \(\approx\) |
F12 | 1.4551E+06 | 9.0130E+06 | \(+\) | 5.4526E+08 | 4.5993E+08 | \(+\) | 1.7481E+04 | 1.0061E+04 | \(+\) | 3.8420E+05 | 2.0437E+05 | \(+\) |
F13 | 1.3000E+04 | 1.1576E+04 | \(+\) | 3.1522E+07 | 3.9151E+07 | \(+\) | 6.0343E+02 | 3.0963E+02 | \(\approx\) | 1.4072E+03 | 9.0899E+02 | \(+\) |
F14 | 1.7482E+02 | 9.3844E+01 | \(+\) | 3.3118E+03 | 6.8091E+03 | \(+\) | 9.4497E+01 | 3.4725E+01 | \(+\) | 3.0181E+01 | 2.6405E+01 | – |
F15 | 2.4609E+03 | 1.4848E+03 | \(+\) | 7.8233E+03 | 8.6978E+03 | \(+\) | 2.7223E+02 | 1.5065E+02 | \(+\) | 1.3021E+02 | 1.9794E+02 | – |
F16 | 8.2981E+01 | 8.5457E+01 | \(+\) | 4.7874E+02 | 1.0948E+02 | \(+\) | 3.6972E+01 | 3.0550E+01 | \(+\) | 1.7444E+00 | 3.3481E−01 | \(+\) |
F17 | 4.9245E+01 | 2.5084E+01 | \(+\) | 1.4140E+02 | 4.5116E+01 | \(+\) | 3.5911E+01 | 5.6231E+00 | \(+\) | 3.4226E+00 | 9.1089E−01 | – |
F18 | 1.3725E+04 | 9.4649E+03 | \(+\) | 1.8568E+07 | 2.1789E+07 | \(+\) | 5.9357E+02 | 3.5131E+02 | – | 1.8131E+03 | 1.2739E+03 | – |
F19 | 7.0635E+02 | 1.2398E+03 | \(+\) | 5.4020E+05 | 1.2020E+06 | \(+\) | 1.0529E+02 | 7.1417E+01 | \(+\) | 1.3103E+02 | 4.0827E+02 | – |
F20 | 7.3304E+01 | 3.4593E+01 | \(+\) | 2.7456E+02 | 6.1862E+01 | \(+\) | 6.0629E+01 | 1.1664E+01 | \(+\) | 1.0306E+00 | 4.7335E−01 | – |
F21 | 1.2272E+02 | 2.9913E+01 | \(+\) | 1.6398E+02 | 4.0283E+01 | \(+\) | 1.0779E+02 | 3.4707E+00 | \(+\) | 1.3544E+02 | 4.0624E+01 | \(+\) |
F22 | 9.6542E+01 | 2.0003E+01 | \(\approx\) | 9.2294E+02 | 3.1656E+02 | \(+\) | 9.5384E+01 | 2.0281E+01 | \(+\) | 1.0022E+02 | 3.3708E−01 | – |
F23 | 3.1782E+02 | 8.1816E+00 | \(+\) | 3.9339E+02 | 1.1618E+01 | \(+\) | 3.1227E+02 | 4.1383E+01 | \(+\) | 3.0940E+02 | 2.0349E+00 | – |
F24 | 3.0932E+02 | 5.3478E+01 | \(+\) | 4.3425E+02 | 3.5143E+01 | \(+\) | 1.2101E+02 | 1.2597E+01 | \(+\) | 2.4729E+02 | 1.0364E+02 | \(+\) |
F25 | 4.1516E+02 | 2.2212E+01 | – | 9.8654E+02 | 2.7705E+02 | \(+\) | 3.8129E+02 | 4.7239E+01 | – | 4.0716E+02 | 1.5669E+01 | – |
F26 | 3.0000E+02 | 5.6522E−08 | \(+\) | 1.3374E+03 | 3.1668E+02 | \(+\) | 2.2046E+02 | 4.7471E+01 | – | 3.0000E+02 | 3.0368E−11 | \(+\) |
F27 | 3.9557E+02 | 2.9103E+00 | \(\approx\) | 4.3282E+02 | 1.0592E+01 | \(+\) | 3.9480E+02 | 5.5676E+00 | \(\approx\) | 3.9145E+02 | 2.0970E+00 | – |
F28 | 3.0000E+02 | 1.2696E−08 | \(+\) | 8.9230E+02 | 1.7949E+02 | \(+\) | 3.2666E+02 | 1.9394E+01 | \(+\) | 3.2223E+02 | 6.5358E+01 | \(+\) |
F29 | 2.6964E+02 | 2.2067E+01 | \(\approx\) | 4.6404E+02 | 8.0907E+01 | \(+\) | 2.9019E+02 | 1.5425E+01 | \(+\) | 2.6856E+02 | 8.2235E+00 | \(\approx\) |
F30 | 2.8853E+05 | 8.4205E+05 | \(+\) | 3.9376E+06 | 7.7581E+06 | \(+\) | 2.1289E+04 | 2.9084E+04 | \(+\) | 1.5277E+04 | 9.5982E+03 | \(+\) |
W/T/L | 22/5/2 | 29/0/0 | 23/2/4 | 13/4/12 | ||||||||
