Scientific Programming

Research Article

A Hybrid Algorithm Framework with Learning and Complementary Fusion Features for Whale Optimization Algorithm

Table 5

Comparison of optimization results obtained for the multimodal benchmark functions.


AL	Runs = 30, N = 30, and T = 1000
AL		DE	BBO	BSA	PSO	HS	GSA	WOA	WOA-DE	WOA-BSA

F₈	Best	−12569.48662	−9288.859115	−12319.8552	−8562.082036	−12569.47154	−3257.715637	−8674.343572	−8985.661303	−10577.83345
	Mean	−12565.53865	−8297.653051	−11982.40966	−6118.655936	−12568.80513	−2434.32431	−6658.410197	−7600.141184	−7657.917372
	Std	21.623781	497.301094	222.38506	1574.830603	0.909111	355.937095	863.971262	861.181057	1183.224502

F₉	Best	44.307253	29.244538	16.598597	14.992992	17.949826	14.924381	0	0	0
	Mean	56.783475	53.437985	25.788633	46.538306	25.466452	25.802587	0	3.672248	0
	Std	6.266486	14.333435	4.03673	14.270927	4.772648	6.630417	0	20.113732	0

F₁₀	Best	4.23E − 07	0.226971	0.007536	3.00E − 06	1.080969	5.45E − 09	8.88E − 16	8.88E − 16	8.88E − 16
	Mean	7.02E − 07	0.349244	0.020786	0.08323	2.063648	7.70E − 09	4.44E − 15	4.20E − 15	3.97E − 15
	Std	1.52E − 07	0.070199	0.011333	0.317495	0.425232	1.50E − 09	2.29E − 15	1.30E − 15	2.03E − 15

F₁₁	Best	1.44E − 11	0.503713	0.001224	7.42E − 12	0.120691	3.009729	0	0	0
	Mean	9.80E − 10	0.766039	0.026851	0.011493	0.618727	8.568461	0.003104	0.001092	0.001282
	Std	3.07E − 09	0.11031	0.029255	0.009701	0.275077	4.065895	0.011816	0.005981	0.007021

F₁₂	Best	2.39E − 13	0.001193	3.00E − 06	3.38E − 14	0.000383	3.75E − 19	0.001352	0.003179	0.002984
	Mean	1.00E − 12	0.002505	6.30E − 05	0.003456	0.000581	0.151627	0.006188	0.013432	0.409258
	Std	1.05E − 12	0.000844	8.00E − 05	0.018927	0.000121	0.210923	0.006962	0.004444	1.60174

F₁₃	Best	1.70E − 12	0.023921	8.00E − 05	5.23E − 12	0.006145	5.94E − 18	0.025211	0.122758	0.080548
	Mean	4.31E − 12	0.044601	0.000741	0.001831	0.013234	0.007331	0.233207	0.468482	0.47475
	Std	2.69E − 12	0.012881	0.000969	0.004165	0.005965	0.025745	0.181659	0.238598	0.224147

F₁₄	Best	0.998004	0.998004	0.998004	0.998004	0.998004	0.998118	0.998004	0.998004	0.998004
	Mean	0.998004	6.813974	0.998004	3.459539	0.998004	4.082277	2.47579	3.454612	2.108894
	Std	0	5.363613	4.12E − 17	2.863628	1.87E − 14	2.690671	2.445473	3.273113	2.790461

F₁₅	Best	0.000352	0.000419	0.000307	0.000308	0.000401	0.001284	0.000308	0.000307	0.000307
	Mean	0.000577	0.006093	0.000329	0.000794	0.004004	0.002584	0.000601	0.000513	0.000468
	Std	0.000109	0.008761	4.10E − 05	0.000233	0.007444	0.002019	0.000315	0.000377	0.000382

F₁₆	Best	−1.031628	−1.031628	−1.031628	−1.031628	−1.031628	−1.031628	−1.031628	−1.031628	−1.031628
	Mean	−1.031628	−0.977217	−1.031628	−1.031628	−1.031628	−1.031628	−1.031628	−1.031628	−1.031628
	Std	6.71E − 16	0.207068	6.12E − 16	6.78E − 16	3.53E − 08	5.13E − 16	3.86E − 11	6.71E − 16	5.98E − 16

F₁₇	Best	0.397887	0.397887	0.397887	0.397887	0.397887	0.397887	0.397887	0.397887	0.397887
	Mean	0.397887	0.397887	0.397887	0.397887	0.397887	0.397887	0.397888	0.397887	0.397887
	Std	0	9.33E − 11	0	0	4.37E − 08	0	9.30E − 07	0	1.25E − 13

F₁₈	Best	3	3	3	3	3	3	3	3	3
	Mean	3	3.9	3	3	3.900137	3	3.000014	3	3
	Std	1.27E − 15	4.929503	2.02E − 15	1.52E − 15	4.929483	2.67E − 15	3.80E − 05	7.82E − 16	1.66E − 15

F₁₉	Best	−3.862782	−3.862782	−3.862782	−3.862782	−3.862782	−3.862782	−3.862782	−3.862782	−3.862782
	Mean	−3.862782	−3.862782	−3.862782	−3.862782	−3.862782	−3.862782	−3.862628	−3.862782	−3.862782
	Std	2.71E − 15	2.29E − 15	2.71E − 15	2.70E − 15	1.00E − 06	2.39E − 15	0.000366	2.66E − 15	2.51E − 15

F₂₀	Best	−3.321995	−3.321995	−3.321995	−3.321995	−3.321994	−3.321995	−3.321989	−3.321995	−3.321995
	Mean	−3.321989	−3.270475	−3.321995	−3.282364	−3.30217	−3.321995	−3.264959	−3.293815	−3.254599
	Std	3.20E − 05	0.059923	1.73E − 14	0.057005	0.045065	1.47E − 15	0.071826	0.051995	0.059944

F₂₁	Best	−10.1532	−10.1532	−10.1532	−10.1532	−10.153199	−10.1532	−10.153181	−10.1532	−10.1532
	Mean	−10.049112	−5.725156	−10.1532	−8.062579	−5.406265	−6.043604	−8.882256	−8.882841	−8.462151
	Std	0.531474	3.327602	1.15E − 09	3.091228	3.673677	3.495936	2.378758	2.378893	2.686626

F₂₂	Best	−10.402941	−10.402941	−10.402941	−10.402941	−10.40294	−10.402941	−10.40292	−10.402941	−10.402941
	Mean	−10.211001	−5.529595	−10.402941	−9.521178	−6.545124	−10.402941	−8.448955	−7.763089	−7.96332
	Std	1.045029	3.343034	1.69E − 14	2.005406	3.686937	1.32E − 15	2.836906	3.128103	2.86345

F₂₃	Best	−10.53641	−10.53641	−10.53641	−10.53641	−10.53641	−10.53641	−10.536335	−10.53641	−10.53641
	Mean	−10.536383	−5.855753	−10.53641	−9.16093	−5.180275	−10.175835	−9.003007	−7.990287	−8.647563
	Std	0.000108	3.655594	1.11E − 09	2.570428	3.329307	1.485436	2.591415	3.225805	2.733822

Rank	Best	1	8	3	2	7	5	6	4	2
	Mean	2	9	1	6	8	7	5	3	4
	Std	1	9	2	6	8	3	7	4	5