Modified Atom Search Optimization Based on Immunologic Mechanism and Reinforcement Learning

Table 4

Test statistical results of functions f₁∼f₂₁.

Functions	Algorithms	Min	Mean	Max	STD
f1(x)	MASO	2.99E − 45	1.82E − 40	3.5E − 39	7.82E − 40
	ASO	7.33E − 27	2.64E − 26	1.05E − 25	3.08E − 26
	FPA	0.006587	0.013937	0.023855	0.004844
	BAT	1.09E − 05	155.0339	891.9841	257.7465
	PSO	3.869551	20.12067	77.25433	16.40947
	CS	0.004142	173.506	1688.332	391.8062
	ACOR	6.98E − 13	2.01E − 12	3.85E − 12	7.74E − 13
	CSA	1.29E − 12	8.89E − 09	7.83E − 08	1.9E − 08
f2(x)	MASO	2.41E − 34	1.63E − 29	1.34E − 28	3.9E − 29
	ASO	2.16E − 13	5.32E − 13	9.67E − 13	2.51E − 13
	FPA	0.677651	1.159856	1.89272	0.340836
	BAT	0.014345	42.11026	124.5488	48.56557
	PSO	6.26162	9.511573	16.34641	2.636531
	CS	0.024397	1.543266	8.754166	2.237715
	ACOR	3.46E − 07	0.000387	0.006348	0.001411
	CSA	9.9E − 07	4.2E − 05	0.000133	3.52E − 05
f3(x)	MASO	1.5E − 47	5.73E − 42	8.19E − 41	1.87E − 41
	ASO	5.73E − 26	1.73E − 25	3.92E − 25	1.03E − 25
	FPA	0.000957	0.001794	0.002651	0.000518
	BAT	0.000105	0.000156	0.000193	2.85E − 05
	PSO	1.681533	8.073872	24.85504	6.662011
	CS	0.138672	63.08822	290.7739	80.09947
	ACOR	7.82E − 14	2.16E − 13	4.42E − 13	1.09E − 13
	CSA	1.27E − 12	1.63E − 09	1.89E − 08	4.14E − 09
f4(x)	MASO	0	0	0	0
	ASO	0	0	0	0
	FPA	0	0	0	0
	BAT	4214	10180.7	16319	3108.576
	PSO	57	191.7	405	90.82435
	CS	9	1558.45	6328	1791.922
	ACOR	0	0	0	0
	CSA	0	0	0	0
f5(x)	MASO	2.34E − 06	1.47E − 05	3.73E − 05	9.24E − 06
	ASO	0.003764	0.00772	0.011135	0.002189
	FPA	0.006508	0.020989	0.029649	0.005437
	BAT	0.218579	0.403253	0.531159	0.080489
	PSO	0.273095	0.62667	1.206052	0.285222
	CS	0.94235	4.144344	13.32052	2.939838
	ACOR	0.016274	0.032401	0.053185	0.009913
	CSA	3.99E − 06	5.29E − 05	0.000164	4.22E − 05
f6(x)	MASO	2.41E − 60	6.82E − 51	1.26E − 49	2.82E − 50
	ASO	2.19E − 52	1.46E − 51	5.24E − 51	1.51E − 51
	FPA	3.28E − 11	3.39E − 10	1.43E − 09	3.33E − 10
	BAT	6.29E − 11	1.02E − 10	1.45E − 10	2.17E − 11
	PSO	0.001601	0.010088	0.038612	0.010173
	CS	0.000168	0.34018	2.726353	0.655415
	ACOR	2.22E − 14	2.49E − 13	1.82E − 12	3.95E − 13
	CSA	3.56E − 31	1.06E − 23	1.02E − 22	2.42E − 23
f7(x)	MASO	2.22E − 14	3.83E − 13	5.36E − 12	1.18E − 12
	ASO	4.35E − 14	7.41E − 14	1.47E − 13	2.97E − 14
	FPA	0.17994	0.618946	1.286978	0.260178
	BAT	10.99074	13.41295	19.95889	1.88331
	PSO	3.213753	5.385464	7.818361	1.282747
	CS	8.631765	13.19331	16.18969	1.882678
	ACOR	2.3E − 07	4.2E − 07	5.56E − 07	1.07E − 07
	CSA	2.84E − 07	1.95E − 05	4.98E − 05	1.54E − 05
f8(x)	MASO	1.11E − 16	1.65E − 15	5.66E − 15	1.65E − 15
	ASO	0	0	0	0
	FPA	0.086888	0.136963	0.230776	0.044916
	BAT	19.19178	53.02061	94.33961	21.77808
	PSO	0.939324	1.178857	1.343306	0.102008
	CS	0.610131	4.621765	33.27042	8.201352
	ACOR	4.88E − 12	0.022291	0.247193	0.069061
	CSA	3.57E − 10	4.25E − 08	4.24E − 07	9.87E − 08
f9(x)	MASO	0	4.97E − 13	3.18E − 12	9.06E − 13
	ASO	9.949591	14.5264	20.89413	3.356089
	FPA	53.45329	72.74809	84.86339	8.314181
	BAT	38.80531	79.29986	169.1439	35.18297
	PSO	75.51275	110.7482	169.454	25.42386
	CS	30.82795	51.57561	84.51111	14.55878
	ACOR	183.2672	214.6805	237.272	12.35372
	CSA	1.76E − 12	2.82E − 09	7.92E − 09	2.7E − 09
f10(x)	MASO	0.000382	0.000382	0.000382	1.35E − 11
	ASO	4106.132	4650.915	5626.077	521.8557
	FPA	3244.947	3698.782	4219.299	237.132
	BAT	5972.237	7099.381	8123.466	576.3557
	PSO	3897.536	5375.416	6743.524	814.7786
	CS	1848.715	2793.954	4026.17	610.2232
	ACOR	1478.361	5880.244	7607.56	360.132
	CSA	0.000382	0.000382	0.000382	7E − 08
f11(x)	MASO	3.44E − 33	1.83E − 05	9.49E − 05	3.11E − 05
	ASO	4.23E − 14	8.76E − 14	1.97E − 13	4.72E − 14
	FPA	3.227243	6.669646	9.146383	1.526006
	BAT	0.834363	3.926399	9.459001	2.576305
	PSO	2.748819	6.186047	10.35792	2.432746
	CS	0.039447	0.854302	2.652989	0.811748
	ACOR	0.034574	0.302129	5.21918	1.157372
	CSA	7.64E − 08	5.4E − 06	2.31E − 05	5.86E − 06
f12(x)	MASO	1.15E − 31	6.29E − 28	1.18E − 26	2.63E − 27
	ASO	2.78E − 27	6.39E − 27	2.1E − 26	5.59E − 27
	FPA	1.055809	2.341216	3.399438	0.776936
	BAT	3.166887	8.757063	15.72329	3.538368
	PSO	1.089431	7.019646	16.18996	3.659522
	CS	9.73E − 09	0.005653	0.104223	0.02323
	ACOR	1.69E − 11	9.04E − 11	4.43E − 10	9.82E − 11
	CSA	1.24E − 13	5.21E − 11	4.53E − 10	1.13E − 10
f13(x)	MASO	2.22E − 16	9.1E − 16	5.33E − 15	1.14E − 15
	ASO	0.111929	0.171566	0.247934	0.04255
	FPA	0.164183	0.23541	0.288289	0.035616
	BAT	0.493711	0.498347	0.499585	0.001342
	PSO	0.213232	0.336072	0.442089	0.05312
	CS	0.470226	0.493116	0.498482	0.006452
	ACOR	0.018867	0.036172	0.046726	0.006468
	CSA	1.33E − 15	4.72E − 12	2.33E − 11	6.56E − 12
f14(x)	MASO	1.78E − 15	1.44E − 14	4.09E − 14	1.04E − 14
	ASO	12	18.70114	35.0005	6.254622
	FPA	58.21314	75.72416	92.27162	10.98064
	BAT	142.0001	225.425	320.25	43.94807
	PSO	51.61432	97.72042	154.1477	27.68505
	CS	22.03245	65.54915	124.1758	23.27628
	ACOR	159.1282	193.0334	215.8543	15.42857
	CSA	5.74E − 11	5.9E − 09	2.73E − 08	8.01E − 09
f15(x)	MASO	9.9E − 13	7.77E − 09	8.32E − 08	2E − 08
	ASO	13.22012	14.72793	15.51416	0.63515
	FPA	23.19234	26.9907	29.18676	1.618782
	BAT	0.095172	18.58866	73.64575	30.2613
	PSO	305.1689	574.6557	1477.057	316.5058
	CS	92.87089	4700.287	44298.9	10461.76
	ACOR	24.81962	25.10222	25.36799	0.158517
	CSA	2.24E − 11	1.35E − 09	5.65E − 09	1.97E − 09
f16(x)	MASO	3.12E − 32	1.12E − 29	2.02E − 28	4.51E − 29
	ASO	3.53E − 29	8.02E − 29	1.31E − 28	3.4E − 29
	FPA	0.795144	1.715761	2.908752	0.550818
	BAT	0.519122	11.97827	28.55629	9.510974
	PSO	3.03539	7.040329	11.93878	2.719544
	CS	7.55E − 08	0.019757	0.366427	0.081707
	ACOR	9.06E − 09	2.61E − 07	2.1E − 06	4.61E − 07
	CSA	2.13E − 14	3.58E − 11	1.54E − 10	4.63E − 11
f17(x)	MASO	1.26E − 32	2.26E − 31	1.86E − 30	4.61E − 31
	ASO	6.55E − 28	2.14E − 27	5.54E − 27	1.53E − 27
	FPA	0.075634	0.159202	0.383942	0.076028
	BAT	28.02352	59.78051	75.72239	10.54314
	PSO	3.843779	71.81837	922.9239	200.6675
	CS	3.36E − 22	8.9E − 22	1.77E − 21	4.31E − 22
	ACOR	3.17E − 08	8.81E − 07	6.2E − 06	1.69E − 06
	CSA	9.23E − 13	2.67E − 10	8.43E − 10	2.34E − 10
f18(x)	MASO	2.22E − 14	1.37E − 13	8.96E − 13	2.02E − 13
	ASO	4.71E − 14	9.18E − 14	1.75E − 13	3.96E − 14
	FPA	1.169421	1.510639	2.147189	0.28967
	BAT	12.84255	14.03792	16.05812	0.884501
	PSO	3.803351	5.659908	7.923137	0.986783
	CS	14.20253	16.00982	17.58788	1.084051
	ACOR	3.56E − 07	8.04E − 07	1.54E − 06	3.01E − 07
	CSA	6.75E − 07	2.1E − 05	5.99E − 05	1.93E − 05
f19(x)	MASO	0	5E − 13	4.66E − 12	1.05E − 12
	ASO	8.954632	16.01884	19.89918	3.492605
	FPA	59.36055	76.74372	100.1136	10.82332
	BAT	31.84028	84.16432	198.9923	34.74113
	PSO	68.84252	109.8572	159.7136	22.65396
	CS	65.6753	109.6626	154.4972	25.50229
	ACOR	202.1726	219.1054	230.5434	8.423366
	CSA	6.31E − 12	5.12E − 09	3.79E − 08	1.01E − 08
f20(x)	MASO	0	1.46E − 15	8.99E − 15	1.99E − 15
	ASO	0	0	0	0
	FPA	0.053602	0.127282	0.226275	0.05216
	BAT	30.91076	58.58257	119.8216	20.73163
	PSO	0.744758	1.095692	1.310903	0.116757
	CS	0.868257	3.707799	19.43772	4.708682
	ACOR	8.4E − 10	0.00311	0.062197	0.013908
	CSA	2.6E − 13	3.35E − 08	1.31E − 07	4.47E − 08
f21(x)	MASO	1140.688	2142.009	2596.788	363.9108
	ASO	4046.881	4920.749	5321.667	391.9594
	FPA	2480.206	2843.365	3076.285	173.4922
	BAT	3444.441	5378.728	7371.65	998.61
	PSO	1906.464	4896.28	6944.173	1242.374
	CS	3032.338	4125.281	5258.021	555.419
	ACOR	2691.551	3918.742	5687.103	463.1189
	CSA	7233.943	8295.476	9227.791	487.6725

The best results are shown in bold.