Table 3.

Comparisons of the statistical results obtained from the different variants of the ATOA algorithms on various benchmark functions.

Function	Global Minima	Measure	AOA	ATOAs	ATOAc	ATOAt	ATOAsc	ATOAcs
F1	0	Mean	0.0000	4.46$\times {10}^{-33}$	9.89$\times {10}^{-28}$	0.0000	0.0000	0.0000
		Best	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
		Worst	0.0000	2.10$\times {10}^{-31}$	4.95$\times {10}^{-26}$	0.0000	0.0000	0.0000
		Std. Dev	0.0000	2.97$\times {10}^{-32}$	7.00$\times {10}^{-27}$	0.0000	0.0000	0.0000
F2	0	Mean	0.0000	8.65$\times {10}^{-139}$	4.13$\times {10}^{-86}$	1.67$\times {10}^{-241}$	5.95$\times {10}^{-55}$	0.0000
		Best	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
		Worst	0.0000	4.32$\times {10}^{-137}$	2.06$\times {10}^{-84}$	8.36$\times {10}^{-240}$	2.97$\times {10}^{-53}$	0.0000
		Std. Dev	0.0000	6.11$\times {10}^{-138}$	2.92$\times {10}^{-85}$	0.0000	4.21$\times {10}^{-54}$	0.0000
F3	0	Mean	0.0000	2.03$\times {10}^{-19}$	1.07$\times {10}^{-10}$	0.0000	0.0000	0.0000
		Best	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
		Worst	0.0000	1.02$\times {10}^{-17}$	5.35$\times {10}^{-9}$	0.0000	0.0000	0.0000
		Std. Dev	0.0000	1.44$\times {10}^{-18}$	7.57$\times {10}^{-10}$	0.0000	0.0000	0.0000
F4	0	Mean	5.1652	7.7668	6.9754	23.2905	6.8636	7.8511
		Best	4.6129	6.9030	5.2892	8.2527	5.2644	6.6216
		Worst	5.8078	8.1612	7.9103	547.483	8.4399	8.6583
		Std. Dev	0.2658	0.3346	0.499	79.9482	0.7056	0.4639
F5	0	Mean	0.0143	0.0018	0.3024	0.032	0.0726	0.0014
		Best	0.0054	0.0004	0.1694	0.0161	0.0279	0.0007
		Worst	0.0232	0.003	0.4172	0.0531	0.1604	0.0027
		Std. Dev	0.0041	5.23$\times {10}^{-4}$	0.056	0.0105	0.0318	0.0005
F6	0	Mean	0.0000	3.69$\times {10}^{-4}$	7.94$\times {10}^{-5}$	0.0211	6.11$\times {10}^{-4}$	0.0022
		Best	0.0000	2.91$\times {10}^{-5}$	3.22$\times {10}^{-6}$	7.45$\times {10}^{-4}$	5.37$\times {10}^{-5}$	0.0000
		Worst	0.0001	0.0018	2.29$\times {10}^{-4}$	0.0926	0.0018	0.0173
		Std. Dev	0.0000	3.83$\times {10}^{-4}$	5.79$\times {10}^{-5}$	0.0204	4.12$\times {10}^{-4}$	0.0029
F7	−418.9829 × n	Mean	−3475.0000	−3.22$\times {10}^3$	−3069.4000	−2.18$\times {10}^3$	−3.08$\times {10}^3$	−3178.2000
		Best	−4071.3000	−4.00$\times {10}^3$	−3670.2000	−3.04$\times {10}^3$	−3.58$\times {10}^3$	−3794.9000
		Worst	−2914.5000	−2.44$\times {10}^3$	−2341.0000	−1.36$\times {10}^3$	−2.52$\times {10}^3$	−2620.2000
		Std. Dev	252.5026	262.4675	284.0105	368.5892	228.423	311.1536
F8	0	Mean	0.0000	6.32$\times {10}^{-12}$	1.98$\times {10}^{-9}$	8.88$\times {10}^{-16}$	8.88$\times {10}^{-16}$	0.0000
		Best	0.0000	8.88$\times {10}^{-16}$	8.88$\times {10}^{-16}$	8.88$\times {10}^{-16}$	8.88$\times {10}^{-16}$	0.0000
		Worst	0.0000	3.16$\times {10}^{-10}$	9.91$\times {10}^{-8}$	8.88$\times {10}^{-16}$	8.88$\times {10}^{-16}$	0.0000
		Std. Dev	0.0000	4.47$\times {10}^{-11}$	1.40$\times {10}^{-8}$	0.0000	0.0000	0.0000
F9	0	Mean	0.5947	8.85$\times {10}^6$	5.88$\times {10}^6$	0.9595	0.4228	1.0271
		Best	0.4953	1.0361	3.2475	0.8689	0.3078	0.9451
		Worst	0.6443	1.67$\times {10}^8$	9.76$\times {10}^7$	1.0228	0.523	1.0944
		Std. Dev	0.0337	2.94$\times {10}^7$	1.52$\times {10}^7$	0.0315	0.0453	0.0317
F10	0	Mean	0.7823	0.1646	0.1116	0.9423	0.1327	0.8997
		Best	0.3881	0.0294	0.0554	0.5441	0.059	0.6860
		Worst	0.9948	0.4447	0.1734	0.9907	0.2184	0.9868
		Std. Dev	0.1563	0.1028	0.0245	0.0795	0.0376	0.0639
F11	1	Mean	8.8855	5.9519	5.5608	8.666	6.1167	5.6534
		Best	0.9980	0.998	0.998	0.998	0.998	0.9980
		Worst	12.6705	11.7187	12.6705	12.6705	12.6705	11.7187
		Std. Dev	3.9075	3.3638	3.1317	3.7149	3.6442	3.2853
F12	0.0003	Mean	0.0104	0.0016	0.0043	0.008	0.0111	0.0078
		Best	0.0003	4.02$\times {10}^{-4}$	3.92$\times {10}^{-4}$	6.66$\times {10}^{-4}$	3.13$\times {10}^{-4}$	0.0004
		Worst	0.0863	0.0245	0.0234	0.0569	0.0566	0.0572
		Std. Dev	0.0153	0.0037	0.0062	0.011	0.0133	0.0111
F13	−1.0316	Mean	−1.0316	−1.0313	−1.0316	−1.0238	−1.0316	−1.0313
		Best	−1.0316	−1.0316	−1.0316	−1.0311	−1.0316	−1.0316
		Worst	−1.0316	−1.0303	−1.0316	−1.0063	−1.0316	−1.0301
		Std. Dev	0.0000	3.27$\times {10}^{-4}$	9.91$\times {10}^{-6}$	0.0066	8.16$\times {10}^{-6}$	0.0003
F14	0.398	Mean	1.1631	0.4233	0.3979	0.4025	0.3979	0.4088
		Best	0.4123	0.3979	0.3979	0.398	0.3979	0.3979
		Worst	2.8698	0.7824	0.398	0.4356	0.3981	0.5690
		Std. Dev	0.5939	0.0758	2.16$\times {10}^{-5}$	0.0067	3.86$\times {10}^{-5}$	0.0259
F15	3	Mean	6.2400	3.0005	3.0002	8.3111	6.253	3.0005
		Best	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000
		Worst	30.0000	3.0028	3.0006	91.5641	84.6034	3.0027
		Std. Dev	8.8630	6.54$\times {10}^{-4}$	1.50$\times {10}^{-4}$	21.2206	16.0971	0.0006
F16	−3.86	Mean	−3.8544	−3.8563	−3.8261	−3.8502	−3.8445	−3.8559
		Best	−3.8608	−3.8625	−3.854	−3.8616	−3.8548	−3.8614
		Worst	−3.8498	−3.8515	−3.0119	−3.8324	−3.8202	−3.8503
		Std. Dev	0.0023	0.0026	0.1181	0.0052	0.009	0.0024
F17	−0.32	Mean	−3.1259	−3.1647	−2.9368	−3.0409	−2.8466	−3.1710
		Best	−3.2643	−3.2551	−3.2132	−3.1812	−3.117	−3.2631
		Worst	−2.9339	−2.9039	−2.2038	−2.4088	−2.3154	−3.0584
		Std. Dev	0.0650	0.0595	0.1899	0.1378	0.2263	0.0473
F18	−10.1532	Mean	−4.1529	−8.2176	−7.7996	−6.5441	−5.3207	−4.7763
		Best	−8.0049	−10.1471	−10.0933	−10.1441	−10.1435	−10.1413
		Worst	−1.9980	−2.6275	−2.6046	−2.6249	−2.6102	−2.6253
		Std. Dev	1.2108	3.251	3.1242	3.2365	2.9748	2.7482
F19	−959.6407	Mean	−800.3974	−863.6167	−850.7201	−706.1923	−859.4495	−856.7580
		Best	−941.2958	−959.6407	−959.6407	−932.4704	−959.6406	−959.6407
		Worst	−644.2784	−559.7869	−559.7868	−474.3529	−545.6967	−575.2190
		Std. Dev	89.8151	97.5246	114.0678	116.7106	112.0762	86.4870
F20	−19.2085	Mean	−18.7179	−18.8544	−18.9728	−18.7377	−18.9140	−18.9721
		Best	−19.2083	−19.2084	−19.2085	−19.2085	−19.2085	−19.2084
		Worst	−15.8160	−16.2678	−16.2678	−16.2678	−16.2678	−16.2678
		Std. Dev	0.7836	0.9649	0.8057	1.0889	0.8910	0.8055
F21	−186.7309	Mean	−116.2292	−176.7813	−178.8541	−186.6622	−176.8742	−181.4309
		Best	−184.3297	−186.7222	−186.7259	−186.7258	−186.7211	−186.7294
		Worst	−50.6600	−79.3989	−123.4528	−186.5087	−64.6800	−123.0804
		Std. Dev	38.0811	25.0093	20.6675	0.0496	29.1740	17.3085
F22	−1.9133	Mean	−1.8773	−1.9132	−1.9132	−1.8676	−1.9132	−1.9131
		Best	−1.9132	−1.9132	−1.9132	−1.9128	−1.9132	−1.9132
		Worst	−1.4783	−1.9131	−1.9132	−1.6836	−1.9131	−1.9127
		Std. Dev	0.1158	3.84$\times {10}^{-6}$	3.64$\times {10}^{-7}$	0.0507	7.44$\times {10}^{-6}$	9.84$\times {10}^{-5}$
F23	−1.0316	Mean	−1.0091	−1.0316	−1.0314	−1.0316	−1.0314	−1.0316
		Best	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316
		Worst	−0.9990	−1.0314	−1.0310	−1.0316	−1.0309	−1.0314
		Std. Dev	0.0135	4.68$\times {10}^{-5}$	1.50$\times {10}^{-4}$	3.78$\times {10}^{-9}$	1.65$\times {10}^{-4}$	4.25$\times {10}^{-5}$
F24	−1	Mean	−0.0532	−0.9400	−0.8598	−0.1601	−0.9998	−0.9638
		Best	−0.9963	−1.0000	−1.0000	−1.0000	−1.0000	−1.0000
		Worst	−2.92$\times {10}^{-9}$	−8.11$\times {10}^{-5}$	−8.09$\times {10}^{-5}$	−8.11$\times {10}^{-5}$	−0.9994	−0.9265
		Std. Dev	0.1934	0.2399	0.3504	0.3703	1.46$\times {10}^{-4}$	0.0169
F25	0.3978	Mean	0.7698	0.3989	0.3980	0.4073	0.3981	0.3993
		Best	0.3985	0.3979	0.3979	0.3982	0.3979	0.3979
		Worst	1.8135	0.4045	0.3984	0.4645	0.4000	0.4142
		Std. Dev	0.3683	0.0011	9.50$\times {10}^{-5}$	0.0103	4.03$\times {10}^{-4}$	0.0027
F26	3	Mean	21.7513	3.0005	3.0002	6.2811	3.0002	3.0004
		Best	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000
		Worst	156.2760	3.0035	3.0010	85.6932	3.0008	3.0016
		Std. Dev	27.7673	7.45$\times {10}^{-4}$	2.02$\times {10}^{-4}$	16.2291	1.71$\times {10}^{-4}$	4.55$\times {10}^{-4}$
F27	−39.1659	Mean	−72.2967	−78.3312	−78.3323	−76.3317	−76.3532	−76.9175
		Best	−78.3276	−78.3323	−78.3323	−78.3312	−78.3323	−78.3323
		Worst	−61.4160	−78.3283	−78.3322	−64.1633	−64.1956	−64.1936
		Std. Dev	5.2366	8.32$\times {10}^{-4}$	2.35$\times {10}^{-5}$	4.9494	4.9551	4.2841
F28	0	Mean	1.4327	0.1547	0.1249	0.0135	0.0647	0.0955
		Best	0.2886	0.0016	6.65$\times {10}^{-5}$	2.44$\times {10}^{-5}$	3.00$\times {10}^{-5}$	3.75$\times {10}^{-4}$
		Worst	2.0000	0.6345	0.4463	0.1169	0.1157	0.1469
		Std. Dev	0.5954	0.1213	0.1332	0.0335	0.0548	0.0438
F29	0	Mean	38.8803	8.2303	5.4673	3.7098	3.2438	3.5159
		Best	6.4751	0.4230	0.1357	0.0792	0.2594	0.1630
		Worst	42.0000	35.4304	28.2788	8.2784	8.2392	8.8946
		Std. Dev	9.3207	8.8018	6.1918	3.0627	2.6992	2.6469
F30	−3.8627	Mean	−3.8545	−3.8560	−3.8128	−3.8366	−3.8257	−3.8558
		Best	−3.8603	−3.8621	−3.8576	−3.8616	−3.8562	−3.8612
		Worst	−3.8482	−3.8491	−3.0234	−3.0861	−3.0025	−3.8497
		Std. Dev	0.0029	0.0029	0.1620	0.1084	0.1194	0.0027
F31	−10.5364	Mean	−4.6271	−5.8656	−4.6245	−5.9376	−4.7527	−6.3397
		Best	−8.3487	−10.5298	−10.4165	−10.5222	−10.4922	−10.5194
		Worst	−2.1589	−1.8533	−1.8495	−1.8526	−1.6908	−1.8526
		Std. Dev	1.3407	3.6248	2.9955	3.2566	2.8037	2.4718
F32	−3.3223	Mean	−2.9401	−2.9589	−2.8212	−2.9047	−2.7873	−2.9603
		Best	−3.0060	−3.0201	−2.9447	−2.9750	−2.9337	−3.0257
		Worst	−2.8450	−2.8800	−2.5712	−2.7635	−2.5398	−2.8884
		Std. Dev	0.0292	0.0268	0.1079	0.0431	0.1204	0.0249
F33	−9.6601	Mean	−3.3874	−3.7494	−2.9064	−3.3626	−2.9680	−3.5913
		Best	−4.1525	−4.5626	−3.3311	−3.8498	−3.3714	−4.2671
		Worst	−2.6242	−2.6367	−2.3804	−2.3524	−2.5597	−2.7726
		Std. Dev	0.3532	0.4382	0.2460	0.3167	0.2131	0.3409