A Novel Selection Approach for Genetic Algorithms for Global Optimization of Multimodal Continuous Functions

Table 5

Statistical results of optimum values for different selection schemes using multimodal benchmark functions.

Selection schemes
Benchmark	Dimension	Statistics	RW	TS	RS	LRS	SWS
Beale	10	Mean	2.5691E + 01	2.3543E + 01	2.5206E + 01	3.0214E + 01	2.5657E + 01
		S.D	5.5649E + 00	4.7657E + 00	5.4125E + 00	5.4887E + 00	5.1936E + 00
		t-test	0.98075	0.10591	0.74302	0.01616
	50	Mean	3.2013E + 02	2.1626E + 02	2.8608E + 02	3.2635E + 02	2.3568E + 02
		S.D	6.3963E + 01	2.3247E + 01	3.9857E + 01	5.1910E + 01	2.5334E + 01
		t-test	0.00000	0.00305	0.00000	0.00000
	100	Mean	9.9730E + 02	6.0705E + 02	9.7269E + 02	1.0248E + 03	6.6155E + 02
		S.D	1.9613E + 02	3.9781E + 01	1.5325E + 02	1.7469E + 02	4.3353E + 01
		t-test	0.00000	0.00000	0.00000	0.00000
Bohachevsky	10	Mean	5.5629E − 01	5.5020E − 07	2.2209E − 01	3.8920E − 01	5.0851E − 07
		S.D	1.1163E + 00	3.9736E − 07	4.9787E − 01	8.0708E − 01	3.6725E − 07
		t-test	0.00838	0.67454	0.01762	0.00000
	50	Mean	88.2869	19.2995	92.1134	93.8072	17.8370
		S.D	29.3399	3.6071	26.7442	31.6492	3.3338
		t-test	0.0000	0.1083	0.0000	0.0000
	100	Mean	251.3332	105.8799	267.6916	274.2843	97.8563
		S.D	41.7324	14.7719	49.6698	45.7011	13.6525
		t-test	0.0000	0.0330	0.0000	0.0000
Drop-wave	10	Mean	−8.4413	−8.3885	−8.4403	−8.4322	−4.4669
		S.D	0.1928	0.2039	0.1480	0.3682	0.5363
		t-test	0.0000	0.0000	0.0000	0.0000
	50	Mean	−36.9639	−37.6504	−37.5574	−37.2521	−17.9598
		S.D	1.7057	1.4806	1.8322	1.9668	1.9511
		t-test	0.0000	0.0000	0.0000	0.0000
	100	Mean	−65.0661	−67.0406	−66.3824	−65.7157	−35.1363
		S.D	6.5688	2.7298	2.7023	4.8334	4.9423
		t-test	0.0000	0.0000	0.0000	0.0000
Egg-holder	10	Mean	−608.5168	−608.5186	−608.5177	−608.5087	−413.0947
		S.D	0.0021	0.0008	0.0017	0.0020	21.3483
		t-test	0.0000	0.0000	0.0000	0.0000
	50	Mean	−3318.3868	−3320.1156	−3318.4740	−3317.6446	−1875.4208
		S.D	1.5810	0.5672	1.2800	1.4306	200.0058
		t-test	0.0000	0.0000	0.0000	0.0000
	100	Mean	−6672.6025	−6694.8362	−6678.1726	−6674.6017	−3541.7138
		S.D	9.4819	4.4452	8.4881	8.9851	251.8305
		t-test	0.0000	0.0000	0.0000	0.0000
Schwefel	10	Mean	−4020.3795	−4062.8059	−4083.1594	−4065.7950	−2898.0973
		S.D	137.7577	114.5713	125.9619	117.8343	249.1698
		t-test	0.0000	0.0000	0.0000	0.0000
	50	Mean	−15391.3838	−14734.6805	−15825.5138	−15622.4744	−8337.1457
		S.D	849.1259	822.2196	766.8707	793.9728	785.8473
		t-test	0.0000	0.0000	0.0000	0.0000
	100	Mean	−24086.2209	−23310.1510	−25169.1100	−24641.6910	−11872.9339
		S.D	1712.0336	1468.7940	1456.0778	1570.0301	1476.5720
		t-test	0.0000	0.0000	0.0000	0.0000