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

Table 4

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

Benchmark	Selection schemes
	Dimension	Statistics	RW	TS	RS	LRS	SWS
Axis parallel hyper ellipsoid	10	Mean	5.0835E − 05	3.2320E − 07	1.5786E − 05	3.3563E − 05	2.9418E − 07
		S.D	7.9800E − 05	2.5215E − 07	3.5962E − 05	5.8134E − 05	2.2951E − 07
		t-test	0.00099	0.64287	0.02170	0.00619
	50	Mean	3.1488E + 02	1.9369E + 01	3.0683E + 02	3.2088E + 02	1.7630E + 01
		S.D	1.8473E + 02	1.0019E + 01	1.9457E + 02	1.9967E + 02	9.1199E + 00
		t-test	0.00000	0.48486	0.00000	0.00000
	100	Mean	3.0516E + 03	7.7562E + 02	2.8707E + 03	3.1637E + 03	7.0599E + 02
		S.D	8.2320E + 02	2.0261E + 02	7.9006E + 02	1.0092E + 03	1.8442E + 02
		t-test	0.00000	0.16921	0.00000	0.00000
Colville	10	Mean	14.1036	1.4075	5.2867	10.0411	1.3926
		S.D	64.1057	1.9450	24.4547	44.6261	1.9245
		t-test	0.2822	0.9763	0.3882	0.0001
	50	Mean	2465.5693	613.2918	2293.8499	2380.0555	606.8044
		S.D	2717.5439	287.6108	1873.1595	2295.6976	284.5685
		t-test	0.0004	0.9303	0.0000	0.0000
	100	Mean	19237.7118	6003.8649	20567.1507	19902.7771	5940.3560
		S.D	6243.5664	1519.1380	8549.9094	7397.0838	1503.0685
		t-test	0.0000	0.8713	0.0000	0.0000
Ellipsoidal	10	Mean	5.8835E − 06	3.6933E − 07	2.1295E − 05	5.1129E − 05	3.3639E − 07
		S.D	8.9408E − 06	4.7645E − 07	8.7336E − 05	5.4186E − 05	4.3396E − 07
		t-test	0.00125	0.78053	0.19389	0.00477
	50	Mean	2408.7101	716.6584	2228.5647	2423.9663	652.7483
		S.D	941.5905	392.6978	784.5990	956.8467	357.6778
		t-test	0.0000	0.5125	0.0000	0.0000
	100	Mean	87681.5841	34427.9838	83676.4636	87696.8404	31357.7650
		S.D	18184.3183	9613.1629	14342.6643	18199.5746	8755.8803
		t-test	0.0000	0.2010	0.0000	0.0000
Rosenbrook	10	Mean	6.9873	7.0153	5.9280	8.2780	6.4416
		S.D	3.6870	1.8204	3.0867	5.2072	1.6715
		t-test	0.0000	0.0000	0.0000	0.0000
	50	Mean	444.0538	263.3415	348.7025	443.6579	241.8055
		S.D	159.6747	47.2798	110.5376	182.3859	43.4132
		t-test	0.0000	0.0712	0.0000	0.0000
	100	Mean	3491.3158	1296.4477	2971.5222	3461.4764	1190.4242
		S.D	1123.5970	230.0574	653.2587	1118.4853	211.2433
		t-test	0.0000	0.0681	0.0000	0.0000
Schaffer	10	Mean	4.5650	4.5645	4.5653	4.5694	4.1455
		S.D	0.0089	0.0084	0.0076	0.0103	0.0076
		t-test	0.0000	0.0000	0.0000	0.0000
	50	Mean	25.5110	25.2177	25.1508	25.5793	22.2471
		S.D	0.1419	0.0708	1.9232	0.1418	0.0643
		t-test	0.0000	0.0000	0.0000	0.0000
	100	Mean	52.6225	51.7076	52.5572	52.9025	45.6164
		S.D	0.2535	0.2024	0.3154	0.3309	0.1838
		t-test	0.0000	0.0000	0.0000	0.0000