Hybrid Biogeography Based Optimization for Constrained Numerical and Engineering Optimization

Table 3

Statistical features of results for twelve benchmark functions obtained by HBBO and other six state-of-the-art algorithms.

Function	Metrics	HBBO	CBO-DM	PSO-DE	CRGA	SAPF	SMES	CDE
G01	Best	−15	−15.000	15.000000	−14.9977	−15.000	−15.000	−15.0000
	Mean	−14.799953	−15.000	−15.000000	−14.9850	−14.552	−15.000	−15.0000
	Worst	−13	−15.000	−15.000000	−14.9467	−13.097	−15.000	−15.0000
G02	Best	−0.8036179	−0.803557	−0.8036145	−0.802959	−0.803202	−0.803601	−0.794669
	Mean	−0.7805965	−0.802774	−0.756678	−0.764494	−0.755798	−0.785238	−0.785480
	Worst	−0.7330360	−0.792576	−0.6367995	−0.722109	−0.745712	−0.751322	−0.779837
G03	Best	−1.0050100	−1.000	−1.0050100	−0.9997	−1.000	−1.000	NA
	Mean	−1.0050100	−1.000	−1.0050100	−0.9972	−0.964	−1.000	NA
	Worst	−1.0050100	−1.000	−1.0050100	−0.9931	−0.887	−1.000	NA
G04	Best	−30665.53867	−30665.539	−30665.539	−30665.520	−30665.401	−30665.539	−30665.539
	Mean	−30665.5387	−30665.539	−30665.539	−30664.398	−30665.922	−30665.539	−30665.536
	Worst	−30665.5387	−30665.539	−30665.539	−30660.313	−30656.471	−30665.539	−30665.509
G05	Best	5126.4842	5126.498	NA	5126.500	5126.907	5126.599	NA
	Mean	5126.4842	5126.498	NA	5507.041	5214.232	5174.492	NA
	Worst	5126.4842	5126.498	NA	6112.075	5564.642	5304.167	NA
G06	Best	−6961.81388	−6961.814	−6961.8139	−6956.251	−6961.046	−6961.814	−6961.814
	Mean	−6961.81388	−6961.814	−6961.8139	−6740.288	−6953.061	−6961.284	−6960.603
	Worst	−6961.81388	−6961.814	−6961.8139	−6077.123	−6943.304	−6952.482	−6901.285
G07	Best	24.3062091	24.326	24.3062091	24.882	24.838	24.327	NA
	Mean	24.3062091	24.345	24.306210	25.746	27.328	24.475	NA
	Worst	24.3062091	24.378	24.3062172	27.381	33.095	24.843	NA
G08	Best	−0.095825	−0.095825	−0.095826	−0.095825	−0.095825	−0.095825	NA
	Mean	−0.095825	−0.095825	−0.095826	−0.095819	−0.095635	−0.095825	NA
	Worst	−0.095825	−0.095825	−0.095826	−0.095808	−0.092697	−0.095825	NA
G09	Best	680.630057	680.630	680.63006	680.726	680.773	680.632	680.771
	Mean	680.630057	680.630	680.63006	681.347	681.246	680.643	681.503
	Worst	680.630057	680.630	680.63006	682.965	682.081	680.719	685.144
G10	Best	7049.248021	7059.802	7049.2480	7114.743	7069.981	7051.903	NA
	Mean	7049.248021	7075.832	7049.2480	8785.149	7238.964	7253.047	NA
	Worst	7049.248021	7098.254	7049.2482	10826.09	7489.406	7638.366	NA
G11	Best	0.74990	0.75	0.749999	0.750	0.749	0.75	NA
	Mean	0.74990	0.75	0.749999	0.752	0.751	0.75	NA
	Worst	0.74990	0.75	0.749999	0.757	0.757	0.75	NA
G12	Best	−1	−1.000000	1.000000	−1.000000	−1.000000	−1.000	−1.000000
	Mean	−1	−1.000000	−1.000000	−1.000000	−0.99994	−1.000	−1.000000
	Worst	−1	−1.000000	−1.000000	−1.000000	−0.999548	−1.000	−1.000000