Solving Constrained Global Optimization Problems by Using Hybrid Evolutionary Computing and Artificial Life Approaches

Table 2

Numerical results of the proposed RGA-PSO and AIA-PSO algorithms for TPs 1–13.

TP number	Global optimum	Methods	Best	Mean	Median	Worst	MAPE %	S.D.	MCCT (sec)	P value
1	24.306	The proposed; RGA-PSO	24.323	24.568	24.521	24.831	1.078	0.13	432.39	0.643
		The proposed; AIA-PSO	24.358	24.579	24.564	24.809	1.125	0.11	417.63
2	−30665.539	The proposed; RGA-PSO	−30665.539	−30665.539	−30665.539	−30665.534	1.34𝐸−06	7.36𝐸−04	349.55	0.478
		The proposed; AIA-PSO	−30665.539	−30665.539	−30665.539	−30665.539	9.95𝐸−07	3.81𝐸−05	349.76
3	680.630	The proposed; RGA-PSO	680.632	680.640	680.639	680.658	1.46𝐸−03	5.55𝐸−03	398.72	0.839
		The proposed; AIA-PSO	680.633	680.640	680.640	680.657	1.50𝐸−03	0.01	398.84
4	−15	The proposed; RGA-PSO	−15	−15	−15	−15	1.27𝐸−06	1.27𝐸−06	466.00	0.294
		The proposed; AIA-PSO	−15	−15	−15	−15	1.20𝐸−10	2.64𝐸−11	450.28
5	1227.1978	The proposed; RGA-PSO	1227.2321	1228.8471	1228.3935	1231.6003	1.34𝐸−01	1.27	481.86	0.002*
		The proposed; AIA-PSO	1227.1598	1228.1800	1227.9762	1229.6590	8.02𝐸−02	0.71	461.30
6	3.9511	The proposed; RGA-PSO	3.9521	3.9577	3.9566	3.9722	1.67𝐸−01	5𝐸−03	439.16	0.682
		The proposed; AIA-PSO	3.9516	3.9581	3.9569	3.9735	1.77𝐸−01	4.79𝐸−03	438.55
7	−5.7398	The proposed; RGA-PSO	−5.7398	−5.7398	−5.7398	−5.7398	2.54𝐸−04	1.13𝐸−05	325.73	0.358
		The proposed; AIA-PSO	−5.7398	−5.7398	−5.7398	−5.7398	2.42𝐸−04	7.74𝐸−06	329.12
8	−83.254	The proposed; RGA-PSO	−83.2497	−83.2497	−83.2497	−83.2497	5.13𝐸−03	1.26𝐸−06	211.91	0.291
		The proposed; AIA-PSO	−83.2497	−83.2497	−83.2497	−83.2497	5.13𝐸−03	6.85𝐸−07	213.01
9	−6.0482	The proposed; RGA-PSO	−6.0441	−5.9883	−5.9968	−5.8051	0.991	0.05	440.72	0.577
		The proposed; AIA-PSO	−6.0467	−5.9828	−5.9979	−5.8274	1.082	0.05	441.42
10	6300	The proposed; RGA-PSO	6299.8374	6299.8412	6299.8419	6299.8428	2.52𝐸−03	1.42𝐸−03	224.13	0.001*
		The proposed; AIA-PSO	6299.8395	6299.8420	6299.8423	6299.8425	2.51𝐸−03	6.83𝐸−04	223.91
11	10122.6964	The proposed; RGA-PSO	10122.4732	10122.4925	10122.4925	10122.6444	2.01𝐸−03	2.26𝐸−02	340.89	0.884
		The proposed; AIA-PSO	10122.4852	10122.4920	10122.4927	10122.4931	2.02𝐸−03	1.85𝐸−03	338.37
12	—	The proposed; RGA-PSO	0.012692	0.012724	0.012721	0.012784	—	1.46𝐸−05	293.67	0.014*
		The proposed; AIA-PSO	0.012667	0.012715	0.012719	0.012778	—	2.00𝐸−05	296.15
13	—	The proposed; RGA-PSO	5885.3018	5895.0381	5885.3326	6005.4351	—	24.33	291.17	0.019*
		The proposed; AIA-PSO	5885.3249	5886.5426	5885.3323	5906.7404	—	4.54	292.16

(The “—” denotes unavailable information, and *represents that the mean values obtained using the RGA-PSO and AIA-PSO algorithms are statistically different.)