Hybrid Multi-Evolutionary Algorithm to Solve Optimization Problems

Figures & data

Figure 1. The hybrid algorithm block diagram.

Table 1. The number of parents and descendants in different types of ESs.

ES	Number of parents	Number of descendants
$\left(\mu +1\right)-\mathrm{E}\mathrm{S}$	20	1
$\left(\mu ,\lambda \right)-\mathrm{E}\mathrm{S}$	30	200
$\left(\mu +\lambda \right)-\mathrm{E}\mathrm{S}$	20	160

Table 2. The value of the algorithm stop criterion and the value of the constant C used for each function.

Function	Stop criterion	constant C
$f_1$	1.596	-
$f_2$	0.999	-
$f_3$	2.50048	-
$f_{Ra}2d$	74.999	75
$f_{Ra}5d$	149.999	150
$f_{Ra}10d$	289.5	290
$f_{ST}2d$	656.663	500
$f_{ST}5d$	1641.6	1250
$f_{ST}10d$	2383.0	2500
$f_{Ro}$	1210120.99	1210121
$f_{Sh}$	271.795	187

Table 3. The average running time and the number of fitness function calls needed to reach the predetermined value of the optimized function.

Algorithm	Number of fitness calls	σ	Running time [s]	σ
Function ${\mathbf{f}}_{\mathbf{1}}$
$\mathrm{G}\mathrm{A}-\left(1+1\right)\mathrm{E}\mathrm{S}$	4394	2548	0.000029	0.000014
$\mathrm{G}\mathrm{A}-\left(\mu +1\right)\mathrm{E}\mathrm{S}$	57500	31841	0.000022	0.000011
$\mathrm{G}\mathrm{A}-\left(\mu ,\lambda \right)ES$	197700	156065	0.000194	0.000106
$\mathrm{G}\mathrm{A}-\left(\mu +\lambda \right)\mathrm{E}\mathrm{S}$	234696	162480	0.000201	0.000113
Function ${\mathbf{f}}_{\mathbf{2}}$
$\mathrm{G}\mathrm{A}-\left(1+1\right)\mathrm{E}\mathrm{S}$	8450	685	0.000048	0.000016
$\mathrm{G}\mathrm{A}-\left(\mu +1\right)\mathrm{E}\mathrm{S}$	205800	26078	0.000069	0.000007
$\mathrm{G}\mathrm{A}-\left(\mu ,\lambda \right)ES$	927700	275635	0.002464	0.000713
$\mathrm{G}\mathrm{A}-\left(\mu +\lambda \right)\mathrm{E}\mathrm{S}$	755601	210262	0.001792	0.000483
Function ${\mathbf{f}}_{\mathbf{3}}$
$\mathrm{G}\mathrm{A}-\left(1+1\right)\mathrm{E}\mathrm{S}$	7137	2405	0.000072	0.000027
$\mathrm{G}\mathrm{A}-\left(\mu +1\right)\mathrm{E}\mathrm{S}$	840450	548589	0.000158	0.000010
$\mathrm{G}\mathrm{A}-\left(\mu ,\lambda \right)ES$	621400	345817	0.000677	0.000348
$\mathrm{G}\mathrm{A}-\left(\mu +\lambda \right)\mathrm{E}\mathrm{S}$	565194	361504	0.000586	0.000336

Table 4. The average time and number of fitness function calls needed to reach the predetermined value of the optimized function.

Function		f1	f2	f3	fRa2d	fRa5d	fRa10d
Number of fitness function calls	SGA	43317	31380	75927	53547	609717	967040
	ES	3459	3136*	16300	750909	*	*
	GA-ES	4427	8320	5990	7092	41730	459019
σ	SGA	35232	3622	49888	28302	298351	306805
	ES	7272	1268*	15662	671132	*	*
	GA-ES	2535	1096	1520	2587	20003	116278
Time [s]	SGA	0.0732	0.0379	0.1506	0.0794	0.7763	1.8962
	ES	0.0309	14.403*	0.0683	0.7091	*	*
	GA-ES	0.0365	0.0614	0.0415	0.0342	0.0997	1.0523
σ	SGA	0.0282	0.0059	0.0778	0.3128	0.3344	0.7065
	ES	0.0074	44.833	0.0399	0.5797	*	*
	GA-ES	0.0069	0.0082	0.0057	0.0064	0.0352	0.2440
Function		fST2d	fST5d	fST10d	fRO	fSH	-
Number of fitness function calls	SGA	60942	193842	857025	651025	42222	-
	ES	46345	*	*	3868	40722	-
	GA-ES	5720	11310	28990	43295	4030	-
σ	SGA	28981	90410	281590	680334	36062	-
	ES	137035	*	*	2522	70610	-
	GA-ES	1397	2297	7606	27897	737	-
Time [s]	SGA	0.0725	0.2098	0.9966	0.3772	0.1182	-
	ES	0.0578	*	*	0.0324	0.1492	-
	GA-ES	0.0398	0.0504	0.0982	0.0641	0.0448	-
σ	SGA	0.0195	0.0807	0.3044	0.3469	0.0623	-
	ES	0.0770	*	*	0.0038	0.1939	-
	GA-ES	0.0056	0.0077	0.0281	0.0184	0.0070	-

*- at least once, the algorithm got stuck to the local optima.

Figure 2. The average running time in the Genetic Algorithm (SGA), the Evolutionary Strategy and the proposed (GA-ES) algorithm.

Figure 3. The number of fitness function calls in the Genetic Algorithm (SGA), the Evolutionary Strategy, and the proposed (GA-ES) algorithm.