Discrete and Continuous Optimization Based on Hierarchical Artificial Bee Colony Optimizer

Table 7

Performance of all algorithms on benchmark continuous functions 𝑓₁--𝑓₁₅.

Function		HABC	CABC	CPSO	CMA-ES	ABC	PSO	GA
𝑓1	Mean	0	9.87𝑒−138	2.14𝑒−014	0	1.93𝑒−015	8.52𝑒−010	3.84𝑒−001
	Std	0	1.390𝑒−3	1.93𝑒−137	0	8.59𝑒−016	1.83𝑒−009	1.50𝑒−001
	Rank	1	3	5	1	4	6	7
𝑓2	Mean	0	5.41	3.81	1.32𝑒−007	1.92𝑒−001	2.98𝑒+001	3.26𝑒+002
	Std	0	6.25	2.56	5.03𝑒−002	2.03𝑒−001	2.48𝑒+001	7.61𝑒+002
	Rank	1	5	4	2	3	6	7
𝑓3	Mean	0	3.80𝑒+001	2.66𝑒+001	0	2.75𝑒+001	2.67𝑒−003	8.55𝑒+001
	Std	0	8.47𝑒+001	3.65𝑒+001	0	4.79	4.45𝑒−003	3.76𝑒+001
	Rank	1	6	4	1	5	3	7
𝑓4	Mean	7.72𝑒−032	2.55𝑒−032	6.65𝑒−009	𝟐.𝟑𝟒𝐞−𝟎𝟑𝟐	8.07𝑒−014	8.27𝑒−001	8.52𝑒−001
	Std	5.75𝑒−032	3.55𝑒−032	1.38𝑒−008	𝟖.𝟕𝟓𝐞−𝟎𝟑𝟒	6.23𝑒−014	6.22𝑒−001	3.54𝑒−001
	Rank	3	2	5	1	4	7	6
	Total Rank1	6	16	18	5	16	22	27
𝑓5	Mean	0	0	2.58	3.61𝑒+001	7.65𝑒−009	3.42𝑒+001	5.29𝑒+001
	Std	0	0	1.55	8.31	5.76𝑒−009	154𝑒+001	4.52
	Rank	1	1	4	6	3	5	7
𝑓6	Mean	2.19	4.50𝑒+002	1.02𝑒+003	2.53𝑒+003	2.31𝑒+001	3.43𝑒+003	1.99𝑒+002
	Std	4.99	1.00𝑒+003	1.06𝑒+002	3.98𝑒+002	5.2𝑒+001	4.39𝑒+002	7.81𝑒+001
	Rank	1	4	5	6	2	7	3
𝑓7	Mean	𝟖.𝟎𝟏𝐞−𝟎𝟏𝟔	3.44𝑒−014	1.15𝑒−004	1.89𝑒+001	2.24𝑒−006	2.42	5.29𝑒+001
	Std	0	6.45𝑒−015	1.22𝑒−004	9.54𝑒−001	1.09𝑒−006	5.43	4.52
	Rank	1	2	5	7	4	6	8
𝑓8	Mean	0	1.20𝑒−008	3.43𝑒−002	1.97𝑒−003	6.23𝑒−002	4.37𝑒−002	2.21
	Std	0	1.17𝑒−008	2.22𝑒−002	4.40𝑒−003	5.05𝑒−002	4.23𝑒−002	1.46𝑒−001
	Rank	1	2	4	3	6	5	7
	Total Rank2	4	9	18	22	15	23	25
𝑓9	Mean	0	1.13𝑒−013	3.37𝑒+002	5.68𝑒−014	2.27𝑒−012	2.06𝑒+001	7.12𝑒+001
	Std	0	4.92𝑒−014	5.84𝑒+002	4.92𝑒−014	1.58𝑒−012	4.61𝑒+001	1.66𝑒+001
	Rank	1	3	7	2	4	5	6
𝑓10	Mean	3.91𝑒+001	2.10𝑒+003	6.14𝑒+003	0	4.50𝑒+003	1.44𝑒+002	8.93𝑒+003
	Std	4.96𝑒+001	1.52𝑒+002	1.17𝑒+004	𝟓.𝟔𝟖𝐞−𝟎𝟏𝟒	1.32𝑒+003	1.10𝑒+002	5.18𝑒+003
	Rank	2	4	6	1	5	3	7
𝑓11	Mean	1.39	1.21𝑒+001	1.33𝑒+006	1.59	6.74	3.20𝑒+001	2.66𝑒+004
	Std	𝟖.𝟗𝟎𝐞−𝟎𝟎𝟏	7.09	1.21𝑒+006	2.18	4.47	3.31𝑒+001	2.11𝑒+004
	Rank	1	4	7	2	3	5	6
𝑓12	Mean	𝟐.𝟒𝟔𝐞−𝟎𝟎𝟑	1.72𝑒+003	1.73𝑒+003	1.72𝑒+003	1.72𝑒+003	4.13𝑒+003	1.92𝑒+003
	Std	𝟓.𝟓𝟏𝐞−𝟎𝟎𝟑	2.39𝑒−011	3.30	3.02𝑒−001	4.41𝑒−008	4.23𝑒+002	4.62𝑒+001
	Rank	1	3	5	2	3	7	6
𝑓13	Mean	𝟐.𝟎𝟔𝐞+𝟎𝟎𝟏	2.07𝑒+001	2.04𝑒+001	𝟐.𝟎𝟏𝐞+𝟎𝟎𝟏	2.07𝑒+001	2.05𝑒+001	2.08𝑒+001
	Std	𝟓.𝟐𝟐𝐞−𝟎𝟎𝟐	1.01𝑒−001	6.10𝑒−002	𝟑.𝟖𝟐𝐞−𝟎𝟎𝟏	4.04𝑒−002	1.40𝑒−001	1.69𝑒−002
	Rank	3	5	2	1	5	4	7
𝑓14	Mean	0	1.01𝑒+003	1.58𝑒+001	4.16𝑒+001	4.26𝑒−002	9.79𝑒+001	3.26𝑒+001
	Std	𝟒.𝟎𝟒𝐞−𝟎𝟏𝟒	3.82𝑒+001	1.15𝑒+001	9.98𝑒+001	9.51𝑒−002	7.66	8.18
	Rank	1	3	4	6	2	7	5
𝑓15	Mean	𝟒.𝟔𝟑𝐞+𝟎𝟎𝟏	1.34𝑒+002	1.87𝑒+002	1.21𝑒+002	1.35𝑒+002	1.79𝑒+002	4.22𝑒+002
	Std	𝟏.𝟐𝟐𝐞+𝟎𝟎𝟏	4.33𝑒+001	5.77𝑒+001	2.47𝑒+001	1.97𝑒+001	5.06𝑒+001	3.62𝑒+001
	Rank	1	3	6	2	4	5	7
Total Rank3		9	25	30	16	26	36	44