Discrete and Continuous Optimization Based on Hierarchical Artificial Bee Colony Optimizer

Table 3

Results of all algorithms on benchmark continuous functions 𝑓₁--𝑓₈ and discrete functions 𝑓₁₆--𝑓₁₉. In bold are the best results.

Function		HABC_SB	HABC_SW	HABC_SM	HABC_SR	HABC_AB	HABC_AW	HABC_AM	HABC_AR
𝑓1	Mean	3.84𝑒−20	9.87𝑒−4	2.14𝑒−14	1.25𝑒−7	4.66𝑒−144	𝟒.𝟓𝟐𝐞−𝟑𝟎𝟎	5.84𝑒−301	5.67𝑒−259
	Std	5.44𝑒−20	1.390𝑒−3	3.03𝑒−14	1.77𝑒−7	6.59𝑒−144	0	0	0
	Rank	5	8	6	7	4	1	2	3
𝑓2	Mean	2.12𝑒−002	2.15𝑒+001	1.0𝑒+001	7.54	3.96𝑒−005	1.26𝑒−004	2.75	𝟏.𝟏𝟎𝐞−𝟎𝟎𝟓
	Std	5.03𝑒−002	1.54𝑒+001	1.39𝑒+001	1.26𝑒+001	7.43𝑒−005	3.38𝑒−004	8.69	𝟏.𝟗𝟐𝐞−𝟎𝟎𝟓
	Rank	4	8	7	6	2	3	5	1
𝑓3	Mean	2.20𝑒−005	2.36𝑒−018	2.99𝑒−007	1.21𝑒−009	3.57𝑒−148	𝟏.𝟎𝟕𝐞−𝟐𝟖𝟐	4.02𝑒−282	6.57𝑒−278
	Std	2.65𝑒−005	4.74𝑒−018	5.99𝑒−007	2.41𝑒−009	7.14𝑒−148	0	0	0
	Rank	8	5	7	6	4	1	2	3
𝑓4	Mean	1.77𝑒−004	4.11𝑒−007	8.44𝑒−007	0	0	0	0	0
	Std	2.27𝑒−004	5.66𝑒−007	1.12𝑒−006	0	0	0	0	0
	Rank	8	6	7	1	1	1	1	1
𝑓5	Mean	1.74𝑒−004	9.39	3.59𝑒−006	9.12𝑒−001	0	0	0	0
	Std	3.87𝑒−004	11.47	6.14𝑒−006	8.61𝑒−001	0	0	0	0
	Rank	6	8	5	7	1	1	1	1
𝑓6	Mean	5.01𝑒−004	3.84𝑒−002	3.87𝑒−004	3.84𝑒−004	4.14𝑒−004	𝟑.𝟖𝟏𝐞−𝟎𝟎𝟒	3.99𝑒−004	3.82𝑒−004
	Std	2.86𝑒−004	5.15𝑒−002	7.92𝑒−006	3.42𝑒−006	3.99𝑒−005	6.43𝑒−012	2.54𝑒−012	0
	Rank	7	8	5	4	6	1	2	3
𝑓7	Mean	1.77𝑒−003	1.21	6.83𝑒−005	2.78𝑒−001	𝟖.𝟖𝟖𝐞−𝟎𝟏𝟔	𝟖.𝟖𝟖𝐞−𝟎𝟏𝟔	𝟖.𝟖𝟖𝐞−𝟎𝟏𝟔	𝟖.𝟖𝟖𝐞−𝟏𝟔
	Std	5.12𝑒−003	1.93	2.13𝑒−004	8.24𝑒−001	0	0	0	0
	Rank	6	8	5	7	1	1	1	1
𝑓8	Mean	1.76𝑒−004	4.10𝑒−007	8.44𝑒−007	0	0	0	0	0
	Std	2.27𝑒−004	5.80𝑒−007	1.19𝑒−006	0	0	0	0	0
	Rank	8	6	7	1	1	1	1	1
Total Rank		52	57	49	39	20	10	15	14
𝑓16	Mean	0.12	0.3	0.34	0.26
	Std	0.10	0.19	0.09	0.16
	Rank	1	3	4	2
𝑓17	Mean	0.07	0.25	0.21	0.26
	Std	0.08	0.18	0.73	0.13
	Rank	1	3	2	4
𝑓18	Mean	1.76𝑒−004	4.11𝑒−007	8.44𝑒−007	0
	Std	2.22𝑒−004	5.81	1.19	0
	Rank	4	2	3	1
𝑓19	Mean	1.94	3.61	2.72	3.86
	Std	0.59	0.95	0.90	1.10
	Rank	1	3	2	4
Total Rank		7	11	11	11