Modified Biogeography-Based Optimization with Local Search Mechanism

Table 4

Comparisons with hybrid algorithms.

	OXDE			DE/BBO			MLBBO
	Mean (std)	MeanFEs	SR	Mean (std)	MeanFEs	SR	Mean (std)	MeanFEs	SR
f1	2.60𝐸−03 (9.04𝐸−04)^+	NaN	0	1.53𝐸−30 (1.48𝐸−30)^+	4.7𝐸+04	30	2.30𝐸−31 (2.74𝐸−31)	2.8𝐸+04	30
f2	1.82𝐸−02 (4.77𝐸−03)^+	NaN	0	2.52𝐸−24 (1.74𝐸−24)^−	6.7𝐸+04	30	1.61𝐸−21 (7.89𝐸−22)	5.8𝐸+04	30
f3	5.43𝐸−01 (2.52𝐸−01)^+	NaN	0	2.60𝐸−03 (1.72𝐸−03)^+	NaN	0	2.16𝐸−20 (3.19𝐸−20)	1.4𝐸+05	30
f4	4.80𝐸−02 (7.19𝐸−02)^+	NaN	0	6.50𝐸−02 (2.23𝐸−01)^~	2.5𝐸+05	18	7.99𝐸−08 (1.38𝐸−07)	3.5𝐸+05	30
f5	3.73𝐸−01 (7.68𝐸−01)^+	NaN	0	2.33𝐸+01 (9.10𝐸+00)^+	NaN	0	2.56𝐸−21 (6.21𝐸−21)	2.3𝐸+05	30
f6	0.00𝐸+00 (0.00𝐸+00)^~	9.3𝐸+04	30	0.00𝐸+00 (0.00𝐸+00)^~	2.2𝐸+04	30	0.00𝐸+00 (0.00𝐸+00)	1.1𝐸+04	30
f7	6.64𝐸−03 (1.97𝐸−03)^+	2.0𝐸+05	30	4.69𝐸−03 (1.54𝐸−03)^+	1.4𝐸+05	30	2.13𝐸−03 (9.10𝐸−04)	6.7𝐸+04	30
f8	1.33𝐸−05 (1.57𝐸−05)^+	NaN	0	0.00𝐸+00 (0.00𝐸+00)^~	1.0𝐸+05	30	6.06𝐸−14 (3.32𝐸−13)	5.5𝐸+04	30
f9	5.32𝐸+01 (6.72𝐸+00)^+	NaN	0	0.00𝐸+00 (0.00𝐸+00)^~	1.8𝐸+05	30	0.00𝐸+00 (0.00𝐸+00)	9.2𝐸+04	30
f10	1.45𝐸−02 (2.75𝐸−03)^+	NaN	0	6.10𝐸−15 (6.49𝐸−16)^~	6.7𝐸+04	30	6.10𝐸−15 (6.49𝐸−16)	5.0𝐸+04	30
f11	1.17𝐸−04 (1.35𝐸−04)^−	NaN	0	0.00𝐸+00 (0.00𝐸+00)^−	5.0𝐸+04	30	3.37𝐸−03 (4.64𝐸−03)	2.8𝐸+04	19
f12	6.97𝐸−05 (3.20𝐸−05)^+	NaN	0	1.68𝐸−31 (1.43𝐸−31)^~	4.3𝐸+04	30	1.63𝐸−25 (8.95𝐸−25)	2.7𝐸+04	30
f13	5.10𝐸−04 (2.19𝐸−04)^+	NaN	0	2.37𝐸−30 (2.69𝐸−30)^+	4.7𝐸+04	30	4.98𝐸−32 (4.19𝐸−32)	2.7𝐸+04	30
F1	1.57𝐸−09 (7.48𝐸−10)^+	2.4𝐸+05	30	0.00𝐸+00 (0.00𝐸+00)^−	4.8𝐸+04	30	2.69𝐸−29 (6.98𝐸−29)	2.9𝐸+04	30
F2	6.08𝐸+02 (1.92𝐸+02)^+	NaN	0	1.08𝐸+01 (1.07𝐸+01)^+	NaN	0	1.60𝐸−12 (1.57𝐸−12)	1.6𝐸+05	30
F3	8.26𝐸+06 (2.15𝐸+06)^+	NaN	0	2.91𝐸+06 (1.01𝐸+06)^+	NaN	0	9.00𝐸+04 (4.99𝐸+04)	NaN	0
F4	2.28𝐸+03 (7.46𝐸+02)^+	NaN	0	1.49𝐸+02 (8.27𝐸+01)^+	NaN	0	6.70𝐸−05 (1.87𝐸−04)	2.9𝐸+05	6
F5	3.56𝐸+02 (9.42𝐸+01)^−	NaN	0	1.17𝐸+03 (2.83𝐸+02)^+	NaN	0	8.28𝐸+02 (3.40𝐸+02)	NaN	0
F6	2.03𝐸+01 (1.74𝐸+00)^+	NaN	0	2.89𝐸+01 (1.61𝐸+01)^+	NaN	0	1.76𝐸−09 (6.58𝐸−09)	1.9𝐸+05	30
F7	3.52𝐸−03 (1.07𝐸−02)^−	2.5𝐸+05	27	7.64𝐸−03 (5.42𝐸−03)^−	1.3𝐸+05	29	1.47𝐸−02 (1.47𝐸−02)	5.0𝐸+04	19
F8	2.09𝐸+01 (5.72𝐸−02)^~	NaN	0	2.09𝐸+01 (4.84𝐸−02)^~	NaN	0	2.10𝐸+01 (4.54𝐸−02)	NaN	0
F9	7.46𝐸+01 (1.03𝐸+01)^+	NaN	0	0.00𝐸+00 (0.00𝐸+00)^~	1.5𝐸+05	30	0.00𝐸+00 (0.00𝐸+00)	7.8𝐸+04	30
F10	1.87𝐸+02 (1.13𝐸+01)^+	NaN	0	7.95𝐸+01 (9.19𝐸+00)^+	NaN	0	3.37𝐸+01 (9.16𝐸+00)	NaN	0
F11	3.96𝐸+01 (9.53𝐸−01)^+	NaN	0	3.09𝐸+01 (1.35𝐸+00)^+	NaN	0	1.83𝐸+01 (8.49𝐸+00)	NaN	0
F12	4.78𝐸+03 (4.49𝐸+03)^+	NaN	0	4.25𝐸+03 (3.87𝐸+03)^+	NaN	0	1.57𝐸+03 (1.87𝐸+03)	NaN	0
F13	1.08𝐸+01 (7.91𝐸−01)^+	NaN	0	2.78𝐸+00 (1.94𝐸−01)^−	NaN	0	3.05𝐸+00 (1.88𝐸−01)	NaN	0
F14	1.34𝐸+01 (1.33𝐸−01)^+	NaN	0	1.29𝐸+01 (1.19𝐸−01)^+	NaN	0	1.28𝐸+01 (3.00𝐸−01)	NaN	0
Better	22			14
Similar	2			8
Worse	3			5

Note: “+”, “−”, and “~” indicate that MLBBO is significantly better, worse, and indifferent, respectively, compared with other algorithms.