Biogeography-Based Optimization with Orthogonal Crossover

Table 2

Comparison of improved BBO algorithms and OXBBO.

	PBBO-G [10]	RCBBO-G [11]	BBO [11]	MOBBO [12]	OXBBO
	Mean (Std)	Mean (Std)	Mean (Std)	Mean (Std)	Mean (Std)
𝑓1	5.61𝐸−11(4.92𝐸−11)	1.39𝐸−03(5.50𝐸−04)	1.63𝐸−03(6.60𝐸−04)	7.13𝐸−50(8.47𝐸−50)	3.96𝐸−28(2.47𝐸−28)
𝑓2	3.10𝐸−07(1.80𝐸−07)	7.99𝐸−02(1.44𝐸−02)	8.04𝐸−02(1.42𝐸−02)	8.64𝐸−42(3.69𝐸−42)	1.12𝐸−19(4.01𝐸−20)
𝑓3	5.93𝐸−02(1.84𝐸−01)	2.27𝐸+01(1.03𝐸+01)	4.80𝐸+01(2.19𝐸+01)	2.00𝐸−03(3.30𝐸−03)	3.21𝐸−03(1.96𝐸−03)
𝑓4	1.28𝐸−02(2.00𝐸−03)	3.09𝐸−02(7.27𝐸−03)	2.68𝐸−02(5.09𝐸−03)	1.67𝐸−03(7.00𝐸−03)	2.61𝐸−15(8.83𝐸−15)
𝑓5	7.55𝐸+00(1.02𝐸+01)	5.54𝐸+01(3.52𝐸+01)	5.27𝐸+01(3.91𝐸+01)	2.10𝐸+01(6.85𝐸+00)	1.89𝐸+01(1.84𝐸−01)
𝑓6	0.00𝐸+00(0.00𝐸+00)	0.00𝐸+00(0.00𝐸+00)	0.00𝐸+00(0.00𝐸+00)	0.00𝐸+00(0.00𝐸+00)	0.00𝐸+00(0.00𝐸+00)
𝑓7	1.16𝐸−02(3.9𝐸−03)	1.75𝐸−02(6.43𝐸−03)	1.87𝐸−02(5.11𝐸−03)	3.49𝐸−03(6.30𝐸−03)	2.01𝐸−03(6.45𝐸−04)
𝑓8	−12569(4.168𝐸−12)	−12569.5(2.20𝐸−05)	−12569.5(1.93𝐸−05)	−12569.5(1.05𝐸−12)	−12596.5(0.00𝐸+00)
𝑓9	1.74𝐸−11(1.12𝐸−11)	2.62𝐸−02(9.76𝐸−03)	2.77𝐸−02(1.02𝐸−02)	0.00𝐸+00(0.00𝐸+00)	0.00𝐸+00(0.00𝐸+00)
𝑓10	3.35𝐸−06(2.31𝐸−06)	2.51𝐸−02(5.51𝐸− 03)	2.89𝐸−02(5.15𝐸−03)	6.22𝐸−15(0.00𝐸+00)	6.79𝐸−15(1.50𝐸−15)
𝑓11	8.40𝐸−03(1.49𝐸−03)	8.49𝐸−02(5.44𝐸−02)	2.99𝐸−02(3.20𝐸−02)	0.00𝐸+00(0.00𝐸+00)	0.00𝐸+00(0.00𝐸+00)
𝑓12	7.81𝐸−14(1.20𝐸−13)	3.28𝐸−05(3.33𝐸−05)	2.73𝐸−05(2.49𝐸−05)	1.57𝐸−32(2.89𝐸−48)	7.06𝐸−30(3.95𝐸−30)
𝑓13	5.33𝐸−12(9.44𝐸−12)	3.72𝐸−04(4.63𝐸−04)	5.84𝐸−04(8.82𝐸−04)	1.42𝐸−32(1.56𝐸−33)	7.45𝐸−29(3.00𝐸−29)
𝑓14	0.9980(3.67𝐸−11)	0.998017(5.23𝐸−05)	0.998(2.74𝐸−05)	0.998(1.96𝐸−16)	0.998004(2.34𝐸−16)
𝑓15	7.28𝐸−04(1.94𝐸−04)	7.86𝐸−04(1.80𝐸−04)	9.00𝐸−04(2.68𝐸−04)	7.26𝑒−04(7.39𝐸−05)	6.29𝐸−04(4.79𝐸−05)
𝑓16	−1.0316(5.92𝐸−07)	−1.03101(9.01𝐸−04)	−1.03095(1.09𝐸−03)	−1.0316(7.17𝐸−14)	−1.03163(4.52𝐸−12)
𝑓17	0.3979(1.03𝐸−06)	0.398414(6.77𝐸−04)	0.398327(4.26𝐸−04)	0.3979(0)	0.397888(2.41𝐸−06)
𝑓18	3.0000(1.76𝐸−08)	3.009504(1.12𝐸−02)	3.007858(9.57𝐸−03)	3.000(9.87𝐸−15)	3.000(8.30𝐸−15)
𝑓19	−3.8628(7.30𝐸−09)	−3.86248(3.65𝐸−04)	−3.86253(2.62𝐸−04)	−3.8628(4.19𝐸−16)	−3.86278(2.40𝐸−15)
𝑓20	−3.2744(0.0614)	−3.31691(2.36𝐸−02)	−3.30741(3.90𝐸−02)	−3.3101(0.0376)	−3.31962(0.0168)
𝑓21	−6.6284(3.1750)	−5.51341(3.35𝐸+00)	−4.49193(3.34𝐸+00)	−6.6598(3.7522)	−7.69866(3.011)
𝑓22	−7.4888(3.8490)	−6.80022(3.52𝐸+00)	−6.73583(3.40𝐸+00)	−9.8714(1.6808)	−10.3598(0.3048)
𝑓23	−6.3403(3.7343)	−7.28480(3.38𝐸+00)	−7.80261(3.29𝐸+00)	−9.8663(2.119)	−10.4040(0.810)

Note: according to the setting, these algorithms use the same maximum generations for each function, but population size of MOBBO is 65∗3=195 while that of other algorithms is 100, so the MAX_NFFEs of MOBBO are twice those of the others.