Wolf Pack Algorithm for Unconstrained Global Optimization

Table 10

Statistical results of 50 runs obtained by GA, PSO, AFSA, ABC, FA, and WPA algorithms.

Function	Global extremum	𝐷	C	Algorithms	Best	Worst	Mean	StdDev	SR/%	Art/𝑠
Rosenbrock	𝑓min(𝑥)=0	2	UN	GA	1.78𝑒−10	0.0373	0.0091	0.0092	10	>759.8323
				PSO	2.26𝑒−11	5.89𝑒−7	1.07𝑒−7	1.30𝑒−7	100	0.7444
				AFSA	𝟏.𝟏𝟎𝐞−𝟏𝟑	1.11𝑒−9	2.34𝑒−10	2.62𝑒−10	100	2.0578
				ABC	5.99𝑒−6	0.0099	8.61𝑒−4	0.0015	0	>391.0297
				FA	6.28𝑒−13	𝟔.𝟐𝟗𝐞−𝟏𝟎	𝟏.𝟖𝟔𝐞−𝟏𝟎	𝟏.𝟔𝟐𝐞−𝟏𝟎	100	33.1256
				WPA	3.49𝑒−11	2.34𝑒−8	5.09𝑒−9	4.34𝑒−9	100	6.6333
Colville	𝑓min(𝑥)=0	4	UN	GA	0.0022	0.3343	0.1272	0.1062	0	>1.22𝑒+3
				PSO	1.29𝑒−6	3.46𝑒−4	5.06𝑒−5	6.71𝑒−5	0	>114.0869
				AFSA	𝟑.𝟔𝟔𝐞−𝟖	8.91𝑒−7	3.16𝑒−7	2.32𝑒−7	100	40.1807
				ABC	0.0103	0.5337	0.1871	0.1232	0	>384.4193
				FA	2.41𝑒−7	3.69𝑒−5	6.62𝑒−6	8.07𝑒−6	8	>3.14𝑒+3
				WPA	4.71𝑒−8	𝟑.𝟕𝟐𝐞−𝟕	𝟏.𝟐𝟓𝐞−𝟕	𝟔.𝟗𝟕𝐞−𝟖	100	27.4054
Sphere	𝑓min(𝑥)=0	200	US	GA	1.56𝑒+5	1.81𝑒+5	1.71𝑒+5	5.78𝑒+3	0	>4.44𝑒+4
				PSO	1.0361	1.5520	1.2883	0.1206	0	>271.9201
				AFSA	5.12𝑒+5	5.79𝑒+5	5.51𝑒+5	1.63𝑒+4	0	>7.41𝑒+3
				ABC	0.0041	1.2521	0.0444	0.1773	0	>442.9045
				FA	0.1432	0.2327	0.1865	0.0199	0	>8.34𝑒+3
				WPA	𝟏.𝟒𝟗𝐞−𝟏𝟕𝟐	𝟐.𝟒𝟏𝐞−𝟏𝟔𝟓	𝟏.𝟓𝟔𝐞−𝟏𝟔𝟔	0	100	6.1729
Sumsquares	𝑓min(𝑥)=0	150	US	GA	5.93𝑒+4	7.15+4	6.63𝑒+4	2.88𝑒+3	0	>3.16𝑒+4
				PSO	39.7098	91.1145	55.9050	10.4165	0	>232.5464
				AFSA	1.43𝑒+5	1.79𝑒+5	1.64𝑒+5	9.58𝑒+3	0	>7.36𝑒+3
				ABC	1.71𝑒−5	0.0017	1.99𝑒−4	3.36𝑒−4	0	>435.1848
				FA	8.9920	99.8861	40.5721	19.2743	0	>6.88𝑒+3
				WPA	𝟐.𝟔𝟖𝐞−𝟏𝟕𝟐	𝟓.𝟒𝟕𝐞−𝟏𝟔𝟔	𝟐.𝟔𝟐𝐞−𝟏𝟔𝟕	0	100	6.5954
Booth	𝑓min(𝑥)=0	2	MS	GA	4.55𝑒−11	4.55𝑒−11	4.55𝑒−11	0	100	1.2621
				PSO	1.22𝑒−12	2.41𝑒−8	2.80𝑒−9	4.52𝑒−9	100	0.2079
				AFSA	3.02𝑒−12	1.45𝑒−9	4.61𝑒−10	4.08𝑒−10	100	4.4329
				ABC	𝟔.𝟎𝟓𝐞−𝟐𝟎	𝟏.𝟒𝟏𝐞−𝟏𝟕	𝟒.𝟔𝟑𝐞−𝟏𝟖	𝟒.𝟏𝟒𝐞−𝟏𝟖	100	0.4175
				FA	1.80𝑒−12	4.39𝑒−9	1.18𝑒−9	1.11𝑒−9	100	37.9191
				WPA	8.22𝑒−15	7.05𝑒−13	1.21𝑒−13	1.19𝑒−13	100	6.9339
Bridge	𝑓max(𝑥)=3.0054	2	MN	GA	3.0054	3.0054	3.0054	1.35𝑒−15	100	0.1927
				PSO	3.0054	3.0054	3.0054	4.84𝑒−8	100	0.0929
				AFSA	3.0054	3.0047	3.0052	1.69𝑒−4	12	>8.01𝑒+3
				ABC	3.0054	3.0054	3.0054	3.59𝑒−15	100	0.0932
				FA	3.0054	3.0054	3.0054	3.11𝑒−10	100	22.7230
				WPA	3.0054	3.0054	3.0054	𝟑.𝟓𝟖𝐞−𝟏𝟓	100	0.1742
Ackley	𝑓min(𝑥)=0	50	MN	GA	11.4570	12.6095	12.1612	0.2719	0	>1.04𝑒+4
				PSO	0.0469	1.7401	0.6846	0.6344	0	>192.5522
				AFSA	20.1600	20.6009	20.4229	0.1009	0	>9.80𝑒+3
				ABC	20.0085	20.0025	20.0061	0.0014	0	>596.3841
				FA	0.0101	0.0209	0.0160	0.0021	0	>4.28𝑒+3
				WPA	𝟖.𝟖𝟖𝐞−𝟏𝟔	𝟒.𝟒𝟒𝐞−𝟏𝟓	𝟏.𝟏𝟎𝐞−𝟏𝟓	𝟖.𝟓𝟐𝐞−𝟏𝟔	100	7.9476
Griewank	𝑓min(𝑥)=0	100	MN	GA	317.4525	399.6376	363.4174	17.2922	0	>2.07𝑒+4
				PSO	0.0029	0.0082	0.0052	0.0011	0	>367.0080
				AFSA	2.05𝑒+3	2.55𝑒+3	2.33𝑒+3	109.6821	0	>6.51𝑒+3
				ABC	8.95𝑒−7	0.0043	2.26𝑒−4	7.81𝑒−4	2	>620.9561
				FA	0.0068	0.0118	0.0091	0.0011	0	>5.72𝑒+3
				WPA	0	0	0	0	100	14.5338