A Multistrategy Artificial Bee Colony Algorithm Enlightened by Variable Neighborhood Search

Table 3

Objective values searched by ABCVNS and ABC for 60 and 200 ( and ) dimensional problems.

No.	Dim	maxFEs	Methods	Best	Worst	Median	Mean	Std.	Sig.
	60	30e4	ABC	2.28e − 055	1.03e − 052	4.89e − 054	1.35e − 053	2.21e − 053	†
			ABCVNS	9.82e − 129	1.24e − 119	2.49e − 123	9.44e − 121	2.92e − 120
	60	30e4	ABC	1.61e − 046	5.23e − 043	7.51e − 045	4.08e − 044	1.01e − 043	†
			ABCVNS	1.62e − 124	1.52e − 114	4.00e − 118	9.47e − 116	3.01e − 115
	60	30e4	ABC	5.28e − 057	4.01e − 054	3.04e − 055	6.76e − 055	9.21e − 055	†
			ABCVNS	2.77e − 128	1.43e − 118	3.45e − 123	7.26e − 120	2.69e − 119
	60	30e4	ABC	4.25e − 060	5.99e − 050	9.36e − 056	2.27e − 051	1.09e − 050	†
			ABCVNS	.00e + 000	3.64e − 177	3.05e − 249	1.21e − 178	0.00e + 000
	60	30e4	ABC	2.54e − 029	6.59e − 028	1.00e − 028	1.32e − 028	1.25e − 028	†
			ABCVNS	6.99e − 066	3.05e − 060	3.57e − 063	1.37e − 061	5.62e − 061
	60	30e4	ABC	4.63e + 000	1.28e + 001	9.95e + 000	9.42e + 000	1.86e + 000	†
			ABCVNS	6.86e − 001	1.30e + 000	1.02e + 000	1.01e + 000	1.62e − 001
	60	30e4	ABC	0	0	0	0	0
			ABCVNS	0	0	0	0	0
	60	30e4	ABC	5.91e − 119	7.61e − 114	1.25e − 115	8.84e − 115	1.90e − 114	†
			ABCVNS	2.69e − 258	1.52e − 239	5.16e − 251	5.07e − 241	0
	60	30e4	ABC	6.29e − 002	1.23e − 001	1.03e − 001	1.00e − 001	1.61e − 002	†
			ABCVNS	2.43e − 002	3.91e − 002	3.08e − 002	3.11e − 002	4.60e − 003
	60	30e4	ABC	1.11e − 003	7.30e − 001	3.65e − 002	7.67e − 002	1.41e − 001	-
			ABCVNS	1.69e − 002	1.43e + 002	7.23e + 001	5.79e + 001	3.71e + 001
	60	30e4	ABC	0	0	0	0	0
			ABCVNS	0	0	0	0	0
	60	30e4	ABC	0	0	0	0	0
			ABCVNS	0	0	0	0	0
	60	30e4	ABC	0	1.27e − 011	0	4.24e − 013	2.32e − 012
			ABCVNS	0	0	0	0	0
	60	30e4	ABC	2.91e − 011	3.64e − 011	2.91e − 011	3.19e − 011	3.40e − 012	†
			ABCVNS	2.91e − 011	2.91e − 011	2.91e − 011	2.91e − 011	0
	60	30e4	ABC	6.48e − 014	1.04e − 013	8.62e − 014	8.69e − 014	9.21e − 015	†
			ABCVNS	1.51e − 014	4.00e − 014	2.93e − 014	2.93e − 014	5.11e − 015
	60	30e4	ABC	7.85e − 033	7.85e − 033	7.85e − 033	7.85e − 033	2.78e − 048
			ABCVNS	7.85e − 033	7.85e − 033	7.85e − 033	7.85e − 033	2.78e − 048
	60	30e4	ABC	1.35e − 032	1.35e − 032	1.35e − 032	1.35e − 032	5.57e − 048
			ABCVNS	1.35e − 032	1.35e − 032	1.35e − 032	1.35e − 032	5.57e − 048
	60	30e4	ABC	2.87e − 010	2.15e − 006	6.63e − 009	1.60e − 007	4.48e − 007	†
			ABCVNS	2.23e − 080	4.22e − 065	2.40e − 067	3.78e − 066	9.53e − 066
	60	30e4	ABC	1.35e − 031	1.35e − 031	1.35e − 031	1.35e − 031	6.68e − 047
			ABCVNS	1.35e − 031	1.35e − 031	1.35e − 031	1.35e − 031	6.68e − 047
	60	30e4	ABC	0	2.84e − 014	1.42e − 014	1.47e − 014	9.50e − 015	†
			ABCVNS	0	0	0	0	0
	60	30e4	ABC	4.30e − 001	4.89e − 001	4.80e − 001	4.76e − 001	1.21e − 002	†
			ABCVNS	3.46e − 001	4.72e − 001	4.52e − 001	4.44e − 001	2.83e − 002
	200	30e4	ABC	−78.33233	−78.33233	−78.33233	−78.33233	6.27e − 013	†
			ABCVNS	−78.33233	−78.33233	−78.33233	−78.33233	6.67e − 014
	200	30e4	ABC	−192.8520	−191.0752	−192.0762	−192.0878	4.09e − 001	†
			ABCVNS	−1.99e + 002	−1.99e + 002	−1.99e + 002	−1.99e + 002	7.25e − 002
	60	30e4	ABC	0	0	0	0	0
			ABCVNS	0	0	0	0	0
	60	30e4	ABC	0	0	0	0	0
			ABCVNS	0	0.0823	0	0.0027	0.0150
	60	30e4	ABC	0	1.11e − 016	0	3.70e − 018	2.03e − 017
			ABCVNS	0	0	0	0	0
	60	30e4	ABC	7.55e − 014	9.33e − 014	8.62e − 014	8.49e − 014	6.56e − 015	†
			ABCVNS	5.06e − 014	7.55e − 014	6.48e − 014	6.19e − 014	6.26e − 015
	60	30e4	ABC	2.01e − 009	2.80e − 005	3.86e − 007	3.37e − 006	6.57e − 006	†
			ABCVNS	1.56e − 016	2.06e − 015	2.17e − 016	4.13e − 016	4.21e − 016

“†” indicates ABCVNS is superior to ABC by the Wilcoxon signed rank test at . “-” means that ABCVNS is inferior to ABC. “” means that there is no significant difference between ABC and ABCVNS.