A Free Search Krill Herd Algorithm for Functions Optimization

Table 2

Simulation results for 𝐹₁∼𝐹₉ in low dimension.

Benchmark function	Result	Method
		PSO	DE	KH II	FSKH	HS	FS	BA
𝐹1(𝐷=30)	Best	1.8151𝐸−02	1.2075𝐸−01	1.1148𝐸−04	𝟎	1.2072𝐸+01	4.5486𝐸−02	9.4991𝐸−04
	Mean	4.5954𝐸−01	2.4943𝐸−01	1.6110𝐸−03	𝟔.𝟏𝟐𝟓𝟒𝐄−𝟑𝟏𝟔	3.4612𝐸+01	5.9487𝐸−02	1.4434𝐸−03
	Worst	1.6052𝐸+00	3.7078𝐸−01	6.4270𝐸−03	𝟏.𝟏𝟓𝟗𝟗𝐄−𝟑𝟏𝟒	5.1108𝐸+00	7.0424𝐸−02	1.7796𝐸−03
	Std.	3.2409𝐸−01	6.1704𝐸−02	1.6590𝐸−03	𝟎	1.7649𝐸+01	6.7912𝐸−03	1.8339𝐸−04
	Rank	6	5	3	𝟏	7	4	2
𝐹2(𝐷=30)	Best	7.9000𝐸+01	2.1000𝐸+01	𝟎	𝟎	4.0710𝐸+03	1.6000𝐸+01	3.2450𝐸+04
	Mean	2.4706𝐸+02	8.8740𝐸+01	2.8200𝐸+00	𝟎	7.8597𝐸+03	2.3780𝐸+01	5.9598𝐸+04
	Worst	6.2900𝐸+02	2.7900𝐸+02	8.4000𝐸+01	𝟎	1.8688𝐸+03	3.0000𝐸+01	7.2507𝐸+04
	Std.	1.2618𝐸+02	5.3506𝐸+01	1.1845𝐸+01	𝟎	7.1600𝐸+03	2.9017𝐸+00	6.9512𝐸+03
	Rank	5	4	3	𝟏	7	2	6
𝐹3(𝐷=30)	Best	2.8953𝐸+01	3.0199𝐸+01	2.6378𝐸+01	2.8703𝐸+01	3.1922𝐸+02	2.2637𝐸+01	𝟏.𝟎𝟗𝟔𝟏𝐄+𝟎𝟏
	Mean	6.2541𝐸+01	3.6796𝐸+01	2.7980𝐸+01	2.8912𝐸+01	5.8759𝐸+02	𝟐.𝟔𝟗𝟎𝟔𝐄+𝟎𝟏	3.0216𝐸+01
	Worst	4.5307𝐸+02	4.9489𝐸+01	2.9430𝐸+01	𝟐.𝟖𝟗𝟗𝟏𝐄+𝟎𝟏	1.0549𝐸+03	2.9389𝐸+01	8.3654𝐸+01
	Std.	6.0248𝐸+01	3.7560𝐸+00	8.1537𝐸−01	𝟔.𝟏𝟐𝟖𝟕𝐄−𝟎𝟐	1.5737𝐸+02	1.4196𝐸+00	1.3356𝐸+01
	Rank	5	6	3	4	7	2	𝟏
𝐹4(𝐷=30)	Best	3.0162𝐸−02	8.6779𝐸−03	1.1579𝐸−02	𝟐.𝟐𝟑𝟗𝟑𝐄−𝟎𝟓	2.7531𝐸−01	6.3723𝐸−01	8.6847𝐸−02
	Mean	8.0366𝐸−02	2.5195𝐸−02	4.4238𝐸−02	𝟕.𝟔𝟏𝟏𝟓𝐄−𝟎𝟒	7.3426𝐸−01	9.7518𝐸−01	1.6032𝐸−01
	Worst	1.7465𝐸−01	4.5602𝐸−02	9.5740𝐸−02	𝟐.𝟏𝟔𝟐𝟒𝐄−𝟎𝟑	1.1753𝐸+00	1.2684𝐸+00	2.4878𝐸−01
	Std.	3.3563𝐸−02	7.4389𝐸−03	2.0059𝐸−02	𝟓.𝟎𝟕𝟐𝟎𝐄−𝟎𝟒	2.1038𝐸−01	1.3336𝐸−01	3.4310𝐸−02
	Rank	4	2	3	𝟏	7	6	5
𝐹5(𝐷=30)	Best	4.3907𝐸+01	1.2231𝐸+02	7.2421𝐸+00	𝟎	8.8691𝐸+01	1.3828𝐸+02	1.4257𝐸+02
	Mean	8.8471𝐸+01	1.6497𝐸+02	2.1243𝐸+01	𝟎	1.3383𝐸+02	1.9791𝐸+02	2.0262𝐸+02
	Worst	1.4702𝐸+02	1.8931𝐸+02	4.0592𝐸+01	𝟎	1.7953𝐸+02	2.4035𝐸+02	2.8786𝐸+02
	Std.	2.3543𝐸+01	1.4949𝐸+01	6.7613𝐸+00	𝟎	1.5616𝐸+01	2.2151𝐸+01	2.8986𝐸+01
	Rank	6	3	2	𝟏	4	5	7
𝐹6(𝐷=30)	Best	3.4508𝐸+00	3.1954𝐸+00	8.0527𝐸−02	𝟖.𝟖𝟖𝟏𝟖𝐄−𝟏𝟔	1.3509𝐸+01	1.9172𝐸+01	1.8667𝐸+01
	Mean	5.8518𝐸+00	4.2090𝐸+00	3.1470𝐸+00	𝟖.𝟖𝟖𝟏𝟖𝐄−𝟏𝟔	1.4935𝐸+01	1.9908𝐸+01	1.9148𝐸+01
	Worst	8.4440𝐸+00	5.2716𝐸+00	6.8219𝐸+00	𝟖.𝟖𝟖𝟏𝟖𝐄−𝟏𝟔	1.6955𝐸+01	2.0247𝐸+01	1.9503𝐸+01
	Std.	1.2143𝐸+00	5.3016𝐸−01	1.1229𝐸+00	𝟎	8.6116𝐸−01	2.2907𝐸−01	1.9351𝐸−01
	Rank	7	4	6	𝟏	5	3	2
𝐹7(𝐷=30)	Best	3.5384𝐸+03	5.6171𝐸+01	7.0436𝐸+03	𝟎	1.3848𝐸+05	1.4414𝐸+04	1.8780𝐸+04
	Mean	2.1959𝐸+04	1.4336𝐸+02	3.2982𝐸+04	𝟒.𝟑𝟐𝟓𝟎𝐄−𝟏𝟖𝟎	4.6076𝐸+05	2.3871𝐸+04	1.7660𝐸+05
	Worst	1.7426𝐸+05	4.3311𝐸+02	1.7281𝐸+05	𝟐.𝟏𝟔𝟐𝟓𝐄−𝟏𝟕𝟖	1.3967𝐸+06	3.6638𝐸+04	7.5561𝐸+05
	Std.	2.7411𝐸+04	7.5588𝐸+01	2.7450𝐸+04	𝟎	2.4220𝐸+05	5.4373𝐸+03	1.6793𝐸+05
	Rank	4	2	5	𝟏	7	3	6
𝐹8(𝐷=30)	Best	1.0973𝐸+00	1.1225𝐸+00	2.0611𝐸−02	𝟎	5.0342𝐸+01	1.1446𝐸+00	3.7618𝐸+02
	Mean	2.0579𝐸+00	1.6510𝐸+00	1.2319𝐸−01	𝟎	7.3791𝐸+01	1.2110𝐸+00	5.3973𝐸+00
	Worst	3.9297𝐸+00	2.5709𝐸+00	5.1307𝐸−01	𝟎	1.2387𝐸+02	1.2616𝐸+00	6.61453177+02
	Std.	5.5487𝐸−01	3.5593𝐸−01	8.0788𝐸−02	𝟎	1.4831𝐸+01	2.4008𝐸−02	6.4097𝐸+01
	Rank	5	4	3	𝟏	6	2	7
𝐹9(𝐷=30)	Best	𝟎	1.9429𝐸−16	1.0908𝐸−09	𝟎	5.3806𝐸−05	2.8243𝐸−08	1.2522𝐸−09
	Mean	9.1221𝐸−41	1.7080𝐸−13	6.3405𝐸−08	𝟑.𝟕𝟐𝟑𝟏𝐄−𝟏𝟔𝟖	3.6484𝐸−02	1.4745𝐸−06	1.2981𝐸−06
	Worst	1.9689𝐸−39	1.1459𝐸−12	2.5349𝐸−07	𝟖.𝟗𝟓𝟗𝟎𝐄−𝟏𝟔𝟕	3.9778𝐸−01	1.3508𝐸−05	2.5807𝐸−05
	Std.	3.4654𝐸−40	2.5917𝐸−13	6.0990𝐸−08	𝟎	6.7720𝐸−02	2.4162𝐸−06	3.7830𝐸−06
	Rank	2	3	4	𝟏	7	5	6
𝐹10(𝐷=30)	Best	4.7818𝐸−01	2.7822𝐸−01	4.0704𝐸−02	𝟎	6.2320𝐸+02	1.6179𝐸−01	3.2702𝐸−03
	Mean	3.5156𝐸+00	9.6176𝐸−01	1.5103𝐸+00	𝟖.𝟔𝟓𝟏𝟎𝐄−𝟑𝟏𝟖	8.5918𝐸+02	2.7912𝐸−01	5.2075𝐸−03
	Worst	3.1501𝐸+01	3.1197𝐸+00	9.2295𝐸+00	𝟑.𝟑𝟕𝟑𝟗𝐄−𝟑𝟏𝟔	1.2293𝐸+03	3.8473𝐸−01	7.4336𝐸−03
	Std.	4.3846𝐸+00	5.1776𝐸−01	1.9643𝐸+00	𝟎	1.0523𝐸+03	4.0208𝐸−02	1.0260𝐸−03
	Rank	6	4	5	𝟏	7	3	2

Note: In this paper, both KH and KH II represent the KH with crossover operator.