Theoretical and Empirical Analyses of an Improved Harmony Search Algorithm Based on Differential Mutation Operator

Table 2

The statistical results for 50 runs tested on sixteen benchmark functions.

Function name	Algorithm	Best	Mean	Worst	Std
𝐹1; Ackley function	CHS	2.8576𝐸−02	1.5581𝐸−01	3.9193𝐸−01	7.6256𝐸−02
	HSDE	8.8818𝐸−16	2.5301𝐸−03	2.8793𝐸−02	6.2847𝐸−03
	IHSDE	9.0683𝐸−13	6.7733𝐸−11	1.0303𝐸−09	1.5937𝐸−10
	NGHS	4.4409𝐸−15	6.3594𝐸−15	1.5099𝐸−14	2.6087𝐸−15
𝐹2; Dixon and price function	CHS	8.8618𝐸−06	1.2597𝐸−01	6.1434𝐸−01	1.5982𝐸−01
	HSDE	2.5471𝐸−08	2.3591𝐸−01	6.6820𝐸−01	2.7753𝐸−01
	IHSDE	1.6000𝐸−13	1.8755𝐸−01	6.6667𝐸−01	3.0089𝐸−01
	NGHS	5.0758𝐸−18	1.8658𝐸−01	6.6667𝐸−01	3.0222𝐸−01
𝐹3; Levy function	CHS	6.8353𝐸−07	3.2798𝐸−04	2.2443𝐸−03	4.5606𝐸−04
	HSDE	1.4998𝐸−32	1.0481𝐸−06	2.3110𝐸−05	3.8915𝐸−06
	IHSDE	2.2192𝐸−27	8.1646𝐸−22	1.2126𝐸−20	2.2009𝐸−21
	NGHS	1.4998𝐸−32	1.4998𝐸−32	1.4998𝐸−32	1.3823𝐸−47
𝐹4; Michalewics function	CHS	−4.6877𝐸+00	−4.6859𝐸+00	−4.6745𝐸+00	2.4486𝐸−03
	HSDE	−4.6877𝐸+00	−4.6873𝐸+00	−4.6778𝐸+00	1.4840𝐸−03
	IHSDE	−4.6877𝐸+00	−4.6877𝐸+00	−4.6876𝐸+00	3.1708𝐸−06
	NGHS	−4.6877𝐸+00	−4.6615𝐸+00	−4.4959𝐸+00	4.4690𝐸−02
𝐹5; Perm function	CHS	3.8305𝐸−01	3.0003𝐸+02	2.7696𝐸+03	5.6432𝐸+02
	HSDE	1.1911𝐸−02	1.3767𝐸+02	4.9112𝐸+03	6.9678𝐸+02
	IHSDE	7.7908𝐸−02	1.1573𝐸+02	1.1205𝐸+03	1.8330𝐸+02
	NGHS	2.5956𝐸−02	3.0299𝐸+01	3.0618𝐸+02	6.7526𝐸+01
𝐹6; Powell function	CHS	4.9848𝐸−06	1.5997𝐸−02	1.1282𝐸−01	1.8387𝐸−02
	HSDE	2.3029𝐸−07	5.6853𝐸−03	3.3194𝐸−02	7.9504𝐸−03
	IHSDE	2.7878𝐸−07	9.6453𝐸−06	5.0805𝐸−05	1.2526𝐸−05
	NGHS	3.4739𝐸−08	1.3536𝐸−06	3.0482𝐸−06	6.9536𝐸−07
𝐹7; Rastrigin function	CHS	1.2141𝐸−03	3.8574𝐸−02	3.1431𝐸−01	5.0927𝐸−02
	HSDE	0.0000𝐸+00	4.0904𝐸−03	8.3710𝐸−02	1.3816𝐸−02
	IHSDE	4.2633𝐸−14	2.5944𝐸+00	8.1800𝐸+00	2.6713𝐸+00
	NGHS	0.0000𝐸+00	9.9496𝐸−02	9.9496𝐸−01	3.0152𝐸−01
𝐹8; Rosenbrock function	CHS	4.3234𝐸−03	1.7703𝐸+00	7.2601𝐸+00	1.9667𝐸+00
	HSDE	2.2513𝐸−04	1.9541𝐸+00	3.3526𝐸+00	1.0636𝐸+00
	IHSDE	9.5471𝐸−03	1.1625𝐸+00	2.1110𝐸+00	3.9876𝐸−01
	NGHS	1.0784𝐸−03	9.3133𝐸−01	3.5246𝐸+00	6.5434𝐸−01
𝐹9; Sphere function	CHS	3.9500𝐸−06	2.2513𝐸−04	1.3357𝐸−03	2.6271𝐸−04
	HSDE	1.2298𝐸−48	1.1425𝐸−06	1.7339𝐸−05	3.5860𝐸−06
	IHSDE	4.9171𝐸−28	2.5756𝐸−23	5.8627𝐸−22	9.2448𝐸−23
	NGHS	8.0887𝐸−67	1.2697𝐸−54	6.1078𝐸−53	8.6366𝐸−54
𝐹10; Trid Function	CHS	−3.0000𝐸+01	−2.9875𝐸+01	−2.9156𝐸+01	2.3411𝐸−01
	HSDE	−3.0000𝐸+01	−2.9962𝐸+01	−2.9826𝐸+01	4.7768𝐸−02
	IHSDE	−3.0000𝐸+01	−3.0000𝐸+01	−3.0000𝐸+01	1.4565𝐸−10
	NGHS	−3.0000𝐸+01	−3.0000𝐸+01	−3.0000𝐸+01	9.1416𝐸−08
𝐹11; Zakharov Function	CHS	2.3830𝐸−05	1.9843𝐸−02	1.4567𝐸−01	3.0974𝐸−02
	HSDE	2.2372𝐸−22	4.1294𝐸−03	1.2299𝐸−01	1.8304𝐸−02
	IHSDE	2.1141𝐸−15	6.1236𝐸−10	2.9812𝐸−08	4.2138𝐸−09
	NGHS	2.5239𝐸−29	1.0790𝐸−18	3.7173𝐸−17	5.3803𝐸−18
𝐹12; Beale function	CHS	4.3647𝐸−10	5.9406𝐸−03	9.3633𝐸−02	1.7594𝐸−02
	HSDE	1.4369𝐸−12	3.6930𝐸−03	3.9410𝐸−02	6.6767𝐸−03
	IHSDE	0.0000𝐸+00	9.5965𝐸−05	3.5321𝐸−03	5.0902𝐸−04
	NGHS	0.0000𝐸+00	4.3172𝐸−01	9.3929𝐸+00	1.3416𝐸+00
𝐹13; Easom function	CHS	−9.9999𝐸−01	−6.3682𝐸−01	−7.6526𝐸−05	4.7875𝐸−01
	HSDE	−1.0000𝐸+00	−9.5984𝐸−01	−7.9071𝐸−05	1.9790𝐸−01
	IHSDE	−1.0000𝐸+00	−1.0000𝐸+00	−1.0000𝐸+00	0.0000𝐸+00
	NGHS	−1.0000𝐸+00	−7.8001𝐸−01	0.0000𝐸+00	4.1842𝐸−01
𝐹14; Goldstein and Price function	CHS	3.0000𝐸+00	3.0016𝐸+00	3.0277𝐸+00	4.1190𝐸−03
	HSDE	3.0000𝐸+00	3.0003𝐸+00	3.0063𝐸+00	1.1281𝐸−03
	IHSDE	3.0000𝐸+00	3.0000𝐸+00	3.0000𝐸+00	1.6324𝐸−07
	NGHS	3.0000𝐸+00	1.5420𝐸+01	8.4000𝐸+01	2.5122𝐸+01
𝐹15; Hump function	CHS	4.6668𝐸−08	2.8159𝐸−05	3.5591𝐸−04	5.9106𝐸−05
	HSDE	4.6510𝐸−08	1.7936𝐸−06	4.2616𝐸−05	8.1164𝐸−06
	IHSDE	4.6510𝐸−08	3.3050𝐸−07	1.3125𝐸−05	1.8532𝐸−06
	NGHS	4.6510𝐸−08	4.6510𝐸−08	4.6510𝐸−08	1.0318𝐸−16
𝐹16; My function	CHS	1.3297𝐸−08	4.7659𝐸−01	3.4872𝐸+00	7.9292𝐸−01
	HSDE	0.0000𝐸+00	1.5107𝐸−01	1.5166𝐸+00	4.5650𝐸−01
	IHSDE	0.0000𝐸+00	1.4701𝐸−07	7.1040𝐸−06	1.0045𝐸−06
	NGHS	0.0000𝐸+00	2.1788𝐸+00	7.3673𝐸+00	2.5506𝐸+00