An Improved Sparrow Search Algorithm and Its Application in HIFU Sound Field

Table 4

Minimum, average, and standard deviation of eight algorithms on benchmark functions.

Func	Algorithm	Fmin	Avg	Std
F1	GA	0.1726	0.3688	0.1058
	PSO	0.1152	0.3618	0.2198
	GWO	7.4055E − 43	4.2803E − 41	8.5127E − 41
	WOA	2.4924E − 105	9.1573E − 97	2.7034E − 96
	HHO	5.0860E − 120	3.6083E − 106	1.3799E − 105
	SSA	0	0	0
	ISSA	0	0	0
	SNS	4.5158E − 80	3.3939E − 78	5.7930E − 78
F2	GA	1.3102	2.2009	0.3276
	PSO	1.3232	3.1762	1.3121
	GWO	7.9563E − 25	5.6060E − 24	4.0470E − 24
	WOA	1.5696E − 64	2.1231E − 57	7.3635E − 57
	HHO	2.2888E − 64	5.0350E − 56	1.4616E − 55
	SSA	0	5.9657E − 246	0
	ISSA	0	0	0
	SNS	1.1772E − 41	1.2192E − 40	8.8551E − 41
F3	GA	35.0637	113.5643	98.2013
	PSO	9.3528	31.9008	17.2877
	GWO	1.0519E − 14	1.7352E − 11	6.7481E − 11
	WOA	3.0453E + 03	1.3447E + 04	6.1948E + 03
	HHO	1.9320E − 112	1.9745E − 95	4.6188E − 95
	SSA	0	1.6209E − 303	0
	ISSA	0	0	0
	SNS	4.2048E − 30	2.8538E − 23	7.9304E − 23
F4	GA	0.8578	4.0991	2.0845
	PSO	1.0475	5.0234	2.7751
	GWO	1.5166E − 11	1.4806E − 10	1.1322E − 10
	WOA	6.8169E − 12	20.9694	23.8119
	HHO	8.1970E − 59	1.4348E − 54	4.9994E − 54
	SSA	0	9.1203E − 237	0
	ISSA	0	0	0
	SNS	3.1845E − 36	1.5309E − 35	1.2401E − 35
F5	GA	132.8354	328.3109	172.5915
	PSO	46.0601	244.4676	309.2162
	GWO	24.9588	26.1575	0.7470
	WOA	26.2030	26.8192	0.2945
	HHO	3.2743E − 05	0.0012	0.0016
	SSA	9.8449E − 09	1.2272E − 05	2.8983E − 05
	ISSA	2.5419E − 09	7.6049E − 09	1.3492E − 05
	SNS	25.2683	26.1652	0.3588
F6	GA	0.2511	0.4112	0.1158
	PSO	0.0927	0.3191	0.1521
	GWO	1.5281E − 05	6.3023E − 05	0.1987
	WOA	0.0023	0.0043	0.0017
	HHO	5.8671E − 08	7.3739E − 06	1.1985E − 05
	SSA	6.6045E − 11	7.0626E − 07	1.0864E − 07
	ISSA	3.9263E − 13	4.8787E − 11	7.7172E − 10
	SNS	4.0101E − 07	1.4531E − 05	2.8176E − 05
F7	GA	0.4491	0.8592	0.3148
	PSO	0.1019	0.2901	0.1288
	GWO	1.6605E − 04	1.0398E − 03	2.6688E − 04
	WOA	6.5593E − 05	5.2027E − 04	8.3366E − 03
	HHO	2.7958E − 06	3.7936E − 06	3.9055E − 05
	SSA	8.5655E − 06	1.0812E − 04	8.7271E − 05
	ISSA	9.4541E − 07	6.7345E − 06	5.4023E − 05
	SNS	2.8520E − 05	4.8744E − 05	1.0430E − 04
F8	GA	−1.8296E + 03	−1.4652E + 03	170.2547
	PSO	−8.6206E + 03	−8.1473E + 03	764.9701
	GWO	−8.5730E + 03	−6.7798E + 03	543.8278
	WOA	−1.2569E + 04	−1.2524E + 04	1.2405E + 03
	HHO	−1.2621E + 04	−1.2569E + 04	627.8604
	SSA	−9.6550E + 03	−9.2471E + 03	661.9266
	ISSA	−1.2569E + 04	−1.1637E + 04	1.9303E + 03
	SNS	−9.1063E + 03	−8.5178E + 03	274.8565
F9	GA	22.7386	43.8443	18.1360
	PSO	35.6856	63.8049	15.2725
	GWO	0	157.723	2.8713
	WOA	0	0	0
	HHO	0	0	0
	SSA	0	0	0
	ISSA	0	0	0
	SNS	0	0	0
F10	GA	0.5610	1.9370	1.2596
	PSO	1.7736	4.5744	3.1795
	GWO	2.2204E − 14	2.8362E − 14	3.3553E − 15
	WOA	8.8818E − 16	8.8818E − 16	2.4567E − 15
	HHO	8.8818E − 16	8.8818E − 16	0
	SSA	8.8818E − 16	8.8818E − 16	0
	ISSA	8.8818E − 16	8.8818E − 16	0
	SNS	8.8818E − 16	8.8818E − 16	0
F11	GA	0.0110	0.0803	0.0507
	PSO	0.4939	1.3628	0.8000
	GWO	0	0	0
	WOA	0	0	0
	HHO	0	0	0
	SSA	0	0	0
	ISSA	0	0	0
	SNS	0	0	0
F12	GA	0.0017	0.0083	0.0157
	PSO	5.4913	13.3800	4.7394
	GWO	1.3595E − 06	0.0167	0.0133
	WOA	1.6101E − 04	0.0013	0.0024
	HHO	3.5147E − 08	1.1030E − 06	1.3409E − 06
	SSA	3.6602E − 13	4.4157E-11	1.1270E − 08
	ISSA	5.4643E − 17	1.9518E − 13	1.2207E − 11
	SNS	1.8399E − 08	5.5631E − 08	8.9952E − 08
F13	GA	0.0367	0.0882	0.0265
	PSO	0.0637	0.4392	2.0113
	GWO	1.4168E − 05	0.1006	0.1266
	WOA	0.0055	0.0068	0.0287
	HHO	2.5137E − 08	4.0881E − 06	1.2922E − 05
	SSA	4.5725E − 11	2.7827E − 09	1.1367E − 07
	ISSA	1.7253E − 14	1.3480E − 12	3.0559E − 10
	SNS	4.7620E − 07	0.0110	0.0401
F14	GA	0.9980	0.9980	1.1822E − 05
	PSO	0.9980	2.9821	1.2826
	GWO	0.9980	1.1206	1.9074
	WOA	0.9980	1.1641	0.3622
	HHO	0.9980	0.9983	3.4100E − 11
	SSA	0.9980	14.7720	16.9265
	ISSA	0.9980	1.1084	1.9682
	SNS	0.9980	0.9980	0
F15	GA	9.2281E − 04	1.1875E − 03	4.9309E − 04
	PSO	3.0749E − 04	1.2237E − 03	3.8690E − 04
	GWO	3.0749E − 04	5.0112E − 04	6.8127E − 04
	WOA	3.0772E − 04	7.1811E − 04	4.3624E − 04
	HHO	3.0749E − 04	5.3561E − 04	2.3027E − 04
	SSA	3.0749E − 04	3.0801E − 04	8.1578E − 07
	ISSA	3.0749E − 04	3.0749E − 04	1.6916E − 08
	SNS	3.0749E − 04	3.0921E − 04	1.0241E − 05
F16	GA	−1.0308	−1.0093	0.0269
	PSO	−1.0316	−1.0314	6.7752E − 15
	GWO	−1.0316	−1.0315	1.4321E − 05
	WOA	−1.0316	−1.0315	1.5771E − 11
	HHO	−1.0316	−1.0316	3.5948E − 14
	SSA	−1.0316	−1.0315	6.6486E − 15
	ISSA	−1.0316	−1.0316	6.4539E − 16
	SNS	−1.0316	−1.0315	6.7752E − 15
F17	GA	0.3979	0.4099	0.0154
	PSO	0.3979	0.3979	0
	GWO	0.3979	0.3979	3.4787E − 07
	WOA	0.3979	0.3979	9.4224E − 08
	HHO	0.3979	0.3979	5.5899E − 10
	SSA	0.3979	0.3979	0
	ISSA	0.3979	0.3979	0
	SNS	0.3979	0.3979	0
F18	GA	3.0191	4.2707	20.0975
	PSO	3.0000	3.0000	1.9110E − 15
	GWO	3.0000	3.0000	4.5137E − 10
	WOA	3.0000	3.0000	7.6160E − 06
	HHO	3.0000	3.0000	1.0883E − 10
	SSA	3.0000	3.0000	2.0748E − 14
	ISSA	3.0000	3.0000	1.1778E − 15
	SNS	3.0000	3.0000	2.2296E − 14
F19	GA	−3.8125	−1.7296	1.4953
	PSO	−3.8244	−3.8244	1.3550E − 15
	GWO	−3.8244	−3.8244	1.2697E − 06
	WOA	−3.8244	−3.8244	3.3715E − 04
	HHO	−3.8244	−3.8243	1.8163E − 04
	SSA	−3.8244	−3.8244	1.3550E − 15
	ISSA	−3.8244	−3.8244	1.3550E − 15
	SNS	−3.8244	−3.8244	1.3550E − 15
F20	GA	−3.2285	−2.3482	0.9614
	PSO	−3.3220	−3.2625	0.0605
	GWO	−3.3220	−3.2590	0.0711
	WOA	−3.3220	−3.2532	0.0798
	HHO	−3.3179	−3.2014	0.0784
	SSA	−3.3220	−3.2507	0.0592
	ISSA	−3.3220	−3.2744	0.0592
	SNS	−3.3220	−3.2520	0.0624
F21	GA	−9.8954	−4.4387	4.6735
	PSO	−10.1532	−2.6828	3.4455
	GWO	−10.1531	−5.0415	1.7986
	WOA	−10.1532	−9.9830	0.9307
	HHO	−10.1500	−5.0550	1.2921
	SSA	−10.1532	−10.1531	1.5634E − 04
	ISSA	−10.1532	−9.5262	1.4992
	SNS	−10.1532	−10.1531	7.1207E − 05
F22	GA	−10.2468	−3.3236	6.0573
	PSO	−10.4029	−10.3015	1.5411
	GWO	−10.4029	−10.0491	1.3432
	WOA	−10.4029	−10.0132	1.3732
	HHO	−10.3999	−5.7947	1.8339
	SSA	−10.4029	−10.4028	4.0757E − 03
	ISSA	−10.4029	−10.4027	2.0212E − 05
	SNS	−10.4029	−10.3021	1.1427E − 03
F23	GA	−10.4686	−2.0108	3.1604
	PSO	−10.5364	−10.5181	0.6198
	GWO	−10.5363	−5.1064	1.7620
	WOA	−10.5364	−9.1405	2.6164
	HHO	−10.5358	−5.1278	1.8662
	SSA	−10.5364	−10.5363	1.6540E − 04
	ISSA	−10.5364	−10.3659	2.5198E − 04
	SNS	−10.5364	−10.3445	1.8067E − 03