Table 2. Comparison results with other excellent algorithms on rPSP index.

PSP is the inverse of PSP. Therefore, the smaller metric is the better the performance. The algorithm runs on 13 test functions and independently runs 30 times, which gets the average and standard deviation. The data in the table are the average of the 30 times. The experiment tested the data obtained by all algorithms for significant differences using the Wilson test. The “+” and “-” in the table indicate statistically superior or inferior to the comparison algorithm, respectively, while “=” indicates statistically similar results. In Table 1, Mean represents the average of 30 results. Std represents the standard deviation on the test problem.

		MO_Ring _PSO_SCD	DN-NSGAII	Omni-optimizer	TriMOEA-TA&R	MMODE_ICD	MMODE_ES
MMF1	Mean	0.0488+	0.0969+	0.0975+	0.0735+	0.0493+	0.0417
	± Std	0.0019	0.0145	0.013	0.0108	0.0030	0.0013
MMF2	Mean	0.0465+	0.1742+	0.1643+	0.0931+	0.0247+	0.0075
	± Std	0.0147	0.1368	0.1305	0.0574	0.0054	0.0010
MMF3	Mean	0.0335+	0.1172+	0.1351+	0.0871+	0.0209+	0.0068
	± Std	0.0099	0.0759	0.0994	0.0275	0.0037	0.0007
MMF4	Mean	0.0273+	0.0771+	0.0840+	0.1537+	0.0257+	0.0217
	± Std	0.0020	0.0137	0.0238	0.2262	0.0031	0.0010
MMF5	Mean	0.0869+	0.1773+	0.1789+	0.1132+	0.0853+	0.0711
	± Std	0.0060	0.0217	0.0245	0.0126	0.0039	0.0035
MMF6	Mean	0.0733+	0.1427+	0.1523+	0.0958+	0.0713+	0.0637
	± Std	0.0043	0.0150	0.0185	0.0123	0.0043	0.0004
MMF7	Mean	0.0267+	0.0553+	0.0511+	0.0672+	0.0263+	0.0206
	± Std	0.0015	0.0151	0.0127	0.0520	0.0046	0.0023
MMF8	Mean	0.0678+	0.2799+	0.3149+	0.3974+	0.1303+	0.0504
	± Std	0.0042	0.0911	0.1326	0.1572	0.0352	0.0048
MMF9	Mean	0.0082+	0.0219+	0.0316+	0.0031-	0.0047-	0.0061
	± Std	0.0008	0.0078	0.0269	0.0001	0.0003	0.0004
MMF10	Mean	0.1708-	1.3420+	3.1260+	0.2014=	0.2011=	0.2035
	± Std	0.0231	2.3280	3.3420	0.0001	0.0010	0.0024
SYM-PART simple	Mean	0.1776+	5.4511+	6.4608+	0.0210-	0.0427-	0.0441
	± Std	0.0226	2.576	3.0139	0.0021	0.0055	0.0042
SYM-PART rotated	Mean	0.2784-	4.4542+	5.4757+	2.0975+	0.0892+	0.0536
	± Std	0.2500	1.4104	3.4185	1.4432	0.0153	0.0033
Omni-test	Mean	0.4279+	1.5563+	1.7939+	0.7547+	0.0512+	0.0430
	± Std	0.0954	0.2878	0.6207	0.2166	0.0036	0.0020
+/=/-		11/0/2	13/0/0	13/0/0	10/1/2	10/1/2

Notes.

rPSP results and rHV results obtained by these comparison algorithms are underlined and shown in bold when the rPSP results and rHV results obtained by MMODE_ES are better than the comparison algorithms under the same test function.