Different Transfer Functions for Binary Particle Swarm Optimization with a New Encoding Scheme for Discounted {0-1} Knapsack Problem

Table 3

Comparison of 8 BPSO algorithms on SDKP1–SDKP10.

		BPSO1	BPSO2	BPSO3	BPSO4	BPSO5	BPSO6	BPSO7	BPSO8
SDKP1	Best	91046	90103	90067	89558	94300	94421	94414	94363
	AVE	90484.4	89446.6	88655.5	88449.9	94089.3	94112.7	94174.4	94181.6
	Worst	89964	88772	87835	87847	93687	93807	93920	93897
	Std. dev	240.44	312.35	513.73	422.86	152.07	128.84	103.15	113.74
	Gap	4.21	5.31	6.14	6.36	0.39	0.37	0.30	0.29
SDKP2	Best	152080	149910	148440	148320	160470	160490	160430	160570
	AVE	150627.7	148857.0	147307.0	146598.3	160058.0	160033.3	160127.0	160136.0
	Worst	149590	148000	146410	145830	159730	159380	159480	159780
	Std. dev	591.17	505.77	535.13	595.93	190.92	237.35	214.82	182.33
	Gap	6.33	7.43	8.39	8.83	0.46	0.48	0.42	0.42
SDKP3	Best	225340	222940	220310	218570	237580	237350	237430	237370
	AVE	223456.7	221130.0	218620.0	217727.3	236629.7	236821.0	236829.3	236778.3
	Worst	222120	219740	217530	217020	235770	236390	236110	235960
	Std. dev	617.52	777.22	567.24	435.05	359.95	265.44	352.00	314.48
	Gap	6.21	7.18	8.24	8.61	0.68	0.60	0.60	0.62
SDKP4	Best	317610	313630	310850	310020	338360	338480	338520	338240
	AVE	315521.0	311837.3	308411.3	306979.7	337381.0	337525.0	337545.0	337433.0
	Worst	314020	310180	306940	305490	336270	336540	336440	336210
	Std. dev	938.08	789.99	1010.63	1074.09	501.50	524.94	540.16	495.98
	Gap	7.21	8.29	9.30	9.72	0.78	0.74	0.73	0.76
SDKP5	Best	438720	431560	427340	425190	459900	459890	460190	459970
	AVE	435066.7	429523.7	424714.0	422803.3	458421.7	458479.3	458505.0	458292.0
	Worst	432570	427170	422560	421120	456570	456980	457430	457130
	Std. dev	1412.53	1057.97	1127.73	833.64	724.35	669.22	548.81	802.76
	Gap	6.04	7.24	8.28	8.69	1.00	0.98	0.98	1.02
SDKP6	Best	428070	424110	417840	415960	463040	462310	463070	462650
	AVE	424916.3	420600.0	415690.3	413662.3	461229.3	461288.0	461370.0	461310.7
	Worst	423230	418080	414020	412230	459870	459390	459610	459880
	Std. dev	1121.73	1370.31	1020.89	994.05	759.15	708.60	789.20	680.64
	Gap	8.84	9.76	10.81	11.25	1.04	1.03	1.01	1.03
SDKP7	Best	580940	574630	567460	562990	615640	615810	615100	614830
	AVE	578574.3	571778.7	564752.3	561783.7	613474.3	613542.7	613242.0	613182.7
	Worst	576180	569410	563080	560270	611790	611970	611180	610700
	Std. dev	1210.37	1118.88	1034.42	708.71	955.25	1000.74	953.90	984.83
	Gap	6.75	7.84	8.98	9.45	1.12	1.11	1.16	1.17
SDKP8	Best	619660	613680	605780	602220	665540	665340	665070	665230
	AVE	616572.0	609950.0	603096.3	600201.3	663894.7	663703.3	663633.7	663681.7
	Worst	614660	607940	601290	598950	662710	661970	661310	661500
	Std. dev	1288.63	1507.53	1244.25	876.08	772.79	865.64	806.70	925.22
	Gap	8.07	9.06	10.08	10.51	1.01	1.04	1.05	1.05
SDKP9	Best	679360	672620	664590	661320	733260	732790	732800	732910
	AVE	676278.0	668958.7	661978.0	659159.0	731433.3	731057.0	731107.0	730950.3
	Worst	671400	667010	659530	657170	730050	729290	728590	728990
	Std. dev	1620.02	1566.35	1313.95	1201.62	645.22	869.96	892.30	1024.07
	Gap	8.50	9.49	10.44	10.82	1.04	1.09	1.08	1.11
SDKP10	Best	691200	681980	677280	672420	758000	757900	758090	758110
	AVE	687130.3	679407.7	672908.3	669316.7	756100.0	756291.0	756452.3	756337.3
	Worst	684090	676750	669860	667010	754400	754250	754150	753730
	Std. dev	1737.15	1527.54	1694.79	1452.26	837.73	993.51	1023.71	1038.86
	Gap	10.22	11.23	12.07	12.54	1.20	1.18	1.16	1.17