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

Table 2

Comparison of 8 BPSO algorithms on IDKP1–IDKP10.

		BPSO1	BPSO2	BPSO3	BPSO4	BPSO5	BPSO6	BPSO7	BPSO8
IDKP1	Best	67982	65735	66130	65215	69940	69944	69905	69950
	AVE	66360.4	65219.1	64523.1	64206.4	69650.5	69689.3	69652.8	69761.0
	Worst	65724	64686	63911	63583	69345	69069	69348	69521
	Std. dev	450.89	285.19	468.09	367.69	127.49	184.26	151.25	109.15
	Gap	5.34	6.97	7.96	8.42	0.65	0.59	0.65	0.49
IDKP2	Best	111220	109980	107920	106690	117840	117700	117960	118000
	AVE	109885.7	108308.3	106567.0	105800.3	117157.7	117265.3	117237.7	117418.0
	Worst	108670	107420	105580	104920	116530	116690	116880	117040
	Std. dev	526.83	583.77	527.62	375.53	317.69	220.23	250.72	228.83
	Gap	7.09	8.42	9.89	10.54	0.94	0.85	0.87	0.72
IDKP3	Best	223100	220370	217660	217270	234550	234600	234510	234540
	AVE	221673.7	219018.7	216489.0	215686.7	234234.0	234193.3	234192.3	234253.7
	Worst	220330	218260	215340	214360	233870	233740	233820	233800
	Std. dev	655.53	522.10	621.85	588.95	170.00	194.82	196.90	192.52
	Gap	5.59	6.72	7.80	8.14	0.24	0.26	0.26	0.23
IDKP4	Best	262250	261600	256030	254560	281630	281460	281860	281560
	AVE	260578.7	257426.7	254213.0	253443.3	280915.7	280937.0	280926.7	280953.3
	Worst	259350	255980	253070	251990	280230	280210	280310	280220
	Std. dev	680.61	1036.33	669.28	545.32	370.76	298.83	381.13	328.28
	Gap	7.79	8.90	10.04	10.31	0.59	0.59	0.59	0.58
IDKP5	Best	306710	304080	300290	299740	334080	334460	334370	334330
	AVE	305651.3	302176.0	298676.3	297292.7	333196.0	333540.0	333450.7	333380.7
	Worst	304530	300500	297810	296300	331670	332140	331700	332480
	Std. dev	627.16	837.40	572.50	822.63	558.47	545.39	595.10	479.64
	Gap	8.92	9.96	11.00	11.41	0.71	0.61	0.64	0.66
IDKP6	Best	424860	420840	414340	414270	451530	451120	451180	451390
	AVE	422738.0	418002.7	412954.3	410614.0	450461.3	450450.0	450227.7	450455.0
	Worst	420920	416490	411800	409440	449550	449230	449230	448980
	Std. dev	1094.93	1134.30	671.58	922.85	476.18	491.61	468.12	563.12
	Gap	6.57	7.62	8.73	9.25	0.44	0.44	0.49	0.44
IDKP7	Best	445470	440660	437100	434590	485500	485610	485970	485920
	AVE	443578.0	438952.7	434153.7	432156.0	484351.7	484123.3	484322.3	484205.7
	Worst	441760	437450	432760	430820	482930	482580	482450	482400
	Std. dev	973.40	1031.28	1060.95	965.40	741.76	739.59	938.96	879.02
	Gap	9.32	10.26	11.24	11.65	0.98	1.03	0.99	1.01
IDKP8	Best	476350	469590	465180	462400	528810	528720	528910	528450
	AVE	473369.7	467065.0	462243.3	460488.7	526964.3	526535.3	526150.3	526356.7
	Worst	470860	465310	459400	459070	524250	524640	523860	524150
	Std. dev	1361.11	1059.54	1294.50	853.73	1377.91	1070.18	1437.35	1268.72
	Gap	11.33	12.51	13.41	13.74	1.29	1.37	1.44	1.40
IDKP9	Best	467480	460890	456610	452870	521210	521390	520960	522140
	AVE	463983.7	458418.0	453862.3	451666.0	516689.3	517894.3	518507.0	518834.7
	Worst	460970	456540	451430	450200	513400	514390	514080	515480
	Std. dev	1575.60	1013.39	1205.04	700.42	1976.73	1723.29	1353.10	1726.89
	Gap	12.15	13.20	14.06	14.48	2.17	1.94	1.82	1.76
IDKP10	Best	512140	505660	503250	500520	571780	572400	572250	572260
	AVE	509844.0	503661.7	498110.3	495816.3	568518.7	569094.3	568045.0	569364.7
	Worst	506800	501750	495910	493320	564150	564620	563900	566140
	Std. dev	1072.29	996.78	1548.61	1635.38	1711.13	1681.26	1724.57	1446.41
	Gap	12.28	13.35	14.30	14.70	2.19	2.09	2.27	2.04