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

Table 5

Comparison of 8 BPSO algorithms on UDKP1–UDKP10.

	Algorithm	BPSO1	BPSO2	BPSO3	BPSO4	BPSO5	BPSO6	BPSO7	BPSO8
UDKP1	Best	79070	77244	75113	74482	85728	85740	85728	85740
	AVE	77771.0	75977.2	74212.6	73635.3	85526.1	85555.0	85532.2	85590.0
	Worst	76454	74811	72928	72824	85193	85282	85026	85347
	Std. dev	618.95	667.48	534.68	487.81	135.12	130.18	184.05	102.30
	Gap	9.29	11.39	13.44	14.12	0.25	0.22	0.24	0.17
UDKP2	Best	145300	143840	138350	136930	163560	163500	163630	163560
	AVE	143420.3	139566.3	136063.0	134346.3	162678.3	162878.7	163002.0	163061.7
	Worst	141530	137850	134630	132520	161520	161760	162360	162260
	Std. dev	1032.82	1243.82	741.11	1018.84	466.99	399.78	304.90	314.45
	Gap	12.41	14.77	16.91	17.95	0.65	0.53	0.45	0.42
UDKP3	Best	236780	231160	227280	224910	268270	267810	268450	268430
	AVE	233404.0	228742.0	224682.7	222441.0	266421.0	266729.3	266814.3	267008.7
	Worst	231530	226860	222570	220600	264740	265150	264980	265560
	Std. dev	1247.53	1053.30	1101.78	1077.08	864.86	715.80	805.32	681.78
	Gap	13.36	15.09	16.60	17.43	1.10	0.99	0.96	0.89
UDKP4	Best	301870	295140	290750	288990	345340	345110	345980	346120
	AVE	299048.3	292933.7	287865.0	285816.7	343429.7	343510.3	343958.3	343789.0
	Worst	296850	290590	285580	284460	341160	341330	341970	341800
	Std. dev	1338.88	1104.72	1178.90	1126.14	999.47	848.45	1036.98	924.93
	Gap	13.97	15.73	17.18	17.77	1.20	1.18	1.05	1.10
UDKP5	Best	377980	370780	363930	362700	436440	437610	437610	437870
	AVE	374958.3	367368.0	361415.3	358738.3	434067.0	434640.7	435638.0	435053.0
	Worst	372200	363700	358340	354560	430540	431110	432930	431500
	Std. dev	1687.23	1465.29	1430.94	1788.70	1415.51	1410.72	1362.97	1572.94
	Gap	15.29	17.01	18.35	18.96	1.94	1.81	1.58	1.71
UDKP6	Best	460830	452640	441270	439260	528730	529110	528940	530600
	AVE	455015.3	446125.3	438852.3	435647.0	525352.3	525774.0	525666.0	527004.7
	Worst	451680	443210	435620	432550	521500	522460	519710	523880
	Std. dev	2176.80	1998.82	1368.33	1766.47	1682.21	1691.98	2340.59	1877.81
	Gap	15.20	16.86	18.21	18.81	2.09	2.01	2.03	1.78
UDKP7	Best	545040	537910	525410	521590	624860	624300	626480	627350
	AVE	542066.3	531413.7	522129.3	518659.0	620906.0	620467.7	622543.3	621875.3
	Worst	539210	527820	518860	515290	616570	615210	618950	617290
	Std. dev	1425.63	2665.99	1579.61	1717.16	1898.05	2135.62	2122.03	2120.14
	Gap	14.75	16.43	17.89	18.43	2.35	2.42	2.09	2.20
UDKP8	Best	551650	540430	532270	533240	638060	638320	641500	637900
	AVE	546316.3	536630.3	528103.7	524527.7	633762.3	635553.7	635667.3	634388.3
	Worst	543810	532280	524740	521420	630280	630450	630150	629600
	Std. dev	1938.41	1996.29	1807.58	2466.31	2124.68	1784.87	2357.17	2121.45
	Gap	15.98	17.47	18.78	19.33	2.53	2.25	2.24	2.43
UDKP9	Best	594880	582140	570220	565930	702810	705110	704700	704350
	AVE	588958.7	577306.7	566914.3	561907.7	699517.7	699452.3	700655.0	699156.3
	Worst	584970	574490	561620	558820	692160	693090	694810	693320
	Std. dev	2264.91	1811.95	1962.56	1931.90	2335.85	2628.18	2570.03	3132.54
	Gap	18.03	19.65	21.10	21.80	2.65	2.66	2.49	2.70
UDKP10	Best	633620	619750	604900	598780	763810	760920	762640	767550
	AVE	627361.3	612962.3	598703.3	593618.7	756328.3	757261.7	757706.3	756768.3
	Worst	620570	608500	594060	589560	751310	752420	749190	749330
	Std. dev	3489.47	2565.51	2702.42	2415.98	3078.53	2348.74	3148.56	3865.48
	Gap	19.51	21.36	23.19	23.84	2.97	2.85	2.79	2.91