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

Table 4

Comparison of 8 BPSO algorithms on WDKP1–WDKP10.

		BPSO1	BPSO2	BPSO3	BPSO4	BPSO5	BPSO6	BPSO7	BPSO8
WDKP1	Best	80619	79846	79486	78556	83086	83083	83083	83090
	AVE	80013.7	79208.4	78425.3	78067.9	82977.3	83001.4	82995.0	83014.1
	Worst	79143	78770	77952	77505	82783	82876	82866	82649
	Std. dev	344.73	308.46	352.70	246.35	81.16	59.47	61.93	82.14
	Gap	3.71	4.68	5.62	6.05	0.15	0.12	0.12	0.10
WDKP2	Best	129590	127850	126590	126010	138020	138040	138060	138110
	AVE	128602.7	126549.3	124836.0	124143.0	137769.7	137740.0	137814.3	137830.0
	Worst	127680	125580	123800	123260	137390	137420	137430	137270
	Std. dev	534.56	561.11	643.57	620.55	154.53	128.04	171.26	161.59
	Gap	6.95	8.44	9.68	10.18	0.32	0.34	0.29	0.28
WDKP3	Best	243760	240740	239080	237110	256120	256320	256330	256360
	AVE	242457.7	239491.0	236892.0	235980.0	255743.0	255603.0	255831.3	255778.7
	Worst	241370	238570	235560	235110	254750	254910	254930	254850
	Std. dev	661.12	581.07	749.34	481.79	330.32	377.36	351.57	325.67
	Gap	5.52	6.67	7.69	8.04	0.34	0.39	0.31	0.33
WDKP4	Best	296760	293210	289800	288090	315040	315120	315020	315040
	AVE	294886.7	291169.0	287984.7	286589.7	314392.0	314444.7	314413.3	314340.7
	Worst	293860	289950	286820	285510	313470	313900	313160	313360
	Std. dev	660.30	638.33	728.30	541.86	360.52	318.05	423.15	471.89
	Gap	6.58	7.76	8.77	9.21	0.40	0.38	0.39	0.42
WDKP5	Best	402060	397880	393220	392120	427390	427280	427420	427390
	AVE	400608.7	395759.3	391217.7	389422.0	426337.3	426314.7	426490.7	426349.7
	Worst	398530	394270	389620	387990	425140	424980	425170	424950
	Std. dev	909.19	915.60	995.13	928.49	570.29	569.34	507.68	640.64
	Gap	6.51	7.64	8.70	9.12	0.50	0.51	0.47	0.50
WDKP6	Best	432220	427900	421630	419290	464810	464440	464430	464040
	AVE	429596.3	424036.7	419191.3	417385.0	463214.7	463253.3	463201.7	463228.3
	Worst	428160	421740	417910	416010	461760	461920	462000	461980
	Std. dev	1087.14	1200.48	892.42	762.16	666.55	627.89	555.16	487.82
	Gap	7.82	9.01	10.05	10.44	0.61	0.60	0.61	0.61
WDKP7	Best	506860	503130	495310	494200	545820	545570	545570	545270
	AVE	505569.3	499531.0	493412.7	491195.3	544327.0	544235.3	544057.0	544006.7
	Worst	503840	496860	491740	489660	543130	542560	542170	542140
	Std. dev	786.27	1319.85	847.95	1281.14	620.46	884.96	783.31	779.24
	Gap	7.69	8.79	9.91	10.31	0.61	0.63	0.66	0.67
WDKP8	Best	521960	515930	509890	508840	574630	574580	573780	574290
	AVE	519367.7	513180.0	507473.7	505650.3	572645.7	572659.0	572513.7	572652.7
	Worst	515830	510670	505790	503570	571260	570240	570270	571520
	Std. dev	1441.06	1205.03	1069.10	1391.39	797.36	771.39	705.13	693.20
	Gap	9.98	11.05	12.04	12.36	0.75	0.75	0.77	0.75
WDKP9	Best	584160	577510	571210	568630	646730	647010	646690	646910
	AVE	580843.3	574534.3	568415.0	566222.3	645331.7	645445.0	645101.0	645133.0
	Worst	576840	572340	566130	564130	643900	643790	642660	642980
	Std. dev	1659.97	1273.19	1136.32	1116.77	811.74	901.38	900.69	923.83
	Gap	10.73	11.70	12.64	12.98	0.82	0.80	0.85	0.85
WDKP10	Best	592210	585730	580200	578010	675100	675790	675310	675460
	AVE	589632.0	582542.0	576737.3	574541.0	672859.7	673530.0	673296.3	673135.3
	Worst	586780	580160	574670	572440	669350	671700	671090	670990
	Std. dev	1281.47	1245.80	1511.33	1320.04	1249.33	1078.37	959.27	1171.04
	Gap	13.15	14.20	15.05	15.38	0.90	0.80	0.83	0.86