Research Article
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 |
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