计算机科学与探索 ›› 2022, Vol. 16 ›› Issue (2): 468-479.DOI: 10.3778/j.issn.1673-9418.2007047
收稿日期:
2020-07-17
修回日期:
2020-10-15
出版日期:
2022-02-01
发布日期:
2020-11-05
通讯作者:
+ E-mail: heyichao@hgu.edu.cn作者简介:
张发展(1995—),男,山东潍坊人,硕士研究生,主要研究方向为演化算法及其应用。基金资助:
ZHANG Fazhan, HE Yichao+(), LIU Xuejing, WANG Zekun
Received:
2020-07-17
Revised:
2020-10-15
Online:
2022-02-01
Published:
2020-11-05
About author:
ZHANG Fazhan, born in 1995, M.S. candidate. His research interests include evolutionary algorithm and its applications.Supported by:
摘要:
折扣{0-1}背包问题(D{0-1}KP)是0-1背包问题(0-1KP)的一种更复杂的扩展形式。为了利用离散差分演化高效求解D{0-1}KP,首先提出了一个新V型转换函数(NV),通过NV将个体的实向量映射为一个二进制向量,与已有的S型和V型转换函数相比,NV计算复杂度更低,求解效率更高。然后,基于新V型转换函数给出了一种新的离散差分演化算法(NDDE),并利用NDDE提出了求解D{0-1}KP的一个新的高效方法。最后,为了验证NDDE求解D{0-1}KP的性能,利用它求解四类大规模D{0-1}KP实例,并与基于群论的优化算法(GTOA)、基于环理论的演化算法(RTEA)、混合教学优化算法(HTLBO)和鲸鱼优化算法(WOA)等已有算法的最好计算结果进行比较,比较结果表明,NDDE不仅求解精度更高,而且算法的稳定性佳,非常适于求解大规模D{0-1}KP实例。
中图分类号:
张发展, 贺毅朝, 刘雪静, 王泽昆. 新颖的离散差分演化算法求解D{0-1}KP问题[J]. 计算机科学与探索, 2022, 16(2): 468-479.
ZHANG Fazhan, HE Yichao, LIU Xuejing, WANG Zekun. Novel Discrete Differential Evolution Algorithm for Solving D{0-1}KP Problem[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(2): 468-479.
S型转换函数 | V型转换函数 | ||
---|---|---|---|
名称 | 计算公式 | 名称 | 计算公式 |
S1 | | V1 | |
S2 | | V2 | |
S3 | | V3 | |
S4 | | V4 | |
表1 S型转换函数和V型转换函数
Table 1 S-shaped and V-shaped families of transfer functions
S型转换函数 | V型转换函数 | ||
---|---|---|---|
名称 | 计算公式 | 名称 | 计算公式 |
S1 | | V1 | |
S2 | | V2 | |
S3 | | V3 | |
S4 | | V4 | |
Instances | | |
---|---|---|
UDKP | 500 | 1.90E-24 |
WDKP | 500 | 3.10E-20 |
SDKP | 500 | 1.10E-27 |
IDKP | 500 | 1.96E-2 |
表2 秩和检验结果
Table 2 Results of Kruskal-Wallis test
Instances | | |
---|---|---|
UDKP | 500 | 1.90E-24 |
WDKP | 500 | 3.10E-20 |
SDKP | 500 | 1.10E-27 |
IDKP | 500 | 1.96E-2 |
Algorithm | | | | Other parameters | ||
---|---|---|---|---|---|---|
NDDE | 50 | | 100 | | ||
GTOA | 20 | | 100 | | ||
RTEA | 20 | | 100 | | ||
HTLBO2 | 50 | | 30 | | ||
DWOA | 40 | | 50 | | ||
FirEGA | 50 | | 30 | | ||
SecEGA | 50 | | 30 | |
表3 算法的参数设置
Table 3 Setting of algorithm parameters
Algorithm | | | | Other parameters | ||
---|---|---|---|---|---|---|
NDDE | 50 | | 100 | | ||
GTOA | 20 | | 100 | | ||
RTEA | 20 | | 100 | | ||
HTLBO2 | 50 | | 30 | | ||
DWOA | 40 | | 50 | | ||
FirEGA | 50 | | 30 | | ||
SecEGA | 50 | | 30 | |
Algorithm | Result | UDKP1 | UDKP2 | UDKP3 | UDKP4 | UDKP5 | UDKP6 | UDKP7 | UDKP8 | UDKP9 | UDKP10 |
---|---|---|---|---|---|---|---|---|---|---|---|
| 85 740 | 163 744 | 269 393 | 347 599 | 442 644 | 536 578 | 635 860 | 650 206 | 718 532 | 779 460 | |
NDDE | | 85 740* | 163 744* | 269 393* | 347 599* | 442 644* | 536 578* | 635 854 | 650 198 | 718 528 | 779 459 |
| 85 732.9 | 163 741.0 | 269 355.0 | 347 571.0 | 442 614.0 | 536 546.0 | 635 807.0 | 650 110.0 | 718 401.0 | 779 387.0 | |
| 85 687 | 163 703 | 269 231 | 347 426 | 442 506 | 536 387 | 635 598 | 649 558 | 717 773 | 779 059 | |
| 14.52 | 7.35 | 31.36 | 26.23 | 22.83 | 38.46 | 35.64 | 110.09 | 128.95 | 73.17 | |
GTOA | | 85 740* | 163 744* | 269 393* | 347 582 | 442 605 | 536 563 | 635 311 | 649 514 | 717 243 | 778 576 |
| 85 684.8 | 163 727.0 | 269 305.0 | 347 535.0 | 442 514.0 | 536 177.0 | 634 648.0 | 648 719.0 | 715 941.0 | 777 781.0 | |
| 85 489 | 163 640 | 269 148 | 347 462 | 442 332 | 535 573 | 633 941 | 647 259 | 714 110 | 776 820 | |
| 53.87 | 23.16 | 57.48 | 27.65 | 51.50 | 215.42 | 323.17 | 416.97 | 533.29 | 370.86 | |
RTEA | | 85 740* | 163 744* | 269 393* | 347 574 | 442 627 | 536 503 | 635 481 | 649 514 | 716 760 | 778 692 |
| 85 661.0 | 163 710.0 | 269 273.0 | 347 507.0 | 442 499.0 | 536 067.0 | 634 402.0 | 647 934.0 | 714 787.0 | 777 524.0 | |
| 85 477 | 163 592 | 269 089 | 347 336 | 442 303 | 534 370 | 631 836 | 645 934 | 711 259 | 775 519 | |
| 63.48 | 30.53 | 60.12 | 45.08 | 66.74 | 440.87 | 614.55 | 659.77 | 903.15 | 573.73 | |
HTLBO2 | | 85 721 | 163 730 | 269 292 | 347 508 | 442 217 | 535 929 | 635 267 | 649 705 | 717 584 | 777 990 |
| 85 686.0 | 163 647.2 | 269 094.8 | 347 288.2 | 441 792.5 | 535 434.1 | 634 671.4 | 649 020.9 | 717 045.5 | 777 004.1 | |
| 85 634 | 163 440 | 268 835 | 346 978 | 441 253 | 534 909 | 633 721 | 648 347 | 716 468 | 776 224 | |
| 12.90 | 64.80 | 110.80 | 129.20 | 255.40 | 258.00 | 297.70 | 339.10 | 296.50 | 431.20 | |
DWOA | | 84 983 | 160 946 | 266 407 | 343 345 | 435 048 | 528 335 | 628 467 | 639 426 | 708 207 | 763 000 |
| 84 862.6 | 160 625.8 | 266 238.4 | 343 167.9 | 434 859.9 | 528 168.0 | 628 234.2 | 639 075.9 | 708 035.6 | 762 830.2 | |
| 84 791 | 160 495 | 266 177 | 342 986 | 434 521 | 527 895 | 628 071 | 638 912 | 707 808 | 762 621 | |
| — | — | — | — | — | — | — | — | — | — | |
FirEGA | | 80 593 | 152 969 | 244 291 | 319 680 | 403 908 | 483 350 | 564 656 | 590 237 | 652 354 | 708 744 |
| 79 325.3 | 151 045.2 | 241 061.2 | 316 503.4 | 399 525.2 | 478 779.5 | 559 815.4 | 584 264.3 | 646 592.2 | 703 947.8 | |
| 78 499 | 149 732 | 239 114 | 313 141 | 396 937 | 474 558 | 555 763 | 580 258 | 642 965 | 700 702 | |
| — | — | — | — | — | — | — | — | — | — | |
SecEGA | | 77 970 | 148 042 | 230 485 | 306 358 | 375 619 | 447 231 | 531 192 | 560 932 | 619 444 | 689 248 |
| 76 808.3 | 146 310.5 | 225 232.4 | 301 700.8 | 371 688.8 | 442 556.4 | 523 809.2 | 555 100.7 | 615 990.4 | 684 872.4 | |
| 75 624 | 144 113 | 222 118 | 299 059 | 368 445 | 438 762 | 517 579 | 545 509 | 609 077 | 662 742 | |
| — | — | — | — | — | — | — | — | — | — |
表4 NDDE和其他算法求解UDKP类实例的计算结果
Table 4 Calculation results by NDDE and other algorithms for UDKP instances
Algorithm | Result | UDKP1 | UDKP2 | UDKP3 | UDKP4 | UDKP5 | UDKP6 | UDKP7 | UDKP8 | UDKP9 | UDKP10 |
---|---|---|---|---|---|---|---|---|---|---|---|
| 85 740 | 163 744 | 269 393 | 347 599 | 442 644 | 536 578 | 635 860 | 650 206 | 718 532 | 779 460 | |
NDDE | | 85 740* | 163 744* | 269 393* | 347 599* | 442 644* | 536 578* | 635 854 | 650 198 | 718 528 | 779 459 |
| 85 732.9 | 163 741.0 | 269 355.0 | 347 571.0 | 442 614.0 | 536 546.0 | 635 807.0 | 650 110.0 | 718 401.0 | 779 387.0 | |
| 85 687 | 163 703 | 269 231 | 347 426 | 442 506 | 536 387 | 635 598 | 649 558 | 717 773 | 779 059 | |
| 14.52 | 7.35 | 31.36 | 26.23 | 22.83 | 38.46 | 35.64 | 110.09 | 128.95 | 73.17 | |
GTOA | | 85 740* | 163 744* | 269 393* | 347 582 | 442 605 | 536 563 | 635 311 | 649 514 | 717 243 | 778 576 |
| 85 684.8 | 163 727.0 | 269 305.0 | 347 535.0 | 442 514.0 | 536 177.0 | 634 648.0 | 648 719.0 | 715 941.0 | 777 781.0 | |
| 85 489 | 163 640 | 269 148 | 347 462 | 442 332 | 535 573 | 633 941 | 647 259 | 714 110 | 776 820 | |
| 53.87 | 23.16 | 57.48 | 27.65 | 51.50 | 215.42 | 323.17 | 416.97 | 533.29 | 370.86 | |
RTEA | | 85 740* | 163 744* | 269 393* | 347 574 | 442 627 | 536 503 | 635 481 | 649 514 | 716 760 | 778 692 |
| 85 661.0 | 163 710.0 | 269 273.0 | 347 507.0 | 442 499.0 | 536 067.0 | 634 402.0 | 647 934.0 | 714 787.0 | 777 524.0 | |
| 85 477 | 163 592 | 269 089 | 347 336 | 442 303 | 534 370 | 631 836 | 645 934 | 711 259 | 775 519 | |
| 63.48 | 30.53 | 60.12 | 45.08 | 66.74 | 440.87 | 614.55 | 659.77 | 903.15 | 573.73 | |
HTLBO2 | | 85 721 | 163 730 | 269 292 | 347 508 | 442 217 | 535 929 | 635 267 | 649 705 | 717 584 | 777 990 |
| 85 686.0 | 163 647.2 | 269 094.8 | 347 288.2 | 441 792.5 | 535 434.1 | 634 671.4 | 649 020.9 | 717 045.5 | 777 004.1 | |
| 85 634 | 163 440 | 268 835 | 346 978 | 441 253 | 534 909 | 633 721 | 648 347 | 716 468 | 776 224 | |
| 12.90 | 64.80 | 110.80 | 129.20 | 255.40 | 258.00 | 297.70 | 339.10 | 296.50 | 431.20 | |
DWOA | | 84 983 | 160 946 | 266 407 | 343 345 | 435 048 | 528 335 | 628 467 | 639 426 | 708 207 | 763 000 |
| 84 862.6 | 160 625.8 | 266 238.4 | 343 167.9 | 434 859.9 | 528 168.0 | 628 234.2 | 639 075.9 | 708 035.6 | 762 830.2 | |
| 84 791 | 160 495 | 266 177 | 342 986 | 434 521 | 527 895 | 628 071 | 638 912 | 707 808 | 762 621 | |
| — | — | — | — | — | — | — | — | — | — | |
FirEGA | | 80 593 | 152 969 | 244 291 | 319 680 | 403 908 | 483 350 | 564 656 | 590 237 | 652 354 | 708 744 |
| 79 325.3 | 151 045.2 | 241 061.2 | 316 503.4 | 399 525.2 | 478 779.5 | 559 815.4 | 584 264.3 | 646 592.2 | 703 947.8 | |
| 78 499 | 149 732 | 239 114 | 313 141 | 396 937 | 474 558 | 555 763 | 580 258 | 642 965 | 700 702 | |
| — | — | — | — | — | — | — | — | — | — | |
SecEGA | | 77 970 | 148 042 | 230 485 | 306 358 | 375 619 | 447 231 | 531 192 | 560 932 | 619 444 | 689 248 |
| 76 808.3 | 146 310.5 | 225 232.4 | 301 700.8 | 371 688.8 | 442 556.4 | 523 809.2 | 555 100.7 | 615 990.4 | 684 872.4 | |
| 75 624 | 144 113 | 222 118 | 299 059 | 368 445 | 438 762 | 517 579 | 545 509 | 609 077 | 662 742 | |
| — | — | — | — | — | — | — | — | — | — |
Algorithm | Result | WDKP1 | WDKP2 | WDKP3 | WDKP4 | WDKP5 | WDKP6 | WDKP7 | WDKP8 | WDKP9 | WDKP10 |
---|---|---|---|---|---|---|---|---|---|---|---|
| 83 098 | 138 215 | 256 616 | 315 675 | 428 490 | 566 050 | 547 683 | 576 959 | 650 660 | 678 967 | |
NDDE | | 83 098* | 138 215* | 256 616* | 315 657* | 428 490* | 466 050* | 547 683* | 576 958 | 650 660* | 678 967* |
| 83 087.6 | 138 213.0 | 256 610.0 | 315 653.0 | 428 486.0 | 466 044.0 | 547 672.0 | 576 944.0 | 650 656.0 | 678 960.0 | |
| 83 083 | 138 213 | 256 589 | 315 638 | 428 446 | 466 022 | 547 633 | 576 894 | 650 640 | 678 910 | |
| 5.26 | 0.63 | 5.82 | 3.93 | 4.76 | 5.28 | 7.41 | 9.73 | 3.46 | 8.16 | |
GTOA | | 83 098* | 138 215* | 256 616* | 315 657* | 428 487 | 466 050* | 547 675 | 576 954 | 650 648 | 678 945 |
| 83 083.9 | 138 202.0 | 256 605.0 | 315 635.0 | 428 469.0 | 466 036.0 | 547 647.0 | 576 904.0 | 650 596.0 | 678 857.0 | |
| 83 058 | 138 157 | 256 575 | 315 596 | 428 431 | 466 008 | 547 572 | 576 820 | 650 450 | 678 551 | |
| 7.00 | 11.31 | 9.30 | 12.93 | 12.69 | 8.39 | 17.00 | 28.43 | 33.57 | 63.58 | |
RTEA | | 83 098* | 138 215* | 256 616* | 315 655 | 428 485 | 466 045 | 547 682 | 576 946 | 650 646 | 678 957 |
| 83 085.0 | 138 195.0 | 256 592.0 | 315 622.0 | 428 452.0 | 466 023.0 | 547 631.0 | 576 893.0 | 650 592.0 | 678 836.0 | |
| 83 036 | 138 130 | 256 523 | 315 566 | 428 377 | 465 983 | 547 553 | 576 813 | 650 391 | 678 500 | |
| 7.88 | 15.42 | 15.87 | 18.44 | 18.22 | 13.18 | 23.03 | 25.77 | 39.17 | 78.57 | |
Algorithm | Result | WDKP1 | WDKP2 | WDKP3 | WDKP4 | WDKP5 | WDKP6 | WDKP7 | WDKP8 | WDKP9 | WDKP10 |
HTLBO2 | | 83 090 | 138 213 | 256 612 | 315 606 | 428 385 | 466 012 | 547 623 | 576 847 | 650 538 | 678 848 |
| 83 085.2 | 138 182.7 | 256 566.4 | 315 556.2 | 428 321 | 465 962.1 | 547 542.3 | 576 708.0 | 650 465.9 | 678 738.1 | |
| 83 049 | 138 120 | 256 501 | 315 492 | 428 233 | 465 886 | 547 426 | 576 565 | 650 395 | 678 629 | |
| 28.00 | 22.30 | 28.00 | 30.70 | 39.90 | 35.00 | 49.50 | 70.80 | 37.30 | 58.20 | |
DWOA | | 82 887 | 137 859 | 255 891 | 314 994 | 427 645 | 464 806 | 546 477 | 575 494 | 648 699 | 677 428 |
| 82 831.9 | 137 787.3 | 255 828.3 | 314 929.4 | 427 607.1 | 464 759.9 | 546 378.6 | 575 428.7 | 648 674.8 | 677 372.5 | |
| 82 786 | 137 747 | 255 816 | 314 925 | 427 566 | 464 733 | 546 297 | 575 374 | 648 623 | 677 332 | |
| — | — | — | — | — | — | — | — | — | — | |
FirEGA | | 82 722 | 137 712 | 254 234 | 314 107 | 426 783 | 463 870 | 544 059 | 574 201 | 647 012 | 677 359 |
| 82 539.6 | 137 225.8 | 253 294.4 | 312 343.1 | 424 384.2 | 460 750.4 | 541 505.3 | 571 594.9 | 644 298.2 | 673 776 | |
| 82 454 | 136 983 | 252 909 | 310 665 | 421 584 | 455 201 | 535 551 | 565 119 | 639 241 | 660 332 | |
| — | — | — | — | — | — | — | — | — | — | |
SecEGA | | 82 664 | 137 712 | 254 315 | 313 227 | 426 783 | 463 435 | 544 776 | 574 862 | 648 402 | 677 118 |
| 81 118.3 | 135 951.3 | 247 826.2 | 308 628.9 | 421 415.1 | 461 779.8 | 537 821.2 | 567 507.7 | 643 444.6 | 672 912.3 | |
| 80 284 | 134 490 | 236 444 | 293 697 | 391 633 | 446 741 | 501 094 | 527 204 | 589 018 | 610 343 | |
| — | — | — | — | — | — | — | — | — | — |
表5 NDDE和其他算法求解WDKP类实例的计算结果
Table 5 Calculation results by NDDE and other algorithms for WDKP instances
Algorithm | Result | WDKP1 | WDKP2 | WDKP3 | WDKP4 | WDKP5 | WDKP6 | WDKP7 | WDKP8 | WDKP9 | WDKP10 |
---|---|---|---|---|---|---|---|---|---|---|---|
| 83 098 | 138 215 | 256 616 | 315 675 | 428 490 | 566 050 | 547 683 | 576 959 | 650 660 | 678 967 | |
NDDE | | 83 098* | 138 215* | 256 616* | 315 657* | 428 490* | 466 050* | 547 683* | 576 958 | 650 660* | 678 967* |
| 83 087.6 | 138 213.0 | 256 610.0 | 315 653.0 | 428 486.0 | 466 044.0 | 547 672.0 | 576 944.0 | 650 656.0 | 678 960.0 | |
| 83 083 | 138 213 | 256 589 | 315 638 | 428 446 | 466 022 | 547 633 | 576 894 | 650 640 | 678 910 | |
| 5.26 | 0.63 | 5.82 | 3.93 | 4.76 | 5.28 | 7.41 | 9.73 | 3.46 | 8.16 | |
GTOA | | 83 098* | 138 215* | 256 616* | 315 657* | 428 487 | 466 050* | 547 675 | 576 954 | 650 648 | 678 945 |
| 83 083.9 | 138 202.0 | 256 605.0 | 315 635.0 | 428 469.0 | 466 036.0 | 547 647.0 | 576 904.0 | 650 596.0 | 678 857.0 | |
| 83 058 | 138 157 | 256 575 | 315 596 | 428 431 | 466 008 | 547 572 | 576 820 | 650 450 | 678 551 | |
| 7.00 | 11.31 | 9.30 | 12.93 | 12.69 | 8.39 | 17.00 | 28.43 | 33.57 | 63.58 | |
RTEA | | 83 098* | 138 215* | 256 616* | 315 655 | 428 485 | 466 045 | 547 682 | 576 946 | 650 646 | 678 957 |
| 83 085.0 | 138 195.0 | 256 592.0 | 315 622.0 | 428 452.0 | 466 023.0 | 547 631.0 | 576 893.0 | 650 592.0 | 678 836.0 | |
| 83 036 | 138 130 | 256 523 | 315 566 | 428 377 | 465 983 | 547 553 | 576 813 | 650 391 | 678 500 | |
| 7.88 | 15.42 | 15.87 | 18.44 | 18.22 | 13.18 | 23.03 | 25.77 | 39.17 | 78.57 | |
Algorithm | Result | WDKP1 | WDKP2 | WDKP3 | WDKP4 | WDKP5 | WDKP6 | WDKP7 | WDKP8 | WDKP9 | WDKP10 |
HTLBO2 | | 83 090 | 138 213 | 256 612 | 315 606 | 428 385 | 466 012 | 547 623 | 576 847 | 650 538 | 678 848 |
| 83 085.2 | 138 182.7 | 256 566.4 | 315 556.2 | 428 321 | 465 962.1 | 547 542.3 | 576 708.0 | 650 465.9 | 678 738.1 | |
| 83 049 | 138 120 | 256 501 | 315 492 | 428 233 | 465 886 | 547 426 | 576 565 | 650 395 | 678 629 | |
| 28.00 | 22.30 | 28.00 | 30.70 | 39.90 | 35.00 | 49.50 | 70.80 | 37.30 | 58.20 | |
DWOA | | 82 887 | 137 859 | 255 891 | 314 994 | 427 645 | 464 806 | 546 477 | 575 494 | 648 699 | 677 428 |
| 82 831.9 | 137 787.3 | 255 828.3 | 314 929.4 | 427 607.1 | 464 759.9 | 546 378.6 | 575 428.7 | 648 674.8 | 677 372.5 | |
| 82 786 | 137 747 | 255 816 | 314 925 | 427 566 | 464 733 | 546 297 | 575 374 | 648 623 | 677 332 | |
| — | — | — | — | — | — | — | — | — | — | |
FirEGA | | 82 722 | 137 712 | 254 234 | 314 107 | 426 783 | 463 870 | 544 059 | 574 201 | 647 012 | 677 359 |
| 82 539.6 | 137 225.8 | 253 294.4 | 312 343.1 | 424 384.2 | 460 750.4 | 541 505.3 | 571 594.9 | 644 298.2 | 673 776 | |
| 82 454 | 136 983 | 252 909 | 310 665 | 421 584 | 455 201 | 535 551 | 565 119 | 639 241 | 660 332 | |
| — | — | — | — | — | — | — | — | — | — | |
SecEGA | | 82 664 | 137 712 | 254 315 | 313 227 | 426 783 | 463 435 | 544 776 | 574 862 | 648 402 | 677 118 |
| 81 118.3 | 135 951.3 | 247 826.2 | 308 628.9 | 421 415.1 | 461 779.8 | 537 821.2 | 567 507.7 | 643 444.6 | 672 912.3 | |
| 80 284 | 134 490 | 236 444 | 293 697 | 391 633 | 446 741 | 501 094 | 527 204 | 589 018 | 610 343 | |
| — | — | — | — | — | — | — | — | — | — |
Algorithm | Result | SDKP1 | SDKP2 | SDKP3 | SDKP4 | SDKP5 | SDKP6 | SDKP7 | SDKP8 | SDKP9 | SDKP10 |
---|---|---|---|---|---|---|---|---|---|---|---|
| 94 459 | 160 805 | 238 248 | 340 027 | 463 033 | 466 097 | 620 446 | 670 697 | 739 121 | 765 317 | |
NDDE | | 94 447 | 160 805* | 238 248* | 340 027* | 463 012 | 466 079 | 620 374 | 670 649 | 739 048 | 765 245 |
| 94 407.6 | 160 787.0 | 238 205.0 | 339 963.0 | 462 912.0 | 466 024.0 | 620 290.0 | 670 548.0 | 738 971.0 | 765 183.0 | |
| 94 338 | 160 716 | 238 137 | 339 863 | 462 814 | 465 934 | 620 177 | 670 452 | 738 858 | 765 019 | |
| 26.15 | 33.48 | 18.58 | 26.32 | 36.34 | 28.76 | 40.77 | 35.04 | 35.04 | 40.28 | |
GTOA | | 94 452 | 160 805* | 238 243 | 340 025 | 462 980 | 466 074 | 620 219 | 670 304 | 738 458 | 764 586 |
| 94 425.5 | 160 769.0 | 238 204.0 | 339 995.0 | 462 868.0 | 466 010.0 | 619 978.0 | 670 071.0 | 738 107.0 | 764 159.0 | |
| 94 329 | 160 701 | 238 144 | 339 933 | 462 717 | 465 881 | 619 623 | 669 791 | 737 667 | 763 609 | |
| 22.81 | 19.99 | 21.14 | 17.60 | 48.45 | 33.87 | 109.38 | 124.22 | 179.95 | 197.93 | |
RTEA | | 94 449 | 160 786 | 238 215 | 339 988 | 462 929 | 466 036 | 620 238 | 670 431 | 738 707 | 764 821 |
| 94 333.0 | 160 702.0 | 238 133.0 | 339 918.0 | 462 766.0 | 465 903.0 | 620 048.0 | 670 151.0 | 738 162.0 | 764 263.0 | |
| 94 132 | 160 588 | 237 992 | 339 801 | 462 593 | 465 712 | 619 741 | 669 702 | 737 660 | 763 665 | |
| 67.88 | 45.52 | 43.12 | 38.25 | 74.40 | 61.21 | 108.31 | 151.77 | 219.06 | 236.47 | |
HTLBO2 | | 94 435 | 160 702 | 238 165 | 339 728 | 462 524 | 465 415 | 619 849 | 669 609 | 738 003 | 764 051 |
| 94 397.9 | 160 590.3 | 238 041.6 | 339 456.4 | 462 335.3 | 465 077.8 | 619 686.9 | 669 350.1 | 737 678.6 | 763 834.2 | |
| 94 361 | 160 440 | 237 929 | 339 088 | 462 093 | 464 682 | 619 402 | 669 031 | 737 415 | 763 660 | |
| 25.10 | 52.80 | 53.50 | 136.10 | 124.60 | 154.80 | 110.50 | 157.40 | 134.10 | 101.10 | |
DWOA | | 94 258 | 159 902 | 236 431 | 336 851 | 460 095 | 460 966 | 615 866 | 664 653 | 731 654 | 756 144 |
| 94 218.9 | 159 728.0 | 236 227.1 | 336 717.6 | 459 937.2 | 460 776.2 | 615 692.9 | 664 519.5 | 731 522.3 | 755 976.5 | |
| 94 159 | 159 636 | 236 068 | 336 601 | 459 804 | 460 660 | 615 510 | 664 338 | 731 418 | 755 878 | |
| — | — | — | — | — | — | — | — | — | — | |
FirEGA | | 93 316 | 159 116 | 235 372 | 336 369 | 451 184 | 459 236 | 607 200 | 661 104 | 728 443 | 755 189 |
| 93 192.8 | 158 936.7 | 235 204.4 | 335 844.7 | 447 335.9 | 458 746.1 | 602 797.7 | 659 844.6 | 727 364.5 | 752 931.0 | |
| 93 064 | 158 798 | 235 015 | 335 524 | 44 252 | 458 427 | 600 496 | 659 120 | 726 872 | 749 879 | |
| — | — | — | — | — | — | — | — | — | — | |
SecEGA | | 93 009 | 159 107 | 235 474 | 336 597 | 444 748 | 458 509 | 598 648 | 662 465 | 730 036 | 756 662 |
| 91 684.0 | 156 557.3 | 231 287.4 | 330 437.4 | 435 933.3 | 453 973.9 | 592 672.4 | 653 459.8 | 726 324.1 | 750 716.4 | |
| 90 256 | 154 241 | 224 872 | 318 638 | 415 923 | 430 286 | 571 469 | 610 664 | 671 623 | 697 520 | |
| — | — | — | — | — | — | — | — | — | — |
表6 NDDE和其他算法求解SDKP类实例的计算结果
Table 6 Calculation results by NDDE and other algorithms for SDKP instances
Algorithm | Result | SDKP1 | SDKP2 | SDKP3 | SDKP4 | SDKP5 | SDKP6 | SDKP7 | SDKP8 | SDKP9 | SDKP10 |
---|---|---|---|---|---|---|---|---|---|---|---|
| 94 459 | 160 805 | 238 248 | 340 027 | 463 033 | 466 097 | 620 446 | 670 697 | 739 121 | 765 317 | |
NDDE | | 94 447 | 160 805* | 238 248* | 340 027* | 463 012 | 466 079 | 620 374 | 670 649 | 739 048 | 765 245 |
| 94 407.6 | 160 787.0 | 238 205.0 | 339 963.0 | 462 912.0 | 466 024.0 | 620 290.0 | 670 548.0 | 738 971.0 | 765 183.0 | |
| 94 338 | 160 716 | 238 137 | 339 863 | 462 814 | 465 934 | 620 177 | 670 452 | 738 858 | 765 019 | |
| 26.15 | 33.48 | 18.58 | 26.32 | 36.34 | 28.76 | 40.77 | 35.04 | 35.04 | 40.28 | |
GTOA | | 94 452 | 160 805* | 238 243 | 340 025 | 462 980 | 466 074 | 620 219 | 670 304 | 738 458 | 764 586 |
| 94 425.5 | 160 769.0 | 238 204.0 | 339 995.0 | 462 868.0 | 466 010.0 | 619 978.0 | 670 071.0 | 738 107.0 | 764 159.0 | |
| 94 329 | 160 701 | 238 144 | 339 933 | 462 717 | 465 881 | 619 623 | 669 791 | 737 667 | 763 609 | |
| 22.81 | 19.99 | 21.14 | 17.60 | 48.45 | 33.87 | 109.38 | 124.22 | 179.95 | 197.93 | |
RTEA | | 94 449 | 160 786 | 238 215 | 339 988 | 462 929 | 466 036 | 620 238 | 670 431 | 738 707 | 764 821 |
| 94 333.0 | 160 702.0 | 238 133.0 | 339 918.0 | 462 766.0 | 465 903.0 | 620 048.0 | 670 151.0 | 738 162.0 | 764 263.0 | |
| 94 132 | 160 588 | 237 992 | 339 801 | 462 593 | 465 712 | 619 741 | 669 702 | 737 660 | 763 665 | |
| 67.88 | 45.52 | 43.12 | 38.25 | 74.40 | 61.21 | 108.31 | 151.77 | 219.06 | 236.47 | |
HTLBO2 | | 94 435 | 160 702 | 238 165 | 339 728 | 462 524 | 465 415 | 619 849 | 669 609 | 738 003 | 764 051 |
| 94 397.9 | 160 590.3 | 238 041.6 | 339 456.4 | 462 335.3 | 465 077.8 | 619 686.9 | 669 350.1 | 737 678.6 | 763 834.2 | |
| 94 361 | 160 440 | 237 929 | 339 088 | 462 093 | 464 682 | 619 402 | 669 031 | 737 415 | 763 660 | |
| 25.10 | 52.80 | 53.50 | 136.10 | 124.60 | 154.80 | 110.50 | 157.40 | 134.10 | 101.10 | |
DWOA | | 94 258 | 159 902 | 236 431 | 336 851 | 460 095 | 460 966 | 615 866 | 664 653 | 731 654 | 756 144 |
| 94 218.9 | 159 728.0 | 236 227.1 | 336 717.6 | 459 937.2 | 460 776.2 | 615 692.9 | 664 519.5 | 731 522.3 | 755 976.5 | |
| 94 159 | 159 636 | 236 068 | 336 601 | 459 804 | 460 660 | 615 510 | 664 338 | 731 418 | 755 878 | |
| — | — | — | — | — | — | — | — | — | — | |
FirEGA | | 93 316 | 159 116 | 235 372 | 336 369 | 451 184 | 459 236 | 607 200 | 661 104 | 728 443 | 755 189 |
| 93 192.8 | 158 936.7 | 235 204.4 | 335 844.7 | 447 335.9 | 458 746.1 | 602 797.7 | 659 844.6 | 727 364.5 | 752 931.0 | |
| 93 064 | 158 798 | 235 015 | 335 524 | 44 252 | 458 427 | 600 496 | 659 120 | 726 872 | 749 879 | |
| — | — | — | — | — | — | — | — | — | — | |
SecEGA | | 93 009 | 159 107 | 235 474 | 336 597 | 444 748 | 458 509 | 598 648 | 662 465 | 730 036 | 756 662 |
| 91 684.0 | 156 557.3 | 231 287.4 | 330 437.4 | 435 933.3 | 453 973.9 | 592 672.4 | 653 459.8 | 726 324.1 | 750 716.4 | |
| 90 256 | 154 241 | 224 872 | 318 638 | 415 923 | 430 286 | 571 469 | 610 664 | 671 623 | 697 520 | |
| — | — | — | — | — | — | — | — | — | — |
Algorithm | Result | IDKP1 | IDKP2 | IDKP3 | IDKP4 | IDKP5 | IDKP6 | IDKP7 | IDKP8 | IDKP9 | IDKP10 |
---|---|---|---|---|---|---|---|---|---|---|---|
| 70 106 | 118 268 | 234 804 | 282 591 | 335 584 | 452 463 | 489 149 | 533 841 | 528 144 | 581 244 | |
NDDE | | 70 106* | 118 268* | 234 804* | 282 591* | 335 584* | 452 463* | 489 149* | 533 841* | 528 144* | 581 244* |
| 70 106.0 | 118 268.0 | 234 803.0 | 282 590.0 | 335 583.0 | 452 462.0 | 489 144.0 | 533 835.0 | 528 140.0 | 581 238.0 | |
| 70 106 | 118 268 | 234 802 | 282 585 | 335 580 | 452 441 | 489 137 | 533 824 | 528 136 | 581 232 | |
| 0 | 0 | 0.52 | 0.60 | 0.40 | 6.21 | 4.29 | 2.79 | 0.87 | 1.33 | |
GTOA | | 70 106* | 118 268* | 234 804* | 282 583 | 335 580 | 452 463* | 489 144 | 533 834 | 528 144* | 581 234 |
| 70 089.3 | 118 239.0 | 234 792.0 | 282 560.0 | 335 549.0 | 452 438.0 | 489 109.0 | 533 807.0 | 528 103.0 | 581 200.0 | |
| 70 048 | 118 168 | 234 731 | 282 497 | 335 449 | 452 372 | 489 019 | 533 762 | 527 998 | 581 086 | |
| 15.18 | 23.66 | 12.56 | 16.26 | 27.90 | 18.51 | 23.97 | 14.42 | 28.67 | 24.90 | |
RTEA | | 70 106* | 118 268* | 234 804* | 282 583 | 335 580 | 452 450 | 489 142 | 533 833 | 528 144* | 581 238 |
| 70 082.0 | 118 235.0 | 234 770.0 | 282 557.0 | 335 528.0 | 452 405.0 | 489 102.0 | 533 800.0 | 528 099.0 | 581 197.0 | |
| 70 037 | 118 146 | 234 703 | 282 485 | 335 314 | 452 311 | 488 948 | 533 724 | 528 015 | 581 091 | |
| 21.55 | 26.72 | 23.34 | 18.05 | 39.20 | 32.80 | 26.99 | 20.75 | 27.95 | 33.39 | |
HTLBO2 | | 70 106* | 118 268* | 234 804* | 282 591* | 335 584* | 452 462 | 489 149* | 533 836 | 528 140 | 581 238 |
| 70 094.3 | 118 266.8 | 234 800.9 | 282 584.2 | 335 582.4 | 452 453.2 | 489 141 | 533 830.1 | 528 136.2 | 581 236.0 | |
| 70 074 | 118 232 | 234 795 | 282 565 | 335 580 | 452 447 | 489 121 | 533 816 | 528 124 | 581 229 | |
| 8.30 | 11.40 | 2.60 | 5.60 | 1.90 | 4.40 | 5.60 | 4.50 | 3.20 | 2.20 | |
DWOA | | 70 106* | 118 268* | 234 739 | 282 591 | 335 584 | 452 418 | 489 137 | 533 817 | 528 140 | 581 238 |
| 70 101.2 | 118 247.0 | 234 718.9 | 282 568.4 | 335 580.3 | 452 387.1 | 489 102.7 | 533 816.0 | 528 118.6 | 581 226.7 | |
| 70 090 | 118 232 | 234 663 | 282 565 | 335 580 | 452 359 | 489 063 | 533 816 | 528 109 | 581 225 | |
| — | — | — | — | — | — | — | — | — | — | |
FirEGA | | 70 106* | 118 034 | 234 508 | 281 804 | 335 068 | 451 498 | 487 675 | 531 872 | 525 460 | 578 897 |
| 70 078.0 | 117 544.3 | 233 896.3 | 280 536.6 | 332 180.2 | 449 781.0 | 484 305.8 | 529 372.8 | 522 243.5 | 575 128.5 | |
| 70 022 | 117 249 | 233 447 | 278 179 | 328 661 | 446 456 | 475 476 | 524 404 | 501 428 | 551 772 | |
| — | — | — | — | — | — | — | — | — | — | |
SecEGA | | 70 101 | 118 232 | 234 698 | 282 484 | 335 580 | 452 016 | 48 840 | 533 686 | 527 942 | 580 910 |
| 70 010.4 | 116 921.7 | 228 166.1 | 273 731.2 | 331 886.1 | 447 358.4 | 483 569.7 | 531 844.8 | 523 597.8 | 572 178.7 | |
| 69 947 | 115 384 | 218 621 | 259 485 | 309 964 | 412 539 | 445 999 | 508 560 | 476 760 | 520 255 | |
| — | — | — | — | — | — | — | — | — | — |
表7 NDDE和其他算法求解IDKP类实例的计算结果
Table 7 Calculation results by NDDE and other algorithms for IDKP instances
Algorithm | Result | IDKP1 | IDKP2 | IDKP3 | IDKP4 | IDKP5 | IDKP6 | IDKP7 | IDKP8 | IDKP9 | IDKP10 |
---|---|---|---|---|---|---|---|---|---|---|---|
| 70 106 | 118 268 | 234 804 | 282 591 | 335 584 | 452 463 | 489 149 | 533 841 | 528 144 | 581 244 | |
NDDE | | 70 106* | 118 268* | 234 804* | 282 591* | 335 584* | 452 463* | 489 149* | 533 841* | 528 144* | 581 244* |
| 70 106.0 | 118 268.0 | 234 803.0 | 282 590.0 | 335 583.0 | 452 462.0 | 489 144.0 | 533 835.0 | 528 140.0 | 581 238.0 | |
| 70 106 | 118 268 | 234 802 | 282 585 | 335 580 | 452 441 | 489 137 | 533 824 | 528 136 | 581 232 | |
| 0 | 0 | 0.52 | 0.60 | 0.40 | 6.21 | 4.29 | 2.79 | 0.87 | 1.33 | |
GTOA | | 70 106* | 118 268* | 234 804* | 282 583 | 335 580 | 452 463* | 489 144 | 533 834 | 528 144* | 581 234 |
| 70 089.3 | 118 239.0 | 234 792.0 | 282 560.0 | 335 549.0 | 452 438.0 | 489 109.0 | 533 807.0 | 528 103.0 | 581 200.0 | |
| 70 048 | 118 168 | 234 731 | 282 497 | 335 449 | 452 372 | 489 019 | 533 762 | 527 998 | 581 086 | |
| 15.18 | 23.66 | 12.56 | 16.26 | 27.90 | 18.51 | 23.97 | 14.42 | 28.67 | 24.90 | |
RTEA | | 70 106* | 118 268* | 234 804* | 282 583 | 335 580 | 452 450 | 489 142 | 533 833 | 528 144* | 581 238 |
| 70 082.0 | 118 235.0 | 234 770.0 | 282 557.0 | 335 528.0 | 452 405.0 | 489 102.0 | 533 800.0 | 528 099.0 | 581 197.0 | |
| 70 037 | 118 146 | 234 703 | 282 485 | 335 314 | 452 311 | 488 948 | 533 724 | 528 015 | 581 091 | |
| 21.55 | 26.72 | 23.34 | 18.05 | 39.20 | 32.80 | 26.99 | 20.75 | 27.95 | 33.39 | |
HTLBO2 | | 70 106* | 118 268* | 234 804* | 282 591* | 335 584* | 452 462 | 489 149* | 533 836 | 528 140 | 581 238 |
| 70 094.3 | 118 266.8 | 234 800.9 | 282 584.2 | 335 582.4 | 452 453.2 | 489 141 | 533 830.1 | 528 136.2 | 581 236.0 | |
| 70 074 | 118 232 | 234 795 | 282 565 | 335 580 | 452 447 | 489 121 | 533 816 | 528 124 | 581 229 | |
| 8.30 | 11.40 | 2.60 | 5.60 | 1.90 | 4.40 | 5.60 | 4.50 | 3.20 | 2.20 | |
DWOA | | 70 106* | 118 268* | 234 739 | 282 591 | 335 584 | 452 418 | 489 137 | 533 817 | 528 140 | 581 238 |
| 70 101.2 | 118 247.0 | 234 718.9 | 282 568.4 | 335 580.3 | 452 387.1 | 489 102.7 | 533 816.0 | 528 118.6 | 581 226.7 | |
| 70 090 | 118 232 | 234 663 | 282 565 | 335 580 | 452 359 | 489 063 | 533 816 | 528 109 | 581 225 | |
| — | — | — | — | — | — | — | — | — | — | |
FirEGA | | 70 106* | 118 034 | 234 508 | 281 804 | 335 068 | 451 498 | 487 675 | 531 872 | 525 460 | 578 897 |
| 70 078.0 | 117 544.3 | 233 896.3 | 280 536.6 | 332 180.2 | 449 781.0 | 484 305.8 | 529 372.8 | 522 243.5 | 575 128.5 | |
| 70 022 | 117 249 | 233 447 | 278 179 | 328 661 | 446 456 | 475 476 | 524 404 | 501 428 | 551 772 | |
| — | — | — | — | — | — | — | — | — | — | |
SecEGA | | 70 101 | 118 232 | 234 698 | 282 484 | 335 580 | 452 016 | 48 840 | 533 686 | 527 942 | 580 910 |
| 70 010.4 | 116 921.7 | 228 166.1 | 273 731.2 | 331 886.1 | 447 358.4 | 483 569.7 | 531 844.8 | 523 597.8 | 572 178.7 | |
| 69 947 | 115 384 | 218 621 | 259 485 | 309 964 | 412 539 | 445 999 | 508 560 | 476 760 | 520 255 | |
| — | — | — | — | — | — | — | — | — | — |
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