Research Article
A Heuristic Approach Based on Clarke-Wright Algorithm for Open Vehicle Routing Problem
Table 3
Computational results between the proposed CW and CW for data sets A, B, P, E, C, and F.
| Number |
Instance | Solution | Percentage improvement | CPU time (seconds) | CW | CW-1 | | CW-2 | CW-3 | CW-1 | CW-2 | CW-3 |
| 1 | A-n32-k5 | 612.99 | 535.92 | 1.7 | 493.18 | 487.31 | 12.571 | 19.544 | 20.503 | 4.819 | 2 | A-n33-k5 | 496.61 | 458.52 | 1.5 | 425.54 | 424.54 | 7.669 | 14.311 | 14.512 | 5.493 | 3 | A-n33-k6 | 566.16 | 546.18 | 1.7 | 464.10 | 462.43 | 3.528 | 18.026 | 18.321 | 5.289 | 4 | A-n34-k5 | 547.27 | 520.09 | 1.1 | 519.51 | 508.17 | 4.966 | 5.071 | 7.143 | 5.675 | 5 | A-n36-k5 | 689.48 | 571.70 | 1.8 | 535.23 | 519.46 | 17.083 | 22.373 | 24.660 | 6.134 | 6 | A-n37-k5 | 579.74 | 541.64 | 1.9 | 511.12 | 486.24 | 6.572 | 11.836 | 16.128 | 6.689 | 7 | A-n37-k6 | 712.86 | 612.76 | 1.9 | 616.26 | 581.07 | 14.041 | 13.551 | 18.487 | 6.825 | 8 | A-n38-k5 | 552.83 | 520.78 | 1.6 | 531.49 | 498.00 | 5.796 | 3.859 | 9.918 | 7.092 | 9 | A-n39-k5 | 704.55 | 590.71 | 1.2 | 554.80 | 549.68 | 16.158 | 21.254 | 21.980 | 7.568 | 10 | A-n39-k6 | 720.55 | 558.86 | 1.9 | 547.57 | 533.07 | 22.439 | 24.007 | 26.019 | 7.494 | 11 | A-n44-k6 | 757.90 | 724.33 | 1.1 | 641.77 | 617.39 | 4.430 | 15.323 | 18.540 | 10.323 | 12 | A-n45-k6 | 734.97 | 643.56 | 2.0 | 716.52 | 648.67 | 12.437 | 2.511 | 11.742 | 10.414 | 13 | A-n45-k7 | 829.44 | 759.40 | 1.6 | 699.86 | 685.16 | 8.444 | 15.622 | 17.395 | 10.754 | 14 | A-n46-k7 | 738.00 | 645.24 | 1.1 | 593.57 | 583.54 | 12.569 | 19.570 | 20.930 | 11.259 | 15 | A-n48-k7 | 789.39 | 756.58 | 1.1 | 726.54 | 669.83 | 4.156 | 7.962 | 15.146 | 12.534 | 16 | A-n53-k7 | 816.38 | 702.85 | 1.5 | 665.39 | 655.18 | 13.907 | 18.495 | 19.746 | 15.624 | 17 | A-n54-k7 | 951.51 | 808.42 | 1.5 | 723.60 | 709.27 | 15.038 | 23.952 | 25.458 | 16.224 | 18 | A-n55-k9 | 802.06 | 736.91 | 1.9 | 696.52 | 669.06 | 8.123 | 13.158 | 16.582 | 17.324 | 19 | A-n62-k8 | 979.51 | 851.15 | 1.3 | 815.21 | 783.18 | 13.105 | 16.774 | 20.044 | 22.959 | 20 | A-n65-k9 | 844.35 | 800.41 | 1.2 | 783.42 | 728.59 | 5.205 | 7.217 | 13.710 | 24.524 | 21 | A-n69-k9 | 942.87 | 798.64 | 2.0 | 773.17 | 757.76 | 15.296 | 17.998 | 19.632 | 28.600 |
| | Average percentage improvement of data set A | 10.644 | 14.877 | 17.933 | 11.601 |
| 1 | B-n31-k5 | 383.68 | 367.00 | 0.9 | 364.80 | 362.73 | 4.347 | 4.923 | 5.463 | 4.365 | 2 | B-n34-k5 | 541.34 | 506.26 | 0.9 | 459.59 | 458.76 | 6.480 | 15.103 | 15.254 | 5.556 | 3 | B-n35-k5 | 599.16 | 595.15 | 1.4 | 567.34 | 557.33 | 0.669 | 5.311 | 6.982 | 5.981 | 4 | B-n38-k6 | 500.64 | 483.14 | 1.4 | 450.72 | 445.63 | 3.495 | 9.972 | 10.989 | 7.109 | 5 | B-n39-k5 | 382.11 | 354.03 | 1.4 | 334.70 | 322.54 | 7.350 | 12.409 | 15.590 | 7.602 | 6 | B-n41-k6 | 539.15 | 507.25 | 1.8 | 493.34 | 483.07 | 5.917 | 8.496 | 10.402 | 8.549 | 7 | B-n43-k6 | 483.21 | 481.57 | 1.6 | 432.30 | 428.17 | 0.340 | 10.536 | 11.391 | 9.422 | 8 | B-n44-k7 | 575.52 | 560.54 | 1.9 | 512.64 | 501.31 | 2.603 | 10.927 | 12.895 | 10.244 | 9 | B-n45-k5 | 601.71 | 512.73 | 1.9 | 509.56 | 488.07 | 14.788 | 15.315 | 18.887 | 10.357 | 10 | B-n45-k6 | 459.90 | 430.83 | 1.1 | 431.54 | 403.81 | 6.323 | 6.168 | 12.197 | 10.573 | 11 | B-n50-k7 | 537.23 | 491.93 | 1.9 | 446.07 | 437.15 | 8.433 | 16.969 | 18.629 | 13.265 | 12 | B-n51-k7 | 703.81 | 683.31 | 2.0 | 656.01 | 625.14 | 2.913 | 6.792 | 11.178 | 13.509 | 13 | B-n52-k7 | 482.90 | 465.17 | 1.8 | 450.07 | 441.19 | 3.672 | 6.798 | 8.637 | 14.586 | 14 | B-n56-k7 | 497.15 | 474.13 | 1.5 | 463.06 | 420.48 | 4.629 | 6.856 | 15.420 | 17.409 | 15 | B-n63-k10 | 950.48 | 950.48 | 1.0 | 857.90 | 837.07 | 0.000 | 9.740 | 11.931 | 23.724 | 16 | B-n64-k9 | 572.41 | 541.70 | 1.2 | 581.72 | 520.47 | 5.364 | −1.627 | 9.074 | 23.764 | 17 | B-n68-k9 | 830.48 | 777.68 | 1.1 | 758.69 | 701.72 | 6.357 | 8.644 | 15.504 | 27.874 |
| | Average percentage improvement of data set B | 4.922 | 9.020 | 12.378 | 12.582 |
| 1 | P-n16-k8 | 235.89 | 235.89 | 0.3 | 235.06 | 235.06 | 0.000 | 0.352 | 0.352 | 1.173 | 2 | P-n19-k2 | 198.25 | 172.93 | 1.9 | 168.57 | 168.57 | 12.771 | 14.972 | 14.972 | 1.565 | 3 | P-n20-k2 | 210.01 | 184.50 | 1.9 | 170.28 | 170.28 | 12.147 | 18.918 | 18.918 | 1.712 | 4 | P-n21-k2 | 209.92 | 180.27 | 1.8 | 168.15 | 163.88 | 14.124 | 19.897 | 21.933 | 1.729 | 5 | P-n22-k2 | 206.00 | 183.58 | 1.8 | 171.46 | 167.19 | 10.883 | 16.765 | 18.840 | 1.888 | 6 | P-n22-k8 | 370.26 | 345.53 | 1.6 | 352.14 | 345.87 | 6.679 | 4.892 | 6.588 | 2.330 | 7 | P-n23-k8 | 309.74 | 307.28 | 1.7 | 304.83 | 302.87 | 0.794 | 1.586 | 2.219 | 2.608 | 8 | P-n40-k5 | 420.65 | 395.73 | 1.5 | 370.64 | 349.55 | 5.924 | 11.889 | 16.902 | 7.749 | 9 | P-n45-k5 | 459.29 | 442.23 | 1.5 | 396.64 | 391.81 | 3.714 | 13.641 | 14.692 | 10.159 | 10 | P-n50-k7 | 468.42 | 447.69 | 1.5 | 440.56 | 397.38 | 4.426 | 5.948 | 15.167 | 13.237 | 11 | P-n55-k7 | 513.45 | 464.90 | 1.4 | 452.69 | 411.58 | 9.455 | 11.833 | 19.840 | 16.627 | 12 | P-n55-k8 | 505.72 | 476.43 | 1.8 | 442.21 | 412.55 | 5.793 | 12.558 | 18.423 | 16.349 | 13 | P-n55-k10 | 555.25 | 502.56 | 1.6 | 488.65 | 444.31 | 9.490 | 11.995 | 19.981 | 17.387 | 14 | P-n60-k10 | 584.40 | 539.43 | 1.8 | 503.38 | 482.09 | 7.694 | 13.864 | 17.507 | 20.499 | 15 | P-n65-k10 | 649.31 | 592.35 | 1.6 | 531.57 | 522.50 | 8.772 | 18.133 | 19.529 | 29.626 |
| | Average percentage improvement of data set P | 7.511 | 11.816 | 15.058 | 9.642 |
| 1 | E-n22-k4 | 286.91 | 260.61 | 1.3 | 252.61 | 252.61 | 9.169 | 11.955 | 11.955 | 3.753 | 2 | E-n23-k3 | 497.18 | 456.86 | 1.2 | 444.29 | 442.98 | 8.111 | 10.638 | 10.901 | 4.156 | 3 | E-n33-k4 | 633.04 | 576.29 | 1.4 | 518.04 | 511.26 | 8.965 | 18.165 | 19.236 | 12.223 | 4 | E-n51-k5 | 493.02 | 477.78 | 2.0 | 452.67 | 416.06 | 3.091 | 8.185 | 15.610 | 36.154 | 5 | E-n76-k10 | 697.03 | 641.48 | 1.9 | 587.35 | 567.14 | 7.971 | 15.736 | 18.635 | 68.917 | 6 | E-n101-k8 | 807.33 | 724.48 | 2.0 | 694.88 | 642.36 | 10.262 | 13.929 | 20.433 | 82.387 |
| | Average percentage improvement of data set E | 7.928 | 13.101 | 16.128 | 34.598 |
| 1 | F-n45-k4 | 615.12 | 535.85 | 2.0 | 478.40 | 463.90 | 12.886 | 22.227 | 24.584 | 10.365 | 2 | F-n72-k4 | 208.29 | 191.18 | 2.0 | 187.67 | 177.00 | 8.212 | 9.899 | 15.021 | 42.554 | 3 | F-n135-k7 | 1033.24 | 856.33 | 1.9 | 816.30 | 775.80 | 17.122 | 20.996 | 24.916 | 112.258 |
| | Average percentage improvement of data set F | 12.740 | 17.707 | 21.507 | 55.059 |
|
|
Italic number indicates the infeasible solution (the number of vehicles used is inadequate).
|