1 Introduction
2 Problem description and mathematical formulations
2.1 Explicit formulation
2.2 Implicit vehicle index formulation
2.3 Compact formulation with loading variables
2.4 Compact formulation with disaggregated loading variables
2.5 Capacity-indexed formulation
3 Theoretical insights
4 Solution algorithms
5 Computational experiments
5.1 Implementation details
5.2 Description of the instances
Instance | References | Origin | # Depots | # Customers |
\({\hat{Q}}\)
|
---|---|---|---|---|---|
4-55-100 |
Salhi and Sari (1997) |
Perl and Daskin (1985) | 4 | 55 | 100 |
3-85-100 |
Salhi and Sari (1997) |
Perl and Daskin (1985) | 3 | 85 | 100 |
3-85-160 |
Salhi and Sari (1997) |
Perl and Daskin (1985) | 3 | 85 | 160 |
4-50-80 |
Salhi and Sari (1997) |
Gillett and Johnson (1976) | 4 | 50 | 80 |
4-50-160 |
Salhi and Sari (1997) |
Gillett and Johnson (1976) | 4 | 50 | 160 |
5-75-140 |
Salhi and Sari (1997) |
Gillett and Johnson (1976) | 5 | 75 | 140 |
2-100-100 |
Salhi and Sari (1997) |
Gillett and Johnson (1976) | 2 | 100 | 100 |
2-100-200 |
Salhi and Sari (1997) |
Gillett and Johnson (1976) | 2 | 100 | 200 |
3-100-100 |
Salhi and Sari (1997) |
Gillett and Johnson (1976) | 3 | 100 | 100 |
4-100-100 |
Salhi and Sari (1997) |
Gillett and Johnson (1976) | 4 | 100 | 100 |
2-80-60 |
Salhi and Sari (1997) |
Chao et al. (1993) | 2 | 80 | 60 |
4-160-60 |
Salhi and Sari (1997) |
Chao et al. (1993) | 4 | 160 | 60 |
6-240-60 |
Salhi and Sari (1997) |
Chao et al. (1993) | 6 | 240 | 60 |
9-360-60 |
Salhi and Sari (1997) |
Chao et al. (1993) | 9 | 360 | 60 |
2-10-60 | New |
Salhi and Sari (1997) | 2 | 10 | 60 |
2-15-60 | New |
Salhi and Sari (1997) | 2 | 15 | 60 |
3-20-80 | New |
Salhi and Sari (1997) | 3 | 20 | 80 |
3-25-80 | New |
Salhi and Sari (1997) | 3 | 25 | 80 |
3-30-80 | New |
Salhi and Sari (1997) | 3 | 30 | 80 |
2-10-60* | New |
Salhi and Sari (1997) | 2 | 10 | 60 |
2-15-60* | New |
Salhi and Sari (1997) | 2 | 15 | 60 |
3-20-100 | New |
Salhi and Sari (1997) | 3 | 20 | 100 |
3-25-100 | New |
Salhi and Sari (1997) | 3 | 25 | 100 |
3-30-100 | New |
Salhi and Sari (1997) | 3 | 30 | 100 |
5.3 Computational experiments
# | Formulation name | Origin | O.F. and constraints |
---|---|---|---|
F1 | Explicit vehicle index formulation |
Vidal et al. (2014) with introducing new assignment variables \(y_{i}^{{td}}\)
| |
F2 | Implicit vehicle index formulation | F1 with vehicle index denoting vehicle type instead of individual vehicles | |
F3 | Compact formulation with loading variables | Salhi et al. (2014) with introducing new assignment variables \(y_{i}^{{kd}}\)
| |
F4 | Compact formulation with disaggregated loading variables | F3 with disaggregating the continuous loading variables \(z_{{ij}}\)
| |
F5 | Capacity-indexed formulation | Pessoa et al. (2009) |
Instance | Before preprocessing | After preprocessing |
---|---|---|
4-55-100 | 7,031,620 | 417,720 |
3-85-100 | 11,732,160 | 696,960 |
3-85-160 | 18,701,760 | 998,976 |
4-50-80 | 4,723,920 | 4548,960 |
4-50-160 | 9,389,520 | 9,214,560 |
5-75-140 | 22,560,000 | 22,400,000 |
2-100-100 | 10,508,040 | 10,508,040 |
2-100-200 | 20,912,040 | 20,912,040 |
3-100-100 | 16,072,635 | 16,072,635 |
4-100-100 | 21,848,320 | 21,848,320 |
2-80-60 | 4,101,640 | 4,101,640 |
4-160-60 | 32,813,120 | 32,813,120 |
6-240-60 | 110,744,280 | 110,744,280 |
9-360-60 | Out of memory | Out of memory |
Average | 22,395,311.92 | 19,636,711.62 |
5.3.1 Linear programming relaxation
Instance | Formulation F1 | Formulation F2 | Formulation F3 | Formulation F4 | Formulation F5 | |||||
---|---|---|---|---|---|---|---|---|---|---|
Value | Time (s) | Value | Time (s) | Value | Time (s) | Value | Time (s) | Value | Time (s) | |
4-55-100 | NF | 7200 | 574.32 | 17 | 1313.54 | 29 | 1354.41 | 58 |
1359.99
| 56 |
3-85-100 | NF | 7200 | 841.57 | 43 | 2027.98 | 66 | 2079.01 | 456 |
2094.42
| 157 |
3-85-160 | NF | 7200 | 631.65 | 44 | 1347.39 | 77 | 1411.31 | 505 |
1435.19
| 357 |
4-50-80 | NF | 7200 | 642.90 | 23 | 1322.30 | 21 | 1381.63 | 58 |
1416.05
| 4348 |
4-50-160 | NF | 7200 | 496.07 | 10 | 807.47 | 17 |
890.99
| 89 | NF | 7200 |
5-75-140 | NC | 7200 | 707.52 | 62 | 1362.47 | 102 |
1478.41
| 530 | NF | 7200 |
2-100-100 | NC | 7200 | 966.10 | 53 | 2095.84 | 94 |
2191.73
| 816 | NF | 7200 |
2-100-200 | NC | 7200 | 749.49 | 54 | 1262.37 | 80 |
1382.34
| 1021 | NF | 7200 |
3-100-100 | NC | 7200 | 952.52 | 84 | 1990.05 | 143 |
2106.65
| 1074 | NF | 7200 |
4-100-100 | NC | 7200 | 951.76 | 166 | 1993.18 | 197 |
2103.29
| 1422 | NF | 7200 |
2-80-60 | NC | 7200 | 1186.35 | 48 | 1573.03 | 24 | 1763.45 | 287 |
1790.17
| 6861 |
4-160-60 | NC | 7200 | NF | 7200 |
3063.71
| 1090 | NF | 7200 | NF | 7200 |
6-240-60 | NC | 7200 | NF | 7200 | NF | 7200 | NF | 7200 | NF | 7200 |
9-360-60 | NC | 7200 | NF | 7200 | NF | 7200 | NF | 7200 | NC | 7200 |
2-10-60 | 268.51 | 5 | 268.58 | 0 | 399.87 | 0 | 409.55 | 0 |
422.33
| 3 |
2-15-60 | 355.68 | 50 | 356.02 | 0 | 600.23 | 0 | 627.41 | 1 |
650.23
| 9 |
3-20-80 | 333.43 | 540 | 334.30 | 0 | 594.47 | 1 | 626.40 | 2 |
640.66
| 117 |
3-25-80 | 395.63 | 1880 | 397.93 | 1 | 706.59 | 1 | 744.38 | 6 |
759.54
| 218 |
3-30-80 | NF | 7200 | 438.25 | 2 | 828.87 | 3 | 866.25 | 7 |
884.23
| 393 |
2-10-60* | 245.29 | 3 | 246.38 | 0 | 448.59 | 0 | 459.77 | 0 |
468.59
| 0 |
2-15-60* | 316.37 | 43 | 318.53 | 0 | 646.36 | 0 | 659.91 | 1 |
681.01
| 0 |
3-20-100 | 255.55 | 206 | 255.41 | 1 | 525.82 | 1 | 544.05 | 2 |
544.71
| 2 |
3-25-100 | 303.23 | 1299 | 303.47 | 1 | 631.16 | 1 | 658.49 | 4 |
660.80
| 3 |
3-30-100 | 382.75 | 2216 | 383.20 | 1 | 784.34 | 2 | 820.20 | 9 |
824.98
| 5 |
5.3.2 Comparison of upper and lower bounds
Instance | Formulation F1 | Formulation F2 | Formulation F3 | Formulation F4 | Formulation F5 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
UB | LB | Gap (%) | UB | LB | Gap (%) | UB | LB | Gap (%) | UB | LB | Gap (%) | UB | LB | Gap (%) | |
4-55-100 | 1612.30 | 552.52 | 65.73 | 1606.63 | 575.62 | 64.17 | 1438.50 | 1328.99 | 7.61 | 1408.25 | 1361.85 | 3.29 |
1398.30
|
1384.55
| 0.98 |
3-85-100 | NF | NF | – | 2258.22 | 842.50 | 62.69 | 2209.95 | 2040.45 | 7.67 | 2163.63 | 2086.32 | 3.57 |
2131.46
|
2116.12
| 0.72 |
3-85-160 | NF | NF | – | 1698.58 | 633.22 | 62.72 | 1675.05 | 1360.58 | 18.77 | 1663.01 | 1423.30 | 14.41 |
1483.51
|
1451.86
| 2.13 |
4-50-80 | 1770.66 | 602.44 | 65.98 | 1770.66 | 621.48 | 64.90 | 1573.68 | 1348.51 | 14.31 |
1565.27
| 1390.51 | 11.16 | 1733.20 |
1416.09
| 18.30 |
4-50-160 | 1217.74 | 485.68 | 60.12 | 1216.52 | 499.13 | 58.97 |
1021.59
| 832.21 | 18.54 | 1122.08 |
907.71
| 19.10 | 1217.74 | 902.18 | 25.91 |
5-75-140 | NC | NC | – | 1962.67 | 708.06 | 63.92 | 1831.02 | 1388.41 | 24.15 |
1828.73
|
1483.11
| 18.92 | NF | NF | – |
2-100-100 | NC | NC | – | 2757.95 | 968.67 | 64.88 |
2660.50
| 2116.36 | 20.45 | 2725.87 | 2201.16 | 19.25 | 2757.95 |
2236.91
| 18.89 |
2-100-200 | NC | NC | – | 1826.28 | 750.27 | 58.92 |
1818.69
| 1280.61 | 29.59 | 1825.91 |
1396.05
| 23.54 | NF | NF | – |
3-100-100 | NC | NC | – | 2728.89 | 953.12 | 65.07 |
2648.17
| 2013.95 | 23.95 | 2696.05 |
2109.31
| 21.76 | NF | NF | – |
4-100-100 | NC | NC | – | 2691.82 | 952.05 | 64.63 |
2626.33
| 2003.95 | 23.70 | 2684.89 |
2104.36
| 21.63 | NF | NF | – |
2-80-60 | NC | NC | – | 2573.82 | 1186.35 | 53.91 | 2566.13 | 1683.51 | 34.39 |
2565.53
| 1788.35 | 30.29 | 2571.35 |
1794.38
| 30.22 |
4-160-60 | NC | NC | – | 5172.65 | 2199.05 | 57.49 |
5157.49
| 3094.10 | 40.01 | 5172.65 |
3506.89
| 32.20 | NF | NF | – |
6-240-60 | NC | NC | – | 7758.97 | 3182.68 | 58.98 |
7758.97
| 4564.18 | 41.18 |
7758.97
|
5243.12
| 32.43 | NF | NF | – |
9-360-60 | NC | NC | – | 11638.50 | 4441.25 | 61.84 |
11638.50
| 0.00 | 100.00 |
11638.50
|
7852.44
| 32.53 | NC | NC | – |
Average gap | 90.99 | 61.65 | 28.88 | 20.63 | 56.94 | ||||||||||
2-10-60 |
441.59
|
441.59
| 0.00 |
441.59
| 441.57 | 0.00 |
441.59
|
441.59
| 0.00 |
441.59
|
441.59
| 0.00 |
441.59
|
441.59
| 0.00 |
2-15-60 | 679.18 | 611.92 | 9.90 | 679.18 | 489.20 | 27.97 |
674.32
| 674.26 | 0.01 |
674.32
| 674.26 | 0.01 |
674.32
|
674.32
| 0.00 |
3-20-80 | 784.97 | 354.03 | 54.90 | 757.34 | 405.73 | 46.43 |
666.02
| 665.95 | 0.01 |
666.02
| 665.97 | 0.01 |
666.02
|
666.02
| 0.00 |
3-25-80 | 845.41 | 409.43 | 51.57 | 839.13 | 429.99 | 48.76 | 794.46 | 745.08 | 6.21 |
787.37
| 787.29 | 0.01 |
787.37
|
787.30
| 0.01 |
3-30-80 | 1009.93 | 448.00 | 55.64 | 1005.05 | 448.83 | 55.34 | 932.13 | 870.77 | 6.58 | 932.13 |
890.71
| 4.44 |
940.49
| 886.95 | 5.69 |
2-10-60* |
482.09
|
482.09
| 0.00 |
482.09
| 482.05 | 0.01 |
482.09
|
482.09
| 0.00 |
482.09
|
482.09
| 0.00 |
482.09
|
482.09
| 0.00 |
2-15-60* |
690.38
|
690.38
| 0.00 |
690.38
| 393.98 | 42.93 |
690.38
| 690.31 | 0.01 |
690.38
| 690.31 | 0.01 |
690.38
|
690.38
| 0.00 |
3-20-100 | 597.44 | 543.12 | 9.09 | 605.95 | 293.85 | 51.51 |
563.19
| 563.13 | 0.01 |
563.19
| 563.15 | 0.01 |
563.19
|
563.19
| 0.00 |
3-25-100 | 727.54 | 651.16 | 10.50 | 717.86 | 321.37 | 55.23 |
676.59
| 674.68 | 0.28 |
676.59
| 676.53 | 0.01 |
676.59
|
676.59
| 0.00 |
3-30-100 | 994.17 | 735.29 | 26.04 | 994.17 | 397.68 | 60.00 |
839.98
| 812.71 | 3.25 |
839.98
| 836.77 | 0.38 |
839.98
|
839.98
| 0.00 |
Average gap/ time (s) | 21.76/8590.10 | 38.82/8811.400 | 1.64/5138.40 | 0.49/1832.56 | 0.57/2390.00 |
5.3.3 Comparison against the best known solutions
Instance |
Vidal et al. (2014) |
Salhi et al. (2014) | Formulation F3 | Formulation F4 | Formulation F5 | F3, F4, F5 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
UB | UB | LB | Gap (%) | UB | LB | Gap (%) | UB | LB | Gap (%) | UB | LB | Gap (%) | Best gap (%) | |
4-55-100 | – | 1621.90 | 1299.80 | 19.86 | 1438.50 | 1328.99 | 7.61 | 1408.25 | 1361.85 | 3.29 |
1398.30
|
1384.55
| 0.98 | 0.98 |
3-85-100 | – | 2677.80 | 1996.00 | 25.46 | 2209.95 | 2040.45 | 7.67 | 2163.63 | 2086.32 | 3.57 |
2131.46
|
2116.12
| 0.72 | 0.72 |
3-85-160 | – | 2516.00 | 1326.10 | 47.29 | 1675.05 | 1360.58 | 18.77 | 1663.01 | 1423.30 | 14.41 |
1483.51
|
1451.86
| 2.13 | 2.13 |
4-50-80 |
1477.73
| 1725.20 | 1318.50 | 23.57 | 1573.68 | 1348.51 | 14.31 | 1565.27 | 1390.51 | 11.16 | 1733.20 |
1416.09
| 18.30 | 9.53 |
4-50-160 |
957.73
| 1378.90 | 799.70 | 42.00 | 1021.59 | 832.21 | 18.54 | 1122.08 |
907.71
| 19.10 | 1217.74 | 902.18 | 25.91 | 11.14 |
5-75-140 |
1569.67
| 2561.10 | 1341.40 | 47.62 | 1831.02 | 1388.41 | 24.15 | 1828.73 |
1483.11
| 18.92 | NF | NF | – | 18.90 |
2-100-100 |
2292.64
| 3039.70 | 2079.30 | 31.59 | 2660.50 | 2116.36 | 20.45 | 2725.87 | 2201.16 | 19.25 | 2757.95 |
2236.91
| 18.89 | 15.92 |
2-100-200 |
1453.64
| 2265.60 | 1228.30 | 45.78 | 1818.69 | 1280.61 | 29.59 | 1825.91 |
1396.05
| 23.54 | NF | NF | – | 23.23 |
3-100-100 |
2208.66
| 3390.30 | 1970.20 | 41.89 | 2648.17 | 2013.95 | 23.95 | 2696.05 |
2109.31
| 21.76 | NF | NF | – | 20.34 |
4-100-100 |
2198.91
| 3609.90 | 1971.30 | 45.39 | 2626.33 | 2003.95 | 23.70 | 2684.89 |
2104.36
| 21.63 | NF | NF | – | 19.87 |
2-80-60 |
2072.18
| 2846.40 | 1607.60 | 43.52 | 2566.13 | 1683.51 | 34.39 | 2565.53 | 1788.35 | 30.29 | 2571.35 |
1794.38
| 30.22 | 30.05 |
4-160-60 |
3973.47
| NF | 3032.60 | – | 5157.49 | 3094.10 | 40.01 | 5172.65 |
3506.89
| 32.20 | NF | NF | – | 32.00 |
6-240-60 |
5887.43
| NF | NF | – | 7758.97 | 4564.18 | 41.18 | 7758.97 |
5243.12
| 32.43 | NF | NF | – | 32.42 |
9-360-60 |
8709.26
| NF | NF | – | 11638.50 | 0.00 | 100.00 | 11638.50 |
7852.44
| 32.53 | NC | NC | – | 32.53 |
Average gap | 51.00 | 28.88 | 20.63 | 56.94 | 17.84 |
Instances | Formulation F1 | Formulation F2 | Formulation F3 | Formulation F4 | Formulation F5 | |||||
---|---|---|---|---|---|---|---|---|---|---|
Without VI | With VI | Without VI | With VI | Without VI | With VI | Without VI | With VI | Without VI | With VI | |
Existing | 94.66 | 90.99 | 61.79 | 61.65 | 29.45 | 28.88 | 25.52 | 20.29 | 62.43 | 56.94 |
New | 57.68 | 21.76 | 40.76 | 38.82 | 2.05 | 1.64 | 0.69 | 0.49 | 1.18 | 0.57 |