Figure 1.
Relationship between the HVRPSPD and its variants
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2021, 17(3): 1069-1100.
doi: 10.3934/jimo.2020012
A mathematical formulation and heuristic approach for the heterogeneous fixed fleet vehicle routing problem with simultaneous pickup and delivery
| 1. | Deparment of Industrial Engineering, School of Engineering, Başkent University, Baǧlıca Kampüsü Fatih Sultan Mahallesi, Eskişehir Yolu 18.km, Etimesgut, Ankara, TR 06790, Türkiye |
| 2. | Deparment of Industrial Engineering, School of Engineering, Gazi University, Eti Mahallesi, Yükseliş Sokak, No:5, Maltepe, Ankara, TR 06570, Türkiye |
This study considers a variant of the vehicle routing problem (VRP) called the heterogeneous VRP with simultaneous pickup and delivery (HVRPSPD). The HVRPSPD may broadly be defined as identifying the minimum cost routes and vehicle types. To solve the HVRPSPD, first, we propose a polynomial-size mixed integer programming formulation. Because the HVRPSPD is an NP-hard problem, it is difficult to determine the optimal solution in a reasonable time for moderate and large-size problem instances. Hence, we develop a hybrid metaheuristic approach based on the simulated annealing and local search algorithms called SA-LS. We conduct a computational study in three stages. First, the performance of the mathematical model and SA-LS are investigated on small and medium-size HVRPSPD instances. Second, we compare SA-LS with the constructive heuristics, nearest neighborhood and Clarke-Wright savings algorithms, adapted for the HVRPSPD. Finally, the performance of SA-LS is evaluated on the instances of the heterogeneous VRP (HVRP), which is a special case of the HVRPSPD. Computational results demonstrate that the mathematical model can solve small-size instances optimally up to 35 nodes; SA-LS provides good quality solutions for medium and large-size problems. Moreover, SA-LS is superior to simple constructive heuristics and can be a preferable solution method to solve HVRP and VRPSPD instances successfully.
Keywords:
Vehicle routing problem,
simultaneous pickup-delivery,
heterogeneous fixed fleet,
mathematical model,
simulated annealing algorithm.
Mathematics Subject Classification:
Primary: 90B06, 90C11, 90C59; Secondary: 90C27.
Citation:
Bariş Keçeci, Fulya Altıparmak, İmdat Kara. A mathematical formulation and heuristic approach for the heterogeneous fixed fleet vehicle routing problem with simultaneous pickup and delivery. Journal of Industrial and Management Optimization,
2021, 17
(3)
: 1069-1100.
doi: 10.3934/jimo.2020012
![]()
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show all references
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|
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|
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|
| [18] |
J. Dethloff,
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doi: 10.1057/jors.2009.83. |
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M. Gendreau, G. Laporte, C. Musaraganyi and É. D. Taillard,
A tabu search heuristic for the heterogeneous fleet vehicle routing problem, Computers & Operations Research, 26 (1999), 1153-1173.
doi: 10.1016/S0305-0548(98)00100-2. |
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A successive approximations method for the heterogeneous vehicle routing problem: Analysing different fleet configurations, Eur. J. Ind. Eng, 8 (2014), 762-788.
doi: 10.1504/EJIE.2014.066934. |
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I. Kara and T. Derya,
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doi: 10.1063/1.3636940. |
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B. Keçeci, F. Altıparmak and I. Kara, The heterogeneous vehicle routing problem with simultaneous pickup and delivery: A hybrid heuristic approach based on simulated annealing, CIE44 & IMSS'14 Proceedings, (2014). |
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B. Keçeci, F. Altiparmak and I. Kara,
Heterogeneous vehicle routing problem with simultaneous pickup and delivery: Mathematical formulations and a heuristic algorithm, Journal of the Faculty of Engineering and Architecture of Gazi University, 30 (2015), 185-195.
|
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Figure 2.
Representation of the HVRPSPD solution with 10 customers and three routes
Figure 3.
Inter-route neighborhood structures
Figure 4.
Intra-route neighborhood structures
Figure 5.
Pseudo code of the LS
Figure 6.
Pseudo code of SA-LS
Figure 7.
The convergence of SA-LS with different initial temperature values
Figure 8.
The convergence of SA-LS with different final temperature values
Figure 9.
The convergence of SA-LS with different cooling rate values
Figure 10.
Pseudo code of the NN
Figure 11.
Pseudo code of the PCWS
Table 1.
Characteristics of the test problems
Table 2.
Computational results of SA-LS on small and medium-size test problems
| n | Type | SA-LS | MIP Formulation | MatH-LS* | |||||||
| Gap% | Imp% | #OpSol | CPU (s) | Gap% | #OpSol | CPU (s) | Gap% | #OpSol | CPU (s) | ||
| 20 | X | 4.50 | 5.50 | 6 | 1.90 | 1.90 | 20 | 306.01 | 5.38 | 8 | 16.53 |
| Y | 4.40 | 5.30 | 7 | 2.00 | 1.51 | 20 | 321.93 | 5.49 | 7 | 13.6 | |
| W | 4.00 | 7.10 | 7 | 1.90 | 0.49 | 23 | 290.63 | 4.38 | 12 | 13.24 | |
| Z | 6.30 | 3.30 | 3 | 1.80 | 3.75 | 13 | 295.44 | 5.23 | 11 | 12.33 | |
| (Total)Average | 4.80 | 5.30 | (23) | 1.90 | 1.91 | (76) | 303.50 | 5.12 | (38) | 13.93 | |
| 25 | X | 5.10 | 3.80 | 3 | 3.50 | 2.39 | 13 | 1174.24 | 9.35 | 5 | 18.48 |
| Y | 5.00 | 3.50 | 2 | 2.70 | 2.38 | 15 | 1362.22 | 8.87 | 7 | 17.52 | |
| W | 6.10 | 3.80 | 3 | 3.00 | 2.65 | 14 | 644.11 | 7.21 | 7 | 16.74 | |
| Z | 8.10 | 3.20 | 1 | 3.00 | 2.78 | 13 | 580.30 | 9.69 | 7 | 15.25 | |
| (Total)Average | 6.08 | 3.58 | (9) | 3.05 | 2.55 | (55) | 940.22 | 8.78 | (26) | 17 | |
| 30 | X | 8.75 | 4.30 | 2 | 5.30 | 5.91 | 7 | 2133.27 | 13.06 | 1 | 21.95 |
| Y | 11, 20 | 2.80 | 4 | 4.10 | 7.08 | 5 | 2154.08 | 12.95 | 2 | 28.12 | |
| W | 6.55 | 6.30 | 0 | 5.90 | 6.26 | 4 | 1672.95 | 11.11 | 5 | 22.58 | |
| Z | 6.18 | 4.90 | 3 | 5.20 | 4.39 | 9 | 1694.82 | 11.56 | 4 | 18.93 | |
| (Total)Average | 8.17 | 4.58 | (9) | 5.13 | 5.91 | (25) | 1913.78 | 12.17 | (12) | 22.89 | |
| 35 | X | 8.32 | 11, 70 | 0 | 7.90 | 10, 76 | 0 | 3984.36 | 26.08 | 1 | 32.78 |
| Y | 9.29 | 5.72 | 0 | 7.50 | 9.00 | 1 | 3724.80 | 16.14 | 0 | 31.31 | |
| W | 7.97 | 3.99 | 1 | 7.40 | 9.90 | 4 | 3741.48 | 14.92 | 0 | 25.42 | |
| Z | 9.88 | 3.67 | 1 | 8.20 | 9.59 | 6 | 3242.11 | 13.69 | 1 | 26.26 | |
| (Total)Average | 8.87 | 6.27 | (2) | 7.75 | 9.81 | (11) | 3673.19 | 17.71 | (2) | 28.94 | |
| 40 | X | 16.60 | 5.44 | 0 | 11, 70 | 20.54 | 0 | 5833.66 | 27.66 | 0 | 41.16 |
| Y | 18, 10 | 8.10 | 0 | 11, 60 | 19, 06 | 0 | 6216.06 | 19.28 | 0 | 46.15 | |
| W | 9.08 | 4.93 | 0 | 11, 30 | 13.82 | 1 | 5685.47 | 15.68 | 0 | 28.66 | |
| Z | 8.36 | 6.66 | 2 | 11, 50 | 10, 68 | 6 | 4810.54 | 11.87 | 0 | 28.79 | |
| (Total)Average | 13, 04 | 6.28 | (2) | 11, 53 | 16, 02 | (7) | 5636.43 | 18.62 | (0) | 36.19 | |
| Overall(Total) | 8.19 | 5.20 | (45) | 5.87 | 7.24 | (174) | 2493.42 | 12.48 | (78) | 23.79 | |
| *MatH-LS is the algorithm proposed by Kececi et al. [32]. | |||||||||||
| n | Type | SA-LS | MIP Formulation | MatH-LS* | |||||||
| Gap% | Imp% | #OpSol | CPU (s) | Gap% | #OpSol | CPU (s) | Gap% | #OpSol | CPU (s) | ||
| 20 | X | 4.50 | 5.50 | 6 | 1.90 | 1.90 | 20 | 306.01 | 5.38 | 8 | 16.53 |
| Y | 4.40 | 5.30 | 7 | 2.00 | 1.51 | 20 | 321.93 | 5.49 | 7 | 13.6 | |
| W | 4.00 | 7.10 | 7 | 1.90 | 0.49 | 23 | 290.63 | 4.38 | 12 | 13.24 | |
| Z | 6.30 | 3.30 | 3 | 1.80 | 3.75 | 13 | 295.44 | 5.23 | 11 | 12.33 | |
| (Total)Average | 4.80 | 5.30 | (23) | 1.90 | 1.91 | (76) | 303.50 | 5.12 | (38) | 13.93 | |
| 25 | X | 5.10 | 3.80 | 3 | 3.50 | 2.39 | 13 | 1174.24 | 9.35 | 5 | 18.48 |
| Y | 5.00 | 3.50 | 2 | 2.70 | 2.38 | 15 | 1362.22 | 8.87 | 7 | 17.52 | |
| W | 6.10 | 3.80 | 3 | 3.00 | 2.65 | 14 | 644.11 | 7.21 | 7 | 16.74 | |
| Z | 8.10 | 3.20 | 1 | 3.00 | 2.78 | 13 | 580.30 | 9.69 | 7 | 15.25 | |
| (Total)Average | 6.08 | 3.58 | (9) | 3.05 | 2.55 | (55) | 940.22 | 8.78 | (26) | 17 | |
| 30 | X | 8.75 | 4.30 | 2 | 5.30 | 5.91 | 7 | 2133.27 | 13.06 | 1 | 21.95 |
| Y | 11, 20 | 2.80 | 4 | 4.10 | 7.08 | 5 | 2154.08 | 12.95 | 2 | 28.12 | |
| W | 6.55 | 6.30 | 0 | 5.90 | 6.26 | 4 | 1672.95 | 11.11 | 5 | 22.58 | |
| Z | 6.18 | 4.90 | 3 | 5.20 | 4.39 | 9 | 1694.82 | 11.56 | 4 | 18.93 | |
| (Total)Average | 8.17 | 4.58 | (9) | 5.13 | 5.91 | (25) | 1913.78 | 12.17 | (12) | 22.89 | |
| 35 | X | 8.32 | 11, 70 | 0 | 7.90 | 10, 76 | 0 | 3984.36 | 26.08 | 1 | 32.78 |
| Y | 9.29 | 5.72 | 0 | 7.50 | 9.00 | 1 | 3724.80 | 16.14 | 0 | 31.31 | |
| W | 7.97 | 3.99 | 1 | 7.40 | 9.90 | 4 | 3741.48 | 14.92 | 0 | 25.42 | |
| Z | 9.88 | 3.67 | 1 | 8.20 | 9.59 | 6 | 3242.11 | 13.69 | 1 | 26.26 | |
| (Total)Average | 8.87 | 6.27 | (2) | 7.75 | 9.81 | (11) | 3673.19 | 17.71 | (2) | 28.94 | |
| 40 | X | 16.60 | 5.44 | 0 | 11, 70 | 20.54 | 0 | 5833.66 | 27.66 | 0 | 41.16 |
| Y | 18, 10 | 8.10 | 0 | 11, 60 | 19, 06 | 0 | 6216.06 | 19.28 | 0 | 46.15 | |
| W | 9.08 | 4.93 | 0 | 11, 30 | 13.82 | 1 | 5685.47 | 15.68 | 0 | 28.66 | |
| Z | 8.36 | 6.66 | 2 | 11, 50 | 10, 68 | 6 | 4810.54 | 11.87 | 0 | 28.79 | |
| (Total)Average | 13, 04 | 6.28 | (2) | 11, 53 | 16, 02 | (7) | 5636.43 | 18.62 | (0) | 36.19 | |
| Overall(Total) | 8.19 | 5.20 | (45) | 5.87 | 7.24 | (174) | 2493.42 | 12.48 | (78) | 23.79 | |
| *MatH-LS is the algorithm proposed by Kececi et al. [32]. | |||||||||||
Table 3.
Comparison of SA-LS with NN and CWS on large-size test problems
| n | Type | CPU(s) | |||
| 50 | X | -36.69 | -14.23 | 6.71 | 19.62 |
| Y | -35.79 | -18.23 | 5.57 | 20.75 | |
| W | -42.96 | -7.83 | 8.79 | 23.65 | |
| Z | -42.43 | -9.62 | 9.26 | 24.42 | |
| Average | -39.47 | -12.48 | 7.58 | 22.11 | |
| 75 | X | -58.39 | -16.50 | 11.96 | 72.23 |
| Y | -56.69 | -18.08 | 11.79 | 58.14 | |
| W | -59.49 | -18.32 | 15.11 | 82.00 | |
| Z | -59.14 | -12.37 | 17.55 | 57.12 | |
| Average | -58.43 | -16.32 | 14.10 | 67.37 | |
| 100 | X | -60.04 | -20.91 | 15.65 | 210.87 |
| Y | -60.46 | -20.23 | 16.17 | 220.00 | |
| W | -63.42 | -16.71 | 12.20 | 223.16 | |
| Z | -63.30 | -16.32 | 14.95 | 221.59 | |
| Average | -61.81 | -18.54 | 14.74 | 218.90 | |
| n | Type | CPU(s) | |||
| 50 | X | -36.69 | -14.23 | 6.71 | 19.62 |
| Y | -35.79 | -18.23 | 5.57 | 20.75 | |
| W | -42.96 | -7.83 | 8.79 | 23.65 | |
| Z | -42.43 | -9.62 | 9.26 | 24.42 | |
| Average | -39.47 | -12.48 | 7.58 | 22.11 | |
| 75 | X | -58.39 | -16.50 | 11.96 | 72.23 |
| Y | -56.69 | -18.08 | 11.79 | 58.14 | |
| W | -59.49 | -18.32 | 15.11 | 82.00 | |
| Z | -59.14 | -12.37 | 17.55 | 57.12 | |
| Average | -58.43 | -16.32 | 14.10 | 67.37 | |
| 100 | X | -60.04 | -20.91 | 15.65 | 210.87 |
| Y | -60.46 | -20.23 | 16.17 | 220.00 | |
| W | -63.42 | -16.71 | 12.20 | 223.16 | |
| Z | -63.30 | -16.32 | 14.95 | 221.59 | |
| Average | -61.81 | -18.54 | 14.74 | 218.90 | |
Table 4.
Comparison of SA-LS with HVRP test instances
| Inst. | n | Salhi and Rand[52] | MRPERT Osman and Salhi[43] | TSVFM Osman and Salhi[43] | Taillard[58] | Gendreau et al.[20] | Wassan and Osman[65] | Renaud and Boctor[49] | Choi and Tcha[14] | Brandao[10] | Prins[48] | Liu et al.[37] | Penna et al.[45] | Subramanian et al. [56] | Simeonova et al. [53] | SA-LS | Dev% | CPU (s) |
| G_3 | 20 | 1003 | 971.95 | 971.24 | 961.03 | 961.03 | 961.03 | 963.61 | 961.03 | 961.03 | 961.03 | 961.03 | 961.03 | 961.03 | 961.03 | 961.03 | 0.00 | 0.46 |
| G_4 | 20 | 6447 | 6447.80 | 6445.10 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 0.00 | 0.87 |
| G_5 | 20 | 1015 | 1015.13 | 1009.15 | 1008.59 | 1007.05 | 1007.05 | 1007.96 | 1007.05 | 1007.05 | 1007.05 | 1007.05 | 1007.05 | 1007.05 | 1007.05 | 1007.05 | 0.00 | 0.47 |
| G_6 | 20 | 6516 | 6516.56 | 6516.56 | 6516.47 | 6516.47 | 6516.47 | 6537.74 | 6516.47 | 6516.47 | 6516.47 | 6516.47 | 6516.47 | 6516.47 | 6516.47 | 6516.47 | 0.00 | 0.86 |
| G_13 | 50 | 2493 | 2462.01 | 2471.07 | 2413.78 | 2408.41 | 2422.10 | 2406.43 | 2406.36 | 2406.36 | 2406.36 | 2406.36 | 2408.41 | 2406.36 | 2406.36 | 2412, 36 | 0.25 | 5.21 |
| G_14 | 50 | 9153 | 9141.69 | 9125.65 | 9119.03 | 9119.03 | 9119.86 | 9122.01 | 9119.03 | 9119.03 | 9119.03 | 9119.03 | 9119.03 | 9119.03 | 9119.03 | 9121, 71 | 0.03 | 11.86 |
| G_15 | 50 | 2623 | 2600.31 | 2606.72 | 2586.37 | 2586.37 | 2586.37 | 2618.03 | 2586.37 | 2586.84 | 2586.37 | 2586.37 | 2586.37 | 2586.37 | 2586.37 | 2588, 77 | 0.09 | 7.02 |
| G_16 | 50 | 2765 | 2745.04 | 2745.01 | 2741.50 | 2741.50 | 2730.08 | 2761.96 | 2720.43 | 2728.14 | 2729.08 | 2724.22 | 2724.22 | 2720.43 | 2720.43 | 2779, 72 | 2.18 | 6.97 |
| G_17 | 75 | 1767 | 1766.81 | 1762.05 | 1747.24 | 1749.50 | 1755.10 | 1757.21 | 1744.83 | 1736.09 | 1746.09 | 1734.53 | 1734.53 | 1734.53 | 1734.53 | 1770, 09 | 2.05 | 31.30 |
| G_18 | 75 | 2439 | 2439.40 | 2412.56 | 2373.63 | 2381.43 | 2385.52 | 2413.39 | 2371.49 | 2376.89 | 2369.65 | 2369.65 | 2371.48 | 2369.65 | 2369.65 | 2434, 78 | 2.75 | 16.43 |
| G_19 | 100 | 8751 | 8704.20 | 8685.71 | 8661.81 | 8675.16 | 8659.74 | 8687.31 | 8664.29 | 8667.26 | 8665.12 | 8662.94 | 8662.86 | 8661.81 | 8667.26 | 9173, 19 | 5.93 | 60.70 |
| G_20 | 100 | 4187 | 4166.03 | 4188.73 | 4047.55 | 4086.76 | 4061.64 | 4094.54 | 4039.49 | 4048.09 | 4044.78 | 4038.46 | 4037.9 | 4032.81 | 4038.45 | 4153, 74 | 3.00 | 56.60 |
| Avg. | 1.36 | 16.56 | ||||||||||||||||
| MRPERT: Modified version of procedure RPERT proposed by Salhi and Rand [52], TSVFM: Tabu search for the vehicle fleet mix problem (VFM). | ||||||||||||||||||
| Inst. | n | Salhi and Rand[52] | MRPERT Osman and Salhi[43] | TSVFM Osman and Salhi[43] | Taillard[58] | Gendreau et al.[20] | Wassan and Osman[65] | Renaud and Boctor[49] | Choi and Tcha[14] | Brandao[10] | Prins[48] | Liu et al.[37] | Penna et al.[45] | Subramanian et al. [56] | Simeonova et al. [53] | SA-LS | Dev% | CPU (s) |
| G_3 | 20 | 1003 | 971.95 | 971.24 | 961.03 | 961.03 | 961.03 | 963.61 | 961.03 | 961.03 | 961.03 | 961.03 | 961.03 | 961.03 | 961.03 | 961.03 | 0.00 | 0.46 |
| G_4 | 20 | 6447 | 6447.80 | 6445.10 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 6437.33 | 0.00 | 0.87 |
| G_5 | 20 | 1015 | 1015.13 | 1009.15 | 1008.59 | 1007.05 | 1007.05 | 1007.96 | 1007.05 | 1007.05 | 1007.05 | 1007.05 | 1007.05 | 1007.05 | 1007.05 | 1007.05 | 0.00 | 0.47 |
| G_6 | 20 | 6516 | 6516.56 | 6516.56 | 6516.47 | 6516.47 | 6516.47 | 6537.74 | 6516.47 | 6516.47 | 6516.47 | 6516.47 | 6516.47 | 6516.47 | 6516.47 | 6516.47 | 0.00 | 0.86 |
| G_13 | 50 | 2493 | 2462.01 | 2471.07 | 2413.78 | 2408.41 | 2422.10 | 2406.43 | 2406.36 | 2406.36 | 2406.36 | 2406.36 | 2408.41 | 2406.36 | 2406.36 | 2412, 36 | 0.25 | 5.21 |
| G_14 | 50 | 9153 | 9141.69 | 9125.65 | 9119.03 | 9119.03 | 9119.86 | 9122.01 | 9119.03 | 9119.03 | 9119.03 | 9119.03 | 9119.03 | 9119.03 | 9119.03 | 9121, 71 | 0.03 | 11.86 |
| G_15 | 50 | 2623 | 2600.31 | 2606.72 | 2586.37 | 2586.37 | 2586.37 | 2618.03 | 2586.37 | 2586.84 | 2586.37 | 2586.37 | 2586.37 | 2586.37 | 2586.37 | 2588, 77 | 0.09 | 7.02 |
| G_16 | 50 | 2765 | 2745.04 | 2745.01 | 2741.50 | 2741.50 | 2730.08 | 2761.96 | 2720.43 | 2728.14 | 2729.08 | 2724.22 | 2724.22 | 2720.43 | 2720.43 | 2779, 72 | 2.18 | 6.97 |
| G_17 | 75 | 1767 | 1766.81 | 1762.05 | 1747.24 | 1749.50 | 1755.10 | 1757.21 | 1744.83 | 1736.09 | 1746.09 | 1734.53 | 1734.53 | 1734.53 | 1734.53 | 1770, 09 | 2.05 | 31.30 |
| G_18 | 75 | 2439 | 2439.40 | 2412.56 | 2373.63 | 2381.43 | 2385.52 | 2413.39 | 2371.49 | 2376.89 | 2369.65 | 2369.65 | 2371.48 | 2369.65 | 2369.65 | 2434, 78 | 2.75 | 16.43 |
| G_19 | 100 | 8751 | 8704.20 | 8685.71 | 8661.81 | 8675.16 | 8659.74 | 8687.31 | 8664.29 | 8667.26 | 8665.12 | 8662.94 | 8662.86 | 8661.81 | 8667.26 | 9173, 19 | 5.93 | 60.70 |
| G_20 | 100 | 4187 | 4166.03 | 4188.73 | 4047.55 | 4086.76 | 4061.64 | 4094.54 | 4039.49 | 4048.09 | 4044.78 | 4038.46 | 4037.9 | 4032.81 | 4038.45 | 4153, 74 | 3.00 | 56.60 |
| Avg. | 1.36 | 16.56 | ||||||||||||||||
| MRPERT: Modified version of procedure RPERT proposed by Salhi and Rand [52], TSVFM: Tabu search for the vehicle fleet mix problem (VFM). | ||||||||||||||||||
Table 5.
Comparison of SA-LS with VRPSPD test instances
| Instance | n | Q | Salhi and Nagy [51] | Goksal et al. [22] | SA-LS | ||||
| Min | Avg | Max | Dev% | CPU (s) | |||||
| CMT1X | 50 | 160 | 601 | 466.77 | 487 | 494.4 | 503 | 4.33 | 2.65 |
| CMT1Y | 50 | 160 | 603 | 466.77 | 467 | 473.0 | 486 | 0.05 | 2.48 |
| CMT2X | 75 | 140 | 873 | 668.77 | 698 | 709.0 | 726 | 4.37 | 4.58 |
| CMT2Y | 75 | 140 | 924 | 663.25 | 672 | 683.6 | 699 | 1.32 | 5.32 |
| CMT3X | 100 | 200 | 923 | 721.27 | 731 | 740.8 | 753 | 1.35 | 19.00 |
| CMT3Y | 100 | 200 | 923 | 721.27 | 714 | 721.6 | 739 | -1.01 | 21.66 |
| CMT4X | 150 | 200 | 1178 | 852.46 | 884 | 891.0 | 899 | 3.70 | 50.35 |
| CMT4Y | 150 | 200 | 1178 | 852.46 | 849 | 853.2 | 857 | -0.41 | 50.06 |
| CMT5X | 199 | 200 | 1509 | 1029.25 | 1115 | 1134.0 | 1151 | 8.33 | 86.07 |
| CMT5Y | 199 | 200 | 1477 | 1029.25 | 1021 | 1042.8 | 1060 | -0.80 | 90.89 |
| CMT11X | 120 | 200 | 1500 | 833.92 | 916 | 937.2 | 978 | 9.84 | 34.47 |
| CMT11Y | 120 | 200 | 1500 | 830.39 | 785 | 803.0 | 835 | -5.47 | 38.87 |
| CMT12X | 100 | 200 | 820 | 644.7 | 674 | 700.0 | 725 | 4.54 | 14.87 |
| CMT12Y | 100 | 200 | 873 | 659.52 | 632 | 655.0 | 671 | -4.17 | 15.02 |
| Overall Avg. | 1063.00 | 745.72 | 760.36 | 774.19 | 791.57 | 1.86 | 31.16 | ||
| Instance | n | Q | Salhi and Nagy [51] | Goksal et al. [22] | SA-LS | ||||
| Min | Avg | Max | Dev% | CPU (s) | |||||
| CMT1X | 50 | 160 | 601 | 466.77 | 487 | 494.4 | 503 | 4.33 | 2.65 |
| CMT1Y | 50 | 160 | 603 | 466.77 | 467 | 473.0 | 486 | 0.05 | 2.48 |
| CMT2X | 75 | 140 | 873 | 668.77 | 698 | 709.0 | 726 | 4.37 | 4.58 |
| CMT2Y | 75 | 140 | 924 | 663.25 | 672 | 683.6 | 699 | 1.32 | 5.32 |
| CMT3X | 100 | 200 | 923 | 721.27 | 731 | 740.8 | 753 | 1.35 | 19.00 |
| CMT3Y | 100 | 200 | 923 | 721.27 | 714 | 721.6 | 739 | -1.01 | 21.66 |
| CMT4X | 150 | 200 | 1178 | 852.46 | 884 | 891.0 | 899 | 3.70 | 50.35 |
| CMT4Y | 150 | 200 | 1178 | 852.46 | 849 | 853.2 | 857 | -0.41 | 50.06 |
| CMT5X | 199 | 200 | 1509 | 1029.25 | 1115 | 1134.0 | 1151 | 8.33 | 86.07 |
| CMT5Y | 199 | 200 | 1477 | 1029.25 | 1021 | 1042.8 | 1060 | -0.80 | 90.89 |
| CMT11X | 120 | 200 | 1500 | 833.92 | 916 | 937.2 | 978 | 9.84 | 34.47 |
| CMT11Y | 120 | 200 | 1500 | 830.39 | 785 | 803.0 | 835 | -5.47 | 38.87 |
| CMT12X | 100 | 200 | 820 | 644.7 | 674 | 700.0 | 725 | 4.54 | 14.87 |
| CMT12Y | 100 | 200 | 873 | 659.52 | 632 | 655.0 | 671 | -4.17 | 15.02 |
| Overall Avg. | 1063.00 | 745.72 | 760.36 | 774.19 | 791.57 | 1.86 | 31.16 | ||
Table 6.
Comparison of SA-LS on set 1 test instances
| Intance | n | B | Avci and Topaloglu [5] | SA-LS | ||||||
| Min | Avg | CPU (s) | Min | Avg | Max | Dev% | CPU (s) | |||
| 1 | 10 | 2 | 620.2 | 620.2 | 17.2 | 607.7 | 618.3 | 626.1 | -2.02 | 0.0 |
| 2 | 10 | 2 | 588.5 | 588.5 | 14.7 | 585.6 | 587.5 | 590.7 | -0.49 | 0.0 |
| 3 | 15 | 3 | 445.1 | 445.1 | 22.7 | 415.3 | 433.1 | 445.5 | -6.71 | 0.1 |
| 4 | 15 | 4 | 437.1 | 437.1 | 24.5 | 417.1 | 435.3 | 441.5 | -4.57 | 0.1 |
| 5 | 20 | 3 | 494 | 498.9 | 27.1 | 480.3 | 493.3 | 503.4 | -2.77 | 0.2 |
| 6 | 20 | 4 | 542.7 | 551.9 | 26.7 | 539.4 | 561.8 | 578.1 | -0.60 | 0.2 |
| 7 | 35 | 3 | 1108.2 | 1123.4 | 56.6 | 1126.6 | 1140.2 | 1158.3 | 1.66 | 1.0 |
| 8 | 35 | 3 | 1586.5 | 1601.2 | 54.7 | 1614.5 | 1673.8 | 1711.6 | 1.76 | 0.7 |
| 9 | 50 | 3 | 964.4 | 990.2 | 91.4 | 943.9 | 948.3 | 960.5 | -2.12 | 2.3 |
| 10 | 50 | 2 | 1197.7 | 1228.6 | 95.8 | 1177.7 | 1192.2 | 1206.3 | -1.67 | 2.0 |
| 11 | 75 | 3 | 1642.2 | 1673.9 | 143.8 | 1538.4 | 1572.9 | 1609.8 | -6.32 | 7.4 |
| 12 | 75 | 2 | 973.1 | 1002.5 | 164.9 | 969.5 | 982.1 | 1007.3 | -0.37 | 6.9 |
| 13 | 100 | 2 | 1299.5 | 1353.5 | 288.5 | 1262.3 | 1279.7 | 1311.0 | -2.86 | 22.7 |
| 14 | 100 | 2 | 1658.2 | 1678.2 | 310.3 | 1525.6 | 1545.0 | 1580.0 | -8.00 | 19.2 |
| Overall Avg. | 968, 4 | 985, 2 | 95.6 | 943.1 | 961.7 | 980.7 | -2.51 | 4.5 | ||
| Intance | n | B | Avci and Topaloglu [5] | SA-LS | ||||||
| Min | Avg | CPU (s) | Min | Avg | Max | Dev% | CPU (s) | |||
| 1 | 10 | 2 | 620.2 | 620.2 | 17.2 | 607.7 | 618.3 | 626.1 | -2.02 | 0.0 |
| 2 | 10 | 2 | 588.5 | 588.5 | 14.7 | 585.6 | 587.5 | 590.7 | -0.49 | 0.0 |
| 3 | 15 | 3 | 445.1 | 445.1 | 22.7 | 415.3 | 433.1 | 445.5 | -6.71 | 0.1 |
| 4 | 15 | 4 | 437.1 | 437.1 | 24.5 | 417.1 | 435.3 | 441.5 | -4.57 | 0.1 |
| 5 | 20 | 3 | 494 | 498.9 | 27.1 | 480.3 | 493.3 | 503.4 | -2.77 | 0.2 |
| 6 | 20 | 4 | 542.7 | 551.9 | 26.7 | 539.4 | 561.8 | 578.1 | -0.60 | 0.2 |
| 7 | 35 | 3 | 1108.2 | 1123.4 | 56.6 | 1126.6 | 1140.2 | 1158.3 | 1.66 | 1.0 |
| 8 | 35 | 3 | 1586.5 | 1601.2 | 54.7 | 1614.5 | 1673.8 | 1711.6 | 1.76 | 0.7 |
| 9 | 50 | 3 | 964.4 | 990.2 | 91.4 | 943.9 | 948.3 | 960.5 | -2.12 | 2.3 |
| 10 | 50 | 2 | 1197.7 | 1228.6 | 95.8 | 1177.7 | 1192.2 | 1206.3 | -1.67 | 2.0 |
| 11 | 75 | 3 | 1642.2 | 1673.9 | 143.8 | 1538.4 | 1572.9 | 1609.8 | -6.32 | 7.4 |
| 12 | 75 | 2 | 973.1 | 1002.5 | 164.9 | 969.5 | 982.1 | 1007.3 | -0.37 | 6.9 |
| 13 | 100 | 2 | 1299.5 | 1353.5 | 288.5 | 1262.3 | 1279.7 | 1311.0 | -2.86 | 22.7 |
| 14 | 100 | 2 | 1658.2 | 1678.2 | 310.3 | 1525.6 | 1545.0 | 1580.0 | -8.00 | 19.2 |
| Overall Avg. | 968, 4 | 985, 2 | 95.6 | 943.1 | 961.7 | 980.7 | -2.51 | 4.5 | ||
Table 7.
Comparison of SA-LS on set 2 test instances
| Intance | n | B | Avci and Topaloglu [5] | SA-LS | ||||||
| Min | Avg | CPU (s) | Min | Avg | Max | Dev% | CPU (s) | |||
| 1 | 150 | 3 | 1499.4 | 1624.5 | 592.5 | 1468.4 | 1493.1 | 1520.3 | -2.07 | 157.8 |
| 2 | 150 | 3 | 2144.8 | 2152.5 | 548.9 | 2105.9 | 2168.6 | 2241.4 | -1.81 | 118.1 |
| 3 | 200 | 3 | 3673.1 | 3688.8 | 831.6 | 3188.5 | 3302.3 | 3335.5 | -13.19 | 116.0 |
| 4 | 200 | 2 | 2485.3 | 2682.5 | 919.4 | 2169.6 | 2181.8 | 2207.9 | -12.70 | 313.8 |
| 5 | 250 | 3 | 2639.6 | 2810.0 | 1448.1 | 2373.7 | 2428.6 | 2541.9 | -10.08 | 483.9 |
| 6 | 250 | 2 | 2549.8 | 2605.6 | 1521.3 | 2192.4 | 2219.6 | 2272.0 | -14.02 | 311.8 |
| 7 | 300 | 3 | 3205.0 | 3431.1 | 2440.4 | 2502.0 | 2523.7 | 2549.2 | -21.94 | 775.7 |
| 8 | 300 | 2 | 3252.8 | 3364.6 | 2518.2 | 2459.7 | 2533.5 | 2573.6 | -24.38 | 908.8 |
| 9 | 350 | 3 | 3457.9 | 3637.0 | 3770.1 | 3270.1 | 3380.9 | 3434.2 | -5.43 | 2385.5 |
| 10 | 350 | 2 | 3760.9 | 3897.5 | 3988.7 | 2262.0 | 2332.3 | 2409.0 | -39.85 | 1766.4 |
| 11 | 400 | 2 | 5809.5 | 6018.9 | 5662.4 | 3188.3 | 3236.3 | 3276.6 | -45.12 | 4466.4 |
| 12 | 400 | 2 | 4045.9 | 4447.0 | 5791.5 | 2871.9 | 2964.2 | 3048.6 | -29.02 | 3306.5 |
| 13 | 500 | 2 | 11008.8 | 12062.5 | 7200.0 | 7889.3 | 7948.0 | 8056.6 | -28.34 | 7789.4 |
| 14 | 550 | 2 | 12762.0 | 13046.3 | 7200.0 | 8987.9 | 9041.7 | 9089.8 | -29.57 | 7620.6 |
| Overall Avg. | 4449.6 | 4676.3 | 3173.8 | 3352.1 | 3411.1 | 3468.3 | -19.82 | 2180.0 | ||
| Intance | n | B | Avci and Topaloglu [5] | SA-LS | ||||||
| Min | Avg | CPU (s) | Min | Avg | Max | Dev% | CPU (s) | |||
| 1 | 150 | 3 | 1499.4 | 1624.5 | 592.5 | 1468.4 | 1493.1 | 1520.3 | -2.07 | 157.8 |
| 2 | 150 | 3 | 2144.8 | 2152.5 | 548.9 | 2105.9 | 2168.6 | 2241.4 | -1.81 | 118.1 |
| 3 | 200 | 3 | 3673.1 | 3688.8 | 831.6 | 3188.5 | 3302.3 | 3335.5 | -13.19 | 116.0 |
| 4 | 200 | 2 | 2485.3 | 2682.5 | 919.4 | 2169.6 | 2181.8 | 2207.9 | -12.70 | 313.8 |
| 5 | 250 | 3 | 2639.6 | 2810.0 | 1448.1 | 2373.7 | 2428.6 | 2541.9 | -10.08 | 483.9 |
| 6 | 250 | 2 | 2549.8 | 2605.6 | 1521.3 | 2192.4 | 2219.6 | 2272.0 | -14.02 | 311.8 |
| 7 | 300 | 3 | 3205.0 | 3431.1 | 2440.4 | 2502.0 | 2523.7 | 2549.2 | -21.94 | 775.7 |
| 8 | 300 | 2 | 3252.8 | 3364.6 | 2518.2 | 2459.7 | 2533.5 | 2573.6 | -24.38 | 908.8 |
| 9 | 350 | 3 | 3457.9 | 3637.0 | 3770.1 | 3270.1 | 3380.9 | 3434.2 | -5.43 | 2385.5 |
| 10 | 350 | 2 | 3760.9 | 3897.5 | 3988.7 | 2262.0 | 2332.3 | 2409.0 | -39.85 | 1766.4 |
| 11 | 400 | 2 | 5809.5 | 6018.9 | 5662.4 | 3188.3 | 3236.3 | 3276.6 | -45.12 | 4466.4 |
| 12 | 400 | 2 | 4045.9 | 4447.0 | 5791.5 | 2871.9 | 2964.2 | 3048.6 | -29.02 | 3306.5 |
| 13 | 500 | 2 | 11008.8 | 12062.5 | 7200.0 | 7889.3 | 7948.0 | 8056.6 | -28.34 | 7789.4 |
| 14 | 550 | 2 | 12762.0 | 13046.3 | 7200.0 | 8987.9 | 9041.7 | 9089.8 | -29.57 | 7620.6 |
| Overall Avg. | 4449.6 | 4676.3 | 3173.8 | 3352.1 | 3411.1 | 3468.3 | -19.82 | 2180.0 | ||
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