A Novel Spatial-Temporal Voronoi Diagram-Based Heuristic Approach for Large-Scale Vehicle Routing Optimization with Time Constraints
Abstract
:1. Introduction
2. Literature Review
2.1. Heuristic Algorithms for the VRPTW
2.2. Integration of Vehicle Routing Optimization and GIS
3. The VRPTW and the Spatial-Temporal Voronoi Diagram
3.1. The VRPTW
3.2. The Spatial-Temporal Voronoi Diagram for VRPTWs
4. The Spatial-Temporal Voronoi Diagram-Based Heuristic
4.1. The Construction Algorithm
4.2. Local Search
- 1-0 exchange move injects a node from the original place and inserts it after another. As Figure 5a shows, node b is removed from its current position and inserted before node i.
- 1-1 exchange move swaps positions of a pair of nodes. As Figure 5b shows, the places of nodes b and i are swapped.
- 2-Opt move swaps the ends of two routes after the positions of a pair of nodes. As Figure 5c illustrates, the partial routes after b and i are swapped.
4.2.1. The Spatial-Temporal Distance Decay Strategy
4.2.2. Time Warp Operation
4.2.3. Acceptance Criterion
4.3. Spatial-Temporal Features-Guided Search
4.3.1. Spatial-Temporal Features
- Time-window violation nodes
- Longest-distance nodes
- The smallest route’s nodes
4.3.2. Removal Algorithm
4.3.3. Insertion Algorithm
4.4. An Extension of the Solving Algorithm for MDVRPTW
5. Experiment and Comparison
5.1. Test VRPTW and MDVRPTW Problem Dataset
5.2. Parameter Tuning
Notation | Parameter | Setting |
---|---|---|
Nmax | the maximum number of iterations | 5000N |
Zmax | the maximum number of iterations to start the spatial-temporal features guided search | 100N |
kmax | the maximum searching Voronoi neighbors | 1, 2, 3, 4, 5 |
α | Voronoi distance decaying speed | (0, 4) |
5.3. The Results of the Solomon [15] and Gehring and Homberger [53] VRPTW Benchmarks
Instance | N | AGEA | HGSDAC | VRPEJ | STVDH | STVDH |
---|---|---|---|---|---|---|
1 run | Best 5 run | Best 10 run | AVG 10 run | Best 10 run | ||
R1 | 100 | 11.92|1211.43 | 11.92|1210.69 | 11.92|1214.67 | 11.92|1213.85 | 11.92|1212.65 |
R2 | 100 | 2.73|954.05 | 2.73|951.51 | 2.73|954.10 | 2.73|954.01 | 2.73|953.20 |
C1 | 100 | 10.0|828.38 | 10.0|828.38 | 10.0|828.38 | 10.0|828.38 | 10.0|828.38 |
C2 | 100 | 3.0|589.86 | 3.0|589.36 | 3.0|589.86 | 3.0|589.86 | 3.0|589.86 |
RC1 | 100 | 11.50|1384.83 | 11.5|1384.17 | 11.5|1387.81 | 11.5|1386.93 | 11.5|1386.01 |
RC2 | 100 | 3.25|1121.26 | 3.25|1119.24 | 3.25|1127.38 | 3.25|1126.49 | 3.25|1125.72 |
CNV | 405 | 405 | 405 | 405 | 405 | |
CTD | 57,254.73 | 57,195 | 57,366.96 | 57,341.97 | 57,305.10 | |
T(/min) | 180 | 2.68 | 3.62 | 0.91 | 0.9 | |
T2(/min) | 52.3 | 1.56 | 3.62 | 0.91 | 0.9 |
Instance | N | AGEA | HGSDAC | VRPEJ | STVDH | STVDH |
---|---|---|---|---|---|---|
1 run | Best 5 run | Best 10 run | AVG 10 run | Best 10 run | ||
R1 | 200 | 18.2|3640.11 | 18.2|3613.16 | 18.2|3664.28 | 18.2|3653.24 | 18.2|3617.54 |
R2 | 200 | 4.0|2941.99 | 4.0|2929.41 | 4.0|2938.53 | 4.0|2967.87 | 4.0|2938.53 |
C1 | 200 | 18.9|2721.90 | 18.9|2718.41 | 18.9|2749.18 | 18.9|2758.67 | 18.9|2721.47 |
C2 | 200 | 6.0|1833.36 | 6.0|1831.59 | 6.0|1880.47 | 6.0|1858.34 | 6.0|1835.52 |
RC1 | 200 | 18.0|3224.63 | 18.0|3180.48 | 18.0|3205.81 | 18.0|3218.83 | 18.0|3179.89 |
RC2 | 200 | 4.3|2554.33 | 4.3|2536.20 | 4.3|2574.92 | 4.3|2584.37 | 4.3|2544.44 |
CNV | 694 | 694 | 694 | 694 | 694 | |
CTD | 169,163 | 168,092 | 176,440.8 | 170,415.5 | 168,373.7 | |
T(/min) | 90 | 8.40 | 5.4 | 1.22 | 1.2 | |
T2(/min) | 26.7 | 4.9 | 5.4 | 1.22 | 1.2 | |
R1 | 400 | 36.4|8514.11 | 36.4|8402.57 | 36.4|8615.29 | 36.4|8598.39 | 36.4|8446.36 |
R2 | 400 | 8.0|6258.82 | 8.0|6152.92 | 8.0|6274.20 | 8.0|6279.13 | 8.0|6160.84 |
C1 | 400 | 37.6|7273.90 | 37.6|7170.47 | 37.6|7339.88 | 37.6|7514.01 | 37.6|7186.10 |
C2 | 400 | 11.7|3941.70 | 11.6|3950.95 | 11.7|4024.82 | 11.7|3995.21 | 11.7|3951.71 |
RC1 | 400 | 36.0|8088.46 | 36.0|7907.14 | 36.0|8107.86 | 36.0|8098.32 | 36.0|7952.00 |
RC2 | 400 | 8.40|5516.59 | 8.5|5215.21 | 8.4|5394.54 | 8.4|5393.49 | 8.4|5292.92 |
CNV | 1381 | 1381 | 1381 | 1381 | 1381 | |
CTD | 395,936 | 388,013 | 397,565.9 | 398,785.5 | 389,385.5 | |
T(/min) | 180 | 34.1 | 9.8 | 2.30 | 2.3 | |
T2(/min) | 53.3 | 19.8 | 9.8 | 2.30 | 2.3 | |
R1 | 600 | 54.5|18,781.79 | 54.5|18,023.18 | 54.5|18,620.73 | 54.5|18,587.90 | 54.5|18,237.74 |
R2 | 600 | 11.0|12,804.60 | 11.0|12,352.38 | 11.0|12,615.07 | 11.0|12,612.60 | 11.0|12,343.51 |
C1 | 600 | 57.3|14,236.86 | 57.4|14,058.46 | 57.3|14,605.53 | 57.3|14,585.55 | 57.3|14,271.58 |
C2 | 600 | 17.4|7729.80 | 17.4|7594.41 | 17.4|7748.47 | 17.4|7728.74 | 17.4|7589.10 |
RC1 | 600 | 55.0|16,767.72 | 55.0|16,097.05 | 55.0|16,529.63 | 55.0|16,524.77 | 55.0|16,203.93 |
RC2 | 600 | 11.4|11,311.81 | 11.5|10,511.86 | 11.4|10,879.26 | 11.4|10,824.00 | 11.4|10,626.35 |
CNV | 2066 | 2068 | 2068 | 2066 | 2066 | |
CTD | 816,326 | 786,793 | 809,986.9 | 808,635.65 | 792,722.1 | |
T(/min) | 270 | 99.4 | 16.2 | 3.91 | 3.9 | |
T2(/min) | 80.0 | 57.8 | 16.2 | 3.91 | 3.9 | |
R1 | 800 | 72.8|32,734.57 | 72.8|31,311.38 | 72.8|32,281.48 | 72.8|32,108.01 | 72.8|31,540.28 |
R2 | 800 | 15.0|20,618.21 | 15.0|19,933.39 | 15.0|20,448.88 | 15.0|20,339.05 | 15.0|19,969.61 |
C1 | 800 | 75.2|25,911.44 | 75.4|24,876.93 | 75.2|26,097.53 | 75.1|25,972.63 | 75.1|25,490.85 |
C2 | 800 | 23.4|11,835.72 | 23.3|11,475.05 | 23.4|11,897.31 | 23.3|11,826.41 | 23.3|11,621.87 |
RC1 | 800 | 72.0|33,975.61 | 72.0|29,404.32 | 72.0|31,071.16 | 72.0|30,904.01 | 72.0|30,390.41 |
RC2 | 800 | 15.5|17,536.54 | 15.4|16,495.82 | 15.5|16,878.69 | 15.4|16,733.78 | 15.4|16,467.01 |
CNV | 2739 | 2739 | 2739 | 2736 | 2736 | |
CTD | 1,424,321 | 1,334,963 | 1,386,750.4 | 1,378,838.8 | 1,354,800 | |
T(/min) | 360 | 215 | 24.8 | 5.82 | 5.8 | |
T2(/min) | 106.6 | 125.1 | 24.8 | 5.82 | 5.8 | |
R1 | 91.9|51,414.26 | 91.9|47,759.66 | 91.9|49,741.43 | 91.9|49,396.91 | 91.9|48,523.49 | |
R2 | 19.0|30,804.79 | 19.0|29,076.45 | 19.0|29,871.68 | 19.0|29,595.30 | 19.0|29,092.01 | |
C1 | 94.2|43,111.60 | 94.1|41,572.86 | 94.1|43,089.45 | 94.1|42,682.27 | 94.1|41,977.06 | |
C2 | 29.3|16,810.22 | 28.8|16,796.45 | 29.0|117,340.13 | 28.9|17,174.73 | 28.8|16,877.68 | |
RC1 | 90.0|46,753.61 | 90.0|44,333.40 | 90.0|46,152.97 | 90.0|45,824.65 | 90.0|44,974.63 | |
RC2 | 18.4|25,588.52 | 18.2|24,131.12 | 18.4|24,951.31 | 18.3|24,655.48 | 18.2|24,248.11 | |
CNV | 3428 | 3420 | 3421 | 3419 | 3419 | |
CTD | 2,144,830 | 2,036,700 | 2,111,469.7 | 2,093,293.5 | 2,056,930 | |
T(/min) | 450 | 349 | 34.5 | 7.75 | 7.7 | |
T2(/min) | 133.3 | 203.1 | 34.5 | 7.75 | 7.7 |
5.4. The Results of Large Scale VRPTW and MDVRPTW Problem in Shanghai, China
Instance | Type | N | M | D | Q | STVDH | |||
---|---|---|---|---|---|---|---|---|---|
Savg 10 (/km) | Tavg 10 (/min) | SBest 10(/km) | TBest 10 (/min) | ||||||
Sh1a | VRPTW | 2000 | 1 | 200 | 2000 | 59.3|3024.870 | 15.7 | 58|2996.104 | 15.9 |
Sh2a | VRPTW | 4000 | 1 | 300 | 2000 | 119.8|4674.122 | 28.3 | 118|4598.592 | 30.6 |
Sh3a | VRPTW | 6000 | 1 | 400 | 2000 | 171.0|6145.245 | 60.0 | 169|6052.084 | 65.9 |
Sh4a | VRPTW | 8000 | 1 | 500 | 2000 | 220.3|7511.361 | 93.6 | 218|7372.345 | 98.9 |
Sh5a | VRPTW | 10,000 | 1 | 600 | 2000 | 276.8|8829.127 | 120.8 | 274|8681.006 | 128.8 |
CNV | 847.1 | 837 | |||||||
CTD(/km) | 30,184.725 | 29,701.132 | |||||||
Gap to Savg | −1.60% | ||||||||
Total Time (/min) | 318.4 | 340.1 | |||||||
Sh1b | MDVRPTW | 2000 | 2 | 200 | 2000 | 70.2|3045.601 | 16.1 | 69|2999.616 | 16.9 |
Sh2b | MDVRPTW | 4000 | 3 | 300 | 2000 | 127.9|4265.365 | 29.8 | 126|4174.144 | 31.5 |
Sh3b | MDVRPTW | 6000 | 4 | 400 | 2000 | 172.1|6247.152 | 62.2 | 169|5260.810 | 69.6 |
Sh4b | MDVRPTW | 8000 | 5 | 500 | 2000 | 237.4|6849.848 | 98.4 | 235|6238.438 | 105.4 |
Sh5b | MDVRPTW | 10,000 | 6 | 600 | 2000 | 298.8|6845.606 | 138.9 | 293|6784.751 | 139.5 |
CNV | 906.4 | 892 | |||||||
CTD (/km) | 26,353.572 | 26,289.418 | |||||||
Gap to Savg | −1.73% | ||||||||
Total Time (/min) | 345.4 | 362.9 |
Instance | Type | N | M | STVDH | ArcGIS | VRPEJ | |||
---|---|---|---|---|---|---|---|---|---|
SBest 10(/km) | TBest 10 (/min) | SBest 10(/km) | TBest 10 (/min) | SBest 10(/km) | TBest 10 (/min) | ||||
Sh1a | VRPTW | 1 | 2000 | 58|2996.104 | 15.9 | 62|3296.104 | 60.5 | 59|3285.404 | 50.8 |
Sh2a | VRPTW | 1 | 4000 | 118|4598.592 | 30.6 | 120|4022.248 | 125.4 | 126|5234.168 | 122.2 |
Sh3a | VRPTW | 1 | 6000 | 169|6052.084 | 65.9 | 173|6738.745 | 197.2 | 190|6948.372 | 242.7 |
Sh4a | VRPTW | 1 | 8000 | 218|7372.345 | 98.9 | 223|8045.327 | 254.8 | 242|8438.982 | 290.8 |
Sh5a | VRPTW | 1 | 10,000 | 274|8681.006 | 128.8 | 278|9874.135 | 407.2 | 312|9814.247 | 428.8 |
Sh1b | MDVRPTW | 2 | 2000 | 69|2999.616 | 16.9 | 72|3308.245 | 60.5 | 74|3374.126 | 58.1 |
Sh2b | MDVRPTW | 3 | 4000 | 126|4174.144 | 31.5 | 124|4610.283 | 125.4 | 132|4878.785 | 142.3 |
Sh3b | MDVRPTW | 4 | 6000 | 169|5260.810 | 69.6 | 174|6645.397 | 197.2 | 180|7013.522 | 248.5 |
Sh4b | MDVRPTW | 5 | 8000 | 235|6238.438 | 105.4 | 239|6982.136 | 254.8 | 260|6989.971 | 280.5 |
Sh5b | MDVRPTW | 6 | 10,000 | 293|6784.751 | 139.5 | 299|7525.189 | 568.1 | 339|7881.788 | 490.6 |
CNV | 1729 | 1764 | 1893 | ||||||
CTD (/km) | 55,990.550 | 62,047.809 | 63,659.365 | ||||||
Gap to STVDH | 10.82% | 13.70% | |||||||
Total Computing Time (/min) | 703.0 | 2141.1 | 2355.3 |
6. Discussion
6.1. Impact of the Spatial-Temporal Voronoi Diagram
6.2. Spatial-Temporal Proximity Analysis on the Best Found Results
- The number of larger Voronoi distances is only a small proportion of the best found results. As Figure 11 displays, the percentages of Voronoi distances greater than three in the VRPTW and MDRPTW solutions are only 1.77% and 1.58%, respectively. Such a distribution agrees with the setting of parameter (3) in Section 5.2. Therefore, the spatial-temporal Voronoi neighborhood is very typical in the found solution.
- The percentage decreases sharply as the Voronoi distance increases. For the large-scale VRPTW dataset (Sh1a–Sh5a), the percentages for Voronoi distances 1, 2, 3, and >= 4 are 59.44%, 30.13%, 8.66%, and 1.77%, respectively, which is similar to the distribution in the MDVRPTW solution. This verifies that there is a spatial-temporal local compact structure in the routes of best found solutions. Compared with the theoretical searching intensity (as Equation (7) in Section 4.2.1 where ) in the Voronoi distance decay strategy, these percentages systematically slightly shift to small Voronoi distances. This result demonstrates the effectiveness of the Voronoi distance-decay strategy, which searches more on near neighbors in the local search but still spends necessary efforts on far neighbors.
7. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Tu, W.; Li, Q.; Fang, Z.; Zhou, B. A Novel Spatial-Temporal Voronoi Diagram-Based Heuristic Approach for Large-Scale Vehicle Routing Optimization with Time Constraints. ISPRS Int. J. Geo-Inf. 2015, 4, 2019-2044. https://doi.org/10.3390/ijgi4042019
Tu W, Li Q, Fang Z, Zhou B. A Novel Spatial-Temporal Voronoi Diagram-Based Heuristic Approach for Large-Scale Vehicle Routing Optimization with Time Constraints. ISPRS International Journal of Geo-Information. 2015; 4(4):2019-2044. https://doi.org/10.3390/ijgi4042019
Chicago/Turabian StyleTu, Wei, Qingquan Li, Zhixiang Fang, and Baoding Zhou. 2015. "A Novel Spatial-Temporal Voronoi Diagram-Based Heuristic Approach for Large-Scale Vehicle Routing Optimization with Time Constraints" ISPRS International Journal of Geo-Information 4, no. 4: 2019-2044. https://doi.org/10.3390/ijgi4042019