The Shortest Path Problems in Battery-Electric Vehicle Dispatching with Battery Renewal
Abstract
:1. Introduction
2. The Shortest Path Problems in Electric Vehicle Dispatching
2.1. The Shortest Path Problem in Electric Transit Bus Scheduling
2.2. The Shortest Path Problem in Electric Truck Routing with Time Windows
3. The Label-Correcting Algorithm
3.1. The Label Operations for Electric Bus Scheduling
3.2. The Label Operations for Electric Truck Routing
4. Computational Experiments
4.1. The Shortest Path in Electric Bus Scheduling with Battery Renewal
4.2. The Shortest Path in Electric Truck Routing with Battery Renewal
5. Conclusions and Future Research
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Objective Function and Restrictions | Variable Range | Equation No. |
---|---|---|
(a1) | ||
st | ||
(a2) | ||
(a3) | ||
(a4) | ||
(a5) | ||
(a6) | ||
(a7) | ||
Objective Function and Restrictions | Variable Range | Equation No. |
---|---|---|
(b1) | ||
st | ||
(a2) − (a7) | (b2) | |
(b3) | ||
(b4) | ||
(b5) | ||
(b6) | ||
(b7) |
Instance | # of Trips | # of Time-Expanded Nodes | Description |
---|---|---|---|
S Sa Sb Sc | 242 | 952 951 943 930 | published weekend schedules randomly-generated based S same as above same as above |
W Wa Wb Wc | 947 | 1132 1136 1136 1130 | published weekday schedules randomly-generated based W same as above same as above |
W1 W1a W1b W1c | 463 | 1073 1087 1074 1084 | half of W randomly-generated based W1 same as above same as above |
W2 W2a W2b W2c | 484 | 1132 1131 1124 1126 | the other half of W randomly-generated based W2 same as above same as above |
Instance | Dual | Max Distance 120 km | Max Distance 150 km | ||||||
---|---|---|---|---|---|---|---|---|---|
Optimal Objective Value | CPLEX | Label Correcting | Optimal Objective Value | CPLEX | Label Correcting | ||||
CPU (s) | Labels | CPU (s) | CPU (s) | Labels | CPU (s) | ||||
S Sa Sb Sc | 1 2 1 2 1 2 1 2 | –822,351 –819,804 –707,440 –705,225 –712,234 –724,879 –751,972 –790,796 | 117.94 1167.49 26.77 39.38 45.25 26.09 101.99 33.72 | 2875 2668 1735 2065 1917 2134 2305 2283 | 0.72 0.72 0.53 0.52 0.54 0.54 0.57 0.56 | –824,405 –829,648 –707,440 –705,225 –712,234 –724,879 –755,469 –790,796 | 141.92 61.65 27.79 32.6 38.13 29.23 42.92 40.46 | 2897 2739 1735 2066 1919 2134 2511 2302 | 0.74 0.78 0.55 0.58 0.56 0.6 0.6 0.58 |
W1 W1a W1b W1c | 1 2 1 2 1 2 1 2 | –1,175,252 –1,185,467 –1,190,403 –1,140,363 –1,180,257 –1,133,505 –1,226,202 –1,160,532 | 2927.46 338.88 262.11 328.97 738.82 249.87 248.59 + | 3239 3012 3123 3075 3431 2702 3116 4126 | 1.64 1.62 1.5 1.46 1.46 1.35 1.69 1.53 | –1,175,252 –1,185,467 –1,190,403 –1,140,363 –1,181,390 –1,133,505 –1,226,202 –1,162,385 | + 273.81 273.04 256.57 298.17 808.2 251.33 1052.89 | 3287 2987 3123 3078 3639 2717 3118 4513 | 1.74 1.71 1.58 1.56 1.69 1.41 1.8 1.65 |
W2 W2a W2b W2c | 1 2 1 2 1 2 1 2 | –1,241,010 –1,250,571 –1,236,314 –1,293,550 –1,271,817 –1,228,527 –1,321,741 –1,279,684 | 1142.48 339.46 314.43 421.19 3565.16 1645.09 376.29 274.28 | 3060 3133 3492 2899 4166 3831 3967 3681 | 1.69 1.65 1.61 1.62 1.53 1.56 1.81 1.84 | –1,241,010 –1,250,571 –1236,314 –1293,550 –1286,476 –1235,273 –1,321,741 –1,279,684 | + 593.26 481.85 1513.48 505.13 424.58 362.13 311.64 | 3195 3205 3548 2924 4906 4133 3990 3742 | 1.82 1.74 1.66 1.68 1.58 1.67 1.92 1.97 |
W Wa Wb Wc | 1 2 1 2 1 2 1 2 | –1,614,504 –1,574,276 –1,465,572 –1,513,933 –1,484,716 –1,469,219 –1,490,073 –1,518,417 | * * * * * * * * | 8333 7751 7953 8779 13,329 8644 8952 12,600 | 8.58 8.13 7.47 8.04 8.24 7.99 8.04 8.72 | –1,628,219 –1,590,012 –1,471,389 –1,518,442 –1,495,817 –1,502,780 –1,498,017 –1,526,627 | * * * * * * * * | 9104 8420 8228 9101 16,182 10,332 10,500 14,393 | 9.65 8.96 8.21 9.07 9.42 8.74 9.36 9.74 |
Instance | # of Customers | # of Time-Expanded Nodes | Instance | # of Customers | # of Time-Expanded Nodes | |||
---|---|---|---|---|---|---|---|---|
Time Windows | Time Windows | |||||||
[5, 20] min | [5, 40] min | [5, 60] min | [5, 20] min | [5, 40] min | ||||
50a | 50 | 620 | 665 | 685 | 150a | 150 | 662 | 695 |
50b | 50 | 585 | 666 | 687 | 150b | 150 | 636 | 677 |
50c | 50 | 623 | 661 | 681 | 150c | 150 | 652 | 716 |
50d | 50 | 597 | 641 | 665 | 150d | 150 | 638 | 696 |
50e | 50 | 547 | 644 | 653 | 150e | 150 | 665 | 686 |
100a | 100 | 644 | 656 | 662 | 200a | 200 | 667 | 678 |
100b | 100 | 634 | 639 | 663 | 200b | 200 | 664 | 680 |
100c | 100 | 655 | 693 | 696 | 200c | 200 | 633 | 680 |
100d | 100 | 645 | 680 | 700 | 200d | 200 | 645 | 660 |
100e | 100 | 658 | 683 | 697 | 200e | 200 | 656 | 694 |
Instance | Dual | Max Distance 100 km | Max Distance 130 km | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Optimal Objective Value | CPLEX | State Relaxation | Optimal Objective Value | CPLEX | State Relaxation | ||||||
Optimal Gap | CPU (s) | Labels | CPU (s) | Optimal Gap | CPU (s) | Labels | CPU (s) | ||||
50a | 1 | −6220 | 0.00 | 0.63 | 1744 | 0.02 | −6220 | 0.00 | 0.70 | 1780 | 0.02 |
2 | −5314 | 0.00 | 0.63 | 1491 | 0.02 | −5314 | 0.00 | 0.61 | 1539 | 0.02 | |
50b | 1 | −8742 | 0.00 | 0.57 | 2896 | 0.01 | −8742 | 0.00 | 0.57 | 2904 | 0.01 |
2 | −8320 | 0.00 | 0.55 | 3204 | 0.02 | −8320 | 0.00 | 0.57 | 3208 | 0.02 | |
50c | 1 | −7195 | 0.00 | 0.50 | 2657 | 0.01 | −7195 | 0.00 | 0.50 | 2771 | 0.02 |
2 | −7487 | 0.00 | 0.53 | 2629 | 0.01 | −7487 | 0.00 | 0.49 | 2824 | 0.01 | |
50d | 1 | −12,234 | 0.00 | 0.54 | 3412 | 0.02 | −12,234 | 0.00 | 0.54 | 3846 | 0.02 |
2 | −11,807 | 0.00 | 0.59 | 4405 | 0.02 | −11,807 | 0.00 | 0.56 | 4841 | 0.02 | |
50e | 1 | −7986 | 0.00 | 0.47 | 2163 | 0.01 | −7986 | 0.00 | 0.49 | 2404 | 0.01 |
2 | −7962 | 0.00 | 0.48 | 2287 | 0.01 | −7962 | 0.00 | 0.45 | 2536 | 0.01 | |
100a | 1 | −11,773 | 0.00 | 1.66 | 5088 | 0.04 | −11,773 | 0.00 | 1.45 | 5483 | 0.04 |
2 | −12,385 | 0.00 | 1.37 | 5364 | 0.05 | −12,385 | 0.00 | 1.35 | 5754 | 0.04 | |
100b | 1 | −12,292 | 0.00 | 3.84 | 9008 | 0.09 | −13,195 | 0.00 | 1.55 | 9588 | 0.1 |
2 | −10,774 | 0.00 | 5.41 | 9615 | 0.10 | −11,494 | 0.00 | 1.59 | 10,277 | 0.11 | |
100c | 1 | −11,655 | 0.00 | 1.27 | 6211 | 0.05 | −11,655 | 0.00 | 1.23 | 6446 | 0.06 |
2 | −9025 | 0.00 | 1.33 | 5253 | 0.04 | −9025 | 0.00 | 1.38 | 5539 | 0.04 | |
100d | 1 | −13,405 | 0.00 | 1.70 | 9543 | 0.08 | −13,405 | 0.00 | 1.55 | 9816 | 0.09 |
2 | −12,576 | 0.00 | 1.53 | 10,192 | 0.10 | −12,576 | 0.00 | 1.67 | 10,391 | 0.1 | |
100e | 1 | −13,473 | 0.00 | 1.61 | 10,546 | 0.10 | −13,473 | 0.00 | 1.50 | 11,208 | 0.11 |
2 | −14,937 | 0.00 | 1.57 | 8134 | 0.08 | −14,937 | 0.00 | 1.53 | 8628 | 0.08 | |
150a | 1 | −14,980 | 0.00 | 4.99 | 20,942 | 0.84 | −16,159 | 0.00 | 3.48 | 21,548 | 0.84 |
2 | −16,327 | 0.00 | 6.80 | 18,039 | 0.75 | −16,883 | 0.00 | 4.66 | 18,726 | 0.75 | |
150b | 1 | −15,493 | 0.00 | 4.56 | 12,588 | 0.14 | −15,493 | 0.00 | 4.54 | 12,913 | 0.15 |
2 | −14,026 | 0.00 | 6.85 | 9966 | 0.13 | −14,026 | 0.00 | 5.86 | 10,167 | 0.14 | |
150c | 1 | −17,062 | 0.00 | 3.18 | 14,051 | 0.62 | −17,062 | 0.00 | 5.24 | 14,118 | 0.65 |
2 | −16,292 | 0.00 | 4.13 | 18,626 | 1.50 | −16,292 | 0.00 | 4.32 | 18,738 | 1.58 | |
150d | 1 | −18,158 | 0.00 | 10.70 | 29,828 | 3.36 | −19,656 | 0.00 | 4.61 | 31,441 | 2.34 |
2 | −17,744 | 0.00 | 12.90 | 19,941 | 1.48 | −18,587 | 0.00 | 5.99 | 26,233 | 1.87 | |
150e | 1 | −17,925 | 0.00 | 43.71 | 23,911 | 0.42 | −19,214 | 0.00 | 11.70 | 26,744 | 0.44 |
2 | −19,397 | 0.00 | 24.29 | 23,281 | 0.38 | −21,270 | 0.00 | 6.96 | 25,491 | 0.39 | |
200a | 1 | −17,718 | 0.00 | 15.93 | 25,200 | 2.40 | −19,265 | 0.00 | 13.09 | 24,390 | 0.53 |
2 | −18,891 | 0.00 | 24.28 | 30,553 | 2.68 | −18,891 | 0.00 | 10.14 | 26,201 | 2.37 | |
200b | 1 | −21,300 | 0.00 | 13.07 | 26,969 | 0.53 | −22,216 | 0.00 | 7.18 | 27,031 | 0.54 |
2 | −22,300 | 0.00 | 12.21 | 20,303 | 0.36 | −23,112 | 0.00 | 6.95 | 20,396 | 0.38 | |
200c | 1 | −18,336 | 0.00 | 14.00 | 23,512 | 1.29 | −19,260 | 0.00 | 6.19 | 25,316 | 1.43 |
2 | −17,194 | 0.00 | 14.90 | 25,744 | 2.46 | −18,212 | 0.00 | 6.54 | 26,967 | 2.46 | |
200d | 1 | −19,536 | 0.00 | 27.92 | 46,037 | 4.80 | −20,742 | 0.00 | 67.16 | 48,768 | 4.93 |
2 | −20,315 | 0.00 | 92.37 | 46,088 | 2.80 | −20,607 | 0.00 | 58.97 | 46,342 | 5.51 | |
200e | 1 | −20,541 | 0.00 | 14.92 | 30,201 | 0.62 | −20,541 | 0.00 | 18.21 | 30,808 | 0.65 |
2 | −21,171 | 0.00 | 36.28 | 20,917 | 1.00 | −22,303 | 0.00 | 21.47 | 21,413 | 0.98 |
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Huang, M.; Li, J.-Q. The Shortest Path Problems in Battery-Electric Vehicle Dispatching with Battery Renewal. Sustainability 2016, 8, 607. https://doi.org/10.3390/su8070607
Huang M, Li J-Q. The Shortest Path Problems in Battery-Electric Vehicle Dispatching with Battery Renewal. Sustainability. 2016; 8(7):607. https://doi.org/10.3390/su8070607
Chicago/Turabian StyleHuang, Minfang, and Jing-Quan Li. 2016. "The Shortest Path Problems in Battery-Electric Vehicle Dispatching with Battery Renewal" Sustainability 8, no. 7: 607. https://doi.org/10.3390/su8070607