Simplified Swarm Optimization for the Heterogeneous Fleet Vehicle Routing Problem with Time-Varying Continuous Speed Function
Abstract
:1. Introduction
2. Literature Review
2.1. Time Dependent Vehicle Routing Problem with Time-Varying Speeds
2.2. Heterogeneous Fixed Fleet Vehicle Routing Problem
3. Problem Statement
3.1. Time Dependent Vehicle Routing Problem with Time-Varying Speeds of Continuous Function
3.2. Heterogeneous Fixed Fleet Vehicle Routing Problem
- The depot has a demand equal to zero
- Each customer location is serviced from only one vehicle
- Each customer’s delivery must arrive within the time window
- The number of each type of vehicle in routing cannot exceed each type of vehicle; the maximum available number is
- Each vehicle shall not exceed its maximum load capacity
- The total delivery time of each vehicle shall not exceed 9 h
3.3. Fitness Function and Mathematical Model
4. Simplified Swarm Optimization
4.1. Simplified Swarm Optimization
4.2. Example for Code and Decode
4.3. SSO Algorithm Pseudo Code
STEP 0. Initialize parameters; let , randomly initial solution . |
Each variable for is , there are in total vehicles, subscript n stands for n customer nodes, subscript m stands for m particle size, stands for customer i needs, stands for customer i time window. |
STEP 1. Let , first row of solution matrix. |
STEP 1.1 Let |
STEP 1.2 Find |
STEP 1.3.1 Rank get , then read the distance link node i and node j, add the vehicle fix cost; |
STEP 1.3.2 calculate arrive time for each node, and departure time check whether the arrival time is in the time window. |
STEP 1.3.3 calculate variable cost for each node, |
STEP 1.3.4 check total weight not over |
STEP 1.4 Calculate for vehicle |
STEP 1.5 |
if |
, go back to STEP 1.2 |
Otherwise |
Calculate |
If |
Otherwise |
STEP 2. |
if |
, go back to STEP 1.1 |
Otherwise |
If |
Otherwise |
STEP 3. |
if and CPU time is not met, randomly |
switch |
Case1 |
Case2 |
Case3 |
go back to STEP1 |
Otherwise |
Halt |
5. Computational Experiments
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
D | Dem | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 83 | 80 | 32 | 86 | 79 | 57 | 57 | 44 | 86 | 61 | 0 | 0 | 9 |
82 | 0 | 25 | 88 | 135 | 9 | 87 | 47 | 97 | 96 | 29 | 403 | 1.45 | 7.16 | |
80 | 24 | 0 | 90 | 113 | 31 | 98 | 28 | 80 | 111 | 37 | 411 | 0.73 | 6.12 | |
32 | 88 | 90 | 0 | 117 | 82 | 48 | 75 | 74 | 77 | 64 | 596 | 0.75 | 6.29 | |
86 | 135 | 112 | 116 | 0 | 139 | 140 | 91 | 44 | 169 | 125 | 582 | 0.24 | 6.1 | |
78 | 9 | 32 | 82 | 139 | 0 | 79 | 49 | 98 | 87 | 23 | 212 | 0.26 | 5.73 | |
58 | 87 | 98 | 48 | 140 | 79 | 0 | 95 | 98 | 39 | 65 | 330 | 0.96 | 6.28 | |
57 | 47 | 28 | 75 | 91 | 49 | 94 | 0 | 55 | 113 | 38 | 687 | 0.45 | 5.91 | |
43 | 97 | 78 | 73 | 44 | 98 | 97 | 55 | 0 | 126 | 83 | 210 | 0.89 | 6.55 | |
87 | 96 | 111 | 77 | 169 | 86 | 39 | 113 | 127 | 0 | 78 | 330 | 0.88 | 6.19 | |
59 | 30 | 38 | 62 | 124 | 22 | 63 | 39 | 83 | 76 | 0 | 697 | 0.84 | 6.51 |
D | Dem | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 93 | 53 | 48 | 34 | 63 | 80 | 99 | 51 | 16 | 31 | 0 | 0 | 9 |
93 | 0 | 46 | 55 | 106 | 106 | 139 | 9 | 140 | 109 | 124 | 187 | 0.96 | 6.47 | |
53 | 46 | 0 | 33 | 73 | 92 | 114 | 47 | 98 | 67 | 84 | 806 | 0.33 | 6.12 | |
48 | 54 | 33 | 0 | 53 | 62 | 86 | 59 | 96 | 63 | 77 | 672 | 1.16 | 6.62 | |
34 | 106 | 73 | 53 | 0 | 38 | 50 | 110 | 67 | 40 | 41 | 465 | 1.01 | 6.42 | |
63 | 106 | 92 | 61 | 38 | 0 | 34 | 114 | 103 | 76 | 77 | 628 | 1.16 | 6.94 | |
80 | 139 | 113 | 86 | 50 | 34 | 0 | 143 | 113 | 87 | 86 | 735 | 1.1 | 6.55 | |
99 | 9 | 47 | 59 | 110 | 114 | 143 | 0 | 144 | 113 | 129 | 207 | 1.23 | 6.7 | |
51 | 140 | 98 | 96 | 67 | 103 | 113 | 145 | 0 | 35 | 29 | 824 | 0.3 | 5.9 | |
16 | 108 | 67 | 63 | 40 | 76 | 87 | 113 | 35 | 0 | 18 | 348 | 0.97 | 6.19 | |
31 | 124 | 83 | 77 | 42 | 77 | 87 | 130 | 29 | 19 | 0 | 672 | 0.23 | 5.79 |
D | Dem | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 48 | 51 | 34 | 95 | 59 | 32 | 71 | 14 | 97 | 59 | 0 | 0 | 9 |
48 | 0 | 91 | 14 | 120 | 71 | 16 | 95 | 38 | 119 | 101 | 691 | 1.49 | 6.92 | |
51 | 91 | 0 | 78 | 137 | 51 | 77 | 113 | 64 | 150 | 24 | 692 | 0.67 | 6.48 | |
34 | 14 | 78 | 0 | 106 | 65 | 3 | 81 | 25 | 106 | 88 | 190 | 0.85 | 6.04 | |
95 | 119 | 138 | 106 | 0 | 150 | 103 | 25 | 84 | 30 | 136 | 613 | 0.65 | 6.28 | |
58 | 71 | 51 | 64 | 152 | 0 | 66 | 127 | 66 | 152 | 72 | 528 | 0.28 | 5.62 | |
32 | 16 | 77 | 3 | 104 | 66 | 0 | 78 | 22 | 103 | 87 | 375 | 1.3 | 6.56 | |
71 | 94 | 113 | 81 | 25 | 124 | 78 | 0 | 59 | 35 | 111 | 718 | 1.15 | 6.59 | |
14 | 38 | 64 | 25 | 84 | 66 | 22 | 59 | 0 | 85 | 73 | 203 | 1.34 | 6.46 | |
97 | 118 | 149 | 105 | 30 | 151 | 102 | 35 | 85 | 0 | 151 | 414 | 0.86 | 6.35 | |
59 | 100 | 25 | 88 | 136 | 73 | 86 | 111 | 73 | 150 | 0 | 168 | 0.95 | 6.31 |
D | Dem | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 50 | 24 | 34 | 49 | 86 | 24 | 28 | 52 | 73 | 93 | 0 | 0 | 9 |
51 | 0 | 36 | 19 | 33 | 115 | 39 | 32 | 45 | 115 | 124 | 169 | 0.56 | 5.91 | |
24 | 36 | 0 | 22 | 31 | 83 | 6 | 20 | 48 | 84 | 92 | 508 | 0.92 | 6.22 | |
35 | 19 | 22 | 0 | 30 | 102 | 27 | 13 | 33 | 99 | 112 | 702 | 1.09 | 6.55 | |
49 | 33 | 31 | 30 | 0 | 105 | 35 | 36 | 61 | 111 | 116 | 406 | 0.66 | 6.17 | |
86 | 117 | 84 | 103 | 106 | 0 | 80 | 101 | 128 | 59 | 12 | 269 | 0.39 | 5.63 | |
25 | 40 | 6 | 27 | 36 | 79 | 0 | 25 | 52 | 80 | 88 | 279 | 1.02 | 6.62 | |
28 | 32 | 20 | 13 | 35 | 100 | 25 | 0 | 31 | 96 | 109 | 659 | 1.36 | 6.76 | |
51 | 45 | 48 | 33 | 60 | 128 | 52 | 31 | 0 | 123 | 137 | 215 | 0.19 | 5.62 | |
73 | 114 | 84 | 99 | 111 | 58 | 80 | 96 | 123 | 0 | 66 | 589 | 0.71 | 6.19 | |
94 | 126 | 93 | 112 | 117 | 13 | 89 | 109 | 137 | 66 | 0 | 541 | 1.32 | 6.97 |
D | Dem | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 76 | 53 | 84 | 35 | 76 | 63 | 31 | 80 | 63 | 74 | 0 | 0 | 9 |
77 | 0 | 95 | 26 | 55 | 141 | 22 | 50 | 92 | 78 | 148 | 575 | 0.37 | 6.12 | |
53 | 94 | 0 | 113 | 42 | 111 | 91 | 76 | 128 | 32 | 97 | 675 | 1.13 | 6.57 | |
84 | 26 | 113 | 0 | 73 | 142 | 22 | 53 | 72 | 101 | 153 | 194 | 1.33 | 6.92 | |
35 | 55 | 43 | 73 | 0 | 108 | 52 | 41 | 97 | 35 | 99 | 438 | 0.63 | 6.05 | |
76 | 141 | 111 | 143 | 108 | 0 | 122 | 92 | 94 | 131 | 39 | 147 | 1.09 | 6.26 | |
63 | 25 | 91 | 26 | 51 | 123 | 0 | 33 | 73 | 79 | 130 | 651 | 0.82 | 6.51 | |
31 | 50 | 76 | 53 | 41 | 91 | 33 | 0 | 58 | 75 | 97 | 712 | 1 | 6.77 | |
80 | 92 | 127 | 72 | 97 | 94 | 70 | 58 | 0 | 130 | 110 | 533 | 0.93 | 6.46 | |
64 | 77 | 32 | 101 | 34 | 131 | 79 | 75 | 130 | 0 | 119 | 298 | 0.7 | 6.07 | |
74 | 149 | 97 | 154 | 99 | 40 | 132 | 97 | 111 | 118 | 0 | 125 | 0.41 | 5.95 |
D | Dem | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 94 | 29 | 65 | 71 | 87 | 99 | 100 | 42 | 73 | 18 | 0 | 0 | 9 |
94 | 0 | 76 | 150 | 150 | 10 | 9 | 44 | 69 | 46 | 82 | 697 | 1.09 | 6.54 | |
29 | 76 | 0 | 74 | 98 | 69 | 81 | 92 | 49 | 69 | 34 | 816 | 0.2 | 5.89 | |
65 | 149 | 73 | 0 | 94 | 144 | 155 | 164 | 105 | 137 | 81 | 110 | 0.58 | 6.16 | |
71 | 150 | 98 | 93 | 0 | 142 | 159 | 135 | 82 | 112 | 72 | 413 | 0.6 | 6.15 | |
87 | 10 | 69 | 144 | 142 | 0 | 18 | 43 | 61 | 38 | 75 | 343 | 0.6 | 6.12 | |
100 | 9 | 81 | 155 | 159 | 19 | 0 | 44 | 78 | 56 | 91 | 146 | 1.14 | 6.48 | |
100 | 44 | 92 | 163 | 135 | 43 | 44 | 0 | 61 | 28 | 83 | 335 | 0.51 | 6.02 | |
41 | 69 | 49 | 104 | 82 | 61 | 78 | 61 | 0 | 35 | 24 | 366 | 0.81 | 6.57 | |
73 | 46 | 69 | 137 | 112 | 38 | 55 | 28 | 35 | 0 | 57 | 146 | 0.83 | 6.07 | |
18 | 83 | 34 | 81 | 72 | 75 | 90 | 84 | 24 | 57 | 0 | 668 | 0.84 | 6.73 |
D | Dem | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 54 | 97 | 71 | 88 | 48 | 68 | 48 | 79 | 73 | 38 | 0 | 0 | 9 |
54 | 0 | 139 | 109 | 139 | 98 | 94 | 39 | 97 | 75 | 54 | 629 | 0.52 | 6.01 | |
96 | 139 | 0 | 134 | 43 | 50 | 58 | 138 | 163 | 92 | 133 | 114 | 0.62 | 5.69 | |
71 | 110 | 135 | 0 | 109 | 87 | 135 | 76 | 39 | 143 | 55 | 475 | 1.39 | 7.15 | |
88 | 139 | 43 | 109 | 0 | 42 | 76 | 132 | 137 | 109 | 118 | 370 | 0.72 | 6.35 | |
48 | 98 | 51 | 88 | 41 | 0 | 51 | 94 | 116 | 75 | 83 | 596 | 0.5 | 6.3 | |
69 | 94 | 58 | 136 | 77 | 52 | 0 | 109 | 146 | 36 | 106 | 679 | 0.38 | 6.15 | |
48 | 39 | 139 | 75 | 132 | 94 | 108 | 0 | 63 | 102 | 21 | 512 | 1.26 | 6.6 | |
79 | 97 | 164 | 39 | 137 | 114 | 146 | 63 | 0 | 150 | 45 | 781 | 0.34 | 5.9 | |
73 | 75 | 92 | 143 | 109 | 75 | 36 | 102 | 149 | 0 | 105 | 392 | 1.38 | 6.82 | |
38 | 55 | 133 | 55 | 118 | 82 | 105 | 21 | 45 | 105 | 0 | 771 | 0.72 | 6.39 |
D | Dem | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 91 | 45 | 36 | 97 | 89 | 72 | 63 | 74 | 95 | 75 | 0 | 0 | 9 |
91 | 0 | 111 | 87 | 145 | 36 | 121 | 117 | 32 | 49 | 144 | 486 | 1.32 | 6.78 | |
45 | 111 | 0 | 24 | 41 | 91 | 38 | 19 | 104 | 94 | 31 | 682 | 0.59 | 6.2 | |
35 | 87 | 24 | 0 | 63 | 70 | 39 | 32 | 86 | 75 | 61 | 533 | 1.46 | 7.1 | |
97 | 145 | 41 | 63 | 0 | 113 | 26 | 49 | 147 | 111 | 10 | 556 | 1.37 | 6.78 | |
89 | 36 | 90 | 69 | 112 | 0 | 89 | 85 | 58 | 15 | 111 | 257 | 0.56 | 6.01 | |
73 | 122 | 37 | 39 | 26 | 89 | 0 | 24 | 123 | 93 | 23 | 227 | 0.75 | 6.15 | |
63 | 117 | 19 | 33 | 49 | 85 | 24 | 0 | 117 | 88 | 47 | 128 | 1.08 | 6.61 | |
74 | 32 | 104 | 86 | 147 | 58 | 123 | 117 | 0 | 72 | 146 | 365 | 0.27 | 5.56 | |
95 | 49 | 94 | 75 | 111 | 14 | 92 | 88 | 72 | 0 | 115 | 126 | 1.49 | 7.08 | |
75 | 144 | 31 | 61 | 10 | 112 | 23 | 47 | 145 | 115 | 0 | 404 | 0.92 | 6.16 |
D | Dem | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 57 | 65 | 38 | 100 | 82 | 59 | 93 | 38 | 86 | 59 | 0 | 0 | 9 |
57 | 0 | 9 | 80 | 122 | 107 | 54 | 39 | 45 | 126 | 101 | 431 | 1.23 | 6.95 | |
66 | 9 | 0 | 88 | 130 | 115 | 57 | 33 | 54 | 135 | 110 | 236 | 1.38 | 6.86 | |
37 | 79 | 88 | 0 | 84 | 62 | 75 | 116 | 54 | 49 | 22 | 426 | 0.27 | 5.73 | |
100 | 122 | 130 | 84 | 0 | 22 | 79 | 141 | 77 | 61 | 89 | 550 | 1.16 | 6.76 | |
82 | 107 | 115 | 62 | 23 | 0 | 70 | 129 | 64 | 42 | 66 | 687 | 0.72 | 6.14 | |
59 | 54 | 57 | 75 | 79 | 70 | 0 | 60 | 22 | 100 | 96 | 578 | 0.19 | 6.06 | |
94 | 39 | 33 | 117 | 138 | 127 | 60 | 0 | 70 | 155 | 138 | 403 | 1.48 | 6.86 | |
38 | 45 | 53 | 54 | 77 | 63 | 22 | 70 | 0 | 83 | 74 | 409 | 0.43 | 5.71 | |
86 | 127 | 136 | 49 | 62 | 42 | 100 | 159 | 83 | 0 | 41 | 597 | 1.43 | 7.15 | |
59 | 101 | 109 | 21 | 89 | 67 | 95 | 138 | 74 | 41 | 0 | 441 | 1.07 | 6.73 |
Dataset | Object | Runtime | Dataset | Object | Runtime | Dataset | Object | Runtime |
---|---|---|---|---|---|---|---|---|
UK10_1 | 4579.4 | 28.02 | UK10_5 | 3800.175 | 28.66 | UK10_9 | 4404.37 | 31.04 |
4578.21 | 29.21 | 3626.59 | 30.17 | 4694.27 | 30.46 | |||
4857.73 | 26.86 | 3916.33 | 27.38 | 4167.22 | 29.78 | |||
4650.555 | 30.03 | 3636.265 | 29.52 | 4125.075 | 28.61 | |||
5056.05 | 29.67 | 3952.075 | 29.86 | 3996.045 | 30.7 | |||
UK10_2 | 4990.565 | 29.78 | UK10_6 | 4277.8 | 31.49 | UK10_10 | 5374.835 | 27.83 |
4423.505 | 30.36 | 4618.615 | 29.93 | 5444.72 | 30.14 | |||
4848.19 | 28.71 | 4776.03 | 27.87 | 5243.825 | 29.49 | |||
4918.935 | 28.97 | 4426.16 | 29.12 | 5399.75 | 29.27 | |||
4930.825 | 30.55 | 4233.62 | 28.89 | 5596.515 | 28.14 | |||
UK10_3 | 5283.625 | 29.06 | UK10_7 | 3665.455 | 28.96 | |||
5532.12 | 27.72 | 3859.02 | 30.25 | |||||
5106.665 | 27.83 | 3727.06 | 28.77 | |||||
5178.215 | 31.24 | 3979.495 | 29.8 | |||||
4676.6 | 30.5 | 3666.08 | 29.37 | |||||
UK10_4 | 4438.18 | 31.17 | UK10_8 | 5821.58 | 30.58 | |||
4925.145 | 29.44 | 5944.335 | 29.17 | |||||
4617.705 | 30.74 | 5556.985 | 29.87 | |||||
4536.81 | 30.38 | 5550.52 | 29.34 | |||||
4576.975 | 30.16 | 5803.02 | 30.17 |
Dataset | Object | Runtime | Dataset | Object | Runtime | Dataset | Object | Runtime |
---|---|---|---|---|---|---|---|---|
UK10_1 | 5299.55 | 20.34 | UK10_5 | 3942.56 | 18.93 | UK10_9 | 4372.21 | 18.34 |
4991.555 | 20.92 | 4660.105 | 19.12 | 4416.71 | 18.67 | |||
5005.5 | 19.68 | 3983.105 | 20.31 | 4248.505 | 19.03 | |||
5130.775 | 20.55 | 3864.295 | 19.56 | 4342.815 | 20.29 | |||
4999.415 | 21.06 | 3919.825 | 19.42 | 4365.025 | 19.74 | |||
UK10_2 | 5045.55 | 21.65 | UK10_6 | 4537.515 | 18.97 | UK10_10 | 5539.805 | 20.31 |
5104.19 | 20.92 | 4452.69 | 19.77 | 5476.845 | 19.72 | |||
4849.96 | 19.68 | 4477.275 | 20.05 | 5426.95 | 19.86 | |||
4748.23 | 20.55 | 4464.92 | 19.94 | 5621.52 | 18.81 | |||
5049.005 | 21.06 | 4562.27 | 20.35 | 5636.43 | 19.83 | |||
UK10_3 | 5412.62 | 22.27 | UK10_7 | 3978.84 | 20.33 | |||
5416.275 | 19.76 | 3863.26 | 18.67 | |||||
5451.84 | 21.66 | 4011.7 | 20.1 | |||||
5221.17 | 20.46 | 4107.485 | 19.85 | |||||
5176.115 | 19.94 | 4051.87 | 19.74 | |||||
UK10_4 | 4942.085 | 21.52 | UK10_8 | 5564.86 | 19.76 | |||
4788.575 | 20.12 | 5762.02 | 20.3 | |||||
4947.73 | 22.02 | 5663.74 | 19.45 | |||||
5062.86 | 19.97 | 5657.115 | 19.23 | |||||
5057.325 | 20.75 | 5760.365 | 19.55 |
Dataset | Object | Runtime | Dataset | Object | Runtime | Dataset | Object | Runtime |
---|---|---|---|---|---|---|---|---|
UK10_1 | 4729.38 | 19.05 | UK10_5 | 3862.26 | 19.82 | UK10_9 | 3998.045 | 19.12 |
4664.14 | 17.68 | 3851.745 | 20.58 | 3998.045 | 19.36 | |||
4664.14 | 19.25 | 3877.2 | 18.21 | 3998.045 | 19.07 | |||
4664.14 | 18.73 | 3877.2 | 18.23 | 3786.95 | 19.42 | |||
4948.695 | 17.68 | 3862.26 | 19.67 | 3998.045 | 18.33 | |||
UK10_2 | 4423.505 | 18.53 | UK10_6 | 4357.945 | 18.97 | UK10_10 | 5393.96 | 20.34 |
4554.775 | 20.1 | 4319.66 | 20.1 | 5316.535 | 18.78 | |||
4753.095 | 19.58 | 4316.145 | 19.03 | 5320.675 | 19.23 | |||
4423.505 | 18.79 | 4444.69 | 18.42 | 5554.535 | 19.36 | |||
4708.27 | 19.17 | 4290.77 | 18.5 | 5329.32 | 18.73 | |||
UK10_3 | 4876.6 | 18.11 | UK10_7 | 3727.06 | 18.67 | |||
4876.6 | 19.51 | 3727.06 | 18.39 | |||||
4768.49 | 18.76 | 3727.06 | 17.89 | |||||
4876.6 | 19.29 | 3777.7 | 19.02 | |||||
4885.03 | 18.01 | 3727.06 | 18.37 | |||||
UK10_4 | 4639.595 | 18.11 | UK10_8 | 5655.395 | 19.02 | |||
4858.115 | 19.51 | 5624.36 | 17.88 | |||||
4854.055 | 18.76 | 5724.765 | 19.27 | |||||
4676.96 | 20.29 | 5610.88 | 18.76 | |||||
4762.76 | 18.01 | 5750.38 | 19.41 |
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Paper | Speed | HF | TW | Test Problem | Veh Type | Tp | Objective | Solution Method | ||
---|---|---|---|---|---|---|---|---|---|---|
D | C | F | V | |||||||
Çimen and Soysal [30] | √ | - | - | - | - | Pollution-Routing Problem Instance Library | homo | 4 | carbon | Approximate Dynamic Programming (ADP) based heuristic algorithm |
Soysal and Çimen [31] | √ | - | - | - | - | Pollution-Routing Problem InstanceLibrary | homo | 4 | carbon | Simulation Based Restricted Dynamic Programming (RDP) algorithm |
Sun, Veelenturf, Dabia and Van Woensel [32] | √ | - | - | - | √ | Instances proposed by Ropke et al. [47] | homo | 5 | max profit | Tailored labeling algorithm |
Wang, Assogba, Fan, Xu, Liu and Wang [14] | √ | - | - | - | √ | Pollution-Routing Problem Instance Library | homo | 5 | carbon and cost | Clarke and Wright Saving, Sweep algorithm, and multi-objective PSO |
Adriano, Montez, Novaes and Wangham [12] | - | - | - | - | √ | Self-generated | homo | not TD | cost | Dynamic milk-run vehicle routing solution |
Liu, Kou et al. [33] | √ | - | - | - | √ | Solomon dataset | homo | 3 | cost | Ant colony algorithm |
Pan, Zhang et al. [34] | √ | - | - | - | √ | Solomon dataset | homo | 5 | time | Tabu search |
Pan, Zhang et al. [35] | √ | - | - | - | √ | Solomon dataset | homo | 5 | dis | Tabu search |
Gmira, Gendreau et al. [36] | √ | - | - | - | √ | NEWLET coming from [48] | homo | 5 | dis | Tabu search |
Afshar-Nadjafi and Afshar-Nadjafi [42] | √ | - | - | √ | √ | Self-generated | 4 | 3 | cost | Simulated annealing |
Vincent, Redi et al. [43] | - | - | - | √ | - | Pollution-Routing Problem Instance Library | 3 | not TD | cost | Simulated annealing |
Wang, Qi et al. [44] | - | - | - | √ | - | Instances proposed by [49] | 10 | not TD | cost | Linear solution provided by the column-and-cut generation with local search |
Soman and Patil [46] | - | - | - | √ | - | Instances proposed by [50] | 2 | not TD | cost | Scatter search |
De and Giri [45] | - | - | - | √ | - | Self-generated | 3 | not TD | carbon and cost | Mixed integer linear programming |
Cao, Liao et al. [22] | - | - | √ | √ | - | Self-generated | 3 | not TD | cost | Genetic algorithm |
Huang, Jiang et al. [38] | - | √ | - | √ | √ | Self-generated | 3 | 5 | cost | Simplified swarm optimization |
Xu, Elomri et al. [21] | - | √ | - | √ | √ | Instances proposed by [51] | 12 | 4 | fuel | Genetic algorithm |
Fan, Zhang et al. [23] | - | √ | √ | √ | √ | MDVRP by [52], MDVRPTW by [53] | 3 | 4 | cost | Genetic algorithm |
Our study | - | √ | √ | √ | √ | Pollution-Routing Problem Instance Library | 3 | 5 | cost | Simplified swarm optimization |
Sets and Indices | |
V | are customers |
i,j | |
A | is the arcs set linking node i and node j |
the set of vehicles with m types | |
m | |
Parameters | |
demand of i customer | |
time window of i customer | |
service time of i customer | |
T | time period |
the arrive time of node i | |
the departure time from node i to node j | |
the travel time from node i to node j | |
the distance linking node i and node j | |
time-varying speed function of m type vehicle | |
time-varying travel time function of m type vehicle | |
each type of vehicle, maximum available number | |
maximum load capacity of m type vehicle | |
fixed cost of m type vehicle | |
variable cost of m type vehicle | |
amount carried using type m vehicle from i to j | |
Decision variable | |
one if a type m vehicle travels from node i to j; otherwise, zero |
Notation | Description | Typical Values of a Type of m Vehicle | ||
---|---|---|---|---|
Vehicle Type 1 | Vehicle Type 2 | Vehicle Type 3 | ||
Maximum load capacity of m type vehicle | 1000 | 2000 | 3650 | |
Variable cost of m type vehicle (£/m) | 0.0001 | 0.00015 | 0.0002 |
D | Demand | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 41,150 | 25,680 | 23,000 | 32,450 | 22,500 | 0 | 0 | 22,400 | 0 |
40,660 | 0 | 51,980 | 40,000 | 23,000 | 32,000 | 900 | 752 | 21,289 | 200 | |
25,010 | 51,780 | 0 | 30,000 | 32,000 | 23,000 | 727 | 270 | 24,050 | 2000 | |
20,000 | 30,000 | 300,000 | 0 | 23,000 | 25,000 | 800 | 250 | 22,500 | 1500 | |
32,500 | 23,000 | 32,000 | 23,000 | 0 | 30,000 | 580 | 700 | 28,000 | 200 | |
22,500 | 32,000 | 23,000 | 25,000 | 30,000 | 0 | 600 | 300 | 27,000 | 300 |
1.2 | 2.1 | 5.3 | 5.2 | 5.1 |
From to | Distance | Travel time | Arrive T | TW | Service Time | Departure T |
---|---|---|---|---|---|---|
0–2 | 25,680 | 10,272 | 10,272 | 2000 | 12,272 | |
2–5 | 23,000 | 920 | 13,192 | 300 | 13,492 | |
5–1 | 32,000 | 1280 | 14,772 | 200 | 14,972 | |
1–4 | 23,000 | 920 | 15,892 | 200 | 16,092 | |
4–3 | 23,000 | 920 | 17,012 | 1500 | 18,512 | |
3–0 | 20,000 | 800 | 19,312 | 0 | / |
Notation | Description | Typical Values of a Type of m Vehicle | ||
---|---|---|---|---|
Vehicle Type 1 | Vehicle Type 2 | Vehicle Type 3 | ||
Maximum load capacity of m type vehicle | 1000 | 2000 | 3650 | |
) | 0.00002 | 0.000015 | 0.00001 | |
Fixed cost of m type vehicle | 50 | 100 | 180 |
From to | Distance | Total Dis | Variable Cost | Demand | Total VC |
---|---|---|---|---|---|
0–5 | 22,500 | - | 0.00002 | 600 | 270 |
5–0 | 22,500 | - | 1 | 0.45 | |
0–2 | 25,680 | - | 0.000015 | 727 | 280.0404 |
2–0 | 25,010 | - | 1 | 0.3752 | |
0–4 | 32,450 | - | 0.000015 | 580 | 282.315 |
4–0 | 32,500 | - | 1 | 0.4875 | |
0–3 | 23,000 | - | 0.00001 | 800 | 184 |
3–1 | 30,000 | 53,000 | 900 | 477 | |
1–0 | 40,660 | - | 1 | 0.4066 | |
1924.6995 |
D | Dem | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D | 0 | 41 | 26 | 54 | 95 | 16 | 89 | 74 | 26 | 88 | 66 | 0 | 0 | 9 |
41 | 0 | 52 | 33 | 100 | 42 | 76 | 64 | 24 | 72 | 26 | 721 | 0.60 | 6.15 | |
25 | 52 | 0 | 62 | 74 | 13 | 69 | 53 | 43 | 73 | 77 | 814 | 0.18 | 5.85 | |
54 | 33 | 62 | 0 | 77 | 52 | 43 | 32 | 49 | 40 | 30 | 620 | 0.29 | 5.67 | |
95 | 100 | 74 | 77 | 0 | 81 | 56 | 46 | 112 | 62 | 106 | 311 | 1.42 | 6.73 | |
16 | 43 | 13 | 52 | 81 | 0 | 78 | 61 | 34 | 82 | 68 | 167 | 0.65 | 6.03 | |
89 | 76 | 69 | 43 | 55 | 78 | 0 | 17 | 92 | 7 | 69 | 513 | 1.02 | 6.70 | |
73 | 63 | 52 | 32 | 46 | 61 | 17 | 0 | 76 | 21 | 61 | 568 | 1.22 | 6.96 | |
26 | 24 | 44 | 50 | 112 | 34 | 91 | 76 | 0 | 89 | 49 | 763 | 0.97 | 6.76 | |
88 | 72 | 73 | 39 | 61 | 82 | 7 | 21 | 88 | 0 | 64 | 558 | 1.04 | 6.68 | |
65 | 26 | 77 | 29 | 106 | 67 | 69 | 61 | 49 | 64 | 0 | 636 | 1.47 | 7.28 |
GA | PSO | SSO | ||||
---|---|---|---|---|---|---|
Average | Runtime | Average | Runtime | Average | Runtime | |
1 | 4744.389 | 28.758 | 5085.359 | 20.51 | 4734.099 | 18.478 |
2 | 4822.404 | 29.674 | 4959.387 | 20.772 | 4572.63 | 19.234 |
3 | 5155.445 | 29.27 | 5335.604 | 20.818 | 4856.664 | 18.736 |
4 | 4618.963 | 30.378 | 4959.715 | 20.876 | 4758.297 | 18.936 |
5 | 3786.287 | 29.118 | 4073.978 | 19.468 | 3866.133 | 19.302 |
6 | 4466.445 | 29.3425 | 4498.934 | 19.816 | 4345.842 | 19.004 |
7 | 3779.422 | 29.43 | 4002.631 | 19.738 | 3737.188 | 18.468 |
8 | 5735.288 | 29.826 | 5681.62 | 19.658 | 5673.156 | 18.868 |
9 | 4277.396 | 30.118 | 4349.053 | 19.214 | 3955.826 | 19.06 |
10 | 5411.929 | 28.9 | 5540.31 | 19.706 | 5383.005 | 19.288 |
average | 4679.797 | 29.48145 | 4845.599 | 20.0576 | 4589.337 | 18.9374 |
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Yeh, W.-C.; Tan, S.-Y. Simplified Swarm Optimization for the Heterogeneous Fleet Vehicle Routing Problem with Time-Varying Continuous Speed Function. Electronics 2021, 10, 1775. https://doi.org/10.3390/electronics10151775
Yeh W-C, Tan S-Y. Simplified Swarm Optimization for the Heterogeneous Fleet Vehicle Routing Problem with Time-Varying Continuous Speed Function. Electronics. 2021; 10(15):1775. https://doi.org/10.3390/electronics10151775
Chicago/Turabian StyleYeh, Wei-Chang, and Shi-Yi Tan. 2021. "Simplified Swarm Optimization for the Heterogeneous Fleet Vehicle Routing Problem with Time-Varying Continuous Speed Function" Electronics 10, no. 15: 1775. https://doi.org/10.3390/electronics10151775
APA StyleYeh, W.-C., & Tan, S.-Y. (2021). Simplified Swarm Optimization for the Heterogeneous Fleet Vehicle Routing Problem with Time-Varying Continuous Speed Function. Electronics, 10(15), 1775. https://doi.org/10.3390/electronics10151775