Bi-Objective Modelling for Hazardous Materials Road–Rail Multimodal Routing Problem with Railway Schedule-Based Space–Time Constraints
Abstract
:1. Introduction
- (1)
- Customer demands, including origins, destinations, volumes, release times, and due dates.
- (2)
- Multiple hazardous materials flows but single type of hazardous materials, i.e., multiple origin–destination pairs.
- (3)
- Capacitated schedule-based rail services and uncapacitated time-flexible road services [26]. In particular, the restrictions of railway schedules on the routing decision are formulated as the railway schedule-based space–time constraints.
- (4)
- Environmental risk constraint to lower and balance the environmental risk under a given threshold.
- (5)
- Bi-objective optimization, including minimizing the total generalized costs of transporting the multiple hazardous materials and the social risk along the planned routes.
2. Transportation Scenario Description
3. Risk Evaluation and Modelling
3.1. Social Risk Evaluation
3.2. Environmental Risk Evaluation
4. Mathematical Model
4.1. Notations
- Indices
- Sets
- Parameters
- Decision Variables
4.2. A Node–Arc-Based Model
- Objective 1
- Objective 2
- Subject to
- (i)
- If = 0, (which is always satisfied), which means the arrival time of the hazardous materials at the node is not restricted by the unselected rail services.
- (ii)
- If = 1 and = 1, according to Equation (16), = 0, i.e., the hazardous materials arrive at node by road service, then according to Equation (21), , which matches the description of Case 1.1 in Section 2.
- (iii)
- If = 1 and = 1, just contrary to Equation (2), there is , which matches the description of Case 1.2 in Section 2.
- (i)
- If the following transportation is by road service (, and according to Equation (15), there exists = 0); we are far more concerned about the unloading start time instead of , because the following operations cannot be conducted until . Therefore, in such a case, , which is more effective than .
- (ii)
- If the following transportation is by rail service (, and according to Equation (16), there exists =0), for similar reasons instead of .
5. Solution Strategy for the Bi-Objective Nonlinear Programming
5.1. Improved Linear Reformulation
- Objective 1
- Objective 2
- Subject to
5.2. Normalized Weighted Sum Method for the Bi-Objective Optimization
6. Empirical Case Study from the Beijing–Tianjin–Hebei Region in China
6.1. Case Description and Parameter Setting
6.2. Optimization Results and Discussions
6.2.1. Simulation Environment
6.2.2. Multimodal Routes Illustration
6.2.3. Single-Objective Scenarios
6.2.4. Bi-Objective Scenario
6.2.5. Sensitivity of the Pareto Solutions with Respect to the Environmental Risk Threshold
7. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A. Model Linear Reformulation
- Reformulation 1. By using a non-negative variable to replace the nonlinear function and adding two linear Equations (A2) and (A3), nonlinear Component (11) in Objective 1 can be reformulated as a linear function (A1).
- (i)
- Hazardous materials get transshipped between rail services at node , i.e., = 1 and = 1, in this case, = 1.
- (ii)
- Hazardous materials get transshipped between road service and rail service at node , i.e., =1 and = 0 or = 0 and = 1, in this case, = 0.
- (iii)
- Node is not covered in the route of hazardous materials , i.e., = 0 and = 0, in this case, = 0.
- Reformulation 2. By using a non-negative decision variable to replace the nonlinear function and adding two linear Equations (A5) and (A6), the nonlinear Component (12) in Objective 1 can be reformulated as a linear function (A4).
- (i)
- If = 0, i.e., hazardous materials arrive at node by the rail service, there are and according to Equations (A5) and (A6), and the minimization of Component (A4) will restrict = 0, which matches the setting that there are no inventory costs at the node if the hazardous materials are transshipped between different rail services.
- (ii)
- If = 1, i.e., hazardous materials arrive at node by road service, there are and , and minimization of Component (A4) will restrict , which matches the setting that inventory costs will be charged at the node according to the charged inventory time if the hazardous materials are transshipped from road service to rail service.
- Reformulation 3. Nonlinear Equation (21) is equivalent to linear Equation (A7) as follows.
- Reformulation 4. Nonlinear Equations (23), (24) and (26) are separately equivalent to linear Equations (A8)–(A13). For detailed proofs of the above three equivalencies, readers can refer to our previous study [26].
Appendix B
Node | Population Exposure (Unit: 104 People) | Node | Population Exposure (Unit: 104 People) | Node | Population Exposure (Unit: 104 People) |
---|---|---|---|---|---|
1 | 3.44 | 16 | 2.87 | 31 | 1.74 |
2 | 0.66 | 17 | 1.32 | 32 | 1.74 |
3 | 2.29 | 18 | 0.58 | 33 | 0.50 |
4 | 2.29 | 19 | 0.58 | 34 | 2.48 |
5 | 2.31 | 20 | 0.66 | 35 | 3.56 |
6 | 2.13 | 21 | 2.48 | 36 | 2.13 |
7 | 2.84 | 22 | 2.48 | 37 | 2.84 |
8 | 3.56 | 23 | 2.32 | 38 | 2.84 |
9 | 0.32 | 24 | 2.29 | 39 | 2.87 |
10 | 2.13 | 25 | 2.29 | 40 | 0.58 |
11 | 2.13 | 26 | 2.29 | 41 | 1.12 |
12 | 0.66 | 27 | 2.29 | 42 | 0.43 |
13 | 2.29 | 28 | 2.29 | 43 | 2.32 |
14 | 2.13 | 29 | 2.29 | 44 | 1.74 |
15 | 1.32 | 30 | 1.69 | 45 | 0.82 |
Arc | Distance (Unit: km) | Time (Unit: h) | Population Exposure (Unit: 104 People) | Environmental Capacity (Unit: 104 Ton) | Arc | Distance (Unit: km) | Time (Unit: h) | Population Exposure (Unit: 104 People) | Environmental Capacity (Unit: 104 Ton) |
---|---|---|---|---|---|---|---|---|---|
(1, 5) | 129 | 1.8 | 57.17 | 7.47 | (4, 26) | 80 | 1 | 56.72 | 4.64 |
(1, 6) | 153 | 2.3 | 81.36 | 8.86 | (4, 29) | 108 | 1.8 | 76.57 | 6.26 |
(1, 7) | 162 | 2 | 96.92 | 9.39 | (4, 45) | 215 | 2.8 | 117.47 | 12.46 |
(1, 8) | 148 | 1.8 | 62.31 | 8.57 | (6, 37) | 8 | 0.2 | 6.74 | 0.46 |
(1, 9) | 90 | 1.5 | 52.42 | 5.27 | (6, 38) | 10 | 0.3 | 8.42 | 0.58 |
(1, 15) | 243 | 2.8 | 123.84 | 14.08 | (8, 35) | 3 | 0.2 | 3.99 | 0.17 |
(1, 20) | 235 | 3.3 | 57.28 | 13.62 | (10, 36) | 9 | 0.4 | 4.39 | 0.52 |
(1, 23) | 174 | 2.5 | 61.69 | 10.08 | (11, 36) | 8 | 0.3 | 3.90 | 0.46 |
(1, 24) | 291 | 3.4 | 77.37 | 16.86 | (12, 42) | 152 | 2 | 72.41 | 8.81 |
(1, 26) | 308 | 3.7 | 76.43 | 17.84 | (13, 42) | 15 | 0.6 | 6.48 | 0.87 |
(1, 33) | 403 | 5 | 98.22 | 23.35 | (14, 15) | 101 | 1.4 | 49.23 | 5.85 |
(1, 34) | 248 | 3.3 | 63.74 | 14.37 | (14, 16) | 123 | 1.7 | 31.67 | 7.13 |
(1, 36) | 238 | 3.2 | 137.11 | 13.79 | (14, 36) | 114 | 1.8 | 75.78 | 6.60 |
(1, 39) | 262 | 3.3 | 156.74 | 15.18 | (14, 37) | 9 | 0.4 | 5.58 | 0.52 |
(1, 40) | 382 | 4.2 | 169.28 | 22.13 | (14, 38) | 9 | 0.4 | 5.58 | 0.52 |
(1, 41) | 333 | 3.8 | 162.32 | 19.29 | (14, 39) | 119 | 1.9 | 65.92 | 6.89 |
(1, 42) | 164 | 2.3 | 83.58 | 9.50 | (14, 40) | 239 | 2.7 | 119.68 | 13.85 |
(1, 43) | 171 | 2.3 | 64.41 | 9.91 | (14, 41) | 191 | 2.4 | 63.48 | 11.07 |
(1, 44) | 434 | 4.9 | 211.56 | 25.15 | (16, 39) | 6 | 0.2 | 3.99 | 0.35 |
(1, 45) | 445 | 5.4 | 226.78 | 25.78 | (16, 40) | 143 | 1.8 | 22.18 | 8.29 |
(2, 6) | 150 | 1.8 | 76.44 | 8.69 | (16, 41) | 95 | 1.5 | 43.36 | 5.50 |
(2, 7) | 151 | 1.7 | 76.95 | 8.75 | (18, 39) | 71 | 1.6 | 15.10 | 4.11 |
(2, 8) | 265 | 6 | 131.52 | 15.35 | (18, 40) | 56 | 0.8 | 9.93 | 3.24 |
(2, 12) | 3 | 0.3 | 0.40 | 0.17 | (18, 41) | 45 | 1 | 13.96 | 2.61 |
(2, 15) | 150 | 2.1 | 66.47 | 8.69 | (19, 40) | 75 | 1.1 | 13.29 | 4.35 |
(2, 20) | 377 | 4.5 | 125.30 | 21.84 | (19, 41) | 161 | 2.1 | 46.84 | 8.75 |
(2, 33) | 577 | 6.6 | 145.74 | 33.43 | (20, 33) | 215 | 2.6 | 100.04 | 12.46 |
(2, 34) | 422 | 5.2 | 114.07 | 24.45 | (20, 34) | 59 | 1 | 41.83 | 3.42 |
(2, 35) | 265 | 3 | 131.52 | 15.35 | (21, 33) | 168 | 2.2 | 72.21 | 9.73 |
(2, 36) | 237 | 3 | 119.52 | 13.73 | (21, 34) | 4 | 0.1 | 2.39 | 0.21 |
(2, 37) | 155 | 1.9 | 78.99 | 8.98 | (22, 33) | 113 | 1.9 | 48.57 | 6.55 |
(2, 38) | 140 | 1.8 | 71.35 | 8.11 | (22, 34) | 23 | 0.5 | 15.29 | 1.33 |
(2, 41) | 256 | 2.8 | 113.44 | 14.83 | (23, 43) | 8 | 0.3 | 3.37 | 0.46 |
(3, 6) | 439 | 4.8 | 184.81 | 25.43 | (23, 44) | 266 | 3.1 | 126.13 | 15.41 |
(3, 12) | 376 | 4.4 | 159.96 | 21.78 | (23, 45) | 278 | 4.2 | 123.19 | 16.11 |
(3, 24) | 23 | 0.6 | 17.84 | 1.33 | (26, 44) | 137 | 1.6 | 77.71 | 7.94 |
(3, 25) | 10 | 0.3 | 7.76 | 0.58 | (26, 45) | 149 | 2.2 | 81.41 | 8.63 |
(3, 26) | 27 | 0.7 | 19.14 | 1.56 | (27, 44) | 140 | 1.7 | 79.41 | 8.11 |
(3, 28) | 17 | 0.4 | 16.87 | 0.98 | (27, 45) | 152 | 2.2 | 83.05 | 8.81 |
(3, 29) | 42 | 1 | 32.57 | 2.43 | (28, 44) | 126 | 1.6 | 76.77 | 7.30 |
(3, 31) | 144 | 1.9 | 73.38 | 8.34 | (28, 45) | 138 | 2.1 | 70.33 | 8.00 |
(3, 35) | 201 | 2.7 | 172.80 | 11.65 | (29, 44) | 101 | 1.5 | 58.95 | 5.85 |
(3, 44) | 142 | 1.8 | 80.55 | 8.23 | (29, 45) | 101 | 2.2 | 61.54 | 5.85 |
(3, 45) | 154 | 2.3 | 84.14 | 8.92 | (30, 31) | 97 | 1.5 | 59.10 | 5.62 |
(4, 6) | 412 | 4.4 | 173.45 | 23.87 | (30, 44) | 95 | 1.3 | 57.89 | 5.50 |
(4, 12) | 359 | 4.4 | 152.72 | 20.80 | (30, 45) | 107 | 1.7 | 59.27 | 6.20 |
(4, 24) | 59 | 0.8 | 41.83 | 3.42 | (31, 44) | 7 | 0.3 | 6.20 | 0.41 |
(4, 25) | 74 | 1 | 52.47 | 4.29 | (31, 45) | 131 | 1.7 | 101.59 | 7.59 |
(4, 31) | 171 | 2.1 | 96.99 | 9.91 | (32, 44) | 14 | 0.5 | 12.41 | 0.81 |
(4, 35) | 204 | 2.8 | 175.38 | 11.82 | (32, 45) | 142 | 1.8 | 100.68 | 8.23 |
(4, 44) | 169 | 1.9 | 145.29 | 9.79 |
Train No. | 47501 | 47503 | 47505 | 47507 | 21018 | 21020 | 21022 | 21024 | 21026 | 34029 | 34035 |
---|---|---|---|---|---|---|---|---|---|---|---|
Origin | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
Loading start time | 20.4 | 2.8 | 10.6 | 15.1 | 1.6 | 0.8 | 3.1 | 6 | 10.3 | 1.5 | 5.8 |
Loading cutoff time | 20.9 | 3.8 | 11.5 | 15.8 | 3.1 | 2.3 | 4.6 | 7.5 | 11.8 | 3 | 7.3 |
Classification start time | 20.7 | 3.5 | 11.3 | 15.6 | 2.6 | 2.8 | 4.5 | 6.5 | 10.8 | 2 | 6.3 |
Classification cutoff time | 21.2 | 4 | 12.2 | 16.2 | 3.6 | 3.8 | 5.1 | 8 | 12.3 | 3.5 | 7.8 |
Departure time | 21.4 | 4.3 | 12.6 | 16.5 | 4.1 | 4.3 | 5.6 | 8.5 | 12.8 | 4 | 8.3 |
Destination | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 10 | 10 |
Arrival time | 22.3 | 5.1 | 13.4 | 17.4 | 18 | 18.4 | 18.7 | 18.8 | 19.5 | 7.5 | 10.8 |
Disassembly start time | 22.5 | 5.3 | 13.6 | 17.7 | 18.5 | 18.9 | 19.2 | 18.3 | 19 | 7 | 10.3 |
Disassembly cutoff time | 23 | 5.7 | 14.2 | 18.4 | 20 | 20.5 | 20.7 | 20 | 21 | 8.5 | 12 |
Unloading start time | 22.8 | 5.5 | 14 | 18 | 19 | 19.4 | 19.7 | 19.8 | 20.5 | 8.5 | 11.8 |
Unloading cutoff time | 23.5 | 6 | 14.5 | 19 | 20.5 | 21 | 21.2 | 21.3 | 22 | 10 | 13.3 |
Distance (unit: km) | 17 | 17 | 17 | 17 | 273 | 273 | 273 | 273 | 273 | 115 | 115 |
Capacity (unit: ton) | 706 | 894 | 994 | 762 | 1973 | 1354 | 1181 | 1541 | 1668 | 799 | 994 |
Population exposure (unit: 104 people) | 9.86 | 9.86 | 9.86 | 9.86 | 146.59 | 146.59 | 146.59 | 146.59 | 146.59 | 64.22 | 64.22 |
Environmental capacity (unit: 104 ton) | 13.67 | 13.67 | 13.67 | 13.67 | 219.45 | 219.45 | 219.45 | 219.45 | 219.45 | 92.44 | 92.44 |
Train No. | 34047 | 34055 | 34001 | 34045 | 47551 | 47555 | 47557 | 32101 | 32105 | 32109 | 32119 |
Origin | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
Loading start time | 9.3 | 11.8 | 16.1 | 8.6 | 17.9 | 5.5 | 11.7 | 16.3 | 17.1 | 17.3 | 2.8 |
Loading cutoff time | 10.8 | 13.3 | 17.6 | 10.1 | 18.4 | 6.2 | 12.3 | 17.8 | 18.6 | 18.8 | 4.3 |
Classification start time | 9.8 | 12.3 | 16.6 | 9.1 | 18.3 | 6 | 11.9 | 16.8 | 17.6 | 17.8 | 3.3 |
Classification cutoff time | 11.3 | 13.8 | 18.1 | 10.6 | 18.6 | 6.5 | 12.4 | 18.3 | 19.1 | 19.3 | 4.8 |
Departure time | 11.8 | 14.3 | 18.6 | 11.1 | 18.9 | 6.9 | 12.7 | 18.8 | 19.6 | 19.8 | 5.3 |
Destination | 10 | 10 | 11 | 11 | 14 | 14 | 14 | 17 | 17 | 17 | 17 |
Arrival time | 14.3 | 16.8 | 21 | 13.5 | 19.1 | 7 | 12.8 | 21.4 | 22.5 | 22.9 | 7.5 |
Disassembly start time | 13.8 | 16.3 | 20.5 | 13 | 19.4 | 7.5 | 13.2 | 20.9 | 22 | 22.4 | 7 |
Disassembly cutoff time | 15.5 | 18 | 22.3 | 14.7 | 19.9 | 8 | 13.7 | 22.4 | 23.8 | 24 | 8.7 |
Unloading start time | 15.3 | 17.8 | 22 | 14.5 | 20 | 7.8 | 13.5 | 22.4 | 23.5 | 23.9 | 8.5 |
Unloading cutoff time | 16.8 | 19.3 | 23.5 | 16 | 21 | 8.5 | 14 | 23.9 | 25 | 25.4 | 10 |
Distance (unit: km) | 115 | 115 | 107 | 107 | 9 | 9 | 9 | 162 | 162 | 162 | 162 |
Capacity (unit: ton) | 1066 | 1747 | 1062 | 1094 | 874 | 1080 | 972 | 1680 | 1181 | 1145 | 1786 |
Population exposure (unit: 104 people) | 64.22 | 64.22 | 59.75 | 59.75 | 4.25 | 4.25 | 4.25 | 80.03 | 80.03 | 80.03 | 80.03 |
Environmental capacity (unit: 104 ton) | 92.44 | 92.44 | 86.01 | 86.01 | 7.23 | 7.23 | 7.23 | 130.22 | 130.22 | 130.22 | 130.22 |
Train No. | 32123 | 30002 | 30004 | 30006 | 34005 | 34011 | 34037 | 34041 | 34049 | 34013 | 34025 |
Origin | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
Loading start time | 6.8 | 16.7 | 17.4 | 18.3 | 18.3 | 20.1 | 6.6 | 7.8 | 10.3 | 21.4 | 1.6 |
Loading cutoff time | 8.3 | 18.2 | 18.9 | 19.8 | 19.8 | 21.6 | 8.1 | 9.3 | 11.8 | 22.9 | 3.1 |
Classification start time | 7.3 | 17.2 | 17.9 | 18.8 | 18.8 | 20.6 | 7.1 | 8.3 | 10.8 | 21.9 | 2.1 |
Classification cutoff time | 8.8 | 18.7 | 19.4 | 20.3 | 20.3 | 22.1 | 8.6 | 9.8 | 12.3 | 23.4 | 3.6 |
Departure time | 9.3 | 19.2 | 19.9 | 20.8 | 20.8 | 22.6 | 9.1 | 10.3 | 12.8 | 23.9 | 4.1 |
Destination | 17 | 12 | 12 | 12 | 10 | 10 | 10 | 10 | 10 | 11 | 11 |
Arrival time | 11.9 | 21.5 | 22.2 | 23.1 | 23 | 0.8 | 11.3 | 12.4 | 15 | 2 | 6.1 |
Disassembly start time | 11.4 | 21 | 21.7 | 22.6 | 22.5 | 0.3 | 10.8 | 11.9 | 14.5 | 1.5 | 5.6 |
Disassembly cutoff time | 13 | 22.8 | 23.5 | 24.3 | 24.6 | 2 | 12.5 | 13.5 | 16.7 | 3.3 | 7.3 |
Unloading start time | 12.9 | 22.5 | 23.2 | 24.1 | 24 | 1.8 | 12.3 | 13.4 | 16 | 3 | 7.1 |
Unloading cutoff time | 14.4 | 24 | 24.7 | 25.6 | 25.5 | 3.3 | 13.8 | 14.9 | 17.5 | 4.5 | 8.6 |
Distance (unit: km) | 162 | 121 | 121 | 121 | 107 | 107 | 107 | 107 | 107 | 99 | 99 |
Capacity (unit: ton) | 1001 | 1505 | 1109 | 1508 | 1987 | 828 | 1134 | 1727 | 1836 | 940 | 1620 |
Population exposure (unit: 104 people) | 80.03 | 53.69 | 53.69 | 53.69 | 59.75 | 59.75 | 59.75 | 59.75 | 59.75 | 55.28 | 55.28 |
Environmental capacity (unit: 104 ton) | 130.22 | 97.26 | 97.26 | 97.26 | 86.01 | 86.01 | 86.01 | 86.01 | 86.01 | 79.58 | 79.58 |
Train No. | 34031 | 33912 | 33914 | 21001 | 21017 | 21019 | 21023 | 21025 | 21027 | 33039 | 33041 |
Origin | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |
Loading start time | 3.6 | 20.4 | 0.4 | 19.1 | 2.5 | 3.5 | 4.1 | 6.8 | 11.1 | 18.5 | 20.5 |
Loading cutoff time | 5.1 | 21.9 | 1.1 | 20.6 | 4 | 5 | 5.6 | 8.3 | 12.6 | 20 | 22 |
Classification start time | 4.1 | 20.9 | 0.1 | 19.6 | 3 | 4 | 4.6 | 7.3 | 11.6 | 19 | 21 |
Classification cutoff time | 5.6 | 22.4 | 1.6 | 21.1 | 4.5 | 5.5 | 6.1 | 8.8 | 13.1 | 20.5 | 22.5 |
Departure time | 6.1 | 22.9 | 2.1 | 21.6 | 5 | 6 | 6.6 | 9.3 | 13.6 | 21 | 23 |
Destination | 11 | 13 | 13 | 6 | 6 | 6 | 6 | 6 | 6 | 20 | 20 |
Arrival time | 8.2 | 3.3 | 5.2 | 5.7 | 11.3 | 13.5 | 15.3 | 20.5 | 0.7 | 22.9 | 0.9 |
Disassembly start time | 7.7 | 2.8 | 4.7 | 5.2 | 10.8 | 13 | 14.8 | 20 | 0.2 | 22.4 | 0.4 |
Disassembly cutoff time | 9.5 | 4.6 | 6.3 | 7 | 12.7 | 14.5 | 16.6 | 21.6 | 2 | 23.9 | 2 |
Unloading start time | 9.2 | 4.3 | 6.2 | 6.7 | 12.3 | 14.5 | 16.3 | 21.5 | 1.7 | 23.9 | 1.9 |
Unloading cutoff time | 10.7 | 5.8 | 7.7 | 8.2 | 13.8 | 16 | 17.8 | 23 | 3.2 | 25.4 | 3.4 |
Distance (unit: km) | 99 | 185 | 185 | 273 | 273 | 273 | 273 | 273 | 273 | 121 | 121 |
Capacity (unit: ton) | 1469 | 1174 | 1253 | 1840 | 1990 | 1146 | 1075 | 1393 | 1495 | 1170 | 922 |
Population exposure (unit: 104 people) | 55.28 | 75.49 | 75.49 | 146.59 | 146.59 | 146.59 | 146.59 | 146.59 | 146.59 | 33.26 | 33.26 |
Environmental capacity (unit: 104 ton) | 79.58 | 148.71 | 148.71 | 219.45 | 219.45 | 219.45 | 219.45 | 219.45 | 219.45 | 97.26 | 97.26 |
Train No. | 33057 | 33061 | 33067 | 35547 | 35553 | 35557 | 35505 | 38001 | 38011 | 38027 | 40103 |
Origin | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 |
Loading start time | 8 | 10.1 | 14.5 | 12.7 | 13.6 | 14.9 | 18.6 | 16.3 | 1 | 13.7 | 15.9 |
Loading cutoff time | 9.5 | 11.6 | 16 | 14.2 | 15.1 | 16.4 | 20.1 | 17.8 | 2.5 | 15.2 | 16.4 |
Classification start time | 8.5 | 10.6 | 15 | 13.2 | 14.1 | 15.4 | 19.1 | 16.8 | 1.5 | 14.2 | 16.1 |
Classification cutoff time | 10 | 12.1 | 16.5 | 14.7 | 15.6 | 16.9 | 20.6 | 18.3 | 3 | 15.7 | 16.9 |
Departure time | 10.5 | 12.6 | 17 | 15.2 | 16.1 | 17.4 | 21.1 | 18.8 | 3.5 | 16.2 | 17.3 |
Destination | 20 | 20 | 21 | 23 | 23 | 23 | 23 | 26 | 26 | 26 | 12 |
Arrival time | 12.4 | 14.4 | 20 | 16.8 | 18.6 | 19.2 | 23.7 | 21.3 | 6 | 21 | 20.6 |
Disassembly start time | 11.9 | 13.9 | 19.5 | 16.3 | 18.1 | 18.7 | 23.2 | 20.8 | 5.5 | 20.5 | 21 |
Disassembly cutoff time | 13.5 | 15.7 | 21.2 | 18 | 20 | 20.5 | 24.7 | 22.5 | 7 | 22.3 | 21.6 |
Unloading start time | 13.4 | 15.4 | 21 | 17.8 | 19.6 | 20.2 | 24.7 | 22.3 | 7 | 22 | 21.4 |
Unloading cutoff time | 14.9 | 16.9 | 22.5 | 19.3 | 21.1 | 21.7 | 26.2 | 23.8 | 8.5 | 23.5 | 22 |
Distance (unit: km) | 121 | 121 | 171 | 113 | 113 | 113 | 113 | 262 | 262 | 262 | 127 |
Capacity (unit: ton) | 1546 | 994 | 1102 | 944 | 1239 | 1462 | 1009 | 856 | 1484 | 1135 | 864 |
Population exposure (unit: 104 people) | 33.26 | 33.26 | 49.21 | 82.52 | 82.52 | 82.52 | 82.52 | 225.09 | 225.09 | 225.09 | 94.54 |
Environmental capacity (unit: 104 ton) | 97.26 | 97.26 | 137.46 | 90.83 | 90.83 | 90.83 | 90.83 | 210.61 | 210.61 | 210.61 | 102.09 |
Train No. | 30001 | 30003 | 30005 | 40081 | 47405 | 47406 | 39101 | 39109 | 39111 | 39113 | 39115 |
Origin | 12 | 12 | 12 | 15 | 16 | 17 | 17 | 17 | 17 | 17 | 17 |
Loading start time | 23.1 | 5.5 | 7.1 | 17.7 | 22.5 | 14.2 | 17.3 | 5.5 | 8.2 | 11.5 | 14.7 |
Loading cutoff time | 24.6 | 7 | 8.6 | 19.2 | 24 | 15.7 | 18.8 | 7 | 9.7 | 13 | 16.2 |
Classification start time | 23.6 | 6 | 7.6 | 18.2 | 23 | 14.7 | 17.8 | 6 | 8.7 | 12 | 15.2 |
Classification cutoff time | 25.1 | 7.5 | 9.1 | 19.7 | 24.5 | 16.2 | 19.3 | 7.5 | 10.2 | 13.5 | 16.7 |
Departure time | 25.6 | 8 | 9.6 | 20.2 | 25 | 16.7 | 19.8 | 8 | 10.7 | 14 | 17.2 |
Destination | 6 | 6 | 6 | 17 | 17 | 16 | 18 | 18 | 18 | 18 | 18 |
Arrival time | 28.1 | 10.8 | 12.1 | 22.3 | 25.9 | 17.6 | 21.2 | 9.4 | 12.2 | 15.4 | 18.7 |
Disassembly start time | 27.6 | 10.3 | 11.6 | 21.8 | 25.4 | 17.1 | 20.7 | 8.9 | 11.7 | 14.9 | 18.2 |
Disassembly cutoff time | 29.5 | 12 | 13.7 | 23.5 | 27 | 18.8 | 22.2 | 10.5 | 13.6 | 16.4 | 20 |
Unloading start time | 29.1 | 11.8 | 13.1 | 23.3 | 26.9 | 18.6 | 22.2 | 10.4 | 13.2 | 16.4 | 19.7 |
Unloading cutoff time | 30.6 | 13.3 | 14.6 | 24.8 | 28.4 | 20.1 | 23.7 | 11.9 | 14.7 | 17.9 | 21.2 |
Distance (unit: km) | 121 | 12 | 12 | 67 | 52 | 52 | 44 | 44 | 44 | 44 | 44 |
Capacity (unit: ton) | 1901 | 884 | 1392 | 857 | 1893 | 1107 | 2011 | 1892 | 1469 | 1242 | 1058 |
Population exposure (unit: 104 people) | 55.41 | 5.49 | 5.49 | 31.66 | 20.10 | 20.10 | 11.34 | 20.79 | 20.79 | 20.79 | 20.79 |
Environmental capacity (unit: 104 ton) | 97.26 | 9.65 | 9.65 | 53.86 | 41.80 | 41.80 | 35.37 | 35.37 | 35.37 | 35.37 | 35.37 |
Train No. | 39181 | 39183 | 39185 | 33201 | 33203 | 46721 | 46723 | 82114/3 | 86662/1 | 46915 | 46947 |
Origin | 17 | 17 | 17 | 20 | 20 | 24 | 24 | 24 | 24 | 25 | 25 |
Loading start time | 22.7 | 1.5 | 13.7 | 18.3 | 1.9 | 22.5 | 4.8 | 2.5 | 7.9 | 8.6 | 5 |
Loading cutoff time | 24.2 | 3 | 15.2 | 19.8 | 3.4 | 23 | 5.4 | 4 | 9.4 | 9.2 | 5.7 |
Classification start time | 23.2 | 2 | 14.2 | 18.8 | 2.4 | 23 | 5.2 | 3 | 8.4 | 8.9 | 5.5 |
Classification cutoff time | 24.7 | 3.5 | 15.7 | 20.3 | 3.9 | 23.5 | 6 | 4.5 | 9.9 | 9.6 | 6.2 |
Departure time | 25.2 | 4 | 16.2 | 20.8 | 4.4 | 23.8 | 6.4 | 5 | 10.4 | 10 | 6.5 |
Destination | 19 | 19 | 19 | 22 | 22 | 28 | 28 | 29 | 30 | 26 | 26 |
Arrival time | 34.8 | 14.7 | 0.7 | 22.2 | 5.8 | 1.4 | 7 | 7 | 12.8 | 10.5 | 7 |
Disassembly start time | 34.3 | 14.2 | 0.2 | 21.7 | 5.3 | 1.8 | 7.5 | 6.5 | 12.3 | 10.8 | 7.2 |
Disassembly cutoff time | 36 | 15.9 | 1.9 | 23.4 | 7 | 2.8 | 8.1 | 8.2 | 14 | 11.5 | 7.8 |
Unloading start time | 35.8 | 15.7 | 1.7 | 23.2 | 6.8 | 2.1 | 7.8 | 8 | 13.8 | 11.4 | 7.5 |
Unloading cutoff time | 37.3 | 17.2 | 3.2 | 24.7 | 8.3 | 3 | 8.5 | 9.5 | 15.3 | 12 | 8.1 |
Distance (unit: km) | 203 | 203 | 203 | 75 | 75 | 26 | 26 | 49 | 69 | 13 | 13 |
Capacity (unit: ton) | 722 | 1284 | 1675 | 1156 | 1786 | 1023 | 1111 | 1987 | 1509 | 1160 | 1283 |
Population exposure (unit: 104 people) | 52.32 | 52.32 | 52.32 | 45.10 | 45.10 | 18.99 | 18.99 | 35.78 | 35.57 | 9.49 | 9.49 |
Environmental capacity (unit: 104 ton) | 163.18 | 163.18 | 163.18 | 60.29 | 60.29 | 20.90 | 20.90 | 39.39 | 55.47 | 10.45 | 10.45 |
Train No. | 46951 | 46944 | 46821 | 46829 | 46833 | 46835 | 46839 | 43093 | 46981 | 46983 | 46981 |
Origin | 25 | 26 | 28 | 28 | 28 | 28 | 28 | 29 | 31 | 31 | 31 |
Loading start time | 21 | 1.2 | 18.5 | 3.9 | 7.5 | 8.6 | 11 | 10.9 | 20 | 7.5 | 19.5 |
Loading cutoff time | 21.7 | 2 | 19.2 | 4.5 | 8.3 | 9.4 | 12 | 11.6 | 21 | 8.6 | 21 |
Classification start time | 21.9 | 1.8 | 19 | 4.2 | 8 | 9 | 11.5 | 11.2 | 20.8 | 8.4 | 20.7 |
Classification cutoff time | 22.3 | 2.4 | 19.5 | 5 | 8.8 | 10 | 12.4 | 12 | 21.7 | 9.2 | 21.6 |
Departure time | 22.5 | 2.7 | 19.7 | 5.4 | 8.1 | 10.3 | 12.6 | 12.6 | 22 | 9.6 | 22 |
Destination | 26 | 27 | 30 | 30 | 30 | 30 | 30 | 31 | 32 | 32 | 32 |
Arrival time | 23 | 3.1 | 20.4 | 6.2 | 8.9 | 11 | 13.3 | 14 | 22.7 | 12.4 | 22.7 |
Disassembly start time | 23.3 | 3.5 | 21 | 6.5 | 9.1 | 11.4 | 13.7 | 14.5 | 30 | 12.8 | 23 |
Disassembly cutoff time | 23.9 | 4.1 | 21.5 | 7.1 | 9.8 | 12.3 | 14.4 | 15.2 | 30.8 | 14 | 23.8 |
Unloading start time | 23.7 | 3.9 | 21.4 | 6.7 | 9.5 | 11.7 | 14 | 14.8 | 30.4 | 13.4 | 23.4 |
Unloading cutoff time | 24.2 | 4.6 | 22.2 | 7.5 | 10.4 | 12.7 | 15 | 15.8 | 31.4 | 15 | 24.2 |
Distance (unit: km) | 13 | 8 | 53 | 53 | 53 | 53 | 53 | 88 | 10 | 10 | 10 |
Capacity (unit: ton) | 879 | 1930 | 1036 | 2143 | 778 | 1456 | 1220 | 1696 | 1987 | 972 | 762 |
Population exposure (unit: 104 people) | 9.49 | 5.84 | 27.32 | 27.32 | 27.32 | 27.32 | 27.32 | 45.36 | 4.94 | 4.94 | 4.94 |
Environmental capacity (unit: 104 ton) | 10.45 | 6.43 | 42.60 | 42.60 | 42.60 | 42.60 | 42.60 | 70.74 | 8.04 | 8.04 | 8.04 |
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No. | Origin | Destination | Volume (Unit: Ton) | Release Time | Due Date |
---|---|---|---|---|---|
1 | 1 | 33 | 270 | 0 | 9.5 |
2 | 1 | 34 | 120 | 8 | 14 |
3 | 1 | 36 | 60 | 5 | 26 |
4 | 1 | 39 | 180 | 14 | 40 |
5 | 1 | 40 | 120 | 6 | 25 |
6 | 1 | 41 | 90 | 14 | 39 |
7 | 1 | 42 | 120 | 28 | 47 |
8 | 1 | 43 | 210 | 32.5 | 40 |
9 | 1 | 44 | 270 | 24.5 | 48 |
10 | 1 | 45 | 90 | 1 | 25 |
11 | 2 | 33 | 60 | 0 | 58 |
12 | 2 | 34 | 180 | 25 | 60 |
13 | 2 | 35 | 270 | 8 | 19 |
14 | 2 | 36 | 250 | 13.9 | 39 |
15 | 2 | 37 | 90 | 21.5 | 45.5 |
16 | 2 | 38 | 180 | 27 | 41 |
17 | 2 | 41 | 210 | 30 | 42 |
18 | 3 | 35 | 150 | 3.2 | 19 |
19 | 3 | 44 | 120 | 23 | 41 |
20 | 3 | 44 | 210 | 28 | 42 |
21 | 3 | 45 | 180 | 8.5 | 23 |
22 | 4 | 35 | 120 | 18 | 30.5 |
23 | 4 | 44 | 210 | 12.5 | 31 |
24 | 4 | 45 | 240 | 0 | 9 |
25 | 4 | 45 | 270 | 5 | 15 |
Total Variables | Integer Variables | Constraints |
---|---|---|
22,403 | 8925 | 60,006 |
No. | Multimodal Routes | Arrival Time at Destination |
---|---|---|
1 | 1 33 | 5 |
2 | 1 34 | 11.3 |
3 | 1 36 | 8.2 |
4 | 1 9 12 16 39 | 36 |
5 | 1 6 17 18 40 | 23 |
6 | 1 9 12 16 41 | 37.3 |
7 | 1 42 | 30.3 |
8 | 1 43 | 34.8 |
9 | 1 8 26 44 | 47.6 |
10 | 1 8 26 45 | 24.5 |
11 | 2 6 8 20 22 33 | 8.7 |
12 | 2 8 21 34 | 45.1 |
13 | 2 35 | 11 |
14 | 2 36 | 16.9 |
15 | 2 37 | 23.4 |
16 | 2 38 | 28.8 |
17 | 2 12 16 41 | 37.3 |
18 | 3 35 | 5.9 |
19 | 3 29 31 44 | 39.1 |
20 | 3 29 31 44 | 39.1 |
21 | 3 45 | 10.8 |
22 | 4 35 | 20.8 |
23 | 4 44 | 14.4 |
24 | 4 45 | 2.8 |
25 | 4 45 | 7.8 |
Performance | Scenario 1: Min Objective 1 | Scenario 2: Min Objective 2 |
---|---|---|
Objective 1 | 850,192 * | 906,037 |
Objective 2 | 621,099 | 506,362 * |
Environmental risk | 0.553 | 0.459 |
Solver state | Global opt | Global opt |
Computational time | 2 min 51 s ** | 2 min 38 s ** |
© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Sun, Y.; Lang, M.; Wang, D. Bi-Objective Modelling for Hazardous Materials Road–Rail Multimodal Routing Problem with Railway Schedule-Based Space–Time Constraints. Int. J. Environ. Res. Public Health 2016, 13, 762. https://doi.org/10.3390/ijerph13080762
Sun Y, Lang M, Wang D. Bi-Objective Modelling for Hazardous Materials Road–Rail Multimodal Routing Problem with Railway Schedule-Based Space–Time Constraints. International Journal of Environmental Research and Public Health. 2016; 13(8):762. https://doi.org/10.3390/ijerph13080762
Chicago/Turabian StyleSun, Yan, Maoxiang Lang, and Danzhu Wang. 2016. "Bi-Objective Modelling for Hazardous Materials Road–Rail Multimodal Routing Problem with Railway Schedule-Based Space–Time Constraints" International Journal of Environmental Research and Public Health 13, no. 8: 762. https://doi.org/10.3390/ijerph13080762