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Aiming at route optimization problem of hazardous materials transportation in uncertain environment, this paper presents a multiobjective robust optimization model by taking robust control parameters into consideration. The objective of the model is to minimize not only transportation risk but also transportation time, and a robust counterpart of the model is introduced through applying the Bertsimas-Sim robust optimization theory. Moreover, a fuzzy C-means clustering-particle swarm optimization (FCMC-PSO) algorithm is designed, and the FCMC algorithm is used to cluster the demand points. In addition the PSO algorithm with the adaptive archives grid is used to calculate the robust optimization route of hazmat transportation. Finally, the computational results show the multiobjective route robust optimization model with 3 centers and 20 demand points’ sample studied and FCMC-PSO algorithm for hazmat transportation can obtain different robustness Pareto solution sets. As a result, this study will provide basic theory support for hazmat transportation safeguarding.

Hazardous materials (hazmat) refer to products with flammable, poisonous, and corrosive properties that can cause casualties, damage to properties, and environmental pollution and require special protection in the process of transportation, loading, unloading, and storage. In recent years, the demand for hazmat has increased, its freight volume has increased year by year, and the potential transportation risk is also expanding. Practice has proved that the optimization of the transportation route of hazmat can effectively reduce the transportation risk, and it has significant influence to ensure the safety of people along route and protect the surrounding ecological environment.

Many scholars have studied the transportation route optimization problem for hazmat. Rhyne (1994) conducted a statistical analysis of the hazmat transport accident using the diffusion formula [_{.} Wang et al. (2009) established a hazmat transportation path optimization model based on a geographic information system [

Ben-Tal and Neimirovski (1998) proposed robust optimization theory based on ellipsoid uncertainty set [

The rest of this paper is structured as follows: Section

Hazmat transportation route robust optimization for multidistribution center is defined as follows: there are several hazmat distribution centers, and each distribution center owns enough hazmat transport vehicles; meanwhile, multiple need points exist which should be assigned to the relevant hazmat distribution center. Vehicles from distribution center will service the corresponding demand points. Each vehicle can service several customer demand points while each customer demand point only can be serviced by one vehicle. After completing the transport mission, the vehicles must return to the distribution center [

Uncertainty of hazmat transport refers to the uncertainty of the transportation time and transportation risk, which may be caused by the traffic accident, weather, and traffic density of the road. Compared to ordinary goods transport, hazmat transport is more complicated, and it needs more security demands. Therefore, it is needed to set the goal of minimizing the total hazmat transportation risk. In the process of hazmat transportation, transport time reduction is also necessary. As a consequence, this paper will target minimizing the total transportation time. In conclusion, the scientific transportation routes should be found to guarantee the hazmat transported safely and quickly.

There are a few assumptions in this study:

(1) Multiple hazmat distribution centers are existent

(2) The supply of hazmat distribution centers is adequate

(3) Vehicle loading capacity is provided and the demand of each customer is specified

(4) Multiple vehicles of the distribution center can service the customer

(5) The transportation risk and transportation time are identified among the customer demand points, but they are uncertain number as interval number

_{i}^{r}: the set of column subscripts

Γ_{i}^{t}: parameter

_{i}^{t}: the set of column subscript

_{max} set by decision makers. Constraint (_{max} set by decision makers. Constraint (

Each objective function of the above multiobjective robust model corresponds to parameter Γ. The purpose is to control the degree of conservatism of the solution. Objective functions (_{vrp} to satisfy all constraints, and robust discrete optimization criterion is used to transform the multiobjective route robust optimization model, and a new robust counterpart of the model is as follows [

Objective function is

Constraint condition is

Then, the optimal objective function value can be obtained as

In this section, we propose FCMC-PSO algorithm to solve the multiobjective route optimization problem of hazmat transportation in uncertain environment. The demand points is clustered by the fuzzy C means algorithm, and the transportation route for each demand points is determined based on the adaptive archives grid multiobjective particle swarm optimization [

Suppose that n data samples are _{k} (abbreviated as _{i} and the

Formulas (_{i} for the category_{k} and C clustering centers

Let

Using formulas (

Particle swarm algorithm is derived from the study of the predatory behavior of birds. It is used to solve the problem of path optimization. Each particle in the algorithm represents a potential solution, and the fitness value for each particle is determined by the fitness function, and the value of fitness determines the pros and cons of the particle. The particle moves in the N-dimensional solution space and updates the individual position by the individual extremum and the group extremum. In the algorithm, the velocity, position, and fitness value are used to represent the characteristics of the particle. The velocity of the particle determines the direction and distance of the particle movement, and the velocity is dynamically adjusted with the moving experience of its own and other particles. Once the position of the particle is updated, the fitness value will be calculated, and the individual extremum and the population extremum are updated by the fitness values of the new particles, the individual extremum, and the population extremum. Multiobjective particle swarm optimization algorithm is a method based on particle swarm optimization algorithm to solve multiobjective problem. At the same time, the best location of multiple populations exists in the population, and the optimal positions of multiple particles themselves are also found in the iterative process. Therefore,

Crossover operation.

The new individuals need to be adjusted if there is a duplicate position, and the adjustment method is to replace the repeated demand points by using the absence of demand points in individuals. For the new individual 1, there are mappings about 2 to 3, 9 to 6, and 5 to 7. The specific adjustments process can be seen in Figure

Adjustment operation.

The strategy of retaining outstanding individuals is used for the owned new individuals, and the particles are updated only when the new particle fitness is better than the old particles.

The variation is also related to the way the particle is encoded. Based on the lease point and the dispatch center, there are many methods about the coarranged coding and the variation. In this paper, the variation method adopts the individual internal exchange method. For example, for an individual [9 2 3 4 1 5 6 7 8], at first, the mutated positions pos1 and pos2 are selected randomly; and then the positions of two variants are swapped. Assuming that the selected mutation positions are 2 and 6, the mutation operation process can be seen in Figure

Mutation operation.

The strategy of retaining outstanding individuals is used for the owned new individuals, and the particles are updated only when the new particle fitness is better than the old particles [

Create and initialize a group so that the

Evaluate all particles and add the noninferior solution to the external file.

Maintain external files according to the adaptive grid method.

Select

Updating the velocity and position of the particles according to the speed formula and the position formula of the particle swarm.

Make sure the particles exist in the search space.

If the termination condition is satisfied, the output result algorithm is terminated; if it is not satisfied, Step

There are 3 distribution centers and 20 demand points, the maximum load for each transport vehicle is 8 ton, and each distribution center has adequate hazmat. The distribution centers are marked as a, b, and c, and the hazmat demand points are marked as 1, 2,..., 20. The demand amount of each demand point is shown in Table

Customer demands.

Demand | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

| ||||||||||||||||||||

Demand | 2 | 1.5 | 4.5 | 3 | 1.5 | 4 | 2.5 | 3 | 3 | 4.5 | 2 | 1.5 | 2.5 | 3.5 | 2 | 2 | 2.5 | 3.5 | 2.5 | 3 |

The nominal transportation risk value of hazmat.

| a | b | c | 1 | 2 | 3 | …… | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|

a | 0 | 39 | 71 | 59 | …… | 38 | 80 | 52 | ||

b | 0 | 30 | 30 | 49 | …… | 74 | 67 | 78 | ||

c | 0 | 77 | 57 | 37 | …… | 49 | 34 | 64 | ||

1 | 39 | 30 | 77 | 0 | 32 | 30 | …… | 35 | 60 | 49 |

2 | 71 | 30 | 57 | 32 | 0 | 67 | …… | 33 | 65 | 55 |

3 | 59 | 49 | 37 | 30 | 67 | 0 | …… | 39 | 39 | 71 |

…… | …… | …… | …… | …… | …… | …… | …… | …… | …… | …… |

18 | 38 | 74 | 49 | 35 | 33 | 39 | …… | 0 | 55 | 34 |

19 | 80 | 67 | 34 | 60 | 65 | 39 | …… | 55 | 0 | 67 |

20 | 52 | 78 | 64 | 49 | 55 | 71 | …… | 34 | 67 | 0 |

The nominal transportation time value of hazamt.

| a | b | c | 1 | 2 | 3 | …… | 18 | 19 | 20 |
---|---|---|---|---|---|---|---|---|---|---|

a | 0 | 92 | 103 | 43 | …… | 107 | 97 | 44 | ||

b | 0 | 37 | 36 | 88 | …… | 95 | 57 | 115 | ||

c | 0 | 41 | 35 | 101 | …… | 40 | 74 | 64 | ||

1 | 92 | 37 | 41 | 0 | 75 | 102 | …… | 64 | 101 | 80 |

2 | 103 | 36 | 35 | 75 | 0 | 62 | …… | 108 | 84 | 95 |

3 | 43 | 88 | 101 | 102 | 62 | 0 | …… | 103 | 77 | 85 |

…… | …… | …… | …… | …… | …… | …… | …… | …… | …… | …… |

18 | 107 | 95 | 40 | 64 | 108 | 103 | …… | 0 | 52 | 43 |

19 | 97 | 57 | 74 | 101 | 84 | 77 | …… | 52 | 0 | 86 |

20 | 44 | 115 | 64 | 80 | 95 | 85 | …… | 43 | 86 | 0 |

We use FCMC algorithm to calculate the demand points clustering results, and the results can be seen in Table

Demand points clustering result.

Distribution center | Demand points |
---|---|

a | 3,9,14,15,16,20 |

b | 1,2,6,8,10,11,12 |

c | 4,5,7,13,17,18,19 |

Based on the cluster results, we use the multiobjective PSO to solve the robust optimization problem for each distribution center. The parameters of the algorithm are set as follows: population size is 100, maximum evolution generation is 1000, inertia weight is 0.6, accelerated factor is 1.7, crossover rate is 0.95, and mutation rate is 0.09. The Pareto solution set with different robust control parameters can be obtained by calculation, which are showed in Tables

Pareto solution set of robust control parameters

Encoding | Decoding | Total risk | Total time |
---|---|---|---|

9-20-14-3-15-16 | a-9-20-a-14-3-a-15-16 | 237 | 536 |

9-16-14-3-15-20 | a-9-16-a-14-3-a-15-20 | 261 | 448 |

15-20-9-14-3-16 | a-15-20-9-a-14-3-a-16 | 247 | 520 |

14-3-16-9-20-15 | a-14-3-a-16-9-20-a-15 | 253 | 469 |

15-14-16-9-3-20 | a-15-14-16-a-9-3-a-20 | 291 | 404 |

14-3-15-9-16-20 | a-14-3-a-15-9-16-a-20 | 285 | 425 |

Pareto solution set of robust control parameters

Encoding | Decoding | Total risk | Total time |
---|---|---|---|

3-14-20-9-16-15 | a-3-14-a-20-9-16-a-15 | 317.817 | 637.81 |

9-20-16-14-3-15 | a-9-20-16-a-14-3-a-15 | 319.893 | 621.749 |

9-20-14-16-3-15 | a-9-20-a-14-16-a-3-15 | 377.505 | 520.018 |

20-9-16-14-3-15 | a-20-9-16-a-14-3-a-15 | 337.903 | 532.141 |

9-3-15-14-16-20 | a-9-3-a-15-14-16-a-20 | 385.542 | 463.803 |

9-3-20-15-14-16 | a-9-3-a-20-15-a-14-16 | 354.151 | 527.306 |

9-20-14-3-15-16 | a-9-20-a-14-3-a-15-16 | 305.49 | 657.577 |

20-15-14-3-9-16 | a-20-15-a-14-3-a-9-16 | 359.107 | 525.349 |

Pareto solution set of robust control parameters

Encoding | Decoding | Total risk | Total time |
---|---|---|---|

3-14-20-9-16-15 | a-3-14-a-20-9-16-a-15 | 323.453 | 761.423 |

9-3-15-20-14-16 | a-9-3-a-15-20-a-14-16 | 381.844 | 560.087 |

9-20-16-14-3-15 | a-9-20-16-a-14-3-a-15 | 357.98 | 644 |

9-16-20-14-3-15 | a-9-16-20-a-14-3-a-15 | 391.378 | 558.043 |

9-20-14-16-3-15 | a-9-20-a-14-16-a-3-15 | 415.592 | 542.269 |

9-3-15-14-16-20 | a-9-3-a-15-14-16-a-20 | 423.629 | 486.054 |

9-20-15-14-3-16 | a-9-20-15-a-14-3-a-16 | 378.69 | 606.48 |

3-14-9-20-16-15 | a-3-14-a-9-20-16-a-15 | 316.995 | 866.785 |

9-3-20-15-14-16 | a-9-3-a-20-15-a-14-16 | 392.239 | 549.557 |

3-14-16-9-20-15 | a-3-14-a-16-9-20-a-15 | 336.135 | 750.943 |

9-20-14-3-15-16 | a-9-20-a-14-3-a-15-16 | 343.577 | 679.828 |

Pareto solution set of robust control parameters

Encoding | Decoding | Total risk | Total time |
---|---|---|---|

12-10-11-1-2-6-8 | b-12-10-11-b-1-2-6-b-8 | 289 | 582 |

11-10-1-12-6-2-8 | b-11-10-b-1-12-6-b-2-8 | 303 | 471 |

12-6-10-11-1-2-8 | b-12-6-b-10-11-b-1-2-8 | 302 | 530 |

12-10-11-1-6-8-2 | b-12-10-11-b-1-6-b-8-2 | 293 | 573 |

1-12-10-2-6-8-11 | b-1-12-10-b-2-6-b-8-11 | 296 | 572 |

8-6-12-10-11-1-2 | b-8-6-b-12-10-11-b-1-2 | 299 | 549 |

2-1-6-12-10-11-8 | b-2-1-6-b-12-10-11-b-8 | 284 | 627 |

8-6-1-12-2-10-11 | b-8-6-b-1-12-2-b-10-11 | 363 | 458 |

Pareto solution set of robust control parameters

Encoding | Decoding | Total risk | Total time |
---|---|---|---|

2-6-8-12-1-11-10 | b-2-6-b-8-12-1-b-11-10 | 415.472 | 552.167 |

1-6-2-11-10-12-8 | b-1-6-2-b-11-10-12-b-8 | 348.014 | 799.695 |

2-8-6-1-12-11-10 | b-2-8-b-6-1-12-b-11-10 | 399.364 | 631.219 |

1-12-6-11-10-8-2 | b-1-12-6-b-11-10-b-8-2 | 375.264 | 660.062 |

2-12-1-6-8-11-10 | b-2-12-1-b-6-8-b-11-10 | 485.04 | 533.589 |

2-8-6-12-1-11-10 | b-2-8-b-6-12-1-b-11-10 | 402.203 | 557.881 |

1-6-2-12-10-8-11 | b-1-6-2-b-12-10-b-8-11 | 346.521 | 872.417 |

1-6-12-11-10-8-2 | b-1-6-12-b-11-10-b-8-2 | 356.381 | 724.274 |

2-6-1-11-10-12-8 | b-2-6-1-b-11-10-12-b-8 | 381.268 | 645.953 |

Pareto solution set of robust control parameters

Encoding | Decoding | Total risk | Total time |
---|---|---|---|

2-6-1-12-10-8-11 | b-2-6-1-b-12-10-b-8-11 | 417.863 | 778.297 |

1-6-2-12-10-8-11 | b-1-6-2-b-12-10-b-8-11 | 352.157 | 1028.237 |

2-6-1-11-10-12-8 | b-2-6-1-b-11-10-12-b-8 | 419.356 | 677.732 |

1-6-2-11-10-12-8 | b-1-6-2-b-11-10-12-b-8 | 353.65 | 955.516 |

2-8-6-1-12-11-10 | b-2-8-b-6-1-12-b-11-10 | 437.452 | 662.997 |

2-8-6-12-1-11-10 | b-2-8-b-6-12-1-b-11-10 | 440.291 | 589.659 |

2-6-8-12-1-11-10 | b-2-6-b-8-12-1-b-11-10 | 453.559 | 583.946 |

1-6-12-11-10-8-2 | b-1-6-12-b-11-10-b-8-2 | 362.017 | 880.094 |

1-12-6-11-10-8-2 | b-1-12-6-b-11-10-b-8-2 | 380.9 | 815.883 |

2-12-1-6-8-10-11 | b-2-12-1-b-6-8-b-10-11 | 532.501 | 576.115 |

Pareto solution set of robust control parameters

Encoding | Decoding | Total risk | Total time |
---|---|---|---|

17-4-19-7-13-18-5 | c-17-4-19-c-7-13-c-18-5 | 278 | 678 |

7-5-18-17-4-19-13 | c-7-5-18-c-17-4-19-c-13 | 294 | 569 |

19-4-13-7-5-18-17 | c-19-4-13-c-7-5-18-c-17 | 298 | 566 |

7-4-13-17-19-5-18 | c-7-4-13-c-17-19-5-c-18 | 382 | 524 |

7-5-18-17-19-4-13 | c-7-5-18-c-17-19-4-c-13 | 305 | 545 |

17-4-13-19-5-7-18 | c-17-4-13-c-19-5-7-c-18 | 361 | 539 |

17-18-19-4-13-7-5 | c-17-18-c-19-4-13-c-7-5 | 320 | 544 |

18-5-19-17-4-13-7 | c-18-5-19-c-17-4-13-c-7 | 359 | 543 |

19-5-7-17-18-13-4 | c-19-5-7-c-17-18-c-13-4 | 379 | 530 |

Pareto solution set of robust control parameters

Encoding | Decoding | Total risk | Total time |
---|---|---|---|

18-17-13-4-19-7-5 | c-18-17-c-13-4-19-c-7-5 | 433.965 | 640.855 |

18-5-7-17-4-19-13 | c-18-5-7-c-17-4-19-c-13 | 374.262 | 714.495 |

4-19-13-7-5-18-17 | c-4-19-13-c-7-5-18-c-17 | 328.774 | 1068.644 |

18-17-19-4-13-7-5 | c-18-17-c-19-4-13-c-7-5 | 432.616 | 649.645 |

18-17-13-4-19-5-7 | c-18-17-c-13-4-19-c-5-7 | 442.263 | 632.41 |

5-18-7-17-4-19-13 | c-5-18-7-c-17-4-19-c-13 | 361.711 | 813.376 |

18-17-7-5-19-13-4 | c-18-17-c-7-5-19-c-13-4 | 565.935 | 629.522 |

18-5-13-17-4-19-7 | c-18-5-13-c-17-4-19-c-7 | 369.796 | 741.803 |

13-4-18-17-19-5-7 | c-13-4-c-18-17-c-19-5-7 | 538.139 | 632.006 |

18-17-13-19-4-7-5 | c-18-17-c-13-19-4-c-7-5 | 415.204 | 661.122 |

4-19-13-17-18-5-7 | c-4-19-13-c-17-18-5-c-7 | 345.113 | 937.323 |

18-17-13-19-4-5-7 | c-18-17-c-13-19-4-c-5-7 | 421.737 | 652.677 |

4-19-13-17-5-18-7 | c-4-19-13-c-17-5-18-c-7 | 333.451 | 975.738 |

4-19-13-17-18-7-5 | c-4-19-13-c-17-18-c-7-5 | 351.435 | 928.3 |

18-5-7-19-4-17-13 | c-18-5-7-c-19-4-17-c-13 | 388.454 | 682.357 |

Pareto solution set of robust control parameters

Encoding | Decoding | Total risk | Total time |
---|---|---|---|

18-5-13-17-4-19-7 | c-18-5-13-c-17-4-19-c-7 | 418.386 | 797.49 |

13-4-19-17-18-7-5 | c-13-4-19-c-17-18-c-7-5 | 565.673 | 677.462 |

13-4-19-17-18-5-7 | c-13-4-19-c-17-18-5-c-7 | 558.373 | 686.485 |

13-4-19-18-17-7-5 | c-13-4-19-c-18-17-c-7-5 | 588.331 | 674.181 |

18-17-13-19-4-7-5 | c-18-17-c-13-19-4-c-7-5 | 463.794 | 716.809 |

4-19-13-17-5-18-7 | c-4-19-13-c-17-5-18-c-7 | 339.087 | 1301.559 |

18-17-19-4-13-7-5 | c-18-17-c-19-4-13-c-7-5 | 481.205 | 705.332 |

4-19-13-17-18-5-7 | c-4-19-13-c-17-18-5-c-7 | 350.749 | 1263.143 |

18-17-13-4-19-7-5 | c-18-17-c-13-4-19-c-7-5 | 482.554 | 696.543 |

18-17-13-19-4-5-7 | c-18-17-c-13-19-4-c-5-7 | 470.326 | 708.364 |

4-19-13-18-17-7-5 | c-4-19-13-c-18-17-c-7-5 | 379.729 | 1250.84 |

4-19-13-17-18-7-5 | c-4-19-13-c-17-18-c-7-5 | 357.071 | 1254.121 |

4-19-13-7-5-18-17 | c-4-19-13-c-7-5-18-c-17 | 334.41 | 1394.464 |

4-17-19-18-5-7-13 | c-4-17-19-c-18-5-7-c-13 | 405.101 | 1245.591 |

18-5-7-17-19-4-13 | c-18-5-7-c-17-19-4-c-13 | 449.149 | 738.192 |

18-17-13-4-19-5-7 | c-18-17-c-13-4-19-c-5-7 | 490.853 | 688.097 |

5-18-7-17-4-19-13 | c-5-18-7-c-17-4-19-c-13 | 409.798 | 895.155 |

18-5-7-17-4-19-13 | c-18-5-7-c-17-4-19-c-13 | 422.852 | 770.182 |

Pareto optimal solution distribution of robust control parameters

Pareto optimal solution distribution of robust control parameters

Pareto optimal solution distribution of robust control parameters

In Table

When the robustness control parameters

Pareto optimal solutions for distribution center a when

| Distribution center a | |
---|---|---|

Risk optimal route | Time optimal route | |

0 | a-9-20-a | a-15-14-16-a |

a-14-3-a | a-9-3-a | |

a-15-16-a | a-20-a | |

| ||

10 | a-9-20-a | a-15-14-16-a |

a-14-3-a | a-9-3-a | |

a-15-16-a | a-20-a | |

| ||

20 | a-9-20-16-a | a-15-14-16-a |

a-14-3-a | a-9-3-a | |

a-15-a | a-20-a |

Pareto optimal solutions for distribution center b when

| Distribution center b | |
---|---|---|

Risk optimal route | Time optimal route | |

0 | b-2-1-6-b | b-8-6-b |

b-12-10-11-b | b-1-12-2-b | |

b-8-b | b-10-11-b | |

| ||

10 | b-1-6-2-b | b-8-6-b |

b-12-10-b | b-2-12-1-b | |

b-8-11-b | b-10-11-b | |

| ||

20 | b-1-6-2-b | b-8-6-b |

b-12-10-b | b-2-12-1-b | |

b-8-11-b | b-10-11-b |

Pareto optimal solutions for distribution center c when

| Distribution center c | |
---|---|---|

Risk optimal route | Time optimal route | |

0 | c-17-4-19-c | c-7-4-13-c |

c-7-13-c | c-17-19-5-c | |

c-18-5-c | c-18-c | |

| ||

10 | c-4-19-13-c | c-13-4-c |

c-7-5-18-c | c-7-5-19-c | |

c-17-c | c-18-17-c | |

| ||

20 | c-4-19-13-c | c-13-4-19-c |

c-7-5-18-c | c-7-5-c | |

c-17-c | c-18-17-c |

For solution robustness, as the robustness control parameters get bigger, the corresponding Pareto solution robustness should be enhanced in theory. After a large number of analyses for the actual situation and basic data, we can determine the robustness control parameter value and obtain the corresponding candidate route set.

The strength Pareto genetic algorithm (SPEA) is used to test the efficiency of the FCMC-PSO algorithm. The parameters of the SPEA are set as follows: population size is 50, maximum evolution generation is 300, crossover rate is 0.8, and mutation rate is 0.05. The solution set can be obtained by calculation. The results are shown in Table

Performance comparison between FCMC-PSO algorithm and SPEA.

Index | FCMC-PSO | SPEA | ||||
---|---|---|---|---|---|---|

| 0 | 20 | 40 | 0 | 20 | 40 |

Convergence Iterations | 33 | 39 | 51 | 58 | 72 | 97 |

Run time/(s) | 21 | 33 | 49 | 40 | 52 | 88 |

Hazmat transportation route optimization is an important link to ensure transportation safety of hazmat. In this paper, we take the hazmat transportation route problem with multidistribution center as the research object, considering the transportation risk and transportation time. In addition, an adjustable robustness transportation route multiobjective robust optimization model is established in the end. Speaking of the solution, the FCMC-PSO algorithm is designed in this research. The demand points were assigned through FCMC algorithm in which transportation time and transportation risk are considered. The multiobjective route robust optimization model is solved by multiobject PSO algorithm based on adaptive archives grid. In the end, the example shows that the robust optimization model and FCMC-PSO algorithm can obtain different robustness Pareto solution sets. The robust optimization transportation routes of hazmat will provide basic theory support for safeguarding the transportation safety of hazmat.

In this paper, we only consider the two uncertainties including transportation risk and transportation time. However, there may be some other uncertainties such as customer demand and service time window in the real world. According to this situation, we need to establish the corresponding robust model for future study. Although most of the hazmats are transported by road transportation, the hazmat transportation risks induced by other modes cannot be ignored. The optimization research of multimodal transport modes for the hazmat needs to be further studied.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was funded by National Natural Science Foundation of China (no. 71861023 and no. 51808057), the Program of Humanities and Social Science of Education Ministry of China (no. 18YJC630118), Lanzhou Jiaotong University (no. 201804), and Hunan Key Laboratory of Smart Roadway and Cooperative Vehicle-Infrastructure Systems (no. 2017TP1016).