# Routing for Hazardous Materials Transportation in Urban Areas

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## Abstract

**:**

## 1. Introduction

## 2. An Optimization Model

#### 2.1. Problem Hypotheses

**Hypothesis**

**1:**

**Hypothesis**

**2:**

**Hypothesis**

**3:**

**Hypothesis**

**4:**

**Hypothesis**

**5:**

#### 2.2. Model Establishment

- $S$:
- Set of HAZMAT shipments, indexed by $s$, and the number of shipments is $\lambda $.
- ${R}^{s}$:
- Auxiliary decision variable, maximum local risk caused by the closest link of shipment $s$ to population centers, unit: persons per kilometer.

- ${n}^{s}$:
- Transportation demand of shipment $s$, measured by the number of standardized vehicles;
- $i,\text{}j$:
- Nodes of road transportation network in the urban area, and the number of nodes is $\epsilon $;
- $A$:
- Set of links of road transportation network in the urban area, $(i,\text{}j)\in A$, and the number of links is $\phi $;
- ${N}^{c}$:
- Set of population centers, indexed by $c$;
- ${\rho}^{c}$:
- Population quantity of center $c$, unit: persons, and the number of centers is $\mu $;
- ${d}_{cs}^{ij}$:
- Euclidean distance between center $c$ and its closest point on link $(i,\text{}j)$ of shipment $s$, illustrated in Figure 1, unit: kilometers;
- ${v}_{cs}^{ij}$:
- Decision variable, which is equal to 1 if link $(i,\text{}j)$ used for HAZMAT shipment $s$ is the closest link to $c$, 0 otherwise.

- ${x}_{s}^{ij}$:
- Decision variable, which is equal to 1 if link $(i,\text{}j)$ is used for HAZMAT shipment $s$, 0 otherwise.

- ${v}_{cs}^{ij}$:
- This notation has been explained in Formula (2).

- ${v}_{cs}^{mn}$:
- Referring to the definition of ${v}_{cs}^{ij}$;
- ${d}_{cs}^{mn}$:
- Referring to the definition of ${d}_{cs}^{ij}$.

- $o(s)$:
- Origin of HAZMAT shipment $s$;
- $d(s)$:
- Destination of HAZMAT shipment $s$.

- ${R}^{s}$:
- Nonnegative constraint for auxiliary decision.

- ${x}_{s}^{ij}$:
- Constrained to be binary.

- ${v}_{cs}^{ij}$:
- Constrained to be binary.

## 3. Algorithm Design

**Step 1:**- Initialization. List the structure of the transportation network $\left(N,A\right)$ including nodes and links, OD pairs for HAZMAT shipments and population centers in the area.
**Step 1-1:**- Set the counter $t=0$, and determine the risk radius ${\gamma}^{s}$ according to the type of HAZMAT for all shipments.
**Step 1-2:**- $\forall \text{}s\in S,\text{}c\in {N}^{c},\text{}(i,j)\in A|{d}_{cs}^{ij}\le {\gamma}^{s}$, compute the local risk along the route of shipment $s$ as ${\omega}_{cs}^{ij}$. While $\forall \text{}s\in S,\text{}c\in {N}^{c}$, $(i,j)\in A|{d}_{cs}^{ij}>{\gamma}^{s}$, set ${\omega}_{cs}^{ij}=0$.
**Step 1-3:**- For all $s\in S,\text{}(i,j)\in A$ in the HAZMAT transportation network, compute the maximum local risk along the route of shipment $s$ as ${\omega}_{s}^{ij}$.

**Step 2:**- Initial Route. For all $s\in S$, the Dijkstra algorithm is used to search for a shortest route between $o(s)$ and $d(s)$ [23], and the route is denoted as ${F}_{s}^{0}$ for shipment $s$. Thereafter, the maximum local risk to population centers along route ${F}_{s}^{0}$ is calculated as ${\omega}^{{F}_{s}^{0}}$.
**Step 2-1:**- If ${\omega}^{{F}_{s}^{0}}=0$, ${F}_{s}^{0}$ is an optimal solution route for shipment $s$, and ${R}^{s}=0$. The computation for shipment $s$ stops, turn to next shipment until all shipments are checked.
**Step 2-2:**- If ${\omega}^{{F}_{s}^{0}}>0$, there is possible to search for a better route. For the corresponding $s\in S$ turn to Step 3 for computation.

**Step 3:**- Route update. The initial route is improved by the following sub-steps.
**Step 3-1:**- $\forall \text{}s\in S$, the set of links is obtained as ${A}_{s}^{t}=A-\left\{(i,j)\in A|{\omega}_{s}^{ij}\ge {\omega}^{{F}_{s}^{t}}\right\}$. When $t=0$, ${A}_{s}^{t-1}=A$. Denote ${N}_{s}^{t}$ as the set of relevant nodes, and the candidate graph in ${t}_{th}$ iteration for shipment $s$ is $\left({N}_{s}^{t},{A}_{s}^{t}\right)$. Thereafter, the counter $t=t+1$.
**Step 3-2:**- $\forall \text{}s\in S$, search for a feasible route ${F}_{s}^{t}$ over $\left({N}_{s}^{t},{A}_{s}^{t}\right)$ between $o(s)$ and $d(s)$.
**Step 3-2-1:**- If no feasible route ${F}_{s}^{t}$ is found, the candidate graph in ${t}_{th}$ iteration for shipment $s$ is disconnected, ${F}_{s}^{t-1}$ is an optimal solution route for shipment $s$, and ${R}^{s}={\omega}^{{F}_{s}^{t-1}}$. The computation for shipment $s$ stops, turn to next shipment until all shipments are checked.
**Step 3-2-2:**- If a feasible route ${F}_{s}^{t}$ is found, turn to Step 3-1.

**Step 4:**- Scheme check. In the completely connected transportation network, any shipment is able to be assigned an optimal route, and all optimal routes are the final output.

## 4. Case Study

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Center | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 |

Population | 4270 | 4910 | 5180 | 7100 | 6850 | 5190 | 6230 | 10020 |

Center | C9 | C10 | C11 | C12 | C13 | C14 | C15 | C16 |

Population | 9470 | 5960 | 8030 | 12030 | 7310 | 7210 | 5540 | 14640 |

Shipment | S1 | S2 | S3 | S4 | S5 | S6 |

OD pair | N1-N226 | N2-N305 | N3-N271 | N4-N225 | N5-N193 | N6-N236 |

Demand | 10 | 15 | 20 | 25 | 10 | 15 |

Shipment | S7 | S8 | S9 | S10 | S11 | S12 |

OD pair | N7-N164 | N8-N322 | N9-N150 | N10-N297 | N11-N148 | N12-N249 |

Demand | 20 | 25 | 10 | 15 | 20 | 25 |

**Table 3.**Comparison between deterministic solving and the heuristic algorithm. (Risk radius = 1.0 km).

Index | Shipment Set | Deterministic Solving by CPLEX 12.8 | The Proposed Heuristic Algorithm | Gap |
---|---|---|---|---|

Optimal objective (persons/km) | S1–S4 | 0.6335 × 10^{6} | 0.7156 × 10^{6} | 12.97% |

S1–S8 | 1.3371 × 10^{6} | 1.5075 × 10^{6} | 12.74% | |

S1–S12 | 1,9268 × 10^{6} | 2.1619 × 10^{6} | 12.20% | |

Max R^{s} (persons/km) | S1–S4 | 2.5083 × 10^{5} | 2.6241 × 10^{5} | 4.62% |

S1–S8 | 2.5083 × 10^{5} | 2.6241 × 10^{5} | 4.62% | |

S1–S12 | 2.7358 × 10^{5} | 2.7358 × 10^{5} | 0.00% | |

Run time (s) | S1–S4 | 1.8729 × 10^{4} | 0.8461 × 10^{4} | 121.36% |

S1–S8 | 3.6114 × 10^{4} | 1.3715 × 10^{4} | 163.32% | |

S1–S12 | 5.3782 × 10^{4} | 1.9295 × 10^{4} | 178.73% |

Risk Radius (km) | Optimal Objective (persons/km) | Max R^{s} (persons/km) |
---|---|---|

0.5 | 2.2348 × 10^{6} | 3.0155 × 10^{5} |

1.0 | 2.1619 × 10^{6} | 2.7358 × 10^{5} |

1.5 | 2.0862 × 10^{6} | 2.5183 × 10^{5} |

2.0 | 2.0047 × 10^{6} | 2.4924 × 10^{5} |

2.5 | 1.9490 × 10^{6} | 2.4418 × 10^{5} |

3.0 | 1.9205 × 10^{6} | 2.3561 × 10^{5} |

3.5 | 1.9205 × 10^{6} | 2.3561 × 10^{5} |

4.0 | 1.9205 × 10^{6} | 2.3561 × 10^{5} |

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**MDPI and ACS Style**

Zhang, L.; Feng, X.; Yang, Y.; Ding, C.
Routing for Hazardous Materials Transportation in Urban Areas. *Symmetry* **2019**, *11*, 1091.
https://doi.org/10.3390/sym11091091

**AMA Style**

Zhang L, Feng X, Yang Y, Ding C.
Routing for Hazardous Materials Transportation in Urban Areas. *Symmetry*. 2019; 11(9):1091.
https://doi.org/10.3390/sym11091091

**Chicago/Turabian Style**

Zhang, Lukai, Xuesong Feng, Yan Yang, and Chuanchen Ding.
2019. "Routing for Hazardous Materials Transportation in Urban Areas" *Symmetry* 11, no. 9: 1091.
https://doi.org/10.3390/sym11091091