# A Distributionally Robust Chance-Constrained Approach for Modeling Demand Uncertainty in Green Port-Hinterland Transportation Network Optimization

^{*}

## Abstract

**:**

_{2}emissions is contrary to that in total network costs. Additionally, both network costs and network emissions increase significantly with the growth of the lower bound of probability for chance constraint. The higher the probability level grows, the greater the trade-offs between two objectives are influenced, which indicates that the operation capacity of inland intermodal terminals should be increased to meet the high probability level. These findings can help provide decision supports for the green development strategy of the port-hinterland container transportation network, which meanwhile faces a dynamic planning problem caused by stochastic demands in real life.

## 1. Introduction

_{2}pricing, terminal network configuration, and hub-service networks in freight transport optimization model and used the case of hinterland container transport in the Netherlands to calibrate and validate the model. On top of that, there were also a few articles analyzing the trade-off relationship between multiple objectives (such as cost, time, emissions, and so on) in the port-hinterland freight logistics field. Lam and Gu [15] analyzed the trade-offs between cost and time in multimodal transport network optimization under different limitations on total network emission. Demir et al. [16] discussed the modeling of transportation planning incorporating environment criteria and present a bi-objective hinterland intermodal transportation model. However, the trade-offs analysis has been insufficient and can be further explored.

## 2. The Distributionally Robust Chance Constrained Bi-Objective Modeling

#### 2.1. ProblemStatement

#### 2.2. Notations

#### 2.3. Model Formulation

_{2}emissions of the network, respectively. The former includes the transportation costs on the transportation routes and the handling as well as storage costs at the inland intermodal terminals. The latter consists of the corresponding route emissions and terminal handling CO

_{2}emissions. A bi-objective optimization model for the port-hinterland container intermodal transportation network with chance constraint can be constructed as follows:

_{2}emissions of the network, respectively. Constraint (3) is the chance constraint, which ensures that the total amount of goods transported from inland city to all seaports meets the worst distribution of the transportation demand of each city. Constraint (4) indicates that the quantity of containers routed from the city to the assigned intermodal terminal is the sum of the volume through road–rail or road–waterway intermodal transportation and the volume through inter-terminal transportation. Constraint (5) ensures the balance of goods entering and leaving the inland intermodal terminals. Constraints (6) and (7) are the container handling capacity limitations of inland intermodal terminals and gateway seaports, respectively. Constraints (8)–(10) are non-negative integer constraints of decision variables.

#### 2.4. Model Transformation

**Lemma**

**1.**

**Proof.**

- (1)
- $f({D}_{i})\ge 0$, for any ${D}_{i}$;
- (2)
- $f({D}_{i})\ge 1$, for any ${D}_{i}$, which satisfies $\sum _{s\in S}{Q}_{is}^{}}+{\displaystyle \sum _{j\in H}{\displaystyle \sum _{s\in S}{Q}_{ijs}^{}}}+{\displaystyle \sum _{j\in H}{\displaystyle \sum _{k\ne j\in H}{\displaystyle \sum _{s\in S}{Q}_{ijks}^{}}}}\ge {D}_{i$.

**Theorem**

**1.**

**Proof.**

- (1)
- $\sqrt{\frac{1-\beta}{\beta}}\sqrt{{x}^{\mathrm{T}}{\mathsf{\Gamma}}_{i}x}-{u}^{\mathrm{T}}x\le \gamma $;
- (2)
- there is asymmetric matrix $M$ and $\tau \in {\mathbb{R}}^{+}$, which means that$\begin{array}{c}\langle M,\sum \rangle \le \tau \beta ,\\ \tau \ge 0,\\ M\succcurlyeq 0,\\ M+\left[\begin{array}{cc}0& x\\ x& 2\gamma -\tau \end{array}\right]\succcurlyeq 0,\end{array}$

#### 2.5. Model Solution

## 3. Case Study

#### 3.1. CaseDescription and Data Collection

_{2}emissions rate varies from country to country and the work of Die Zhang [29], which reflects the emissions situation of inland transportation activities in China, is referred in this paper. The carbon emissions factor of each transportation mode is shown in Table 6. The carbon emissions factor for transshipment at inland intermodal terminals is estimated as 5.8 kg/TEU, according to China Port Yearbook.

#### 3.2. Experimental Results

#### 3.2.1. Results in Different Objective Optimization and Trade-Off Relationship Analysis ($\alpha =0.90$)

_{2}emissions that the transportation network could achieve is 3.238 million tons, which is approximately 27.2% lower than the network emissions in the lowest cost model, and the corresponding total network cost is 3511.2 million dollars.

#### 3.2.2. Sensitivity Analysis of Probability Levels of Chance Constraint

## 4. Discussion

_{2}minimum emissions case. It is an interesting finding for intermodal freight transport, because it implies that there is competition in hybrid intermodal transportation alternatives, rather than the only competition in unimodal and single intermodal options in most intermodal transport research articles (such as the works of Crainic et al. [5], Santoset al. [13], and Bouchery and Fransoo [30]). This finding might provide a different perspective for the port-hinterland intermodal transportation network optimization with the greenness consideration.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The proposed port-hinterland intermodal transportation network. FRS, freight rail station; IRP, inland river port; IC, inland city; GP, gateway seaport.

**Figure 3.**Trends of Pareto frontier between total costs and total emissions under different settings on $\alpha $.

Description | |
---|---|

I | Set of inland cities, indexed by i |

S | Set of gateway seaports, indexed by s |

H | Set of inland intermodal terminals, indexed by j, k, H = H_{W}∪H_{R} |

H_{W} | Set of inland river ports, indexed by j, k |

H_{R} | Set of dry ports, indexed by j, k |

M | Set of transportation modes, indexed by m, m′∈{1,2,3}, {1} = truck, {2} = barge, {3} = rail |

Description | |
---|---|

Q_{is} | TEU flows from inland city i to gateway seaport s directly by road, ∀i∈I, ∀s∈S |

Q_{ijs} | TEU flows from inland city i to gateway seaport s, only transshipping at inland intermodal terminal j with the long-haul travel by barge or rail, ∀i∈I, ∀s∈S, ∀j∈H |

Q_{ijks} | TEU flows from inland city i, firstly collected to inland intermodal terminal j and then routed through terminal k, and finally arrived at gateway seaport s, ∀i∈I, ∀s∈S, ∀j,k∈H |

Description | |
---|---|

D_{i} | Transportation demand of city i |

${\mu}_{i}$ | The mean of transportation demand for city i |

${\sigma}_{i}{}^{2}$ | The variance of transportation demand forcity i |

p | A probability distribution ofparameter of transportation demand |

Γ | Set of probability distributions p |

α | The lower bound of probability for chance constraint |

X_{ij} | Total TEU volume of goods from city i to the assigned intermodalterminal j |

Y_{ks} | Total TEU volume of goods from the final intermodalterminal k to gateway seaport s |

C_{is}^{m}, d_{is}^{m} | Unit transport cost, transport distance from inland city i to gateway seaport s by transportation mode m |

C_{ij}^{m}, d_{ij}^{m} | Unit transport cost, transport distance from inland city i to inland intermodal terminal j by transportation mode m |

C_{jk}^{m}, d_{jk}^{m} | Unit transport cost, transport distance between inland intermodal terminal j and k by transportation mode m |

C_{ks}^{m}, d_{ks}^{m} | Unit transport cost, transport distance from inland intermodal terminal k to gateway seaport s by transportation mode m |

HC_{j}^{mm’} | Unit container handling cost at inland intermodal terminal j, for transshipment between mode m and mode m′ |

SC_{j} | Unit container storage cost at inland intermodal terminal j |

U_{k} | Container handling capacity of inland intermodal terminal k |

U_{s} | Container handling capacity of gateway seaport s |

e^{m} | CO_{2} emission rate of transportation mode m |

e_{j}^{mm’} | CO_{2} emission rate of handling a TEU at inland intermodal terminal j for transshipment between mode m and mode m′ |

ε | Limitation on network carbon emissions |

**Table 4.**The mean and standard deviation of transportation demand of inland cities (unit: twenty-foot equivalent unit (TEU)).

City No. | Mean | Standard Deviation | City No. | Mean | Standard Deviation | City No. | Mean | Standard Deviation | City No. | Mean | Standard Deviation | City No. | Mean | Standard Deviation |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1,817,448 | 56,795 | 16 | 597,304 | 26,132 | 31 | 9335 | 584 | 46 | 6586 | 494 | 61 | 1027 | 52 |

2 | 410,016 | 12,813 | 17 | 255,286 | 11,168 | 32 | 21,575 | 1348 | 47 | 5458 | 409 | 62 | 1398 | 70 |

3 | 97,103 | 3035 | 18 | 222,058 | 11,103 | 33 | 23,724 | 1483 | 48 | 4870 | 366 | 63 | 1054 | 53 |

4 | 71,051 | 2220 | 19 | 26,332 | 1316 | 34 | 272,821 | 20,461 | 49 | 9308 | 698 | 64 | 1324 | 66 |

5 | 9482 | 297 | 20 | 89,750 | 4488 | 35 | 30,337 | 2275 | 50 | 285,409 | 21,406 | 65 | 1810 | 90 |

6 | 35,681 | 1115 | 21 | 35,255 | 1762 | 36 | 48,450 | 3634 | 51 | 161,931 | 12,144 | 66 | 63,857 | 3193 |

7 | 16,311 | 509 | 22 | 26,737 | 1337 | 37 | 38,854 | 2914 | 52 | 16,103 | 1208 | 67 | 3674 | 183 |

8 | 54,174 | 1693 | 23 | 23,352 | 1167 | 38 | 15,862 | 1190 | 53 | 12,975 | 973 | 68 | 17,183 | 859 |

9 | 891,286 | 38,994 | 24 | 28,405 | 1420 | 39 | 17,518 | 1314 | 54 | 7389 | 554 | 69 | 47 | 2 |

10 | 1,134,601 | 49,639 | 25 | 10,737 | 537 | 40 | 19,103 | 1433 | 55 | 4296 | 322 | 70 | 2918 | 146 |

11 | 83,857 | 3669 | 26 | 48,063 | 3004 | 41 | 21,209 | 1591 | 56 | 2925 | 220 | 71 | 12,509 | 625 |

12 | 209,270 | 9155 | 27 | 14,802 | 925 | 42 | 82,092 | 6157 | 57 | 4640 | 348 | 72 | 4249 | 213 |

13 | 172,693 | 7555 | 28 | 27,896 | 1743 | 43 | 10,752 | 806 | 58 | 1918 | 144 | |||

14 | 120,057 | 5252 | 29 | 19,144 | 1197 | 44 | 7105 | 533 | 59 | 54,022 | 2701 | |||

15 | 554,651 | 24,266 | 30 | 13,564 | 848 | 45 | 12,747 | 956 | 60 | 5910 | 295 |

Inland River Port | Capacity | Inland Railway Container Station | Capacity | Gateway Seaport | Capacity |
---|---|---|---|---|---|

Suzhou port | 400 | Yiwu | 650 | Shanghai | 5000 |

Nanjing port | 500 | Hefei | 260 | Ningbo-Zhoushan | 5000 |

Wuhu port | 420 | Bengbu | 50 | ||

Jiujiang port | 200 | Nanchang | 130 | ||

Wuhan port | 500 | Wuhan | 200 | ||

Yueyang port | 200 | Xiangyang | 50 | ||

Chongqing port | 400 | Changsha | 150 | ||

Luzhou port | 100 | Chongqing | 50 | ||

Yibin port | 350 | Chengdu | 200 | ||

Guiyang | 60 | ||||

Kunming | 150 |

Road | Waterway | Railway | |
---|---|---|---|

Carbon emissions factor | 2.189 | 0.423 | 0.094 |

**Table 7.**Results of total costs, total emissions, and flow distribution with different optimization objectives ($\alpha =0.90$).

Total Costs/Million US$ | Total Emissions/Million Tons | Flow Distribution | ||||
---|---|---|---|---|---|---|

Direct Road | Waterway/Road Transshipment | Rail/Road Transshipment | Inter-Terminal Transshipment | |||

Cost minimization | 3255.6 | 4.448 | 61.0% | 23.8% | 9.3% | 5.9% |

Bi-objective optimization | 3320.7 | 3.781 (−15.0%) ^{1} | 60.8% | 21.3% | 15.2% | 2.7% |

CO_{2} emissions minimization | 3511.2 | 3.238 (−27.2%) ^{1} | 60.6% | 20.1% | 19.30% | 0.0% |

^{1}The value in bracket indicates the percentage of emissions reduction compared to cost minimization model.

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**MDPI and ACS Style**

Dai, Q.; Yang, J.
A Distributionally Robust Chance-Constrained Approach for Modeling Demand Uncertainty in Green Port-Hinterland Transportation Network Optimization. *Symmetry* **2020**, *12*, 1492.
https://doi.org/10.3390/sym12091492

**AMA Style**

Dai Q, Yang J.
A Distributionally Robust Chance-Constrained Approach for Modeling Demand Uncertainty in Green Port-Hinterland Transportation Network Optimization. *Symmetry*. 2020; 12(9):1492.
https://doi.org/10.3390/sym12091492

**Chicago/Turabian Style**

Dai, Qian, and Jiaqi Yang.
2020. "A Distributionally Robust Chance-Constrained Approach for Modeling Demand Uncertainty in Green Port-Hinterland Transportation Network Optimization" *Symmetry* 12, no. 9: 1492.
https://doi.org/10.3390/sym12091492