# Robust Optimization of Transport Organization for China Railway Express
## 3 Optimizing China-Europe Freight Train Logistics Amidst Uncertain Transportation Demand
### A Robust Transformation Model for Uncertain Cargo Transportation Needs
#### 3.4.1 Uncertain Description of Goods Transportation Demand
The robust regulation parameter \(\Gamma\) is introduced to adjust the conservative degree of the robust model, \(\Gamma\in[0,n]\), where \(n\) is the number of nodes with uncertainty of freight demand, \(n\in[0,N_{1}\cup N_{3}]\), and \(\lfloor\Gamma\rfloor\) is the largest integer not greater than \(\Gamma\). When the freight demand with a model maximum of \(\lfloor\Gamma\rfloor\) nodes in the model changes in the uncertainty set of the box, the transformed robust model must be able to obtain a feasible solution. When the number of nodes changing the freight demand exceeds \(\lfloor\Gamma\rfloor\), if the disturbance range is \((\Gamma-\lfloor\Gamma\rfloor)\cdot\overline{Q_{i}}\cdot\epsilon_{i}\), the transformed robust model still has a high probability of obtaining a feasible solution. In the established model of transport organization optimization in a definite environment, only the uncertainty parameters exist in the objective function, so the uncertainty parameters in the objective function are changed to the uncertainty parameters in the constraint. Original objective function:  
\[\text{min}F=\sum_{i\in N}\sum_{j\in N}\sum_{v\in V}\left[f_{ij}^{v}\cdot y_{ij }\cdot\left(C_{v}+C_{ijv}\cdot d_{ij}\right)\right]+\sum_{i\in N_{3}}\left[z_ {i}\cdot Q_{i}\cdot C_{i}^{3}\right] \tag{12}\]  
Minimizing the objective function to maximizing it:  
\[\text{max}-F=-\sum_{i\in N}\sum_{j\in N}\sum_{v\in V}\left[f_{ij}^{v}\cdot y_ {ij}\cdot\left(C_{v}+C_{ijv}\cdot d_{ij}\right)\right]-\sum_{i\in N_{3}}\left[ z_{i}\cdot Q_{i}\cdot C_{i}^{3}\right] \tag{13}\]  
Let \(F_{tr}=-F\), then Formula (13) can be converted to the following form:  
\[\text{max}F_{tr} \tag{14}\]  
\[F_{tr}+\sum_{i\in N}\sum_{j\in N}\sum_{v\in V}\left[f_{ij}^{v}\cdot y_{ij} \cdot\left(C_{v}+C_{ijv}\cdot d_{ij}\right)\right]+\sum_{i\in N_{3}}\left[z_ {i}\cdot Q_{i}\cdot C_{i}^{3}\right]\leq 0 \tag{15}\]  
The [PERSON] robust model of Formula (15) is as follows:  
\[F_{tr}+\sum_{i\in N}\sum_{j\in N}\sum_{v\in V}\left[f_{ij}^{v} \cdot y_{ij}\cdot\left(C_{v}+C_{ijv}\cdot d_{ij}\right)\right]+\sum_{i\in N _{3}}\left[z_{i}\cdot\overline{Q_{i}}\cdot C_{i}^{3}\right] \tag{16}\] \[+\max_{\{S\cup\{t\}|S\subseteq N,|S|=\lfloor\Gamma\rfloor,t=N\setminus S \}}\left\{\sum_{i\in S}\left[\overline{Q_{i}}\cdot\epsilon_{i}\cdot W_{i} \cdot C_{i}^{3}\right]+\left[\left(\Gamma-\lfloor\Gamma\rfloor\right)\cdot \overline{Q_{t}}\cdot\epsilon_{t}\cdot W_{t}\cdot C_{i}^{3}\right]\right\} \leq 0\] \[-W_{i}\leq z_{i}\leq W_{i}\quad\forall i\in S \tag{17}\]When \(\Gamma\) takes an integer, Formula (16) can be reduced to the following form:  
\[ \begin{array}{l}F_{tr}+\sum\limits_{i\in N}\sum\limits_{j\in N}\sum\limits_{v\in V }\left[f_{ij}^{v}\cdot y_{ij}\cdot\left(C_{v}+C_{ijv}\cdot d_{ij}\right)\right]+ \sum\limits_{i\in N_{3}}\left[z_{i}\cdot\overline{Q_{i}}\cdot C_{i}^{3}\right] \\ +\max\limits_{\{S\cup\{t\}|S\subset N,|S|=|\Gamma|,t=N\setminus S\}}\left\{\sum \limits_{i\in S}\left[\overline{Q_{i}}\cdot\epsilon_{i}\cdot W_{i}\cdot C_{i}^ {3}\right]\right\}\leq 0\end{array}  \tag{18}\]  
Formula (18) is robustly transformed by [PERSON] as follows:  
\[F_{tr}+\sum\limits_{i\in N}\sum\limits_{j\in N}\sum\limits_{v\in V}\left[f_{ij }^{v}\cdot y_{ij}\cdot\left(C_{v}+C_{ijv}\cdot d_{ij}\right)\right]+\sum \limits_{i\in N_{3}}\left[z_{i}\cdot\overline{Q_{i}}\cdot C_{i}^{3}\right]+ \lambda*\Gamma+\sum\limits_{i\in S}\sigma_{i}\leq 0 \tag{19}\]  
\[\lambda+\sigma_{i}\geq\overline{Q_{i}}\cdot\epsilon_{i}\cdot W_{i}\cdot C_{i} ^{3}\quad\forall i\in S \tag{20}\]  
\[\lambda\geq 0 \tag{21}\]  
\[\sigma_{i}\geq 0\quad\forall i\in S \tag{22}\]  
To sum up, the optimization model of China-Europe freight train transportation organization with robust and controllable cargo transportation demand is as follows:  
\[\text{max}F_{tr} \tag{23}\]  
s.t.