# Optimization of Multimodal Transport Paths Considering a Low-Carbon Economy Under Uncertain Demand
## 3 Materials and Methods
### Model Building  
Due to the erratic need for cargo transportation, the scenario method in robust optimization is used. In this paper, the discrete scenarios with known probabilities show the uncertainty in the parameters. The probability of occurrence of each scenario is determined, and a multimodal transit path strategy under uncertainty is established by using the scenario analysis method in robust optimization as follows:  
\[C_{1}=\sum_{m\in M}\sum_{i=s}^{N}\sum_{j=s}^{N}c_{ij}^{m}d_{i,j}^{m}x_{(i,j)}^{m }q+\sum_{m\in M}\sum_{i\in N}\sum_{j\in N}c_{ij}^{m,n}y_{j}^{m,n}q \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Meaning** \\ \hline \(q\) & Demand for goods, kg \\ \(c_{i,n}^{m,n}\) & Cost of unit transit from \(m\) node \(n\) to \(i\), Yuan \\ \(d_{i,j}^{m}\) & Transportation distance from node \(m\) to \(n\) via \(i\) \\ \(e_{ij}^{m}\) & Carbon emissions from node \(i\) to \(j\) for transportation mode \(m\), kg \\ \(e_{i,n}^{m}\) & Carbon emissions per unit when switching from transportation mode \(m\) to \(n\) at node \(i\) \\ \(c_{m}\) & Unit waiting costs for waiting for buses to dispatch \\ \(p_{s}\) & Probability of scenario \(S\) occurring \\ \(q^{s}\) & Uncertain demand under Scenario \(S\) \\ \(C_{s}\) & Feasible solutions of the objective function under scenario \(S\) \\ \(C_{s*}\) & Scenario \(S\) determines the optimal solution of the objective function under \\ \(\gamma\) & Maximum regret value allowed under scenario \(S\) \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 1: Diagrammatic representation of the multimodal transportation system. \[C_{2} =\sum_{m\in M}\sum_{n\in M}\sum_{j\in N}\Bigl{[}t^{\prime}_{j(m,n)}-(t^{ \prime}_{j(m)}+t^{m,n}_{j})\Bigr{]}c_{w}q\gamma^{m,n}_{j} \tag{2}\] \[E =\sum_{i\in N}\sum_{j\in N}\sum_{k\in M}qe^{m}_{ij}x^{m}_{(i,j)}+ \sum_{i\in N}\sum_{k\in M}\sum_{l\in M}qe^{m,n}_{i}y^{m,n}_{i} \tag{3}\]  
Equation (1) is the cost of transportation and the cost of transit between various forms of transportation; Equation (2) is the cost of waiting at the transit node; and Equation (3) is the overall quantity of greenhouse gases released from the conversion of each road section and node. \[ \begin{array}{l}\min C^{\prime}=\sum\limits_{s=1}^{S}p_{s}C_{s}\\ =\sum\limits_{s=1}^{S}p_{s}\left\{\begin{array}{l}\sum\limits_{m\in M}\sum \limits_{i=s}^{N}\sum\limits_{j=s}^{C}c^{m}_{ij}d^{m}_{ij}x^{m}_{(i,j)}q^{s}+ \sum\limits_{m\in M}\sum\limits_{i\in N}\sum\limits_{j\in N}c^{m,n}_{j}y^{m,n }_{j}q^{s}+\\ \sum\limits_{m\in M}\sum\limits_{n\in M}\sum\limits_{j\in N}\Bigl{[}t^{\prime }_{j(m,n)}-(t^{\prime}_{j(m)}+t^{m,n}_{j})\Bigr{]}c_{w}q^{s}y^{m,n}_{j}\end{array} \right\}\end{array}  \tag{4}\]  
Equation (4) is the robust optimization's objective function of multimodal transportation under uncertain demand. The constraints used to describe the path optimization for multimodal transportation are as follows:  
\[C_{s}\leq(1+\gamma)C_{s*} \tag{5}\]  
\[\sum\limits_{s=1}^{S}p_{s}=1 \tag{6}\]  
\[\sum\limits_{m\in M}\sum\limits_{j=s,j\
eq i}^{N}x^{m}_{(i,j,t_{i})}-\sum \limits_{m\in M}\sum\limits_{j=d,j\
eq i}^{N}x^{m}_{(i,j,t_{i})}=\left\{ \begin{array}{l}1,\forall i=O\\ 0,\forall i\in N\backslash\{O,D\}\\ -1,\forall i\in D\end{array} \right. \tag{7}\]  
\[\sum\limits_{m,n\in M}x^{m}_{(i,j,t_{i})}=1,\forall i,j\in N \tag{8}\]  
\[x^{m}_{(i,j,t_{i})}+x^{n}_{(j,k,t_{j})}\geq 2y^{m,n}_{j} \tag{9}\]  
\[\sum\limits_{m\in M}\sum\limits_{n\in M}y^{m,n}_{j}\leq 1,\forall j\in N \tag{10}\]  
\[x^{m}_{(i,j,t_{i})}\in\{0,1\},\forall m\in M;\forall i,j\in N \tag{11}\]  
\[y^{m}_{j}\in\{0,1\},\forall m\in M;\forall i\in N \tag{12}\]  
According to the principle of flow conservation, Equation (5) is a robust constraint, which indicates what the achievable solution's goal for the optimal objective function under any scenario should be less than. Equation (6) indicates that the sum of the probabilities of all scenarios occurring is 1.