# Multi-Attribute Collaborative Optimization for Multimodal Transportation Based on User Preferences
## 3 Problem Description and Model Construction
### Model Construction  
Considering that railway transportation is influenced by train scheduling, there will be waiting time associated with transportation mode transfers involving railway transportation. When cargo is transferred at a certain node, the total transfer time is determined, as illustrated in Figure 2. If the subsequent transportation mode is railway transportation, there will be a waiting time \(\Delta t_{34}\). Based on this, a multi-objective route optimization model is established, as shown in Equations (1)-(18). Equations (1)-(5) represent the objective functions. Equation (1) aims to minimize the total cost, total time, total carbon emissions, and total risk. \[min\{C,T,E,R\} \tag{1}\]  
Equation (2) represents the total cost, which includes transportation costs, transfer costs, and waiting costs. \[C=\sum_{i\in V}\sum_{j\in V}\sum_{m\in D}qc_{ij}^{m}x_{ij}^{m}+\sum_{i\in V} \sum_{m\in D}\sum_{n\in D}y_{i}^{mn}q(c_{i}^{mn}+c_{i}max(T_{j}^{\rm L}-\delta _{j},0)) \tag{2}\]  
Equation (3) represents the total time, which includes transportation time, loading/unloading and marshalling time, and waiting time. \[T=\sum_{i\in V}\sum_{j\in V}\sum_{m\in D}x_{ij}^{m}\frac{q_{ij}^{mn}}{v_{ij}^{ m}}+\sum_{i\in V}\sum_{j\in V}\sum_{m\in D}\sum_{n\in D}y_{i}^{mn}(qt_{i}^{mn}+ max(T_{j}^{\rm L}-\delta_{j},0)) \tag{3}\]  
Equation (4) represents the total carbon emissions, which include transportation carbon emissions and transfer carbon emissions. \[E=\sum_{i\in V}\sum_{j\in V}\sum_{m\in D}qe_{ij}^{m}d_{ij}^{m}x_{ij}^{m}+\sum _{i\in V}\sum_{m\in D}\sum_{n\in D}qe_{i}^{mn}y_{i}^{mn} \tag{4}\]  
Equation (5) represents the total risk, which includes transportation risk and transfer risk. \[R=\sum_{i\in V}\sum_{j\in V}\sum_{m\in D}r_{ij}^{m}x_{ij}^{m}+\sum_{i\in V} \sum_{m\in D}\sum_{n\in D}r_{i}^{mn}y_{i}^{mn} \tag{5}\]  
Equations (6)-(17) represent the constraints. Equation (6) indicates the time when the cargo departs from node \(i\) and arrives at the next node j via transportation mode m.  
\[T_{j}^{\rm A}=T_{i}^{\rm L}+t_{ij}^{m},i,j\in V;m\in D \tag{6}\]  
Equation (7) represents the moment when the transfer, loading, and unloading operations are completed. Figure 2: Illustration of transfer time under departure schedule constraints. \[\delta_{j}=T_{j}^{\text{A}}+t_{j}^{mn},j\in V;m,n\in D \tag{7}\]  
In Equation (8), \(\omega(\delta_{j},n)\) represents the nearest departure time for adopting transportation mode \(n\) in the next stage of the journey at moment \(\delta_{j}\). \[T_{j}^{\text{L}}=\omega(\delta_{j},n),j\in V;n\in D \tag{8}\]  
Equation (9) states that the total travel time for the cargo to reach the destination should not exceed the maximum time allowed by the consignee. \[min\{C,T,E,R\} \tag{1}\]0  
Equation (10) is the flow balance constraint. \[\sum\limits_{j\in V}\sum\limits_{m\in D}x_{ij}^{m}-\sum\limits_{j\in V}\sum \limits_{m\in D}x_{ji}^{m}=\left\{ \begin{array}{l}-1,i\in o\\ 0,i\in V\backslash\{o,d\}\\ 1,i\in d\end{array} \right. \tag{10}\]  
Equation (11) ensures the continuity of transportation. If \(y_{i}^{mn}=1\), then \(x_{ki}^{m}=1\), \(x_{ij}^{n}=1\) and \(\sum\limits_{i,j\in V}\sum\limits_{l\in D}x_{lj}^{l}=1\),  
\[l,m,n\in D;k,i,j\in V;i\in U_{k},j\in U_{i}\]  
Equation (12) indicates that, at most, one transportation mode can be selected between adjacent nodes. \[\sum\limits_{j\in V}\sum\limits_{m\in D}x_{ij}^{m}\leq 1,i\in V \tag{12}\]  
Equation (13) states that the cargo can be transferred, at most, once at any node. \[\sum\limits_{m,n\in D}y_{i}^{mn}\leq 1,i\in V \tag{13}\]  
Equation (14) indicates that the cargo volume cannot exceed the transportation capacity of the selected transportation mode between two nodes. \[qx_{ij}^{m}\leq Q_{ij}^{m},m\in D;i,j\in V \tag{14}\]  
Equation (15) states that the cargo volume cannot exceed the transfer capacity of the transfer node. \[qy_{i}^{mn}\leq Q_{i}^{mn},m,n\in D;i\in V \tag{15}\]  
Equation (16) indicates that no transfer occurs between the same transportation modes.