## Methodology
### Model Formulation
#### 3.2.2 Formulation  
The proposed problem can be formulated as the following mixed-integer program (MIP), which requires minimization of  
\[\min f=c_{1}\cdot\sum_{\forall i,j\in I\cup D\cup M}\sum_{\forall k\in K}x_{ij}^ {k}\cdot\mathbf{d}_{ij}-c_{2}\cdot\sum_{\forall i\in I}\sum_{\forall k\in K}y_{ i}^{k}\cdot q_{i}\cdot g(t_{M}^{k}-t_{i}^{k}) \tag{1}\]  
subject to  
\[\sum_{\forall i\in I}y_{i}^{k}\geq 1,\,\forall k\in K \tag{2}\]  
\[\sum_{\forall k\in K}y_{i}^{k}=1,\,\forall i\in I \tag{3}\]  
\[\sum_{\forall j\in I}x_{jm}^{k}=1,\,\forall k\in K,\,\,\forall m\in M \tag{4a}\]  
\[\sum_{\forall j\in I}x_{mj}^{k}=0,\,\forall k\in K,\,\,\forall m\in M \tag{4b}\]  
\[\sum_{\forall j\in I}x_{jm}^{k}=0,\,\forall k\in K,\,\,\forall m\in D \tag{5a}\]  
\[\sum_{\forall j\in I}x_{mj}^{k}=1,\,\forall k\in K,\,\,\forall m\in D \tag{5b}\]  
\[\sum_{\forall j\in I\cup D\cup M}x_{ij}^{k}=\sum_{\forall j\in I\cup D\cup M }x_{ji}^{k}=y_{i}^{k},\,\forall k\in K,\,\,\forall i\in I \tag{6}\]  
\[U_{ik}-U_{jk}+\left|I\cup D\cup M\right|\cdot x_{ij}^{k}\geq\left|I\cup D\cup M \right|-1,\,\,\forall k\in K,\,\,\forall i,j\in I\cup D\cup M \tag{7}\]  
\[t_{i}^{k}+t_{ij}-(1-x_{ij}^{k})\cdot H\leq t_{j}^{k},\forall k\in K,\,\, \forall i,j\in I\cup D\cup M \tag{8a}\]  
\[\sum_{\forall i\in I}y_{i}^{k}\geq 1,\,\forall k\in K \tag{2}\]0  
\[\sum_{\forall i\in I}y_{i}^{k}\geq 1,\,\forall k\in K \tag{2}\]1  
\[\sum_{\forall i\in I}y_{i}^{k}\geq 1,\,\forall k\in K \tag{2}\]2  
\[\sum_{\forall i\in I}y_{i}^{k}\geq 1,\,\forall k\in K \tag{2}\]3  
\[\sum_{\forall i\in I}y_{i}^{k}\geq 1,\,\forall k\in K \tag{2}\]4  
\[\sum_{\forall i\in I}y_{i}^{k}\geq 1,\,\forall k\in K \tag{2}\]5  
\[\sum_{\forall i\in I}y_{i}^{k}\geq 1,\,\forall k\in K \tag{2}\]6  
where \(g(t_{M}^{k}-t_{i}^{k})\) is a function used to calculate the satisfaction of passengers at the demand point \(\forall i\in I\) based on his/her ride time \(t_{M}^{k}-t_{i}^{k}\), with the arrival time for the vehicle visiting the demand point \(i\) satisfying one of its multiple travel time windows. A shorter ride time results in higher passenger satisfaction. At present, many researchers mainly use linear and nonlinear functions to describe the coupling relationship between customer satisfaction and service time in the VRPs. Considering that the former is incompatible with reality and the latter cannot be quickly and accurately solved, this paper introduces the fuzzy gradient function to describe passenger satisfaction. Similar to the problem that considers the time window and customer demands in VRPs [7; 39], this can be expressed as follows:\[g(t_{M}^{k}-t_{i}^{k})=\left\{ \begin{array}{l}1,t_{M}^{k}-t_{i}^{k}\leq Sf_{i} \\ \frac{Lt_{i}-[t_{M}^{k}-t_{i}^{k}]}{Lt_{i}-St_{i}},St_{i}\leq t_{M}^{k}-t_{i}^{k} \leq Lt_{i}\\ 0,t_{M}^{k}-t_{i}^{k}\geq Lt_{i}\end{array} \right.. \tag{14}\]  
In this formulation, the objective functions are given by Equation (1) to minimize the loss of reduction in passenger satisfaction and the total mileage cost of designed feeder routes. Constraint (2) indicates that each vehicle visits at least one demand point. Constraint (3) guarantees that each demand point is only covered by one vehicle. Constraints (4a) and (4b) guarantee that each vehicle eventually started at the dispatch center. Constraints (5a) and (5b) guarantee that each vehicle eventually ends at the rail transit station. Constraint (6) sets all demand points (except rail station and dispatch center) being served by the vehicle to have exactly the same incoming and outgoing arcs. Constraint (7) is used for the sub-tour elimination in the vehicle routing. Constraints (8a) and (8b) are used for calculating the arrival times of adjacent vehicular points \(i\) and \(j\) covered by vehicles \(k\). Constraint (9) guarantees the time that vehicle \(k\) reaches demand point \(i\) satisfies one of its multiple travel time windows. Constraints (10a) and (10b) are used for calculating the load capacity of adjacent vehicular points \(i\) and \(j\) covered by vehicles \(k\).