# Data-Driven Modular Vehicle Scheduling in Scenic Areas
## 3 Problem Statement and Solution Method
### M2: Data-Driven Model  
\(T_{1}\) and \(T_{2}\) are considered as two datasets split from set \(T\) in a 4:1 ratio, sharing the same data distribution as they both originate from \(T\). In practice, we cannot precisely predict future traffic demand at time \(t^{\prime}\) in \(T_{2}\); however, we do have access to the demand in \(T_{1}\). This allows us to use \(T_{1}\) to develop a data-driven model named M2. The schematic of M2 is shown in Figure 3 and can be divided into five stages, namely stages (a), (b), (c), (d), and (e). Firstly, in stage (a), \(T_{1}\) is used to generate \(F\) scenarios of uncertain passenger numbers for the future \(T_{2}-t^{\prime}\) period, which serves as simulated values for future data. Each scenario is represented by \(f,f\ \in\ \{1,...,\ F\}\). In this study, random sampling is employed to create future scenarios by selecting data from \(T_{1}\) at random across \(T_{2}-t^{\prime}\) periods, repeating this process \(F\) times to generate \(F\) scenarios. Next, in stage (b), the average cost of these \(F\) scenarios is used as the estimated future cost. This value is then combined with the current cost in stage (c), where the optimization model computes the optimal solution. Here, we determine whether to dispatch vehicles at the current time and make a decision at time \(t^{\prime}\) (stage (d)). Finally, in stage (e), we incorporate the data at \(t^{\prime}\) into the training set \(T_{1}\) to form an updated \(T_{1}\) set. The process then proceeds to the next time step \(t^{\prime}+1\) where the optimal solution is recalculated, continuing in this iterative manner. \[\text{min}z_{1}=x_{t^{\prime}}(k_{1}+k_{0}(v_{t^{\prime}}-1))+(1-x_{t^{\prime}})p_{ t^{\prime}}N_{t^{\prime}}+\frac{1}{F}\sum_{s=1}^{f}\sum_{j=1}^{T^{\prime}}\left[x_{j}^{f} \Big{(}k_{1}+k_{0}\Big{(}v_{j}^{f}-1\Big{)}+\Big{(}1-x_{j}^{f}\Big{)}p_{j}^{f}N_ {j}^{f}\Big{]}\right] \tag{6}\]  
subject to  
\[v_{t^{\prime}}\geq\frac{n_{t^{\prime}}+N_{t^{\prime}}}{c}x_{t^{\prime}} \tag{7}\]  
\[v_{t^{\prime}}\leq Mx_{t^{\prime}} \tag{8}\]  
\[v_{j}^{f}\geq\frac{n_{j}^{f}+N_{j}^{f}}{c}x_{j}^{f},\forall j\in\{1,\cdots,T^ {\prime}\},\forall f\in\{1,\cdots,F\} \tag{9}\]  
\[v_{j}^{f}\leq Mx_{j}^{f},\forall j\in\{1,\cdots,T^{\prime}\},\forall f\in\{1,\cdots,F\} \tag{10}\]  
\[N_{j}^{f}=(N_{j-1}^{f}+n_{j-1}^{f})\Big{(}1-x_{j-1}^{f}\Big{)},\forall j\in\{ 2,\cdots,T^{\prime}\},\forall f\in\{1,\cdots,F\} \tag{11}\]  
\[N_{1}^{f}=(n_{t^{\prime}}+N_{t^{\prime}})(1-x_{t^{\prime}}),\forall f\in\{1, \cdots,F\} \tag{12}\]  
\[x_{T^{\prime}}^{f}\geq\|\Big{(}\Big{(}n_{T^{\prime}}^{f}+N_{T^{\prime}}^{f} \geq 1\Big{)},\forall f\in\{1,\cdots,F\}\Big{)} \tag{13}\]  
\[x_{\mathcal{T}_{2}}\geq\|\big{(}(n_{\mathcal{T}_{2}}+N_{\mathcal{T}_{2}}\geq 1 )\big{)} \tag{14}\]  
\[x_{t^{\prime}}\in\{0,1\},\forall t^{\prime}\in\{1,2,\cdots,\mathcal{T}_{2}\} \tag{15}\]  
\[v_{t^{\prime}}\geq\frac{n_{t^{\prime}}+N_{t^{\prime}}}{c}x_{t^{\prime}} \tag{7}\]0  
\[v_{t^{\prime}}\geq\frac{n_{t^{\prime}}+N_{t^{\prime}}}{c}x_{t^{\prime}} \tag{7}\]1 The schematic diagram of M2. \[v_{t^{\prime}}\geq\frac{n_{t^{\prime}}+N_{t^{\prime}}}{c}x_{t^{\prime}} \tag{7}\]2  
\[v_{t^{\prime}}\geq\frac{n_{t^{\prime}}+N_{t^{\prime}}}{c}x_{t^{\prime}} \tag{7}\]3  
The specific definitions of the parameters for M2 are shown in \[v_{t^{\prime}}\geq\frac{n_{t^{\prime}}+N_{t^{\prime}}}{c}x_{t^{\prime}} \tag{7}\]4  
The objective function (6) aims to minimize the total cost, where \(x_{t^{\prime}}(k_{1}+k_{0}(v_{t^{\prime}}-1))+(1-x_{t^{\prime}})p_{t^{\prime}} N_{t^{\prime}}\) is the cost for the current \(t\prime\) period, and \(\frac{1}{t}\sum_{f=1}^{F}\sum_{j=1}^{T^{\prime}}\left[x_{j}^{f}\left(k_{1}+k_ {0}\left(v^{f}_{j}-1\right)\right.+\left(1-x_{t^{\prime}}^{f}\right)p_{j}^{f} N_{j}^{f}\right.\) is the simulated approximate cost for the future \(T^{\prime}\) period.