## 2 Model Formulation
### Platform Exit Width Range
#### 2.2.2 Exit Width Range  
To improve passenger safety, it is necessary to scientifically and reasonably determine the width of the exits. Specific parameters on a platform are marked in Figure 4. As shown, a high-speed rail platform typically contains two exits with the same length and width. The exits are symmetrically designed with platform width \(\lambda\) and platform length \(\phi\). \(\frac{\lambda-x}{2}\)refers to the width of the bottleneck. When the exit is excessively large, passengers can quickly pass through them from node B to D or E. However, the shrinkage of the bottleneck is likely to cause passengers to be unable to reach node B from nodes A or C within the stipulated time, resulting in congestion in the bottleneck. Conversely, a small exit may ensure passengers successfully pass though the bottleneck and reach node B within the stipulated time while causing congestion on node B. The above two situations all increase the risk of falling off the platform. Next, we provide the following discussions about these two scenarios, respectively. (1) When \(p=3,4,7,8\), it corresponds to passengers traveling from node A or C to node B, passing through the bottleneck area. According to classic flow-density theory [44], the relationship between speed and density is described by Equation (18). \[v_{p}^{T}=v_{f^{\prime}}(1-\frac{k_{p}^{T}}{k_{f}}) \tag{18}\]  
where \(k_{f}\) is the maximum passenger flow density and \(g_{p}^{T}(t)\) is the occupied area by each passenger. \[k_{p}^{T}=(\frac{\lambda-x}{2}\cdot\phi)/g_{p}^{T},p=3,4,7,8,\forall T \tag{19}\]  
In Figure 5, initially, due to fewer passengers entering the bottleneck, the area occupied by each passenger is extensive, resulting in a lower passenger flow density. As more passengers go through the bottleneck, the occupied area gradually drops, which means the passenger flow density increases. It remains stable once reaching the maximum passenger flow density (the minimum area occupied). As the number of passengers decreases, the occupied area increases, approaching the end. Therefore, the occupied area \(g_{p}^{T}(t)\) is a function of decision variable travel time \(t\). Figure 4: Parameters for typical platform (platform plan with main parameter based on actual data by Microsoft Visio). Every link is constrained by the maximum allowable travel time, thus obtaining the passenger flow \(Q_{p}^{T}\) passing through link \(p\) under the maximum allowable travel time \(\theta_{p}^{T}\). \[Q_{p}^{T}=\int\limits_{0}^{\theta_{p}^{T}}v_{f}\cdot(1-\frac{(\lambda-x_{p}^{T} )\cdot\phi}{2\cdot g_{p}^{T}(t)\cdot k_{f}})\cdot(\frac{(\lambda-x_{p}^{T}) \cdot\phi}{2\cdot g_{p}^{T}(t)})dt,p=\{3,4,7,8\},\forall T \tag{20}\]  
The range of exit widths must ensure that \(Q_{p}^{T}\) meets Equation (21). \[Q_{p}^{T}(x_{p}^{T})\geq f(p,T) \tag{21}\]  
Because \(\forall x_{p}^{T}>0\), \(\frac{dQ_{p}^{T}}{dx_{p}^{T}}=-\int\limits_{0}^{\theta_{p}^{T}}\frac{1}{4 \cdot g_{p}^{T}(t)}dt\cdot\frac{2\cdot\phi^{2}\cdot v_{f}\cdot x_{p}^{T}}{4 \cdot k_{f}}-\frac{2\cdot\phi^{2}\cdot\lambda\cdot v_{f}}{4g_{p}^{T}(t)\cdot k _{f}}<0\), and the passenger flow \(Q_{p}^{T}\) decreases monotonically as the exit width increases. Thus, the boundary value of the exit width is \(x_{p}^{T}(*)\). \[x_{p}^{T}(*)=x_{p}^{T}\Big{|}Q_{p}^{T}(x_{p}^{T})=f(p,T),p=\{3,4,7,8\},\forall T \tag{22}\]  
The exit width range is \(y_{p}^{T}=(0,x_{p}^{T}(*)],p=3,4,7,8\). (2) When \(p=11,12\), this describes the process of passengers traveling from node B to exits D or E. The width of the exit directly determines the time taken by passengers to pass through the exits. The density of passenger flow is given as below.