## 2 Model Formulation
### Platform Exit Width Range  
#### 2.2.1 Passenger Flow Assignment Based on Path Utility  
In Figure 1, passengers disembarking within \(I_{AB}\) will gather at node A, that is, moving from nodes A1 and A2 to node A. Similarly, passengers getting off within \(I_{B}\) will arrive at node B. Due to the unidirectional platform exits, passengers at node A as well as node C will first move to node B, and then from node B to exit D or exit E. Passengers disembarking from nodes B1 and B2 can choose exits D or E directly. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Microscopic Model**} & \\ \hline
**Cellular Automata** & **Social Force Model** & \
\begin{tabular}{c} **Game-Theoretic** \\ **Model** \\ \end{tabular} &
\begin{tabular}{c} **Mathematical** \\ **Optimization Model** \\ \end{tabular} & **Macroscopic Model** \\ \hline
[10, 27, 28] & [29, 30, 31] & [32, 33] & [16, 34, 35, 36] & [14, 15, 20, 22, 23, 24, 25] \\  & & & This paper & This paper \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of modeling techniques (summary based on the above literature). The above can be illustrated in Figure 2. Considering the sequential order of passengers from different initial nodes arriving at node B, the passenger choice process is depicted in Figure 3, which simplifies the process into two periods. In period 1 (\(T=1\)), passengers from nodes A1(C1) and A2(C2) go through the bottleneck to node A and then to node B. At the same time, passengers from nodes B1 and B2 move directly to node B and then proceed to exit D or exit E. In period 2 (\(T=2\)), passengers who have already departed from nodes A(C) and are arriving at node B then travel to exit D or E.  
Figure 3 describes a time-space path network \(G(N,W)\) on a high-speed railway platform. \(N\) represents a set of nodes and \(W\) represents a set of paths. Passengers make choices based on path utility, which is negatively correlated with travel time. The utility of path \(m\) linking two adjacent nodes \((r,s)\) is \(U_{rs}^{m}\) with a random error \(\varepsilon\) that follows the Gamble probability distribution. \[U_{rs}^{m}=\alpha\cdot t_{rs}^{m}+\varepsilon,\forall N,S \tag{1}\]  
According to the utility theory [38] and MNL model [42], the probability of passengers choosing path \(m\) between node pair \((r,s)\) is computed by Equation (2). Figure 3: The path selection of passengers (considering the travel order of passengers on the platform based on Figure 2 by Microsoft Visio). Figure 2: A diagram of passenger paths on the platform (abbreviated according to Figure 1 by Microsoft Visio). \[\text{Pro}b_{rs}^{m}=\frac{\exp(-\beta\cdot U_{rs}^{m})}{\sum\limits_{m\in W_{rs}} \exp(-\beta\cdot U_{rs}^{m})}, \tag{2}\]  
where utility can be seen as the opposite of generalized cost. We take the minimized generalized cost of all passengers as the objective function to propose a passenger flow assignment model on a high-speed railway platform. \[\text{min}Z(f(p,T))=\sum\limits_{(r,s)}\sum\limits_{m\in W_{rs}}U_{rs}^{m}.f_{ rs}^{m}+\frac{1}{\theta}\sum\limits_{(r,s)}\sum\limits_{m\in W_{rs}}f^{m}(\ln f _{rs}^{m}-1) \tag{3}\]  
\[\sum\limits_{m\in W_{rs}}f_{rs}^{m}=q_{rs},\forall(r,s) \tag{4}\]  
\[\sum\limits_{(r,s)}\sum\limits_{m\in W_{rs}}\delta_{p}^{m}.f_{rs}^{m}\leq \sigma_{p}^{T} \tag{5}\]  
where \(\frac{1}{\theta}\sum\limits_{(r,s)}\sum\limits_{m\in W_{rs}}f_{rs}^{m}(\ln f _{rs}^{m}-1)\) is the penalty entropy, which represents the congestion caused by an increase in passenger flow. This congestion exacerbates the generalized cost in the entire network. At the same time, the penalty entropy ensures that all passenger flows are not concentrated on the path with the lowest cost to achieve equilibrium. Thus, all paths being selected by passengers guarantees consistency with the actual circumstances. \(\sigma_{p}^{T}\) is the maximum allowable passenger flow, resulting from the limited space on the platform.