# Congestion Evaluation of Pedestrians in Metro Stations Based on Normal-Cloud Theory
## 4 Methodology
### NC Model  
The NC (normal-cloud) model was proposed by [PERSON] et al. [40], which included qualitative and quantitative mapping mainly through the combination of the fuzzy and random sets of an object to study the universal law among uncertainties. The NC model currently provides the best solutions to many problems in effectiveness evaluation, decision making, risk assessment, and condition assessment of bridges [41; 42]. In this study, the NC model was used to evaluation the GPC in metro stations. The steps to construct a cloud model were as follows. 1. Cloud model: Let \(U\) be a quantitative domain and let \(C\) be a qualitative concept on \(U\). If a certain value \(x\in U\), then \(x\) is a random implementation of \(C\). The determinacy of \(x\) to \(C\) is a random number with a stable tendency \(\mu(x)\): \(U\)\(\rightarrow\)[0, 1], \(\forall\)\(x\in U\), \(\mathrm{x}\)\(\rightarrow\)\(\mu(x)\). Then, the distribution of \(x\) on the domain \(U\) is called a cloud, and each x is described as a cloud droplet. Three parameters, the expectation \(Ex\), the entropy \(En\), and the super-entropy He, are used to characterize the normal-cloud model, as shown in Figure 5.
2. NC generator: If the definition of \(x\) in the cloud satisfies \(\mathrm{x}\)\(\rightarrow\)\(N(Ex\), \(En^{\prime})\) and \(En^{\prime}\)\(\sim\)\(N(En\), \(He^{2})\), the determinacy of \(x\) to \(C\) satisfies the following: \[\mu(x)=Exp\frac{-(x-Ex)^{2}}{2(En^{\prime})^{2}}.\] (1)  
The NC model is widely used in solving probabilistic and ambiguous problems, which combines the normal distribution function with the bell-shaped membership function. All the cloud models used in this study referred to the NC model. By definition, the quantitative data obtained from cloud digital eigenvalues were forward NC generators, while those obtained from the quantitative data were backward NC generators. In this study, the forward NC generator was used to solve the NC model, see Figure 6.  
Figure 5: Basic normal-cloud (NC) model. 3. Cloud synthesis: The synthesis of clouds was to combine the clouds of the same nature with a parent cloud. The parent cloud \(C\) (\(Ex\), \(Ent\), \(He\)) was synthesized using \(n\) child clouds \(Cn\) (\(Ex_{n}\), \(Ent_{n}\), \(He_{n}\)), expressed as follows: \[C=C_{1}\circ C_{2}\circ C_{3}\circ\cdots\circ C_{n},\] (2) \[\left\{ \begin{array}{l}Ex=\frac{\sum\limits_{i=1}^{n}Ex_{i}y \times En_{i}\times w_{i}}{\sum\limits_{i=1}^{n}En_{i}\times w_{i}}\\ En=\sum\limits_{i=1}^{n}En_{i}\times w_{i}\end{array} \right.,\] (3) \[He=\frac{\sum\limits_{i=1}^{n}He_{i}y\times En_{i}y\times n_{i}}{\sum \limits_{i=1}^{n}En_{i}\times w_{i}}\] where \"\(\circ\)\" refers to the process of cloud synthesis and \(w_{i}\) to the weight of the \(i\)-th child cloud. 4. Model of GPC: The process of the model followed five steps:  
Step 1: Determining the digital features of the cloud  
Let \(X=(x_{1}\), \(x_{2}\),..., \(x_{j}\),..., \(x_{n-1})\) be the threshold vector of an index in which \(x_{j}\) denotes the threshold of the index at \(j\)-th GPC. As the synthetic clouds were conducted in the same domain, the index needed to be standardized before calculating the digital features. Using the bigger-is-better and the smaller-is-better rules, the standardizations were as shown in Equations (4) and (5), respectively. Bigger is better,  
\[x_{j}^{\star}=\frac{x_{j}-\min\{x_{j}\}}{\max\{x_{j}\}-\min\{x_{j}\}}\quad(j=1, 2,\ldots,n-1). \tag{4}\]  
Smaller is better,  
\[x_{j}^{\star}=\frac{\max\{x_{j}\}-x_{j}}{\max\{x_{j}\}-\min\{x_{j}\}}\quad(j=1, 2,\ldots,n-1), \tag{5}\]  
where \(x_{j}^{\star}\) is the standardized value of \(x_{j}\), and \(\max\{x_{j}\}\) and \(\min\{x_{j}\}\) are the maximum and minimum values for the threshold \(j\), respectively. The GPC \(j\) (\(j=1\), \(2\),..., \(n-1\)) is indicated by NCs in the semi-rising and semi-descending states, leading to the three digital features in Equation (6).