# Research of Park and Ride Parking Spaces Tiered Pricing Methodology Based on Subway Ride Distance
## 2 Materials and Methods
### Statistical Analysis  
The SP questionnaire data provided information related to individual respondents such as daily travel habits, individual characteristics and information related to travel modes such as distance travelled and cost of travel. Based on this information and the factors identified in previous studies as having a serious impact on travel mode choice behavior, the model variables were selected as shown in Table 2.  
Given that the independent variables encompass two types, the most suitable statistical approach is the mixed logit model. Therefore, both the FP model and the TP model in this paper are modeled using the mixed logit model, and the difference between the two models lies in the different travel attributes in the corresponding scenario of the questionnaire; the setting rules are shown in Table 1.  
In different choice scenarios, each respondent \(n\) (\(n\) = 1, 2, \(\ldots\), \(N\)) can preferentially choose one out of four modes of travel \(m\) (\(m\) = 1, 2, \(\ldots\), \(M\)). These individual preferences culminate in a preferred combination \(C_{n}=\{m_{1},\,m_{2},\,\ldots,\,m_{M}\}\). Each element in combination \(C_{n}\), denoted as \(m_{M}\), represents a binary choice. If mode m is selected by traveler \(n\), \(m\) = 1; otherwise, \(m\) = 0. Hence, this paper establishes two preference models for mode selection based on fixed and tiered pricing methods. These models predict the probabilities of different modes of travel being chosen by travelers across various travel distances. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Variables**} & \multicolumn{1}{c}{**Code**} & \multicolumn{1}{c}{**Ratios**} \\ \hline \multirow{2}{*}{Mode attributes} & Travel cost per kilometer (Pkm) & Scene Setting Values & — \\  & Walking distance & Scene Setting Values & — \\ \hline \multirow{8}{*}{Individual characteristics} & Gender & 0 = Feale & 48.0\% \\  & & 1 = Male & 52.0\% \\  & & 0 = Less than 24 & 25.8\% \\  & Age & 1 = 25–44 years & 59.2\% \\  & & 2 = Older than 45 & 15.0\% \\  & Purpose & 0 = Official business and study & 53.2\% \\  & & 1 = Leisure and entertainment & 46.8\% \\  & Occupation & 0 = Student or no stable job & 43.5\% \\  & & 1 = Have a stable job & 56.5\% \\ \hline \multirow{8}{*}{Daily travel habits} & Car & 0 = not & 72.1\% \\  & & 1 = yes & 27.9\% \\  & Public transportation (PT) & 0 = not & 43.0\% \\  & & 1 = yes & 57.0\% \\  & & 0 = not & 91.1\% \\  & Taxi & 1 = yes & 8.9\% \\  & Cycling or Walking & 0 = not & 93.8\% \\  & & 1 = yes & 6.2\% \\ \hline \multirow{8}{*}{Travel consideration} & Prices in public parking lots (PP group) & 0 = non-consider & 23.6\% \\  & & 1 = consider & 76.3\% \\ \cline{1-1}  & Traveling comfort (CF group) & 0 = non-consider & 51.5\% \\ \cline{1-1}  & & 1 = consider & 48.5\% \\ \cline{1-1}  & Season and weather (SW group) & 0 = non-consider & 52.5\% \\ \cline{1-1}  & & 1 = consider & 47.5\% \\ \cline{1-1}  & Availability of spaces in the P\&R parking lot & 0 = non-consider & 13.1\% \\ \cline{1-1}  & (PS group) & 1 = consider & 86.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Variables and their codes. Defining the utility function \(U_{nm}^{C_{n}}\), representing respondent \(n\)'s preference for travel mode \(m\), can be expressed as follows:  
\[U_{nm}^{C_{n}}=V_{nm}^{C_{n}}(\beta)+\varepsilon_{nm}^{C_{n}}=\beta X_{nm}+ \gamma Z_{nm}+\varepsilon_{nm} \tag{1}\]  
In this equation, \(V_{nm}^{C_{n}}(\beta)\) represents the deterministic term, which depends on parameters \(\beta\), and \(\beta\) conforms to a general normal distribution (GND), used to indicate the heterogeneity of each traveler [36], while \(\varepsilon_{nm}^{C_{n}}\) denotes the stochastic term. The deterministic term's utility function \(V_{nm}^{C_{n}}(\beta)\) comprises \(\beta X_{nm}\) and \(\gamma Z_{nm}\), and \(X_{nm}\) are model attribute explanatory variables whose coefficients are the stochastic parameters \(\beta\).