# A Traffic Equilibrium Model for Multi-Modal Networks with Uncertain Demands
## 6 Numerical Example
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Link Node**} & \multirow{2}{*}{**Bus Line**} & \multirow{2}{*}{**Subway Line**} & \multirow{2}{*}{**Frequency (min)**} & \multirow{2}{*}{**Seat Number**} & \multicolumn{2}{c}{**Vehicle Capacity**} \\  & & & & & \(\boldsymbol{A}_{j}\) (Passengers/Vehicle) \\ \hline
1–2 & 1 & – & 6 & 29 & 50 \\
2–3 & 1 & – & 6 & 29 & 50 \\
3–6 & 1 & – & 6 & 29 & 50 \\
6–9 & 1 & – & 6 & 29 & 50 \\
2–5 & 2 & – & 10 & 39 & 70 \\
5–8 & 2 & – & 10 & 39 & 70 \\
8–11 & 2 & – & 10 & 39 & 70 \\
4–7 & 3 & – & 15  
To highlight the differences between using the ETC as the travel mode and route choice criterion in the ETC equilibrium model and the acceptable travel cost as the route choice criterion in the traditional equilibrium model (TEM), a comparison and analysis between the two models are, respectively, illustrated in Tables 6 and 7. It can be seen from the tables that the distribution results of both the ETC equilibrium model and the TEM are consistent in the following. On the one hand, the summation of the traffic volume of each travel mode is equal to the total travel demands between \(OD\) pair \(w\), and the summation of passenger flows of each link is also equal to the total travel demands between \(OD\) pair \(w\) when the equilibrium state is reached. This indicates that the distribution result is correct and effective. On the other hand, no matter which criterion is adopted, the bicycle travel mode always results in the least traffic volume, which is also far lower than that of the combined travel modes of \"bicycle + bus or bicycle + subway\". This indicates that the bicycle travel mode is rarely adopted. However, as a complementary transport mode, bicycles are more commonly combined with other transport modes. The difference in the results of modal splits is not significant, and the most obvious differences are mainly concentrated in links 1-5, 4-7, 7-8, and 8-9, which are links of the transport bus line 3 and subway. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**OD Pair**} & \multirow{2}{*}{**Travel Mode**} & \multicolumn{2}{c}{**Mode Splits (Passengers-\(h^{-1}\))**} \\ \cline{3-4}  & & **ETC** & **TEM** \\ \hline \multirow{5}{*}{\(OD_{1,9}\)} & Bicycle & [0, 0.01] & [0, 0.01] \\  & Bus & [3338.52, 3689.43] & [2121.24, 2344.51] \\  & Subway & [27.05, 30.96] & [23.65, 26.14] \\ \cline{1-1}  & Auto & [1605.41, 1774.02] & [1011.42, 1179.13] \\ \cline{1-1}  & Bicycle + Bus & [419.47, 463.68] & [631.12, 697.55] \\ \cline{1-1}  & Bicycle + Subway & [14.69, 16.83] & [12.62, 13.95] \\ \hline \multirow{5}{*}{\(OD_{1,12}\)} & Bicycle & [0, 0.01] & [0, 0] \\ \cline{1-1}  & Bus & [821.42, 907.88] & [1156.34, 1278.06] \\ \cline{1-1}  & Auto & [1751.08, 1935.38] & [1750.91, 1935.28] \\ \cline{1-1}  & Bicycle + Bus & [5426.27, 5998.56] & [5316.83, 5876.52] \\ \cline{1-1}  & Bicycle + Subway & [0, 0] & [0, 0] \\ \cline{1-1}  & Bus + Subway & [1498.45, 1656.64] & [1276.02, 1410.31] \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Resultant modal splits. Figure 4: Convergence curve of the SMA algorithm. It can be seen from Figures 5 and 6 that as the deviation value \(\omega\) increases from 0.1 to 1, the distribution results show only slight fluctuations on the whole except when \(\omega=0.2\) and \(\omega=0.3\). For the results of the modal split, the traffic volume of cars increases rapidly at \(\omega=0.2\), then decreases rapidly at \(\omega=0.3\). On the contrary, the traffic volume of the \"bicycle + bus\" travel mode decreases rapidly at \(\omega=0.2\), then increases rapidly at \(\omega=0.3\). Regarding link flow distribution, the passenger flows of both links 5-8 and links 7-10 decrease rapidly at \(\omega=0.3\). In addition, the value of the perception variable \(\theta\) also has significant effects on travellers' route and mode choice behaviors. It can be seen from Figures 5 and 6 that as the deviation value \(\omega\) increases from 0.1 to 1, the overall trend of the distribution result increases.