# A Traffic Equilibrium Model for Multi-Modal Networks with Uncertain Demands
## 6 Numerical Example  
To demonstrate the effectiveness of the model and algorithm, a multi-modal network was designed with four transport modes, as shown in Figure 1, with interval demands of [3800, 4200] and [9500, 10,500] passengers per hour from the origin, Node 1, to the destination, Nodes 9 and 12, respectively. The determination for crowd discomfort of transport mode i per time unit (\(u^{0}\)), average crowd discomfort per time unit for transfer (\(u^{h^{0}}\)), travel fare of transport mode i per distance unit (\(m^{0}\)), and average travel fare per distance unit for transfer have been fully demonstrated [23], so these parameters were adopted directly. \(f,g\) are, respectively, the passenger equivalents for modes 2 and 4, passenger equivalents for transport mode 2 (bus) \(f\), supposing that the average number of passengers on a bus is 20 (_passengers_) and the average number of passengers in a car is 2 (_passengers_), we can obtain values of 0.05 and 0.5. Then the other dimensionless parameter in the formula of travel cost is shown in Table 1. According to the actual situation, some dimensional parameters related to the network were set. The length of each roadway link is shown in Table 2, and the free-flow travel time and capacity of each roadway link are shown in Table 3. The subway section has a fixed travel time, and the travel time of the roadway link in subway mode is shown in Table 4. The operative information of the bus and subway is shown in Table 5. The walking time by subway \(t^{h3}\) is assumed to be 5 min. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(\theta\) & \(\alpha\) & \(\beta\) & \(a^{\prime}\) & \(\beta^{\prime}\) & \(\lambda_{t}\) & \(\lambda_{u}\) & \(\lambda_{m}\) & \(\gamma\) & \(\pi\) \\ \hline
0.4 & 0.15 & 4 & 0.02 & 1.8 & 0.5 & 0.3 & 0.2 & 0.5 & 0.5 \\ \hline \(f\) & \(g\) & \(u^{10}\) & \(u^{20}\) & \(u^{30}\) & \(u^{40}\) & \(u^{h^{0}}\) & \(m^{10}\) & \(m^{20}\) & \(m^{30}\) \\ \hline
0.5 & 0.05 & 0.6 & 0.5 & 0.2 & 0.1 & 0.15 & 0.15 & 0.1 & 0.2 \\ \hline \(m^{40}\) & \(m^{h2}\) & & & & & & & & \\ \hline
0.8 & 2 & & & & & & & & \\ \hline \end{tabular}
\end{table} 
Table 1: Modal parameters. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline Link & 1–2 & 1–4 & 1–5 & 2–3 & 2–5 & 3–6 & 4–5 & 4–7 & 5–6 & 5–8 \\ \hline Length/km & 2 & 1 & 5 & 2 & 2 & 3 & 1 & 2 & 1 & 3 \\ \hline Link & 5–9 & 6–9 & 7–8 & 7–10 & 8–9 & 8–11 & 9–12 & 10–11 & 11–12 \\ \hline Length/km & 6 & 1 & 3 & 8 & 2 & 7 & 6 & 4 & 3 \\ \hline \end{tabular}
\end{table} 
Table 2: Link length. The convergence characteristics of the proposed solution algorithm are illustrated in Figure 4. It can be seen that the equilibrium condition at a relative gap of \(10_{-3}(e=0.001)\) was obtained after 7200 iterations (for \(\theta=0.4\), \(\omega=0.1\) ). This result indicates that the proposed MSA solution algorithm can solve the problem with an acceptable accuracy level. \begin{table}
\begin{tabular}{c c c} \hline Link node & 1–5 & 5–9 \\ \hline Travel time (min) & 4 & 6 \\ \hline \end{tabular}
\end{table} 
Table 4: Travel time of roadway link in subway mode. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Link Node**} & \multicolumn{3}{c}{**Free-Flow Travel Time (min)**} & \multicolumn{3}{c}{**Capacity\((pcu\cdot h^{-1})\)**} \\ \cline{2-5}  & **Bicycle** & \(\left(\boldsymbol{t}_{l}^{\mathfrak{t}^{\mathfrak{t}}}\right)\) & **Bus** & \(\left(\boldsymbol{t}_{l}^{\mathfrak{g}^{\mathfrak{g}}}\right)\) & **Auto** & \(\left(\boldsymbol{t}_{l}^{\mathfrak{g}}\right)\) & **Bus and Auto** & **Bike** & \(\left(\boldsymbol{B}_{l}\right)\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: Free-flow travel time and capacity of each roadway link.