# Highway Travel-Time Forecasting with Greenshields Model-Based Cascaded Fuzzy Logic Systems
## 3 Cascaded Fuzzy Systems Based on Greenshields Models for Vehicle Speed and Travel-Time Prediction
### Input and Output Membership Functions' Setup for Greenshields Model-Based Fuzzy Logic Systems
The fuzzy domain of the output membership function for vehicle speed includes the following levels: Extremely Slow (ES), Very Slow (VS), Slow (S), Steady (St), Quite Slow (QS), Moderate Slow (MS), Moderate (M), Moderate Fast (MF), Quite Fast (QF), Speedy (Sp), Fast (F), Very Fast (VF), and Extremely Fast (EF), as presented in Table 3. The membership function and its range for vehicle speed are depicted in Figure 11.  
For the non-congested mode, the fuzzy domain of the membership functions for traffic flow is divided into the following levels: Extremely Low (EL), Very Low (VL), Low (L), Sparse (Sp), Quite Low (QL), Moderately Low (ML), Moderate (M), Moderate High (MH), Quite High (QH), Dense (De), High (H), Very High (VH), and Extremely High (EH), as presented in Table 4. The membership function and range values for traffic flow in the non-congested mode are illustrated in Figure 12. \begin{table}
\begin{tabular}{c c} \hline
**Traffic Density (veh/km)** & **Percentage of Density (\%)** \\ \hline Moderate (M) & 50–60 \\ Moderate High (MH) & 57–68 \\ Quite High (QH) & 66–76 \\ Dense (De) & 73–82 \\ High (H) & 80–88 \\ Very High (VH) & 87–95 \\ Extremely High (EH) & 92–100 \\ \hline \end{tabular}
\end{table} 
Table 2: Membership function of traffic density and its range values in congested mode. Figure 10: Membership function of traffic density and its range values in congested mode. Similarly, the fuzzy domain of the membership function for traffic density in the non-congested mode is segmented into seven levels: Extremely Low (EL), Very Low (VL), Low (L), Sparse (Sp), Quite Low (QL), Moderately Low (ML), and Moderate (M), as detailed in Table 5. The membership function and range values for traffic density are shown in Figure 13. The fuzzy domain of the output membership function for vehicle speed in the non-congested mode is defined by the following levels: Extremely Slow (ES), Very Slow (VS), Slow (S), Steady (St), Quite Slow (QS), Moderate Slow (MS), Moderate (M), Moderate Fast (MF), Quite Fast (QF), Speedy (Sp), Fast (F), Very Fast (VF), and Extremely Fast (EF), as presented in Table 6. The membership function and its corresponding range for vehicle speed are depicted in Figure 14. \begin{table}
\begin{tabular}{c c} \hline
**Traffic Flow (veh/h)** & **Percentage of Traffic Flow (\%)** \\ \hline Extremely (EL) & 0–12 \\ Very Low (VL) & 8–20 \\ Low (L) & 16–27 \\ Sparse (Sp) & 23–32 \\ Quite Low (QL) & 30–40 \\ Moderately Low (ML) & 36–48 \\ Moderate (M) & 44–57 \\ Moderate High (MH) & 52–65 \\ Quite High (QH) & 60–73 \\ Dense (De) & 69–80 \\ High (H) & 76–87 \\ Very High (VH) & 86–95 \\ Extremely High (EH) & 92–100 \\ \hline \end{tabular}
\end{table} 
Table 4: Membership function of traffic flow and its range values in non-congested mode. Figure 11: Membership function of traffic speed and its range values in congested mode. Figure 12: Membership function of traffic flow and its range values in non-congested mode. \begin{table}
\begin{tabular}{c c} \hline \hline
**Vehicle Speed (km/h)** & **Mapping Interval (km)** \\ \hline Extremely Slow (ES) & 0–13 \\ Very Slow (VS) & 8–23 \\ Slow (S) & 19–32 \\ Steady (St) & 29–43 \\ Quite Slow (QS) & 39–52 \\ Moderate Slow (MS) & 49–62 \\ Moderate (M) & 59–73 \\ Moderate Fast (MF) & 69–83 \\ Quite Fast (QF) & 78–92 \\ Speedy (Sp) & 89–102 \\ Fast (F) & 100–114 \\ Very Fast (VF) & 111–123 \\ Extremely Fast (EF) & 121–130 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Membership function of vehicle speed and its range values in non-congested mode. Figure 14: Membership function of vehicle speed and its range values in non-congested mode. Figure 13: Membership function of traffic density and its range values in non-congested mode.