# 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  
In fuzzy logic systems, the choice of membership function type depends on the specific application, data characteristics, and desired system behavior. There is no universally optimal function; rather, the goal is to select one that best captures the linguistic terms and relationships within the system. After several trials, we found that trapezoidal membership functions offer the best balance. Regarding the number of levels used in the membership functions for traffic density and flow, there is no single correct value. The selection process is problem-specific and often relies on practical experience and experimentation to achieve the best fit. The number of linguistic variables is typically determined by the level of resolution required in the system's behavior. After testing several configurations, we identified an appropriate number of levels that achieves a trade-off between model complexity and parameter tuning effort, while closely aligning with the ideal model performance. The proposed fuzzy logic system is designed to account for two traffic conditions: congested and non-congested. In each condition, traffic flow and traffic density serve as input parameters, while vehicle speed is the output parameter. Traffic flow is defined as  
Figure 8: Architecture of a fuzzy logic system for vehicle speed prediction. the number of vehicles passing through a road section per hour (veh/h), with its range expressed as a percentage of the specified full flow. Vehicle density refers to the number of vehicles per kilometer on a road section (veh/km), also expressed as a percentage of the specified full density. Vehicle speed represents the rate at which vehicles travel on the road (km/h), with a parameter range from 0 km/h to 130 km/h. Trigonometric or trapezoidal membership functions are chosen for the fuzzy sets, with ambiguous regions specified based on the domain range of collected data and constraints of each highway section. The fuzzy system, with two input parameters and one output parameter, operates in two modes: a congested mode and non-congested mode. In the congested mode, and to align with the Greenshields models, the input membership function for traffic flow is subdivided into thirteen levels to ensure adequate accuracy of the fuzzy logic model. These levels are 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 1. The membership function and its range for traffic flow in the congested mode are illustrated in Figure 9.  
For traffic density in the congested mode, the fuzzy domain is divided into the following levels: Moderate (M), Moderate High (MH), Quite High (QH), Dense (De), High (H), Very High (VH), and Extremely High (EH), as detailed in Table 2. The corresponding membership function and range are shown in Figure 10. \begin{table}
\begin{tabular}{c c} \hline \hline
**Traffic Flow (veh/h)** & **Percentage of Traffic Flow (\%)** \\ \hline Extremely Low (EL) & 0–11 \\ Very Low (VL) & 8–20 \\ Low flow (L) & 17–26 \\ Sparse (Sp) & 24–34 \\ Quite Low (QL) & 31–40 \\ Moderately Low (ML) & 38–48 \\ Moderate (M) & 46–57 \\ Moderate High (MH) & 53–64 \\ Quite High (QH) & 60–71 \\ Dense (De) & 69–78 \\ High (H) & 76–86 \\ Very High (VH) & 83–92 \\ Extremely High (EH) & 89–100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Membership function of traffic flow and its range values in congested mode. Figure 9: Membership function of traffic flow and its range values in congested mode.