## 2 Multi-Class Traffic Assignment
### Travel Cost Function  
\(C_{a}\) capacity (veh/hour) of link \(a\).  
\(C_{a}^{H}\) and \(C_{a}^{C}\) capacity of HV and CAV on link \(a\), respectively.  
\(h_{a}\) headway (hour unit) of link \(a\).  
\(h_{a}^{H}\) and \(h_{a}^{C}\) headway (hour unit) of HV and CAV, respectively on link \(a\).  
\(v_{a}^{H}\) and \(v_{a}^{H}\) link flows of HV and CAV, respectively on link \(a\).  
\(t_{a}^{m}(\cdot)\) travel time on link \(a\) of class \(m\).  
\(p^{m}\) travel time weight parameter of class \(m\).  
\(t_{a}^{0}\) free-flow travel time of link \(a\).  
\(v_{a}^{m}\) flow on link \(a\) of class \(m\).  
\(C_{a}^{m}\) capacity on link \(a\) of class \(m\).  
\(a_{a}\) and \(\beta_{a}\) parameters of travel time function on link \(a\).  
\(c_{km}^{rs}\) cost of route \(k\) between origin \(r\) and destination \(s\) (O-D pair \(rs\)) in class \(m\).  
\(c_{km}^{rs}\) travel time on route \(k\) and shortest route \(\overline{k}\) between O-D pair \(rs\), respectively.  
\(f_{km}^{rs}\) route \(k\) flow between O-D pair \(rs\).  
\(g_{km}^{rs}\) route-link indicator, 1 if link \(a\) is on route \(k\) between O-D pair \(rs\) and 0 otherwise.  
\(n\) iteration number.  
step size.u  
\(s_{km}^{rs}\) diagonal, positive definite scaling matrix on route \(k\) between O-D pair \(rs\).  
Vehicle interactions occur when multiple vehicle classes interact in the same roadway space. These interactions typically occur between vehicles with varied maneuvering and operational characteristics. Due to the lower reaction times described in the introduction, CAVs have a smaller headway than HVs. With this assumption, headway in mixed flows can be described as a function of the proportion of HVs and CAVs [7]. The function of headway based on the proportion of mixed flows is shown in Equation (1).  
\[\overline{h}_{a}=\frac{v_{a}^{H}}{v_{a}^{H}+v_{a}^{C}}h_{a}^{H}+\frac{v_{a}^{C }}{v_{a}^{H}+v_{a}^{C}}h_{a}^{C} \tag{1}\]  
The capacity of a road refers to its maximum traffic density, which can be expressed as follows in terms of headway.  
\[C_{a}=\frac{1}{h_{a}} \tag{2}\]  
If only a single class (e.g., HV or CAV) is present, we can obtain Equation (3), and Equation (1) can be written as Equation (4) using Equation (3).  
\[h_{a}^{H}=\frac{1}{C_{a}^{H}}\text{ and }h_{a}^{C}=\frac{1}{C_{a}^{C}} \tag{3}\]  
\[\overline{h}_{a}=\frac{v_{a}^{H}}{v_{a}^{H}+v_{a}^{C}}\frac{1}{C_{a}^{H}}+ \frac{v_{a}^{C}}{v_{a}^{H}+v_{a}^{C}}\frac{1}{C_{a}^{C}} \tag{4}\]  
With the derived headway in Equation (4), the capacity for mixed flows can be expressed as Figure 1 with Equation (2).  
Figure 2 illustrates an example of travel time and capacity change based on the proportion of HVs and CAVs.  
Figure 2a illustrates a typical link travel time function (i.e., a Bureau of Public Roads (BPR) function) for the TAP and the used capacity for HVs and CAVs. Using the input data in Figure 2a, Figure 2b illustrates travel time change and capacity change as a function of the proportion of CAVs. As CAVs have a smaller headway than HVs, the total capacity for mixed flows increased as the proportion of CAVs increased. However, the increasing rate was not a linear pattern. With increased capacity, travel time was intuitively decreased, and this was also not a linear trend. Hence, the travel time function was asymmetric with regard to HVs and CAVs because the travel time of each mode was affected by the other modes. The following link travel time function in Equation (5) was used to model the asymmetric interactions between various modes in a general situation involving a variety of vehicle types.  
Figure 1: Example of the headway in mixed traffic flows.  
\[t_{a}^{m}(\cdot)=p^{m}t_{a}^{0}\Bigg{[}1+\alpha_{a}\left(\frac{\sum\limits_{m\in \mathrm{M}}v_{a}^{m}}{1\bigg{/}\left(\frac{1}{\sum\limits_{m\in\mathrm{M}}\frac{ \sum\limits_{m\in\mathrm{M}}^{m}}{\mathrm{C}_{a}^{m}}}\right)\right)^{\beta_{a}} \Bigg{]} \tag{5}\]