# A Framework to Determine the Utilization of Vacant Taxis on HOV Lanes with the Optimal Deployment
## 5 Optimal Deployment of HOV Lanes  
Allowing or forbidding vacant taxis from using HOV lanes means changing the topology of the network with the demand pattern. It is worth detecting the optimal design of HOV lanes with or without access of vacant taxis to HOV lanes. However, we know that adding new links may lead to a Braess paradox. In this paper we assume that an HOV lane is separated from a link. The construction cost is negligible since the cost of separating a lane is low.  
We use a binary variable \(\zeta_{a}\) to denote the determination of the deployment of the HOV lanes. If we deploy a HOV lane on link \(a\), \(\zeta_{a}=1\), and \(\zeta_{a}=0\) otherwise. Then the capacities of the lanes would alter, which leads to various travel times. When no HOV lane is on the link, the travel time on \(a\in A\) consisting of GP lanes can be expressed as:  
\[TT_{a}(v_{a},C^{\prime}{}_{a}),\ C^{\prime}{}_{a}=C_{a},\ a\in A \tag{42}\]  
When an HOV lane is separated from the link, the travel time on \(a\in A\) is denoted as:  
\[tt_{a}(v_{a},C^{\prime}{}_{a}),\ C^{\prime}{}_{a}=e_{a}\cdot C_{a},\ a\in A \tag{43}\]  
and the travel time on \(a\in\overline{A}\) is:  
\[Tt_{a}(v_{a},C^{\prime}),\ C^{\prime}=(1-e_{a})\cdot C_{a},\ a\in\overline{A} \tag{44}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Social Welfare** & \begin{tabular}{c} **Solo-Driving** \\ **Demand** \\ \end{tabular} & \
\begin{tabular}{c} **Carpooling** \\ **Demand** \\ \end{tabular} &
\begin{tabular}{c} **Occupied Taxi** \\ **Demand** \\ \end{tabular} \\ \hline
**Case 1** & 413,638 & 11,849 & 909 & 1399 \\
**Case 2** & 421,983 & 11,808 & 939 & 1443 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The network performance in case 1 and case 2.  
Figure 2: Ratio of flow on the HOV lane to the capacity versus the HOV lane.  
where \(e_{a}\) is the proportion of the link capacity and \(0<e_{a}<1\).  
Thus, the travel time on \(a\in A\) is calculated by:  
\[t_{a}=(1-\zeta_{a})\cdot TT_{a}+\zeta_{a}\cdot tt_{a},\,a\in A \tag{45}\]  
and the travel time on \(a\in\overline{A}\) is:  
\[t_{a}=Tt_{a},\,\,a\in\overline{A} \tag{46}\]  
Based on the calculation above, we propose a bi-level programming for optimal design. The lower level is the VI above, describing the travel mode and route choice, whereas the upper level aims at maximizing the social welfare. With or without the accesses of vacant taxis to HOV lanes, the upper level can be expressed as \(\max Z_{1}\) and \(\max Z_{2}\), respectively. We use max (Z) to denote upper level aiming at \(\max Z_{1}\) or \(\max Z_{2}\), and the bi-level model is written as:  
\[\max Z(\mathbf{v},\zeta)\]  
The flow patterns \(\mathbf{v}=\mathbf{v}(\zeta)\) are defined by the VI above.  
In this procedure, the decision variable is \(\zeta_{a}\). In terms of the variation of \(\zeta_{a}\), the capacity change and the link travel time change, which has an impact on the topology of the network. With or without the access of vacant taxis to HOV lanes in the network, the social welfare varies. Therefore, the deployment design aims to find the optimal value of the binary variable \(\zeta_{a}\) to determine whether the HOV lane is supposed to be constructed on link \(a\) for the purpose of maximal social welfare.