# A Framework to Determine the Utilization of Vacant Taxis on HOV Lanes with the Optimal Deployment
## 8 Conclusions  
This paper tries to state the issue of whether vacant taxis are supposed to use HOV lanes. We propose a framework including a VI describing the travel mode and route choice of travelers for the purpose of exploring the network performance with/without the access of vacant taxis to HOV lanes, as well as bi-level programming for optimal deployment of HOV lanes as a bi-level programming for maximum social welfare. Due to the binary variables and the expensive computation, the genetic algorithm combined with a surrogate assistant approach is adopted to solve the bi-level model. The models and the algorithms are applied to numerical examples to demonstrate the feasibility and effectiveness. The main findings are listed as follows:  
1. A paradox similar to the Braess paradox is revealed. Given HOV lane deployment, allowing vacant taxis to use HOV lanes may be negative for maximization of social welfare with intensive usage of HOV lanes. This paradox may derive from the congestion in GP lanes and the capacity limits of the bottlenecks, instead of the congestion in HOV lanes. Allowing vacant taxis to use HOV lanes can intensify the utilization of the HOV lanes while decreasing social welfare.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Social Welfare**} & **Solo-Driving** & **Carpooling** & **Occupancy Taxi** \\  & **Demand** & **Demand** & **Demand** \\ \hline
**Case 1** & 160,739 & 6822 & 189 & 3030 \\
**Case 2** & 162,542 & 6820 & 188 & 3036 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The social welfare of case 1 and case 2 in the Sioux Falls network.  
Figure 7: The optimal deployment of HOV lanes.  
2. Considering optimization of HOV lane deployment, allowing vacant taxis to use HOV lanes is more likely negative to social welfare due to the paradox, although the optimal deployments are different in two cases.
3. The paradox depends on the topology and the parameters of the network, e.g., the flag-fall price of taxis. If the paradox is inexistent, the scheme may be beneficial to the traffic system. Therefore, specific information and analysis are necessary to determine whether to forbid or allow vacant taxis to use HOV lanes.
4. The models and the algorithms can be used in reality.  
This study offers a framework for decision-makers to determine the access of vacant taxis to HOV lanes. Road managers can apply this framework to determine whether vacant taxis are supposed to use HOV lanes in specific situations. They also may notice the possible paradox and the network performances displayed in our results. A more efficient algorithm would be helpful for the model, and real traffic network and data will be tested in further research.  
Conceptualization, D.D.; methodology, F.G. and D.D.; software, F.G. and K.L.; validation, F.G. and D.D.; formal analysis, F.G.; investigation, F.G.; resources, D.D.; writing--original draft preparation, F.G. and D.D.; writing--review and editing, F.G., K.L. and D.D.; visualization, F.G.; supervision, K.L. and D.D.; project administration, D.D.; funding acquisition, D.D. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Special Postdoctoral Science Fund Project for basic research and frontier exploration of Chongqing Science and Technology Bureau, grant number cstc2021 jcyj-bsh0210, and jointly supported by the National Social Science Foundation of China, grant number 22 BGL276; the National Natural Science Foundation of China: 52262045; and the Natural Science Foundation of Gansu Province, China, grant number 20 JR5 RA398.  
Not applicable.  
The authors thank the anonymous reviewers and editors for their valuable and constructive suggestions for improving this article.  
The authors declare no conflict of interest.