# A Framework to Determine the Utilization of Vacant Taxis on HOV Lanes with the Optimal Deployment
## 3 Formulations
### Variational Inequality  
Since the cost functions are asymmetric, namely, \(\frac{\partial c_{s}^{al}}{\partial x_{s}^{2}}\
eq\frac{\partial\bar{c}_{s}^{ s}}{\partial x_{s}^{4}}\), \(\frac{\partial\bar{c}_{s}^{al}}{\partial x_{s}^{2}}\
eq\frac{\partial\bar{c}_ {s}^{a}}{\partial x_{s}^{4}}\), \(\frac{\partial\bar{c}_{s}^{al}}{\partial x_{s}^{4}}\
eq\frac{\partial\bar{c}_ {s}^{a}}{\partial x_{s}^{4}}\), a VI is used to describe the traffic system as below:  
\[ \begin{array}{l}\sum\limits_{r\in R}\sum\limits_{s\in S}\sum\limits_{p\in P }\sum\limits_{m\in M}c_{rs,p}^{m}(f^{*})\Big{(}f_{rs,p}^{m}-f_{rs,p}^{m,s}\Big{)} +\sum\limits_{s\in S}\sum\limits_{r\in R}\sum\limits_{p_{v}\in P_{v}}c_{sr,p_ {v}}^{v}(f^{*})\Big{(}f_{sr,p_{v}}^{v}-f_{sr,p_{v}}^{v,s}\Big{)}\\ +\sum\limits_{r\in R,s\in S}\sum\limits_{m\in M}(\frac{1}{p}\ln\frac{d_{rs}^{ m,s}}{d_{rs}}-k_{rs}^{m})\big{(}d_{rs}^{m}-d_{rs}^{m,s}\big{)}+\sum\limits_{s\in S \in S\in P}\frac{1}{p}\ln d_{sr}^{p,s}(d_{sr}^{v}-d_{sr}^{v,s})\\ +\frac{1}{m_{s}}\sum\limits_{r\in R,s\in S}\ln\frac{d_{rs}}{d_{rs}}(d_{rs}-d_{ rs}^{s})\geq 0\\ s.t.\ (8)\sim(16),\ (19)\end{array} \]  
The VI above can be proven to be equivalent to a logit mode split and the traffic assignment following the UE principle simultaneously. Note that in the situations that allow or forbid vacant taxis from using HOV lanes, the VI above can be used to describe the traffic systems, just with different \(P_{v}\) sets. The constrains (8)-(16) and (19) are non-negative and linear, whereas the decision variables are non-negative, which implies that at least one solution to the VI above exists [8]. Several approaches to solving the VI problem have emerged, such as a projection algorithm, a diagonalization algorithm, an approach with a gap function to transfer a VI problem into equivalent mathematical programming, and so on. In this paper we adopt the block Gauss-Seidel decomposition approach combined with the method of successive averages and a diagonalization algorithm to solve the VI above. The flow patterns and demands can be captured. Readers may be interested in [38] for the proof and the steps of this algorithm. According to [39], the overall computational complexity of the block Gauss-Seidel decomposition approach combined with the method of successive averages and diagonalization algorithm is \(O(\frac{2|N|_{3}+2|W||M|+|W||A|}{\epsilon^{2}_{p}})\) and \(O(\frac{2|N|_{3}+2|W||M|+|W||A|}{\epsilon^{2}_{p}\epsilon^{2}_{p}})\) for the projection method, in which \(\varepsilon\) and \(\varepsilon_{p}\) are the convergence error for the two algorithms, respectively. Therefore, the projection method is more effective than the solving method we adopt, especially in the large-scale network. However, the approach we adopt is still effective in solving the VI and easier to implement. In this paper we still choose to use the block Gauss-Seidel decomposition algorithm for simplicity. If differentiating \(d_{sr}^{v}\), we can obtain the following equations:  
\[d_{sr}^{v}=\exp[-\sigma(u_{sr}^{v}+\rho_{s}+\theta_{r})],\ r\in R,s\in S \tag{26}\] \[d_{sr}^{v}=D_{s}^{o}\frac{\exp[-\sigma(u_{sr}^{v}+\theta_{r})]}{ \sum\limits_{i\in R}\exp\left[-\sigma(u_{si}^{v}+\theta_{i})\right]},\ s\in S,r\in R \tag{27}\]  
then we know that \(\theta_{r}=\lambda_{v}w_{r}^{v}\). Note that \(\rho_{s}\) and \(\theta_{r}\) are Lagrange multipliers, and [PERSON] et al. pointed out that \(\theta_{r}\) is not unique since the expression \((\rho_{s}-\varphi)+(\theta_{r}+\varphi)\) exists. However, \(\theta_{r}\) should be unique due to the time conservation equation for vacant taxi movements [37]. Therefore, [PERSON] et al. proposed the methods to calculate the searching and waiting time based on the equations above on the condition of single taxi mode [10]. In this paper, referring to [38], we adopt this method in the multi-mode condition. We represent \(\lambda_{v}w_{r}^{v}\) in Equation (17) as:  
\[\lambda_{v}w_{r}^{v}=\theta_{r}+\varphi,\ r\in R \tag{28}\]  
\(\varphi\) is an arbitrary constant.