## 2 Materials and Methods
### Model Formulation
#### 2.2.1 Objective Function
Meanwhile, in view of the fact that a conventional phase sequence (sequence 1 in Figure 6) is not always adopted in traffic signal optimization practices, 0/1 variables are introduced to optimize the phase sequences, which contributes to better control effects and improves the applicability of the presented model. Suppose an intersection is numbered (\(p\), \(q\)) and (\(r\), \(s\)) in the W-E and S-N arterials, respectively. The expressions of \(\psi^{\text{SW}}_{(p,q);(r,s)}\), \(\psi^{\text{WN}}_{(p,q);(r,s)}\), \(\psi^{\text{NE}}_{(p,q);(r,s)}\), and \(\psi^{\text{ES}}_{(p,q);(r,s)}\) are as follows:  
\[\psi^{\text{SW}}_{(p,q);(r,s)}=\left\{ \begin{array}{l}\Big{(}2y_{(p,q);(r,s) }-u_{(p,q);(r,s)}\Big{)}\Big{(}0.5r^{\text{SNT}}_{(r,s)}-0.5r^{\text{WET}}_{(p,q)}+g^{\text{SNT}}_{(r,s)}\Big{)}+\\ \Big{(}2z_{(p,q);(r,s)}-v_{(p,q);(r,s)}\Big{)}\Big{(}0.5r^{\text{SNT}}_{(r,s)}+ 0.5r^{\text{WET}}_{(p,q)}-g^{\text{WEL}}_{(p,q)}\Big{)},g^{\text{SNL}}_{(r,s) }\leq g^{\text{WEL}}_{(p,q)}\\ \Big{(}2y_{(p,q);(r,s)}-u_{(p,q);(r,s)}\Big{)}\Big{(}0.5r^{\text{SNT}}_{(r,s)}- 0.5r^{\text{WET}}_{(p,q)}+g^{\text{SNL}}_{(r,s)}\Big{)}+\\ \Big{(}v_{(p,q);(r,s)}-2z_{(p,q);(r,s)}\Big{)}\Big{(}0.5r^{\text{SNT}}_{(r,s)}+ 0.5r^{\text{WET}}_{(p,q)}-g^{\text{SNL}}_{(r,s)}\Big{)},g^{\text{SNL}}_{(r,s) }>g^{\text{WEL}}_{(p,q)}\end{array} \right., \tag{19}\]  
(20)  
\(u_{(p,q);(r,s)}\), \(v_{(p,q);(r,s)}\), \(y_{(p,q);(r,s)}\), and \(z_{(p,q);(r,s)}\) are 0/1 variables, which are employed to optimize the phase sequences. \(u_{(p,q);(r,s)}\), \(v_{(p,q);(r,s)}\), \(y_{(p,q);(r,s)}\), and \(z_{(p,q);(r,s)}\) should satisfy the following constraints:  
\[\left\{ \begin{array}{l}u_{(p,q);(r,s)}+v_{(p,q);(r,s)}=1\\ u_{(p,q);(r,s)}-y_{(p,q);(r,s)}\geq 0\\ v_{(p,q);(r,s)}-z_{(p,q);(r,s)}\geq 0\end{array} \right.. \tag{23}\]  
The relationship between \(u_{(p,q);(r,s)}\), \(v_{(p,q);(r,s)}\), \(y_{(p,q);(r,s)}\), and \(z_{(p,q);(r,s)}\) and phase sequences are shown in Table 1.  
\(\bullet\) Bandwidth ratio constraints  
To balance the outbound and inbound bandwidth, the bandwidth ratio constraints for cars and buses are expressed as  
\[\left(1-k^{\text{car}}_{(i,j),(i,j+1)}\right)\overline{b}^{\text{ car}}_{(i,j),(i,j+1)}-\left(1-k^{\text{car}}_{(i,j),(i,j+1)}\right)k^{\text{ car}}_{(i,j),(i,j+1)}b^{\text{car}}_{(i,j),(i,j+1)}\geq 0, \tag{24}\]  
\[\left(1-k^{\text{bus}}_{(i,j),(i,j+1)}\right)\overline{b}^{\text{bus}}_{(i,j),(i,j+1)}-\left(1-k^{\text{bus}}_{(i,j),(i,j+1)}\right)k^{\text{bus}}_{(i,j),(i,j+1)}b^{\text{bus}}_{(i,j),(i,j+1)}\geq 0, \tag{25}\]  
where \(k^{\text{car}}_{(i,j),(i,j+1)}\) [\(k^{\text{bus}}_{(i,j),(i,j+1)}\)] is the car [bus] ratio of inbound to outbound bandwidth between \((i,j)\) and \((i,j+1)\). \(\bullet\)  
Figure 6: Six phase sequences when the symmetrical phase scheme is used. The ratio of passengers per hour in the inbound through cars [buses] at (\(i\), \(j\)) to passengers per hour in the outbound through cars [buses] at (\(i\), \(j+1\)) can be used as the value of \(k^{\text{car}}_{(i,j),(i,j+1)}\) [\(k^{\text{bus}}_{(i,j),(i,j+1)}\)]. Additionally, the values of \(k^{\text{car}}_{(i,j),(i,j+1)}\) and \(k^{\text{bus}}_{(i,j),(i,j+1)}\) can be specified by traffic engineers according to the control needs. Equations (24) and (25) can be updated to the following constraints by relaxing them:  
\[\left(1-k^{\text{car}}_{(i,j),(i+1)}\right)\overline{b}^{\text{ car}}_{(i,j),(i,j+1)}-\left(1-k^{\text{car}}_{(i,j),(i+1)}\right)k^{\text{ car}}_{(i,j),(i,j+1)}b^{\text{car}}_{(i,j),(i,j+1)}+x^{\text{car}}_{(i,j),(i,j+1)}M \geq 0, \tag{26}\] \[\left(1-k^{\text{bus}}_{(i,j),(i,j+1)}\right)\overline{b}^{\text {bus}}_{(i,j),(i,j+1)}-\left(1-k^{\text{bus}}_{(i,j),(i,j+1)}\right)k^{\text {bus}}_{(i,j),(i,j+1)}b^{\text{bus}}_{(i,j),(i,j+1)}+x^{\text{bus}}_{(i,j),(i, j+1)}M \geq 0. \tag{27}\]  
* Mutual offset constraints The mutual offset constraint is due to the fact that cars and buses are controlled by the same traffic signal system. Consequently, car bands and bus bands should have an identical offset.