## 2 Materials and Methods
### Model Formulation
#### 2.2.1 Objective Function
 \tag{14}\]  
By relaxing Equation (14), the equality constraint shown in Equation (14) can be equivalently transformed into two inequality constraints, which are Equations (15) and (16). \[ \begin{split}& 0.5\Big{(}r_{(m,n)}^{\text{CNT}}+r_{(e+1,h)}^{ \text{WET}}+r_{(m+1,k)}^{\text{CNT}}+r_{(e,f)}^{\text{WET}}\Big{)}-0.5\Big{(}r _{(m,n+1)}^{\text{CNT}}+r_{(e+1,k+1)}^{\text{WET}}+r_{(m+1,k+1)}^{\text{CNT}}+r _{(e,f+1)}^{\text{WET}}\Big{)}+\\ &\Big{(}w_{(m,n)}^{\text{car}}+w_{(e+1,h)}^{\text{car}}+\overline {w}_{(m+1,k),(m+1,k+1)}^{\text{car}}+\overline{w}_{(e,f),(e,f+1)}^{\text{car}} \Big{)}-\Big{(}w_{(m,n),(m,n+1)}^{\text{car}}+w_{(e+1,h),(e+1,h+1)}^{\text{car} }+\overline{w}_{(m+1,k+1)}^{\text{car}}+\overline{w}_{(e,f+1)}^{\text{car}} \Big{)}+\\ &\Big{(}r_{(m,n),(m,n+1)}^{\text{car}}+r_{(e+1,h),(e+1,h+1)}^{ \text{car}}+\overline{w}_{(1+h,(m+1,k+1)),(m+1,k+1)}^{\text{car}}+\overline{ w}_{(e,f),(e,f+1)}^{\text{car}}\Big{)}z-\psi_{(e+1,h);(m,n+1)}^{\text{SW}}- \psi_{(e+1+h);(m+1,k+1)}^{\text{WN}}-\\ &\psi_{(e,f+1);(m+1,k)}^{\text{NE}}-\psi_{(e,f);(m,n)}^{\text{ES}} \leq n_{(e,f),(m+1,k+1)}^{\text{car}}+\Big{(}x_{(m,n),(m+1)}^{\text{car}}+x_{(e+1,h),(e+1,h+1)}^{\text{car}}+x_{(m+1,k),(m+1,k+1)}^{\text{car}}+x_{(e,f),(e,f+1)} ^{\text{car}}\Big{)}M\\ & 0.5\Big{(}r_{(m,n)}^{\text{CNT}}+r_{(e+1,h)}^{\text{WET}}+r_{(m+1,k)}^{ \text{CNT}}+r_{(e,f)}^{\text{WET}}\Big{)}-0.5\Big{(}r_{(m,n+1)}^{\text{CNT}}+r_{(e+1,h)}^{\text{WET}}+r_{(m+1,k+1)}^{\text{CNT}}+r_{(e,f+1)}^{\text{WET}}\Big{)}+\\ &\Big{(}w_{(m,n)}^{\text{car}}+w_{(e+1,h)}^{\text{car}}+\overline {w}_{(m+1,k),(m+1,k+1)}^{\text{car}}+\overline{w}_{(e,f),(e,f+1)}^{\text{car}} \Big{)}-\Big{(}w_{(m,n),(m,n+1)}^{\text{car}}+w_{(e+1,h),(e+1,h+1)}^{\text{car}}+ \overline{w}_{(m+1,k+1)}^{\text{car}}+\overline{w}_{(e,f+1)}^{\text{car}} \Big{)}+\\ &\Big{(}r_{(m,n),(m+1)}^{\text{car}}+r_{(e+1,h),(e+1,h+1)}^{\text{ car}}+\overline{w}_{(1+1,k),(m+1,k+1)}^{\text{car}}+\overline{w}_{(e,f),(e,f+1)}^{\text{ car}}\Big{)}z-\psi_{(e+1,h);(m,n+1)}^{\text{SW}}-\psi_{(e+1,h);(m+1,k+1)}^{\text{WN}}-\\ &\psi_{(e,f+1);(m+1,k)}^{\text{NE}}-\psi_{(e,f);(m,n)}^{\text{ES}} \geq n_{(e,f),(m+1,k+1)}^{\text{car}}-\Big{(}x_{(m,n),(m+1)}^{\text{car}}+x_{(e+1,h),(e+1,h+1)}^{\text{car}}+x_{(m+1,h),(m+1,k+1)}^{\text{car}}+x_{(e,f),(e,f+1)} ^{\text{car}}\Big{)}M\\ \end{split}  \tag{16}\]  
Figure 5: Geometry of a network-level loop integer constraint. where \(x^{\text{car}}_{(m,n),(m,n+1)}\), \(x^{\text{car}}_{(c+1,h),(c+1,h+1)}\), \(x^{\text{car}}_{(m+1,k),(m+1,k+1)}\), and \(x^{\text{car}}_{(e,f),(e,f+1)}\) are 0/1 variables. When all these four binary variables equal to 0, Equations (15) and (16) will transform into Equation (14). Otherwise, Equation (14) is relaxed. Similarly, the network-level loop integer constraint with relaxation for buses is expressed as  
(17)  
(18)  
where \(x^{\text{bus}}_{(m,n),(m,n+1)}\), \(x^{\text{bus}}_{(e+1,h),(c+1,h+1)}\), \(x^{\text{bus}}_{(m+1,k),(m+1,k+1)}\), and \(x^{\text{bus}}_{(e,f),(e,f+1)}\) are 0/1 variables, and \(u^{\text{bus}}_{(e,f),(m+1,k+1)}\) is an integer variable representing an integer multiple of the common cycle time. Based on the definitions of \(\psi^{\text{SW}}_{(e+1,h);(m,n+1)}\), \(\psi^{\text{WN}}_{(e+1,h+1);(m+1,k+1)}\), \(\psi^{\text{NE}}_{(e,f+1);(m+1,k)}\), and \(\psi^{\text{ES}}_{(e,f);(m,n)}\), the expressions of these intranode offset variables are related to the phase scheme and phase sequence adopted at an intersection. The symmetrical phase scheme (see Figure 6) has the advantages of simple implementation and low management cost and is widely used in traffic signal optimization practices in China. Based on this, the expressions of these intranode offset variables under the symmetrical phase scheme are given in the following discussion.