## 2 Materials and Methods
### Constraints
#### 2.3.4 Loop Integer Constraint
Similarly, constraint (15) can be updated as:  
\[\theta_{i}+\overline{n}_{i,m,\overline{p}}-\overline{r}_{i,\overline{p}}^{\rm{ E,R}}-\overline{w}_{i,m,\overline{p}}^{\rm{E}}\leq\theta_{i+1}-\overline{r}_{i+1, \overline{p}}^{\rm{S,R}}-\overline{w}_{i+1,m,\overline{p}}^{\rm{S}}+\overline {t}_{i,m,\overline{p}}z+M\big{(}1-\overline{x}_{i,m,\overline{p}}\big{)}\ i=1,2, \ldots,N-1;m=1,2,3;\overline{p}=1,2,3,4, \tag{16}\]  
\[\theta_{i}+\overline{n}_{i,m,\overline{p}}-\overline{r}_{i,\overline{p}}^{\rm{ E,R}}-\overline{w}_{i,m,\overline{p}}^{\rm{E}}\geq\theta_{i+1}-\overline{r}_{i+1, \overline{p}}^{\rm{S,R}}-\overline{w}_{i+1,m,\overline{p}}^{\rm{S}}+\overline {t}_{i,m,\overline{p}}z-M\big{(}1-\overline{x}_{i,m,\overline{p}}\big{)}\ i=1,2, \ldots,N-1;m=1,2,3;\overline{p}=1,2,3,4. \tag{17}\]  
Similarly, the values of \(\overline{r}_{i,\overline{p}}^{\rm{E,R}}\) or \(\overline{r}_{i+1,\overline{p}}^{\rm{S,R}}\) are closely related to the phase sequences and the coordinated paths. \(\overline{r}_{i,\overline{p}}^{\rm{E,R}}\) and \(\overline{r}_{i+1,\overline{p}}^{\rm{S,R}}\) are expressed by:  
\[\overline{r}_{i,\overline{p}}^{\rm{E,R}}=1-\overline{r}_{i,\overline{p}}^{\rm{ E,L}}-\overline{\delta}_{i,\overline{p}}^{\rm{E}}=1-\sum_{u=1}^{4}y_{i,u,v}g_{i,u}- \overline{\delta}_{i,\overline{p}}^{\rm{E}}, \tag{18}\]  
\[\overline{r}_{i+1,\overline{p}}^{\rm{S,R}}=1-\overline{r}_{i+1,\overline{p}}^{ \rm{S,L}}-\overline{\delta}_{i+1,\overline{p}}^{\rm{S}}=1-\sum_{u=1}^{4}y_{i+1,u,\overline{v}}g_{i+1,u}-\overline{\delta}_{i+1,\overline{p}}^{\rm{S}}, \tag{19}\]  
where \(\overline{r}_{i,\overline{p}}^{\rm{E,L}}\) (\(\overline{r}_{i+1,\overline{p}}^{\rm{S,L}}\)) represents the total red time on the left of the green wave bandwidth for coordinated path \(\overline{p}\) at the ending intersection \(i\) (starting intersection \(i+1\)), and \(y_{i,u,\overline{p}}\) (\(y_{i+1,u,\overline{p}}\)) represents the sequence between phase \(\overline{v}\) that the coordinated path \(\overline{p}\) passes through and phase \(u\) at intersection \(i\) (intersection \(i+1\)). If phase \(u\) is before phase \(\overline{v}\) at intersection \(i\) (intersection \(i+1\)), \(y_{i,u,\overline{v}}\) (\(y_{i+1,u,\overline{v}}\)) equals \(1\); otherwise, \(y_{i,u,\overline{v}}\) (\(y_{i+1,u,\overline{v}}\)) equals \(0\). Note that variables \(y_{i,u,v}\), \(y_{i+1,u,v}\), \(y_{i,u,\overline{v}}\), and \(y_{i+1,u,\overline{v}}\) must also meet the following additional constraints:  
\[\theta_{i}+r_{i,p}^{\rm S,L}+w_{i,m,p}^{\rm S}+t_{i,m,p}z\leq\theta_{i+1}+k_{i +1,m,p}+r_{i+1,p}^{\rm E,L}+w_{i+1,m,p}^{\rm E}+M\big{(}1-x_{i,m,p}\big{)}\;i=1,2,\ldots,N-1;m=1,2,3;p=1,2,3,4, \tag{11}\]0  
\[\theta_{i}+r_{i,p}^{\rm S,L}+w_{i,m,p}^{\rm S}+t_{i,m,p}z\leq\theta_{i+1}+k_{i +1,m,p}+r_{i+1,p}^{\rm E,L}+w_{i+1,m,p}^{\rm E}+M\big{(}1-x_{i,m,p}\big{)}\;i=1,2,\ldots,N-1;m=1,2,3;p=1,2,3,4, \tag{11}\]1  
\[\theta_{i}+r_{i,p}^{\rm S,L}+w_{i,m,p}^{\rm S}+t_{i,m,p}z\leq\theta_{i+1}+k_{i +1,m,p}+r_{i+1,p}^{\rm E,L}+w_{i+1,m,p}^{\rm E}+M\big{(}1-x_{i,m,p}\big{)}\;i=1,2,\ldots,N-1;m=1,2,3;p=1,2,3,4, \tag{11}\]2  
where \(r\), \(s\), and \(t\) are the phase numbers at intersection \(k\). Equation (20) indicates that there is no sequence relationship between the same phase. Equation (21) shows that if phase \(r\) is before phase \(s\), then phase \(s\) is after phase \(r\); conversely, if phase \(r\) is after phase \(s\), then phase \(s\) is before phase \(r\). Equation (22) is used to eliminate unreasonable combinations of \(0/1\) variables to ensure the optimized phase sequence is correct. As shown in Table 1, the three \(0/1\) variables \(y_{k,3,2}\), \(y_{k,2,4}\), and \(y_{k,3,4}\) can form eight different \(0/1\) combinations. However, as can be seen from Figure 3, there are only six types of phase sequences. At this point, among the eight different \(0/1\) combinations, there are two invalid combinations: 1 and 2. When \(y_{k,3,2}=0\), \(y_{k,2,4}=0\), and \(y_{k,3,4}=1\), it is the invalid combination 1.