## 2 Materials and Methods
### Constraints
#### 2.3.1 Location of Green Wave Band  
According to the definition of the green wave band, it can be seen that the green wave band of each traffic mode must be limited within the green time. Based on Figure 4, the constraint for the location of the green wave band can be expressed as:  
\[\left\{\begin{array}{l}w_{i,m,p}^{\text{S}}+b_{i,m,p}\leq S_{i,p}^{\text{S}}\\ w_{i+1,m,p}^{\text{E}}+b_{i,m,p}\leq S_{i+1,p}^{\text{E}}\end{array}\right.i=1,2,\cdots,N-1;m=1,2,3;p=1,2,3,4, \tag{2}\]  
\[\left\{\begin{array}{l}\overline{w}_{i,m,\overline{p}}^{\text{E}}+\overline{ b}_{i,m,\overline{p}}\leq\overline{S}_{i,\overline{p}}^{\text{E}}\\ \overline{w}_{i+1,m,\overline{p}}^{\text{S}}+\overline{b}_{i,m,\overline{p}} \leq\overline{S}_{i+1,\overline{p}}^{\text{S}}\end{array}\right.i=1,2,\cdots, N-1;m=1,2,3;\overline{p}=1,2,3,4, \tag{3}\]  
where \(w_{i,m,p}^{\text{S}}\) (\(w_{i+1,m,p}^{\text{E}}\)) represents the time from the beginning of the outbound green time at the starting intersection \(i\) (ending intersection \(i+1\)) to the left side of the green wave bandwidth \(b_{i,m,p}\), \(\overline{w}_{i,m,\overline{p}}^{\text{E}}\) (\(\overline{w}_{i+1,m,\overline{p}}^{\text{S}}\)) represents the time from the end of the inbound green time at the ending intersection \(i\) (starting intersection \(i+1\)) to the right side of the green wave bandwidth \(\overline{b}_{i,m,\overline{p}}\), \(S_{i,p}^{\text{S}}\) (\(S_{i+1,p}^{\text{E}}\)) represents the outbound green time that the coordinate path \(p\) has at the starting intersection \(i\) (ending intersection \(i+1\)), and \(\overline{S}_{i,\overline{p}}^{\text{E}}\) (\(\overline{S}_{i+1,\overline{p}}^{\text{S}}\)) represents the inbound green time that the coordinated path \(\overline{p}\) has at the ending intersection \(i\) (starting intersection \(i+1\)).  
Because conflicts exist among different paths, it is highly challenging to ensure that every path for each traffic mode can receive a green wave band. In other words, when the path \(p\) does not obtain a green wave band, constraint (2) can be violated; when the path \(\overline{p}\) does not obtain a green wave band, constraint (3) can be violated. To solve this problem, \(0/1\) variables \(x_{i,m,p}\) and \(\overline{x}_{i,m,\overline{p}}\), along with a sufficiently large positive number \(M\), are introduced into constraints (2) and (3). As a result, constraints (2) and (3) are transformed into the following constraints (4) and (5).  
\[\left\{\begin{array}{l}w_{i,m,p}^{\text{S}}+b_{i,m,p}\leq S_{i,p}^{\text{S}}+ M(1-x_{i,m,p})\\ \overline{w}_{i+1,m,p}^{\text{E}}+b_{i,m,p}\leq S_{i+1,p}^{\text{E}}+M(1-x_{i,m,p})\end{array}\right.i=1,2,\cdots,N-1;m=1,2,3;p=1,2,3,4, \tag{4}\]  
\[\left\{\begin{array}{l}\overline{w}_{i,m,\overline{p}}^{\text{E}}+\overline{ b}_{i,m,\overline{p}}\leq\overline{S}_{i,\overline{p}}^{\text{E}}+M(1-\overline{x}_{i,m,\overline{p}})\\ \overline{w}_{i+1,m,\overline{p}}^{\text{S}}+\overline{b}_{i,m,\overline{p}} \leq\overline{S}_{i+1,\overline{p}}^{\text{S}}+M(1-\overline{x}_{i,m, \overline{p}})\end{array}\right.i=1,2,\cdots,N-1;m=1,2,3;\overline{p}=1,2,3,4, \tag{5}\]  
where \(x_{i,m,p}\) and \(\overline{x}_{i,m,\overline{p}}\) are \(0/1\) variables, and \(M\) is a sufficiently large positive number.  
\(x_{i,m,p}=1\) (\(\overline{x}_{i,m,\overline{p}}=1\)) indicates that the path \(p\) (\(\overline{p}\)) of traffic mode \(m\) between intersections \(i\) and \(i+1\) is coordinated. In this case, constraints (4) and (5) are transformed into constraints (2) and (3), respectively. \(x_{i,m,p}=0\) (\(\overline{x}_{i,m,\overline{p}}=0\)) indicates that the path \(p\) (\(\overline{p}\)) of  
Figure 4: Arterial time–space diagram for multi-path.  
traffic mode \(m\) between intersections \(i\) and \(i+1\) is not coordinated. In this case, constraints (4) and (5) are relaxed.