## 2 Materials and Methods
### Model Formulation
#### 2.2.1 Objective Function
The mutual offset constraints are formulated as  
\[\left(w^{\text{car}}_{(i,j)}+t^{\text{car}}_{(i,j),(i,j+1)}z-w^{\text{car}}_{( i,j),(i,j+1)}\right)+m_{(i,j),(i,j+1)}-\left(w^{\text{bus}}_{(i,j)}+t^{\text{ bus}}_{(i,j),(i,j+1)}z-w^{\text{bus}}_{(i,j),(i,j+1)}\right)=0, \tag{28}\]  
where \(m_{(i,j),(i,j+1)}\) is an integer variable representing an integer multiple of the common cycle time. Equation (28) can be equivalently transformed into two inequality constraints by relaxing it, as shown in Equations (29) and (30). \[\left(w^{\text{car}}_{(i,j)}+t^{\text{car}}_{(i,j),(i,j+1)}z-w^{\text{car}}_{ (i,j),(i,j+1)}\right)+m_{(i,j),(i,j+1)}-\left(w^{\text{bus}}_{(i,j)}+t^{\text {bus}}_{(i,j),(i,j+1)}z-w^{\text{bus}}_{(i,j),(i,j+1)}\right)\leq\left(x^{ \text{car}}_{(i,j),(i,j+1)}+x^{\text{bus}}_{(i,j),(i,j+1)}\right)M, \tag{29}\] \[\left(w^{\text{car}}_{(i,j)}+t^{\text{car}}_{(i,j),(i,j+1)}z-w^{ \text{car}}_{(i,j),(i,j+1)}\right)+m_{(i,j),(i,j+1)}-\left(w^{\text{bus}}_{(i,j)}+t^{\text{bus}}_{(i,j),(i,j+1)}z-w^{\text{bus}}_{(i,j),(i,j+1)}\right) \geq-\left(x^{\text{car}}_{(i,j),(i,j+1)}+x^{\text{bus}}_{(i,j),(i,j+1)} \right)M. \tag{30}\]  
* Minimum bandwidth constraints If the bandwidth is too small (for example, the bandwidth is 1 s), vehicles cannot actually get the green wave experience. Therefore, a green wave bandwidth that is too small is meaningless. To avoid the above situation, it is necessary to meet the following minimum bandwidth constraints. \[\left\{ \begin{array}{l}b^{\text{car}}_{(i,j),(i,j+1)}\geq b^{\text{ car,min}}_{(i,j),(i,j+1)}z-x^{\text{car}}_{(i,j),(i,j+1)}M\\ \overline{b}^{\text{car}}_{(i,j),(i,j+1)}\geq\overline{b}^{\text{car}}_{(i,j),(i,j+1)}z-x^{\text{car}}_{(i,j),(i,j+1)}M\\ b^{\text{bus}}_{(i,j),(i,j+1)}\geq\overline{b}^{\text{bus,min}}_{(i,j),(i,j+1 )}z-x^{\text{bus}}_{(i,j),(i,j+1)}M\\ \overline{b}^{\text{bus}}_{(i,j),(i,j+1)}\geq\overline{b}^{\text{bus,min}}_{( i,j),(i,j+1)}z-x^{\text{bus}}_{(i,j),(i,j+1)}M\\ \end{array} \right.,\] (31) where \(b^{\text{car,min}}_{(i,j),(i,j+1)}\), \(\overline{b}^{\text{car,min}}_{(i,j),(i,j+1)}\), \(b^{\text{bus,min}}_{(i,j),(i+1)}\), and \(\overline{b}^{\text{bus,min}}_{(i,j),(i,j+1)}\) are the minimum for the corresponding bandwidths. * Constraints that bandwidths are forced to be 0 When there is no green wave band on a road segment, the bandwidth is 0, and the following constraints should be satisfied: \[\left\{ \begin{array}{l}b^{\text{car}}_{(i,j),(i,j+1)}+x^{\text{car}}_{(i,j),(i,j+1)}\leq 1\\ \overline{b}^{\text{car}}_{(i,j),(i,j+1)}+x^{\text{car}}_{(i,j),(i,j+1)}\leq 1 \\ b^{\text{bus}}_{(i,j),(i,j+1)}+x^{\text{bus}}_{(i,j),(i+1)}\leq 1\\ \overline{b}^{\text{bus}}_{(i,j),(i,j+1)}+x^{\text{bus}}_{(i,j),(i,j+1)}\leq 1 \\ b^{\text{car}}_{(i,j),(i,j+1)},\overline{b}^{\text{car}}_{(i,j),(i,j+1)},b^{ \text{bus}}_{(i,j),(i,j+1)}\overline{b}^{\text{bus}}_{(i,j),(i,j),(i,j+1)} \geq 0\end{array} \right..\] (32)When there is no green wave band on a road segment, Equation (32) can force the corresponding bandwidth to be 0, which ensures that the objective optimization function expression is correct. * Common cycle time constraints To optimize the common signal cycle, the following constraints should be satisfied: \[\frac{1}{C_{\max}}\leq z\leq\frac{1}{C_{\min}},\] (33) where \(C_{\min}\) and \(C_{\max}\) are the lower and upper limits, respectively, on the common cycle time. * Additional constraints on interference variables It is obvious that the interference variables \(w^{\rm car}_{(i,j)}\), \(w^{\rm car}_{(i,j),(i,j+1)}\), \(\overline{w}^{\rm car}_{(i,j),(i,j+1)}\), \(\overline{w}^{\rm car}_{(i,j+1)}\), \(w^{\rm bus}_{(i,j),(i,j+1)}\), \(\overline{w}^{\rm bus}_{(i,j),(i,j+1)}\), and \(\overline{w}^{\rm bus}_{(i,j+1)}\) also need to satisfy the following constraints: \[\left\{