## 2 Materials and Methods
### Model Formulation
#### 2.2.1 Objective Function
\right., \tag{4}\]  
where \(w^{\text{bus}}_{(i,j)}\) [\(\overline{w}^{\text{bus}}_{(i,j),(i,j+1)}\)] is the time from the beginning [end] of the green time at \((i,j)\) to the left [right] side of the bus outbound [inbound] bandwidth, \(w^{\text{bus}}_{(i,j),(i,j+1)}\) [\(\overline{w}^{\text{bus}}_{(i,j+1)}\)] is the time from the beginning [end] of the green time at \((i,j+1)\) to the left [right] side of the bus outbound [inbound] bandwidth, and \(\overline{\varepsilon}^{\text{bus}}_{(i,j)}\) denotes the initial queue clearance time corresponding to the bus inbound bandwidth at \((i,j)\). When \(x^{\text{bus}}_{(i,j),(i,j+1)}\) equals to 0, it indicates that there are bus green wave bands between \((i,j)\) and \((i,j+1)\). When \(x^{\text{bus}}_{(i,j),(i,j+1)}\) equals to 1, it indicates that there are no bus green wave bands between \((i,j)\) and \((i,j+1)\). * Arterial-level loop integer constraints Since all intersections are operated with the same cycle time (common cycle time), there is an arterial-level loop integer constraint for cars, according to Figure 2, and it can be expressed as \[\overline{\phi}^{\text{car}}_{(i,j),(i,j+1)}+\phi^{\text{car}}_{(i,j),(i,j+1) }=m^{\text{car}}_{(i,j),(i,j+1)},\] (5) where \(\phi^{\text{car}}_{(i,j),(i,j+1)}\) [\(\overline{\phi}^{\text{car}}_{(i,j),(i,j+1)}\)] denotes the time from the center of an outbound [inbound] red time at \((i,j)\) to the center of the nearest outbound [inbound] red time at \((i,j+1)\), the centers of an outbound [inbound] red time at \((i,j)\) and \((i,j+1)\) are situated closest to the left [right] side of an outbound [inbound] green wave band for cars, and \(m^{\text{car}}_{(i,j),(i,j+1)}\) is an integer variable, representing an integer multiple of the common cycle time. As shown in Figure 2, \(\phi^{\text{car}}_{(i,j),(i,j+1)}\) and \(\overline{\phi}^{\text{car}}_{(i,j),(i,j+1)}\) can be further formulated by  
\[\phi^{\text{car}}_{(i,j),(i,j+1)}=0.5r^{D}_{(i,j)}+w^{\text{car}}_{(i,j)}+t^{ \text{car}}_{(i,j),(i,j+1)}z-w^{\text{car}}_{(i,j),(i,j+1)}-0.5r^{D}_{(i,j+1)}, \tag{6}\]  
\[\overline{\phi}^{\text{car}}_{(i,j),(i,j+1)}=0.5r^{D}_{(i,j)}+\overline{w}^{ \text{car}}_{(i,j),(i,j+1)}+t^{\text{car}}_{(i,j),(i,j+1)}z-\overline{w}^{ \text{car}}_{(i,j+1)}-0.5r^{D}_{(i,j+1)}, \tag{7}\]  
where \(r^{D}_{(i,j)}\) [\(r^{D}_{(i,j+1)}\)] is the red time for the coordinated movement \(D\) at \((i,j)\) [\((i,j+1)\)], \(t^{\text{car}}_{(i,j),(i,j+1)}\) [\(\overline{t}^{\text{car}}_{(i,j),(i,j+1)}\)] is the car travel time between \((i,j)\) and \((i,j+1)\) in the outbound [inbound] direction, and \(z\) is the reciprocal of the common cycle time. Combining Equations (5)-(7), Equation (5) can be updated as  
\[\left(r^{D}_{(i,j)}-r^{D}_{(i,j+1)}\right)+\left(w^{\text{car}}_{(i,j)}+\overline{ w}^{\text{car}}_{(i,j),(i,j+1)}\right)+\left(t^{\text{car}}_{(i,j),(i,j+1)}+ \overline{t}^{\text{car}}_{(i,j),(i+1)}\right)\!z-\left(w^{\text{car}}_{(i,j),( i,j+1)}+\overline{w}^{\text{car}}_{(i,j+1)}\right)=m^{\text{car}}_{(i,j),(i,j+1)}. \tag{8}\]  
By relaxing Equation (8), the equality constraint shown in Equation (8) can be equivalently transformed into two inequality constraints, which are Equations (9) and (10). \[\left(r^{D}_{(i,j)}-r^{D}_{(i,j+1)}\right)+\left(w^{\text{car}}_{(i,j)}+ \overline{w}^{\text{car}}_{(i,j),(i+1)}\right)+\left(t^{\text{car}}_{(i,j),(i +1)}+\overline{t}^{\text{car}}_{(i,j),(i+1)}\right)\!z-\left(w^{\text{car}}_{( i,j),(i+1)}+\overline{w}^{\text{car}}_{(i,j+1)}\right)\!z\!-\left(w^{\text{car}}_{(i,j),(i +1)}+\overline{w}^{\text{car}}_{(i,j+1)}\right)\!z\!-\!\left(w^{\text{car}}_{ (i,j),(i,j+1)}+\overline{w}^{\text{car}}_{(i,j+1)}\right)\geq m^{\text{car}}_{( i,j),(i+1)}-\chi^{\text{car}}_{(i,j),(i+1)}M. \tag{9}\]  
When \(x^{\text{car}}_{(i,j),(i,j+1)}\) equals to 0, Equations (9) and (10) will transform into Equation (8). When \(x^{\text{car}}_{(i,j),(i,j+1)}\) equals to 1, Equation (8) is relaxed.