## 2 Materials and Methods
### Model Formulation  
#### 2.2.1 Objective Function  
Taking the sum of the weighted bandwidths on each road segment as the optimization goal, the objective function of the green wave control model for passenger cars and buses in closed road networks can be formulated as  
\[\max\!\sum_{i=1}^{A}\sum_{j=1}^{N_{i}-1}\left(a_{(i,j),(i,j+1)}^{\text{car}}b_{( i,j),(i,j+1)}^{\text{car}}+\overline{a}_{(i,j),(i,j+1)}^{\text{car}}\overline{b }_{(i,j),(i,j+1)}^{\text{bus}}+a_{(i,j),(i,j+1)}^{\text{bus}}b_{(i,j),(i,j+1)}^ {\text{bus}}+\overline{a}_{(i,j),(i,j+1)}^{\text{bus}}\overline{b}_{(i,j),(i, j+1)}^{\text{bus}}\right), \tag{1}\]  
where \(A\) is the number of arterials in the road network, \(N_{i}\) is the number of intersections on the \(i\)th arterial, \(b_{(i,j),(i+1)}^{\text{car}}\) [\(\overline{b}_{(i,j),(i,j+1)}^{\text{car}}\)] is the outbound [inbound] car bandwidth between \((i,j)\) and \((i,j+1)\), \(b_{(i,j),(i,j+1)}^{\text{bus}}\) [\(\overline{b}_{(i,j),(i,j+1)}^{\text{bus}}\)] is the outbound [inbound] bus bandwidth between \((i,j)\) and \((i,j+1)\), \(a_{(i,j),(i,j+1)}^{\text{car}}\) [\(\overline{a}_{(i,j),(i,j+1)}^{\text{car}}\)] is the weighting factor for \(b_{(i,j),(i,j+1)}^{\text{car}}\) [\(\overline{b}_{(i,j),(i,j+1)}^{\text{car}}\)], and \(a_{(i,j),(i,j+1)}^{\text{bus}}\) [\(\overline{a}_{(i,j),(i,j+1)}^{\text{bus}}\)] is the weighting factor for \(b_{(i,j),(i,j+1)}^{\text{bus}}\) [\(\overline{b}_{(i,j),(i,j+1)}^{\text{bus}}\)]. The number of passengers per hour in the outbound [inbound] through cars at \((i,j+1)\) [\((i,j)\)] can be used as the weighting factor \(\alpha_{(i,j),(i,j+1)}^{\text{car}}\) [\(\overline{a}_{(i,j),(i,j+1)}^{\text{car}}\)]. The number of passengers per hour in the outbound [inbound] through buses at \((i,j+1)\) [\((i,j)\)] can be used as the weighting factor \(\alpha_{(i,j),(i,j+1)}^{\text{bus}}\). In addition, the weighting factors can also be specified by traffic engineers according to the control needs. Figure 1: Topology of the traffic network studied in this paper. \(x^{\text{bus}}_{(i,j),(i,j+1)}\) and a sufficiently large positive number \(M\) are introduced to relax some of the constraints in the presented model. With the introduction of \(x^{\text{car}}_{(i,j),(i,j+1)}\) and \(M\), Equation (2) can be updated as  
\[\left\{ \begin{array}{l}w^{\text{car}}_{(i,j)}+b^{\text{car}}_{(i,j),(i,j+1)} \leq g^{D}_{(i,j)}+x^{\text{car}}_{(i,j),(i,j+1)}M\\ w^{\text{car}}_{(i,j),(i,j+1)}+b^{\text{car}}_{(i,j),(i,j+1)}\leq g^{D}_{(i,j+1 )}+x^{\text{car}}_{(i,j),(i,j+1)}M\\ \overline{w}^{\text{car}}_{(i,j),(i,j+1)}+\overline{b}^{\text{car}}_{(i,j),(i, j+1)}\leq g^{D}_{(i,j)}-\overline{\varepsilon}^{\text{car}}_{(i,j)}+x^{\text{ car}}_{(i,j),(i,j+1)}M\\ \overline{w}^{\text{car}}_{(i,j+1)}+\overline{b}^{\text{car}}_{(i,j),(i,j+1)} \leq g^{D}_{(i,j+1)}+x^{\text{car}}_{(i,j),(i,j+1)}M\\ \end{array} \right.. \tag{3}\]  
When \(x^{\text{car}}_{(i,j),(i,j+1)}\) equals to 0, it indicates that there are car green wave bands between \((i,j)\) and \((i,j+1)\), and Equation (3) will transform into Equation (2). When \(x^{\text{car}}_{(i,j),(i,j+1)}\) equals to 1, it indicates that there are no car green wave bands between \((i,j)\) and \((i,j+1)\), and Equation (2) is relaxed. Similarly, the constraints on the location of the bus green wave bands are formulated as  
\[\left\{ \begin{array}{l}w^{\text{bus}}_{(i,j)}+b^{\text{bus}}_{(i,j),(i,j+1)} \leq g^{D}_{(i,j)}+x^{\text{bus}}_{(i,j),(i,j+1)}M\\ w^{\text{bus}}_{(i,j),(i,j+1)}+b^{\text{bus}}_{(i,j),(i,j+1)}\leq g^{D}_{(i,j +1)}+x^{\text{bus}}_{(i,j),(i,j+1)}M\\ \overline{w}^{\text{bus}}_{(i,j),(i,j+1)}+\overline{b}^{\text{bus}}_{(i,j),(i, j+1)}\leq g^{D}_{(i,j)}-\overline{\varepsilon}^{\text{bus}}_{(i,j)}+x^{ \text{bus}}_{(i,j),(i,j+1)}M\\ \overline{w}^{\text{bus}}_{(i,j+1)}+\overline{b}^{\text{bus}}_{(i,j),(i,j+1)} \leq g^{D}_{(i,j+1)}+x^{\text{bus}}_{(i,j),(i,j+1)}M\\ \end{array}