# Exclusive Bus Lane Allocation Considering Multimodal Traffic Equity Based on Bi-Level Programming
## 3 Methodology
### Bi-Level Programming Model  
#### 3.3.1 The Upper-Level Model  
With the minimum total system cost considering the Gini coefficient as the objective function and the construction investment of EBLs as the main constraint condition, the  
Figure 3: Generalized travel cost diagram of bus.  
optimal EBL setting scheme is determined. The mathematical model is as shown in Equations (10)-(12)  
\[\min Z_{1}(Y)=\min\Biggl{\{}G\times\sum\limits_{a}\sum\limits_{i}V_{i}\Big{[}T_{a, c}(y_{a})X_{a}^{i,c}+\Big{(}t_{o}^{w}+T_{a,b}^{l}(y_{a})+t_{d}^{w}\Big{)}X_{a}^{i,b }\Big{]}+\sum\limits_{a}\Big{(}F_{a}^{c}X_{a}^{c}+P_{b}X_{a}^{b}\Big{)}\Biggr{\}} \tag{10}\]  
\[\sum\limits_{a}y_{a}L_{a}B_{a}\leq B,\forall a\in A \tag{11}\]  
\[0\leq G\leq 1 \tag{12}\]  
where \(Y=(y_{1},\cdots,y_{a},\cdots y_{N})\) is a set of decision variables for whether links need to set up EBLs and is defined as \(y_{a}=1\) if a bus lane is set up on link \(a\); otherwise, \(y_{a}=0\).  
The objective function (10) is to minimize the system cost considering the Gini coefficient. The first term is the Gini coefficient, and the second term is the total system cost. Constraint (11) is the budget constraint of bus lane construction, and constraint (12) is the value constraint of Gini coefficient.  
The Gini coefficient is calculated by the lower-level model as follows:  
* The lower-level model obtains the traffic volume \(X_{a}^{i,m}\) of travelers with different time values in each link when the traffic network is balanced;
* Calculating the total generalized travel cost value as shown in Equation (13): \[C=\sum\limits_{a}\sum\limits_{i}V_{i}\Big{[}T_{a,c}X_{a}^{i,c}+\Big{(}t_{o}^{w }+T_{a,b}^{l}+t_{d}^{w}\Big{)}X_{a}^{i,b}\Big{]}+\sum\limits_{a}\Big{(}F_{a}^ {c}X_{a}^{c}+P_{b}X_{a}^{b}\Big{)}\] (13)
* Calculating the generalized travel costs of travelers with different time values as shown in Equation (14): \[C_{i}=\sum\limits_{a}V_{i}\Big{[}T_{a,c}X_{a}^{i,c}+\Big{(}t_{o}^{w}+T_{a,b}^{ l}+t_{d}^{w}\Big{)}X_{a}^{i,b}\Big{]}+\sum\limits_{a}\Big{(}F_{a}^{c}X_{a}^{c}+P_{b}X _{a}^{b}\Big{)}\] (14)
* Calculating the cumulative percentage of the generalized travel cost of travelers with different time values to the total generalized travel cost \(\sum\limits_{i}\frac{C_{i}}{C}\); calculating the population percentage of travelers with different time values \(\frac{Q_{i}}{Q}\), and calculating the cumulative percentage of population \(\sum\limits_{i}\frac{Q_{i}}{Q}\);
* According to the cumulative percentage of generalized travel costs and the cumulative percentage of the population, the Lorentz curve is drawn, as shown in Figure 4.
* Assuming the area between the actual Lorenz curve and the absolutely equitable line is \(S_{A}\) and the area between the actual Lorenz curve and the absolutely inequitable line is \(S_{B}\)[34]. The Gini coefficient is shown in Equation (15). The range of values is shown in Table 2. \[G=\frac{S_{A}}{S_{A}+S_{B}}\] (15)  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Gini Coefficient Range** & **Level of Equity** \\ \hline \(G<0.2\) & Absolute average \\ \(0.2\leq G<0.3\) & Comparative average \\ \(0.3\leq G<0.4\) & Relatively reasonable \\ \(0.4\leq G<0.6\) & Gap \\ \(G\geq 0.6\) & Large gap \\ \hline \hline \end{tabular}
\end{table}
Table 2: Gini coefficient range and equity.