# Research on Optimization of Multi-Objective Regional Public Transportation Scheduling
## 3 The Regional Public Transportation Scheduling Model
### Operation Analysis  
1. Analysis of bus operation process  
The dwelling time of each bus can be expressed as \(U_{x_{i}s_{i,q}}=a_{0}+\max(a_{1}Q_{x_{i}s_{i,q}}^{s},a_{2}Q_{x_{i}s_{i,q}}^{a})\), where \(a_{0}\) is the time of buses to enter and leave the station, \(a_{1}\) and \(a_{2}\) indicate the average boarding and alighting time for each passenger, respectively. The bus arrival time at the departure station can written as \(t_{x_{i}s_{i,1}}=A_{x_{i}}+t_{s_{i,0}s_{i,1}}\), where \(t_{s_{i,0}s_{i,1}}\) indicates the running time of line \(i\) from the parking lot to the departure station, and the bus departure time can be computed as \(l_{x_{i}s_{i,q}}=t_{x_{i}s_{i,q}}+U_{x_{i}s_{i,q}}\). The total running time of a bus can be calculated by \(l_{x_{i}s_{i_{q}}}-A_{x_{i}}\). (2) Passenger number analysis  
The number of passengers getting off the bus \(Q^{g}_{x_{i}s_{i_{q}}}\) is determined by the number of passengers on the bus, \(C_{x_{i}s_{i_{q}},q}\), and the alighting rate of passengers, \(p^{g}_{x_{i}s_{i_{q}}}\), which can be calculated using  
\[Q^{g}_{x_{i}s_{i_{q}}}=C_{x_{i}s_{i_{q}}}p^{g}_{x_{i}s_{i_{q}}} \tag{1}\]  
There are two aspects to consider in terms of the number of passengers getting on the bus: the remaining capacity of the bus and the number of passengers the who expect to board the bus which includes the number of people arriving randomly, the number of people who need to transfer, and the number of people who fail to get on the previous bus (do not include those who choose other modes of transportation). So, the number of passengers getting on the bus can be formulated as follows:  
\[ \begin{array}{l}Q^{a}_{x_{i}s_{i_{q}}}=\min(V-(C_{x_{i}s_{i_{q}}}-Q^{g}_{x_{i }s_{i_{q}}}),(t_{x_{i}s_{i_{q}}}-t_{x_{i}-1s_{i_{q}}})p^{a}_{x_{i}s_{i_{q}}}\\ +R^{s_{i}s_{i_{q}}}_{x_{i}s_{i_{q}}}Q^{g}_{x_{i}s_{i_{p}}}f_{s_{i_{q}}s_{i_{q} }}+F_{x_{i}-1s_{i_{q}}}-Y_{x_{i}-1s_{i_{q}}})\end{array}  \tag{2}\]  
Then, the number of passengers on the bus at station, \(C_{x_{i}s_{i_{q}}}\), is the number of passengers on the vehicle, \(C_{x_{i}-1s_{i_{q}}}\), minus the number of passengers getting off the vehicle, \(Q^{g}_{x_{i}-1s_{i_{q}}}\), plus the number of passengers getting on the vehicle, which can be obtained:  
\[C_{x_{i}s_{i_{q}}}=C_{x_{i}-1s_{i_{q}}}+Q^{a}_{x_{i}-1s_{i_{q}}}-Q^{g}_{x_{i}- 1s_{i_{q}}} \tag{3}\]  
In addition, the number of passengers who have not achieved easy transfer can be expressed as \(Q^{g}_{x_{i}s_{i_{q}}}f_{s_{i_{p}}s_{i_{q}}}(1-T^{g_{i}p_{i_{q}}s_{i_{q}}}_{x_ {i}s_{i}})\). Thus, the number of passengers who fail to get on the bus is the number of passengers who need to get on the bus in the current period minus the number of passengers who actually get on the vehicle, which can be computed as follows:  
(3) Analysis of passenger waiting time  
\[ \begin{array}{l}F_{x_{i}s_{i_{q}}}=(t_{x_{i}s_{i_{q}}}-I_{x_{i}-1s_{i_{q}}} )p^{a}_{x_{i}s_{i_{q}}}+R^{s_{i}s_{i_{p}}s_{i_{q}}}_{x_{i}s_{i}}Q^{g}_{x_{i}s_ {i_{q}}}\\ \hskip 142.26378 pt+F_{x_{i}-1s_{i_{q}}}-Y_{x_{i}-1s_{i_{q}}}-Q^{a}_{x_{i}s_{i _{q}}}.\end{array}