# Considering the Optimization Design of Urban Bus Network Scheduling
## 2 Synchronized Optimization Model for Network and Scheduling
### Model Construction
#### 2.3.1 The Objective Function  
The optimization objective of the bus network should comprehensively consider the interests of both passengers and bus operators. Passengers seek short travel times, while operators aim for efficient resource utilization and minimized investment costs. Therefore, in the optimization design of the bus network and scheduling model in this paper, the interests of both passengers and bus operators are simultaneously considered. The goal is to reduce passenger travel time and lower bus operation costs, aiming to minimize costs for both passengers and bus operators in the bus network. Passenger travel cost mainly refers to the total travel time for passengers within the network, which includes waiting time and in-vehicle time. (1) Waiting time  
In the model assumptions mentioned earlier, it is assumed that passengers, upon arriving at the station, can board the first bus of the target route for travel, meaning passengers can all catch the next scheduled bus of the target route. Therefore, the waiting time for passengers mainly depends on the headway of the route they are waiting for, with passenger waiting time being half of the headway of the selected route. The passenger waiting time is expressed as:  
\[T_{wij}=\delta_{w}\sum\limits_{r=1}^{R}x_{ij}^{r}\frac{\frac{1}{f_{r}}}{\sum \limits_{r=1}^{R}x_{ij}^{r}\frac{1}{f_{r}}}f_{r}=\delta_{w}\sum\limits_{r=1}^{R }\frac{x_{ij}^{r}}{\sum\limits_{r=1}^{R}x_{ij}^{r}\frac{1}{f_{r}}} \tag{1}\]  
(2) In-vehicle time  
In-vehicle time mainly consists of two parts: inter-stop travel time and dwell time at stops. Ignoring the influence of road and other external factors, travel time is related to distance and travel speed, while dwell time is related to the number of intermediate stops. Therefore, travel time and dwell time are represented as follows:  
\[T_{dij}=\frac{1}{V}\sum\limits_{r=1}^{R}x_{ij}^{r}\frac{\frac{1}{f_{r}}}{\sum \limits_{r=1}^{R}x_{ij}^{r}\frac{1}{f_{r}}}L_{ij}^{r} \tag{2}\]  
\[T_{Sij}=T_{s}\sum\limits_{r=1}^{R}x_{ij}^{r}\frac{\frac{1}{f_{r}}}{\sum \limits_{r=1}^{R}x_{ij}^{r}\frac{1}{f_{r}}}s_{ij}^{r} \tag{3}\]  
Based on the above, the total time cost for passengers can be expressed as follows:  
\[T=\sum\limits_{i=1}^{S}\sum\limits_{j=1}^{S}Q_{ij}\lambda(\delta_{w}\sum \limits_{r=1}^{R}\frac{x_{ij}^{r}}{\sum\limits_{r=1}^{R}x_{ij}^{r}\frac{1}{f_{ r}}}+\frac{1}{V}\sum\limits_{r=1}^{R}x_{ij}^{r}\frac{\frac{1}{f_{r}}}{\sum \limits_{r=1}^{R}x_{ij}^{r}\frac{1}{f_{r}}}L_{ij}^{r}+T_{s}\sum\limits_{r=1}^{ R}x_{ij}^{r}\frac{\frac{1}{f_{r}}}{\sum\limits_{r=1}^{R}x_{ij}^{r}\frac{1}{f_{r}}}s_{ij}^{ r}) \tag{4}\]  
Bus Operation Cost  
The expenses of bus operators mainly include the procurement of buses, construction of stations, wages and salaries of staff, management fees, vehicle insurance, fuel costs, vehicle depreciation expenses, and taxes, among others. In this paper, when considering costs, only the operational costs of buses are taken into account, which are mainly reflected in economic costs. This specifically refers to the vehicle procurement costs and operational costs of each route within the bus network. (1) Vehicle procurement cost  
To minimize vehicle procurement costs and maximize vehicle utilization, it is necessary to reduce the number of vehicles as much as possible. However, it is also important to ensure that there is a bus available at the departure station to meet the next scheduled departure for each route. This requires ensuring that the cumulative headway is greater than the duration of each route, meaning that the minimum number of vehicles required for each route should satisfy the following condition:  
\[n_{min}^{r}\geq 2\frac{L_{r}}{V}.\frac{1}{f_{r}} \tag{5}\]  
Therefore, the minimum vehicle procurement cost is:  
\[M=\sum\limits_{r=1}^{R}n_{min}^{r}\overline{M} \tag{6}\]  
(2) Route operation cost The operational cost of a bus route mainly includes the vehicle's operating cost, which is primarily related to the frequency of trips and the cost per trip.