## 2 Dynamic Optimization Strategies for Public Transportation Operations
### Optimization Model Formulation
#### 2.4.3 Objective Function  
The objective function of the model is to minimize the bus operating costs and passenger travel costs, both of which are used as optimization functions for the bi-objective optimization modeling and solving. The decision variables of the model include the bus departure interval, the location of the bus skipping stop, and the distribution of the optimization interval of the traveling path. 1. Cost of bus operating  
The operating costs of public transportation operators can be roughly divided into two parts: fixed operating costs and variable operating costs. Fixed operating costs refer to the costs of bus station construction, equipment purchases, facility maintenance, and personnel management, which have no obvious correlation with the bus interval and mileage, etc. These costs are not related to the decision-making variables in the process of modeling, and can be treated as a constant not involved in the calculation. Variable operating costs refer to the costs of fuel consumption, salary expenditure, asset depreciation, etc., which are closely related to the departure interval and operating mileage in the process of bus operations, and can be used as intuitive feedback in evaluating the implementation effect of the scheduling program. Therefore, they can be used as the objective function. The calculation of the operating costs of a bus operator can be expressed as follows:  
\[W_{1}=\mu_{1}L\sum_{k=1}^{K}\frac{T_{k}}{x_{k}} \tag{8}\]  
2. Cost of passenger travel  
Among the entire process of a passenger's travel, aspects directly affected by bus scheduling include the waiting process after arriving at the station and the bus operation process after getting on the bus. Therefore, this study focuses on the costs incurred in the above two processes. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Description** \\ \hline \(L\) & Total length of the route (km) \\ \(n\) & Total number of buses on the route \\ \(m\) & Total number of bus stops on the route \\ \(K\) & Total number of time periods \\ \(T_{k}\) & Length of the time period \(k\) (s) \\ \(\mu_{1}\) & Variable operating cost control factor for buses \\ \(\mu_{2}\) & Cost control factor for passenger waiting time \\ \(\mu_{3}\) & Cost control factor for passengers waiting time with satisfaction penalties \\ \(\mu_{4}\) & Cost control factor for passenger in-transit time \\ \(N_{u}^{i,j}\) & Number of passengers embarking when bus \(i\) arrives at stop \(j\) \\ \(N_{d}^{i,j}\) & Number of passengers disembarking when bus \(i\) arrives at stop \(j\) \\ \(N_{i,j}\) & Number of passengers on bus \(i\) from stop \(j\) \\ \(N_{u}^{i,j}\) & Number of passengers waiting for bus \(i\) at stop \(j\) \\ \(N_{max}\) & Authorized number of passengers \\ \(l_{ij}\) & Traveling distance of the bus \(i\) traveling from the stop \(j\) to \(j+1\) (km) \\ \(\overline{v}_{i,j}\) & Traveling speed of the bus \(i\) traveling from the stop \(j\) to \(j+1\) (km) \\ \(v_{min}\) & Minimum speed limit of the buses (m/s) \\ \(v_{max}\) & Maximum speed limit of the buses (m/s) \\ \(x_{min}\) & Minimum departure time interval (s) \\ \(x_{max}\) & Maximum departure time interval (s) \\ \(x_{k}\) & Decision variable, the departure interval during the time period \(k\) \\ \(y_{i,j}\) & 0–1 Decision variable, to control the bus \(i\) whether skip stop \(j\) or not \\ \(z_{i,j}\) & 0–1 Decision variable, to control the bus \(i\) whether adjust the path or not \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._There is a linear correlation between the costs of passenger waiting time and the time interval between the moment of passenger arrival at the bus stop and the moment of the arrival of the latest bus. The larger the interval between departures, the longer the average passenger waiting time at the station and the higher the waiting cost.