## 3 Methodology
### Objectives for Optimizing the High-Speed Railway Network
#### 3.1.1 Micro-Objective for Traveler's Demand  
The proposed micro-objective is to minimize the generalized travel cost of passengers. Based on the interests of travelers, the optimization function for the high-speed railway line was constructed from two aspects: travel time and travel cost. To maximize the benefit for passengers, this section mainly builds on the optimization function at the micro-level. Based on the basic concepts and properties of the uncertainty theory, the proposed model is described as a quantitative linear model.  
1. Passenger travel time function of travelers: The travel time for high-speed railway passengers is defined as the total time between the starting point (origin, O) and the endpoint (destination, D). It includes the time from the travel point to the corresponding train station, the transfer times at different stations, the travel time of the train, and the time from the train station to the travel destination. Figure 1 shows a high-speed railway line. When a passenger travels between \(OD\) (origin and destination are \(O\) and \(D\)), they need to traverse a set of railway stations. The set of stations is denoted as \(S~{}=~{}\{S_{1},S_{2},\cdots,S_{n}\}\). For an \(OD\) pair, their travel time is defined as: \[T^{OD}=t_{o_{1}}+\sum_{j=1}^{|n-1|}t_{i,j+1}+t_{nd}\] (1) where \(t_{O_{1}}\) represents the travel time between the origin and the first railway station, \(t_{i,j+1}\) represents the travel time between stations; \(S_{i}\), \(S_{i+1}\), and \(t_{nd}\) represent the travel time between the last railway station and the destination.
2. Passengers' travel cost function: The travel cost, which is the total fare, includes all the expenses for high-speed railway travelling, and is the main factor for passengers in choosing a method of travel. This paper provides general fares, not the actual fare. The general fare is the comparable fare rate per kilometer per person on different routes, determined by the administrative department. The actual fare is the fare between the starting point and the ending point for each person, and it is uncertain due to seasonal fare discounts and other factors. We assume that the high-speed train journey takes place along an itinerary consisting of m sections along a route and that these are sequential. The lower the fare, the more attractive it is to passengers. Thus, the relative function of the cost per passenger can be described as: \[P=\sum_{i=1}^{m}(\alpha_{i}\cdot d_{i})\] (2) where \(P\) is the fare for a passenger traveling by high-speed railway; \(d_{i}\) is the distance traveled by passengers on the \(i\)-th section of the high-speed railway; \(\alpha_{i}\) is the fare rate per kilometer per person of the \(i\)-th section of the high-speed railway line, and it is determined by the operating speed \(v_{i}\); \(m\) is the number of high-speed railway lines.  
3. Micro-objective model for optimizing the high-speed railway network: The micro-objective is to minimize the generalized travel cost for passengers on high-speed railway travel: their travel expenses and travel time. It is the constraint condition that a high-speed train can go from 0 to 350 km/h in about 10 min, while it also takes about 10 min to decelerate from 350 km/h to 0. A high-speed train runs for at least 20 min, and 20 min is also the constraint condition (\(t_{\text{min}}\)). Therefore, the micro-objective function is constructed based on the two optimization objectives in Equations (1) and (2). \[F_{1}=\text{min}F^{\text{micro}}=\gamma_{1}T^{OD}+P\] (3) where \(F_{1}\) is the micro-objective model for optimizing the high-speed railway network and \(\gamma_{1}\) is the value of time in monetary terms used to solve the dimension problem of the different objective functions.