# Layout Model of Bike-Sharing Facilities in the Transfer-Influenced Area of a Subway Station
## 2 Methodology
### Alternative Facility Location Prediction
#### 2.4.1 Upper Model  
Combined with the prediction results of alternative locations, managers set up different bike-sharing facility planning schemes to adjust the transfer cost, which can play a guiding role in the traveler's choice of transport modes. The upper model is an overall optimal problem of bike-sharing facilities. The purpose of the planning is to attract more people to choose the transfer mode in bike-sharing with limited resources, and to ensure the service level of the bike-sharing system. The objective function of the upper-levelmodel of the bi-level programming model established in this paper is composed of three main parts:  
1. Bike-sharing ride volume \(F_{1}\)(Equation (9)), which refers to the number of people who use shared bikes to transfer to rail transit in each traffic zone in the transfer-influenced area. 2. Facility-operation cost \(F_{2}\) (Equation (10)), which mainly includes the operation cost of bike-sharing facilities and overload penalty. 3. Overall transfer cost \(F_{3}\) (Equation (11)), which mainly includes the cost of walking, bike-sharing and bus transfer to subway station. \[F_{1}=\sum_{i=1}^{I}\sum_{j=1}^{J}N_{ij}^{bike} \tag{9}\]  
\[F_{2}=\sum_{j=1}^{I}\bigl{(}pX_{j}+qW_{j}\bigr{)} \tag{10}\]  
\[F_{3}=\sum C_{walk-R}+\sum C_{bike-R}+\sum C_{bus-R} \tag{11}\]  
where \(i\) is the demand point number of the traffic zone in the influenced area, \(i\in\{1,2,3,\cdots,I\}\); \(j\) is the alternative location point number of the shared bike in the area, \(j\in\{1,2,3,\cdots,I\}\); \(N_{ij}^{bike}\) is the number of people from the demand point \(i\) to the facility \(j\) and riding to the subway station; \(X_{j}\) represents the manager choosing whether to establish a parking spot at the alternative point, (yes, \(X_{j}=1\); no, \(X_{j}=0\)); \(p\) is the construction cost and daily operation and management costs of the alternative point \(j\); \(q\) is the construction and management cost of each bicycle when the number of bicycles exceeds or is less than the limit of the selected facilities; \(W_{j}\) is the number of bicycles exceeding or less than the limit. As a matter of fact, due to the constraints of investment, urban development and the intensity of planned regional development, it is difficult to construct bike-sharing systems in one step. The construction of a bike-sharing system is a gradual and continuous improvement process from sparse to dense and from small to large. In different planning periods, the planning objectives should be different. Therefore, bike-sharing should meet different coverage requirements, that is, for the same traffic zone, with the gradual improvement of bike-sharing systems, the number of facilities that can provide services for the traffic zone should gradually increase. Combined with the idea of progressive coverage, the constraints of the upper model are as follows:  
\[C_{j}=max\frac{\sum_{i=1}N_{ij}^{bike}}{\theta}\leq GX_{j}\quad\forall j\in J \tag{12}\]  
where \(C_{j}\) is the number of shared bikes that should be placed at alternative point \(j\) at the initial stage; \(\theta\) is the turnover rate of shared bike; \(G\) is a large positive number. Equation (12) indicates that when an alternative point is selected, the shared bike can be deployed at that point. \[\sum_{j=1}^{J}X_{j}\leq P \tag{13}\]  
where \(P\) is the maximum number of facilities. Equation (13) represents that the number of facilities actually established shall not exceed the planned number of facilities. \[d_{mn}\geq X_{m}X_{n}d_{min}\quad\forall m,\ n\in J;m\
eq n \tag{14}\]  
where \(d_{mn}\) is the distance between different facilities; \(d_{min}\) is the specified minimum distance. Equation (14) shows that the distance between the facilities shall not be less than the specified minimum distance. \[W_{j}= \begin{cases}C_{j}-U_{max}&C_{j}>U_{max}\\ 0&U_{min}<C_{j}<U_{max}\\ U_{min}-C_{j}&C_{j}<U_{min}\end{cases}