# Layout Model of Bike-Sharing Facilities in the Transfer-Influenced Area of a Subway Station
## 2 Methodology
### Alternative Facility Location Prediction
#### 2.4.2 Lower Model  
The decision makers of the lower model are the travelers, and the upper decision makers formulate the layout and configuration plan of bike-sharing facilities. Travelers choose their transfer modes by considering the transfer cost, thus generating the travel demand for multiple combinations of trips to transfer to rail transit. The generated volume of various combined transfer modes feed back to the layout and configuration plan of the upper model. Travelers tend to choose the transfer mode with the least transfer cost. This process can be abstracted as the problem of dividing the way of multimode combined travel [30]. Based on the above analysis, the Logit model is established based on the principle of minimizing transfer cost. Travelers choose among three modes that transfer to rail transit: walking, bike-sharing, and bus. The probability of the transfer mode \(P_{n}\) is calculated by Equation (22), and \(C_{n}\) is the unit transfer cost of this mode.  
\[P_{n}=\frac{exp(-\beta C_{n})}{\sum_{n}exp(-\beta C_{n})}\quad n=\{bus-R,bike -R,walk-R\} \tag{22}\]  
The part of each index item is divided by the average generalized transfer cost of each mode \(\overline{C}\), where \(\beta\) is the correction coefficient, which is generally taken as [3; 3.5][31]. Therefore, the improved model is shown as Equation (23).  
\[P_{n}=\frac{exp(-\beta C_{n}/\overline{C})}{\sum_{n}exp(-\beta C_{n}/\overline {C})}\quad n=\{bus-R,bike-R,walk-R\} \tag{23}\]  
Combined with the analysis of transfer cost in this section, the total transfer costs of walking, bike-sharing and bus are calculated as the following Equations (24)-(26).  
\[\sum C_{walk-R}=\sum_{i\in I}\sum_{j\in I}C_{walk-R}N_{i}^{walk}=\sum_{i\in I }\sum_{j\in I}\biggl{(}\lambda\times\frac{D_{iR}}{V_{walk}}\biggr{)}\times N_ {i}^{walk} \tag{24}\]  
\[\sum C_{bike-R}=\sum_{i\in I}\sum_{j\in I}C_{bike-R}N_{ij}^{bike}=\sum_{i\in I }\sum_{j\in I}\biggl{[}\lambda\times(\frac{D_{iR}}{V_{bike}}+T_{lock}+\frac{D_{ ij}}{V_{walk}})+\delta_{bike}\biggr{]}\times N_{ij}^{bike} \tag{25}\]  
\[\sum C_{bus-R}=\sum_{i\in I}\sum_{j\in I}C_{bus-R}N_{i}^{bus}=\sum_{i\in I} \sum_{j\in I}\biggl{[}\lambda\times(\frac{D_{iR}}{V_{walk}}+T_{wait}+\frac{D_{ kR}}{V_{bus}})+\delta_{bus}\biggr{]}\times N_{i}^{bus} \tag{26}\]  
where \(D_{iR}\) is the walking distance from the demand point \(i\) to the subway station; \(D_{ij}\) is the walking distance from the demand point \(i\) to the bike-sharing facility \(j\); \(D_{jR}\) is the riding distance from the bike-sharing facility \(j\) to the subway station; \(D_{kR}\) is the distance between the bus station and the subway station; \(N\) represents the number of people traveling from the demand point to the subway station by the three modes; \(V_{walk}\), \(V_{bike}\), \(V_{bus}\) are the travel speed of walking, bike-sharing and bus.  
The constraints of the lower-level planning model are shown as Equations (27) and (28). Equation (27) indicates that the travel demands of each traffic zone to transfer to rail transit were able to be met; Equation (28) shows that bike-sharing travelers only chose shared bike facilities within the coverage area.  
\[\sum\limits_{j=1}^{J}N_{ij}^{n}=O_{i}\times P_{n}=O_{i}\times P_{n}\quad n=\{ walk,bike,bus\} \tag{27}\]  
\[N_{ij}^{bike}\leq Gf\bigl{(}D_{ij}\bigr{)}X_{j} \tag{28}\]  
According to the analysis of objective function and constraint conditions, the lower-level model aims at minimizing the transfer cost of travelers, and the objective function is shown as Equation (29).  
\[minF=\sum C_{walk-R}+\sum C_{bike-R}+\sum C_{bus-R} \tag{29}\]