# Optimal Model for Carsharing Station Location Based on Multi-Factor Constraints
## 3 Electric Carsharing Location Optimization Model
### Mathematical Model
#### 3.3.2 Constraint Condition  
(1) Car purchase cost  
We assume that the purchase cost of cars is linearly related to their number. Then, the car purchase cost can be calculated as follows:  
\[F_{car}=\sum_{u=1}^{m}\sum_{v=1}^{n}b*B_{uv}, \tag{8}\]  
where \(B_{uv}\) is the car purchase plan and denotes the number of cars to be invested at the stations in row \(u\) and column \(v\). \(b\) is the cost of each car.  
(2)  
Charger cost  
At present, the chargers located in cities cannot meet the charging demands of all electric vehicles, and electric carsharing will have demands for charging during the operation process. Thus, electric carsharing operators need to build a certain number of chargers. The construction costs for chargers can be calculated as follows:  
\[F_{charging}=\lambda*\epsilon\sum\limits_{u=1}^{m}\sum\limits_{v=1}^{n}B_{uv}, \tag{9}\]  
where \(e\) is the construction cost of each charger and \(\lambda\) is the corresponding coefficient between the required number of stations and charging. \(\lambda\) is generally less than 1.  
(3) Parking fee cost  
At free stations, parking fees are incurred after cars are used. The parking fee is generally related to the parking price, parking time, and the number of cars in the station. The parking costs for one year can be calculated as follows:  
\[F_{parking}=\sum\limits_{u=1}^{m}\sum\limits_{v=1}^{n}J_{uv}*t*K_{uv}, \tag{10}\]  
where \(K_{uv}\) is the locations of free stations. \(J_{uv}\) is the average parking price for 1 h in row \(u\) and column \(v\). \(t\) is the average parking time of each car in a year.  
(4) Land rent cost  
The rent of each fixed station is related to the average local land price and the area of the station. The car number in the station determines the size of the station. The land rent can be calculated as follows:  
\[F_{rent}=\sum\limits_{u=1}^{m}\sum\limits_{v=1}^{n}\mu*V_{uv}*(B_{uv}-K_{uv}) \tag{11}\]  
where \(V_{uv}\) is the average rent per square meter and per year in row \(u\) and column \(v\). \(\mu\) is the correspondence coefficient between size and location of the station. \(B_{uv}\) is the delivery plan for cars in the entire area. \(K_{uv}\) is the delivery plan for cars at free stations.  
The total cost \(F\) is equal to the accumulation of various costs, as shown in Equation (12).  
\[\begin{array}{ll}F&=F_{cur}+F_{rent}+F_{charging}+F_{parking}\\ &=\sum\limits_{u=1}^{m}\sum\limits_{v=1}^{n}b*B_{uv}+\sum\limits_{u=1}^{m}\sum \limits_{v=1}^{n}V_{uv}*\mu*(B_{uv}-K_{uv})+\sum\limits_{u=1}^{m}\sum\limits_{ v=1}^{n}\lambda*e*B_{uv}+\sum\limits_{u=1}^{m}\sum\limits_{v=1}^{n}J_{uv}*t*K_{uv}\\ &=\sum\limits_{u=1}^{m}\sum\limits_{v=1}^{n}B_{uv}l_{uv}(b+\mu*V_{uv}+\lambda* e)+\sum\limits_{u=1}^{m}\sum\limits_{v=1}^{n}K_{uv}*(J_{uv}*t_{uv}*t-V_{uv}*\mu) \end{array} \tag{12}\]  
(5) Budget constraint  
We suppose that the maximum investment of these companies in these costs is \(W\). Thus, the cost constraint is Equation (13).  
\[F\leq W. \tag{13}\]  
(6) Other constraints  
Constraints on the size of stations are given by Equation (14). From practical considerations, the scale of stations should not be too large or too small. Thus, the number of cars put in is limited.  
\[B_{uv}\in[num1,num2],u\in[1,m],v\in[1,n] \tag{14}\]Constraints on the total number of stations are shown below. The total number of stations to be laid out in the solved layout plan should not exceed the total number of alternative stations. According to Equation (15), \(N\) is the total number of proposed network points.  
\[1<N<mn \tag{15}\]