Assumptions.

1. All shared vehicles are fuel vehicles.
2. Benefits include economic benefits and social benefits.
3. Carsharing are one-way carsharing, and users can pick up and return the cars at any site.
4. Carsharing and private cars share roads on the urban road network.
5. Buses have independent private lanes, do not interfere with private cars and shared cars.
6. Travelers choose travel modes and routes according to the principle of maximizing random utility, and finally make the multi-mode network reach the equilibrium state.
7. The number of cars initially placed at each site is the same as that of planned parking Spaces.
8. The construction cost of carsharing sites is only related to the number of vehicles on the site.
9. The purchase cost of each vehicle is the same.

Notation.

1. A = {A_r, A_e}—Set of real links and virtual links;
2. A_r = {I_sc, I_pc}—Set of carsharing and private cars driving on the road;
3. A,a—Set of links, link;
4. M,m—Set of travel modes, travel mode;
5. P,p—Set of travel path, travel path;
6. I—Set of carsharing alternative sites, I = {1,2,⋯i,⋯};
7. u—Fixed construction costs for a site;
8. v₁—Construction cost of a single parking space for carsharing, unit: yuan;
9. v₂—Acquisition cost of shared vehicle, unit: yuan;
10. b—Average number of passengers per shared vehicle;
11. φ₁—Maintenance costs per mileage traveled by shared vehicles, unit: yuan/km;
12. φ₂—Dispatch cost per shared vehicle, unit:yuan/veh;
13. B_i—Number of borrowed cars at i, unit: veh;
14. R_i—Number of returned cars at i, unit: veh;
15. C_a—The travel cost of link a;
16. x_a—Traffic flow on link a, unit: person/h;
17. —Carsharing flow on link a, unit: person/h;
18. L_a—The length of link a,unit: km;
19. —The cost of driving shared vehicle on real link a;
20. α₁—Weight of enterprise economic benefits;
21. α₂—Weight of government social benefits;
22. O_I—Operating income, unit: yuan;
23. OC₁—Maintenance cost, unit: yuan;
24. OC₂—Scheduling cost, unit:yuan;
25. C_max—Maximum number of vehicles in Carsharing site i, unit: vehicle;
26. C^min—Minimum number of vehicles in Carsharing site i, unit: vehicle
27. S—Social benefit, unit: yuan;
28. S_c—Space benefit, unit: yuan;
29. S_e—Environmental benefit, unit: yuan;
30. S_p—Energy benefit, unit: yuan;
31. φ_sc—The amount of exhaust produced by each shared vehicle, unit: kg(veh∙km);
32. ε_sc—The amount of CO₂ produced by each shared vehicle, unit:kg(veh∙km);
33. —The economic loss caused by poisonous waste gas emitted per unit mass, unit: yuan/kg;
34. —Economic loss caused by CO₂ emission per unit mass, unit: yuan/kg;
35. ξ_sc—Fuel consumption per kilometer of a carsharing, unit: L/(veh∙km);
36. ψ—The price per liter of fuel, unit: yuan/L;
37. —Decision variable, mileage rates for carsharing, unit: yuan/km;
38. —Decision variable, time rate for carsharing, unit: yuan/min;
39. x_i—Decision variable, 0–1 variable, if site i builds a carsharing site, x_i = 1; otherwise, it is 0;
40. r_i—Decision variable, the number of vehicles placed in the carsharing site i;
41. q—Total travel between OD pairs;
42. q^m—The number of trips in mode M;
43. h^m—The expected generalized travel cost of mode M, unit: yuan;
44. —The travel volume of mode M on path P, unit: person⁄h;
45. —The generalized travel cost of mode M on path P, unit: yuan;
46. —Traffic flow of mode M on link a, unit: person⁄h;
47. λ₁—The sensitivity of travelers’ mode choice to travel cost, known constant;
48. λ₂—The sensitivity of travelers’ route choice to travel cost, known constant;
49. δ_a,p—Path-link indicator variables for path p,if the path p after link a,δ_a,p = 1, otherwise δ_a,p = 0.

The upper model.

1. Economic benefits of operators

Carsharing enterprises hope to achieve the maximum economic benefits with operating strategies. The economic benefits are related to the cost invested in site construction, the purchase of vehicles, and the operating income and costs in the operation. The economic benefit is the difference between operating revenue and operating cost. Operating income (OI) is the fee paid by users of shared cars, using a dual-rate billing system according to travel time and mileage. The operating income is obtained by multiplying the carsharing traffic of each link with the monetary cost.

(1)

The link cost is calculated from the actual link cost of shared vehicle in the generalized road travel costs, as follows (2)

The carsharing enterprises’ operation cost includes the initial investment cost, maintenance cost, and scheduling cost during operation.

The distribution and scale of carsharing sites are limited by the cost of land, labor, and materials needed in construction. Moreover, the purchase of vehicles is also expensive. Therefore, the initial investment costs contain site construction costs and vehicle purchase costs. To enter the market, carsharing enterprises inevitably need to invest in the initial cost. The investment is significant, and the payback period is long. Therefore, economic benefits only focus on operating profit and take the initial investment cost as the constraint condition of capital investment.

The shared cars in this study are fuel cars, and enterprises need to replenish fuel according to vehicle usage during operation. In addition, vehicle wear occurs when the vehicle is running, and the enterprise is responsible for the maintenance and cleaning of the car, resulting in maintenance costs. Therefore, the maintenance cost caused by the wear and tear of fuel-consuming vehicles needs to be considered. Maintenance costs (OC₁) is positively correlated with mileage, expressed as a function of mileage.

(3)

One-way carsharing allows users to borrow and return cars between different sites. Due to vehicle movement between locations, there is a mismatch between the number of cars at a site and user demand. Therefore, carsharing enterprises should make the demand for borrowing and returning at sites as balanced as possible to reduce scheduling costs. The scheduling cost (OC₂) is related to the difference of the number of vehicles borrowed and returned at sites, and is expressed as a function of the difference of vehicles borrowed and returned.

(4)

Where the number of borrowed vehicles at site i is the sum of the carsharing traffic on the online link and the number of returned vehicles is the sum of the carsharing traffic on the off-grid link.

(5)

(6)

Operating cost (OC) is the sum of maintenance cost and scheduling cost.

(7)

The operator’s economic benefit can be formulated as follows: (8)

2. Social benefit of the government

Carsharing is expected to reduce private car ownership, promote car utilization, guide travelers to use public transportation, alleviate urban traffic congestion, reduce exhaust emissions, and save social energy. Therefore, social benefits are essential to fulfill the needs of stakeholders, such as the government and users, for sustainable development. Social benefits can be expressed as the sum of spatial benefit S_c, environmental benefit S_e and energy benefit S_p.

(9)

Spatial benefits refer to the occupation saving of urban space road when some users choose shared cars instead of private cars.

(10)

Environmental benefits refer to the reduction in pollution caused to the urban environment when some users choose car sharing instead of private car.

(11)

Where φ_c is defined as emissions per kilometer produced by a single carsharing, ε_c as the CO₂ per kilometer for carsharing, and are respectively for the economic loss caused by unit quality toxic exhaust and CO₂ emission.

Since the per capita energy consumption of private cars is higher than that of shared cars, the energy benefit is defined as the reduction in energy consumption when some users choose to travel in shared cars.

(12)

Social benefits can be expressed as (13)

Based on the above analysis, the upper target is the maximum sum of economic benefits and social benefits with different weights. The upper model is defined as (14)

S.t (15) (16) (17) (18) (19) (20) (21) (22) (23)

Constraint (15) indicates that the initial investment of carsharing site construction should not exceed the maximum acceptable cost for operators. Most carsharing sites are rebuilt from existing parking lots, roadside parking spots, and vacant lots. One part of the construction cost is fixed costs such as rent. Another part is the variable cost required to transform each parking spot, which is positively correlated with the size of the parking space. The larger the scale, the more the construction cost. The vehicle purchase cost is proportional to the number of vehicles, and the number of vehicles is supposed to be the same as the number of planned parking spots. The vehicle purchase cost and construction variable cost are expressed together here.

Constraint (16) indicates that the number of vehicles at site i cannot exceed the limit of site size. Constraint (17) indicates that the number of selected sites is greater than or equal to 1 and less than or equal to the number of alternative sites. Constraint (18) indicates that if site i is selected, the site size must be greater than zero. Constraint (19)-(20) indicate that carsharing users can only borrow and return cars at the planned sites. Constraints (21)-(23) represent the value range of decision variables.

The lower model.

1. The impact of carsharing on traffic distribution

• Number of vehicles deployed at sites

The demand for car sharing is increasing, but the number of vehicles available at sites is limited. When users arrive at the sites, their demand for a car may not be met immediately, resulting in queues. Limitations on the number of vehicles placed at sites must be considered in the traffic allocation. Fig 2 shows the relationship between car borrowing and returning demand and link traffic flow of carsharing site. A queuing model can be established to describe the queuing situation of users at sites.

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Fig 2. Relationship of car borrowing and returning demand and link flow of carsharing sites.

https://doi.org/10.1371/journal.pone.0315323.g002

The input of the queuing model is related to the link flow. For a particular carsharing station, the car borrowing demand is the sum of the traffic of all online links connected to the site (The online links indicate that the traveler arrives at the multi-modal network from the origin of the trip). The car returning demand is the sum of the traffic on all the off-grid links connected to the station (The off-grid links indicate that the traveler leaves the multi-modal network to reach the destination).

Assuming that x_a is the traffic volume of the link a, b is the average number of passengers per vehicle, B is the demand for borrowing vehicles at the site, R is the demand for returning vehicles at the site, and r is the number of vehicles released at the site, then (24) (25)

Users adhere to the first-come, first-served queuing rule. The rule is a wait-loss system. When users arrive at the station, they will wait in a queue if there is no available vehicle, and the waiting time will be lost. The carsharing site has multiple vehicles, and each vehicle can be taken as a service station. The queuing model is a multi-service station queuing system. The number of service desks is the number of vehicles, and the distribution of service hours is the distribution of vehicle usage time. In the queuing process, the number change curve of cars borrowed and returned by users is shown in Fig 3.

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Fig 3. Users borrowing and returning car demand curve.

https://doi.org/10.1371/journal.pone.0315323.g003

According to the queuing model, the average queuing length can be expressed as (26) (27) (28)

P_B is the expected wait time of arrival, P₀ is the probability of 0 users in the carsharing system, λ is the average arrival speed of the user, μ is the average vehicle speed of users.

The average waiting time of users arriving at the carsharing site is (29)

• Distance from sites to origins and destinations

One-way carsharing sites have limited coverage, and not all origins and destinations are available for carsharing trips. As a result, the distance from the trip’s origin and destination to the carsharing site must be considered. It is assumed that the maximum acceptable walking distance of carsharing users looking for vehicles is s. A circle is drawn with the starting and ending points as the center and s is taken as the radius to judge whether an OD pair can use carsharing. If there are carsharing sites within the acceptable walking distance from the starting and ending points, they can choose the sites to travel; otherwise, travelers cannot use them.

As shown in Fig 4, for the point {O₁, D₁}, there are carsharing sites within the acceptable walking distance of the starting point O₁, then they can be used for travel. However, there are no carsharing sites within the acceptable walking distance of destination D₁, so carsharing cannot be selected. For the starting and ending points {O₂, D₂}, there are carsharing sites within the acceptable walking distance of the starting and ending points, so travelers can choose carsharing to travel.

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Fig 4. Distance diagram from the starting and ending points to carsharing sites.

https://doi.org/10.1371/journal.pone.0315323.g004

The effective path search algorithm can solve the influence of the distance from sites to origins and destinations on traffic assignment. If the trip origin and destination do not satisfy the distance constraint, the path is removed from the valid paths. The specific steps to check the distance constraint of carsharing sites are as follows. 1) Determine whether the initial link of the path is the online access link of carsharing. If so, delete the path that does not meet the distance constraint; otherwise, retain the path. 2) Determine whether the terminated link of the path is the link of carsharing off the network. If so, delete the path that does not meet the distance constraint; otherwise, retain the path.

2. Lower model building

In multi-mode traffic network, mode selection and traffic distribution are carried out according to Wardrop first equilibrium principle (user equilibrium principle). In the trip, the traveler has to make both the travel mode choice and the travel path choice. When a multi-modal network approaches equilibrium, all travelers are unable to unilaterally modify their travel mode or travel path to minimize their desired generalized travel cost. At this point, all modes and paths have the exact generalized travel cost. Travelers between OD pairs make mode and route selection with a random error term of independent homologous Gumbel distribution. Drawing on the random effects theory and the Logit model, the choice behavior of travelers is given. The multi-modal network mode division and traffic assignment equilibrium model satisfying the stochastic user balance principle can be expressed as (30) (31) (32) (33) (34) (35) (36) (37) (38)

Considering the stochastic user equilibrium principle, the lower model is constructed to be a variational inequality model of travel mode and route choice.

(39)

The feasible region of the model satisfies the constraint conditions (33)-(38), are the optimal solution obtained by traffic assignment.

Theorem 1 The variational inequality model (39) is equivalent to the stochastic user equilibrium principle expressions (30)-(32) in multiple modes.

Proof: From the KKT condition of the variational inequality model (39), we can obtain (40) (41) (42) (43)

Where and l are and qdual variable, respectively.

From Eq (40), >0. Substitute Eqs (42) and (43) into Eq (40), get (44)

Sum the path P, combine with constraint (33), get (45)

It can be introduced that l^m is the minimum expected travel cost for the traveler’s choice of mode M in Eq (41)). Substituting Eqs (45) into (44), we obtain the path choice logit model Eq (32).

Similarly, the logit model of mode choice (30) can be obtained from Eqs (41) and (34).

Theorem 2 The variational inequality model (39) has at least one solution.

Proof: The constraints (33)-(38) of the variational inequality model (39) are a set of linear constraints. Functions are continuous functions, according to Brouwer fixed point theorem, it can be inferred that the model has at least one solution.

NSGA-II algorithm for feasible schemes.

The carsharing strategy is a mixed-integer nonlinear programming problem, which belongs to the NP-hard problem and is solved by a heuristic algorithm. The non-dominated sorting genetic algorithm with elite strategy (NSGA-II) is selected to solve the bi-level optimization problem. This algorithm has the advantages of solid global searchability, good compatibility, and high fault tolerance. The core is to coordinate the relationship of each objective function and find the optimal solution set that makes the objective reach the maximum possible value. The algorithm has two essential mechanisms: non-dominated sorting and congestion calculation.

Since there are both 0–1 variables and real variables in the decision variables, the mixed coding method of binary encoding and real encoding is adopted. Binary encoding is adopted for 0–1 variables and real encoding is adopted for real variables. The algorithm steps are as follows.

1. Step 0: Set the population size as N, randomly generate the initial population P_t, and let the number of iterations t = 0.
2. Step 1: Calculate the lower-level model. Substitute the population into the lower-level model, use MSA algorithm to solve , then substitute into the upper level to find the value of the objective function.
3. Step 2: Non-dominated sort. According to the objective function value, the non-dominated sort is performed on P_t, and the rank value of each individual is recorded.
4. Step 3: Select, cross, and mutate. A selective cross mutation operation is performed on P₀ to generate the first generation of offspring population Q_t, and let the iteration times t = 1.
5. Step 4: Merging parent and child. Populations merging P_t and Q_t to produce combined population R_t = P_t ∪ Q_t.
6. Step 5: Generation of new parent population. According to the objective function value, the rapid non-dominated sort crowding degree of R_t is calculated, and N individuals are selected through the elite retention strategy to form a new parent population P_t.
7. Step 6: Select, cross, and mutate. A selective cross mutation operation is performed on P_t to generate the first generation of offspring population Q_t.
8. Step 7: Stop. Judging t ≤ max (t). If so, make t = t + 1, jump to step Step1; otherwise, end the loop.

Scheme comparison based on entropy weight TOPSIS method.

A certain number of site schemes can be output from the lowest to the highest by solving the model. The site investment intensity and traffic accessibility of the shared car sites can be calculated separately for each option based on the site selection. The scheme comparison will be evaluated by benefit value, site investment intensity, and traffic accessibility of carsharing stations.

Investment intensity is generally defined as fixed asset investment per unit area within the construction project site. Site investment intensity can reflect the regional economic development level, investment, and operation environment. It has a significant impact on the selection of carsharing sites. Site investment intensity can be expressed as (46)

Where C_Z is the investment intensity of alternative sites in cities where carsharing are located, unit: yuan/m²,F is the total amount of fixed assets invested in the shared site in the city, unit: yuan, S is the construction land area of the city where stations are located, unit: m².

When users choose to use a shared car, the site needs to meet the user’s needs for borrowing and returning. Good accessibility is a guarantee to increase the use of car sharing. The layout of carsharing stations should meet a wide range of user demands and be deployed at public transportation interchange points to facilitate the connection with the public. Highly accessible nodes with more convenient transportation are more suitable as a carsharing site.

The accessibility between stations in a multi-modal transportation network is determined by a matrix-based topology approach. Specifically, it is assumed that T is the adjacency matrix that show the connectivity of carsharing between sites. (The value of the element in the matrix is 1, which means connected, and 0, which means disconnected). It can be proved by the rules of matrix operation, in the kth power T^kof the matrix T is the number of nodes j that can be reached from node i using shared car after k(k = 1,2,……, n)steps. The D =T¹ + T² +⋯+Tⁿ, If the number of nodes in the network is N, the total number of nodes that node i can reach directly and indirectly is . Finally, it is concluded that node i adopts the relative accessibility index D_i of shared vehicles.

(47)

After calculating the investment intensity and the accessibility of carsharing sites respectively, the original data matrix Table 1 can be summarized by combining the output results of the optimization model.

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Table 1. Raw data matrix.

https://doi.org/10.1371/journal.pone.0315323.t001

The weight of each index is determined by entropy method, and the steps are as follows.

1. Step 1: Standardize raw data matrix. That is, the negative index in the original matrix X is positive and dimensionless. The jth index value of the standardized scheme p_ji ∈ [0,1], the indicator matrix is the matrix P.
2. Step 2: Calculate the entropy of each index H(x_j): (48)
   Where, k is the adjustment coefficient, .
3. Step 3: Calculate the index difference coefficient h_j: (49)
4. Step 4: Determine the weight coefficient of each index W_j: (50)

The best scheme selection is based on TOPSIS method as follows.

1. Step 1: The normalized matrix is multiplied by the index weight matrix to obtain the index matrix considering the weight Z: (51)
2. Step 2: Find the maximum and minimum values of each index in matrix Z, and construct the optimal solution vector Z_max and the worst solution vector Z_min. (52) (53)
   Among them, z_rj and z_wj are the optimal value and worst value of the jth column index respectively.
3. Step 3: Calculate the distance between each scheme and the optimal solution (the worst solution), respectively (54) (55)
   Finally, the calculation formula of relative closeness, which represents the merits and demerits of the scheme, is (56)

The solution with the highest closeness is the optimal solution under the multi-objective decision of integrated benefit value, station investment intensity and accessibility of carsharing stations.