1.1 Background

The study on ridesharing has received much attention since the rise of e-hailing services (e.g., Uber, Lyft, and DiDi). Using ubiquitous real-time communications, private car drivers and ridesharing service seekers are being automatically matched according to their itineraries and schedules. The ridesharing system is recognized to possess many advantages to participants (both drivers and passengers), to society, and to the environment in terms of saving travel cost, reducing travel time, mitigating traffic congestions, conserving fuel, and reducing air pollution.

In this paper, we focus on an enhanced ridesharing system that respect (i) users preferable time windows and (ii) meet points where passengers are willing to walk to/from for collective pickup or drop-off. The consideration of the above two constraints is necessary and important to furnish a more user-friendly and efficient ridesharing service than the conventional modes. For instance, a preferred departure time window is very commonly associated with commuters (or travelers on business trip) at their origins, from which they won’t be able to be picked up too early on one hand; and on the other hand, they also will refuse to wait too long before served. In addition, a latest arrival time is often specified at the destinations of ridesharing participants. As for including the meet point into the proposed ridesharing system, it is expected to take advantage of constructing routes with smaller detours, while maintaining a satisfactory range of walking for riders. From the theoretical prospective, the conventional door-to-door ridesharing system becomes a special case of the proposed system when each passenger is assigned to his/her own origin/destination as the meet point. The proposed ridesharing problem seeks to determine the optimal route for each vehicle and the optimal assignment of passengers to vehicles as well as to the meet points.

1.2 Literature review

This section reviews the existing literature that is relevant to our work. The ridesharing problem can be mathematically modeled by one of the well-known optimization problems: the vehicle routing problem with pickup and delivery with time windows (VRPPDTW) or simply, pickup and delivery problem with time windows (PDPTW) [1]. The PDPTW has a wide range of applications in the fields of transportation and logistics (see the recent review by Mahmoudi and Zhou [2]). For instance, applications in cargo routing and scheduling include [3–6]; transportation of elderly or handicapped people [7–8]; and school bus routing and scheduling [9–10]. Specifically in the field of passenger transportation, the PDPTW could also be observed as the dial-a-ride (DAR) problem in the literature, which generally concerns designing transportation service for riding requests from specific pickup locations to specific drop-off locations, while imposing certain requirements (e.g., time window and vehicle capacity constraints)[11]. Ridesharing problem adds new constraints from the private driver side to the DARP, such as the maximum detour constraint from the driver’s own itinerary (which may not be a concern to commercial vehicles operated by a company in case of DARP). Other than that, ridesharing problem shares most of the properties with DARP.

DARP literature is extensive and has many variants, which can be classified based on several characteristics, e.g., single vehicle or multiple vehicles, static or dynamic demands, and homogeneous or heterogeneous vehicle fleet. Early studies were mainly about the single-vehicle DARP [12–14], which is a special case of the multiple vehicles DARP [15–16]. The most of cases propose discrete models and have been proved to be NP-hard [12]. For the static DARP (e.g., [17]), all riding demands are known in advance with specified time windows for pickup and/or delivery, while for the dynamic DARP (e.g., [11]), some requests are allowed to emerge or disappear progressively during the service process. Compared to the homogeneous DARP, heterogeneous DARP (H-DARP) [18] recognizes the heterogeneity of vehicles in terms of capacities, performances, and other various resources (e.g., wheelchair spaces). Additional considerations, such as routing properties [19] and stochastic properties [20–21], have also been studied in DARP literature to build models with more realistic characteristics (accounting for transfers among routes and travel time uncertainties, for instance).

When designing DAR systems (ridesharing as well), a commonly used objective is to minimize the operational costs (e.g., in terms of total travel distances) that are subject to full demand satisfaction and constraints [22]. Alternatively, a number of optimizations consider fleet efficiency, such as minimizing fleet size [23] or maximizing vehicle usage efficiency. In a dynamic context, the number of requests served in time may be maximized [24–25], or their total trip time may be minimized [26]. Extending from the DARP with a single objective (which is either cost-based or efficiency-based), many studies introduced multiple objectives to optimize the service quality as well. For instance, the most frequent quality-related objectives include minimizing total user waiting time (caused by deviations from their preference time), minimizing vehicle idle time [27], and maximizing the number of ride-matches. The multi-objective DARP is often formulated into a single weighted-sum objective function to facilitate the solving process.

Solution approaches to DARP can be basically divided into three classes: exact algorithms, heuristics, and meta-heuristics. Due to the NP-hardness of discrete-based DARP models, exact algorithms are promising only for the single vehicle problem or small-size multi-vehicle problems. For instance, Psaraftis [12–13] presented exact dynamic programming (DP) solution algorithm for a single-vehicle DARP to minimize a weighted sum of the total service time and the total waiting time for all customers. A column generation approach is developed by Dumas et al. [28] for a multi-vehicle DARP, of which the objective is to minimize the total travel cost. Cordeau [29] utilizes the branch-and-cut algorithm to solve a medium-size DARP with up to 48 requests. Later only, the branch and cut and price algorithm is further proposed with higher computation capability for the PDPTW [30].

For complex DARP with additional constraints, most research attentions are devoted to developing heuristics and meta-heuristics due to the computational efficiency constraint. For instance, Jaw et al. [31] develop a sequential insertion heuristics by sorting users according to their earliest pickup time and iteratively optimizing the routing plan. Kim and Haghani [32] and Wong et al. [33] apply the sequential insertion heuristics to a time-dependent DARP and a dynamic DARP, respectively. Luo and Schonfeld [34] establish a rolling horizon insertion strategy for dynamic DARP. Another type of heuristics is cluster-first-route-second heuristics, which split the DARP into separate phases: (i) generating clusters and (ii) solving the vehicle routing subproblem. Contributing works of this type include [35–37].

Heuristics are, however, easily be trapped in local optima. In light of this, meta-heuristic algorithms are more often developed nowadays by incorporating these classical heuristics in the solution framework (e.g. to generate an initial solution). A pioneering work is that by Cordeau and Laporte [38], who propose a Tabu Search (TS) algorithm for the DARP. Since then, a variety of meta-heuristics have been developed for contending with many DARP variants. Generally, they can be grouped into two categories: local-search based and population based algorithms. The former includes the Variable Neighborhood Search (VNS) algorithm, Large Neighborhood Search (LNS) algorithm, deterministic annealing, and evolutionary local search. The latter mainly consists of various enhanced genetic algorithms. Detailed reviews are referred to [22].

Previous research has made many important contributions to this challenging problem along with different formulation or solution approaches. The conventional ridesharing system, however, may not work well providing door-to-door service in dense-populated urban areas, where huge waste of time delays may be caused by frequent stops at very closely located pickup and drop-off points in neighborhood streets. Particularly, the door-to-door ridesharing system won’t be feasible in the environment of gated communities, where riders have to gather at some commonly recognized locations (e.g., main gates and major intersections, if there are no designated ones). Moreover, how to efficiently schedule the ride sharing services in the condition of running short of the available dispatching vehicles is also a challenge to both operators and users. Most of the existing works have ignored the location selection problem when neighborhood riders have to share pickup/drop-off points (namely meet points), and some or all of them are willing to walk to and from the meet points to facilitate the pickup and drop off process. Taking this arrangement into account is able to extend the traditional ridesharing system to a more flexible system providing services of door-to-door, door-to-meet-point, meet-point-to-door, and meet-point-to-meet-point. By trading off the walking cost and the benefit of pickup/drop-off saving, the meet-point ridesharing system is expected to minimize the total system cost, particularly in dense urban area with many closely located requests (in peak hours, for instance). It is also expected to better solve the problem of lacking of dispatching vehicles, and to maximize the limited resources.

The setting of meet points imposes new challenge to the modeling efforts in terms of location choice, time coordination, and passenger assignment. So far, very few studies can explicitly account for the meet point setting in ridesharing system optimization. An example is Stiglic et al. [39], who demonstrated the considerable benefits of introducing meet points in a ridesharing system. Our study even complicates the problem by further introducing preferable time windows for ride requests other than the traditional of earliest-pickup and latest-arrival time constraint as the treatment in [39]. This imposes new challenges in matching the riders and ridesharing vehicles as well as riders that can be assigned to the same meeting point within their overlapping time windows. Another example is Daganzo [40], who developed continuum models for a DAR with meet points (called checkpoints). Although system performance metrics (e.g., total travel time) were analytically derived, no routing plans and specific locations of meet points can be generated from these models.

1.3 Research motivation

In light of the above, this study aims to develop a new model to design the appropriate routing plans for the flexible ridesharing system. The contributions of our paper are as follows:

1. (i). The proposed model incorporates meet points in ridesharing system by allowing riders free of choosing to walk to the meet points or to be directly picked up/dropped off at their origins/destinations;
2. (ii). The model simultaneously considers riders’ desired pickup time windows and drop-off time windows as well as the maximum route travel time constraint from the driver side; and
3. (iii). A meta-heuristic algorithm is developed and evaluated in the application on a medium-size network.

The remainder of the paper is organized as follows. Section 2 first describes the ridesharing system in this study, and then formulates the optimization problem as a mixed integer linear program (MILP). Section 3 presents an enhanced Tabu Search (TS) algorithm embedded with another heuristic algorithm that is used to generate high-quality initial solutions. Section 4 summarizes the results of numerical experiments, in which the solution quality and computation efficiency of the proposed meta-heuristic algorithm are demonstrated. The merits of introducing meet points to ridesharing systems are also illustrated. Conclusions are drawn in the final section.

2.1 Description of the ridesharing system

Consider a complete network, G = (V,A)[41], where V is the set of vertices, i,j,m,s ∈ V, and is partitioned as V = V_p ∪ V_d ∪ V_o ∪ V_e, where V_p is the subset of vertices associated with pick-up locations of seekers; V_d is the subset of vertices associated with drop-off locations of onboard passengers; V_o is the subset of vertices associated with the origin locations of drivers; V_e is the subset of vertices associated with the destinations of drivers. Let T be the set of all ridesharing vehicles, P be the set of all demand announcements. Each seeker is associated with an earliest departure time window [] from the corresponding pickup location, and also a latest drop-off time at their corresponding drop-off locations. A maximum trip duration T_t is specific to the driver of vehicle t.

2.2 Notation

Table 1 summarizes the notation used in this paper.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Variables and parameters in the model.

https://doi.org/10.1371/journal.pone.0195927.t001

2.3 Formulation

The proposed problem is formulated into the following mixed integer linear program (MILP). To facilitate the modeling, this study considers the following assumptions:

1. Passenger walking distance, ridesharing car travel time, and travel distance are given;
2. The location of demand points and meet points are given;
3. Number of routes is given; and
4. Each ridesharing car cannot accommodate more passengers than its capacity.

Objective function.

The objective function is expressed in the total system cost, which is the sum of the travel time of the sharing vehicle and users’ walking time to/from the pickup points.

(1)

Contraints.

Constraints (2–5) guarantee that each passenger is assigned to one pick-up and drop off location. These constraints also prevent that vehicle routing from connecting to an unselected potential pick-up or drop-off location: (2) (3) (4) (5)

Constraints (6–8) are used to connect passengers with rider-sharing vehicles, and ensure that each passenger is served by only one vehicle: (6) (7) (8)

Constraints (9–11) prevent any selected pick-up location being connected with multiple vehicles. The constraints further increase efficiency of the entire system: (9) (10) (11)

Constraint (12) is the conservation law and ensures that each selected pick-up or drop off location has the same incoming arc and outgoing arc: (12)

Constraints (13–14) (in the form of inequality for the ease of finding solution) guarantee the feasibility of the schedule for each vehicle trip. Specifically, it is ensured that the travel time on each arc should equal to the corresponding travel time within the given time matrix: (13) (14) Where M is an extreme large value.

Constraint (15) guarantees the number of passengers boarded never exceeds vehicle capacity during the entire trip: (15)

Constraint (16) ensures that passengers are assigned to the route only if this route serves that link: (16)

Constraints (17–18) ensure that the scheduled arrival time at each pick-up point satisfies the preferred service time window of all users served at this pick-up point: (17) (18)

Constraint (19) guarantees that the total travel time of vehicle t does not exceed the maximum travel time that driver would endure.

(19)

Constraints (20–21) ensure that all vehicles start from ridesharing drivers’ origin locations.

(20)(21)

Constraints (22–23) guarantee that all vehicles end at ridesharing drivers’ destinations.

(22)(23)

The proposed mixed integer model is an extension of classic vehicle routing problem (VRP), which is NP-hard to resolve. The model is further complicated by the introduction of (i) meet points associated with passenger assignment and (ii) users’ preferred time windows. Therefore, it is difficult to use exact method to solve such a problem at an acceptable running time, especially for a large-scale instance. To better contend with the issue of computation efficiency, this study further proposes an enhanced Tabu Search algorithm to yield the meta-optimal solutions in a reasonable amount of time at next section.

3.1 Solution framework

The form of Tabu search we followed in this study can be tightly summarized as follows:

1. Start from a solution s₀;
2. s_best ← s₀;
3. Set the Tabu list to the empty list;
4. While a stopping criterion is not satisfied, do:
   1. Generate the neighborhood of N(s_t) through non-Tabu moves (or Tabu moves that lead to solutions that improve s_best) and select the best solution s_t;
   2. If s_t is better than s_best then s_best ← s_t;
   3. s₀ ← s_t;
   4. Update the Tabu list.
5. Output s_best.

3.2 Neighborhood structure

The solutions in the neighborhood of a given solution s ∈ S, denoted as N(s), are the solutions that can be obtained by applying a single-move operation to the current solution. Each solution s is assigned with an attribute set U(s) = {(p,t):passenger p is assigned to vehicle t}. A single-move operator here is to change an attribute (p,t) from U(s) to (p,t′) where t′ ≠ t. And then, the corresponding vertices are also removed from route, in which the new vertices would be inserted. Note that, the insertion of new vertices should follow the rule of minimizing the penalized objective function value. To prevent cycling, the same passenger p who has been moved from vehicle t to vehicle t′ in a given iteration is forbidden to move back to vehicle t, or declared tabu. A tabu move is allowed only when reach a solution of smaller cost than that of the best known solution.

3.3 Diversification/Intensification

In our study, we follow the paper by Lai et al. [44] and integrate two diversification strategies into the Tabu search. Specifically, one is activated during the construction of the neighborhood solutions where any solution, s ∈ S, is penalized proportional to the frequency of a vertex appeared in the incumbent solution. The other is used during the search process where the penalty weights for the violation of constraints (15), (16) and (19) are dynamically adjusted for better exploration of the search space. The penalty weight α and β are initially set to 1 that are recommended by Cordeau et al. [45], and Ho and Gendreau [46]. The values of weights are then adjusted during the search according to whether the solution in the next iteration satisfies the corresponding constraints. Specifically, it is divided by (δ + 1) if there is no violation of the constraint; otherwise, it is multiplied by (δ + 1), where δ is a user-defined parameter.

3.4 Initial population

The quality of the solution found and the convergence speed of using heuristic algorithm, highly depend on the selection of the initial population. In this research, a meta-heuristic algorithm to efficiently generate a decent initial population is proposed. The procedures are explained as following:

Step 1. Define parameters, such as P (the set of passengers), T (the set of vehicles) and V(the set of locations) etc.;

Step 2. Use a backward recursion to generate a feasible routing plan for ride sharing vehicle:

1. 1). Append tag of preferred departure time to each request by using formula:
2. 2). Sort the requests according to the preferred arrival time, and then select the latest T requests, [p_(P−1),…,p_(P−T)], with their preferred arrival and departure time;
3. 2). For each selected request, connect its pick-up location and drop-off location, and then randomly insert another one non-selected request;
4. 3). Repeat 2) until constraint (15), (17) or (19) is violated;

Step 3. Assign passengers who are unable to be directly served to their nearest selected pick-up locations;

Step 6. Generate the initial population s₀ with the results from Step 2 to Step 3.

3.5 Neighborhood evaluation

To evaluate the quality of generated route, we follow up an adapted version of the eight-step evaluation scheme introduced by [38], as recorded in Table 2. Additionally, the eight-step evaluation scheme uses the forward time slack F_i for a node i ∈ N defined by [47]: where w_p denotes the waiting time at node p, q is the last node on the route, and P_t is the ride time of the vehicle. F_i gives the maximum amount of time, by which the departure from a node i can be delayed without violating time windows and passenger ride time.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Eight-step evaluation scheme.

https://doi.org/10.1371/journal.pone.0195927.t002

3.6 Stopping criterion

In our case, two stopping criteria have been applied, i.e., the run time limit and the maximum number of consecutive iterations (2,000).

4.1 Experimental design

4.1.1 Network layout.

The proposed ridesharing system is tested on a manipulated network (as shown in Fig 1), in which 10 ride requests are appended to10 pick-up locations (shown by black solid circles), and another 10 drop-off locations (hollow squares) are also designed. Each ride request is assigned with the preferred service time window of pick-up and drop-off, as well as one drop-off location, as given in Table 3.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Spatial distribution of pick-up and drop-off locations.

https://doi.org/10.1371/journal.pone.0195927.g001

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Ride request information.

https://doi.org/10.1371/journal.pone.0195927.t003

4.1.2 Scenario design.

A total of three scenarios are designed to show the advantage of the proposed model and the Tabu-based meta-heuristic solver. The designed three scenarios will test the service performance of the ridesharing system when the fleet has 3 ridesharing cars, 4 ridesharing cars, and 5 ridesharing cars, respectively. In all scenarios, the proposed MILP model is solved by both the out-of-the-shelf commercial solver CPLEX 12.6 and by the proposed Tabu-based meta-heuristic solver. The program is run on a computer with Windows 8, a 2.8 GHz processor, and an 8 GB of RAM. The results comparison will be further unfolded to show the efficiency of the proposed meta-heuristic algorithm.

To simplify the experiments, the origins and destinations of ridesharing car drivers are randomly selected from the sets of pick-up and drop-off locations. Note that comparing with the scenario of 3-car, the 4-car and 5-car scenarios add new drivers while the information of the existing drivers remains the unchanged (as shown in Table 4).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 4. Ridesharing driver information.

https://doi.org/10.1371/journal.pone.0195927.t004

Other key parameters used in the case study include: the vehicle capacity is set to be 4 persons per vehicle; the maximum allowed travel time is 40 mins per route; and the average walking speed is 5 km/h.

Note that in practice to mitigate impact of travel time uncertainties, the input travel time matrix can be generated from an average travel time of the past three days with using commercial map engine (e.g., Baidu Map). Upon the future research interest, the proposed model is able to be embedded with map engine that reads the real time travel time information, so as to further alleviate the side effects of travel time uncertainty.

4.2 Experiments results

4.2.1 Comparisons between CPLEX and meta-heuristic results.

We run CPLEX and the proposed Tabu-based meta-heuristic algorithm to solve the models under the three scenarios, respectively. The results are summarized in Tables 5–7, in which the meta-heuristic solutions are used to illustrate the effectiveness of the proposed algorithm by comparing with the global optima obtained by CPLEX.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 5. Solution of 3-car scenario.

https://doi.org/10.1371/journal.pone.0195927.t005

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 6. Solution of 4-car scenario.

https://doi.org/10.1371/journal.pone.0195927.t006

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 7. Solution of 5-car Scenario.

https://doi.org/10.1371/journal.pone.0195927.t007

Table 5 gives the solutions of the 3-car scenario by the two solvers. Taking the route of the first vehicle in CPLEX solution as an instance, the driver departs from his/her origin, P1, at 5:58, then picks users at P6 (6:02), P7 (6:04), and P4 (6:06) in sequence, then transport them to their desired drop-off location, D6 (6:11), D7 (6:14), and D4 (6:16), respectively. Finally the first driver terminates the service at his/her own destination, D1, at 6:19. In CPLEX solution, one passenger assignment is observed in the route of vehicle 3, where the user at P5 is suggested to walk to P3 (which takes 10 mins) for the ridesharing service. This assignment benefits in reducing the total system cost. The optimal objective value of 3-car scenario, including in-vehicle time and walking time, is 73 mins. Comparatively, the solution of the proposed algorithm suggests another passenger assignment: the user at P9, whose desired drop-off location is D9, is suggested to alight at D4 (along with users from P4) and then walk to his/her own destination (i.e., D9). The objective value yields 79 mins.

Table 6 presents the solutions of the 4-car ridesharing system by two solvers, respectively. Two passenger assignments are witnessed in the CPLEX solution (i.e., in route of vehicle 3, P4 user is assigned to P5 pick-up location, and P3 users to alight at D1), whereas the meta-heuristic solution suggests no passenger assignment, and thus results in more en-route detour and longer vehicular distance in the ride sharing service (see the total travel time of 85 mins that is higher than that of CPLEX result).

Table 7 reveals the model results of the 5-car ridesharing system. Both of the CPLEX and meta-heuristic solutions suggest that there is no need to assign any passengers walking to other pick-up or drop-off locations when there are 5 cars in service. The difference between the CPLEX solution and our meta-heuristic solution is 11mins (75 mins vs. 86 mins).

From the above results, it is demonstrated that the proposed algorithm is able to provide reasonably good solutions (with in 8%~15% difference from the global optima) under all scenarios. Consistent conclusion is obtained by both the CPLEX and meta-heuristic solutions: The most efficient ridesharing system is the 3-car operation with the minimum objective value (i.e., 73 mins) in serving 10 requests. Thus, the 4-car and 5-car ridesharing systems appear to have redundant fleet assigned in the service. Additionally, the proposed algorithm shows much more advantage in its computation efficiency, which does not increase (from 102 s to 266 s) so rapidly as the CPLEX solver (from 750 s to 4,210 s). The proposed algorithm stably generates the near optimal solutions within 5 minutes, implying the insensitivity to the network complexity.

4.2.2 Comparison between “with” and “without” the setting of meet points.

To illustrate the merits of introducing meet points into the ridesharing system, we compare scenarios “with” and “without” meet points. For the scenario without meet points, the following constraint is added into the proposed model: (24)

Constraint (24) restricts that a passenger can only be assigned to one pickup/drop-off point (i.e., his/her own origin and destination). We conduct the comparisons for 3-car and 4-car ridesharing systems with the global optimal solutions obtained by CPLEX solver. The results are presented in Tables 8 and 9, respectively. It is seen that every passenger is picked up and dropped off at his/her own origin and destination under the scenario without meet points. The optimal routing plans differ from that of the scenario with meeting points. Comparatively, excluding meet points increases the total travel time by 2.7% and 3.8% for the 3-car and 4-car ridesharing systems, respectively. More considerable merits of introducing meet points are expected for ridesharing systems with large size of fleet under day-to-day operation.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 8. Comparison of 3-car scenario with and without meet points.

https://doi.org/10.1371/journal.pone.0195927.t008

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 9. Comparison of 4-car scenario with and without meet points.

https://doi.org/10.1371/journal.pone.0195927.t009