Qualifying the benefits of ride-sharing on reducing fleet size

Reducing the number of operating vehicles in cities has enormous significance on mitigating traffic congestion and environment pollution. Ride-sharing is an efficient way to reduce fleet size in urban areas. In this work, we propose two integer programming models to qualify the benefits of ride-sharing on reducing fleet size. The proposed models are solved by commercial solver Gurobi. Then we conduct a series of instances based on trip records of New York City to test the proposed models. Results indicate that without delaying drop-off times, the fleet size when considering ride-sharing remains almost the same as ride-hailing service for high-density travel demand settings. Whereas the fleet size drops sharply as the demand density decreases. In addition, the number of vehicles required is reduced by nearly 30% regardless of order density under ride-sharing assumptions when a slight delay is allowed.


Introduction
On-demand travel request is one of the most important demands in modern cities. According to the World Bank's forecasts, by 2050, about 5.4 billion will live in urban areas, and the number of vehicles on road will double to 2 billion [1]. These changes will lead to more severe traffic congestion and deteriorated urban environment. Litman proposed that an effective transportation system requires a significant reduction in the use and circulation of vehicles [2]. New York City Council passed regulations to limit the number of vehicles providing ride-hailing service in 2018. Thus, reducing the scale of operating vehicles in cities has enormous significance on improving traffic efficiency.
Previous studies on determining minimum fleet size have made great progress. Danzig & Fulkerson are the first researchers to model the problem of minimum number of tankers as a linear programming problem and use the simplex method to solve it [3]. Vazifeh et al. believed the size of the current taxies in New York City could be reduced by 30% through transforming individual driver decisions to a centralized operation [4]. Yao et al. used a similar approach to conclude that 128,000 shared vehicles are needed to meet travel demands of 3 million cell phone users in Shanghai, China [5].
Ride-sharing is another approach to reducing fleet size in urban areas. The concept of ride-sharing originated during the oil crisis in the 1970s [6]. The studies of Martinez & Viegas [7] and Sun [8] confirmed that sharing mode could increase average vehicle occupancy ratio and reduce urban congestion and greenhouse gas emissions. The development of information and communication technologies has opened the way to share locations and send messages in real time. Through combining multiple similar itineraries into one itinerary, ride-sharing can maximize the use of existing travel resources in society and reduce the number of vehicles in circulation [9]. In this paper, we will qualify the benefits of ride-sharing on reducing fleet size. Different from references [4] and [10], the ideal fleet size in ride-sharing will be compared with the fleet size after optimizing assignment of orders, instead of actual operation fleet size. We believe this is a more intuitive way to reflect the effects of ride-sharing.

Mathematical model
In this section, we will introduce two mathematical models of minimum fleet problem. The minimum fleet problem can be defined as follows: determine the minimum number of vehicles needed to serve all the travel requests specified by origin, destination and desired pick-up time.

Minimum fleet model for ride-hailing service
The model in this subsection aims to acquire the minimum number of vehicles under the situation that each vehicle can only serve the next order after serving an order. Ride-sharing is not allowed.
Each order ∈ is defined as a tuple , , , where represents the desired pick-up time, the drop-off time, the pick-up location and the drop-off location. Here, the pick-up time means the earliest time at which the passenger can be picked up. The drop-off time means the estimated time of dropping off the passenger, calculated using Euclidean distance and estimated realtime speed, assuming the passenger leaves the pick-up location at time . If the set is extracted from a real-world dataset, represents the actual time at which a passenger is picked up.  Figure 1(a) illustrates the construction of vehicle shareability network that enables the minimum fleet problem to be optimally solved. This is a directed network defined as , , where node ∈ corresponds to order ∈ and the directed edge , ∈ exists if and only if (1) ( 2 ) in equation (1) represents the estimated travel time from the drop-off location to the pick-up location . The existence of a link , in the network corresponds to a sequence of orders that can be served by a single vehicle. , as a parameter in equation (2), represents the upper bound of no-load time between two consecutive orders served by a single vehicle. If no-load time of a vehicle is very long, it seems two orders that occur at distant locations or times are irrationally assigned to the vehicle. Therefore, an excessively large no-load time leads to longer travel distances, lower vehicle occupancy ratio, a lot of emissions to the environment. Figure 1(b) indicates the optimal path cover, that is the optimal solution in minimum fleet model. In the optimal path cover, each node is covered by at most one path. The minimum fleet is equal to  figure 1(b), which can be transformed into the problem to include most directed edges covering all orders.
Our goal is to maximize the number of directed edges when each order is only visited once. is the only decision variable in the model. It is a binary decision variable.
1 means the directed edge , ∈ is included in the optimal solution.
The objective function is given by Constraints (4) and (5) ensure that each order is included in at most one path. Orders that are not linked by directed edges will be assigned another separate vehicle to serve. Constraints (6) and (7) guarantee the rationality of directed edges, corresponding to the two conditions for the existence of directed edges. In constraints (8), self-circulation of orders is not allowed. Constraints (9) describes the value range of the decision variable. Compared to serving only one order at a time, ride-sharing can easily cause delays in drop-off time. Let denotes maximum delay that passengers can tolerate, that is, a passenger cannot arrive at the drop-off location minutes later than ride-hailing service. The problem is defined in a directed graph , , ∪ . There are 6 decision variables in the model. , , and are binary decision variables while and are continuous decision variables.

Minimum fleet model for ride-sharing service
1 means vehicle travels from location to carrying the passengers corresponding to order .
1 indicates vehicle travels from location to location .
1 means order is assigned to vehicle to serve. 1 indicates vehicle has not been assigned any order. represents the time when a vehicle arrives at location and represents the time when a vehicle leave location .
Our goal is to maximize the number of unused vehicles. The objective function is given by

Numerical experiments
In this section, we design a lot of instances based on real data and employ Gurobi to solve these instances. The results are shown in this section.

Experimental settings and computational environment
The above two models are solved by Gurobi. All computations are executed on a PC with Intel Core i5-8265U CPU (1.8 GHz) and 8 GB RAM.
The instances to be used are extracted from the dataset composed of taxi trips records originating and ending in Manhattan in January 2016. For each trip, the record reports the order ID, the Global Positioning System (GPS) coordinates of the pickup and drop-off locations, and corresponding times.
In view of complexity of the ride-sharing model and the limited computing space of the PC, instances of 20 orders are tested in the paper.
6 sets of instances are generated and their difference lies in the order density. The first set of instances is drawn from orders happening within 10 minutes on January 30, 2016 and the second set happening within 20 minutes, by that analogy, the sixth set is drawn from orders within 60 minutes.
Each set of the same order density contains 3 subsets with different parameter values. These involved parameters are listed in table 1. The parameter is set to 15 minutes in 3 subsets referring to the literature [4]. These subsets adopt different values of W and . Each subset contains 5 instances of 20 orders to ensure the stability and reliability of results. The experimental results are showed in table 2. 1, 0 (situation 1) represents the situation that ride-sharing is not allowed but assignment of orders is optimized. This part of the results is benchmark for further comparison.
3, 0 (situation 2) represents the situation that a taxi can serve at most 3 orders simultaneously and the service level is the same as the previous situation with no arrival delay.
3, 5 (situation 3) represents the situation that ride-sharing is considered, a taxi can serve at most 3 orders simultaneously but 5 minutes' arrival delay is allowed.