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# On-demand high-capacity ride-sharing via dynamic trip-vehicle assignment

Edited by Michael F. Goodchild, University of California, Santa Barbara, CA, and approved November 22, 2016 (received for review July 20, 2016)

### This article has a correction. Please see:

## Significance

Ride-sharing services can provide not only a very personalized mobility experience but also ensure efficiency and sustainability via large-scale ride pooling. Large-scale ride-sharing requires mathematical models and algorithms that can match large groups of riders to a fleet of shared vehicles in real time, a task not fully addressed by current solutions. We present a highly scalable anytime optimal algorithm and experimentally validate its performance using New York City taxi data and a shared vehicle fleet with passenger capacities of up to ten. Our results show that 2,000 vehicles (15% of the taxi fleet) of capacity 10 or 3,000 of capacity 4 can serve 98% of the demand within a mean waiting time of 2.8 min and mean trip delay of 3.5 min.

## Abstract

Ride-sharing services are transforming urban mobility by providing timely and convenient transportation to anybody, anywhere, and anytime. These services present enormous potential for positive societal impacts with respect to pollution, energy consumption, congestion, etc. Current mathematical models, however, do not fully address the potential of ride-sharing. Recently, a large-scale study highlighted some of the benefits of car pooling but was limited to static routes with two riders per vehicle (optimally) or three (with heuristics). We present a more general mathematical model for real-time high-capacity ride-sharing that (*i*) scales to large numbers of passengers and trips and (*ii*) dynamically generates optimal routes with respect to online demand and vehicle locations. The algorithm starts from a greedy assignment and improves it through a constrained optimization, quickly returning solutions of good quality and converging to the optimal assignment over time. We quantify experimentally the tradeoff between fleet size, capacity, waiting time, travel delay, and operational costs for low- to medium-capacity vehicles, such as taxis and van shuttles. The algorithm is validated with ∼3 million rides extracted from the New York City taxicab public dataset. Our experimental study considers ride-sharing with rider capacity of up to 10 simultaneous passengers per vehicle. The algorithm applies to fleets of autonomous vehicles and also incorporates rebalancing of idling vehicles to areas of high demand. This framework is general and can be used for many real-time multivehicle, multitask assignment problems.

New user-centric services are transforming urban mobility by providing timely and convenient transportation to anybody, anywhere, and anytime. These services have the potential for a tremendous positive impact on personal mobility, pollution, congestion, energy consumption, and thereby quality of life. The cost of congestion in the United States alone is roughly $121 billion per year or 1% of GDP (1), which includes 5.5 billion hours of time lost to sitting in traffic and an extra 2.9 billion gallons of fuel burned. These estimates do not even consider the cost of other potential negative externalities such as the vehicular emissions (greenhouse gas emissions and particulate matter) (2), travel-time uncertainty (3), and a higher propensity for accidents (4). Recently, the large-scale adoption of smart phones and the decrease in cellular communication costs has led to the emergence of a new mode of urban mobility, namely mobility-on-demand (MoD) systems, led by companies such as Uber, Lyft, and Via. These systems are able to provide users with a reliable mode of transportation that is catered to the individual and improves access to mobility to those who are unable to operate a personal vehicle, reducing the waiting times and stress associated with travel.

One of the major inefficiencies of current MoD systems is their capacity limitation, typically restricted to two passengers. Our method applies not only to shared taxis but also to shared vans and minibuses. A recent study in New York City showed that up to 80% of the taxi trips in Manhattan could be shared by two riders, with an increase in the travel time of a few minutes (5). However, the method and analysis of ref. 5 was (*i*) limited to two riders for an optimal allocation (three with heuristics), (*ii*) intractable for larger number of passengers, and (*iii*) did not allow for allocation of additional riders after the start of a trip. There are no studies of this scale that quantify the benefits of larger-scale ride pooling, mainly due to the lack of efficient and scalable algorithms for this problem, both of which we address in this work.

Much of the fleet management literature for MoD systems considers the case of ride-sharing without pooling requests, focusing on fluid approximations (6), queuing based formulations (7), case studies in specific regions [e.g., Singapore (8)], and operational considerations for fleet managers (9). With the growing interest and rapid developments in autonomous vehicles, there is also an increasing focus on autonomous MoD systems (6, 9, 10). However, none of these works considered the ride-pooling problem of servicing multiple rides with a single trip. The ride-pooling problem is more related to the vehicle-routing problem and the dynamic pickup and delivery problem (11⇓⇓⇓–15), where spatiotemporally distributed demand must be picked up and delivered within prespecified time windows. A major challenge when addressing this problem is the need to explore a very large decision space, while computing solutions fast enough to provide users with the experience of real-time booking and service.

Here, we consider the problem of using a fleet of vehicles with varying passenger capacities, and, in contrast to ref. 5, we address both the problems of assigning vehicles to matched passengers and rebalancing—or repositioning—the fleet to service demand. We show how the unified problem of passenger and vehicle assignment can be solved in a computationally efficient manner at a large scale, thereby demonstrating the capability to operate a real-time MoD system with multiple service tiers (shared-taxi, shared-vans, and shared-buses) of varying capacity.

Whereas previous approaches to this problem have focused on heuristic-based solutions (16⇓–18), we present a reactive anytime optimal algorithm. That is, an algorithm that efficiently returns a valid assignment of travel requests to vehicles and then refines it over time, converging to an optimal solution. If enough computational resources are available, the optimal assignment for the current requests and time would be found; otherwise, the best solution found so far is returned.

Traditional approaches that rely on an integer linear program (ILP) formulation, such as ref. 19, also provide anytime guarantees for the multivehicle-routing problem. However, in contrast to our approach, their applicability is limited to small problem instances, which in ref. 19 was 32 requests and 4 vehicles, with a computation cost of several minutes. We also rely on an ILP formulation, but because we do not explicitly model the edges of the road network in the ILP, our approach scales to much larger problem instances. We observe that instances such as New York City, with thousands of vehicles, requests, and road segments, can be solved in real time.

Our approach decouples the problem by first computing feasible trips from a pairwise shareability graph (5) and then assigning trips to vehicles. We show that this assignment can be posed as an ILP of reduced dimensionality. The framework allows for flexibility in terms of prescribing constraints such as (but not limited to) maximum user waiting times and maximum additional delays due to sharing a ride. We also extend the method to proactively rebalance the vehicle fleet by moving idle vehicles to areas of high demand. In summary, we present a framework for solving the real-time ride-pooling problem with (*i*) arbitrary numbers of passengers and trips, (*ii*) anytime optimal rider allocation and routing dependent on the fleet location, and (*iii*) online rerouting and assignment of riders to existing trips.

We quantify experimentally the performance tradeoffs between fleet size, capacity, waiting time, travel delay, and operational costs for low- and medium-capacity vehicles (such as taxis, vans, or minibuses) in a large urban setting. Detailed experimental results are presented for a subset of ∼3 million rides extracted from the New York City taxicab public dataset. We show that 3,000 vehicles with a capacity of 2 and 4 could serve 94 and 98% of the demand with a mean waiting time of 3.2 and 2.7 min, and a mean delay of 1.5 and 2.3 min, respectively. To achieve 98% service rate, with comparable waiting time (2.8 min) and delay (3.5 min), a fleet of just 2,000 vehicles with a capacity of 10 was required. This fleet size is 15% of the active taxis in New York City (Movie S1). We also show that our approach is robust with respect to the density of requests and could therefore be applied to other cities.

Our system runs in real time and is particularly suited to autonomous vehicle fleets that can continuously reroute based on real-time requests. It can also rebalance idle vehicles to areas with high demand and is general enough to be applied to other multivehicle, multitask assignment problems.

## Passenger Assignment and Vehicle Routing

We consider a fleet

We propose an anytime optimal algorithm for batch assignment of a set of requests

Our formulation is flexible with respect to physical and performance-related constraints that might need to be added. In our implementation, we consider the following. (*i*) For each request *ii*) For each passenger or request *iii*) for each vehicle

We define the cost

This constrained optimization problem is solved via four steps (Fig. 1), which are: computing a pairwise request-vehicle shareability graph (RV-graph) (Fig. 1*B*); computing a graph of feasible trips and the vehicles that can serve them (RTV-graph) (Fig. 1*C*); solving an ILP to compute the best assignment of vehicles to trips (Fig. 1*D*); and rebalancing the remaining idle vehicles (Fig. 1*E*).

Given a network graph with travel times, we consider a function *Right* shows the optimal route for a vehicle with four passengers and an additional request.

The RV-graph (Fig. 1*B*) represents which requests and vehicles might be pairwise-shared and builds on the idea of shareability graphs proposed by ref. 5 but also includes the vehicles at their current state. Two requests

Next, the cliques of the RV-graph—or regions for which its induced subgraph is complete—are explored to find feasible trips and compute the RTV-graph (Fig. 1*C*). A trip

This step computes feasible trips. There might be several trips of varying size that can service a particular request. In addition, more than one vehicle might be able to service a trip. The assignment step will later ensure that each request and vehicle are assigned to a maximum of one trip. The RTV-graph contains two types of edges: (*i*) edges *ii*) edges *e*(*T*,*v*).

The algorithm to compute the feasible trips and edges proceeds incrementally in trip size for each vehicle, starting from the request-vehicle edges in the RV-graph (*SI Appendix*, Algorithm 1). For computational efficiency, we rely on the fact that a trip

Next, we compute the optimal assignment *SI Appendix*, Algorithm 2).

The optimization problem is formulated in Algorithm 1. A binary variable

The cost terms

Two types of constraints are included. Line 3 in Algorithm 1 imposes that each vehicle is assigned to one trip at most. Line 4 in Algorithm 1 imposes that each request is assigned to a single vehicle or ignored. In these constraints, three sets appear. The set of trips that can be serviced by a vehicle

This ILP is solved incrementally from the greedy assignment

This method is well suited for online execution to assign incoming requests *i*) new requests are added as they are received and (*ii*) requests are removed when they are either (*a*) picked up by a vehicle or (*b*) could not be successfully matched to any vehicle within the maximum waiting time (they are ignored).

Requests are collected during a time window (e.g., 30 s), after which they are assigned in batch to the different vehicles. If a request is matched to a vehicle at any given iteration, its latest pickup time is reduced to the expected pickup time by that vehicle and the cost

After the assignment, due to fleet imbalances, the set *i*) ignored requests may wait longer and request again, (*ii*) it is likely that more requests occur in the same area where all requests cannot be satisfied, and (*iii*) there are not enough requests in the neighborhood of the idle cars, we propose the following approach to rebalance the fleet by moving only the idle vehicles.

To rebalance the vehicle fleet, after each batch assignment, the vehicles in *SI Appendix*, Algorithm 4). In this approach, if all requests can be satisfied, some vehicles may remain idle, saving fuel and distance traveled, which is the case at nighttime.

### Complexity.

The number of variables in the ILP is equal to the number of edges

### Anytime Optimality.

This method guarantees optimality of the assignment of the currently active requests, while satisfying the constraints **1**, and future requests could be sampled from historical data. The method allows for parallelization in all steps. Proofs are provided in *SI Appendix*, *III*. *Theoretical Guarantees*.

## Results

We assess the performance of a MoD fleet controller using the proposed algorithm, against real data from an arbitrarily chosen representative week, from 0000 hours Sunday, May 5, 2013, to 2359 hours, Saturday May 11, 2013, from the publicly available dataset of taxi trips in Manhattan, New York City (23). This dataset contains for each day the time and location of all of the pickups and drop-offs executed by each of the 13,586 active taxis. From these data, we extract all of the requests (origin and destination within Manhattan) and consider the time of request equal to the time of pickup. We consider the complete road network of Manhattan (4,092 nodes and 9,453 edges), with the travel time on each edge (road segment) of the network given by the daily mean travel time estimate, computed using the method in ref. 5. Shortest paths and travel times between all nodes are then precomputed and stored in a lookup table.

We perform a simulation of the evolution of the taxi fleet, where vehicles are initialized at midnight at sampled positions from a historical demand distribution and continuously travel to pick up and drop off passengers to satisfy the real requests extracted from the dataset. Requests are collected during a time window, 30 s in our experiments, after which they are assigned in batch to the different vehicles. Past requests are kept in the requests pool until picked up and can be reassigned if a better match is found before pickup. Each day contains between 382,779 (Sunday) and 460,700 (Friday) requests, and the running pool of requests contains up to 2,000 requests at any given time. The method is robust both with respect to the chosen time window and the density of demands, as shown in *SI Appendix*, *VI. Robustness Analysis* in results with a time window between 10 and 50 s, and with half/double the amount of requests (

We analyze several metrics, with different vehicle fleet sizes (*SI Appendix*, Table II). Movie S1 shows the evolution of the taxi fleet in New York City for a subset of experiments.

High vehicle occupancy is achieved in times of high demand, with a large number of the trips being shared. In Fig. 2, we observe that many vehicles are located in mid-Manhattan and contain three or four passengers. Fig. 3 shows that the occupancy depends on the fleet size, capacity, and the maximum waiting/delay time. Lower fleet size, larger capacity and longer waiting/delay times increase the possibilities for ride-sharing and lead to higher mean vehicle occupancy. In Fig. 4, we observe that during peak hours, a small fleet of high-capacity vehicles does indeed operate at high occupancy. For a fleet of 1,000 vehicles of capacity 10, we observe that, during peak time (1800 hours) of a Friday, 10% of the vehicles have eight or more passengers, 40% of the vehicles have six or more, 80% have three or more, and 98% have at least one passenger. For a fleet of 2,000 vehicles of capacity four, we observe that, at the same peak time, over 70% of them have at least three passengers onboard.

We observe that the value of fleets with larger passenger capacities increases with larger *A*), a fleet of 1,000 vehicles with a capacity of 10 can satisfy almost 80% of the requests with

As expected, the in-car travel delay does increase with the increase in vehicle capacity (Fig. 5*B*). Nonetheless, that increase seems practically negligible—well below 100 s—once ride-sharing is allowed. Furthermore, the mean waiting time does in fact decrease as vehicle capacity is increased (Fig. 5*C*). For a fleet size of 1,000 vehicles and

We also observed that increasing the vehicle capacity not only increases the service rate but also reduces the mean distance traveled by the vehicles in the fleet (Fig. 5*D*), potentially leading to a reduction in costs, congestion, and pollution. We also observe that, with our online method, about 90% of the rides were shared. The number of shared rides slightly increases with *E*). Finally, we note that our approach is real-time capable (Fig. 5*F*). In our setup, for

## Conclusion

In this paper, we introduced a reactive anytime optimal method with scalable real-time performance for assigning passenger requests to a fleet of vehicles of varying capacity. We quantifyexperimentally the tradeoff between fleet size, capacity, waiting time, travel delay, and operational costs for low- and medium-capacity vehicles, such as taxis or vans in a large-scale city dataset. Under the assumption of one person per ride, we show that 98% of the taxi rides currently served by over 13,000 taxis could be served with just 3,000 taxis of capacity four. We observe that a vehicle capacity of two is sufficient for ride-sharing when a small trip delay of 2 min is imposed. If a maximum delay of 5 min or more (comparable to the time spent retrieving a car from parking) is allowed, higher-capacity vehicles (*i*) increase the service rate significantly, (*ii*) reduce the waiting time, and (*iii*) reduce the distance traveled by each vehicle. Our analysis shows that a ride-pooling service can provide a substantial improvement in urban transportation systems and that the system parameters such as vehicle capacity and fleet size depend on quality of service requirements and demand.

## Acknowledgments

We thank G. Resta, P. Santi, and C. Ratti for sharing the graph of Manhattan and the estimated travel times of ref. 5. This work was supported in part by the Office of Naval ResearchGrant N00014-12-1-1000 and the Massachusetts Institute of Technology–Singapore Alliance on Research and Technology under the Future of Urban Mobility.

## Footnotes

↵

^{1}Present address: Delft Center for Systems and Control, Delft Technical University, 2628 CD, Delft, Netherlands.- ↵
^{2}To whom correspondence should be addressed. Email: J.AlonsoMora{at}tudelft.nl.

Author contributions: J.A.-M., S.S., and D.R. designed research; J.A.-M., S.S., E.F., and D.R. performed research; J.A.-M. and A.W. contributed new reagents/analytic tools; J.A.-M., S.S., A.W., E.F., and D.R. analyzed data; and J.A.-M., S.S., A.W., E.F., and D.R. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1611675114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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