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On-demand high-capacity ride-sharing via dynamic
trip-vehicle assignment
Javier Alonso-Moraa,1,2, Samitha Samaranayakeb, Alex Wallara, Emilio Frazzolic, and Daniela Rusa
aComputer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139; bSchool of Civil and Environmental
Engineering, Cornell University, Ithaca, NY 14853; and cLaboratory for Information and Decision Systems, Massachusetts Institute of Technology,
Cambridge, MA 02139
Edited by Michael F. Goodchild, University of California, Santa Barbara, CA, and approved November 22, 2016 (received for review July 20, 2016)
Ride-sharing services are transforming urban mobility by provid-
ing timely and convenient transportation to anybody, anywhere,
and anytime. These services present enormous potential for pos-
itive societal impacts with respect to pollution, energy consump-
tion, 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 (opti-
mally) or three (with heuristics). We present a more general math-
ematical model for real-time high-capacity ride-sharing that (i)
scales to large numbers of passengers and trips and (ii) dynam-
ically generates optimal routes with respect to online demand
and vehicle locations. The algorithm starts from a greedy assign-
ment and improves it through a constrained optimization, quickly
returning solutions of good quality and converging to the opti-
mal assignment over time. We quantify experimentally the trade-
off 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 mil-
lion 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 frame-
work is general and can be used for many real-time multivehicle,
multitask assignment problems.
ride-sharing |human mobility |vehicle routing |smart cities |
intelligent transportation systems
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, con-
gestion, 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 gal-
lons of fuel burned. These estimates do not even consider the
cost of other potential negative externalities such as the vehic-
ular emissions (greenhouse gas emissions and particulate mat-
ter) (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 reli-
able 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 formula-
tions (7), case studies in specific regions [e.g., Singapore (8)], and
operational considerations for fleet managers (9). With the grow-
ing 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 chal-
lenge 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.
Significance
Ride-sharing services can provide not only a very personalized
mobility experience but also ensure efficiency and sustain-
ability 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.
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.
Freely available online through the PNAS open access option.
1Present address: Delft Center for Systems and Control, Delft Technical University, 2628
CD, Delft, Netherlands.
2To whom correspondence should be addressed. Email: J.AlonsoMora@tudelft.nl.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1611675114/-/DCSupplemental.
462–467 |PNAS |January 17, 2017 |vol. 114 |no. 3 www.pnas.org/cgi/doi/10.1073/pnas.1611675114
COMPUTER SCIENCES
SOCIAL SCIENCES
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 pas-
sengers 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 compu-
tational 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 guaran-
tees 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 feasi-
ble 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 flexi-
bility 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
ABCDE
Fig. 1. Schematic overview of the proposed method for batch assignment of multiple requests to multiple vehicles of capacity ν. The method consists of
several steps leading to an integer linear optimization that provides an anytime optimal assignment. (A) Example of a street network with four requests
(orange human, origin; red triangle, destination) and two vehicles (yellow car, origin; red triangle, destination of passenger). Vehicle 1 has one passenger,
and vehicle 2 is empty. (B) Pairwise shareability RV-graph of requests and vehicles. Cliques of this graph are potential trips. (C) RTV-graph of candidate trips
and vehicles which can execute them. A node (yellow triangle) is added for requests that cannot be satisfied. (D) Optimal assignment given by the solution
of the ILP, where vehicle 1 serves requests 2 and 3 and vehicle 2 serves requests 1 and 4. (E) Planned route for the two vehicles and their assigned requests.
In this case, no rebalancing step is required because all requests and vehicles are assigned.
routing dependent on the fleet location, and (iii) online rerout-
ing and assignment of riders to existing trips.
We quantify experimentally the performance tradeoffs
between fleet size, capacity, waiting time, travel delay, and opera-
tional costs for low- and medium-capacity vehicles (such as taxis,
vans, or minibuses) in a large urban setting. Detailed experimen-
tal 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 Vof mvehicles of capacity ν, the maxi-
mum number of passengers each vehicle can have at any given
time. We address the problems of both optimally assigning online
travel requests to vehicles and finding optimal routes for the vehi-
cle fleet. Each travel request consists of the time of request, a
pickup location and a drop-off location.
We propose an anytime optimal algorithm for batch assign-
ment of a set of requests R={r1,...,rn}to a set of vehicles
V={v1,...,vm}, which minimizes a cost function C, satisfies a
set of constraints Z, and allows for multiple passengers per vehi-
cle. A passenger is a past request that has been picked up by a
vehicle and that is now en route to its destination. We denote by
Pvthe set of passengers for vehicle v∈ V. In a second step, the
method also allows to rebalance the fleet of vehicles by driving
idle vehicles to areas of high demand, where those vehicles are
Alonso-Mora et al. PNAS |January 17, 2017 |vol. 114 |no. 3 |463
likely to be required in the future. A schema of the method is
shown in Fig. 1.
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 r, the waiting time ωr, given by the difference between
the pickup time tp
rand the request time tr
r, must be below a max-
imum waiting time Ω, for example, 2 min. (ii) For each passenger
or request rthe total travel delay δr=td
r−t∗
rmust be lower than
a maximum travel delay ∆, for example, 4 min, where td
ris the
drop-off time and t∗
r=tr
r+τ(or,dr)is the earliest possible time
at which the destination could be reached if the shortest path
between the origin orand the destination drwas followed with-
out any waiting time. The total travel delay δrincludes both the
in-vehicle delay and the waiting time. Finally, (iii) for each vehi-
cle v, we consider a maximum number of passengers, npass
v≤ν,
for example, capacity 10.
We define the cost Cof an assignment as the sum of delays δr
(which includes the waiting time) over all assigned requests and
passengers, plus a large constant cko for each unassigned request.
Given an assignment Σof requests to vehicles, we denote by Rok
the set of requests that have been assigned to some vehicle and
Rko the set of unassigned requests, due to the constraints or the
fleet size. Formally,
C(Σ) = X
v∈V
X
r∈Pv
δr+X
r∈Rok
δr+X
r∈Rko
cko .[1]
This constrained optimization problem is solved via four steps
(Fig. 1), which are: computing a pairwise request-vehicle share-
ability graph (RV-graph) (Fig. 1B); computing a graph of fea-
sible trips and the vehicles that can serve them (RTV-graph)
(Fig. 1C); solving an ILP to compute the best assignment of vehi-
cles to trips (Fig. 1D); and rebalancing the remaining idle vehi-
cles (Fig. 1E).
Given a network graph with travel times, we consider a func-
tion travel(v,Rv)for single-vehicle routing. For a vehicle v, with
passengers Pv, this function returns the optimal travel route σv
to satisfy requests Rv. This route minimizes the sum of delays
P
r∈Pv∪Rv
δrsubject to the constraints Z(waiting time, delay, and
capacity). For low-capacity vehicles, such as taxis, the optimal
path can be computed via an exhaustive search. For vehicles
with larger capacity, heuristic methods such as Lin–Kernighan
(20), Tabu search (21), or simulated annealing (22) may be used.
Fig. 2, Right shows the optimal route for a vehicle with four pas-
sengers and an additional request.
The RV-graph (Fig. 1B) represents which requests and vehi-
cles might be pairwise-shared and builds on the idea of share-
ability graphs proposed by ref. 5 but also includes the vehicles at
their current state. Two requests r1and r2are connected if an
empty virtual vehicle starting at the origin of one of them could
pick up and drop off both requests while satisfying the constraints
Z. A cost δr1+δr2is associated to each edge e(r1,r2). Likewise,
a request rand a vehicle vare connected if the request can be
served by the vehicle while satisfying the constraints Z, as given
by travel(v,r). The edge is denoted by e(r,v).
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. 1C). A trip T={r1,...,rnT}
is a set of nTrequests to be combined in one vehicle. A trip is
feasible if all of the requests can be picked up and dropped off
by some vehicle, while satisfying the constraints Z.
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 assign-
ment step will later ensure that each request and vehicle are
assigned to a maximum of one trip. The RTV-graph contains
Fig. 2. (A) Snapshot: 2,000 vehicles, capacity of 4 (Ω = 5 min, Wednesday,
2000 hours). Vehicle in the fleet are represented at their current positions.
Colors indicate number of passengers (0: light blue; 1: light green; 2: yellow;
3: dark orange; 4: dark red); 39 rebalancing vehicles are displayed in dark
blue—mostly in the upper Manhattan returning to the middle. (B) Close
view of the scheduled path for a vehicle (dark red circle) with four passen-
gers, which drops one off, picks up a new one (blue star), and drops all four.
Drop-off locations are displayed with inverted triangles. See Movie S1 for a
complete simulation.
two types of edges: (i) edges e(r,T), between a request rand
a trip Tthat contains request r(i.e., ∃e(r,T)⇔r∈T), and
(ii) edges e(T,v), between a trip Tand a vehicle vthat can exe-
cute the trip (i.e., ∃e(T,v)⇔travel(v,T)is feasible). The cost
P
r∈Pv∪T
δr, sum of delays, is associated to each edge e(T,v).
The algorithm to compute the feasible trips and edges pro-
ceeds 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 T
only needs to be checked for feasibility if there exists a vehicle v
for which all of its subtrips T0=T\r(obtained by removing
one request) are feasible and have been added as edges e(T0,v)
to the RTV-graph.
Next, we compute the optimal assignment Σoptim of vehicles
to trips. This optimization is formalized as an ILP, initialized
with a greedy assignment obtained directly from the RTV-graph.
To compute the greedy assignment Σgreedy , trips are assigned to
vehicles iteratively in decreasing size of the trip and increasing
cost (sum of travel delays). The idea is the maximize the amount
of requests served while minimizing the cost (SI Appendix,
Algorithm 2).
The optimization problem is formulated in Algorithm 1. A
binary variable i,j∈ {0,1}is introduced for each edge
e(Ti,vj)between a trip Ti∈ T and a vehicle vj∈ V
in the RTV-graph. If i,j= 1, then vehicle vjis assigned
to trip Ti. We denote by ETV the set of {i,j}indices for
which an edge e(Ti,vj)exists in the RTV-graph, i.e., the set
of possible pickup trips. An additional binary variable χk∈
{0,1}is introduced for each request rk∈ R. These vari-
ables are active, i.e., χk= 1, if the associated request rkcan
not be served by any vehicle and is ignored. The set of vari-
ables is then X={i,j, χk;∀e(Ti,vj)edge in RTV-graph and
∀rk∈ R}.
The cost terms ci,jare the sum of delays for trip Tiand vehicle
vjpickup (stored in the e(Ti,vj)edge of the RTV-graph) and
cko is a large constant to penalize ignored requests.
464 |www.pnas.org/cgi/doi/10.1073/pnas.1611675114 Alonso-Mora et al.
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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 sin-
gle vehicle or ignored. In these constraints, three sets appear.
The set of trips that can be serviced by a vehicle j, or edges
e(Ti,vj), is IT
V=j. The set of trips that contain request k, or
edges e(rk,Ti), is IT
R=k. The set of vehicles that can service trip
i, or edges e(Ti,vj), is IV
T=i.
This ILP is solved incrementally from the greedy assignment
Σgreedy , improving the quality of the assignment over time.
Algorithm 1. Optimal assignment
1: Initial guess: Σgreedy
2: Σoptim := arg min
XP
i,j∈ETV
ci,ji,j+P
k∈{1,...,n}
ckoχk
3: s.t. P
i∈IT
V=j
i,j≤1∀vj∈ V
4: P
i∈IT
R=k
P
j∈IV
T=i
i,j+χk=1∀rk∈ R
This method is well suited for online execution to assign
incoming requests r(t)to a fleet of vehicles for which a pool of
requests Ris maintained where (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 vehi-
cle and the cost χko of ignoring it is increased for subsequent
iterations. A request might be rematched to a different vehicle in
subsequent iterations as long as its waiting time does not increase
and until it is picked up by some vehicle. Once a request is picked
up, it remains in that vehicle and cannot be rematched—the vehi-
cle may still pick additional passengers. In each iteration, the new
assignment of requests to vehicles guarantees that the current
passengers are dropped off within the maximum delay constraint.
After the assignment, due to fleet imbalances, the set Rko of
unassigned requests may not be empty, and some empty vehicles
Vidle may still by unassigned to any request. These imbalances
may occur when the idle vehicles are in areas far away from the
area of current requests and due to the maximum waiting time
and delay constraints and vehicle capacity. Under the assump-
tions that (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 Vidle are assigned to requests in Rko to minimize the
sum of travel times, with the constraint that either all requests
or all of the vehicles are assigned. We first compute the travel
time of each individual idle vehicle in Vidle to pick each ignored
request in Rko and then obtain the optimal assignment via a lin-
ear program (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 e(T,v)in the RTV-graph plus the number of
requests. In the worst case, the number of variables is of order
O(mnν)but only reached with complete RV- and RTV-graphs,
where all vehicles can serve all requests and all requests can be
combined with each other. In practice, the number of variables is
orders of magnitudes lower and related to the size of the cliques
in the RV-graph. The number of constraints is n+m.
Anytime Optimality. This method guarantees optimality of the
assignment of the currently active requests, while satisfying the
constraints Z, if all of the steps are executed until termination
and exploration of all possible trips and assignments. In prac-
tice, timeouts can be set both for the amount of time spent gen-
erating candidate trips for each vehicle and for the time spent
exploring the branches of the ILP. A limit on the number of vehi-
cles considered per request, the number of trips per vehicle, or
the optimality gap of the ILP can also be set. These timeouts
trade optimality for tractability, and their values will depend on
the available resources. We note that the method is reactive, in
the sense that it provides anytime-optimality guarantees given
the current state of the system and the current requests. To
inform the assignment and routing about future demand, an
additional cost term could be added to Eq. 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 pub-
licly available dataset of taxi trips in Manhattan, New York
City (23). This dataset contains for each day the time and loca-
tion 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 con-
sider the time of request equal to the time of pickup. We con-
sider 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
Fig. 3. Mean number of passengers per vehicle for four different vehicle
types (capacity one, two, four, and ten). We show four one-week time series
for different fleet sizes and maximum waiting time: (A) 1000 vehicles and
Ω = 2 min; (B) 1000 vehicles and Ω = 7 min; (C) 3000 vehicles and Ω =
2 min; and (D) 3000 vehicles and Ω = 7 min. At night, most vehicles wait,
and during rush hour, the mean occupancy decreases as the fleet gets larger.
Larger maximum waiting time enables more opportunities for ride-sharing.
Alonso-Mora et al. PNAS |January 17, 2017 |vol. 114 |no. 3 |465
Fig. 4. Percentage of vehicles in each state (waiting, rebalancing, and number of passengers) for a representative day (Friday 0000 hours to 2400 hours).
(A) A fleet of 1,000 vehicles of capacity 10 with many opportunities for ride-sharing in high-capacity vehicles. (B) A fleet of 2,000 vehicles of capacity
four, showing the utility of full vehicle-sharing. Additional figures, for varying days and parameters, are in SI Appendix,VIII. Additional Experimental
Figures.
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 (∼220,000/∼880,000 per day) in New York
City.
We analyze several metrics, with different vehicle fleet sizes
(m∈ {1,000, 2,000, 3,000}vehicles), vehicle capacities (χ∈ {1, 2,
4, 10}passengers), and maximum waiting times (Ω∈ {120, 300,
420}s). The maximum trip delay ∆is double the maximum wait-
ing time and includes both the waiting time ωand the inside-
the-vehicle travel delay. Our analysis shows that, thanks to high-
capacity ride-sharing, a reduced fleet of vehicles (below 25%
of the active taxis in New York City) is able to satisfy 99%
of the requests, with a mean waiting time and delay of about
2.5 min. All results in this section include rebalancing of idle
vehicles to unassigned requests; experimentally, we observed
that the rebalancing step contributed an increase in the service
rate of about 20% (SI Appendix, Table II). Movie S1 shows the
evolution of the taxi fleet in New York City for a subset of
experiments.
Fig. 5. Comparison of several performance metrics for varying vehicle capacity (1, 2, 4, and 10 passenger, shown with lines). Each subplot is for a fleet size of
1,000, 2,000, and 3,000 vehicles, and the coordinate axes show increasing maximum waiting time Ωof 2, 5, and 7 min. We analyze service rate (percentage
of requests serviced) (A), average in car delay δ−ω(B), average waiting time ω(C), average distance traveled by each vehicle during a single day (D),
percentage of shared rides (number of passengers who shared a ride divided by the total number of picked-up passengers) (E), and average computational
time for a 30-s iteration of the method (F), in a 24 core 2.5 GHz machine, including computation of the RV-graph, computation of the RTV-graph, ILP
assignment, rebalancing, and data writing (higher levels of parallelization would drastically reduce this computational time). The parameters used in the
simulation are specified in SI Appendix,III.Theoretical Guarantees,C. Heuristics for Real-Time Execution.
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 occu-
pancy depends on the fleet size, capacity, and the maximum wait-
ing/delay time. Lower fleet size, larger capacity and longer wait-
ing/delay times increase the possibilities for ride-sharing and lead
to higher mean vehicle occupancy. In Fig. 4, we observe that dur-
ing peak hours, a small fleet of high-capacity vehicles does indeed
operate at high occupancy. For a fleet of 1,000 vehicles of capac-
ity 10, we observe that, during peak time (1800 hours) of a Fri-
day, 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 Ωand ∆values, as expected,
because passengers are willing to incur a larger personal time
penalty. High-capacity vehicles are also more important when
the fleet size is smaller, because seating capacity might be a
bottleneck with smaller fleets. For instance (Fig. 5A), a fleet of
1,000 vehicles with a capacity of 10 can satisfy almost 80% of
the requests with Ω = 420 s, compared with below 30% for a
single-rider taxi, for a net gain of over 50%. However, with a
larger fleet of 3,000 vehicles and Ω = 120 s, the benefit is only
about 15%. Interestingly, if longer waiting times and delays are
466 |www.pnas.org/cgi/doi/10.1073/pnas.1611675114 Alonso-Mora et al.
COMPUTER SCIENCES
SOCIAL SCIENCES
allowed, Ω = 420 s, a fleet of 3,000 vehicles with a capacity of 2,
4, and 10 could serve 94, 98, and 99% of the demand. To achieve
98% service rate, a fleet of just 2,000 vehicles with a capacity of
10 was required, which represents a reduction of the fleet size to
15% of the active taxi fleet in New York City.
As expected, the in-car travel delay does increase with
the increase in vehicle capacity (Fig. 5B). 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. 5C).
For a fleet size of 1,000 vehicles and ∆ = 420 s, high-capacity
vehicles not only improved the service rate but also achieved
a reduction in mean waiting time of over 100 s, which partially
offsets the increased in-car delay. In particular, we observe 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.
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. 5D), potentially lead-
ing 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
∆and decreases with the fleet size (Fig. 5E). Finally, we note
that our approach is real-time capable (Fig. 5F). In our setup,
for Ω≤300 s, the method is executed in less that 30 s, which is
the period for which requests are collected.
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 quantify
experimentally 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 park-
ing) 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 ser-
vice 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 Research Grant N00014-12-1-
1000 and the Massachusetts Institute of Technology–Singapore Alliance on
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