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Modular Transit: Using Autonomy and Modularity
to Improve Performance in Public Transportation
Zhenhao Zhang1, Amirmahdi Tafreshian2, Neda Masoud2
1Industrial and Operations Engineering, University of Michigan, Ann Arbor
2Civil and Environmental Engineering, University of Michigan, Ann Arbor
Corresponding Author – Email: nmasoud@umich.edu
Abstract
In this paper we investigate a new form of automated public transportation, named ‘modular
transit’, configured to overcome the shortcomings of the traditional bus, including the first-
and last-mile problem, low occupancy, and low levels of comfort, accessibility, and flexibility.
The modular transit system consists of a set of trailer modules who can travel locally to serve
demand and to connect travelers to main modules for long-distance trips. We mathematically
model this system on a time-expanded network, thereby reducing the size of the optimization
problem and rendering the problem amenable to being solved with commercial optimization
engines. We conduct extensive numerical experiments and sensitivity analyses to study
the performance of modular buses under various configurations. Finally, we compare the
modular transit service with a door-to-door shuttle service as benchmark to showcase the
benefits of modular transit.
Keywords: Modular transit, First- and last-mile problem, Public transportation,
Automated bus.
1. Introduction
Traffic congestion has become a serious problem in metropolitan areas in recent years,
mainly due to the rise in travel demand that is not met with the same level of increase in
transportation network capacity. A demand-focused solution to relieve traffic congestion,
especially in populated urban areas, is to reduce the number of vehicles using the
transportation network through offering public transportation, including buses, railways
and subway systems. According to the statistics reported by the United States Bureau of
Transportation Statistics (2018), the number of buses in the United States has been growing
in recent years, reaching over 970,000 buses in 2016—a 10% increase compared to 2015.
Despite providing affordable mobility, traditional bus services suffer from a few drawbacks
that negatively affect the transportation infrastructure as well as the quality of service they
offer to travelers. First, buses typically travel with empty seats, specially during the non-
peak hours. The empty seats traveling in the transportation network contribute to the
pre-mature aging of the infrastructure. Another challenge faced by public transportation
systems that leads to their lower modal share is the so-called first- and last-mile problem.
The first- and last-mile problem refers to lack of accessibility to bus stations (Wang and
Preprint submitted to Transportation Research Part E
Table 1: Difference between modular transit and conventional public transportation system
Modular transit Conventional public transportation
Service rate high Wide range (low/medium/high)
Occupancy high Wide range (low/medium/high)
First- and last-mile service 3 5
Transfer time 5 3
Driver-less 3 5
Odoni,2016), and serves as a deterrent to potential transit users. Waiting times and the
burden associated with making transfers in public transportation is another challenge that
has suppressed its modal share. Transfer waiting times at multi-modal stations have been
shown to be affected by the capacities and headways of the connecting and feeder services
(Hsu (2010), highlighting the importance of employing a door-to-door, system view when
planning public transportation systems. Finally, transit travel times can be much higher
than shortest-path travel times, contributing to transit’s lower level-of-service (LoS).
In this paper, we introduce a new from of public transportation system, named ‘modular
transit’, to overcome the shortcomings of the traditional bus. A modular bus is composed of
a main module, which is larger in size and can travel at higher speeds, and multiple trailer
modules, which are smaller and more fuel efficient. Similar to a traditional bus that has
a high number of seats, a modular bus can transport large numbers of passengers when
needed. Unlike the traditional bus, the modular structure of the modular bus allows for
‘trailer’ modules to travel locally flexibly and address the first- and last-mile trips. Due to
their lower speed and the fact that they cover local demand, trailer modules can be driver-
less (Zellner1 et al.,2016,Moorthy et al.,2017), cutting the operational cost of the system.
The main differences between modular transit and conventional public transportation are
presented in Table 1. Note that the appeal of automated trailers is not merely a function
of their automation peruse, but is of the high frequency and low cost with which they can
serve passengers (Levine et al.,2018). When traveling on major roads with higher speeds,
‘main’ modules drag trailer modules, resembling a traditional bus; however, the size of this
bus is dependent on the level of demand.
The contributions of this paper are as follows. First, we mathematically model the
modular transit system on a time-expanded network. Modeling the problem on a time-
expanded network would allow us to use a subset of constraint sets to trim the feasible region,
thereby rending the problem amenable to be solved by commercial optimization engines
(Masoud and Jayakrishnan,2017c). Moreover, we conduct extensive numerical experiments
to showcase the performance and flexibility of modular transit, present the trade-off in
its performance under different objective functions, and compare the performance of the
modular bus with that of a traditional shuttle service.
The rest of the paper is organized as follows. In Section 2, we provide a literature review
on several alternative door-to-door transportation services, and explain how the modular
transit system can be distinguished from such services. Next, we formally lay out the problem
statement and modeling assumptions in Section 3, followed by the mathematical modeling of
the modular transit system in Section 4. In Section 5, we conduct numerical experiments on
the Sioux Fall network, where we quantify the sensitivity of various individual- and system-
2
level performance measures and LoS metrics with respect to changes in the number of trip
requests, and the number of trailer and main modules. Further on in this section, we apply
our proposed methodology to the New York taxi dataset to show the benefits of using the
modular transit systems in a large-scale transportation network. We finalize this paper by
providing the conclusions and some directions for future research in Section 6.
2. Literature Review
The traditional public transit system is characterized by fixed routes and schedules,
predetermined stops, and large capacity that enables pooling multiple passenger trips into
a single vehicle (Quadrifoglio and Li,2008,Abdolmaleki et al.,2020). A fixed-route transit
system works well under high levels of travel demand. However, it can be ineffective to serve
low demand rates due to its rigid route and schedule structure (Cort´es et al.,2005,Zheng
et al.,2018).
Door-to-door transportation services, including dial-a-ride systems (Cordeau and
Laporte,2007), paratransit services (Kirby et al.,1974), carsharing (Litman,2000,Regue
et al.,2016), and ridesourcing and ridesharing services (Masoud and Jayakrishnan,2017b,c,
Lloret-Batlle et al.,2017,Masoud and Lloret-Batlle,2016) are some of the alternative
Transportation options that can replace transit to make public transportation more flexible.
However, they are not affordable to a large portion of the population, especially to
demographics that depend on public transportation for their daily travel needs.
A number of hybrid solutions have been discussed in the transportation literature and
implemented in practice to combine the affordability, efficiency, and congestion-reliving
effects of transit with comfort and accessibility of door-to-door transportation (Malucelli
et al.,1999). Park-and-ride is one such solution, in which travelers drive their private cars to
park-and-ride locations, where they transfer to public transportation vehicles (Noel,1988).
Park-and-Ride solutions have been shown to overcome motorists’ prejudices against public
transport (Cairns,1998), and are good solutions if the target transit users own private
vehicles.
Bike-sharing is another initiative to improve access to transit (DeMaio,2009,Qin et al.,
2018). The bike-sharing market has witnessed a remarkable growth in recent years, with
2.3 million shared bikes available worldwide in 2016 (Bernard,2018). Bike-sharing provides
flexible access to well-maintained bikes at a relatively low price, and relieves users from
having to bear the cost of purchasing, storing, and maintaining a personal bike as well
as the burden of carrying bikes on transit vehicles. However, bike-sharing is not an all-
season solution for many regions in the United States, and its appeal is limited to narrow
demographics. Additionally, ease and safety of biking is highly dependent on urban design.
The first- and last-mile problem can be addressed by conventional feeder systems, such
as local buses that connect passengers to nearby rapid transit. However, the high operating
cost, largely attributed to driver compensation, tends to limit the frequency of operating
such systems (Burns et al.,2013). Another hybrid solution consists of using door-to-door
services to complement transit, by fulfilling the first and last mile of the trip (Cheng et al.,
2014,Shaheen and Chan,2016,Ma et al.,2019,Nam et al.,2018,Masoud et al.,2017).
This solution makes transit accessible to a larger population, and although it can be more
affordable than seeking door-to-door private transportation, it may still fall outside the
3
Figure 1: A modular vehicle. Passengers can transfer between modules en-route. (The image is taken from
the Next Future Transportation inc. website.)
financial reach of all socioeconomic groups. Therefore, when ridesharing or ridesourcing
companies are formally sought out by transit agencies to offer such services, their fares are
typically subsidized by transit agencies, making this solution limited in scope.
The idea of using autonomous vehicles to replace low-demand bus routes (Shen et al.,
2018) and feed public Transportation (Chong et al.,2011) has been previously explored in the
literature. There is supporting evidence in the literature that shared autonomous vehicles
can constitute a sustainable solution to the first- and last-mile problem (Moorthy et al.,2017,
Masoud and Jayakrishnan,2017a,2016). The modular transit system is an extension of these
concepts: (i) modular transit has a system view when routing and scheduling vehicles; that
is, all modules, main and trailer, are scheduled simultaneously. Unlike many feeder systems,
trailers do not merely play a supporting role to the main modules, but instead form the
backbone of the system; and (ii) unlike other feeder systems, the modular transit system
eliminates the need for a transfer for the last mile of the trip, and the uncertainly and wait
time associated with it.
In addition to academic studies exploring the idea of incorporating autonomous vehicles
into public transportation, such systems have been deployed in practice. The (Next Future
Transportation inc.) has presented and tested this type of modular vehicle in Dubai, United
Emirates. Each module can join and detach with others to serve riders, and riders can walk
from one module to another safely and flexibly while traveling. Figure 1shows the transfer
between modules and the functionality of modular vehicles. Modular vehicles can not only
provide door-to-door services, but also lower traffic congestion on the main roads by pooling
riders with similar itineraries in the same modules. The concept of modular vehicles in the
(Next Future Transportation inc.) has inspired further studies on the modular transit system
in recent years. Chen et al. (2019,2020), for instance, show that the modular transit system
has the potential to reduce the energy cost and passenger’s waiting time due to the capability
of having a time-varying capacity. As another example, Guo et al. (2017) show that such
systems provides an opportunity to switch between fixed and flexible transit services. In
addition, the Dynamic Autonomous Road Transit (DART) proposed by Rau et al. (2019)
also uses the concept of autonomous modular vehicles. In DART, platoons are formed by
varying the number of modular vehicles, which results in having flexible capacities to serve
changing demand.
The modular transit system can be considered as a form of flexible transit. Flexible
transit is a general term that captures a wide spectrum of public transportation services
that many vary in flexibility of their routes and schedules. Fixed-route and schedule transit
4
services and on-demand dial-a-ride systems are two extremes of a flexible transit system.
In practice, a flexible transit system could combine fixed-route and on-demand systems,
and services in between, to best serve the transportation needs of a community. Modular
transit can be viewed as an automated version of a flexible transit system, where modules
are routed in real-time in response to demand, and transfers are made seamless (Errico et al.,
2013,Zheng et al.,2018,Quadrifoglio and Li,2008,Nourbakhsh and Ouyang,2012).
3. Problem Statement
Consider a transportation network modeled as a directed graph G= (S,L), where Sis the
set of stations in the network (where trips can start and end), and Lis the set of links
that connect the stations. We divide the station set Sinto two mutually-exclusive and
collectively-exhaustive sets, namely the set of major stations, denoted by Sm, and the set of
minor stations, denoted by Ss. This classification of stations is based on the characteristics
of the links (i.e., roadway networks) that connect the stations—links incident to at least one
minor station correspond to minor streets with lower speed limits, and are referred to as
minor links, denoted by Ls. Links with major stations at both ends represent the major
streets, and are referred to as major links, denoted by Lm.
Consider a group of riders who wish to complete a set of trips in this network, within
specified time windows. We implement a modular transit system, consisting of a set of main
modules and a set of trailer modules to satisfy the ride requests, where main and trailer
modules can only individually travel on major and minor links, respectively. In addition to
minor links, trailer modules can also travel on major links when accompanied by a main
module. In this paper we seek to develop a mathematical model of the modular transit
system, and use this model to compare the individual- and system-level performance of
modular transit with traditional door-to-door shuttle services. The set of assumptions made
to model the modular transit system are summarized below:
•The modular transit system operates during a time horizon, e.g., a morning/evening
peak hour, that is discretized into short time intervals, e.g., one minute.
•Each rider announces the following information regarding their trip to the system:
earliest departure time, latest arrival time, and origin and destination stations.
•All modules start their trips at time 0 from pre-specified station(s), and finish their
trips at the end of the time horizon at the same or different pre-specified stations.
•The main and trailer modules can only individually travel on major and minor links,
respectively.
•The travel times of main and trailer modules on each link are treated as constants.
•The trailer modules can travel locally through minor links to provide door-to-door
service. However, to travel on major links, a trailer module must be accompanied with
a main module.
•Riders can travel onboard of trailer modules.
•Riders are allowed to transfer between trailer modules to complete their trips. This
transfer can take place when modules are connected, traveling on a major link.
Discretizing both time and space into time intervals and stations, respectively, enables
us to locate an agent (a rider or a module) in the modular transit system using a time-
expanded link. Let us assume a time horizon of length T(in min) that is divided into small
5
Figure 2: An example of the time-expanded network. The left hand side is a road network with 3 stations
A, B and C, where the number on a link denotes its travel time. The right hand side figure shows the
corresponding time-expanded network with 4 time intervals.
time intervals. In this study, we set the length of each time interval to one minute. Thus,
we denote the set of time intervals by T={1,2, ..., T }. The corresponding time-expanded
network is a directed acyclic graph (DAG) G= (N, L) that can be constructed from graph
Gand the set T. Every node n= (s, t)∈Nis a tuple whose first element represents
the station s∈ S, and whose second element denotes the time interval t∈ T . A link
l= (ni, nj)=(si, ti, sj, tj) exists in Lif (si, sj)∈ L and tj−tiis equal to the travel time
from station sito sj. Figure 2demonstrates an example of the time-expanded network with
4 time intervals (on the right) for a road network with 3 stations (on the left).
4. Mathematical Formulations
Let us define a set of agents A, consisting of the set of riders R, the set of trailer modules
P, and the set of main modules M(A=R∪P∪M). Every agent has their own origin and
destination station, denoted by oaand da, respectively. For riders, the origin and destination
stations are the origins and destinations of their trips, while for the main and trailer modules
these stations denote the modular transit system depots. An agent a∈Aspecifies a time
window, [Eda, LAa], within which the agent is present in the network. For a rider, this
time window denotes their earliest departure time from their origin station and their latest
desired arrival time at their destination station. For the main and trailer modules, this time
window encompasses the planning horizon T. A main module has a capacity of Cm, which
is the highest number of trailer modules it can have attached to it. A trailer module has a
capacity of Cp, which indicates its number of available seats.
To model the modular transit system, we construct a time-expanded network G= (N, L)
as described in the previous section. Every agent a∈Ahas a link set La∈L, which contains
all the feasible links on which the agent can travel without violating any of his/her trip’s
parameters (e.g., time window). While the time-expanded link sets for the main and trailer
modules are constructed based on the original link sets Lmand Ls, respectively, the rider
agents’ link sets could include both types of links. Modeling this problem on a time-expanded
network allows us to limit the size of the link sets accessible to each agent, thereby reducing
the size of the optimization problem and rendering the problem amenable to being solved with
6
commercial optimization engines (Masoud and Jayakrishnan,2017c). Table 2summarizes
the notation.
The link sets Lmp =Lm∩Lpand Lrp =Lr∩Lpdefine the set of common time-expanded
links between main module mand trailer module p, and between trailer module pand rider
r, respectively. We provide an example of how these link sets can be generated based on
the time-expanded network from Figure 2. Suppose we set A and B as main stations and
C as a minor station. Let us assume the main module mstarts from station B at time 1
and ends at station A at time 4. Let the trailer module pstart from station B at time 1
and end at station C at time 4. Finally, let the trip request from rider rstart at station
B and end at station C, within the time window [2,4]. We can generate the link set
Lm={(B,1,B,2),(B,2,B,3),(A,2,A,3),(A,3,A,4),(B,1,A,2),(B,2,A,3),(B,3,A,4)}.
Similarly, we can generate the link set Lp={(B,1,C,3),(C,3,C,4),(B,1,B,2),(B,2,C,4),
(B,1,A,2),(A,2,C,3),(A,2,A,3),(A,3,C,4),(B,2,A,3)}. Finally, the list set Lr={
(B,2,A,3),(A,3,C,4),(B,2,C,4),}. We can form the link set Lmp =Lm∩Lp=
{(B,1,B,2),(A,2,A,3),(B,1,A,2),(B,2,A,3)}, and the link set Lrp =Lr∩Lp=
{(B,2,A,3),(A,3,C,4),(B,2,C,4)}.
The modular transit system can be mathematically modeled using the following decision
variables:
xm
l=(1 main module mtravels on link l
0 otherwise (1)
ymp
l=(1 trailer module ptravels on link lwith main module m
0 otherwise (2)
zrp
l=(1 rider rtravels on link lwith trailer module p
0 otherwise (3)
ur=(1 rider ris matched (i.e., served)
0 otherwise (4)
vp=(1 trailer module pis matched (i.e., used in the system)
0 otherwise (5)
Model (6) describes the modular transit system. The objective function in Eq. (6a)
maximizes the number of served rider requests. Equations (6b)-(6d) route the main modules
in the network. Constraint sets (6b) and (6c) direct the main modules out of their
origin depots and into their destination depots (which may be different for each module),
respectively, and the constraint set (6d) enforces the flow conservation for stations other
than the depots.
Analogous to equations (6b)-(6d), equations (6e)-(6g) determine the itineraries of the
trailer modules. Eq. (6e) ensures that a used trailer module (i.e., vp= 1) leaves its origin
depot, and Eq. (6f) ensures that this module returns to its depot before the end of the
planning horizon. Eq. (6g) is the flow conservation constraint. Note that the decision
variable ymp
lby definition requires a trailer module to always be accompanied by a main
module, whereas in practice a trailer modules needs this company only when traveling on
7
Table 2: Table of notations
Notation Definition
SmSet of major stations
SsSet of minor stations
SSet of all stations: S=Sm∪Ss
LmMajor links with major stations at both ends
LsMinor links with a minor station on at least one end
RSet of riders
rIndex to refer to a rider, r∈R
PSet of trailer modules
P0Set of trailer modules including the dummy trailer module
pIndex to refer to a trailer module, p∈P
p0Dummy trailer module, p0∈P0
MSet of main modules
M0Set of main modules including the dummy modules
mIndex to refer to a main module, m∈M
m0, m00 Dummy main modules, m0, m00 ∈M0
ASet of agents: A=R∪P∪M
oaOrigin station of agent a∈A
daDestination station of agent a∈A
EdaEarliest time agent a∈Ais available in the network
LAaLatest time agent a∈Ais available in the network
LaSet of time-expanded links for agent a∈A
TaTravel time window of agent a∈A,Ta= [Eda, LAa]
Laa0Set of links in common for two agents aand a0,Laa0=La∩La0
CmCapacity of a main module
CpCapacity of a trailer module
M∗Maximum number of served ride requests
γlThe cost of traveling on link l(e.g., the length of the link in miles)
ξMin fraction of M∗to be served
a major link. Therefore, we introduce a dummy main module, m0, to travel with trailer
modules on minor links. The set of links for this dummy main module can be defined as
Lm0={(si, ti, sj, tj) : sior sj/∈Sm}. Additionally, trailer modules may have to wait at a
major station before joining a main module. To incorporate this into model (6), we define a
second dummy main module, denoted by m00 , with the link set Lm00 ={(s, t, s, t+1),∀(s, t)∈
Sm× T }. We formally include these two dummy modules into the set of main modules by
introducing the new set M0=M∪m0∪m00 .
Equations (6h)-(6j) route riders, where constraint sets (6h) and (6i) ensure that a served
rider will leave their origin station and arrive at their destination station within their specified
time window, and the constraint set (6j) balances the flow of the rider over the rest of the
stations. Riders’ itineraries are determined by the decision variables zrp
l, which by definition
require that a rider must always be accompanied by a trailer module. Similar to the case of
8
main modules, a dummy trailer module p0is needed for modeling purposes to capture the
waiting of riders at stations. We form the set P0=P∪p0to formally include the dummy
trailer module into the set of trailer modules, and define the link set for the dummy trailer
module as Lp0={(s, t, s, t + 1),∀(s, t)∈ S × T }.
M∗= Max X
r∈R
ur(6a)
Subject to : X
l∈Lm:si=om
xm
l= 1 ∀m∈M(6b)
X
l∈Lm:sj=dm
xm
l= 1 ∀m∈M(6c)
X
si,ti
l=(si,ti,s,t)∈Lm
xm
l=X
sj,tj
l=(s,t,sj,tj)∈Lm
xm
l
∀m∈M
∀t∈Tm
∀s∈Sm\ {om∪dm}
(6d)
X
m∈M0X
l∈Lmp:
si=om
ymp
l−X
m∈M0X
l∈Lmp:
sj=om
ymp
l=vp∀p∈P(6e)
X
m∈M0X
l∈Lmp:
sj=dm
ymp
l=vp∀p∈P(6f)
X
m∈M0X
si,ti
l=(si,ti,s,t)∈Lmp
ymp
l=X
m∈M0X
sj,tj
l=(s,t,sj,tj)∈Lmp
ymp
l
∀p∈P
∀t∈Tp
∀s∈Sp\ {op∪dp}
(6g)
X
p∈P0X
l∈Lrp:
si=or
zrp
l−X
p∈P0X
l∈Lrp:
sj=or
zrp
l=ur∀r∈R(6h)
X
p∈P0X
l∈Lrp:
sj=dr
zrp
l=ur∀r∈R(6i)
X
p∈P0X
si,ti
l=(si,ti,s,t)∈Lrp
zrp
l=X
p∈P0X
sj,tj
l=(s,t,sj,tj)∈Lrp
zrp
l
∀r∈R
∀t∈Tr
∀s∈Sr\ {or∪dr}
(6j)
X
r∈R:
l∈Lrp
zrp
l≤Cpymp
l
∀m∈M0
∀p∈P0
∀l∈Lp
(6k)
X
p∈P:
l∈Lmp
ymp
l≤Cmxm
l
∀m∈M0
∀l∈Lm
(6l)
Constraints (6k) and (6l) enforce consistency between itineraries of main modules, trailer
9
modules, and riders, and ensure that the trailer and main modules’ capacities are not
exceeded. Specifically, constraint (6k) ensures that if a rider is to travel on any time-expanded
link by trailer p, the trailer should also travel on that link. Additionally, this constraint set
ensures that no more than Cppassengers are on board of a trailer module on any time-
expanded link. Similarly, constraint (6l) ensures that a trailer is always accompanied by
a main module on a major link, or a dummy main module on a minor link, and that the
capacity of a main module is respected on all time-expanded links. We set the capacity of
dummy main modules to the number of trailers.
The objective of model (6) is to maximize the number of served trip requests. However,
a transit system may have secondary objectives, such as minimizing the total vehicles-miles-
traveled (VMT) in the network. It is easy to expand model (6) to capture such secondary
objectives, as described in model (7). The objective of model (7) is to minimize the total
VMT, where γlis the length of link l. Note that in Eq. (7a), M0captures both major
links traversed by main and trailer modules and minor links traversed by trailer modules.
The constraint sets in model (7) are similar to those in model (6), except for an additional
constraint (7c), which sets a lower bound on the number of served ride requests. In this
equation, M∗is the maximum number of served ride requests obtained by solving the
optimization problem (6), and ξis a parameter between 0 and 1 that specifies the minimum
fraction of the maximum number of served requests that must to be served in model (7).
Min X
m∈M0
p∈PX
l∈Lmp
γlymp
l(7a)
Subject to : (6b)−(6l) (7b)
X
r∈R
ur≥ξM∗(7c)
The integer programs presented in (6) and (7) are special cases of the general pick-up
and delivery problem introduced by Savelsbergh and Sol (1995), which is NP-hard. As such,
one can use conventional optimization packages to optimally solve small- to medium-seized
instances of this problem. For solving large-scale instances of this problem, however, we
adopt a graph partitioning method from the literature to provide a fast and high-quality,
although not necessarily optimal, solutions. We elaborate on this heuristic method in the
next section.
5. Numerical Experiments
In this section, we conduct a number of numerical experiments to showcase the performance
of the proposed system under two different case studies. In the first case study, we consider
a small part of the well-known Sioux Falls transportation network, and present an extensive
sensitivity analysis on different parameters. The second case study is conducted using
the NYC taxi dataset, where we deploy the modular transit system in the large-scale
transportation network of the Manhattan area. Before presenting the case studies, we define
a number of performance metrics in the next subsection to evaluate the performance of our
system.
10
For all experiments, the optimization problems are solved on a Macbook Pro with Core
i5 3.10GHz and 8 GB of RAM, using the AMPL modeling language and the CPLEX 12.7.0.0
solver with standard tuning. Finally, we used Python 3.7 for preparing the datasets and
analyzing the results.
5.1. System- and Individual-Level Performance Metrics
To evaluate the performance of the modular transit system, we define a set of system-level
performance metrics, namely the solution time, the rider matching rate, the percentage of
used trailer modules, the average empty miles driven, and the average module occupancy.
Furthermore, we define a set of individual-level quality-of-service metrics, namely the average
rider waiting time, and the detour time. In what follows, we elaborate on how a number of
these metrics are computed.
The rider matching rate is the percentage of served rider requests, which is calculated as
follows:
Rider matching rate = P
r∈R
ur
|R|.
The average occupancy of the trailer module pis defined as the total number of rider-miles
covered by p, divided by the total VMT by p:
¯
Op=P
r∈RP
l∈Lrp
γlzrp
l
P
m∈M0P
l∈Lmp
γlymp
l
The average occupancy of the main module mis defined as the ratio of the total trailer
module-miles carried by mto the total VMT by m:
¯
Om=P
p∈PP
l∈Lmp
γlymp
l
P
l∈Lm
γlxm
l
The detour time for ride ris defined as:
Tr
detour =X
p∈PX
l∈Lrp
κlzrp
l−g(or, dr)
where κldenotes the travel time on link l, and g(or, dr) is the shortest-path travel time from
station orto station dr. Finally, the rider average wait time reports the wait time for the
rider’s trailer module to be picked up by a main module for the first time.
5.2. Modified Sioux Falls Case Study
In this section, we implement the modular transit system on a small part of the Sioux Fall
network, displayed in Figure 3. In this network, we identify four major stations, denoted
by H, and five minor stations, denoted by S. Links connecting major stations form the set
of major links. The link travel times are denoted on the links, in number of time intervals.
The origin and destination stations of riders are randomly selected from the set of stations.
11
Accordingly, trips may be completed using only minor links (e.g., from S4 to S5), only major
links (e.g., from H1 to H2), or a combination of minor and major links (e.g., from S4 to
S2). We consider three depots, namely stations S1, S2, and S4, for the trailer modules. A
trailer module’s trip starts and ends from one of these depots, selected following a uniform
distribution. Main modules are assumed to use H1 or H3 as depot, with similar likelihood.
Figure 3: The Sioux Fall transportation network. Numbers on links denote link travel times.
Time windows for trailer and main modules are generated to ensure their availability in
the network for the entirety of the planning horizon, which is set to 60 time intervals in
these experiments. We generate the earliest departure times of riders following a uniform
distribution within the range [0 ,20], and their latest arrival times within the range [40 ,60],
while ensuring that all riders are present in the network for at least the duration of their
shortest-path travel times. To guarantee a high quality service, we assume riders have to be
picked up by at most three time intervals after their earliest departure times. We further
restrict a rider’s waiting time while transferring to a main module to 3 time intervals. (Note
that there is no waiting time when a trailer module detaches from a main module to transport
passengers to their final destinations.) We set the capacity of both trailer and main modules
to four.
We quantify the impact of the modular transit system on the metrics provided in the
previous subsection under different system configurations, where a system configuration
consists of a unique set of values for the number of main modules, trailer modules, and
ride requests. To systematically evaluate the performance of the modular transit system
under various configurations, we generate a ‘base’ scenario with 20 main modules, 50 trailer
modules, and 200 rider requests. We then conduct sensitivity analysis over the value of these
input parameters by creating twelve additional scenarios, each scenario changing the value of
a single parameter in the base scenario. The parameter values of all scenarios are presented in
Table 3. Scenario 1 in this table is the base scenario. The changing parameter for scenarios
2 to 13 is embolded in this table. As such, scenario sets {4,7,10,13},{3,6,9,12}, and
{2,5,8,11}allow us to conduct sensitivity analysis on the number of main modules, trailer
modules, and rider requests, respectively. We generate 10 random instances for each scenario
by randomly generating the set of trips, and solve them using model (6).
Figures 4,5and 6demonstrate how system-level solution properties and performance
metrics (i.e., solution time, rider matching rate, percentage of used trailer modules, average
12
Table 3: Parameter values for different scenarios. Scenario 1 is the base scenario. In Scenarios 2 to 13, we
change one of the three parameter, i.e., the number of main modules, the number of trailer modules, or the
number of rider requests. The parameter changed in each scenario compared to the base scenario is shown
in bold.
Scenario 1 2 3 4 5 6 7 8 9 10 11 12 13
Main modules 20 20 20 30 20 20 12 20 20 620 20 50
Trailer modules 50 50 30 50 50 75 50 50 20 50 50 120 50
Rider requests 200 150 200 200 300 200 200 50 200 200 500 200 200
empty miles driven, and average module occupancy) as well as individual-level quality-of-
service metrics (i.e., average rider waiting time and detour time) are affected by the number
of main modules, trailer modules, and ride requests, respectively. These figures are generated
by solving the optimization problem in model (6). Table 4shows p-values obtained from one-
way ANOVA tests to investigate whether the trends observed in these figures are statistically
significant at the 5% significance level. A p-value lower than 5% rejects the null hypothesis
of the ANOVA test, indicating that there is a statistically-significant difference between the
values of the system performance metrics under different values of the parameter under study.
The significance of the differences between the means of different metrics can be evaluated
by comparing their confidence intervals in Figures A.1,A.2, and A.3. Note that no overlap
between the confidence intervals implies a statistically significant difference.
Table 4: P-values to investigate the significance of performance measures
Group 1 Group 2 Group 3
Scen. {1, 4, 7, 10, 13}Scen. {1, 3, 6, 9, 12}Scen. {1, 2, 5, 8, 11}
Solution time (s) 0.00 0.00 0.00
Rider matching rate 0.00 0.00 0.00
Percentage of matched
trailer modules 0.00 0.00 0.00
Avg. empty miles driven by a
trailer module 0.01 0.00 0.00
Avg. empty miles driven by a
main module 0.00 0.00 0.00
Avg. detour time for riders 0.32 0.81 0.29
Avg. trailer module occupancy 0.04 0.00 0.00
Avg. main module occupancy 0.00 0.00 0.00
Avg. waiting time 0.00 0.69 0.11
Figure 4demonstrates the impact of changing the number of main modules from 20 in
the base scenario to 6 (scenario 10), 12 (scenario 7), 30 (scenario 4) and 50 (scenario 13).
P-values that assess the statistical significance of trends in this figure are reported under
‘Group 1’ in Table 4, and the corresponding confidence intervals are demonstrated in A.1.
Figure 4suggests that the solution time increases with the number of main modules—a trend
that is expected since each additional main module introduces an additional set of decision
variables and constraints into model (6). However, this increase in solution time occurs at a
sublinear rate.
Figure 4suggests that, as expected, the fraction of matched riders increases significantly
13
Figure 4: System performance as a function of the number of main modules. The five columns correspond
to scenarios 1, 4, 7, 10 and 13, respectively. The number on top of each cell shows the average value over 10
randomly generated instances. The numbers in parenthesis in each cell show the range of values across the
10 experiments.
14
Figure 5: System performance as a function of the number of trailer modules. The five columns correspond
to scenarios 1, 3, 6, 9 and 12, respectively. The number on top of each cell shows the average value over 10
randomly generated instances. The numbers in parenthesis in each cell show the range of values across the
10 experiments.
15
Figure 6: System performance as a function of the number of riders. The five columns correspond to
scenarios 1, 2, 5, 8 and 11, respectively. The number on top of each cell shows the average value over 10
randomly generated instances. The numbers in parenthesis in each cell show the range of values across the
10 experiments.
16
as we increase the number of main modules. This trend can be explained by the statistically-
significant increase in the number of used trailer modules, and the reduction in the average
trailer module empty-miles-driven. In Figure 4, the average vehicle occupancy is computed
as the ratio between the occupied and total vehicles-miles-driven. While the average trailer
module occupancy does not follow a specific trend between 12 and 50 main modules, it
experiences a significant increase when increasing the number of main module from 6. The
average main module occupancy is reduced significantly with the number of main modules,
which is partly explained by the higher empty-miles driven. As such, although some system
level performance metrics improve when increasing the number of main modules (e.g., the
matching rate), other metrics deteriorate (e.g., increase in empty-miles driven).
Figure 4displays an increasing trend in the average detour time of served passengers
as we increase the number of main modules; however, Table 4indicates that this increase
in detour time is not statistically significant. This figure also displays a descending trend
in the average waiting time with the number of main modules, which is expected, and is
statistically significant for any pair of scenarios with a difference of larger than 14 between
main modules. As such, Figure 4indicates that a higher level-of-service (LoS) for system
participants can be expected with increasing the number of main modules.
Figure 5demonstrates the impact of changing the number of trailer modules from 50
in the base scenario to 20 (scenario 9), 30 (scenario 3), 75 (scenario 6) and 120 (scenario
12). P-values that assess the statistical significance of trends in this figure are reported
under ‘Group 2’ in Table 4, and their corresponding confidence intervals are demonstrated
in A.2. Similar to main modules, increasing the number of trailer modules increases the size
of the mathematical model in (6) and therefore the solution time sublinearly. As expected,
increasing the number of trailer modules increases the matching rate.
While the average main module occupancy increases, the average occupancy of trailer
modules decreases as we increase the number of main modules. This change in occupancy of
modules is also reflected in their empty-miles-driven: while the average empty-miles-driven of
trailer modules increases with the number of trailer modules, the same metric decreases with
the number of main modules. These trends are expected, as in this set of experiments we
increase the number of trailer modules while keeping the number of main modules unchanged,
thereby increasing the usage of main modules. This figure also shows that the average waiting
and detour times are not affected in a statistically significant manner.
Finally, Figure 6demonstrates the impact of changing the number of ride requests
from 200 in the base scenario to 50 (scenario 8), 150 (scenario 2), 300 (scenario 5), and
500 (scenario 11). P-values that assess the statistical significance of trends in this figure
are reported under ‘Group 3’ in Table 4, and their corresponding confidence intervals are
demonstrated in A.3. Similar to Figures 4and 5, the solution time increases with problem
size; however sublinearly with the number of riders. Although the number of served riders
increases with the density of ride requests, the percentage of served riders decreases, because
this increase in demand is not met with a rise on the supply side. Also, this figure suggests
that the percentage of matched trailer modules increases sublinearly with number of ride
requests. However, the existing supply is used more effectively, as demonstrated by the
higher occupancy rates and lower empty-miles-driven by the trailer and main modules. It is
interesting to note that despite the fact that a rise in demand is not met with a higher level
of supply, rider LoS measures, including the average detour time and the average waiting
17
time, are not affected in a statistically significant manner.
Maximizing the number of served ride requests does not necessarily provide a high-quality
system-level performance. As such, we solve model (7) to minimize the total VMT while
requiring the solution to serve ξM∗number of ride requests, where M∗is the optimal solution
to model (6), and ξis a value between 0 and 1. For each value of ξ, we generate 10 random
instances of the base scenario and solve them using models (6) and (7). Figure 7demonstrates
how system performance is affected by the value of ξ. As expected, the matching rate
decreases as we reduce the value of ξ. The total cost (VMT) decreases drastically when
changing ξfrom 1 to 0.9, which indicates that slightly lowering the minimum target number
of served ride requests can reduce the cost significantly. As displayed in Figure 7, decreasing
the value of ξcan reduce the detour time and empty miles driven, which indicates a higher
LoS. Also the number of used trailer modules decreases with ξ, which has resulted in a
general increase in the trailer module occupancy.
The goal of this paper is to investigate whether, and the degree to which, a modular
transit system can make a more efficient use of empty seats in traditional transit and on-
demand mobility systems (e.g., dial-a-ride systems), by allowing these empty seats to form
independent trailer modules that can detach, serve local demand, and attach to the main
modules only when necessary. To put this hypothesis to test, we compare the base scenario
of the modular transit system with a shuttle service. Recall that the base scenario consists
of 20 main modules and 50 trailer modules, each with capacity of four, providing a total
of 200 seats. Therefore, to make the two systems comparable, we consider 20 shuttles with
capacity 10 (i.e., a total of 200 seats). Similar to trailer modules, we assume shuttles may use
any of the stations S1, S2, and S4 as depots, and thus the origin and destination stations of
each shuttle are selected from these three depots following a uniform distribution. Shuttles
are permitted to travel on any link in the network at any time and similar to the trailer
modules, are available during the entire time horizon of [0, 60].
The results of this comparison are shown in Figure 8. As expected, the rider matching rate
and the vehicle occupancy rate are both higher in the modular transit system, validating
the hypothesis put forward in this paper. Paired t-tests indicate that both changes are
statistically significant at the 1% significance level. Figure 8(c) compares the total vehicle-
miles-driven in the network by the two services. This figure displays two ranges of values for
the shuttle system, one based on the number of shuttles, and the other based on the adjusted
number of shuttles, taking into account that each shuttle accounts for 2.5 trailer modules
(Trailer modules can travel on both major and minor links, and hence we can compare their
total empty miles driven with that of shuttles with the same total number of available seats.
Since the capacity of shuttles is 10/4 = 2.5 times the capacity of trailer modules, each shuttle
accounts for 2.5 trailer modules.). This figure demonstrates that the higher matching rate
for rider requests and the higher utilization rate of vehicles in the modular transit system
come at the cost of a slightly higher total vehicles- miles-driven in the network.
Finally, we perform a sensitivity analysis on the impact of the initial distributions of
system agents on system performance. Result is presented in Appendix B.
18
Figure 7: System performance for different values of ξbased on scenario 1
19
Figure 8: Comparing the performance of the modular transit system with that of a shuttle service
5.3. Manhattan Case Study
In order to show the performance of the modular transit system in a large-scale network,
we run an experiment using the NYC taxi dataset 1in this section. For this case study, we
consider 184 stations scattered around the Manhattan area such that there is at least one
station within 500 meters from each point in the area, as shown in Figure 9. About one third
of these stations are labeled as main stations (blue squares), and the rest are categorized as
minor stations (green circles). We also considered a single depot (blue triangle) at which
all modules start and end their trips. We generate a scenario with 400 main modules, 1000
trailer modules and 4000 riders, and the capacity of both main and trailer modules is set
to 4. The planning time horizon is assumed to be one hour. The information regarding
the origin and destination stations, and earliest departure times of riders are extracted from
the trips served by taxis in the Manhattan area from 19:00 to 20:00 on March 4, 2016. We
further obtained the shortest-path travel time between every pair of stations from the Google
API. For each rider, the latest arrival time is calculated by summing the earliest departure
time, the shortest-path travel time between the origin and destination stations, and a time
flexibility of 3 minutes.
Due to the large size of this problem, we are not able to directly solve the integer programs
in (6) and (7) using CPLEX. Therefore, we adopt a graph partitioning technique from the
literature, proposed by Tafreshian and Masoud (2020). We present a summary of this method
in Appendix C. Using this method, we can partition the original problem into smaller sub-
problems of approximately similar sizes, which can be solved more efficiently in parallel.
The partitioning method clusters rider requests into a small number of partitions prior to
constructing the time-expanded link sets. We equally distribute the main and trailer modules
between these clusters. Finally, we solve the problems in (6) and (7) using CPLEX on each
cluster, separately.
1http://www.nyc.gov/html/tlc/html/about/trip record data.shtml
20
Figure 9: Manhattan transportation network. Blue squares represent the main stations, green circles
represent the minor stations, and the blue triangle marks the depot. The dashed lines represent major links,
and the solid lines represent minor links.
Figure 11 displays the system performance of the modular transit system using 64
partitions, where there are about 6 main modules, 16 trailer modules, and 60 riders in
each partition. This figure indicates that both models can be solved in less than 420 seconds
in all partitions which proves that the modular transit system with 64 partitions can be used
to address on-demand requests. In order to show the impact of the number of partitions on
rider matching rate and solution time, we repeat the experiments with 32 partitions. Figure
11 displays the comparison of rider matching rate and solution time using 32 partitions
with 64 partitions. The results demonstrate that the solution time of solving one partition
is much lower using 64 partitions than 32 partition, while the rider matching rate is only
slightly lower. It is worth mentioning that the graph partitioning takes less than 10 seconds
in both cases.
Note that if one is interested in deploying the modular transit system in a dynamic setting,
21
a rolling time-horizon approach can be adopted, where an optimization problem will be solved
periodically and frequently (e.g., every min) to respond to on-demand requests. Under such
circumstances, the optimization problems need to be solved more quickly; however, the
number of riders in each optimization problem will be much lower, ensuring that a solution
can be obtained in a timely manner.
Figure 10: System performance of solving Model 6and Model 7using 64 partitions
6. Conclusions and Future Work
In this paper we introduced a new type automated public transportation, named modular
transit. Modular transit is designed to overcome the shortcomings of the traditional bus by
22
Figure 11: Rider matching rate and solution time of solving Model 6and Model 7using 32 and 64 partitions
incorporating separable trailer modules that can detach and serve the first- and last-mile
trips. In addition to providing a door-to-door service, modular transit enables en-route
transfers between trailer modules, thereby eliminating the need for a transfer for the last
mile of the trip, and improving the quality of service for travelers.
Our extensive numerical experiments, formulated to assess the performance of modular
transit, indicate that the modular bus can not only be used as a local transportation option,
but also it can better serve long-distance trips by efficiently and conveniently connecting
passengers to main modules that cover the major portion of long-distance trips. Increasing
the number of main and trailer modules, as expected, improves passengers’ level-of-service.
However, higher number of main and trailer modules, keeping all other factors constant,
reduces the occupancy of trailer and main modules, respectively. Despite this reduction in
occupancy, the number of served riders still increases, due to the complementary nature of
main and trailer modules in serving trips. In general, results indicate that it is important
to choose an appropriate ratio of trailer and main modules according to the number of
ride requests to provide high-quality service to passengers and at the same time curb
the additional total vehicle-miles-travelled imposed on the Transportation network. Our
experiments show that slightly lowering the number of served rider requests can provide
higher level-of-service and reduce the VMT significantly. These benefits are mostly realized
when making small compromises in the number of served trips, indicating that there is a
sweet spot where an operator can increase a large portion of requests and at the same time
provide high quality service to customers and reduce VMT.
Our experiments indicate that, compared to a door-to-door shuttle service, modular
23
transit can make a more efficient use of vehicles and serves a larger number of passengers.
However, it can lead to a slightly higher VMT, which stems from flexibility of modules to
roam around and serve demand.
In this paper, we took the initial steps toward making the modular transit system a
viable option for the future by formulating its operation as an integer program, and using
an optimization package to solve it. However, there are a number of potentially promising
research directions. First, our study has shown that the ratio between trailer and main
modules can play an important role on system performance. Thus, for future studies, one
can expand on the proposed formulation to find the optimal number of required modules of
each type. Second, we adopted a heuristic method based on graph partitioning to solve larger
instances of this problem. As such, it seems essential to develop more efficient and/or precise
solution methodologies for solving large-scale problems. Finally, integrating the proposed
modular transit system with other variants of shared-use mobility services may prove useful
due to the existence of some obstacles in relying solely on such systems over the entire
transportation network.
7. Acknowledgment
This study was partially supported by grants from the US DOT Center for Connected
and Automated Transportation (grant #69A3551747105), the Michigan Institute for Data
Science, and the National Science Foundation (grant #1831347).
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Appendix A ANOVA Test Confidence Intervals
Figure A.1: Mean value of system performance with its confidence interval for scenarios 1, 4, 7, 10 and 13
in Group 1.
Figure A.2: Mean value of system performance with its confidence interval for scenarios 1, 3, 6, 9 and 12
in Group 2.
28
Figure A.3: Mean value of system performance with its confidence interval for scenarios 1, 2, 5, 8 and 11
in Group 3.
Appendix B Impact of Initial Distributions of Agents
In order to study the impact of the initial distributions of riders and modules on system
performance, we consider two additional scenarios. Under Conditions 1 and 2, we change the
initial distribution of riders and modules from those in the base scenario, respectively. Recall
that in the base scenario, we have three depots, S1, S2, and S4, from which the origin and
destination of modules are randomly selected. In Condition 1, we consider a single depot and
set the origin and destination of all modules to station S1. Recall that in the base scenario,
the earliest departure time and latest arrival time of each rider is generated following a
uniform distribution on [0, 20], and [40, 60], respectively. In Condition 2, we assume that
the arrival time of riders in the system during [0, 20] and their exit time during [40, 60] both
follow a Poisson distribution with the mean of 10 riders per minute. All parameters except
the ones mentioned above stay the same as in the base scenario.
The system performance of the base scenario and the two scenarios under Conditions
1 and 2 are displayed in Figure B.1. Also, Table B.1 shows the p-value obtained from
paired T-test to investigate whether system performance values in this figure are statistically
significant at the 5% significance level. The figure and table combined indicate that the initial
distributions of both depots and request arrivals have a negligible effect on the solution time.
Appendix C Trip-based Graph Partitioning
In this section, we explain how one can divide |R|riders into Kuniform partitions based
on their trip information. Let us assume that Kis an exponent of 2 (i.e. K= 2κ), and
ε∈(0,1). In order to obtain Kclusters of almost equal size, we recursively implement the
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Figure B.1: System performance of the base Scenario 1 and two conditions with different initial
distributions.
bisection method in Algorithm 1for κtimes. In the first step, riders will be partitioned
into 2 clusters of maximum size d(1 + ε)|R|/2e. Applying the algorithm on each cluster
separately, we obtain 4 new clusters, and so on. Hence, continuing this process for κtimes
yields Kuniform partitions. Prior to using this recursive procedure, one needs to compute
the dissimilarity matrix C= [crk], where crk for r, k ∈Rcan be computed as:
crk =(Edr+LAr)−(Edk+LAk)+1
4
6
X
i=1
ci
rk ,
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Table B.1: P-values to investigate the significance of performance measures in base scenario as well as two
scenarios under Condition 1 and Condition 2
Base Scen. & Cond. 1 Base Scen. & Cond. 2
Solution time (s) 0.53 0.45
Rider matching rate 0.25 0.87
Percentage of matched
trailer modules 0.98 0.35
Avg. empty miles driven by a
trailer module 0.00 0.71
Avg. detour time for riders 0.96 0.47
Avg. trailer module occupancy 0.00 0.77
Avg. main module occupancy 0.10 0.50
Avg. waiting time 0.52 0.22
where
c1
rk =g(or, ok) + g(dk, dr) + g(ok, dk)−g(or, dr),
c2
rk =g(or, ok)+2g(ok, dr) + g(dr, dk)−g(or, dr)−g(ok, dk),
c3
rk = 0 ,
c4
rk =g(ok, or) + g(dr, dk) + g(or, dr)−g(ok, dk),
c5
rk =g(ok, or)+2g(or, dk) + g(dk, dr)−g(or, dr)−g(ok, dk),
c6
rk = 0 .
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Algorithm 1: The ε-uniform partitioning algorithm
Input: Dissimilarity matrix C= [crk],∀r, k ∈R, uniformity parameter ε
Output: 2 uniform clusters of riders
for trials = 1,...,10 do
Randomly select 2 trips denoted by k1and k2as cluster representatives;
repeat
Step1 (Assignment)
Compute δr=|crk1−crk2|r∈R;
Let ¯
Rbe sorted riders descendingly based on δ;
Let Cl1=∅and Cl2=∅;
for r∈¯
Rdo
if crk1≤crk2then
if |Cl1|<(1 + ε)|R|/2then
Add rto Cl1;
else
Add rto Cl2;
else
if |Cl2|<(1 + ε)|R|/2then
Add rto Cl2;
else
Add rto Cl1;
Step 2 (Update Clusters)
Update the cluster representatives:
k1= argmin
k∈Cl1X
r∈Cl1
crk
k2= argmin
k∈Cl2X
r∈Cl2
crk
until Pr∈Cl1crk1+Pr∈C l2crk2do not change;
Take the best result out of 10 trials;
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