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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).

References

Mojtaba Abdolmaleki, Neda Masoud, and Yafeng Yin. Transit timetable synchronization for

transfer time minimization. Transportation Research Part B: Methodological, 131:143–159,

2020.

Z Bernard. The bike-sharing economy is shaking up the transportation market worldwide.

Business Insider, 2018.

Bureau of Transportation Statistics. Bus proﬁle, 2018.

Lawrence D Burns, William C Jordan, and Bonnie A Scarborough. Transforming personal

mobility. The Earth Institute, 431:432, 2013.

Michael R Cairns. The development of park and ride in scotland. Journal of Transport

Geography, 6:295–307, 1998.

Zhiwei Chen, Xiaopeng Li, and Xuesong Zhou. Operational design for shuttle systems with

modular vehicles under oversaturated traﬃc: Discrete modeling method. Transportation

Research Part B: Methodological, 122:1–19, 2019.

Zhiwei Chen, Xiaopeng Li, and Xuesong Zhou. Operational design for shuttle systems

with modular vehicles under oversaturated traﬃc: Continuous modeling method.

Transportation Research Part B: Methodological, 132:76–100, 2020.

24

Shih Fen Cheng, Duc Thien Nguyen, and Hoong Chuin Lau. Mechanisms for arranging

ride sharing and fare splitting for last-mile travel demands. AAMAS ’14: Proceedings of

the 13th International Conference on Autonomous Agents and Multiagent Systems, pages

1505–1506, 2014.

Z. J. Chong, B. Qin, T. Bandyopadhyay, T. Wongpiromsarn, E. S. Rankin, M. H. Ang,

E. Frazzoli, D. Rus, D. Hsu, and K. H. Low. Autonomous personal vehicle for the ﬁrst-

and last-mile transportation services. Proceedings of the 2011 IEEE 5th International

Conference on Cybernetics and Intelligent Systems(CIS), pages 253–260, 2011.

Jean-Fran¸cois Cordeau and Gilbert Laporte. The dial-a-ride problem: models and

algorithms. Annals of operations research, 153(1):29–46, 2007.

Cristi´an E Cort´es, Laia Pag`es, and R Jayakrishnan. Microsimulation of ﬂexible transit

system designs in realistic urban networks. Transportation Research Record, 1923(1):153–

163, 2005.

Paul DeMaio. Bike-sharing: History, impacts, models of provision, and future. Journal of

public transportation, 12(4):3, 2009.

Fausto Errico, Teodor Gabriel Crainic, Federico Malucelli, and Maddalena Nonato. A survey

on planning semi-ﬂexible transit systems: Methodological issues and a unifying framework.

Transportation Research Part C: Emerging Technologies, 36:324–338, 2013.

Qian-Wen Guo, Joseph YJ Chow, and Paul Schonfeld. Stochastic dynamic switching in ﬁxed

and ﬂexible transit services as market entry-exit real options. Transportation research

procedia, 23:380–399, 2017.

Spring C. Hsu. Determinants of passenger transfer waiting time at multi-modal connecting

stations. Transportation Research Part E: Logistics and Transportation Review, 46:404–

413, 2010.

Ronald F Kirby, Kiran U Bhatt, Michael A Kemp, RG McGillivary, and Martin Wohl. Para

transit: Neglected options for urban mobility. Technical report, 1974.

Jonathan Levine, Moira Zellner, Mar´ıa Arquero de Alarc´on, Yoram Shiftan, and Dean

Massey. The impact of automated transit, pedestrian, and bicycling facilities on urban

travel patterns. Transportation planning and technology, 41(5):463–480, 2018.

Todd Litman. Evaluating carsharing beneﬁts. Transportation Research Record, 1702(1):

31–35, 2000.

Roger Lloret-Batlle, Neda Masoud, and Daisik Nam. Peer-to-peer ridesharing with ride-back

on high-occupancy-vehicle lanes: Toward a practical alternative mode for daily commuting.

Transportation Research Record, 2668(1):21–28, 2017.

Tai-Yu Ma, Saeid Rasulkhani, Joseph YJ Chow, and Sylvain Klein. A dynamic

ridesharing dispatch and idle vehicle repositioning strategy with integrated transit

transfers. Transportation Research Part E: Logistics and Transportation Review, 128:

417–442, 2019.

25

Federico Malucelli, Maddalena Nonato, and Stefano Pallottino. Demand adaptive systems:

some proposals on ﬂexible transit. Operational Research in Industry, pages 157–182, 1999.

Neda Masoud and R Jayakrishnan. Formulations for optimal shared ownership and use of

autonomous or driverless vehicles. In Proceedings of the Transportation Research Board

95th Annual Meeting, pages 1–17, 2016.

Neda Masoud and R Jayakrishnan. Autonomous or driver-less vehicles: Implementation

strategies and operational concerns. Transportation research part E: logistics and

transportation review, 108:179–194, 2017a.

Neda Masoud and R Jayakrishnan. A real-time algorithm to solve the peer-to-peer ride-

matching problem in a ﬂexible ridesharing system. Transportation Research part B:

Methodological, 106:218–236, 2017b.

Neda Masoud and R. Jayakrishnan. A decomposition algorithm to solve the multi-hop peer-

to-peer ride-matching problem. Transportation Research Part B: Methodological, 99:1–29,

2017c.

Neda Masoud and Roger Lloret-Batlle. Increasing ridership and user permanence in

ridesharing systems using a novel peer-to-peer exchange mechanism. In 95th Annual

Meeting of the Transportation Research Board, Washington, DC, 2016.

Neda Masoud, Daisik Nam, Jiangboo Yu, and R. Jayakrishnan. Promoting peer-to-peer

ridesharing services as transit system feeders. Transportation Research Record: Journal of

the Transportation Research Board, 2650:74–83, 2017.

Aditi Moorthy, Robert De Kleine, Gregory Keoleian, Jeremy Good, and Geoﬀ Lewis. Shared

autonomous vehicles as a sustainable solution to the last mile problem: A case study of

ann arbor-detroit area. SAE International Journal of Passenger Cars - Electronic and

Electrical Systems 10(2), pages 328–335, 2017.

Daisik Nam, Dingtong Yang, Sunghi An, Jiangbo Gabriel Yu, R Jayakrishnan, and Neda

Masoud. Designing a transit-feeder system using multiple sustainable modes: Peer-to-peer

(p2p) ridesharing, bike sharing, and walking. Transportation Research Record, 2672(8):

754–763, 2018.

Next Future Transportation inc. https://www.next-future-mobility.com/.

Errol C. Noel. Park-and-ride – alive, well and expanding in the united states. Journal of

Urban Planning and Development, 114:2–13, 1988.

Seyed Mohammad Nourbakhsh and Yanfeng Ouyang. A structured ﬂexible transit system

for low demand areas. Transportation Research Part B: Methodological, 46(1):204–216,

2012.

Juan Qin, Stephanie Lee, Xiangbin Yan, and Yong Tan. Beyond solving the last mile

problem: the substitution eﬀects of bike-sharing on a ride-sharing platform. Journal of

Business Analytics, 1(1):13–28, 2018.

26

Luca Quadrifoglio and Xiugang Li. Performance assessment and comparison between

ﬁxed and ﬂexible transit services for diﬀerent urban settings and demand distributions.

Technical report, 2008.

Andreas Rau, Liangyuan Tian, Madhur Jain, Meng Xie, Tao Liu, and Yuan Zhou. Dynamic

autonomous road transit (dart) for use-case capacity more than bus. Transportation

Research Procedia, 41:812–823, 2019.

Robert Regue, Neda Masoud, and Will Recker. Car2work: A shared mobility concept to

connect commuters with workplaces. Transportation Research Record: Journal of the

Transportation Research Board, 2542:102–110, 2016.

Martin WP Savelsbergh and Marc Sol. The general pickup and delivery problem.

Transportation science, 29(1):17–29, 1995.

Susan Shaheen and Nelson Chan. Mobility and the sharing economy: Potential to overcome

ﬁrst- and last-mile public transit connections. Built Environment, 42:573–588, 2016.

Yu Shen, Hongmou Zhang, and Jinhua Zhao. Integrating shared autonomous vehicle in public

transportation system: A supply-side simulation of the ﬁrst-mile service in singapore.

Transportation Research Part A: Policy and Practice, 113:125–136, 2018.

Amirmahdi Tafreshian and Neda Masoud. Trip-based graph partitioning in dynamic

ridesharing. Transportation Research Part C: Emerging Technologies, 114:532–553, 2020.

Hai Wang and Amedeo Odoni. Approximating the performance of a last mile transportation

system. Transportation Science, 50:659–675, 2016.

Moira Zellner1, Dean Massey, Yoram Shiftan, Jonathan Levine, and Arquero Maria Josefa.

Overcoming the last-mile problem with transportation and land-use improvements: An

agent-based approach. International Journal of Transportation, 4:1–26, 2016.

Yue Zheng, Wenquan Li, and Feng Qiu. A methodology for choosing between route

deviation and point deviation policies for ﬂexible transit services. Journal of Advanced

Transportation, 2018, 2018.

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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.

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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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