#  On the Co-Design of AV-Enabled Mobility Systems

Gioele Zardini1,3, Nicolas Lanzetti2,3, Mauro Salazar3,4, Andrea Censi1,
Emilio Frazzoli1, and Marco Pavone3 1Institute for Dynamic Systems and
Control, ETH Zürich<EMAIL_ADDRESS>Control Laboratory, ETH Zürich<EMAIL_ADDRESS>of Aeronautics and
Astronautics, Stanford University<EMAIL_ADDRESS>Systems
Technology Group, Eindhoven University of Technology<EMAIL_ADDRESS>preliminary version of this paper was presented at the 99th Annual Meeting of
the Transportation Research Board [1].This research was supported by the
National Science Foundation under CAREER Award CMMI-1454737, the Toyota
Research Institute (TRI), and ETH Zürich. This article solely reflects the
opinions and conclusions of its authors and not NSF, TRI, or any other entity.

###### Abstract

The design of autonomous vehicles (AVs) and the design of AV-enabled mobility
systems are closely coupled. Indeed, knowledge about the intended service of
AVs would impact their design and deployment process, whilst insights about
their technological development could significantly affect transportation
management decisions. This calls for tools to study such a coupling and co-
design AVs and AV-enabled mobility systems in terms of different objectives.
In this paper, we instantiate a framework to address such co-design problems.
In particular, we leverage the recently developed theory of co-design to frame
and solve the problem of designing and deploying an intermodal Autonomous
Mobility-on-Demand system, whereby AVs service travel demands jointly with
public transit, in terms of fleet sizing, vehicle autonomy, and public transit
service frequency. Our framework is modular and compositional, allowing one to
describe the design problem as the interconnection of its individual
components and to tackle it from a system-level perspective. To showcase our
methodology, we present a real-world case study for Washington D.C., USA. Our
work suggests that it is possible to create user-friendly optimization tools
to systematically assess costs and benefits of interventions, and that such
analytical techniques might gain a momentous role in policy-making in the
future.

##  I Introduction

Arguably, the current design process for AVs largely suffers from the lack of
clear, specific requirements in terms of the service such vehicles will be
providing. Yet, knowledge about their intended service (e.g., last-mile versus
point-to-point travel) might dramatically impact how the AVs are designed,
and, critically, significantly ease their development process. For example, if
for a given city we knew that for an effective on-demand mobility system
autonomous cars only need to drive up to 25 mph and only on relatively easy
roads, their design would be greatly simplified and their deployment could
certainly be accelerated. At the same time, from the system-level perspective
of transportation management, knowledge about the trajectory of technology
development for AVs would certainly impact decisions on infrastructure
investments and provision of service. In other words, the design of the AVs
and the design of a mobility system leveraging AVs are intimately coupled.
This calls for methods to reason about such a coupling, and in particular to
_co-design_ the AVs and the associated AV-enabled mobility system. A key
requirement in this context is the ability to account for a range of
heterogeneous objectives that are often not directly comparable (consider, for
instance, travel time and emissions).

Accordingly, the goal of this paper is to lay the foundations for a framework
through which one can co-design future AV-enabled mobility systems.
Specifically, we show how one can leverage the recently developed mathematical
theory of co-design [2, 3, 4], which provides a general methodology to co-
design complex systems in a modular and compositional fashion. This tool
delivers the set of rational design solutions lying on the Pareto front,
allowing one to reason about costs and benefits of the individual design
options. The framework is instantiated in the setting of co-designing
intermodal AMoD systems [5], whereby fleets of self-driving vehicles provide
on-demand mobility jointly with public transit. Aspects subject to co-design
include fleet size, AV-specific characteristics, and public transit service
frequency.

###  I-A Literature Review

Our work lies at the interface of the design of urban public transportation
services and the design of AMoD systems. The first research stream is reviewed
in [6, 7], and comprises _strategic_ long-term infrastructure modifications
and _operational_ short-term scheduling. The joint design of traffic network
topology and control infrastructure has been presented in [8]. Public
transportation scheduling has been solved jointly with the design of the
transit network in a passengers’ and operators’ cost-optimal fashion in [9],
using demand-driven approaches in [10], and in an energy-efficient way in
[11]. However, these works only focus on the public transit system and do not
consider its joint design with an AMoD system. The research on the design of
AMoD systems is reviewed in [12] and mainly pertains their fleet sizing. In
this regard, studies range from simulation-based approaches [13, 14, 15, 16]
to analytical methods [17]. In [18], the authors jointly design the fleet size
and the charging infrastructure, and formulate the arising design problem as a
mixed integer linear program. The authors of [19] solve the fleet sizing
problem together with the vehicle allocation problem. Finally, [20] co-designs
the AMoD fleet size and its composition. More recently, the joint design of
multimodal transit networks and AMoD systems was formulated in [21] as a
bilevel optimization problem and solved with heuristics. Overall, the problem-
specific structure of existing design methods for AMoD systems is not amenable
to a modular and compositional problem formulation. Moreover, previous work
does not capture important aspects of AV-enabled mobility systems, such as
other transportation modes and AV-specific design parameters (e.g., the level
of autonomy).

###  I-B Statement of Contribution

In this paper we lay the foundations for the systematic study of the design of
AV-enabled mobility systems. Specifically, we leverage the mathematical theory
of co-design [2] to devise a framework to study the design of intermodal AMoD
(I-AMoD) systems in terms of fleet characteristics and public transit service,
enabling the computation of the _rational_ solutions lying on the Pareto front
of minimal travel time, transportation costs, and emissions. Our framework
allows one to structure the design problem in a modular way, in which each
different transportation option can be “plugged in” in a larger model. Each
model has minimal assumptions: Rather than properties such as linearity and
convexity, we ask for very general monotonicity assumptions. For example, we
assume that the cost of automation increases monotonically with the speed
achievable by the AV. We are able to obtain the full Pareto front of
_rational_ solutions, or, given policies, to weigh incomparable costs (such as
travel time and emissions) and to present actionable information to the
stakeholders of the mobility ecosystem. We showcase our methodology through a
real-world case study of Washington D.C., USA. We show how, given the model,
we can easily formulate and answer several questions regarding the
introduction of new technologies and investigate possible infrastructure
interventions.

###  I-C Organization

The remainder of this paper is structured as follows: Section II reviews the
mathematical theory of co-design. Section III presents the co-design problem
for AV-enabled mobility systems. We showcase our approach with real-world case
studies for Washington D.C., USA, in Section IV. Section V concludes the paper
with a discussion and an overview on future research directions.

##  II Background

This paper builds on the mathematical theory of co-design, presented in [2].
In this section, we present a review of the main contents needed for this
work.

###  II-A Orders

We will use basic facts from order theory, which we review in the following.

######  Definition II.1 (Poset).

A partially ordered set (poset) is a tuple
$\langle\mathcal{P},\preceq_{\mathcal{P}}\rangle$, where $\mathcal{P}$ is a
set and $\preceq_{\mathcal{P}}$ is a partial order, defined as a reflexive,
transitive, and antisymmetric relation.

Given a poset, we can formalize the idea of “Pareto front” through antichains.

######  Definition II.2 (Antichains).

A subset $S\subseteq\mathcal{P}$ is an antichain iff no elements are
comparable: For $x,y\in S$, $x\preceq y$ implies $x=y$. We denote by
$\textsf{A}\mathcal{P}$ the set of all antichains in $\mathcal{P}$.

######  Definition II.3 (Directed set).

A subset $S\subseteq\mathcal{P}$ is directed if each pair of elements in $S$
has an upper bound: For all $a,b\in S$, there exists a $c\in S$ such that
$a\preceq c$ and $b\preceq c$.

######  Definition II.4 (Completeness).

A poset is a complete partial order (CPO) if each of its directed subsets has
a supremum and a least element.

For instance, the poset $\langle\mathbb{R}_{+},\leq\rangle$, with
$\mathbb{R}_{+}\coloneqq\\{x\in\mathbb{R}\,|\,x\geq 0\\}$, is not complete, as
its directed subset $\mathbb{R}_{+}\subseteq\mathbb{R}_{+}$ does not have an
upper bound (and therefore a supremum). Nonetheless, we can make it complete
by artificially adding a top element $\top$, i.e., by defining
$\langle\overline{\mathbb{R}}_{+},\leq\rangle$ with
$\overline{\mathbb{R}}_{+}\coloneqq\mathbb{R}_{+}\cup\\{\top\\}$ and
$a\leq\top$ for all $a\in\mathbb{R}_{+}$. Similarly, we can complete
$\mathbb{N}$ to $\overline{\mathbb{N}}$.

In this setting, Scott-continuous maps will play a key role. Intuitively,
Scott-continuity can be understood as a stronger notion of monotonicity.

######  Definition II.5 (Scott continuity).

A map $f:\mathcal{P}\rightarrow\mathcal{Q}$ between two posets
$\langle\mathcal{P},\preceq_{\mathcal{P}}\rangle$ and
$\langle\mathcal{Q},\preceq_{\mathcal{Q}}\rangle$ is Scott-continuous iff for
each directed set $D\subseteq\mathcal{P}$ the image $f(D)$ is directed and
$\sup f(D)=f(\sup D)$.

###  II-B Mathematical Theory of Co-Design

We start by presenting design problems with implementation (DPIs), which can
then be composed and interconnected to form a co-design problem with
implementation (CDPI).

######  Definition II.6 (DPI).

A DPI is a tuple
$\langle\mathcal{F},\mathcal{R},\mathcal{I},\textsf{{exe}},\textsf{{eva}}\rangle$:

  * •

$\mathcal{F}$ is a poset, called functionality space;

  * •

$\mathcal{R}$ is a poset, called resource space;

  * •

$\mathcal{I}$ is a set, called implementation space;

  * •

the map $\textsf{{exe}}:\mathcal{I}\to\mathcal{F}$ maps an implementation to
the functionality it provides;

  * •

the map $\textsf{{eva}}:\mathcal{I}\to\mathcal{R}$, maps an implementation to
the resources it requires.

Given a DPI we can define a map which, given a functionality
$\textsf{{f}}\in\mathcal{F}$, returns all the non-comparable resources (i.e.,
the antichain) which provide f.

######  Definition II.7 (Functionality to resources map).

Given a DPI
$\langle\mathcal{F},\mathcal{R},\mathcal{I},\textsf{{exe}},\textsf{{eva}}\rangle$
define the map $h:\mathcal{F}\to\textsf{{A}}\mathcal{R}$ as

$\displaystyle h:$ $\displaystyle\mathcal{F}$ $\displaystyle\to$
$\displaystyle\textsf{{A}}\mathcal{R}$ (1) f $\displaystyle\mapsto$
$\displaystyle\min_{\preceq_{\mathcal{R}}}\\{\textsf{{eva}}(\textsf{{i}})\,|\,\textsf{{i}}\in\mathcal{I}\wedge\textsf{{f}}\preceq\textsf{{exe}}(\textsf{{i}})\\}.$

In particular, if a functionality is infeasible, then
$h(\textsf{{f}})=\emptyset$. We now turn our attention to “monotone” DPIs.

######  Definition II.8 (Monotone DPI).

We say a DPI
$\langle\mathcal{F},\mathcal{R},\mathcal{I},\textsf{{exe}},\textsf{{eva}}\rangle$
is monotone if:

  1. 1.

The posets $\mathcal{F}$ and $\mathcal{R}$ are CPOs.

  2. 2.

The map $h$ (see Definition II.7) is Scott-continuous.

Individual DPIs can be composed in series (i.e., the functionality of a DPI is
the resource of a second DPI) and in parallel (i.e., two DPIs share the same
resource or functionality) to obtain a CDPI. Notably, such compositions
preserve monotonicity and, thus, all related algorithmic properties. For
further details we refer to [2].

##  III Co-Design of AV-enabled Mobility Systems

###  III-A Intermodal AMoD Framework

####  III-A1 Multi-Commodity Flow Model

The transportation system and its different modes are modeled using the
digraph $\mathcal{G}=\left(\mathcal{V},\mathcal{A}\right)$, shown in Footnote
2.

Figure 1: The I-AMoD network consists of a road, a walking, and a public
transportation digraph. The coloured circles represent stops or intersections
and the black arrows denote road links, pedestrian pathways, or public transit
arcs. Dashed lines are nodes which are close geographically, while grey arrows
denote the mode-switching arcs connecting them222We thank Ms. Sonia Monti for
the illustration..

It is described through a set of nodes $\mathcal{V}$ and a set of arcs
$\mathcal{A}\subseteq\mathcal{V}\times\mathcal{V}$. Specifically, it contains
a road network layer
$\mathcal{G}_{\mathrm{R}}=\left(\mathcal{V}_{\mathrm{R}},\mathcal{A}_{\mathrm{R}}\right)$,
a public transportation layer
$\mathcal{G}_{\mathrm{P}}=\left(\mathcal{V}_{\mathrm{P}},\mathcal{A}_{\mathrm{P}}\right)$,
and a walking layer
$\mathcal{G}_{\mathrm{W}}=\left(\mathcal{V}_{\mathrm{W}},\mathcal{A}_{\mathrm{W}}\right)$.
The road network is characterized through intersections
$i\in\mathcal{V}_{\mathrm{R}}$ and road segments
$(i,j)\in\mathcal{A}_{\mathrm{R}}$. Similarly, public transportation lines are
modeled through station nodes $i\in\mathcal{V}_{\mathrm{P}}$ and line segments
$(i,j)\in\mathcal{A}_{\mathrm{P}}$. The walking network contains walkable
streets $(i,j)\in\mathcal{A}_{\mathrm{W}}$, connecting intersections
$i\in\mathcal{V}_{\mathrm{W}}$. Our model allows mode-switching arcs
$\mathcal{A}_{\mathrm{C}}\subseteq\mathcal{V}_{\mathrm{R}}\times\mathcal{V}_{\mathrm{W}}\cup\mathcal{V}_{\mathrm{W}}\times\mathcal{V}_{\mathrm{R}}\cup\mathcal{V}_{\mathrm{P}}\times\mathcal{V}_{\mathrm{W}}\cup\mathcal{V}_{\mathrm{W}}\times\mathcal{V}_{\mathrm{P}}$,
connecting the road and the public transportation layers to the walking layer.
Consequently,
$\mathcal{V}=\mathcal{V}_{\mathrm{W}}\cup\mathcal{V}_{\mathrm{R}}\cup\mathcal{V}_{\mathrm{P}}$
and
$\mathcal{A}=\mathcal{A}_{\mathrm{W}}\cup\mathcal{A}_{\mathrm{R}}\cup\mathcal{A}_{\mathrm{P}}\cup\mathcal{A}_{\mathrm{C}}$.
Consistently with the structural properties of road and walking networks in
urban environments, we assume the graph $\mathcal{G}$ to be strongly
connected. We represent customer movements by means of travel requests. A
travel request refers to a customer flow starting its trip at a node
$o\in\mathcal{V}$ and ending it at a node $d\in\mathcal{V}$.

######  Definition III.1 (Travel request).

A travel request $\rho$ is a triple
$(o,d,\alpha)\in\mathcal{V}\times\mathcal{V}\times\mathbb{R}_{+}$, described
by an origin node $o\in\mathcal{V}$, a destination node $d\in\mathcal{V}$, and
the request rate $\alpha>0$, namely, the number of customers who want to
travel from $o$ to $d$ per unit time.

To ensure that a customer is not forced to use a given transportation mode, we
assume all requests to lie on the walking digraph, i.e.,
$o_{m},d_{m}\in\mathcal{V}_{\mathrm{W}}$ for all
$m\in\mathcal{M}\coloneqq\\{1,\ldots,M\\}$.

The flow $f_{m}\left(i,j\right)\geq 0$ represents the number of customers per
unit time traversing arc $(i,j)\in\mathcal{A}$ and satisfying a travel request
$m$. Furthermore, $f_{0}\left(i,j\right)\geq 0$ denotes the flow of empty AVs
on road arcs $(i,j)\in\mathcal{A}_{\mathrm{R}}$. This accounts for rebalancing
flows of AVs between a customer’s drop-off and the next customer’s pick-up.
Assuming AVs to carry one customer at a time, the flows satisfy

$\displaystyle\sum_{i:(i,j)\in\mathcal{A}}f_{m}\left(i,j\right)+\mathds{1}_{j=o_{m}}\cdot\alpha_{m}=\sum_{k:(j,k)\in\mathcal{A}}f_{m}\left(j,k\right)+\mathds{1}_{j=d_{m}}\cdot\alpha_{m}$
$\displaystyle\hskip 136.5733pt\forall m\in\mathcal{M},\,j\in\mathcal{V}$ (2a)
$\displaystyle\sum_{i:(i,j)\in\mathcal{A}_{\mathrm{R}}}f_{\mathrm{tot}}\left(i,j\right)=\sum_{k:(j,k)\in\mathcal{A}_{\mathrm{R}}}f_{\mathrm{tot}}\left(j,k\right)\quad\forall
j\in\mathcal{V}_{\mathrm{R}},$ (2b)

where $\mathbb{1}_{j=x}$ denotes the boolean indicator function and
$f_{\mathrm{tot}}\left(i,j\right)\coloneqq
f_{0}\left(i,j\right)+\sum_{m\in\mathcal{M}}f_{m}\left(i,j\right)$.
Specifically, (2a) guarantees flows conservation for every transportation
demand, and (2b) preserves flow conservation for AVs on every road node.
Combining conservation of customers (2a) with the conservation of AVs (2b)
guarantees rebalancing AVs to match the demand.

###  III-B Travel Time and Travel Speed

The variable $t_{ij}$ denotes the time needed to traverse an arc $(i,j)$ of
length $s_{ij}$. We assume a constant walking speed on walking arcs and infer
travel times on public transportation arcs from the public transit schedules.
Assuming that the public transportation system at node $j$ operates with
frequency $\varphi_{j}$, switching from a pedestrian vertex
$i\in\mathcal{V}_{\mathrm{W}}$ to a public transit station
$j\in\mathcal{V}_{\mathrm{P}}$ takes, on average,

$t_{ij}=t_{\mathrm{WS}}+0.5\cdot
1/\varphi_{j}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{W}},$ (3)

where $t_{\mathrm{WS}}$ is a constant sidewalk-to-station travel time. We
assume that the average waiting time for AMoD vehicles is $t_{\mathrm{WR}}$,
and that switching from the road graph and the public transit graph to the
walking graph takes the transfer times $t_{\mathrm{RW}}$ and
$t_{\mathrm{SW}}$, respectively. While each road arc
$(i,j)\in\mathcal{A}_{\mathrm{R}}$ is characterized by a speed limit
$v_{\mathrm{L,V},ij}$, AVs safety protocols impose a maximum achievable
velocity $v_{\mathrm{V,a}}$. In order to prevent too slow and therefore
dangerous driving behaviours, we only consider road arcs through which the AVs
can drive at least at a fraction $\beta$ of the speed limit: Arc
$(i,j)\in\mathcal{A}_{\mathrm{R}}$ is kept in the road network iff

$v_{\mathrm{V,a}}\geq\beta\cdot v_{\mathrm{L,V},ij},$ (4)

where $\beta\in(0,1]$. We set the velocity of all arcs fulfilling condition
(4) to $v_{\mathrm{V},ij}=\min\\{v_{\mathrm{V,a}},v_{\mathrm{L,V},ij}\\}$ and
compute the travel time to traverse them as

$t_{ij}=s_{ij}/v_{\mathrm{V},ij}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{R}}.$
(5)

###  III-C Road Congestion

We capture congestion effects with a threshold model. The total flow on each
road arc $(i,j)\in\mathcal{A}_{\mathrm{R}}$, given by the sum of the AVs flow
$f_{\mathrm{tot}}\left(i,j\right)$ and the baseline usage $u_{ij}$ (e.g.,
private vehicles), must remain below the nominal capacity $c_{ij}$ of the arc:

$f_{\mathrm{tot}}\left(i,j\right)+u_{ij}\leq
c_{ij}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{R}}.$ (6)

###  III-D Energy Consumption

We compute the energy consumption of AVs for each road link considering an
urban driving cycle, scaled so that the average speed $v_{\mathrm{avg,cycle}}$
matches the free-flow speed on the link. The energy consumption is then scaled
as

$e_{ij}=e_{\mathrm{cycle}}\cdot
s_{ij}/s_{\mathrm{cycle}}\quad\forall(i,j)\in\mathcal{A}_{\mathrm{R}}.$ (7)

For the public transportation system, we assume a constant energy consumption
per unit time. This approximation is reasonable in urban environments, as the
operation of the public transportation system is independent from the number
of customers serviced, and its energy consumption is therefore customer-
invariant.

###  III-E Fleet Size

We consider a fleet of $n_{\mathrm{V,max}}$ AVs. In a time-invariant setting,
the number of vehicles on arc $(i,j)\in\mathcal{A}_{\mathrm{R}}$ is expressed
as the product of the total vehicles flow on the arc and its travel time.
Therefore, we constrain the number of used AVs as

$n_{\mathrm{V,u}}=\sum_{(i,j)\in\mathcal{A}_{\mathrm{R}}}f_{\mathrm{tot}}\left(i,j\right)\cdot
t_{ij}\leq n_{\mathrm{V,max}}.$ (8)

###  III-F Discussion

A few comments are in order. First, we assume the demand to be time-invariant
and allow flows to have fractional values. This assumption is in line with the
mesoscopic and system-level planning perspective of our study. Second, we
model congestion effects using a threshold model. This approach can be
interpreted as a municipality preventing AVs to exceed the critical flow
density on road arcs. AVs can be therefore assumed to travel at free-flow
speed [22]. This assumption is realistic for an initial low penetration of
AMoD systems in the transportation market, especially when the AV fleet is of
limited size. Finally, we allow AVs to transport one customer at the time
[23].

###  III-G Co-Design Framework

We integrate the I-AMoD framework presented in Section III-A in the co-design
formalism, allowing one to decompose the CDPI of a complex system in the DPIs
of its individual components in a modular, compositional, and systematic
fashion. We aim at computing the antichain of resources, quantified in terms
of costs, average travel time per trip, and emissions required to provide the
mobility service to a set of customers. In order to achieve this, we decompose
the CDPI in the DPIs of the individual AVs (Section III-G1), of the AV fleet
(Section III-G3), and of the public transportation system (Section III-G2).
The interconnection of the presented DPIs is presented in Section III-G4.

####  III-G1 The Autonomous Vehicle Design Problem

The AV DPI consists of selecting the maximal speed of the AVs. Under the
rationale that driving safely at higher speed requires more advanced sensing
and algorithmic capabilities, we model the achievable speed of the AVs
$v_{\mathrm{V,a}}$ as a monotone function of the vehicle fixed costs
$C_{\mathrm{V,f}}$ (resulting from the cost of the vehicle $C_{\mathrm{V,v}}$
and the cost of its automation $C_{\mathrm{V,a}}$) and of the mileage-
dependent operational costs $C_{\mathrm{V,o}}$ (accounting for maintenance,
cleaning, energy consumption, depreciation, and opportunity costs [24]). In
this setting, the AV DPI provides the functionality $v_{\mathrm{V,a}}$ and
requires the resources $C_{\mathrm{V,f}}$ and $C_{\mathrm{V,o}}$.
Consequently, the functionality space is
$\mathcal{F}_{\mathrm{V}}=\overline{\mathbb{R}}_{+}$, and the resources space
is
$\mathcal{R}_{\mathrm{V}}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$.

####  III-G2 The Subway Design Problem

We design the public transit infrastructure by means of the service frequency
introduced in Section III-B. Specifically, we assume that the service
frequency $\varphi_{j}$ scales linearly with the size of the train fleet
$n_{\mathrm{S}}$ as

$\varphi_{j}/\varphi_{j,\mathrm{base}}=n_{\mathrm{S}}/n_{\mathrm{S,base}}.$
(9)

We relate a train fleet of size $n_{\mathrm{S}}$ to the fixed costs
$C_{\mathrm{S,f}}$ (accounting for train and infrastructural costs) and to the
operational costs $C_{\mathrm{S,o}}$ (accounting for energy consumption,
vehicles depreciation, and train operators’ wages). Given the passenger-
independent public transit operation in today’s cities, we reasonably assume
the operational costs $C_{\mathrm{S,o}}$ to be mileage independent and to only
vary with the size of the fleet. Formally, the number of acquired trains
$n_{\mathrm{S,a}}=n_{\mathrm{S}}-n_{\mathrm{S,base}}$ is a functionality,
whereas $C_{\mathrm{S,f}}$ and $C_{\mathrm{S,o}}$ are resources. The
functionality space is $\mathcal{F}_{\mathrm{S}}=\overline{\mathbb{N}}$ and
the resources space is
$\mathcal{R}_{\mathrm{S}}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$.

####  III-G3 The I-AMoD Framework Design Problem

The I-AMoD DPI considers demand satisfaction as a functionality. Formally,
$\mathcal{F}_{\mathrm{O}}=2^{\mathcal{V}\times\mathcal{V}\times\overline{\mathbb{R}}_{+}}$
with the partial order $\preceq_{\mathcal{F}_{\mathrm{O}}}$ defined by
$\mathcal{D}_{1}\coloneqq\\{(o^{1}_{i},d^{1}_{i},\alpha^{1}_{i})\\}_{i=1}^{M_{1}}\preceq_{\mathcal{F}_{\mathrm{O}}}\\{(o^{2}_{i},d^{2}_{i},\alpha^{2}_{i})\\}_{i=1}^{M_{2}}\eqqcolon\mathcal{D}_{2}$
iff for all $(o^{1},d^{1},\alpha^{1})\in\mathcal{D}_{1}$ there is some
$(o^{2},d^{2},\alpha^{2})\in\mathcal{D}_{2}$ with $o^{1}=o^{2}$,
$d^{1}=d^{2}$, and $\alpha^{2}_{i}\geq\alpha^{1}_{i}$. In other words,
$\mathcal{D}_{1}\preceq_{\mathcal{F}_{\mathrm{O}}}\mathcal{D}_{2}$ if every
travel request in $\mathcal{D}_{1}$ is in $\mathcal{D}_{2}$ too. To
successfully satisfy a given set of travel requests, we require the following
resources: (i) the achievable speed of the AVs $v_{\mathrm{V,a}}$, (ii) the
number of available AVs per fleet $n_{\mathrm{V,max}}$, (iii) the number of
trains $n_{\mathrm{S,a}}$ acquired by the public transportation system, and
(iv) the average travel time of a trip

$t_{\mathrm{avg}}\coloneqq\frac{1}{\alpha_{\mathrm{tot}}}\cdot\sum_{m\in\mathcal{M},(i,j)\in\mathcal{A}}t_{ij}\cdot
f_{m}\left(i,j\right),$ (10)

with $\alpha_{\mathrm{tot}}\coloneqq\sum_{m\in\mathcal{M}}\alpha_{m}$, (v) the
total distance driven by the AVs per unit time

$s_{\mathrm{V,tot}}\coloneqq\sum_{(i,j)\in\mathcal{A}_{\mathrm{R}}}s_{ij}\cdot
f_{\mathrm{tot}}\left(i,j\right),$ (11)

(vi) the total AVs CO2 emissions per unit time

$m_{\mathrm{CO_{2},V,tot}}\coloneqq\gamma\cdot\sum_{(i,j)\in\mathcal{A}_{\mathrm{R}}}e_{ij}\cdot
f_{\mathrm{tot}}\left(i,j\right),$ (12)

where $\gamma$ relates the energy consumption and the CO2 emissions. We assume
that customers’ trips and AMoD rebalancing strategies are chosen to maximize
customers’ welfare, defined through the average travel time
$t_{\mathrm{avg}}$. Hence, we link the functionality and resources of the
I-AMoD DPI through the following optimization problem:

$\begin{split}\min_{\begin{subarray}{c}f_{m}\left(\cdot,\cdot\right)\geq 0,\\\
f_{0}\left(\cdot,\cdot\right)\geq
0\end{subarray}}t_{\mathrm{avg}}&=\frac{1}{\alpha_{\mathrm{tot}}}\sum_{m\in\mathcal{M},(i,j)\in\mathcal{A}}t_{ij}\cdot
f_{m}\left(i,j\right)\\\ &\mathrm{s.t.\ Eq.}\eqref{eq:flowconstotal},\
\mathrm{Eq.}\eqref{eq:capacity},\
\mathrm{Eq.}\eqref{eq:fleetsizecar}.\end{split}$ (13)

Formally, $\mathcal{F}_{\mathrm{O}}=\overline{\mathbb{R}}_{+}$, and
$\mathcal{R}_{\mathrm{O}}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{N}}\times\overline{\mathbb{N}}\times\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$.

######  Remark.

In general, the optimization problem (13) might possess multiple optimal
solutions, making the relation between resources and functionality ill-posed.
To overcome this subtlety, if two solutions share the same average travel
time, we select the one incurring in the lowest mileage.

####  III-G4 The Monotone Co-Design Problem

The functionality of the system is to provide mobility service to the
customers. Formally, the functionality provided by the CDPI is the set of
travel requests. To provide the mobility service, the following three
resources are required. First, on the customers’ side, we require an average
travel time, defined in (10). Second, on the municipality side, the resource
is the total transportation cost of the intermodal mobility system. Assuming
an average vehicles’ life $l_{\mathrm{V}}$, an average trains’ life
$l_{\mathrm{S}}$, and a baseline subway fleet of $n_{\mathrm{S,base}}$ trains,
we express the total costs as

$C_{\mathrm{tot}}=C_{\mathrm{V}}+C_{\mathrm{S}},$ (14)

where $C_{\mathrm{V}}$ is the AV-related cost

$C_{\mathrm{V}}=\frac{C_{\mathrm{V,f}}}{l_{\mathrm{V}}}\cdot
n_{\mathrm{V,max}}+C_{\mathrm{V,o}}\cdot s_{\mathrm{V,tot}},$ (15)

and $C_{\mathrm{S}}$ is the public transit-related cost

$C_{\mathrm{S}}=\frac{C_{\mathrm{S,f}}}{l_{\mathrm{S}}}\cdot
n_{\mathrm{S,a}}+C_{\mathrm{S,o}}.$ (16)

Third, on the environmental side, the resources are the total CO2 emissions

$m_{\mathrm{CO_{2},tot}}=m_{\mathrm{CO_{2},V,tot}}+m_{\mathrm{CO_{2},S}}\cdot
n_{\mathrm{S}},$ (17)

where $m_{\mathrm{CO_{2},S}}$ represents the CO2 emissions of a single train.
Formally, the set of travel requests $\\{\rho_{m}\\}_{m\in\mathcal{M}}$ is the
CDPI functionality, whereas $t_{\mathrm{avg}}$, $C_{\mathrm{tot}}$, and
$m_{\mathrm{CO_{2},tot}}$ are its resources. Consistently, the functionality
space is $\mathcal{F}=\overline{\mathbb{R}}_{+}$ and the resources space is
$\mathcal{R}=\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}}_{+}$.
Note that the resulting CDPI (Figure 2) is indeed monotone, since it consists
of the interconnection of monotone DPIs [2].

####  III-G5 Discussion

A few comments are in order. First, we lump the autonomy functionalities in
its achievable velocity. We leave to future research more elaborated AV
models, accounting for instance for accidents rates [25] and for safety
levels. Second, we assume the service frequency of the subway system to scale
linearly with the number of trains. We inherently rely on the assumption that
the existing infrastructure can homogeneously accommodate the acquired train
cars. To justify the assumption, we include an upper bound on the number of
potentially acquirable trains in our case study design in Section IV. Third,
we highlight that the I-AMoD framework is only one of the many feasible ways
to map total demand to travel time, costs, and emissions. Specifically,
practitioners can easily replace the corresponding DPI with more sophisticated
models (e.g., simulation-based frameworks like AMoDeus [26]), as long as the
monotonicity of the DPI is preserved. In our setting, we conjecture the
customers’ and vehicles’ routes to be centrally controlled by the municipality
in a socially-optimal fashion. Implicitly, we rely on the existence of
effective incentives aligning private and societal interests. The study of
such incentives represents an avenue for future research. Fourth, we assume a
homogenous fleet of AVs. Nevertheless, our model is readily extendable to
capture heterogeneous fleets. Finally, we consider a fixed travel demand, and
compute the antichain of resources providing it. Nonetheless, our
formalization can be readily extended to arbitrary demand models preserving
the monotonicity of the CDPI (accounting for instance for elastic effects). We
leave this topic to future research.

I-AMoDVehicleSubway$\preceq$$\preceq$$\preceq$$\times$$\preceq$$\preceq$$\times$$\preceq$$\preceq$$\times$$\preceq$$\preceq$$\preceq$$+$$\preceq$$+$$\preceq$$\times$$\preceq$$\preceq$$+$$+$$\preceq$$+$$\preceq$$v_{\mathrm{V,a}}$$n_{\mathrm{S,a}}$$s_{\mathrm{V,tot}}$$C_{\mathrm{V,o}}$$C_{\mathrm{V,f}}$co-
design
constraint$C_{\mathrm{S,o}}$$C_{\mathrm{S,f}}$$n_{\mathrm{V,max}}$$C_{\mathrm{tot}}$$\alpha_{\mathrm{tot}}$$m_{\mathrm{CO_{2},V,tot}}$$n_{\mathrm{S}}$$m_{\mathrm{CO_{2},S}}$$m_{\mathrm{CO_{2},tot}}$$t_{\mathrm{avg}}$$l_{\mathrm{V}}$$l_{\mathrm{S}}$total
costaverage travel timetotal emissionstotal request rate

Figure 2: Schematic representation of the CDPI. In solid green the provided
functionalities and in dashed red the required resources. The edges represent
co-design constraints: The resources required by a first design problem are
the lower bound for the functionalities provided by the second one.

##  IV Results

In this section, we leverage the framework presented in Section III to perform
a real-world case study of Washington D.C., USA. Section IV-A details the case
study. We then present numerical results in Sections IV-B and IV-C.

###  IV-A Case Study

We base our studies on a real-world case of the urban area of Washington D.C.,
USA. We import the road network and its features from OpenStreetMap [27]. The
public transit network and its schedules are extracted from the GTFS data
[28]. Demand data is obtained by merging the origin-destination pairs of the
morning peak of May 31st 2017 provided by taxi companies [29] and the
Washington Metropolitan Area Transit Authority (WMATA) [23]. Given the lack of
reliable demand data for the MetroBus system, we focus our studies on the
MetroRail system and its design, inherently assuming MetroBus commuters to be
unaffected by our design methodology. To conform with the large presence of
ride-hailing companies, we scale the taxi demand rate by factor of 5 [30].
Overall, the demand dataset includes 15,872 travel requests, corresponding to
a demand rate of 24.22 $\nicefrac{{requests}}{{s}}$. To account for congestion
effects, we compute the nominal road capacity as in [31] and assume an average
baseline road usage of 93%, in line with [32]. We summarize the main
parameters together with their bibliographic sources in Table I. In the
remainder of this section, we tailor and solve the co-design problem presented
in Section III through the PyMCDP solver [33], and investigate the influence
of different AVs costs on the design objectives and strategies.

| Parameter | Name | Value | Units | Source
---|---|---|---|---|---
| Road usage |  | $u_{ij}$ | 93 | % |  [32]
Vehicle |  |  |  | C1 | C2.1 | C2.2 |  |
Operational cost | $C_{\mathrm{V,o}}$ | 0.084 | 0.084 | 0.062 | $\nicefrac{{USD}}{{mile}}$ |  [34, 35]
Cost | $C_{\mathrm{V}}$ | 32,000 | 32,000 | 26,000 | $\nicefrac{{USD}}{{car}}$ |  [34]
Automation cost | 20 mph | $C_{\mathrm{V,a}}$ | 15,000 | 20,000 | 3,700 | $\nicefrac{{USD}}{{car}}$ |  [35, 36, 37, 38, 39]
25 mph | 15,000 | 30,000 | 4,400 | $\nicefrac{{USD}}{{car}}$ |  [35, 36, 37, 38, 39]
30 mph | 15,000 | 55,000 | 6,200 | $\nicefrac{{USD}}{{car}}$ |  [35, 36, 37, 38, 39]
35 mph | 15,000 | 90,000 | 8,700 | $\nicefrac{{USD}}{{car}}$ |  [35, 36, 37, 38, 39]
40 mph | 15,000 | 115,000 | 9,800 | $\nicefrac{{USD}}{{car}}$ |  [35, 36, 37, 38, 39]
45 mph | 15,000 | 130,000 | 12,000 | $\nicefrac{{USD}}{{car}}$ |  [35, 36, 37, 38, 39]
50 mph | 15,000 | 150,000 | 13,000 | $\nicefrac{{USD}}{{car}}$ |  [35, 36, 37, 38, 39]
Vehicle life | $l_{\mathrm{V}}$ | 5 | 5 | 5 | years |  [34]
CO2 per Joule | $\gamma$ | 0.14 | 0.14 | 0.14 | $\nicefrac{{g}}{{kJ}}$ |  [40]
Time $\mathcal{G}_{\mathrm{W}}$ to $\mathcal{G}_{\mathrm{R}}$ | $t_{\mathrm{WR}}$ | 300 | 300 | 300 | s | -
Time $\mathcal{G}_{\mathrm{R}}$ to $\mathcal{G}_{\mathrm{W}}$ | $t_{\mathrm{RW}}$ | 60 | 60 | 60 | s | -
| Speed fraction | $\beta$ | $\frac{1}{1.3}$ | $\frac{1}{1.3}$ | $\frac{1}{1.3}$ | - | -
Public transit | Operational cost | 100 % | $C_{\mathrm{S,o}}$ | 148,000,000 | $\nicefrac{{USD}}{{year}}$ |  [41]
133 % | 197,000,000 | $\nicefrac{{USD}}{{year}}$ |  [41]
200 % | 295,000,000 | $\nicefrac{{USD}}{{year}}$ |  [41]
Fixed cost | $C_{\mathrm{S,f}}$ | 14,500,000 | $\nicefrac{{USD}}{{train}}$ |  [42]
Train life | $l_{\mathrm{S}}$ | 30 | years |  [42]
Emissions/train | $m_{\mathrm{CO_{2},S}}$ | 140,000 | $\nicefrac{{kg}}{{year}}$ |  [43]
Fleet baseline | $n_{\mathrm{S,base}}$ | 112 | trains |  [42]
Service frequency | $\varphi_{j,\mathrm{base}}$ | $1/6$ | $\nicefrac{{1}}{{min}}$ |  [44]
Time $\mathcal{G}_{\mathrm{W}}$ to $\mathcal{G}_{\mathrm{P}}$ | $t_{\mathrm{WS}}$ | $60$ | s | -
Time $\mathcal{G}_{\mathrm{P}}$ to $\mathcal{G}_{\mathrm{W}}$ | $t_{\mathrm{SW}}$ | $60$ | s | -
TABLE I: Parameters, variables, numbers, and units for the case studies.

###  IV-B Case 1 - Constant Cost of Automation

In line with [35, 36, 37, 38, 39], we first assume an average achievable-
velocity-independent cost of automation. As discussed in Section III, we
design the system by means of subway service frequency, AV fleet size, and
achievable free-flow speed. Specifically, we allow the municipality to (i)
increase the subway service frequency $\varphi_{j}$ by a factor of 0%, 33%, or
100%, (ii) deploy an AVs fleet of size
$n_{\mathrm{V,max}}\in\\{0,500,1000,\ldots,6000\\}$ vehicles, and (iii) design
the single AV achievable velocity
$v_{\mathrm{V,a}}\in\\{20\,\mathrm{mph},25\,\mathrm{mph},\ldots,50\,\mathrm{mph}\\}$.
We assume the AVs fleet to be composed of battery electric BEV-250 mile AVs
[34]. In Figure 3(a), we show the solution of the co-design problem by
reporting the antichain consisting of the total transportation cost, average
travel time, and total CO2 emissions. These solutions are _rational_ (and not
comparable) in the sense that there exists no instance which simultaneously
yields lower cost, average travel time, and emissions.

(a) Left: Three-dimensional representation of antichain elements and their
projection in the cost-time space. Right: Two-dimensional projections.

(b) Results for constant automation costs. On the left, the two-dimensional
representation of the antichain elements: In red are the unfeasible
strategies, in orange the feasible but irrational solutions, and in green the
Pareto front. On the right, the implementations corresponding to the
highlighted antichain elements, quantified in terms of achievable vehicle
speed, AV fleet size, and train fleet size.

Figure 3: Solution of the CDPI: state-of-the art case.

For the sake of clarity, we opt for a two-dimensional antichain
representation, by translating and including the emissions in the total cost.
To do so, we consider the conversion factor 40 $\nicefrac{{USD}}{{kg}}$ [45].
Note that since this transformation preserves the monotonicity of the CDPI it
smoothly integrates in our framework. Doing so, we can conveniently depict the
co-design strategies through the two-dimensional antichain (Figure 3(b), left)
and the corresponding municipality actions (Figure 3(b), right). Generally, as
the municipality budget increases, the average travel time per trip required
to satisfy the given demand decreases, reaching a minimum of about 17.1 min
with an expense of around 43 $\nicefrac{{MilUSD}}{{month}}$. This
configuration corresponds to a fleet of 5,500 AVs able to drive at 50 mph and
to the doubling of the current MetroRail train fleet. On the other hand, the
smallest rational investment of 12.9 $\nicefrac{{MilUSD}}{{month}}$ leads to a
42 % higher average travel time, corresponding to a inexistent autonomous
fleet and an unchanged subway infrastructure. Notably, an expense of 23
$\nicefrac{{MilUSD}}{{month}}$ (48 % lower than the highest rational
investment) only increases the minimal required travel time by 9 %, requiring
a fleet of 4,000 vehicles able to drive at 35 mph and no acquisition of
trains. Conversely, an investment of 15.6 $\nicefrac{{MilUSD}}{{month}}$ (just
2 $\nicefrac{{MilUSD}}{{month}}$ more than the minimal rational investment)
provides a 3 min shorter travel time. Remarkably, the design of AVs able to
exceed 40 mph only improves the average travel time by 6 %, and it is rational
just starting from an expense of 22.8 $\nicefrac{{MilUSD}}{{month}}$. This
suggests that the design of faster vehicles mainly results in higher emission
rates and costs, without substantially contributing to a more time-efficient
demand satisfaction. Finally, it is rational to improve the subway system only
starting from a budget of 28.5 $\nicefrac{{MilUSD}}{{month}}$, leading to a
travel time improvement of just 4 %. This trend can be explained with the high
train acquisition and increased operation costs, related to the subway
reinforcement. We expect this phenomenon to be more marked for other cities,
considering the moderate operation costs of the MetroRail subway system due to
its automation [44] and related benefits [46].

###  IV-C Case 2 - Speed-Dependent Automation Costs

To relax the potentially unrealistic assumption of a velocity-independent
automation cost, we consider a performance-dependent cost structure. The large
variance in sensing technologies and their reported performances [47] suggests
that this rationale is reasonable. Indeed, the technology required today to
safely operate an autonomous vehicle at 50 mph is substantially more
sophisticated, and therefore more expensive, than the one needed at 20 mph. To
this end, we adopt the cost structure reported in Table I. Furthermore, the
frenetic evolution of automation techniques intricates their monetary
quantification. Therefore, we perform our studies with current (2020) costs as
well as with their projections for the upcoming decade (2025) [48, 34].

####  IV-C1 Case 2.1 - 2020

We study the hypothetical case of an immediate AV fleet deployment. We
introduce the aforementioned velocity-dependent automation cost structure and
obtain the results reported in Figure 4(a). Comparing these results with the
state-of-the-art parameters presented in Figure 3 confirms the previously
observed trend concerning high vehicle speeds. Indeed, spending 24.9
$\nicefrac{{MilUSD}}{{month}}$ (55 % lower than the highest rational expense)
only increases the average travel time by 10 %, requiring a fleet of 3,000 AVs
at 40 mph and no subway interventions. Nevertheless, the comparison shows two
substantial differences. First, the budget required to reach the minimum
travel time of 17.1 min is 28 % higher compared to the previous case, and
consists of the same strategy for the municipality, i.e., doubling the train
fleet and having a fleet of 5,500 AVs at 50 mph. Second, the higher vehicle
costs result in an average AV fleet growth of 5 %, an average velocity
reduction of 9 %, and an average train fleet growth of 7 %. This trend
suggests that, compared to Case 1, rational design strategies foster larger
fleets and less performing AVs.

(a) Results for speed-dependent automation costs in 2020.

(b) Results for speed-dependent automation costs in 2025.

Figure 4: Results for the speed-dependent automation costs. On the left, the
two-dimensional representation of the antichain elements: In red are the
unfeasible strategies, in orange the feasible but irrational solutions, and in
green the Pareto front. On the right, the implementations corresponding to the
highlighted antichain elements.

####  IV-C2 Case 2.2 - 2025

Experts forecast a substantial decrease of automation costs (up to 90 %) in
the next decade, mainly due to mass-production of the AVs sensing technology
[48, 49]. In line with this prediction, we inspect the futuristic scenario by
solving the CDPI for the adapted automation costs, and report the results in
Figure 4(b). Two comments are in order. First, the maximal rational budget is
25 % lower than in the immediate adoption case. Second, the reduction in
autonomy costs clearly eases the acquisition of more performant AVs,
increasing the average vehicle speed by 10 %. As a direct consequence, the AV
and train fleets are reduced in size by 5 % and 10 %, respectively.

###  IV-D Discussion

We conclude the analysis of our case study with two final comments. First, the
presented case studies illustrate the ability of our framework to extract the
set of rational design strategies for an AV-enabled mobility system. This way,
stakeholders such as AVs companies, transportation authorities, and policy
makers can get transparent and interpretable insights on the impact of future
interventions. Second, we perform a sensitivity analysis through the variation
of the autonomy cost structures. On the one hand, this reveals a clear
transition from small fleets of fast AVs (in the case of low autonomy costs)
to slow fleets of numerous AVs (in the case of high autonomy costs). On the
other hand, our studies highlight that investments in the public transit
infrastructure are rational only when large budgets are available. Indeed, the
onerous train acquisition and operation costs lead to a comparative advantage
of AV-based mobility.

##  V Conclusion

In this paper, we leveraged the mathematical theory of co-design to propose a
design framework for AV-enabled mobility systems. Specifically, the nature of
our framework allows both for the modular and compositional interconnection of
the DPIs of different mobility options and for multiple objectives. Starting
from the multi-commodity flow model of an I-AMoD system, we optimize the
design of AVs and public transit both from a vehicle-centric and fleet-level
perspective. In particular, we studied the problem of deploying a fleet of AVs
providing on-demand mobility in cooperation with public transit, optimizing
the speed achievable by the vehicles, the fleet size, and the service
frequency of the subway lines. Our framework allows the stakeholders involved
in the mobility ecosystem, from vehicle developers all the way to mobility-as-
a-service companies and governmental authorities, to characterize rational
trajectories for technology and investment development. We showcased our
methodology on a real-world case study of Washington D.C., USA. Notably, our
problem formulation allows for a systematic analysis of incomparable
objectives, providing stakeholders with analytical insights for the socio-
technical design of AV-enabled mobility systems. This work opens the field for
the following future research streams:
_Modeling:_ First, we would like to extend the presented framework to capture
additional modes of transportation, such as micromobility, and heterogeneous
fleets with different self-driving infrastructures, propulsion systems, and
passenger capacity. Second, we would like to investigate variable demand
models. Finally, we would like to analyze the interactions between multiple
stakeholders, characterizing the equilibrium arising from their conflicting
interests.
_Algorithms:_ It is of interest to tailor co-design algorithmic frameworks to
the particular case of transportation DPIs, possibly leveraging their specific
structure.
_Application:_ Finally, we would like to devise a user-friendly web interface
which supports mobility stakeholders to reason about strategic interventions
in urban areas.

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