# Semi-on-Demand Hybrid Transit Route Design with Shared Autonomous Mobility
Services

Max T.M. Ng<EMAIL_ADDRESS>Northwestern University Transportation
Center  Florian Dandl<EMAIL_ADDRESS>Chair of Traffic Engineering and
Control, Technical University of Munich  Hani S. Mahmassani
<EMAIL_ADDRESS>Northwestern University Transportation Center  Klaus
Bogenberger<EMAIL_ADDRESS>Chair of Traffic Engineering and Control,
Technical University of Munich

###### Abstract

This study examines the route design of a semi-on-demand hybrid route
directional service in the public transit network, offering on-demand flexible
route service in low-density areas and fixed route service in higher-density
areas with Shared Autonomous Mobility Service (SAMS). The study develops
analytically tractable cost expressions that capture access, waiting, and
riding costs for users, and distance-based operating and time-based vehicle
costs for operators. Two formulations are presented for strategic and tactical
decisions in flexible route portion, fleet size, headway, and vehicle size
optimization, enabling the determination of route types between fixed, hybrid,
and flexible routes based on demand, cost, and operational parameters. The
practical applications and benefits of semi-on-demand feeders are demonstrated
with numerical examples and a large-scale case study in the Chicago
metropolitan area. Findings reveal scenarios in which flexible route portions
serving passengers located further away reduce total costs, particularly user
costs. Lower operating costs in lower-demand areas favor more flexible routes,
whereas higher demand densities favor more traditional line-based operations.
On two studied lines, a current cost forecast favors smaller vehicles with
flexible routes, but operating constraints and higher operating costs would
favor bigger vehicles with hybrid routes. The study provides an analytical
tool to design SAMS as directional services and transit feeders, and tractable
continuous approximation formulations for future research in transit network
design.

_Keywords_ semi-on-demand $\cdot$ transit feeder $\cdot$ shared autonomous
mobility service $\cdot$ transit design $\cdot$ flexible route

##  1 Introduction

###  1.1 Motivation

Mobility, a cornerstone of socio-economic well-being, is often unevenly
distributed due to public transit systems’ inconsistent availability,
accessibility, and quality. One of the key hurdles limiting transit access
outside city centers is the first-mile-last-mile problem — from a user
perspective, public transit is unattractive if reaching it requires walking
for an excessive distance from one’s origin or to the final destination. From
an operator perspective, line-based transit operations are often cost-
inefficient in low-density areas. Taxis and rideshare services could be
convenient ways to plug this gap but are not economical nor socially desirable
from a congestion or environmental standpoint. Furthermore, long-term trends,
such as suburbanization, and more recent trends associated with the COVID-19
pandemic, such as reduced service and increased prevalence of remote/hybrid
work (Tahlyan et al. 2022), have greatly decreased transit ridership. Poor
accessibility deters many travelers from relying on transit and makes driving
their private vehicles the sole feasible mobility option, despite its higher
costs for users (fuel and maintenance), society (delay due to congestion and
space requirements for parking), and the environment (15% of global carbon
emissions are contributed by road transport (Ritchie 2020)) relative to
transit.

Autonomous vehicles (AVs) could potentially alleviate these mobility concerns
by offering convenient point-to-point journeys (Frei, Hyland, and Mahmassani
2017). However, this convenience is predicted to induce demand, increasing
total traffic volume on the road and leading to greater emissions and
congestion if transit options are not also improved (Xu, Mahmassani, and Chen
2019). This work addresses these problems by exploring semi-on-demand hybrid
route shared autonomous mobility service (SAMS) to plug the first-mile-last-
mile gap, at least for certain situations that we will delineate. Such a
system balances optimally the economies of scale of fixed route public transit
with the accessibility of flexible route taxis or shared autonomous vehicles
(SAVs). Addressing the first-mile-last-mile problem can expand transit
coverage to provide seamless door-to-door transportation options to wider
populations in a cost-effective, green, and equitable manner.

###  1.2 Problem Description

We consider a semi-on-demand hybrid flexible/fixed route service along a
corridor (Figure 1). This entails a vehicle serving an on-demand flexible
route portion for areas further from a train station/downtown (likely with
more scattered demand) and — included seamlessly in the same route — a fixed
route portion for areas close to a train station/downtown (likely with more
concentrated demand). In the flexible route portion, there is a trade-off
between saving in passengers’ access time and additional detours required for
both passengers on board and vehicles. Consequently, the flexible route
portion $x_{f}$ which demarcates flexible and fixed route portions is
determined to minimize the total costs of users and operators at the route
design level. This demarcation of fixed and flexible service areas on a
particular route contributes to a clear and regular service pattern (so riders
along the route would know whether to walk to a fixed stop or be picked up).

We focus on scenarios of directional demand, such as feeder service to train
stations or commuter routes from suburbs to downtown. While our derivation and
results cater to station/downtown-bound services, they equally apply to both
inbound and outbound services. The use of AVs allows real-time change of
routes in the flexible route portion that would pose operational and safety
concerns to traditional human-driven buses. Additionally, this study considers
the effects of operational cost reduction brought by SAVs on the flexible
route portion and fleet size.

Figure 1: Illustration of Fixed Route, On-demand Flexible Route, and Semi-on-
Demand Hybrid (Fixed/Flexible) Route as a Feeder Service

###  1.3 Objectives and Contributions

We present two cost formulations for the semi-on-demand hybrid route service:
one strategic and one tactical. The strategic formulation optimizes the
flexible route portion and fleet size, capturing the additional vehicle
requirement for possible detours of a flexible or hybrid route. In addition to
these two variables, the tactical formulation optimizes headway for new route
planning to balance the effects on the waiting time and vehicle capital cost.
It also considers vehicle types (sizes) in the optimization, resulting in a
comprehensive model to support network planning, select vehicle and fleet
sizes simultaneously, and design flexible route portions and headways.

This paper provides three key contributions.

  * •

Conceptually, we investigate the applicability and benefits of a semi-on-
demand hybrid route form with general cost formulations that also cover fixed
and flexible route forms. This delineates the conditions under which each of
the three route forms is optimal, supplemented with tractable cost
approximations.

  * •

Methodologically, we derive an analytical approach to determine the optimal
route form and, for hybrid routes, the optimal flexible route portion and
fleet size, without the need for computationally expensive simulation-based
optimization. We also conduct joint numerical optimization with the headway
and vehicle size. These allow efficient transit service design and sensitivity
study with respect to demand, geospatial, service, and cost parameters.

  * •

From an application standpoint, we demonstrate the benefits and applications
of semi-on-demand feeders with numerical examples and a large-scale real-world
case study in the Chicago metropolitan area, USA.

The paper is structured as follows: Section 2 reviews relevant literature,
focusing on flexible transit service design and SAMS. We then introduce the
two mathematical formulations for costs and derive optimal values for the
respective decision variables in Section 3. To illustrate these formulations,
we present two numerical examples in Section 4 — the first focuses on the
flexible route portion and fleet size, and the second incorporates headway and
vehicle size. Subsequently, a city-scale case study of feeder services in
Chicago is discussed in Section 5. Finally, we conclude the paper with a
summary of the findings, limitations, and future research directions in
Section 6.

##  2 Background

###  2.1 Flexible Transit Route Design

Previous research efforts mainly focused on flexible/adaptive transit route
designs differentiating usage of fixed route and demand-responsive transit
(DRT) services. Chang and Schonfeld (1991) analytically compared fixed and
one-stop flexible service and optimized fleet and vehicle sizes. Quadrifoglio
and Li (2009) and Li and Quadrifoglio (2010) designed analytical models to
identify the condition for switching between demand-responsive and fixed route
policies that maximize service quality. Nourbakhsh and Ouyang (2012) analyzed
flexible route transit systems in a corridor with continuous approximation.
Rich et al. (2023) compared fixed route and demand-responsive methods as
feeders for light rail transit using agent-based simulation. Other studies
looked into parallel flexible routes (Chen and Nie 2017) and joint design of
transit and DRT with queuing network model (Liu and Ouyang 2021) and path-
based network optimization (Steiner and Irnich 2020). More recently, Martínez
Mori, Speranza, and Samaranayake (2023) showed mathematically the value of
flexibility brought by DRT. Calabrò et al. (2023) chose a fixed route or DRT
feeder and designed the trunk service accordingly. For other recent advances
in demand-responsive systems, see the survey by Vansteenwegen et al. (2022).
Overall, these studies considered decisions to deploy flexible route or fixed
route services. In contrast, this study explores the hybrid route, which is a
continuous spectrum defined by the flexible route portion between the two
service modes.

Research on semi-flexible transit systems involved various optimization
strategies (see a survey by Errico et al. (2013) on earlier planning efforts).
Qiu, Li, and Zhang (2014) used an analytical model and simulation to decide
which curb-to-curb stops are served, aiming to minimize user costs. Galarza
Montenegro, Sörensen, and Vansteenwegen (2021) proposed a demand-responsive
feeder system with mandatory and optional bus stops. Zheng et al. (2019)
introduced a blend of flexible and fixed routes via meeting points. Leffler et
al. (2021) investigated on-demand operational policies for autonomous
vehicles, showing improved waiting times on trunk routes in particular. Mishra
and Mehran (2023) optimized service headway and curb-to-curb service requests
using non-dominated sorting to generate a Pareto set. For stochastic
approaches, Rambha, Boyles, and Waller (2016) formulated an adaptive routing
problem as a finite horizon Markov decision process. Li, Lee, and Lo (2023)
looked into a stochastic problem of zonal on-demand flexible bus routes.
Olivos, Silva, and Vinel (2023) proposed a Markovian continuous approximation-
based semi-flexible system model on a single bus operation. Other similar
service models include "customized bus" with demand-based pre-designed routes
(Huang et al. 2020, Abdelwahed et al. 2023), hybrid first-mile-last-mile
service between shuttle and transportation network companies (TNCs) (Grahn,
Qian, and Hendrickson 2022), and demand-adaptive system (Errico et al. 2021).
While the previously discussed literature explored a wide range of flexible
transit services, this study focuses on the proposed semi-on-demand routes,
which pre-design portions of flexible and fixed routes for consistent service
patterns. It combines the economies of scale and predictability of scheduled
fixed routes and the door-to-door convenience of flexible routes.
Additionally, most previous efforts relied on a mix of simulation, heuristics,
or mixed-integer linear programming, which hinders intuitive sensitive
analysis based on design parameters. This study provides an analytical
approach to total cost minimization for delineating the conditions of the
optimal route form (fixed/hybrid/flexible) with the optimal flexible route
portion and fleet size.

###  2.2 Shared Autonomous Mobility Services (SAMS) in Public Transit Systems

The shared element of SAMS improves service quality and operational efficiency
(Hyland and Mahmassani 2020), with their potential synergies as part of the
public transit systems highlighted by Shen, Zhang, and Zhao (2018), Salazar et
al. (2020), and Cortina, Chiabaut, and Leclercq (2023) in case studies of
Singapore, New York, and Lyon. Badia and Jenelius (2021) designed feeder
systems with AVs and studied cost with continuous approximation. Gurumurthy,
Kockelman, and Zuniga-Garcia (2020) demonstrated the potential of SAVs as
feeders for public transit in Austin, proposing fare benefits for transit
users to increase transit coverage and reduce walking distances. Levin et al.
(2019) looked into the optimal integration of SAVs and transit with continuous
approximation and linear programming. Transit was only used when it reduced
total system travel time. Luo, Samaranayake, and Banerjee (2021) optimized
mobility-on-demand service flow, transit frequency, and pricing jointly. Ng et
al. (2024) redesigned existing multimodal transit networks with SAMS as point-
to-point and feeder services. Some research (e.g., Zhang, Tafreshian, and
Masoud (2020), Liu, Qu, and Ma (2021), Tian et al. (2022)) investigated an
evolving idea of autonomous modular transit and their roles in the overall
network.

A contrasting body of research evaluated SAVs as replacements for existing
transit systems. Sieber et al. (2020) proposed that autonomous mobility on
demand (AMoD) could potentially replace rural trains in Switzerland, analyzing
the difference in costs and service levels. Ng and Mahmassani (2023) and
Volakakis, Ng, and Mahmassani (2023) showed around 20% performance improvement
by replacing fixed bus routes with flexible routes in suburbs. Mo et al.
(2021) analyzed the impact of dynamic adjustable supply strategies and
regulations in SAV fleet size and transit headway changes on system
efficiency. Cao and Ceder (2019) proposed stop-skipping with autonomous
shuttles and evaluated the number of extra vehicles required.

Lastly, several researchers have sought to optimize SAV design variables, such
as vehicle size (Alonso-Mora et al. 2017) and fleet size (Pinto et al. 2020,
Dandl et al. 2021). Pinto et al. (2020) highlighted the need to consider
waiting time in optimization, presenting a bi-level mathematical programming
formulation and solution approach. Sadrani, Tirachini, and Antoniou (2022)
optimized autonomous bus service frequency and vehicle size. Following the
previous work (Ng and Mahmassani 2023) which assessed the attractiveness of
flexible route operation with autonomous minibuses in suburbs, this study
generalizes it to optimize the portion of flexible route with other parameters
and variables. We determine the optimal portion between fixed route and
flexible route in a continuous spectrum of hybrid routes.

##  3 Mathematical Formulations

This section consists of two main models. We start with the optimization of
the flexible route portion $0\leq x_{f}\leq L_{x}$ (in km) and fleet size
$s>0$ (in vehicle, or veh) subject to a fixed headway $H$ (in h), which allows
us to set $s$ as a function of $x_{f}$. We discuss the model first with a
general demand distribution, followed by specific forms under uniform and
triangular distributions. It is worth noting that $x_{f}=0$ implies a fixed
route, $0<x_{f}<L_{x}$ a hybrid route, and $x_{f}=L_{x}$ a flexible route.
Next, we relax the fixed headway assumption and set it as a function
$h(x_{f},s)$ in optimization. This accounts for the trade-off between waiting
times and extra vehicle costs. We also generalize vehicle sizes where the
previously constant operational and vehicle costs are functions of vehicle
sizes under a capacity requirement constraint.

The general assumptions throughout this section are as follows:

  1. 1.

We illustrate the service in a grid network with route running direction as
the x-axis and perpendicular detours as the y-axis. Vehicles travel along the
x-direction and stop by demand points in sequence, i.e., there is no
backtracking.

  2. 2.

The total dwell time and layover time (at the transit station) are constant;
vehicles travel at a constant speed $V_{d}$ (in km/h).

  3. 3.

We consider a static demand in a continuous horizon. Demand is within
capacity.

  4. 4.

All travelers are individuals; group size is not considered other than as a
collection of independent travelers.

The notation is summarized in Table 1. Variables are denoted as small Roman
characters (e.g., $x_{f}$, $s$, $b$, $h$) and constants as capital Roman
characters (e.g., $L_{x}$, $H$), except for cost coefficients $\gamma$ and
total demand $\Lambda$. The demand density function is $f(x)$, with its
derivative as $f^{\prime}(x)$ and cumulative density function as $F(x)$.
Demand and costs are on a per-hour basis.

Table 1: Summary of Notation Symbol | Description
---|---
$\gamma_{a}$ | Access cost factor
$\gamma_{o}$ | Operating cost
$\gamma_{t}$ | Value of time
$\gamma_{v}$ | Vehicle cost
$\gamma_{w}$ | Waiting cost factor
$\Lambda$ | Total demand
$\lambda$ | Demand density
$\lambda_{0}$ | Maximum demand density (for triangular distribution)
$\overline{d_{d,y}}$ | Mean detour
$\overline{t_{a}}$ | Mean fixed route access time
$\rho$ | Capacity buffer over demand
$b$ | Vehicle size
$c$ | Total cost
$c_{o,x}$ | Operator cost (x-directional)
$c_{o,y}$ | Operator cost (y-directional)
$c_{t,x}$ | Riding cost (x-directional)
$c_{t,y}$ | Riding cost (y-directional)
$c_{a}$ | Access cost
$c_{v}$ | Vehicle cost
$c_{w}$ | Waiting cost
$f$ | Demand density function
$F$ | Cumulative demand density function
$f^{\prime}$ | Demand density function derivative
$H$ or $h$ | Headway
$L_{x}$ | Total route length in the x-direction
$s$ | Fleet size
$V_{d}$ | Vehicle speed
$x_{f}$ | Flexible route portion

###  3.1 Optimization of the Flexible Route Portion and Fleet Size given a
Headway

####  3.1.1 General Form.

A general demand distribution along the x-direction is assumed, with density
function $f(x)\geq 0,\forall x\in[0,L_{x}]$ (in passenger/km or pax/km) and
cumulative demand function $F(x)=\int_{0}^{x}f(\tilde{x})d\tilde{x}$ (in pax).
The total demand is denoted as $\Lambda=F(L_{x})$ (in pax), where $L_{x}$ (in
km) is the total route length in the x-direction (without detours), so $0\leq
F(x)\leq\Lambda,\forall x\in[0,L_{x}]$. We conventionally go from low- to
high-density area, i.e., flexible route before fixed route, along the x-axis
(see Figure 1). Demand along the y-axis is also assumed to follow a general
distribution, resulting in a mean fixed route access time $\overline{t_{a}}$
(in h) and a mean detour $\overline{d_{d,y}}$ (in km). $\overline{d_{d,y}}$ is
the average y-directional distance between consecutive request points along
the x-direction (see more detailed discussion and approximation by Ng and
Mahmassani (2023)).

In this formulation, we assume a constant headway $H$ (in h) to maintain
constant waiting time and route capacity. We also ignore the fleet size
integrality. (Non-integer fleet size may be used to represent operational
arrangement, e.g., interlining, where vehicles are shared across routes.
Furthermore, the additional complexity induced by the integrality constraints
may necessitate numerical optimization methods.)

Then, to simplify the minimization problem to univariate, $s$ can be expressed
as a function of $x_{f}$ by considering the cycle time divided by the headway
$H$ in Eq. 3.1.1. $t_{x}$ is the x-directional travel time. $t_{y}(x_{f})$ is
the y-directional total detour time in Eq. 3.1.1, based on the number of
detours, i.e., demand in the flexible route portion
$\int_{0}^{x_{f}}f(x)dx=F(x_{f})$, divided by $H$ for each vehicle trip and
multiplied by the average detour time $\overline{d_{d,y}}/v_{d}$. $T_{l}$ is
the layover time between one-way bus trips.

$\displaystyle s(x_{f})$
$\displaystyle=\frac{2}{H}\left(t_{x}+t_{y}(x_{f})+T_{l}\right)$
$\displaystyle=\frac{2}{H}\left(\frac{L_{x}}{V_{d}}+\frac{H\overline{d_{d,y}}}{V_{d}}F(x_{f})+T_{l}\right)$
(1) $\displaystyle t_{y}(x_{f})$
$\displaystyle=H\frac{\overline{d_{d,y}}}{V_{d}}\int_{0}^{x_{f}}f(x)dx$
$\displaystyle=H\frac{\overline{d_{d,y}}}{V_{d}}F(x_{f})$ (2)

We consider total cost from a societal view from both user and operator
perspectives. The objective is to minimize the total costs $c(x_{f})$ (in $)
as a function of the flexible route portion $x_{f}$. $c(x_{f})$ is composed of
travelers’ costs: access cost $c_{a}(x_{f})$ (for fixed route portion only),
waiting cost $c_{w}$, riding costs (x-direction: $c_{t,x}$, and, for flexible
route portion only, y-directional detour $c_{t,y}(x_{f})$); and distance-based
operator costs: $c_{o,x}$ (in x-direction) and $c_{o,y}(x_{f})$ (y-direction,
for flexible route portion only) and vehicle cost $c_{v}(s)$ as a function of
the fleet size $s>0$. These costs are computed similarly to Ng and Mahmassani
(2023) with reference to Newell (1979) formulation. The cost factors are the
value of time $\gamma_{t}$ (in $/h), access cost factor $\gamma_{a}$ and
waiting cost factor $\gamma_{w}$ (in multiples of riding cost), as well as
operating cost $\gamma_{o}$ (in $/km) and vehicle cost $\gamma_{v}(s)$ (in
$/veh).

The access (walking) cost $c_{a}(x_{f})$ is calculated in Eq. 3 as the product
of the value of access time, $\gamma_{t}\gamma_{a}$, average access time
$\overline{t_{a}}$, and number of passengers served in the fixed route portion
$\Lambda-F(x_{f})$. The waiting cost $c_{w}$ in Eq. 4 is a constant equal to
the product of values of waiting time $\gamma_{t}\gamma_{w}$, total number of
passengers $\Lambda$, and expected waiting time (half of headway $H$).

$\displaystyle c_{a}(x_{f})$
$\displaystyle=\gamma_{t}\gamma_{a}\overline{t_{a}}(\Lambda-F(x_{f}))$ (3)
$\displaystyle c_{w}$ $\displaystyle=\gamma_{t}\gamma_{w}\Lambda\frac{H}{2}$
(4)

The riding cost is separated into x-directional travel and y-directional
detours. For x-directional travel, each passenger getting on the bus at $x$
needs to travel a distance of $L_{x}-x$, resulting in the total riding
distances as the integral $\int_{0}^{L_{x}}f(x)(L_{x}-x)dx$. The constant
x-directional riding cost $c_{t,x}$ is then its product with the value of time
$\gamma_{t}$ divided by the vehicle speed $V_{d}$ in Eq. 5. For y-directional
detour, each passenger getting on the bus at $x$ in the flexible portion rides
the extra detours to pick up remaining passengers between $x$ and $x_{f}$ in
the flexible portion $\int_{x}^{x_{f}}Hf(\tilde{x})d\tilde{x}$ (multiplied by
$H$ to convert hourly demand to number of passengers per vehicle), resulting
in the double integral in Eq. 6. After simplification, it is apparent that the
y-directional riding cost $c_{t,y}(x_{f})$ is proportional to the square of
cumulative demand function $F(x_{f})$.

$\displaystyle c_{t,x}$
$\displaystyle=\gamma_{t}\frac{1}{V_{d}}\int_{0}^{L_{x}}f(x)(L_{x}-x)dx$
$\displaystyle=\gamma_{t}\frac{1}{V_{d}}\int_{0}^{L_{x}}F(x)dx$ (5)
$\displaystyle c_{t,y}(x_{f})$
$\displaystyle=\gamma_{t}\int_{0}^{x_{f}}f(x)\int_{x}^{x_{f}}Hf(\tilde{x})d\tilde{x}dx\frac{\overline{d_{d,y}}}{V_{d}}$
$\displaystyle=\gamma_{t}H\frac{\overline{d_{d,y}}}{2V_{d}}F(x_{f})^{2}$ (6)

The distance-based operating cost is also separated into x- and y-directional
components. The constant x-directional operating cost $c_{o,x}$ in Eq. 7 is
determined by the route length, divided by the headway $H$ to reflect the
hourly frequency. In contrast, for the y-directional detour, $H$ is canceled
out for $c_{o,y}(x_{f})$ in Eq. 8, i.e., the operating cost due to detours
depends on the total number of detours but not the headway.

$\displaystyle c_{o,x}$ $\displaystyle=\gamma_{o}\frac{L_{x}}{H}$ (7)
$\displaystyle c_{o,y}(x_{f})$
$\displaystyle=\gamma_{o}\frac{1}{H}\int_{0}^{x_{f}}Hf(x)dx\overline{d_{d,y}}=\gamma_{o}\overline{d_{d,y}}F(x_{f})$
(8)

The vehicle cost is calculated in Eq. 9 as directly proportional to the hourly
vehicle cost $\gamma_{v}$ (in $/veh), to cover the marginal costs of
additional vehicles.

$\displaystyle c_{v}(x_{f})$
$\displaystyle=\gamma_{v}s(x_{f})=\gamma_{v}\frac{2}{H}\left(\frac{L_{x}}{V_{d}}+\frac{H\overline{d_{d,y}}}{V_{d}}F(x_{f})+T_{l}\right)$
(9)

The total cost $c(x_{f})$ in Eq. 10 is the sum of all costs in Eq. 3-9. It
enables us to determine the optimal route form analytically with Result 1
based on geospatial, cost, and operational parameters.

$\displaystyle c(x_{f})$
$\displaystyle=\gamma_{t}\gamma_{a}\overline{t_{a}}(\Lambda-F(x_{f}))+\gamma_{t}\gamma_{w}\Lambda\frac{H}{2}$
$\displaystyle\quad+\gamma_{t}\frac{1}{V_{d}}\int_{0}^{L_{x}}F(x)dx+\gamma_{t}H\frac{\overline{d_{d,y}}}{2V_{d}}F(x_{f})^{2}$
$\displaystyle\quad+\gamma_{o}\frac{L_{x}}{H}+\gamma_{o}\overline{d_{d,y}}F(x_{f})$
$\displaystyle\quad+\gamma_{v}\frac{2}{H}\left(\frac{L_{x}}{V_{d}}+\frac{H\overline{d_{d,y}}}{V_{d}}F(x_{f})+T_{l}\right)$
(10)

######  Result 1.

Given a fixed fleet size and a fixed headway $H$, the optimal route form that
minimizes the total cost function $c(x_{f})$ in Eq. 10 is one of the
following:

  1. (a)

Fixed route if
$\overline{t_{a}}/\overline{d_{d,y}}\leq\gamma_{o}/(\gamma_{t}\gamma_{a})+2\gamma_{v}/(\gamma_{t}\gamma_{a}V_{d})$;

  2. (b)

Flexible route if $\overline{t_{a}}/\overline{d_{d,y}}\geq
H\Lambda/(\gamma_{a}V_{d})+\gamma_{o}/(\gamma_{t}\gamma_{a})+2\gamma_{v}/(\gamma_{t}\gamma_{a}V_{d})$;
or

  3. (c)

Hybrid route if
$\gamma_{o}/(\gamma_{t}\gamma_{a})+2\gamma_{v}/(\gamma_{t}\gamma_{a}V_{d})<\overline{t_{a}}/\overline{d_{d,y}}<H\Lambda/(\gamma_{a}V_{d})+\gamma_{o}/(\gamma_{t}\gamma_{a})+2\gamma_{v}/(\gamma_{t}\gamma_{a}V_{d})$,
in which case the flexible route portion $x_{f}^{*}$ satisfies Eq. 11.

$\displaystyle F(x_{f}^{*})$
$\displaystyle=\frac{V_{d}}{H}\left(\gamma_{a}\frac{\overline{t_{a}}}{\overline{d_{d,y}}}-\frac{\gamma_{o}}{\gamma_{t}}-2\frac{\gamma_{v}}{\gamma_{t}}\right)$
(11)

###### Proof.

  1. (a)

Fixed route: Assume that the minimum point $x_{f}^{*}>0$, so $\exists
x_{f}\in(0,L_{x}]:c(x_{f})<c(0)$. The condition can be rearranged as
$\gamma_{t}\gamma_{a}\overline{t_{a}}\leq\gamma_{o}\overline{d_{d,y}}+2\gamma_{v}\overline{d_{d,y}}/V_{d}$
, so the first derivative in Eq. 12 $c^{\prime}(x_{f})>0,\forall
x_{f}\in(0,L_{x}]$ as $F(x_{f})>0$. Therefore, $c(x_{f})>c(0),\forall
x_{f}\in(0,L_{x}]$. Contradiction shows that $x_{f}^{*}=0$.

  2. (b)

Flexible route: Similar to Part (a), assume that the minimum point
$x_{f}^{*}<L_{x}$, so $\exists x_{f}\in[0,L_{x}):c(x_{f})<c(L_{x})$. The
condition can be rearranged as
$\gamma_{t}\gamma_{a}\overline{t_{a}}\geq\gamma_{t}H\Lambda\overline{d_{d,y}}/V_{d}+\gamma_{o}\overline{d_{d,y}}+2\gamma_{v}\overline{d_{d,y}}/V_{d}$,
so
$\gamma_{t}\gamma_{a}\overline{t_{a}}>\gamma_{t}H\overline{d_{d,y}}F(x_{f})/V_{d}+\gamma_{o}\overline{d_{d,y}}+2\gamma_{v}\overline{d_{d,y}}/V_{d},\forall
x_{f}\in[0,L_{x})$ as $\Lambda>F(x_{f})$. Therefore,
$c^{\prime}(x_{f})<0,\forall x_{f}\in[0,L_{x})$ in Eq. 12, suggesting
$c(x_{f})>c(L_{x}),\forall x_{f}\in[0,L_{x})$. Contradiction shows that
$x_{f}^{*}=L_{x}$.

  3. (c)

Hybrid route: We minimize the total cost with respect to $x_{f}$ by
considering the optimality condition where $c^{\prime}(x)=0$ in Eq. 12,
resulting in $F(x_{f}^{*})$ in Eq. 11. We note that $F(x_{f})<F(x_{f}^{*})$
implies $c^{\prime}(x_{f})<0$ and $F(x_{f})>F(x_{f}^{*})$ implies
$c^{\prime}(x_{f})>0$. Additionally, the second derivative at this point is
positive from Eq. 14, so the cost function is convex around the optimum point.
Hence, $c(x_{f}^{*})$ is minimal when $x_{f}^{*}$ satisfies Eq. 11. Lastly,
the case condition implies $0<F(x_{f}^{*})<\Lambda$, so $0<x_{f}^{*}<L_{x}$.

$\displaystyle c^{\prime}(x_{f})$
$\displaystyle=-\gamma_{t}\gamma_{a}\overline{t_{a}}f(x_{f})+\gamma_{t}H\frac{\overline{d_{d,y}}}{V_{d}}f(x_{f})F(x_{f})$
$\displaystyle\quad+\gamma_{o}\overline{d_{d,y}}f(x_{f})+2\gamma_{v}\frac{\overline{d_{d,y}}}{V_{d}}f(x_{f})$
(12) $\displaystyle c^{\prime\prime}(x_{f})$
$\displaystyle=-\gamma_{t}\gamma_{a}\overline{t_{a}}f^{\prime}(x_{f})$
$\displaystyle\quad+\gamma_{t}H\frac{\overline{d_{d,y}}}{V_{d}}(f^{\prime}(x_{f})F(x_{f})+(f(x_{f}))^{2})$
$\displaystyle\quad+\gamma_{o}\overline{d_{d,y}}f^{\prime}(x_{f})+2\gamma_{v}\frac{\overline{d_{d,y}}}{V_{d}}f^{\prime}(x_{f})$
(13) $\displaystyle c^{\prime\prime}(x_{f}^{*})$
$\displaystyle=\gamma_{t}H\frac{\overline{d_{d,y}}}{V_{d}}\lambda^{2}(x_{f}^{*})\geq
0$ (14)

∎

We note that if the effect of the flexible route on the fleet size $s$ is
negligible, the vehicle cost can be omitted by setting $\gamma_{v}=0$. This
leads to simpler demarcation criteria from Result 1 as summarized in Figure 2.
The rearranged condition for fixed route
$\gamma_{t}\gamma_{a}\overline{t_{a}}\leq\gamma_{o}\overline{d_{d,y}}$
suggests that for each detour, the access cost saving is not greater than the
extra operational cost brought. For flexible route,
$\gamma_{t}\gamma_{a}\overline{t_{a}}\geq\gamma_{t}H\Lambda\overline{d_{d,y}}/V_{d}+\gamma_{o}\overline{d_{d,y}}$
means that the access cost saving is not smaller than the sum of extra
operational and riding cost (for every other passenger) brought by each
detour.

Figure 2: Illustration of Demarcation of Fixed, Hybrid, and Flexible Route
from Result 1 (Ignoring Additional Vehicle Requirement)

This also delineates the hybrid route case where the access cost saving is
greater than the operational cost but not enough to cover the extra riding
cost for all passengers. Compared to the previous results by Ng and Mahmassani
(2023) (shown by the grey dashes), Result 1 demarcates the previous marginal
cases between fixed and flexible routes where the hybrid route would provide
better services. Eq. 11 effectively means the first $F(x_{f}^{*})$ passengers
should be served with a flexible route, regardless of the specific
x-directional demand distribution. The number increases with the vehicle speed
$V_{d}$, access cost $\gamma_{a}\overline{t_{a}}$ (walking time), and value of
time $\gamma_{t}$, and decreases with the headway $H$, mean detour
$\overline{d_{d,y}}$, operating cost $\gamma_{o}$, and vehicle cost
$\gamma_{v}$. This aligns with the trade-offs of savings in access cost with
extra riding time and operational costs incurred by detours.

The vehicle cost $\gamma_{v}$ suggests extra vehicles needed for the detours
in flexible routes would elevate the cost and limit the extent of the flexible
route at optimum. Determining the optimal fleet size conditioned on the
optimal flexible portion leads to Result 2, where $s^{*}$ increases naturally
with factors that lead to longer flexible route portions.

######  Result 2.

For a hybrid route (with a fixed headway $H$), the optimal fleet size $s^{*}$
can be expressed in Eq. 15, which is independent of the x-directional demand
distribution $f(x)$.

$\displaystyle s^{*}$
$\displaystyle=\frac{2}{H}\left(\frac{L_{x}}{V_{d}}+\gamma_{a}\overline{t_{a}}-\frac{\gamma_{o}}{\gamma_{t}}\overline{d_{d,y}}-2\frac{\gamma_{v}}{\gamma_{t}}\frac{\overline{d_{d,y}}}{V_{d}}+T_{l}\right)$
(15)

###### Proof.

Eq. 15 can be obtained by combining Eq. 3.1.1 and 11. ∎

####  3.1.2 Uniform Demand Distribution.

We now show specific examples of x-directional demand distribution. Assuming
that the demand density follows a uniform distribution with density $\lambda$,
i.e., $f(x)=\lambda$, we get $F(x)=\lambda x$ and $\Lambda=\lambda L_{x}$.
$c(x_{f})$ in Eq. 10 is then simplified in Eq. 16.

$\displaystyle c(x_{f})$
$\displaystyle=\gamma_{t}\gamma_{a}\overline{t_{a}}\lambda(L_{x}-x_{f})+\gamma_{t}\gamma_{w}\lambda
L_{x}\frac{H}{2}$ $\displaystyle\quad+\gamma_{t}\frac{\lambda
L_{x}^{2}}{2V_{d}}+\gamma_{t}\frac{\lambda^{2}}{2}H\frac{\overline{d_{d,y}}}{V_{d}}x_{f}^{2}$
$\displaystyle\quad+\gamma_{o}\frac{L_{x}}{H}+\gamma_{o}\lambda\overline{d_{d,y}}x_{f}$
$\displaystyle\quad+\gamma_{v}\frac{2}{H}\left(\frac{L_{x}}{V_{d}}+\frac{H\overline{d_{d,y}}}{V_{d}}\lambda
x_{f}+T_{l}\right)$ (16)

The optimal flexible route portion under a uniform distribution
$x_{f,uni}^{*}$ in Eq. 17 is then obtained by solving Eq. 11 with
$F(x_{f,uni}^{*})=\lambda x_{f,uni}^{*}$.

$\displaystyle x_{f,uni}^{*}$ $\displaystyle=\frac{V_{d}}{\lambda
H}\left(\gamma_{a}\frac{\overline{t_{a}}}{\overline{d_{d,y}}}-\frac{\gamma_{o}}{\gamma_{t}}-2\frac{\gamma_{v}}{\gamma_{t}}\frac{1}{V_{d}}\right)$
(17)

####  3.1.3 Triangular Demand Distribution.

We may also assume a more likely scenario — increasing demand density closer
to the transit station or downtown (see Figure 1). For a triangular
distribution with demand density $f(x)$ starting with $0$ and increasing
linearly with $x$ to $\lambda_{0}$ at the end, $f(x)=\lambda_{0}/L_{x}\cdot
x$, $F(x)=\lambda_{0}/2L_{x}\cdot x^{2}$, and $\Lambda=\lambda_{0}L_{x}/2$.
The total cost function $c(x_{f})$ and the optimal flexible route portion
under a triangular distribution $x_{f,tri}^{*}$ can be obtained similarly in
Eq. 18 and 19.

$\displaystyle c(x_{f})$
$\displaystyle=\gamma_{t}\gamma_{a}\overline{t_{a}}\frac{\lambda_{0}}{2}\left(L_{x}-\frac{x_{f}^{2}}{L_{x}}\right)$
$\displaystyle\quad+\gamma_{t}\gamma_{w}\frac{\lambda_{0}L_{x}}{2}\frac{H}{2}+\gamma_{t}\frac{\lambda_{0}L_{x}^{2}}{6V_{d}}+\gamma_{t}H\frac{\overline{d_{d,y}}}{V_{d}}\lambda_{0}^{2}\frac{x_{f}^{4}}{8L_{x}^{2}}$
$\displaystyle\quad+\gamma_{o}\frac{L_{x}}{H}+\gamma_{o}\overline{d_{d,y}}\frac{\lambda_{0}x_{f}^{2}}{2L_{x}}$
$\displaystyle\quad+\gamma_{v}\frac{2}{H}\left(\frac{L_{x}}{V_{d}}+\frac{H\overline{d_{d,y}}}{V_{d}}\frac{\lambda_{0}x_{f}^{2}}{2L_{x}}+T_{l}\right)$
(18) $\displaystyle x_{f,tri}^{*}$
$\displaystyle=\sqrt{\frac{2L_{x}V_{d}}{\lambda_{0}H}\left(\gamma_{a}\frac{\overline{t_{a}}}{\overline{d_{d,y}}}-\frac{\gamma_{o}}{\gamma_{t}}-2\frac{\gamma_{v}}{\gamma_{t}}\frac{1}{V_{d}}\right)}$
(19)

The comparison of the results under the two demand distributions leads to
Result 3.

######  Result 3.

For a hybrid route assuming the same total demand $\Lambda$, the optimal
flexible portion under a triangular distribution $x_{f,tri}^{*}$ is always
longer than that under a uniform distribution $x_{f,uni}^{*}$. Specifically,
$x_{f,tri}^{*}=\sqrt{L_{x}x_{f,uni}^{*}}$.

###### Proof.

$x_{f,tri}^{*}=\sqrt{L_{x}x_{f,uni}^{*}}$ is obtained by combining Eq. 11 and
Eq. 19. Then, $x_{f,tri}^{*}>x_{f,uni}^{*}$ for $x_{f,uni}^{*}<L_{x}$. ∎

###  3.2 Joint Optimization of the Flexible Route Portion, Fleet Size, and
Headway with Variable Vehicle Sizes

How would the deployment of CAVs of different vehicle sizes affect the
flexible route portion, fleet size, and headway in semi-on-demand routes? We
examine this problem by considering the required fleet size and respective
operating cost $\gamma_{o}(b)$ and vehicle cost $\gamma_{v}(b)$ of each
vehicle size $b\in\mathcal{B}$ (in pax/veh), where $\mathcal{B}$ is a set of
vehicle sizes. We also relax the assumption of a fixed headway $H$ to allow a
variable headway function $h(x_{f},s)$. A trade-off is expected between
vehicle cost (larger vehicles and less frequent services) and waiting time
(longer headway).

The fleet size $s$ in Eq. 3.1.1 is rewritten with the headway function
$h(x_{f},s)$ in Eq. 20. After re-arranging the terms, we obtain Eq. 21.

$\displaystyle s$
$\displaystyle=\frac{2}{h(x_{f},s)}\left(\frac{L_{x}}{V_{d}}+\frac{\overline{d_{d,y}}}{V_{d}}F(x_{f})h(x_{f},s)+T_{l}\right)$
(20) $\displaystyle h(x_{f},s)$
$\displaystyle=\frac{L_{x}+T_{l}V_{d}}{sV_{d}/2-\overline{d_{d,y}}F(x_{f})}$
(21)

We also need to update several cost components for the variable headway.
Firstly, the waiting cost depends on the headway function in Eq. 22. Next,
riding costs for y-directional detours also vary with the headway in Eq. 23,
as fewer passengers in each vehicle mean fewer detours in the flexible
portion.

$\displaystyle c_{w}(x_{f},s)$
$\displaystyle=\gamma_{t}\gamma_{w}\Lambda\frac{h(x_{f},s)}{2}$ (22)
$\displaystyle c_{t,y}(x_{f},s)$
$\displaystyle=\gamma_{t}\frac{\overline{d_{d,y}}}{2V_{d}}F(x_{f})^{2}h(x_{f},s)$
(23)

For operating costs, the x-component of operating costs depends on
$h(x_{f},s)$ in Eq. 24 because of the service frequency change. However, the
y-component is unaffected due to the constant number of detours per hour.
Lastly, the vehicle cost can no longer be simply expressed as a function of
$x_{f}$ as in Eq. 9, but is still directly proportional to the fleet size in
Eq. 25.

$\displaystyle c_{o,x}(x_{f},s)$
$\displaystyle=\gamma_{o}(b)\frac{L_{x}}{h(x_{f},s)}$ (24) $\displaystyle
c_{v}(s)$ $\displaystyle=\gamma_{v}(b)s$ (25)

As vehicle sizes and corresponding costs are discrete, it provides an
opportunity to enumerate $b$ and solve for the optimal $x_{f}$ and $s$ that
minimize total costs. We impose a capacity requirement constraint that the
provided hourly capacity $b/h(x_{f},s)$ has to be greater than the demand
$\Lambda$ by a buffer of $\rho$ (to avoid over-capacity due to demand
variations), i.e., $\rho b/h(x_{f},s)\geq\Lambda$. Combined with Eq. 21, this
results in a lower bound of the fleet size $s$ for each vehicle capacity $b$
in Eq. 26.

$\displaystyle s\geq\frac{2}{V_{d}}\left(\frac{\Lambda}{\rho
b}(L_{x}+T_{l}V_{d})+\overline{d_{d,y}}F(x_{f})\right)$ (26)

We can formulate a mathematical problem (Eq. 3.2) to minimize the total cost
$c(x_{f},s,b)$ with respect to $x_{f}$, $s$, and $b$, subject to the capacity
requirement constraint in Eq. 26. The analytical expression of optimal $x_{f}$
and $s$ is complicated, so it is more practical to solve the minimization
numerically. For each discrete $b$, we can solve for $(x_{f}^{*},s^{*})$ that
minimize $c(x_{f},s,b)$ and subsequently apply Eq. 21 to obtain the optimal
headway $h(x_{f}^{*},s^{*})$. We note that Result 1 still applies here for a
determined headway, which is useful for conducting analysis but not obtaining
a solution.

$\displaystyle\min_{x_{f},s,b}c(x_{f},s,b)$
$\displaystyle=\gamma_{t}\gamma_{a}\overline{t_{a}}(\Lambda-F(x_{f}))+\gamma_{t}\gamma_{w}\Lambda\frac{h(x_{f},s)}{2}$
$\displaystyle\quad+\gamma_{t}\frac{1}{V_{d}}\int_{0}^{L_{x}}F(x)dx+\gamma_{t}h(x_{f},s)\frac{\overline{d_{d,y}}}{2V_{d}}F(x_{f})^{2}$
$\displaystyle\quad+\gamma_{o}(b)\frac{L_{x}}{h(x_{f},s)}+\gamma_{o}(b)\overline{d_{d,y}}F(x_{f})$
$\displaystyle\quad+\gamma_{v}(b)\frac{2}{h(x_{f},s)}\left(\frac{L_{x}}{V_{d}}+\frac{h(x_{f},s)\overline{d_{d,y}}}{V_{d}}F(x_{f})+T_{l}\right)$
$\displaystyle\text{subject
to}~{}~{}~{}~{}~{}\text{Eq.}~{}(\ref{eq:s_LB}),~{}~{}~{}0\leq x_{f}\leq
L_{x},~{}~{}~{}b\in\mathcal{B}$ (27)

##  4 Numerical Examples

To demonstrate the mathematical models and benefits of the semi-on-demand
routes, we present two sets of numerical examples, each with two Chicago bus
routes illustrated in Figure 3 (also studied by Ng and Mahmassani (2023)). The
first example is the joint optimization of the flexible route portion $x_{f}$
and fleet size $s$ with a given headway $H$ (Section 3.1), and the second is
the joint optimization of flexible route portion $x_{f}$, fleet size $s$,
headway $H$, and vehicle size $b$ (Section 3.2). The two Chicago Transit
Authority (CTA) bus routes studied, CTA126 and CTA84, mainly connect
passengers to downtown and a railway station respectively.

Figure 3: Maps of Bus Routes CTA126 and CTA84 in Chicago (Chicago Transit
Authority 2022a, b)

The scenario details and parameters follow Ng and Mahmassani (2023): a demand
profile with $\Lambda=80$ pax/h is used with a 15-min headway $H$ and
x-directional demand distributions as uniform and triangular distributions
previously discussed in Section 3. The other parameters are as follows: the
value of time $\gamma_{t}$ is $16.5/h, cost factors of access(walking)
$\gamma_{a}$ and waiting $\gamma_{w}$ are 2 and 1.5; the distance-based
operational cost $\gamma_{o}$ is $0.5/km and vehicle time cost $\gamma_{v}$ is
$12/h for a minibus (Tirachini and Antoniou 2020); the vehicle speed $V_{d}$
is 30km/h and layover time $T_{l}$ is 10min; the route length $L_{x}$ is
10.9km and 13.4km for CTA126 and CTA84 respectively; with the assumption of a
uniform distribution of y-directional demand in the catchment areas, the
average access time $\overline{t_{a}}$ is 2.25min for CTA126 and 6.75min for
CTA84 and mean detour $\overline{d_{d,y}}$ is 0.13km and 0.53km respectively.

###  4.1 Joint Optimization of the Flexible Route Portion and Fleet Size given
a Headway

The costs and optimal flexible route portion $x_{f}^{*}$ are obtained with the
analytical formula in Eq. 16-19 with the aforementioned parameters.

Under uniform distribution of demand in the x-direction, Figure 4 shows the
total cost, fleet size, and average cost component of the case CTA126,
respectively. The optimal flexible route portion $x_{f,uni}^{*}=7.91$km serves
$F(x_{f}^{*})=62$pax with the optimal fleet size $s^{*}=4.76$. When the
flexible route portion increases, access cost decreases linearly, while riding
cost increases quadratically. As the mean deviation $\overline{d_{d,y}}$ in
this case is relatively small, the effect of flexible portions on operating
and vehicle costs is minimal, favoring a longer flexible route. Besides, we
note that for fixed route operation, the fleet size $s(0)=4.24$, which if not
shared across routes would also require 5veh, the same as the hybrid route
service. This suggests that the simplified result subject to a fixed fleet
size by setting $\gamma_{v}=0$ suffices given the limited detours required.

Figure 4: Total Costs $c(x_{f})$ (a), Fleet Size $s(x_{f})$ (b), and Average
Cost Components (c) of Case CTA126 with respect to Flexible Route Portion
$x_{f}$ under Uniform Demand Distribution

The results of triangular distribution in Figure 5 favor flexible routes even
more with $x_{f,tri}^{*}=9.28$km. As previously discussed, this is equal to
$\sqrt{L_{x}x_{f,uni}^{*}}$, or 12.6% of the route length longer. The
concentrated demand closer to the train station favors a longer flexible route
portion. On the other hand, access and riding costs change more rapidly closer
to the end as shown in Figure 5(c).

Figure 5: Total Costs $c(x_{f})$ (a), Fleet Size $s(x_{f})$ (b), and Average
Cost Components (c) of Case CTA126 with respect to Flexible Route Portion
$x_{f}$ under Triangular Demand Distribution

Figure 6 shows similar results for the case CTA84 under a uniform demand
distribution. The mean detour $\overline{d_{d,y}}$ and average access time
$\overline{t_{a}}$ are greater than those in the last case, implying higher
potential savings in access cost but more vehicle detours for the flexible
route portion. The optimal flexible portion $x_{f,uni}^{*}$ is 6.90km, serving
$F(x_{f}^{*})=45$pax with the optimal fleet size $s^{*}=6.37$. The smaller
flexible portion is explained by the faster increases in riding, operating,
and vehicle costs.

Figure 6: Total Costs $c(x_{f})$ (a), Fleet Size $s(x_{f})$ (b), and Average
Cost Components (c) of Case CTA84 with respect to Flexible Route Portion
$x_{f}$ under Uniform Demand Distributions

The case of a triangular distribution in demand is illustrated in Figure 7.
While the optimal flexible portion is extended to $x_{f,tri}^{*}=9.61km$, the
flexible route proportion over the route length is still smaller than CTA126
(CTA84 is a longer route).

Figure 7: Total Costs $c(x_{f})$ (a), Fleet Size $s(x_{f})$ (b), and Average
Cost Components (c) of Case CTA84 with respect to Flexible Route Portion
$x_{f}$ under Triangular Demand Distributions

In summary, serving passengers further away with a flexible route portion
lowers the total cost, in particular user cost, for both routes. Smaller mean
detour $\overline{d_{d,y}}$ and an increasing demand gradient favor longer
flexible portions.

###  4.2 Joint Optimization of the Flexible Route Portion, Fleet Size, and
Headway with Variable Vehicle Sizes

The total cost minimization in Eq. 3.2 can be achieved by solving for the
optimal $x_{f}$ and $s$ with numerical solvers (L-BFGS-B in this example) for
each vehicle size $b$. This subsection shows results under uniform
distribution of demand.

Vehicle operating costs $\gamma_{o}$, which are distance-based, and time-based
capital costs $\gamma_{v}$ are derived from different vehicle classes and
capacities as per Tirachini and Antoniou (2020) (under their scenario of 50%
reduction in driving costs). The vehicle types considered are car, van,
20-seater, minibus, and bus, with capacities $b\in$[5,8,20,44,70] (pax/veh),
operating costs $\gamma_{o}\in$ [0.6187, 0.6370, 0.6938, 0.7507, 0.8900]
($/km) and vehicle costs $\gamma_{v}\in$ [2.53, 3.63, 7.59, 11.55, 15.73]
($/h). The capacity buffer is set as $\rho=0.7$.

Figure 8(a) shows the total costs $c(x_{f},s^{*},b)$ under the optimal fleet
size for the CTA126 case. It illustrates the impact of varying the flexible
route portion $x_{f}$ while optimizing the fleet size. For all vehicle types,
the cost curve is monotonically decreasing, indicating only flexible route
services. Figure 8(b) shows the optimal fleet sizes $s^{*}$ under varying
flexible route portions $x_{f}$. To ensure sufficient capacity to meet the
demand, the fleet sizes for cars and vans are the minimum required independent
of $x_{f}$. For other vehicle types, the optimal fleet sizes increase
gradually to serve more detours. The headways shown in Figure 8(c) are much
lower than the 15-minute headway setting in the previous examples, suggesting
that the reduction in waiting time outweighs the increase in operating cost
(Figure 8(d)), if the capital and operating costs of SAVs are as low as
forecasted (relative to the assumed values of time).

Figure 8: Total Costs under Optimal Fleet Size $c(x_{f},s^{*},b)$ (a), Optimal
Fleet Size $s^{*}$ (b), Headway $h(x_{f},s^{*})$ (c), and Total Operator Costs
(d) with respect to Flexible Route Portion $x_{f}$ in Cases CTA126

Figure 9 shows similar results for the case CTA84. The changes with increasing
flexible route portions are more considerable due to the larger detours for
each passenger.

Figure 9: Total Costs under Optimal Fleet Size $c(x_{f},s^{*},b)$ (a), Optimal
Fleet Size $s^{*}$ (b), Headway $h(x_{f},s^{*})$ (c), and Total Operator Costs
(d) with respect to Flexible Route Portion $x_{f}$ in Cases CTA84

Flexible routes are optimal in most cases, different from the previous results
in Section 4.1 where hybrid routes are optimal under a constant 15-minute
headway. This is primarily contributed by the much lower headway, particularly
for smaller vehicle sizes. However, the operator costs required to provide
such low headway are considerably higher than the current human-driven fixed
route operations ($124 and $150 for CTA126 and CTA84 respectively). If a
budget constraint is applied, the optimal results would likely be hybrid route
similar to Section 4.1 where changes in operator costs are much smaller under
fixed headway.

Table 2 lists the detailed results.

Table 2: Results of Joint Optimization of Vehicle Size, Flexible Route Portion, and Fleet Size Bus route | CTA126 | CTA84
---|---|---
Vehicle | Type | Car | Van |  | 20-
---
Seater
| Mini-
---
bus
| Stand-
---
ard
bus
Car | Van |  | 20-
---
Seater
| Mini-
---
bus
| Stand-
---
ard
bus
Size | b |  | pax
---
/veh
5 | 8 | 20 | 44 | 70 | 5 | 8 | 20 | 44 | 70
Optimal variable |  | Flexible
---
portion
$x_{f}^{*}$ | km | 10.90 | 10.90 | 10.90 | 10.90 | 10.90 | 13.40 | 13.40 | 13.40 | 13.40 | 11.04
Fleet size | $s^{*}$ | veh | 24.69 | 15.60 | 10.10 | 8.92 | 7.93 | 29.42 | 18.90 | 15.33 | 13.79 | 11.20
Headway | $h^{*}$ | min | 2.65 | 4.27 | 6.77 | 7.75 | 8.81 | 2.77 | 4.58 | 5.89 | 6.72 | 8.31
Average time | Access | $\overline{t_{a}}$ | min | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.19
Waiting | $\overline{t_{w}}$ | min | 1.33 | 2.14 | 3.39 | 3.87 | 4.40 | 1.38 | 2.29 | 2.95 | 3.36 | 4.16
Riding | $\overline{t_{t}}$ | min | 11.37 | 11.66 | 12.10 | 12.28 | 12.47 | 15.37 | 16.66 | 17.59 | 18.18 | 17.41
Time std. dev. | Access | $\sigma_{t,a}$ | min | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.45
Waiting | $\sigma_{t,w}$ | min | 0.77 | 1.23 | 1.96 | 2.24 | 2.54 | 0.80 | 1.32 | 1.70 | 1.94 | 2.40
Riding | $\sigma_{t,r}$ | min | 6.75 | 6.93 | 7.20 | 7.30 | 7.41 | 9.63 | 10.42 | 10.98 | 11.33 | 11.04
Cost | Access | $c_{a}$ | $ | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 52.24
Waiting | $c_{w}$ | $ | 43.76 | 70.48 | 111.76 | 127.85 | 145.29 | 45.70 | 75.64 | 97.24 | 110.96 | 137.12
| Riding
---
(x-dir.)
$c_{t,x}$ | $ | 239.80 | 239.80 | 239.80 | 239.80 | 239.80 | 294.80 | 294.80 | 294.80 | 294.80 | 294.80
| Riding
---
(y-dir.)
$c_{t,y}$ | $ | 10.37 | 16.71 | 26.49 | 30.31 | 34.44 | 43.33 | 71.72 | 92.20 | 105.20 | 88.30
| Operational
---
(x-dir.)
$c_{o,x}$ | $ | 152.56 | 97.53 | 66.99 | 63.36 | 66.10 | 179.61 | 111.72 | 94.65 | 89.75 | 86.11
| Operational
---
(y-dir.)
$c_{o,y}$ | $ | 6.60 | 6.79 | 7.40 | 8.01 | 9.49 | 26.40 | 27.18 | 29.60 | 32.03 | 31.29
Vehicle | $c_{v}$ | $ | 62.47 | 56.63 | 76.66 | 103.01 | 124.80 | 74.43 | 68.61 | 116.38 | 159.26 | 176.19
Total | $c$ | $ | 515.56 | 487.94 | 529.11 | 572.34 | 619.93 | 664.27 | 649.66 | 724.87 | 792.01 | 866.04

####  4.2.1 Sensitivity to Operator Cost.

To assess the sensitivity to the forecasted operator costs (or user value of
time), Figures 10 and 11 present the results under 200% and 300% of operator
costs (or effectively 50% and 33% of user value of time) for case CTA84. We
can see a reduction in the flexible route portion in subplots (a), where
minibus hybrid route dominates in the 300% case. Additionally, the optimal
flexible route portion (the cost minimum point) decreases with increasing
vehicle sizes. This means smaller vehicles support the deployment of longer
flexible routes with each trip serving fewer passengers and making fewer
detours, whereas larger vehicles may lead to excessive cumulative detours.
Although more smaller vehicles are required to serve the same demand, each
trip involves smaller detours in the flexible route portion alongside shorter
waiting times. This also explains the general reduction in headway with an
increase in the flexible route portion in subplots (c). These findings align
with Result 1, where lower operational costs ($\gamma_{o}$ and $\gamma_{v}$)
and smaller headway ($H$) favor flexible routes.

Figure 10: Total Costs under Optimal Fleet Size $c(x_{f},s^{*},b)$ (a),
Optimal Fleet Size $s^{*}$ (b), Headway $h(x_{f},s^{*})$ (c), and Total
Operator Costs (d) with respect to Flexible Route Portion $x_{f}$ under 200%
Operator Costs for Case CTA84 Figure 11: Total Costs under Optimal Fleet Size
$c(x_{f},s^{*},b)$ (a), Optimal Fleet Size $s^{*}$ (b), Headway
$h(x_{f},s^{*})$ (c), and Total Operator Costs (d) with respect to Flexible
Route Portion $x_{f}$ under 300% Operator Costs for Case CTA84

In short, a balance across vehicle size is important — using 4-passenger cars
would necessitate a lot of vehicles to serve the demand; utilizing larger
buses could result in excessive detours and longer waiting times. The optimal
scenario with the assumed operator costs in both cases is using a van of 8
passengers with only flexible route service. However, before the technology
matures for the cost to fall, bigger vehicles running hybrid routes may be the
optimal solution. This also applies to cases where operators face a strict
budget such that the system-optimally low headway is infeasible.

Table 3 shows the detailed results.

Table 3: Results of Sensitivity Analysis for the Joint Optimization of Vehicle Size, Flexible Route Portion, and Fleet Size Bus route CTA84 | 200% Operator Cost | 300% Operator Cost
---|---|---
Vehicle | Type | Car | Van |  | 20-
---
Seater
| Mini-
---
bus
| Stand-
---
ard
bus
Car | Van |  | 20-
---
Seater
| Mini-
---
bus
| Stand-
---
ard
bus
Size | b |  | pax
---
/veh
5 | 8 | 20 | 44 | 70 | 5 | 8 | 20 | 44 | 70
Optimal variable |  | Flexible
---
portion
$x_{f}^{*}$ | km | 13.40 | 13.40 | 8.39 | 5.87 | 3.89 | 13.40 | 13.40 | 4.69 | 2.75 | 1.14
Fleet size | $s^{*}$ | veh | 29.42 | 18.90 | 9.19 | 7.27 | 5.90 | 29.42 | 18.90 | 6.51 | 5.20 | 4.24
Headway | $h^{*}$ | min | 2.77 | 4.58 | 9.93 | 12.21 | 14.50 | 2.77 | 4.58 | 13.35 | 15.94 | 18.39
Average time | Access | $\overline{t_{a}}$ | min | 0.00 | 0.00 | 2.52 | 3.80 | 4.79 | 0.00 | 0.00 | 4.39 | 5.37 | 6.18
Waiting | $\overline{t_{w}}$ | min | 1.38 | 2.29 | 4.97 | 6.11 | 7.25 | 1.38 | 2.29 | 6.67 | 7.97 | 9.20
Riding | $\overline{t_{t}}$ | min | 15.37 | 16.66 | 16.17 | 15.06 | 14.27 | 15.37 | 16.66 | 14.56 | 13.88 | 13.49
Time std. dev. | Access | $\sigma_{t,a}$ | min | 0.00 | 0.00 | 2.12 | 2.60 | 2.92 | 0.00 | 0.00 | 2.79 | 3.09 | 3.31
Waiting | $\sigma_{t,w}$ | min | 0.80 | 1.32 | 2.87 | 3.52 | 4.19 | 0.80 | 1.32 | 3.85 | 4.60 | 5.31
Riding | $\sigma_{t,r}$ | min | 9.63 | 10.42 | 10.41 | 9.73 | 9.10 | 9.63 | 10.42 | 9.35 | 8.69 | 8.13
Cost | Access | $c_{a}$ | $ | 0.00 | 0.00 | 111.00 | 167.00 | 210.88 | 0.00 | 0.00 | 193.08 | 236.06 | 271.79
Waiting | $c_{w}$ | $ | 45.70 | 75.64 | 163.88 | 201.48 | 239.28 | 45.70 | 75.64 | 220.21 | 263.05 | 303.48
| Riding
---
(x-dir.)
$c_{t,x}$ | $ | 294.80 | 294.80 | 294.80 | 294.80 | 294.80 | 294.80 | 294.80 | 294.80 | 294.80 | 294.80
| Riding
---
(y-dir.)
$c_{t,y}$ | $ | 43.33 | 71.72 | 60.94 | 36.60 | 19.07 | 43.33 | 71.72 | 25.56 | 10.50 | 2.07
| Operational
---
(x-dir.)
$c_{o,x}$ | $ | 359.23 | 223.44 | 112.33 | 98.85 | 98.69 | 538.84 | 335.17 | 125.39 | 113.58 | 116.71
| Operational
---
(y-dir.)
$c_{o,y}$ | $ | 52.80 | 54.36 | 37.08 | 28.04 | 22.02 | 79.19 | 81.54 | 31.07 | 19.72 | 9.67
Vehicle | $c_{v}$ | $ | 148.87 | 137.21 | 139.53 | 167.99 | 185.61 | 223.30 | 205.82 | 148.23 | 180.19 | 200.23
Total | $c$ | $ | 944.71 | 649.66 | 919.55 | 994.77 | 1070.36 | 1225.16 | 1064.68 | 1038.35 | 1117.89 | 1198.76

##  5 Case Study

To investigate the applicability of the described semi-on-demand hybrid routes
in transit feeders, this case study makes use of a real-world transit network
and demand data in the Chicago metropolitan area. We classify analytically
areas served by feeder routes to each railway station into fixed route and
flexible route service areas with analytical formula in Section 3.1.

Demand data are adapted from the activity-based demand model CT-RAMP from
Chicago Metropolitan Agency for Planning (Parsons Brinckerhoff 2011), where
trips originate and end at micro analysis zones (MAZs). Only trips that are
assigned to trains and originate outside walking distance are included, which
equates to approximately 78,000 hourly trips after aggregation into three peak
hours daily. The existing train stations of the commuter rail, Metra, (City of
Chicago 2012) and rapid transit system, Chicago “L”, (Chicago Transit
Authority 2022b) are imported into the Geographic Information System (GIS)
model.

The geospatial pre-processing is as follows. Voronoi zones are created around
each station to assign MAZs to their closest stations. The x-axis in the
previous model is set between the station and the furthest MAZ in each Voronoi
zone. The perpendicular y-axis ($x=0$) then separates the zone into two sub-
zones, each of which contains one or more parallel feeder catchments based on
the maximum walking distance.

We then carry out the analysis in each sub-zone, by calculating the number of
passengers to serve in flexible route portions, i.e., $F(x_{f}^{*})$ in Eq.
11. This suggests serving these passengers with flexible routes would not only
reduce their travel costs, but also total costs of users and operators.
Instead of a uniform or triangular distribution assumption, the analysis
directly captures $f(x)$ from the number of trips in each MAZ (at its center).

The parameters are set as follows: The maximum access time is 15min and the
walking speed is 4km/h, resulting in a catchment width of 2km and average
detour $\overline{d_{d,y}}$ of 0.67km (assuming a uniform distribution of
y-directional demand); other parameters follow the previous numerical example
in Section 4.1. The optimal number of passengers in the flexible route portion
$F(x_{f}^{*})$ is then 35.5 with Eq. 11, equivalent to around 9 pax/trip.

Figure 12 shows which MAZs are served with fixed routes (blue dots) and
flexible routes (in green dots) in each zone (in gray boundary). The flexible
route service areas are mostly found in suburbs and rural areas far from
downtown, with sparser demand and bigger gaps between transit lines.

The detailed results are shown in Table 4. 787 routes (86.6% of the total) are
semi-on-demand hybrid routes, serving 3,562 MAZs (62.1%) with flexible routes.
However, this accounts for only 18,989pax/h (24.3%). The large coverage area
accounting for a small portion of demand highlights the use case of flexible
feeders for low-demand-density areas. The operator cost increases by 49.0%,
arising from the additional mileage and vehicles needed. This, however, is a
small portion of the total costs, based on the SAV cost forecast. Higher
operating and vehicle costs would reduce $x_{f}^{*}$ and the extent of
flexible routes. While hybrid routes only save 7.9% user cost and 6.0% total
cost in general, they make significant differences among travelers who use the
flexible route portion by eliminating the access cost, accounting for a saving
of 29.8% user cost. This enhances the attractiveness of transit systems to
travelers in these zones who are usually further from the train stations. (The
benefits brought by mode shift to transit are however not captured in this
study.)

Figure 12: Case Study Results of Fixed Route and Flexible Route Service Area for Feeders to Train Stations (Chicago Transit Authority 2022b, City of Chicago 2012) Table 4: Case Study Results of Fixed Route and Flexible Route Service Area for Feeders to Train Stations Service area | Among all feeders |  | Among semi-on-demand
---
hybrid route feeders
| Among MAZs in flexible
---
route service areas
Service mode |  | Fixed
---
route
| Semi-on-
---
demand
route
| Percent-
---
age
change
| Fixed
---
route
| Semi-on-
---
demand
route
| Percent-
---
age
change
| Fixed
---
route
| Semi-on-
---
demand
route
| Percent-
---
age
change
Number of feeder routes | 909 | N/A | 787 | N/A | 787 | N/A
Number of MAZs covered | 5,732 | N/A | 5,479 | N/A | 3,562 | N/A
| Number of
---
passengers
pax/h | 78,237 | N/A | 49,921 | N/A | 18,989 | N/A
Average access time | min | 7.50 | 5.68 | -24.3% | 7.50 | 4.65 | -38.0% | 7.50 | 0.00 | -100.0%
Average waiting time | min | 7.50 | 7.50 | 0.0% | 7.50 | 7.50 | 0.0% | 7.50 | 7.50 | 0.0%
Average riding time | min | 4.95 | 6.12 | 23.6% | 5.96 | 7.79 | 30.8% | 7.93 | 12.75 | 60.8%
Average user cost | $ | 8.58 | 7.90 | -7.9% | 8.86 | 7.79 | -12.0% | 9.40 | 6.60 | -29.8%
| Average operator
---
cost
$ | 0.32 | 0.47 | 45.6% | 0.47 | 0.69 | 49.0% | N/A | N/A | N/A
| Average
---
generalized cost
$ | 8.90 | 8.37 | -6.0% | 9.32 | 8.49 | -9.0% | N/A | N/A | N/A
Total access cost | $ | 322,731 | 244,401 | -24.3% | 205,924 | 127,594 | -38.0% | 78,330 | 0 | -100.0%
Total waiting cost | $ | 242,048 | 242,048 | 0.0% | 154,443 | 154,443 | 0.0% | 58,747 | 58,747 | 0.0%
Total riding cost | $ | 106,449 | 131,619 | 23.6% | 81,780 | 106,950 | 30.8% | 41,404 | 66,574 | 60.8%
Total user cost | $ | 671,228 | 618,068 | -7.9% | 442,147 | 388,987 | -12.0% | 178,481 | 125,321 | -29.8%
Total operating cost | $ | 9,849 | 16,179 | 64.3% | 9,414 | 15,744 | 67.2% | N/A | N/A | N/A
Total vehicle cost | $ | 15,151 | 20,215 | 33.4% | 13,827 | 18,891 | 36.6% | N/A | N/A | N/A
Total operator cost | $ | 25,000 | 36,394 | 45.6% | 23,241 | 34,635 | 49.0% | N/A | N/A | N/A
| Total generalized
---
cost
$ | 696,228 | 654,462 | -6.0% | 465,388 | 423,622 | -9.0% | N/A | N/A | N/A

##  6 Conclusion

###  6.1 Summary

This study considers semi-on-demand hybrid route service in public transit
systems. It serves directional demand by first offering passengers further
from a transit station/downtown with on-demand flexible route service and then
continuing with a traditional fixed route. It combines the economies of scale
of fixed route bus service and the accessibility and flexibility of taxi and
shared autonomous mobility service (SAMS).

We develop an analytical approach to delineate the conditions in which each of
the route forms (fixed, hybrid, and flexible) is optimal, with tractable cost
expressions that consider the total costs of users (access, waiting, and
riding) and operators (operating and vehicle) in two formulations to support
strategic and tactical decisions, respectively. Closed-form expressions are
derived to determine the optimal flexible route portion and fleet size for
hybrid routes, and consider the optimal headway with variable vehicle sizes.
Through numerical examples and a case study in the Chicago metropolitan area,
we demonstrate the benefits and applications of semi-on-demand feeders.

The findings demonstrate the general applicability of hybrid routes in transit
networks. Serving passengers located further away with flexible routes lowers
total costs, particularly user costs in access, which could potentially
attract more riders to connect with the main transit system. Besides, demand
gradients favor longer flexible routes, a good fit for cities with urban
sprawl. In the example of joint design of flexible route portion, headway,
fleet size, and vehicle size, vans with flexible routes dominate. However,
minibuses with hybrid routes still play a key role when shared autonomous
vehicle (SAV) costs are still high and the operating budget is limited, or
during transitions to smaller vehicles and flexible routes to introduce the
on-demand service and enhance service attractiveness. Our case study
identifies flexible route areas in suburbs and rural areas further from the
station and with coverage gaps, helping to serve more riders.

###  6.2 Limitations and Future Research Directions

This study develops an analysis tool for semi-on-demand hybrid route services
with tractable and closed-form solutions. The resulting analytical formulation
of cost and optimal flexible route portion and fleet size can aid transit
agencies in planning new routes and making vehicle investment decisions in the
era of SAMS. It provides a continuous approximation alternative to
computationally intensive simulation-based optimization for research in large-
scale transit network design with SAMS. Inherently, it is limited by the
assumptions, e.g., no backtracking or significant impacts brought by headway
variance. Specifically for the joint optimization including headway, the
operator budget and therefore headway may be fixed, favoring hybrid route over
completely flexible services. The model also allows future refinement and
investigation in the following areas.

First, it can be combined with demand- and supply-side models in the transit
network design problem. Demand models to investigate mode shift towards the
transit system with improved first-mile-last-mile connectivity, and its
competition with driving. For the supply side, a joint transit network design
model that explores the implications of transit network design with the
inclusion of SAMS can utilize the analytical expression derived in this study
and consider multiple routes simultaneously. While this model assumes a static
demand, it can be readily applied to problems with multiple horizons and time-
varying design parameters and demand. The methodology can also be generalized
to other network structures by transforming the space with the x-axis as the
shortest path and the y-axis as the detours.

Second, research can focus on the study and forecast of connected and
automated vehicles (CAVs), including cost parameters. Our numerical example
and sensitivity study illustrate the attractiveness of vehicles larger than
sedans to provide hybrid route feeder services, and vehicle size matters in
hybrid route design and headway-fleet size decisions. Following the analytical
approach developed in this paper, more accurate cost forecasts would
facilitate new service planning and deployment.

Third, research with agent-based models and actual fleet control can verify
the model results at a microscopic level and assess the impacts of several
assumptions and design parameters. For example, backtracking is not allowed in
the model, so solving the vehicle routing problem that optimizes pick-
ups/drop-offs could improve the performance of the flexible route portion. A
constant headway is assumed in each case, whereas simulations that account for
the effects of demand and service variances on waiting time could assess
schedule adherence. Vehicle capacity buffer $\rho$ allows for stochastic
demand, with room for future statistical or simulation approaches to evaluate
the effects of vehicle size and demand fluctuation on boarding rejection.

##  7 Acknowledgments

The authors would like to thank Antonios Tsakarestos of the Technical
University of Munich for the valuable comments in the initial stage of the
study.

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