# An adaptive route choice model for integrated fixed and flexible transit
systems

David Lefflera, Wilco Burghouta, Oded Catsa,b and Erik Jeneliusa CONTACT D.
Leffler Email<EMAIL_ADDRESS>aDivision of Transport Planning, KTH Royal
Institute of Technology, Stockholm, Sweden; bDepartment of Transport and
Planning, Delft University of Technology, Delft, The Netherlands

###### Abstract

Over the past decade, there has been a surge of interest in the transport
community in the application of agent-based simulation models to evaluate
flexible transit solutions characterized by different degrees of short-term
flexibility in routing and scheduling. A central modeling decision in the
development of an agent-based simulation model for the evaluation of flexible
transit is how one chooses to represent the mode- and route-choices of
travelers. The real-time adaptive behavior of travelers is intuitively
important to model in the presence of a flexible transit service, where the
routing and scheduling of vehicles is highly dependent on supply-demand
dynamics at a closer to real-time temporal resolution. We propose a utility-
based transit route-choice model with representation of within-day adaptive
travel behavior and between-day learning where station-based fixed-transit,
flexible-transit, and active-mode alternatives may be dynamically combined in
a single path. To enable experimentation, this route-choice model is
implemented within an agent-based dynamic public transit simulation framework.
Model properties are first explored in a choice between fixed- and flexible-
transit modes for a toy network. The framework is then applied to illustrate
level-of-service trade-offs and analyze traveler mode choices within a mixed
fixed- and flexible transit system in a case study based on a real-life
branched transit service in Stockholm, Sweden.

###### keywords:

public transit; flexible transit; agent-based simulation; transit assignment;
route choice

##  1 Introduction

Transit network assignment refers to the process in which an origin-
destination (OD) matrix of traveler demand is assigned to a set of transit
routes that connect each OD pair. This problem plays a vital role in the
modeling and evaluation of transit systems, with rich literature covering both
theoretical developments and a wide array of applications. The route (or, more
generally, the path) choice of transit users is a key element of transit
assignment models. Traditional approaches to route-choice modeling tend to be
based on conventional transportation modes, such as private cars or fixed line
and schedule public transport services (FIX). To accommodate simulation-based
evaluation of novel shared-transport services and demand scenarios, transit
route-choice models have been rapidly developing. This can be attributed to an
increased attention towards resource-sharing and transit systems as a core
component in sustainable urban development, together with innovations within
Intelligent Transport Systems (ITS) and real-time information provision to
both travelers and operators. Furthermore, new data sources with higher
degrees of temporal and spatial richness (e.g., mobile phone, automatic
vehicle location [AVL], automatic passenger counting) have become increasingly
available for tracking the real-time evolution of both supply and demand. This
enables more disaggregate approaches to modeling and forecasting the movements
and interactions of individual travelers and vehicles under alternative
scenario settings and intervention strategies (Ibarra-Rojas et al. 2015;
Koutsopoulos et al. 2019).

The term _flexible transit_ (FLEX) refers to public transit services
characterized by flexible routing and scheduling adapted to traveler needs at
a within-day temporal scale. In the literature, FLEX can refer to many service
types ranging from door-to-door shared taxi-like services (Fagnant and
Kockelman 2014; Cebecauer et al. 2021) to semi-flexible services with
partially fixed routes and timetables that allow for demand-responsive dynamic
fleet management (Errico et al. 2013). In a recent survey, Vansteenwegen et
al. (2022) compiled a wide range of terms referring to such systems, for
example _on-demand_ , _dial-a-ride_ , _demand-adaptive_ , _demand-responsive_
, _flex-route_ , _flexible_ , _hybrid_ , or _variable-type_ to name a few.
Generally, FLEX is found to be most beneficial in scenarios when travel demand
is low (e.g., in more rural or suburban areas) and/or more variable (e.g.,
where there are considerable differences between peak and off-peak hours)
(Potts et al. 2010; Sörensen et al. 2021) but are in practice difficult to
economically sustain due to difficulties in spreading the cost of individual
trips over a greater number of travelers (Ferreira, Charles, and Tether 2007;
Davison et al. 2012). Interest in FLEX systems has grown in the past decade
with the emergence of smartphone-enabled ride-pooling services (e.g., Uber)
and technological progress in the development of automated vehicles (AVs).
With great opportunities for real-time fleet coordination and prospective
lower per-vehicle operational costs (Bösch et al. 2018), many conceptual FLEX
use cases for AVs have been proposed in recent years (Narayanan, Chaniotakis,
and Antoniou 2020).

Agent-based simulation models have been utilized in evaluating a broad FLEX
service design space (both with and without the inclusion of AVs), with
varying levels of integration with FIX, as well as different network
topologies and demand settings at different spatial and temporal scales
(Ronald, Thompson, and Winter 2015; Markov et al. 2021). In relation to public
transit, novel FLEX services evaluated in the literature can be classified as:
(i) independent of traditional public transit (e.g., as a shared trip
alternative to individual-use taxis or private cars) (Fagnant and Kockelman
2014; Martinez, Correia, and Viegas 2014; Bischoff and Maciejewski 2016; Liu
et al. 2017; Alonso-Mora et al. 2017; Markov et al. 2021), (ii) as a
replacement for FIX (Winter et al. 2018; Jäger, Brickwedde, and Lienkamp 2018;
Berrada and Poulhès 2021; Narayan et al. 2021) or as a competing alternative
(Liu et al. 2019; Hörl, Becker, and Axhausen 2021), and (iii) as an
alternative that may be combined with FIX (e.g., feeder/last-mile) to
construct a complete trip from origin to destination (Horn 2002; Scheltes and
de Almeida Correia 2017; Moorthy et al. 2017; Salazar et al. 2018; Shen,
Zhang, and Zhao 2018; Wen et al. 2018; Narayan et al. 2020; Leffler et al.
2021). While performance measures and motivations for these studies vary, they
tend to conclude with similar messages; to achieve a sustainable FLEX service
from passenger-oriented, operator-oriented, or societal perspectives, it is
crucial that this service is efficiently integrated with existing FIX systems
rather than introduced as a directly competing alternative (Narayanan,
Chaniotakis, and Antoniou 2020; Sörensen et al. 2021).

The characteristics of FIX or FLEX services, e.g., in terms of expected in-
vehicle time, waiting time, crowding, and reliability, may inherently be
better suited to the individual preferences of a traveler during different
legs (e.g., first-mile or last-mile) of their trip, and in light of new
information as their trip progresses. In this sense, it is important to
understand how travelers potentially combine FIX and FLEX public transport
services in different ways in the evaluation of such systems. Agent-based
simulation frameworks utilized for this purpose are developing hand-in-hand
with envisioned use cases of AVs and FLEX services. The incorporation of
integrated FIX and FLEX route choice models and dynamic assignment techniques
within these frameworks are, however, still at their earlier stages of
development.

##  2 Literature review

###  2.1 Transit route choice and assignment

Modeling transit route choice involves both representation of the transit
system (e.g., in terms of both tactical and operational policies, vehicles
types, and transit infrastructure) and how users interact and potentially give
rise to congestion (e.g., in the form of in-vehicle crowding, denied boarding,
queuing) within the transit system. More recently developed methodology for
modeling mode/route choice within studies of FLEX systems largely builds on
previous approaches for conventional transit. Liu, Bunker, and Ferreira (2010)
categorize route-choice models within the context of transit assignment into
three groups: (i) static transit assignment, (ii) within-day dynamic transit
assignment, and (iii) emerging approaches.

The first category, static transit assignment, encompasses methods based on
shortest-path estimation (all-or-nothing assignment to the minimal cost route
for a given OD pair) and user equilibrium-based assignment. The equilibrium
assignment problem is often solved through fixed-point iterations to estimate
the travel cost of available routes under capacity constraints, typically
using a variant of the method of successive averages (MSA) (Sheffi and Powell
1982) to update solutions between iterations. Seminal works (see e.g., Dial
(1971); Spiess and Florian (1989)) also find different ways of dealing with
the well-known ‘common lines’ problem of transit assignment (travelers at a
transit stop often choose opportunistically between several competitive
transit lines with potentially different routes) and the task of generating a
‘reasonable’ choice-set of alternative transit routes for each OD (i.e.,
eliminating paths that are highly unlikely to ever be considered by a
traveler) while avoiding the computational intractability of enumerating all
possible paths. To incorporate the behavioral aspects of route choice (for
example different levels of knowledge, familiarity, or preference of travelers
for different routes), these methods are combined with stochastic assignment
and discrete random utility maximization (RUM) choice models.

The adaptive behavior of travelers is intuitively important to model in the
presence of FLEX services where, in comparison with FIX, the routing and
scheduling of vehicles is even more dependent on traveler requests and supply-
demand imbalances at a closer to real-time temporal resolution. Compared to
traditional transit assignment models, a FLEX operator can also be viewed as
more of a within-day decision-maker that anticipates and responds in real-time
to the within-day decision ’events’ of travelers. Within-day dynamic transit
assignment models relax assumptions regarding the static distributions of
travel times and choice behavior of travelers and incorporate the within-day
time dimension in route choices (Cats, West, and Eliasson 2016).

Emerging approaches enhance these methods further by attempting to include
additional complexities in choice behavior, for example, incorporating the
bounded rationality of transit users (Jiang and Ceder 2021). The inclusion of
additional adaptive choice dynamics and the integration of new data sources
into route-choice models is a trend that has continued within the past decade
(Chen et al. 2016). Such studies are often inspired by research originating in
cognitive psychology (Leong and Hensher 2012) and may include how travelers
integrate various information sources into their decision making (e.g.,
smartphone journey-planners, timetables, and both individual and collective
experiences) together with models for habit formation and learning (Bogers,
Bierlaire, and Hoogendoorn 2007; Cats and West 2020).

In simulation-based evaluations of FLEX (whether as an independent mode or as
part of a combined FIX and FLEX service), demand may be modeled exogenously,
i.e., that demand for the FLEX service is independent of the level-of-service
(LoS) provided (e.g., Shen, Zhang, and Zhao (2018); Jäger, Brickwedde, and
Lienkamp (2018)), or endogenously, i.e., that demand for the FLEX service is
dependent on LoS provided (e.g., Atasoy et al. (2015); Archetti, Speranza, and
Weyland (2017); Berrada and Poulhès (2021)). A FLEX system typically does not
exist in isolation, but rather as a part of a public transit system as a
complementary or competing alternative to other modes, and it is reasonable to
model demand for such services endogenously. The vast majority of studies,
however, focus on FLEX systems as an independent mode or replacement to
conventional modes and assume fixed and inelastic demand for such services
Vansteenwegen et al. (2022).

###  2.2 Flexible transit simulation with endogenous demand

Among studies that model demand for FLEX endogenously, however without
integration with FIX, Atasoy et al. (2015) explore the dynamic allocation of a
joint fleet of vehicles to individual use taxi, shared-taxi, and mini-bus
services. Archetti, Speranza, and Weyland (2017) model the choice between a
door-to-door FLEX shuttle system, FIX, and private car for a wide range of
grid-structured road and transit networks and spatio-temporal demand patterns.
Berrada and Poulhès (2021) investigate the socioeconomic and economic impact
of partially replacing FIX services with a stop-based FLEX service with
varying fleet compositions (i.e., number of vehicles and vehicle sizes) based
on a real-life case study of such a scenario in Paris. In modeling predicted
demand for each FLEX service type, Atasoy et al. (2015) adopt a multinomial
logit (MNL) model using projections of fare and perceived utility of travel to
estimate traveler choices in response to re-optimized fleet allocations. The
choice-set presented to each traveler is determined by a set of constraints
based on the capacity of each service and a parameter that defines the
departure and arrival time flexibility of travelers. To incorporate individual
perceptions of ’best route’, Archetti, Speranza, and Weyland (2017) randomize
individual LoS preferences including a FIX user class, however assume
travelers have perfect knowledge of all routes and thus assign travelers
statically to each service based on shortest travel time. Similar to Archetti,
Speranza, and Weyland (2017), Berrada and Poulhès (2021) assume perfect
knowledge of all routes but calculate demand for FIX and FLEX alternatives
using a deterministic function of the ratio of the average generalized travel
costs (GTCs) of the two choice alternatives for each OD stop pair. An
’elasticity’ parameter is also used as an exponent to the GTC ratio to model
the sensitivity to cost differences in resulting mode-split. Both Atasoy et
al. (2015) and Archetti, Speranza, and Weyland (2017) do not consider traffic
and transit congestion effects on travel time, and an all-or-nothing
assignment is used. Berrada and Poulhès (2021) also assume constant speeds for
vehicle and traveler movements. Experienced waiting and in-vehicle times are,
however, modeled as dependent on levels of demand for FLEX at each OD, which
is updated using a simulation framework including an algorithm for individual
FLEX vehicle assignments. The fixed-point assignment equilibrium problem is
solved using an MSA for updating the GTCs.

Several recent studies have also been dedicated to modeling FLEX systems with
endogenous demand that also model their integration with FIX systems (Wen et
al. 2018; Pinto et al. 2020; Narayan et al. 2020).

Wen et al. (2018) evaluate the feedback loop between service performance and
demand for an integrated, on-demand shared automated vehicle (SAV) + FIX
service and alternative conventional modes. A nested-logit model (referred to
as the ’status-quo’ choice model in the paper) is estimated based on a survey
of existing modes (walk, bike, car, taxi, and a combined transit nest
including bus, rail, and first-mile to rail park/kiss and ride) in a major
European city. An SAV to rail service is added as a synthetic mode alternative
within the transit nest, with a utility function based on parameters from the
estimated model and complementary assumptions and operational cost estimates
relating to SAV services. An agent-based simulation platform is used to obtain
the LoS of the SAV + rail alternative with static travel times based on
predicted demand, which is fed into the nested-logit model iteratively. The
equilibrium assignment problem is solved using an MSA update on mode-split
output of the choice model between days. Results from a case study centered
around a high-frequency commuter rail station suggest that allowing for pre-
booked requests, combining fare with transit, and offering an SAV + rail
system are effective measures to attract private car trips to more sustainable
transit alternatives. The authors highlight the need, however, to investigate
combined FIX and SAV + FIX in strategic planning to mitigate poaching demand
from other FIX services.

Pinto et al. (2020) provide an extensive study of the trade-offs between
allocating resources to FIX service patterns (e.g., short-turning) and
frequencies and operating a door-to-door SAV mobility service that can be used
both as a first/last-mile service to FIX and as an independent mode. The
authors formulated a bi-level joint transit network design and SAV fleet
sizing model that incorporates time-dependent mode/route choice in the lower-
level problem of mode choice and assignment. A multi-modal (active modes, bus,
rail) dynamic transit assignment model (Ömer Verbas and Mahmassani 2015) is
used to obtain performance metrics for the FIX alternatives, combined with an
SAV simulator (Hyland and Mahmassani 2018) to obtain performance metrics for
SAVs per OD and time period. The performance metrics for FIX, SAV, and SAV +
FIX services are fed into an MNL mode-choice model in an iterative equilibrium
assignment procedure based on mode and route choices. In a case study of the
Greater Chicago metropolitan area for existing FIX demand, the authors find
that it is possible to improve the LoS provided by the existing transit
network by reallocating resources away from FIX patterns with low utilization
rates toward SAV services and high demand FIX patterns.

Narayan et al. (2020) develop a model for integrated FIX and FLEX route choice
that allows for a combination of FIX and FLEX modes in a single trip. The
model is implemented within the agent-based simulation framework MATSim
(Horni, Nagel, and Axhausen 2016) and applied to a case study of the city of
Amsterdam to study resulting mode-split between private car, walking, biking,
FIX, FLEX, and FIX + FLEX modes and sensitivities to FLEX service design
variables. To limit the choice-set of integrated public transport routes
(combining walking/biking, FIX, and FLEX legs of a trip), a rule-based
algorithm is proposed based on parameters for the catchment areas of FIX stops
by walking and FLEX modes. Furthermore, multi-modal routes are restricted to a
maximum of two transfers, and FLEX trips are either utilized as a direct trip
or as a first/last-mile trip to FIX. Demand for each OD is assigned all-or-
nothing to the generated route with the highest utility, calculated based on
distance-based fare, number of transfers, walking/biking time, in-vehicle
time, and waiting time. The default logit-based equilibrium assignment of
MATSim is used to allow travelers to update their choice of route, mode, and
departure time. Results indicate that LoS improves by offering an integrated
FIX and FLEX system. However, users combining FIX and FLEX modes are primarily
attracted from existing FIX demand.

In general, there are three key components relevant for the incorporation of
mode/route choice into a transit assignment model: (i) the choice-set of
alternatives available to each traveler to travel from each origin to a
destination, (ii) a decision model of which attributes associated with
different alternatives (e.g., travel time, fare, comfort) are considered by a
traveler and estimations of how a traveler evaluates these attributes in
calculating choice probabilities, and, if an equilibrium approach is utilized,
(iii) a day-to-day learning and/or convergence model for how travelers update
their evaluations of alternative paths based on the results of previous
choices and (potentially) unforeseen deviations from prior anticipations (de
Dios Ortúzar and Willumsen 2011). The studies described above apply different
combinations of these components in modeling demand for FLEX endogenously.
Results from case studies of integrated FIX and FLEX services all find that
the addition of such a service can improve the LoS provided to passengers
without increasing operational costs. However, studies have also shown that
the introduction of such services can both encourage or discourage the use of
more sustainable modes.

###  2.3 Synthesis

In all studies seen so far where demand for combined FIX and FLEX is
considered endogenously, integrated FIX and FLEX systems are combined as a
single synthetic mode alternative with predefined paths that may be offered as
an alternative to ’FIX only’ or non-public transit modes such as private car
or taxi services. Route choice decisions are made only once at the origin of
the traveler and thus cannot capture en-route adjustments in response to new
within-day information that may inform anticipated conditions of downstream
paths. In contrast with other approaches, we propose a route choice model
where traveler decisions are made in an ’online’ fashion, i.e., the path used
to reach the traveler’s final destination is constructed at different decision
nodes corresponding to the within-day actions (walking, boarding, alighting,
request-sending) of a traveler. The assignment of travelers to FIX, FLEX, or
combined FIX and FLEX services are thus potentially made dynamically within-
day. The combinatorial possibilities of a multi-modal trip, including FIX,
FLEX, and active-mode legs, can be further extended if needed. Similar to
previous work, we generate a static choice-set as an initialization step and
apply filtering rules which permit the exclusion of certain route combinations
(e.g., transferring from one FLEX leg to another). To model integrated FIX and
FLEX demand and supply interactions in all studies mentioned, LoS is used as a
feedback measure for travelers to re-evaluate their choices. We adopt a logit-
based choice model and MSA approach to include behavioral components in
traveler representation and to solve the equilibrium assignment problem under
congested conditions.

In this study, we develop a transit route choice model for integrated FIX and
FLEX services, i.e., FLEX that can potentially be used as an independent mode
from an origin to a destination but also allows travelers to transfer between
FIX and FLEX in-vehicle legs of a trip. We assume a single centralized fleet
manager of this service that coordinates a fleet of vehicles in response to
traveler requests according to a predetermined operational policy, however
with no synchronization strategy with the co-existing FIX service. This model
is implemented within a dynamic public transit assignment framework
representing FIX and FLEX operations. Using the taxonomy of Vansteenwegen et
al. (2022), the FLEX service that we consider in this study is a stop-based,
many-to-many system that users access in real-time with no pre-bookings, that
is fully-flexible in terms of timetable and semi-flexible in routing. The
assignment of FLEX requests to vehicles is done in a dynamic online fashion,
meaning that the basic schedule for assigned vehicles can change during
operations, even for those already executing a previous assignment.

The contributions of the paper are thus:

  * •

Methodological: A route choice model that allows users to combine walking,
FIX, and FLEX services to perform a trip integrated within a day-to-day
learning framework.

  * •

Substantial: A case study based on a coordinated branched line with
bidirectional demand showing how travelers make trade-offs between FLEX
options and FIX lines and how these choices evolve with day-to-day learning.

The study is divided into two parts. First, the transit simulation framework
and extension to model dynamic transit user route choice in the presence of
FIX, FLEX, and combined FIX and FLEX mode alternatives is described in Section
3. In the second part of the paper, a two-link toy network is used to
illustrate model properties in a simpler setting in Section 4. The combined
simulation framework is then applied in a case study based on a real-world
coordinated branched transit service in Stockholm characterized by a high-
demand trunk section towards the inner-city and lower-demand branches in
Section 5. FIX, and FLEX services are run in parallel on branches, serving as
a feeder/last-mile service to/from a higher frequency trunk segment. The
evolution of experienced LoS and anticipated LoS over time are explored to
study the feedback loop between adaptive traveler choices and real-time
operational vehicle assignment for different traveler groups: those on the
trunk portion, those traveling to or from the branches, and those who travel
between the branches. Conclusions and a discussion of limitations and future
work is presented in Section 6.

##  3 Methodology

This section introduces the simulation framework used to model FIX and FLEX
operations and extensions to the dynamic transit assignment model to include
FLEX alternatives. An overview of the simulation framework is presented in
Section 3.1. Descriptions of the path alternative data structure and choice-
set generation process that lie at the core of the route choice model are
introduced in Section 3.2. Descriptions of traveler states, actions, and state
transitions in the presence of combined FIX + FLEX alternatives are presented
in Section 3.3. The traveler choice model and how path-sets used in choice
probability calculations are associated with different actions are described
in Section 3.4 and Section 3.5. The components of LoS anticipations used in
the evaluation of action choices is presented in Section 3.6. Finally, the
model for traveler learning between days used in this study is described in
Section 3.7 and Section 3.8.

###  3.1 Simulation framework

We propose a transit route choice model in the presence of combined FIX and
FLEX alternatives. The model is implemented within an agent-based public
transit simulation framework, BusMezzo (Toledo et al. 2010), that includes
essential components that enable modeling both FIX and FLEX operations and
traveler behavior. The modeling framework is event-based and embedded within
the mesoscopic traffic simulation model Mezzo (Burghout 2004).

BusMezzo was originally developed to model FIX operations and has been shown
to replicate FIX phenomena such as headway variability propagation and
bunching (Toledo et al. 2010). In the assessment of FIX operations and dynamic
traveler behavior, the model has been applied to evaluate the performance of
different real-time holding strategies (Cats et al. 2011; Laskaris et al.
2018), short-turning strategies (Leffler et al. 2017), and demand-management
strategies with real-time information provision (Drabicki et al. 2020;
Peftitsi, Jenelius, and Cats 2021). A high-level overview of the simulation
framework is displayed in Figure 1.

Figure 1: High-level overview of the simulation framework. Static inputs are
marked with yellow boxes, key transit agents with purple ovals, processes with
blue rectangles, and final and intermediate process outputs with red stadiums.

As shown in Figure 1, the simulation framework consists of _within-day_ and
_day-to-day_ processes and feedback loops. Hitherto, the day-to-day learning
framework described in Cats and West (2020) was limited to FIX services. In
this study, we introduce modeling functionalities to allow for the
representation of FLEX alternatives as part of a dynamic transit assignment
model. Static inputs to the framework (marked with yellow boxes) are the
underlying road network (e.g., links, nodes, turning-servers), FIX service
definitions (e.g., bus lines, stops, schedules, real-time control strategies,
fleet characteristics), FLEX service definitions (e.g., possible service
routes, stops, vehicle-travel request assignment strategy, empty-vehicle
rebalancing strategy, fleet characteristics) and traveler population
definitions (e.g., OD demand rates, value-of-time/learning parameters, choice-
set filtering parameters). As part of the initialization process, a choice-set
generation model is called to generate an initial set of possible path
alternatives for each OD, dependent on the available transit network
(described further in Section 3.2).

Individual traveler agents construct a path from their origin to their
destination dynamically within the day with respect to available choice
alternatives (as also visualized in Figure 2), with intermediate choice
probabilities determined by an expected maximum utility. Anticipations
regarding the downstream utility of a path alternative are updated based on
accumulated experiences learned from day to day.

Travelers choosing to utilize a FLEX service as part of their chosen path will
send a _request_ consisting of a desired stop for pick-up, a desired stop for
drop-off, and a desired time for departure to the FLEX fleet operator. The
FLEX extension to BusMezzo (Leffler et al. 2021) allows for switching between
different greedy nearest-neighbor assignment heuristics to assign traveler
requests made known to a FLEX fleet operator in real-time to vehicles that are
coordinated on-demand. Furthermore, alternative empty-vehicle rebalancing
strategies may be used. Dependent on the fleet coordination strategy employed
by this fleet operator, vehicles are assigned to traveler requests in real-
time. The travel times, paths, and choices of individual passengers and
network-wide LoS and vehicle utilization measurements are produced as model
output for later analysis.

###  3.2 Path alternative and choice-set definitions

The path alternative data structure lies at the core of the transit route
choice model. A path alternative $i$ is defined as an ordered, alternating
sequence of walking links $w$ and transit (i.e., FIX or FLEX) links $l$, each
connecting pairs of stops $s$, or more generally locations, as follows:

$i=\left\\{o,w_{1},s_{2},l_{1},s_{3}\dots
s_{2n},l_{n},s_{2n+1},w_{n+1},d\right\\}.$ (1)

The index of each path component denotes its position in this sequence, with
the exception of the first and last components, $o$ and $d$, corresponding to
the origin and destination of the path, respectively. For example, the stop
$o$ denotes the first stop, or origin, of the path alternative $i$, $w_{1}$
denotes the first walking link connecting stops $o$ and $s_{2}$, $l_{1}$ is
the first transit link connecting stops $s_{2}$ and $s_{3}$, and so on. A path
may include zero-cost walking links that represent the case when a traveler
chooses to remain at the current stop rather than walk (e.g., if $o$ and
$s_{2}$ in an instance of (1) refer to the same stop). The number of
components in a path alternative is thus fully determined by the number of
transit legs $n$ for a given path.

The global set of path alternatives available to all travelers is generated as
part of model initialization. Several filtering rules and logical constraints
may be included to limit the number of relevant paths available to each
traveler for any OD pair, such as a maximum number of transfers, maximum
walking distance, or path dominancy principles. Path components for FIX paths,
with the exception of $o$ and $d$, may also be merged into hyperpaths or, more
specifically, sets (e.g., $S_{j},W_{j},L_{j}$ for component $j$ of a path
alternative) if multiple stops, walking links or transit links that a traveler
is considered to be indifferent to, are available for a given segment in a
path (described in greater detail in Cats (2011)). For example, suppose
several FIX bus lines $l\in L_{j}$ with the same expected travel cost for a
given transit link between a pair of stops are available. In that case, a
traveler may choose to board any of these lines dependent on whichever bus
happens to arrive first (thereby addressing the ’common lines’ problem of
transit assignment).

A FLEX leg of a path differs from a FIX leg in the sense that it represents
one of many possible routes between stops within the service area of the FLEX
operator, which may or may not be realized. In contrast with line-based FIX, a
FLEX service operator serves an area containing possible pick-up and drop-off
locations rather than a specific route. The FLEX service considered in this
study is stop-based, meaning that the set of possible pick-up and drop-off
points in a FLEX service area is fixed. In generating the global set of path
alternatives, the full set of shortest routes in terms of free-flow travel
time between all pairs of stops is generated for the FLEX service area as an
initialization step. This set of routes is then used in the generation of
traveler choice-sets as for paths including FLEX legs. For simplifying
purposes, it is assumed that there are no separately operated FLEX services
with overlapping service areas. For this reason, as well as to mitigate the
combinatorial explosion of explicit enumeration, in the generation of path-
sets, paths with repeating FLEX links are omitted (e.g., if transit link
$l_{j}$ in (1) is a FLEX link, then $l_{j+1}$ cannot also be a FLEX link).
Consequently, FLEX links are always considered distinct from FIX links. This
assumption is considered reasonable as several studies have indicated that
travelers consider FLEX services as a significantly different mode to FIX (Li
and Quadrifoglio 2010).

Let $I^{od}$ denote the set of all path alternatives available to travel
between a location $o$ and a destination $d$ utilizing a sequence of walking-
and transit links as defined in (1). The path that a traveler agent utilizes
is not known or selected upfront. Instead, as displayed in Figure 2, a
traveler makes a sequence of _choices_ between alternative walking, boarding,
and alighting _actions_ at various decision nodes in a trip. Travel
experiences (denoted $t^{x}_{j}$ in Figure 2) for each chosen component $j$ of
a path are accumulated and inform path choices in the following days (more on
this in Section 3.8).

Figure 2: Example of the sequence of within-day decisions made by a traveler
agent and accumulated experiences from the sampled path that inform within-day
decisions the following day. Green arrows indicate a chosen action, and red
arrows indicate evaluated action alternatives. Source: (Cats and West 2020)

As displayed in Figure 2, at each decision node the set of available path
alternatives considered by the traveler will change dependent on their most
recently chosen action. The path-set $I_{a}^{od}(t)\subseteq I^{od}$
considered by a traveler when making a choice of action is dependent on the
context of the traveler in terms of: (1) the current location $o$, (2) the
category of the next action $a$ to be made and (3) the state of the transit
network, as well as the traveler’s knowledge of this state, at time $t$.

###  3.3 Traveler states, actions and state transitions

As described in Section 3.2, a traveler makes a sequence of choices among
alternative actions at various decision nodes (or traveler states) to
construct a complete trip. A diagram of all traveler states and actions that
transition a traveler between these states is displayed in Figure 3.

Figure 3: Traveler state diagram with FIX and FLEX alternatives available.
Traveler states are represented by yellow rounded rectangles, action decision
categories with colored arrows, and diamonds where action decisions branch
conditional on previous actions.

As shown in Figure 3, there are four states a traveler can be in: (1) having
_arrived at a stop_ where a decision is made on how to continue their journey
to their final destination, (2) _waiting at a stop_ where a decision is made
to board either a FIX or (3) FLEX vehicle, and (4) sitting or standing _on-
board a vehicle_ where a decision must be made of which stop to alight at to
continue their journey.

Conditional on the current state of the traveler and simulation events (e.g.,
a vehicle arrives at a stop), five categories of decisions among alternative
actions transition the traveler from one state to another. Three action
categories connect the two traveler states of having arrived to a stop and
waiting for either a FIX or FLEX vehicle to board: (1) a _connection_ decision
to go to a stop within walking distance (including possibly staying at the
current stop) upon arriving to a new location, (2) a _mode_ decision of
whether to use FIX or FLEX for the next trip leg from the traveler’s chosen
connection stop and, conditional on FLEX being chosen, (3) a _drop-off_
decision of which stop to request a ride to. Note that this drop-off stop can
correspond to the traveler’s final destination or an intermediate transfer
stop. The remaining two action categories follow the state of waiting for a
FIX or FLEX vehicle to arrive: (4) a traveler will make a _boarding_ decision
of whether or not to board an approaching transit vehicle, and, once on-board
a vehicle, (5) an _alighting_ decision of which stop to alight at.

Additional assumptions regarding FIX and FLEX service operations and traveler
behavior are embedded in the traveler state definitions in Figure 3. It is
assumed that a traveler commits to using a FIX or FLEX service once a mode
choice has been made (no traveler cancellation). The FLEX service operator
also commits to serving the traveler if a trip request is accepted (no
operator cancellation). A traveler evaluates a potential trip from a FLEX
service operator right after making a connection decision. If FLEX is chosen,
the traveler will instantly send a trip request after deciding to walk to a
stop to wait for the FLEX service.

###  3.4 Connection, mode and drop-off decisions

All action decisions described in Section 3.3 are modeled within the framework
of RUM discrete choice models. An MNL model is used in calculating the
probabilities of action choices. The probability $P_{a,k}$ of traveler $k$ to
choose action $a$ from a set of alternative actions $A$ is:

$\displaystyle P_{a,k}=\frac{e^{v_{a,k}}}{\sum_{a\in A}e^{v_{a,k}}}$

where $v_{a,k}$ is the expected utility of action $a$ for traveler $k$. Each
action $a$ is associated with a downstream path-set, $I_{a}$, that allows the
traveler to reach their final destination conditional on this action being
chosen. The expected utility $v_{a,k}$ is given by the logsum over the
expected utilities of individual path alternatives in $I_{a}$:

$\displaystyle v_{a,k}=ln\sum_{i\in I_{a}}e^{v_{i,k}},$ (2)

where $v_{i,k}$ is the expected utility of path $i\in I_{a}$ for traveler $k$
(more on this in Section 3.6).

The first choice of action a traveler must make is a connection action, i.e.,
the choice of which stop to walk to when initiating their trip. This decision
is also made each time a traveler alights from a vehicle at a new location. As
shown in Figure 3, once a connection decision has been made, it is followed by
a mode decision conditional on the chosen connection stop, and a drop-off
decision conditional on the choice of mode and connection stop.

In evaluating path alternatives for a connection decision, $o$ corresponds to
the current location of the traveler and $d$ the traveler’s final destination.
Recall the general path definition:

$i=\left\\{o,W_{1},S_{2},L_{1},S_{3}\dots
S_{2n},L_{n},S_{2n+1},W_{n+1},d\right\\}.$

A connection decision amounts to choosing between the combined utility of
subsets of $I^{od}(t)$, the set of path alternatives available to the traveler
to move from current location $o$ to final destination $d$ at time $t$. The
path-sets $I_{s^{+}}^{od}\subseteq I^{od}(t)$ associated with each candidate
connection stop $s^{+}$ is then given by

$I_{s^{+}}^{od}=\left\\{i\mid s^{+}\in S_{i,2}\text{ for some }i\in
I^{od}(t)\right\\}.$

The mode decision (i.e., the choice between FIX and FLEX mode actions) is
conditional on the choice of connection stop $s^{+}$. We can define a function
$mode(\cdot)$ that takes a transit link component $L_{i,j}$ of a path $i$ as
an argument and outputs whether or not this is operated as a FIX or a FLEX
service:

$mode(L_{i,j})=\begin{cases}fix&\text{if a fixed route and schedule exists
$\forall l\in L_{i,j}$ of path $i$ and leg $j$}\\\
flex&\text{otherwise}.\end{cases}$

The path-sets $I_{s^{+},m}^{od}\subseteq I_{s^{+}}^{od}$ associated with a
mode decision are defined conditional on the choice of a stop-mode pair, or
action $a=(s^{+},m)$. This action amounts to walking to stop $s^{+}\in
S_{i,2}$, using a walking link $w\in W_{i,1},$ with the intention of using
mode $m\in\\{fix,flex\\}$ for the next transit leg $l\in L_{i,1}$ for some
path $i$. More formally, the path-sets associated with each mode action are:

$I_{s^{+},m}^{od}=\left\\{i\mid m=mode(L_{i,1})\text{ for some }i\in
I^{od}_{s^{+}}\right\\}.$

If a FLEX link is chosen, a traveler must communicate their intention to use a
FLEX service to the operator of this service via a travel request. This
request must include a desired time $t^{+}$ and location $s^{+}$ for pick-up,
and a desired location for drop-off, $s^{-}$. The pick-up point has already
been chosen in the connection decision of the traveler and is associated with
a path-set when choosing to utilize FLEX for the next transit leg, i.e.,
$I_{s^{+},flex}^{od}$. The desired time for pick-up $t^{+}$ is assumed to be
as soon as the traveler plans to arrive to stop $s^{+}$, i.e., the current
time $t$ plus the expected time it takes to walk from the traveler’s current
location $o$ to stop $s^{+}$.

If there are several paths in $I_{s^{+},flex}^{od}$, then a decision has to be
made by the traveler to choose the most beneficial drop-off point to send a
request for. Each drop-off point available after a FLEX transit leg is
associated with its own set of path alternatives
$I^{od}_{s^{+},flex,s^{-}}\subseteq I_{s^{+},flex}^{od}$. The path-set
associated with each candidate drop-off point is given by

$I^{od}_{s^{+},flex,s^{-}}=\left\\{i\mid s^{-}\in S_{i,3}\text{ for some }i\in
I_{s^{+},flex}^{od}\right\\}$

The combined utility of each connection, mode and drop-off action and
resulting probabilities are then calculated by taking the logsum over the
expected utilities of all path alternatives associated with each action (i.e.,
$I^{od}_{s^{+}},I^{od}_{s^{+},flex}$ and $I^{od}_{s^{+},flex,s^{-}}$) as in
(2). The choice structure of these decisions can be visualized as a tree as
displayed in Figure 4.

Figure 4: Connection, mode and drop-off decisions.

Figure 4 displays the event triggering a connection decision and the set of
alternative actions and associated path-sets for each action.

###  3.5 Boarding and alighting decisions

Boarding and alighting decisions are invoked if FIX is chosen by a traveler.
In this paper, if a decision is made to use FIX, the traveler will walk (if
needed) and wait at the chosen stop for the next available FIX service vehicle
from a line deemed relevant to board. The path-set $I_{s^{+},fix}^{od}$,
associated with walking to $s^{+}$ to use $fix$, is divided into path
alternatives associated with choosing to not board $I^{stay}$ versus choosing
to board $I^{board}$ a vehicle arriving from line $l^{arr}(t)$ at time $t$:

$\displaystyle I^{board}(t)$ $\displaystyle=\left\\{i\mid l^{arr}(t)\in
L_{i,1}\text{ for some }i\in I_{s^{+},fix}^{od}\right\\},$ $\displaystyle
I^{stay}(t)$ $\displaystyle=\left\\{i\mid l^{arr}(t)\notin L_{i,1}\text{ for
some }i\in I_{s^{+},fix}^{od}\right\\}$

If a traveler has instead decided to use FLEX (i.e., walk to $s^{+}$ to use
$flex$ to stop $s^{-}$), then the traveler is committed to boarding any FLEX
vehicle that has been assigned to this request. With the assumption that
travelers do not cancel a request once sent, there is thus no boarding
decision invoked for a traveler in the state of waiting for a FLEX trip.

Similarly, FLEX users are assumed committed to alight at the specific drop-off
point for which they sent a request. Again, with the assumption that a
traveler will not re-make their decision once a FLEX vehicle has been boarded,
no alighting decision is invoked for a traveler in the state of being on-board
a FLEX vehicle. Instead, the on-board traveler will alight whenever the chosen
drop-off point is reached. For FIX users, the choice of alighting stop is made
immediately after a boarding decision has been made. The downstream
alternative path-sets for an alighting decision $I^{alight}_{s^{-}}$ are then
determined by the line that was just boarded:

$I^{alight}_{s^{-}}(t)=\left\\{i\mid s^{-}\in S_{i,3}\text{ for some }i\in
I^{board}(t)\right\\}.$

###  3.6 Expected utility of a path

As described in Section 3.4 and Section 3.5, the probability of choosing an
action $a$ is determined by the logsum (2) over the expected utilities of all
path alternatives associated with that action. The expected utility of a path
alternative is calculated based on the anticipated GTC associated with
traversing stop, walking link and transit link path components. More formally,
the expected utility $v_{i,k}$ of path $i\in I_{a}$ for traveler $k$ is given
by the weighted sum of anticipated waiting ($\widehat{t}_{j,k}^{wait}$), in-
vehicle ($\widehat{t}_{j,k}^{ivt}$), and walking ($\widehat{t}_{j,k}^{walk}$)
times over all legs $j\in J_{i}$ of path $i$, and the number of transfers of
path $i$, $n_{i}^{trans}$:

$v_{i,k}=\sum_{j\in
J_{i}}\left[\beta_{j,k}^{wait}\widehat{t}_{j,k}^{wait}+\beta_{j,k}^{ivt}\widehat{t}_{j,k}^{ivt}+\beta_{j,k}^{walk}\widehat{t}_{j,k}^{walk}\right]+\beta_{k}^{trans}n_{i}^{trans},$
(3)

where $\beta^{wait},\beta^{ivt},\beta^{walk}$ and $\beta^{trans}$ are value-
of-time parameters corresponding to each GTC component. The number of
transfers for a given path is considered static and known by all travelers.
The anticipated walking time associated with walking links is assumed static
and uncapacitated, based only on distance and the traveler’s average walking
speed (realized walking time variability is simulated with stochastic error
terms however). The associated cost of waiting- and in-vehicle time at
different stages of a trip are dynamic, however, due to confrontations between
prevailing demand flows and capacitated supply as well as supply and demand
uncertainty (e.g., vehicle occupancy restrictions, on-board crowding, vehicle
travel time delays).

Note in the formulation of (3) that the value-of-time parameters
$\beta^{wait},\beta^{ivt},\beta^{walk}$ and $\beta^{trans}$ allows for
differentiating between accessing and utilizing transit legs dependent on if
it is run by a FIX or FLEX service. In other words, walking or transferring
from/to FLEX or FIX transit alternatives, as well as waiting time and in-
vehicle time when using FLEX or FIX alternatives, may be perceived differently
in the decision-making processes of a traveler.

###  3.7 Traveler waiting- and in-vehicle time anticipation

In Section 3.6 the anticipated GTC of a path used in the choice model for
selecting among alternative actions is described. In this study, the
anticipations of waiting- and in-vehicle times (i.e.,
$\widehat{t}_{j,k}^{wait}$ and $\widehat{t}_{j,k}^{ivt}$ in (3), respectively)
of traveler $k$ for transit link $j$ of a path are based on two information
sources: (1) the accumulated experience ($t_{j,k}^{x}$) of utilizing transit
link $j$, and (2) prior knowledge ($t_{j}^{p}$) corresponding to static LoS
expectations of transit link $j$.

Prior knowledge anticipations are assumed always known by the traveler before
any decision is made (i.e., can inform future decision-making from day 1 and
onward). Accumulated experiences are collected within-day, however, only after
a decision has already been made. Accumulated experiences are thus only
utilized to inform future decisions day-to-day when previously utilized paths
are re-evaluated.

Prior knowledge for FIX transit links is based on the predefined schedule of
the service. In the absence of a predefined schedule, an optimistic estimate
is used for the anticipated waiting- and in-vehicle times based on prior
knowledge for FLEX legs. This could either correspond to a static service
guarantee from the operator (e.g., a maximum waiting- and in-vehicle time
independent of current fleet and demand status) or a ’best case’ evaluation
from the traveler’s perspective (e.g., immediate service with zero waiting
time and the most direct route available with free-flow in-vehicle time).
These estimates can be viewed as an exploration parameter used for the first
iterations of a traveler’s learning process that favors choosing paths that
include unexplored FLEX legs. For both FIX and FLEX links, we assume that
$t_{j}^{p}$ is only used in the absence of any accumulated waiting- or in-
vehicle time experience for a given leg. More precisely, the anticipated
waiting- or in-vehicle time $\widehat{t}_{j,k}$ of path component $j$ for
traveler $k$ is given by:

$\widehat{t}_{j,k}=\begin{cases}t^{p}_{j}&\text{if no experience has been
gathered by traveler $k$ for leg $j$}\\\
t^{x}_{j,k}&\text{otherwise}.\end{cases}$ (4)

###  3.8 Accumulated travel time experiences including dynamic congestion
effects

In Section 3.7 the prior knowledge and accumulated experience information
sources that inform the anticipated GTC calculations are introduced. In this
section, we describe the learning model for how accumulated travel time
experiences between days is estimated and how additional dynamic congestion
effects, such as the discomfort a passenger may experience if denied boarding
a full vehicle and the experience of crowding on-board vehicles (dependent on
vehicle type and the number of standing/seated passengers) are also considered
in the model.

Learned travel time component anticipations for all decisions in the iterative
day-to-day learning process in this study are based directly on the
accumulated travel time experiences as defined in (4). The accumulated travel
time experience of path component $j$ for traveler $k$ at the beginning of day
$d$, $t^{x}_{j,k,d}$ is updated in an MSA formulation. The most recent
experiences, $t_{j,k,d-1}$, are discounted as a function of the number of days
elapsed (with a discount factor of 1 for the first day). More precisely, the
accumulated travel time experiences used in traveler LoS anticipations is
given by:

$\displaystyle
t^{x}_{j,k,d}=t^{x}_{j,k,d-1}+\frac{1}{d}(t_{j,k,d-1}-t^{x}_{j,k,d-1}).$ (5)

To include the additional discomfort induced by denied boarding, a distinction
is made between waiting time until the first vehicle the passenger wishes to
board ($t_{j,k}^{wait,nominal}$) and additional (unexpected) waiting time
($t_{j,k}^{wait,denied}$) until their next opportunity to board if denied
boarding. In registering a waiting time experience, the portion of waiting
time due to denied boarding is weighted with an additional penalty
$\alpha^{denied}$ for the discomfort experienced. The most recent waiting time
experience, $t^{wait}_{j,k,d-1}$, used in the learning update (5) is then
given by the weighted sum of these, i.e.,
$t^{wait}_{j,k,d-1}=t_{j,k,d-1}^{wait,nominal}+\alpha^{denied}\cdot
t_{j,k,d-1}^{wait,denied}$.

The total nominal in-vehicle time ($t^{ivt,nominal}_{j,k}$) of passenger $k$
for an in-vehicle leg $j$ of their trip is comprised of the sum of all sub-
intervals $h$ of in-vehicle times ($t^{ivt,nominal}_{j,k,h}$) between pairs of
stops along the passenger’s trip. The additional discomfort experienced when
on-board a crowded vehicle is taken into account by multiplying each in-
vehicle sub-interval by weights ($\alpha_{h}$) that are a function of crowding
level for that segment of the complete trip, i.e.,
$t^{ivt}_{j,k,d-1}=\sum_{h}\alpha_{h}\cdot t^{ivt,nominal}_{j,k,h,d-1}$. The
crowding level of a given in-vehicle sub-interval may for example be estimated
by the ratio of passengers on-board, and the seated and standing capacity of
the vehicle.

Since the number of transfers and walking time anticipations of each path are
considered static and known beforehand, the learned path component travel time
experiences correspond to the average total waiting time (adjusted with
additional weight for any waiting time due to denied boarding) and the average
crowding-weighted in-vehicle time of different legs of a public transit trip.

##  4 Demonstration: Toy network

To demonstrate how travelers learn of the trade-offs between waiting time
(service headway versus vehicle-travel request assignment) and in-vehicle
crowding levels (larger buses versus smaller shuttles) for the FIX and FLEX
service types and how this is reflected in resulting mode choices, we
construct the following scenario using the setting described in Figure 5. The
physical network consists of two stops (A and B) and two links (AB and BA).
Two types of services are available to transport travelers from stop A to stop
B: (1) a FIX even-headway (10 min) service run by 100 passenger capacity
buses, and (2) a FLEX service with 10 passenger capacity shuttles that assigns
vehicle trips to traveler trip requests. FLEX vehicles are assigned to
traveler requests every second without rebalancing idling vehicles.
Access/egress by walking links is the same for FIX and FLEX services. Both
FIX, and FLEX use the same links with the same constant nominal in-vehicle
time (30 min) in either direction.

Figure 5: Two stop and two link network. Two services are available to
travelers at stop A traveling to stop B: one fixed headway (10 min) service
run with 100 passenger capacity buses (colored green), and one on-demand
service run by 10 passenger capacity shuttles (colored blue). Both services
use the same links with the same in-vehicle time (30 min) in either direction.

###  4.1 Experimental design

The value-of-time parameters used for calculating the GTC of a path as
described in (3), as well as in simulation outputs, are generally set in
relation to the value of in-vehicle time $\beta^{ivt}$. The weight of
perceived waiting time is set to double that of in-vehicle time,
$\beta^{wait}=2\cdot\beta^{ivt}$, based on the study of Wardman (2004). The
weight $\alpha^{denied}$ introduced in Section 3.8 to penalize any additional
waiting time due to denied boarding is set to 3.5 as detailed in Cats, West,
and Eliasson (2016).

The in-vehicle crowding factors described in Section 3.8 that are used in this
paper are based on the values reported in the meta-study by Wardman and Whelan
(2011). The load factor is given by the ratio of passengers on-board a
vehicle, $q^{onboard}_{h}$, to the maximum seat capacity of that vehicle,
$\gamma^{seats}_{h}$, for a given time-interval $h$. Based on this load factor
an in-vehicle time multiplier is used in weighting passenger in-vehicle time
experiences for this time-interval dependent on whether the passenger is
seated or standing. Travelers queuing at stops board vehicles in a first-in-
first-out manner. The in-vehicle time multiplier for seated passengers,
$\alpha^{sit}$, ranges between 0.95 to 1.71 and all passengers that board are
assumed to sit if there are unoccupied seats available. The in-vehicle time
multiplier for standing passengers, $\alpha^{stand}$, used when all seats are
occupied, ranges between 1.78 to 2.69.

To simplify for the sake of demonstration we assume that both FIX and FLEX
vehicle types have seats available for each passenger for the toy network
scenario. Dwell times ($t^{dwell}$, in seconds) at stops for both FLEX and FIX
vehicles are calculated using a linear function of the number of boarding
passengers $n^{board}$ and alighting passengers $n^{alight}$ at that stop as
estimated in Dueker et al. (2004) for buses without lift operations:

$\displaystyle t^{dwell}=5.14+3.48n^{board}+1.7n^{alight}.$ (6)

Besides affecting vehicle running times, $t^{dwell}$ is registered as
additional in-vehicle time in traveler experiences.

100 travelers are generated at stop A simultaneously with destination stop B.
The arrival time of these travelers is 1 s after the departure time of a FIX
bus. Since the capacity of the FIX buses is equal to the total number of
travelers, any traveler that chooses FIX will thus experience a waiting time
of close to 600 s, the full headway of the service. To simulate varying
capacity limitations for the FLEX service, we initialize 1-7 vehicles at stop
A before the travelers arrive. If there is capacity at A available for a
traveler using FLEX, then the traveler will experience a waiting time of 1 s.
If there is no capacity immediately available at A, a vehicle must be sent
from stop B to A to serve the traveler. The number of FLEX vehicles
initialized at stop B for all scenarios is always sufficient to serve all
travelers meaning that, in the worst case, a traveler choosing to use FLEX
will experience a waiting time corresponding to the full in-vehicle time it
takes for a reactive FLEX vehicle to traverse the link BA, i.e., 30 min.

Mode-choice decisions between FIX and FLEX are mainly dependent on differences
in the accumulated experience of waiting time in our demonstration. There are
particularly large variations in LoS due to waiting times dependent on how
many FLEX vehicles are initialized at stop A and how many passengers choose
FLEX on a given day. The influence of anticipated crowding-weighted in-vehicle
time in this scenario becomes relevant in conditions when the anticipated
waiting time between FIX and FLEX are closer to equal. The system optimum
solution for the mode-split over all travelers in terms of resulting total GTC
is always that the capacity of the FLEX fleet at A is filled precisely with no
overflow with the remaining collection of travelers choosing FIX. In other
words, 10% choosing FLEX is optimal for 1 FLEX vehicle at A, 20% for 2
vehicles, etc..

Finally, with no prior experience of either service, the initial anticipated
waiting time for FIX is set to half the headway of the service (i.e., 5 min)
and for FLEX an initial exploration parameter of 0 s. The anticipated in-
vehicle time for both services is set to the shortest-path nominal in-vehicle
time (i.e., 30 min). There is thus an inherent preference towards choosing
FLEX over FIX on the first day to stimulate exploration. Each instance is
simulated until 75 days have passed with 20 replications per scenario.

###  4.2 Results

To demonstrate the evolution of average passenger LoS anticipations and long-
term convergence behavior, Figure 6 displays the average (over the 20
replications) learning trajectory of waiting times (top row), crowding-
weighted in-vehicle times (middle row) and resulting mode-split (bottom row)
for different initial FLEX supply conditions (columns labeled by initial FLEX
capacity at stop A).

Figure 6: Per-mode (blue for FLEX and red for FIX) anticipated and average
experienced waiting time (top row), per-mode anticipated and average
experienced (weighted for crowding) in-vehicle time (middle row), and the
average share of all travelers that chose FLEX or FIX over days (bottom row).
Columns correspond to the size of the FLEX fleet initialized at stop A.

As can be seen in Figure 6, even in this simple setup there is considerable
variation and interaction between LoS experiences. The effects of the strong
MSA discounting (5) of newer experiences is also apparent, where initial
accumulated experiences are retained over a long period of time, and result in
smooth learning curves with little oscillation. After the first 7-10 days
variations in both mode-split and experiences start to stabilize, which
together with quick discounting of newer experiences result in very small
changes in LoS anticipations in later days.

Initial conditions translate into a mode-split of roughly 72% of travelers
choosing FLEX and 28% choosing FIX on the first day for all FLEX supply
variations. For the smallest fleet size of one FLEX vehicle (leftmost column)
this means that, on day 1, 10 FLEX passengers experience a waiting time of 0
s, and the remaining 62 must wait 1800 s for 7 vehicles to be sent from stop
B. With the exception of the last vehicle to arrive, all of these vehicles are
filled to their maximum capacity and the highest crowding factor is used to
weight the in-vehicle time of these passengers. FIX passengers always
experience a waiting time of 600 s and receive a small discount to their in-
vehicle time due to less than half of the seats of the FIX bus being occupied.
Since all 28 passengers board the same bus, rather than separate vehicles, a
slightly longer dwell time is experienced and registered as in-vehicle time
for the FIX passengers compared to FLEX.

The extreme conditions of the first day are followed by a strong increase in
anticipated waiting time and perceived in-vehicle time for FLEX and a shift
towards FIX the following day (on average 97% of travelers). The few
passengers that now chose FLEX experience a 0 s waiting time and a discounted
in-vehicle time, close to the lowest GTC possible for this network. The
remaining travelers choosing FIX experience a 600 s waiting time and the
highest crowding factor for in-vehicle time. The poor experience of FLEX from
the first day is retained over the following days where slowly more
passengers, on average 4% and then 11%, start to choose FLEX. This is also
reflected in a gradually decreasing anticipated waiting time and in-vehicle
time which finally converge towards the stabilizing average experiences. In-
vehicle times are consistently underestimated with slow convergence to average
experiences, however, due to the rapid discounting of new experiences, mode
choices converge to a roughly 22% FLEX and 78% FIX mode-split. Individual runs
oscillate around this within a $\pm 6\%$ band due to the random draws of
choices.

In the second column of Figure 6 results for three FLEX vehicles initialized
at stop A are displayed. Mode-split and LoS anticipations follow a similar
pattern as for the single vehicle case. However, waiting time anticipations
converge to an approximately equal level between FIX and FLEX. With similar
waiting time experiences, the crowding levels on-board the two vehicle types
have a greater impact on choices. The FIX buses are never more than 75% full,
compared to the FLEX shuttles, which are more or less always filled to
capacity, resulting in a higher anticipated crowded in-vehicle time for FLEX
compared to FIX. Close to equal waiting time anticipation and higher in-
vehicle expectations when using FLEX are reflected in the mode split, which
converges to roughly 43% choosing FLEX.

Initial conditions for the 5-7 FLEX vehicle scenarios again result in an
average 72% split in favor of FLEX. This time, however, most FLEX passengers
experience a 0 s waiting time instead of 1800 s which results in less extreme
oscillation the following days. Waiting time anticipations converge to a lower
anticipated waiting time for FLEX compared to FIX. Crowding-weighted in-
vehicle times are still consistently higher for the smaller FLEX vehicles when
compared to FIX with several vehicles filled to their maximum capacity.
However, due to greater differences in waiting times, further weighted at
twice the importance in traveler decisions compared to in-vehicle time, the
mode-split for the 5 vehicle and 7 vehicle cases converge to close to the
capacity of FLEX vehicles at stop A at 60% and 70%, respectively.

##  5 Application: Stockholm coordinated branched line

To further investigate model properties at a larger scale and demonstrate how
the framework can be applied to evaluate a mixed FIX and FLEX system design,
we construct a study based on two existing bus lines in Stockholm: lines 176
and 177. The two lines run with 10 min even-headway during peak hours and form
a fork-like trunk-and-branches network that connects rural parts of the Ekerö
islands to the more central parts of Stockholm. As displayed in Figure 7, the
lines run between the branches (colored red and blue) to/from stops Solbacka
and Skärvik to the west in the Ekerö region, and form a common corridor
(colored purple) to/from Mörby in the northeast. The timetables of the two
lines are coordinated such that they run on the corridor section with an even
5 min headway. However, bunching regularly occurs and vehicle utilization on
the branch sections is low. This network has been previously studied to
evaluate multi-line holding control strategies targeted at reducing bunching
(Laskaris et al. 2018, 2021) and shortening the FIX service to the corridor
and running a FLEX service on the branches (Leffler et al. 2021). Rather than
evaluate the replacement of the FIX services with a FLEX service on branches,
as in Leffler et al. (2021), in this paper we investigate the potential of
only partially replacing FIX service on branches with a FLEX service.
Conceptually, this could provide a more adaptive service on the branches,
while improving service regularity on the corridor where demand is higher with
more direct connections to the inner-city. This scenario also allows us to
further evaluate model properties for a more complex scenario with mixed FIX
and FLEX paths.

Figure 7: Lines 176 (red and purple segments) and 177 (blue and purple
segment) in Stockholm. The purple segment is common for both lines. The lines
merge/split at the stop Malmvik. Figure adapted from Laskaris et al. (2018),
Figure 2.

###  5.1 Traveler population definitions

Table 5.1 displays the total demand rates and distribution of demand for the
AM (07-10) peak for lines 176 and 177. Given the headway-based operations, the
temporal distribution of simulated passenger arrivals in our study is assumed
Poisson, i.e., passengers are assumed to arrive at stops independently of
anticipations of the timing of vehicle arrivals. The demand rates of these
distributions as well as the spatial distribution of arrivals are estimated
from aggregated tap-in data obtained from the Stockholm public transport
authority (SLL).

Total demand rates in average passengers per hour for 3-hour AM peak with
directional distribution. The total demand and share of demand per passenger
OD categories (corridor-to-corridor [C2C], corridor-to-branch [C2B], branch-
to-corridor [B2C], branch-to-branch [B2B]) are also presented for the
Eastbound direction toward Mörby centrum and Westbound direction from Mörby
centrum. Rates and demand shares per passenger OD category are further divided
by bus line (i.e., 176, 177).  Share of total demand Line Direction Total
demand [pass./h] C2C C2B B2C B2B Total 4651 48% 16% 25% 12% Both Eastbound
2485 43% 0% 46% 11% Westbound 2166 54% 34% 0% 12% Total 2477 45% 15% 25% 15%
176 Eastbound 1369 41% 0% 46% 13% Westbound 1109 49% 34% 0% 16% 1-7 Total 2174
52% 16% 24% 8% 177 Eastbound 1117 46% 0% 46% 8% Westbound 1057 59% 33% 0% 8%
In describing the demand profile of the two lines in Table 5.1, as well as for
later reporting of results, we define the following passenger categories that
are spatially grouped based on ODs: corridor-to-corridor (C2C), corridor-to-
branch (C2B), branch-to-corridor (B2C) and branch-to-branch (B2B). The
directionality of demand for lines 176 and 177 is reasonably balanced. As
displayed in Table 5.1, however, in general there is higher demand on line 176
(which is also longer) compared with line 177. For both lines, the majority of
demand (around 50%) in both directions (towards and away from Mörby) is in the
C2C category (i.e., where both the passenger’s origin and destination are on
the common corridor section of the two lines). The smallest proportion of
travelers are within the B2B passenger category, those that have both origin
and destination on a branch. The proportion of B2B demand is almost twice as
high for line 176 than for line 177.

In a combined FIX and FLEX system there are a multitude of possible path
choice compositions from a passenger’s origin to their final destination that
could be constructed. Besides potential computational constraints, many of
these compositions are highly improbable to ever be considered an alternative
(for example those that include a great deal of backtracking or transfers),
yet will still maintain a probability of being chosen and could potentially
influence choice behavior and resulting assignment if included in a
passenger’s choice set. Additionally, it is well known that correlations
between route alternatives can induce bias into discrete choice models (e.g.,
the independence of irrelevant alternatives property of MNL random utility
models). To focus our analysis on a select group of paths that are
characteristic for each of the passenger categories we have defined, we
restrict the choice set of each passenger category as displayed in Table 5.1.

Path type alternatives per passenger category. A one indicates that the path
type is available for the corresponding passenger category, and a zero if it
is not available.  Passenger OD category Path type C2C C2B B2C B2B FIX 1 1 1 1
FLEX 0 0 0 1 FIX-FLEX 0 1 0 0 FLEX-FIX 0 0 1 0

As seen in Table 5.1, for C2C passengers, only FIX paths without transfers are
allowed. Similarly, only direct FIX or FLEX paths are made available to B2B
passengers. A direct FIX path is also always available to C2B and B2C
passengers if one of the lower frequency lines is chosen. Alternatively, the
C2B passengers may transfer from FIX to FLEX and the B2C passengers may
transfer from FLEX to FIX. To simplify the choice set further, the first
common stop (i.e., Malmvik, as seen in Figure 7) between lines 177 and 176 is
designated as the only available transfer stop for these passenger categories.

To speed up learning, LoS experiences of alternative paths are shared between
travelers that share the same OD. In the learning update described in (5) at
the beginning of a day $d$, $t_{j,k,d-1}$ will then correspond to the average
experience of a chosen path component $j$ over all travelers $k$ sharing the
same OD that chose to use that path component. Similarly, $t^{x}_{j,k,d-1}$
corresponds to the so-far accumulated experience for travelers sharing the
same OD of that path component.

###  5.2 Flexible transit operations

The FLEX service in this study employs a nearest-neighbor heuristic (described
in greater detail in Leffler et al. (2021)) that prioritizes serving stops
with the largest cumulative waiting times of passengers when assigning FLEX
vehicles to requests and a rule-based rebalancing strategy that aims to
maintain an even supply of vehicles at all stops on branches. Note that the
aim is not to necessarily optimize the performance of the FLEX service
provided but rather represent characteristics of FLEX operations and observe
learning output in serving existing public transit users with no pre-bookings.
It is common in studies of FLEX assignment algorithms and simulation-based
evaluations of FLEX services to include LoS constraints where more costly
requests (e.g., low-demand that requires relatively high detouring to serve)
may be rejected by the fleet operator to improve assignment efficiency, or to
model a maximum acceptable LoS threshold that travelers might tolerate before
switching modes or planned activities altogether. To evaluate the alternative
system design in our study as a true replacement for a public transit service,
however, LoS constraints are not included in determining which requests to
accept or reject. Furthermore, once a vehicle is assigned to a FLEX traveler
request, the traveler owning this request is also assumed only to board the
vehicle assigned to it, i.e., not opportunistically board other vehicles if
found available.

When traveler requests are made known to the FLEX fleet operator, they are
bundled into trip-plans (consisting of a sequence of stop visits and a route
to serve these requests) and assigned to FLEX vehicles dynamically. In this
study, if there are no existing trip-plans when a new request is received, a
new trip-plan that serves the request directly from origin to destination is
generated. The insertion of an additional request into an existing trip-plan
is considered feasible if both the stop for pick-up and the stop for drop-off
of that request does not require an assigned vehicle to backtrack from already
planned stop visits downstream towards the final destination of that trip-
plan. Furthermore, a request is only inserted into an existing trip plan if
the forecasted load of an assigned vehicle based on already assigned requests
does not exceed its maximum passenger capacity.

Assignment calls are made in ’real-time’, i.e., every 5 s in this study. For
each assignment call, all existing trip-plans are sorted based on the current
cumulative waiting time over all requests that have been grouped into each
plan. Once sorted, the highest-ranking trip plan (i.e., with the largest
cumulative waiting time) is assigned to the nearest on-call vehicle. The next
highest-ranking trip-plan is then assigned to the nearest on-call vehicle and
so on until there are either no more unassigned trip-plans or no on-call
vehicles available.

If on-call vehicles are available after all requests have been assigned to
vehicles, then the excess supply may be rebalanced to other stops in
anticipation of future demand. In our study, a simple rebalancing strategy is
applied. At 10 min intervals throughout the simulation, a rebalancing call is
performed. When rebalancing, if available, on-call vehicles are redistributed
to all stops (again, the closest on-call vehicle to the stop with the lowest
supply) on branches such that an equal supply (i.e., the total number of on-
call vehicles at the stop plus the number of vehicles en-route to this stop)
at these stops is maintained.

###  5.3 Parameter set-up

In the proposed mixed FIX and FLEX system design, we increase the headway of
the two FIX lines on branches from 10 min to 30 min. A third line that only
services the corridor with a 7.5 min headway is added, which, in combination
with the two original lines, maintains the current combined service frequency
of 5 min on the corridor (if headways along the corridor are perfectly
synchronized). Reducing the frequency of the FIX lines on the branches means
that the FIX bus fleet can be reduced by six buses, each with a maximum
capacity of 100 passengers and 44 seats (i.e., an urban articulated bus). For
the alternative service design, we exchange the total maximum capacity of
these vehicles for 60 smaller FLEX shuttles operating by the policy described
in Section 5.2 that only run on the branches, each with a maximum capacity of
10 passengers with 5 seats.

Access/egress times at stops are considered the same for both FIX and FLEX.
For transferring passengers, a FIX cost per transfer $\beta^{trans}$ is set to
be roughly equal to 5 minutes of in-vehicle time in line with the study of
Balcombe et al. (2004). To estimate on-board comfort, the in-vehicle crowding
factors for standing and seated passengers described in Section 4 are used.

The running times of both FIX and FLEX vehicles are drawn from log-normal
distributions to characterize the variability of vehicle running times, with
parameters calculated from AVL data. To model dwell times, we use the same
dwell time function (6) as in the toy network demonstration.

All presented results are averaged over 20 simulation replications per
scenario for 100 days of learning. The relative standard errors of the mean
for all final day results are then smaller than 2% for all reported KPIs.
Passengers are generated according to the AM peak demand pattern presented in
Table 5.1 over 3 simulated hours. All results reported are based on choices,
travel time experiences, and vehicle utilization results registered within
this period. Before the first passenger arrival, a warm-up period is included
to distribute FIX vehicles with an even headway along all FIX lines. The FLEX
vehicles are initialized as on-call uniformly distributed over all branch
stops.

###  5.4 Results

To explore how evolving LoS expectations translate into mode-choice decisions
for the larger case study, learning curves for waiting time including weighted
waiting time due to denied boarding, crowding-weighted in-vehicle time, and
resulting mode-split in a decision between FIX only paths or paths including a
FLEX leg are presented in Figure 8. Note that in Figure 8, unlike the toy
network results in Figure 6, the blue ’FLEX’ results (as also labeled in the
legend) refers to any path that includes a FLEX leg as part of the complete
trip, in other words, the FLEX path alternative for each passenger category
presented in Table 5.1.

Figure 8: Learning results from the mixed-mode scenario and AM peak demand.
Per path type (FIX- for the C2C category, FIX-FLEX for C2B, FLEX-FIX for B2C,
and FLEX- for B2B) waiting time (weighted for denied boarding)
anticipation/experience (top row), in-vehicle time (weighted for crowding)
anticipation/experience (middle row), and the percentage of all travelers that
chose a path that included a FLEX leg or a FIX only path over days (bottom
row). Columns correspond to the passenger OD categories.

As observed in Figure 8, mode-split convergence (bottom row), after initial
oscillations, starts to level out after the first 7-10 days with a $\pm 6\%$
band due to the random draws of choices. Notably, not all individual mode-
split trajectories converge as quickly, however all display a similar long-
term trend. The combination of optimistic initial LoS anticipations and rapid
discounting of new experiences can be observed in the convergence of the
learning curves for waiting time (top row) and in-vehicle time (middle row).
This is perhaps most apparent for the B2C passenger learning curves using the
FIX-only paths. The initially optimistic anticipations for both path types are
retained over successive days. The long-term trend of anticipated experiences
moves toward average realized experiences after consistently underestimating
them in earlier days, both for waiting times and in-vehicle times. In general,
there are higher degrees of variation in waiting time experiences for
transferring passengers (in groups C2B and B2C).

In the second row of Figure 8, all FIX paths converge to higher weighted in-
vehicle times when compared to paths including FLEX for the C2B, B2C, B2B
passenger categories, particularly for the transferring passengers. The ratio
of passenger-kilometers traveled (PKT) over vehicle-kilometers traveled (VKT)
provides a measure of average traveler flow per FIX or FLEX path types.
Dividing this by the seat capacity for FIX versus FLEX vehicles indicates
crowding levels per path type chosen. The PKT/VKT ratio for FIX trips is
38.74, which, divided by the seat capacity of 44 of FIX buses, would give a
load factor of 77%. For FLEX vehicles, the PKT/VKT ratio is 2.77, which with a
seat capacity of 5, yields a load factor of 55% resulting in higher crowding-
weighted in-vehicle times for passengers using FIX.

With perfect even-headway synchronization, the combined frequency of the three
lines serving the C2C passengers (leftmost column) would result in an average
waiting time of 150 seconds (half the combined headway) for Poisson
distributed passengers. This is also set as the initial waiting time
anticipation. Similarly, anticipated in-vehicle times are set to the average
shortest-path nominal (i.e., uncrowded) in-vehicle times for each OD within
this category. However, the actual average waiting time and in-vehicle time
experiences are higher, with the anticipation of waiting time converging to
roughly 600 seconds. This indicates a high level of bunching of FIX vehicles
along the corridor and a higher frequency of denied boarding. Passengers learn
to anticipate the consistent deviations from the planned headway and crowding
levels.

For the C2B passenger category (second column from the left), travelers have
the option of either choosing FIX directly to their final destination (with an
even headway of 1800 s) or using any of the FIX lines (with a combined headway
of 600 s with perfect synchronization) to the transfer stop Malmvik and take
FLEX to their final destination on a branch. The initially optimistic
expectations towards FIX-FLEX (300 s waiting time for FIX to the transfer stop
+ 0 s waiting time for FLEX to the final destination) versus FIX (900 s
expected waiting time) result in an initial preference towards the FIX-FLEX
path (68% of all passengers within this category) on the first day. Waiting
time anticipations between the two alternative path types converge to close to
the same level. Passengers opting to choose FIX experience more crowding for a
longer duration of their trip, resulting in higher average in-vehicle times
when compared to the FIX-FLEX path. This is reflected in path/mode
preferences, which, after initial oscillation, converge slowly together with
travel time anticipations to a close to equal number of passengers choosing
each path type with a slight preference towards the FIX-FLEX (53%) path with
less crowding but an additional transfer.

For the B2C passenger category (third column from the left), the choice
scenario is between the FIX path from the branches (again with an even headway
of 1800 s, however starting on the lower demand branches) and the FLEX-FIX
path (again with an initial 0 s expected waiting time for FLEX to the transfer
stop Malmvik and 300 s waiting time to transfer to one of the three FIX lines
available). Initial anticipations result in a 67% mode-split in favor of the
FLEX-FIX path on the first day. After some initial oscillation in path
choices, the combined FLEX-FIX path results in a lower weighted in-vehicle
time that compensates for a higher waiting time when compared to FIX only.
This results in convergence towards a 66% preference for the FLEX-FIX path.

Finally, for the B2B passenger category (the rightmost column), the choice is
between a FIX direct line with an initial anticipated waiting time of 900 s
and a direct FLEX line with an initial anticipated waiting time of 0 s. With a
larger difference in initial waiting time expectations, the initial mode-split
heavily favors the FLEX paths (81% of all passengers within this category) on
the first day. However, crowding-weighted in-vehicle times converge to similar
levels between FIX and FLEX alternatives, with slightly less crowding on-board
FLEX vehicles. Average waiting times are consistently far lower for FLEX
versus FIX (300 s versus 900 s), and mode-split preferences converge after
initial oscillation to 82% of all passengers choosing FLEX over FIX.

The final day waiting times adjusted for denied boarding and the in-vehicle
times adjusted for crowding are presented per passenger category in Figure 9.
LoS performance is similar between FIX-only paths and paths including FLEX for
transferring passengers (C2B, B2C), with a preference towards FLEX mainly due
to lower crowding on-board the smaller FLEX shuttles. The B2B passenger
category benefits the most from the FLEX service in terms of waiting times
compared to FIX-only paths.

Figure 9: Final day average waiting time (weighted for denied boarding) and
in-vehicle time (weighted for crowding) per passenger OD category (C2C, C2B,
B2C, B2B) and per path type (FIX- for the C2C category, FIX-FLEX for C2B,
FLEX-FIX for B2C, and FLEX- for B2B).

##  6 Conclusions and Discussion

In this paper, a combined FIX (i.e., fixed line and schedule public transit
service) and FLEX (i.e., public transit service with flexible routing and
scheduling) route choice model is developed and integrated into an existing
public transit simulation framework containing core components for modeling
FIX and FLEX operations and adaptive traveler behavior. To solve the
equilibrium assignment problem and represent traveler day-to-day learning, an
MSA approach, where new experiences are discounted by the inverse of the
number of days elapsed, is used to guide traveler waiting time and in-vehicle
time anticipations towards average realized experiences. A two-link toy
network is used to diagnose the implementation of the model and evaluate the
evolution of waiting time (including a penalty if denied boarding) and
crowding-weighted in-vehicle time trade-offs in mode-split convergence. A case
study based on two existing coordinated trunk-and-branches bus lines in
Stockholm is then performed to evaluate model properties for a more complex
case while demonstrating the usefulness of the combined simulation framework
in appraising a broader range of emerging mixed FIX and FLEX service designs
and scenarios.

The two-link toy network intuitively displays a ’worst-case’ scenario for
travelers utilizing FLEX, with imbalanced and limited supply availability.
Large waiting time discrepancies between individual travelers for the
capacitated FLEX service are reflected in mode split convergence as the
capacity for this service increases. In cases where waiting time anticipations
between FIX and FLEX services are similar, differences in crowding-weighted
in-vehicle times due to differences in vehicle sizes are also reflected in
mode-split, where the larger FIX bus is preferred on average over the smaller
FLEX shuttles. The Stockholm case study highlights many important LoS dynamics
for first/last-mile mixed FIX and FLEX service design, for example, the
uncertainty of waiting times when transferring between congested FIX and FLEX
services and potential trade-offs in in-vehicle discomfort due to crowding,
dependent on fleet composition. A high degree of bunching and denied boarding
was observed for FIX transit users. In particular, for the transferring B2C
passengers, high variability in waiting times is observed for travelers
transferring between FLEX and FIX and FIX users that have their origin on one
of the branches. Both the lack of transfer synchronization to an already
congested corridor and the possibility of being denied boarding one of the
lower-frequency lines can substantially reduce the LoS experienced by this
traveler group.

In the Stockholm case study, it is observed that crowding-weighted in-vehicle
times are consistently lower when utilizing multiple smaller shuttles when
compared to larger buses. In the absence of FLEX shuttle-specific crowding
factors, crowding factors based on the meta-study by Wardman and Whelan (2011)
were applied for both FIX and FLEX passenger in-vehicle experiences. This
meta-study was based on experiments for larger vehicles (crowding preferences
based on the seated and standing capacity on-board train cars). The crowding
effect per passenger or square meter on-board a vehicle with lower passenger
capacity is intuitively different. For example, it is questionable if an
increase from 3 to 4 seated passengers on-board a ten passenger capacity FLEX
shuttle, which would increase the crowding factor for this vehicle from 0.95
to 1.05, is directly comparable to the same increase in crowding level from 33
to 34 passengers for the 100 passenger capacity FIX buses. In future studies,
this emphasizes the need for vehicle- and service-type specific value-of-time
valuations. The case studies also highlight the need for more explicit
consideration of the discontinuous increases in travel cost in LoS
anticipations for capacity-restricted FLEX operations. The waiting time
distributions of FLEX services can be characterized as being heavily skewed,
i.e., travelers tend to either experience the lowest possible travel time for
the scenario if FLEX vehicles are available on-call or the longest possible
for the network with the learned long-term anticipation converging to the
average over these. An interesting line of future work is incorporating
waiting time variability more explicitly into the learning processes of the
model, for example, through the inclusion of within-day real-time information
provision in informing traveler decisions, or more directly in LoS variability
valuations (see for instance Alonso-González et al. (2020); Geržinič et al.
(2022)) within traveler GTC functions.

The importance of both initial conditions combined with day-to-day learning
dynamics is seen in both the toy network and larger case study. Using the
inverse of the number of days elapsed in MSA discounting of new experiences
often results in slow convergence that consistently underestimates waiting and
in-vehicle times. A dynamic discount factor (for example, the self-regulated
averaging method suggested by Liu, He, and He (2007)) that instead induces
more aggressive exploration when there are small differences in LoS
anticipations between days and greater discounting when LoS anticipations
diverge or taking into consideration additional behavioral factors of learning
(such as the distinction between reinforcement-based and belief-based learning
in route-choice as suggested by Bogers, Bierlaire, and Hoogendoorn (2007))
could perhaps be used in future work to improve convergence speed while still
pushing anticipations towards an equilibrium.

It is well known that correlations between route alternatives within discrete-
choice models can result in inaccurate substitution patterns. Traveler choice-
sets were restricted to a predefined set of path types in this study. Although
certain correlation structures can be considered within the choice-set
generation step, for example, by merging comparable FIX lines into hyperpaths,
correlations between routes are not explicitly accounted for. There is a lack
of behavioral knowledge on how overlap and similarities among mode-route
combinations of fixed and flexible services are perceived. Future research may
shed light on the choice-set composition and choice preferences based on
empirical behavioral data.

The study presented in this paper leaves several avenues open for future work.
The developed simulation framework is expressive in that it allows for
experimentation with real-time operational policies and real-time information
provided to travelers for individual legs of a complete trip, along with a
wide range of behavioral parameters. To simplify the interpretability of
results at this stage of model development, traveler adaptation was limited to
day-to-day information sources (prior knowledge of each service and
accumulated experiences). To further evaluate integrated FIX and FLEX designs
for the Stockholm case, experimentation with real-time transfer
synchronization strategies (Laskaris et al. 2018, 2021), combined with real-
time information provision to travelers (Cats and West 2020), is an
interesting line of future work. Furthermore, combining the developed
simulation framework with day-to-day supply-side re-planning (e.g., trading
off fleet size for the FLEX service with the frequency of the individual FIX
lines as in Pinto et al. (2020)) could help find a better balance between
service types.

## Acknowledgements

The authors would like to thank Matej Cebecauer for his help and instructive
advice in acquiring the demand data used in the second case study and Giorgos
Laskaris for sharing the underlying network and fixed transit operations data
used in the second case study.

## Funding

This paper is part of the SMART (Simulation and Modeling of Autonomous Road
Transport) project (grant number TRV-2019/27044), financed by the Swedish
Transport Administration Trafikverket. The work of the third author was
supported also by the CriticalMaaS project (grant number 804469), which is
financed by the European Research Council and the Amsterdam Institute for
Advanced Metropolitan Solutions.

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