The document proposes a frequency-based transit assignment model that accounts for online information and strict capacity constraints. It first applies an unconstrained transit assignment procedure and then handles only overloaded transit line segments, reassigning surplus passengers. The model assumes passengers receive online information of predicted arrival times and vehicle occupancy. Two cases are considered: 1) passengers know occupancy and may change routes, 2) passengers have no occupancy information and must choose later alternatives if boarding is denied.

1. A frequency based transit assignment model that considers online
information and strict capacity constraints
Nurit Oliker *
, Shlomo Bekhor
Department of Civil and Environmental Engineering, Technion – Israel Institute of Technology, Haifa, 32000, Israel
A R T I C L E I N F O
Keywords:
Transit assignment
Capacity constraints
Frequency based
Hyperpaths
Online information
A B S T R A C T
This paper proposes a frequency based transit assignment model that accounts for online information and strict
capacity constraints. A heuristic is proposed to solve the problem, which first applies an unconstrained transit
assignment procedure and then handles only the over-loaded transit line segments, re-assigning the surplus
passengers. The developed procedure is efficient, requiring very short running times compared to existing ca-
pacity constrained transit assignment models. The model assumes passengers receive online information of both
predicted arrival time and occupancy condition. Two cases of occupancy information are considered: (1) pas-
sengers are informed of the vehicle occupancy, and may change their route selection accordingly, (2) passengers
have no occupancy information and in case their boarding is denied, they are enforced to choose a later departing
alternative. The model is applied for the Winnipeg network, and as expected the inclusion of capacity constraints
increases the average travel time compared to the unconstrained model. However, prior knowledge of the oc-
cupancy condition was found to reduce the additional travel time. This result emphasizes the potential benefits of
providing occupancy information to passengers.
1. Introduction
Transit assignment is the mapping of passenger demand over a given
transit network; this procedure yields the estimated distribution of pas-
sengers in a network and enables its assessment. Therefore, transit
assignment plays a key role in both planning and management of transit
networks.
Models are broadly divided to schedule and frequency based transit
assignment. Schedule based assignment modify passenger flow in time,
and frequency based assignment estimate the average distribution of
passengers over time. Therefore, schedule based models are commonly
used for management purposes, enabling detailed flow simulation, while
frequency based models are commonly used for planning purposes,
enabling to efficiently model large scale networks.
The transit assignment problem has been studied since the late 1960s
(Kepaptsoglou and Karlaftis, 2009). Dial (1967) developed a transit
assignment model based on a shortest path algorithm, analogous to the
common method for private car assignment. He considered waiting,
walking and transferring times in the total travel time of the paths,
assuming a passenger chooses the path with the minimal total travel
time. Later, Chriqui and Robillard (1975) introduced the ‘common line
dilemma’, suggesting that a passenger may not pre-determine her path,
but board the first arriving vehicle among a set of pre-determined
attractive lines. Spiess and Florian (1989) elaborated this idea to an
efficient assignment model, called ’optimal strategies’; reducing the ex-
pected total travel time by exploiting the combined frequency of lines in
the attractive path set. Their seminal approach was widely adopted in
frequency based assignment models and implemented in a popular shelf
software (INRO Consultants, 1999). Schm€
ocker et al. (2013) describe an
’hyperpath’ as a choice set of a transit traveler. This choice set usually
consist several paths out of which any could be optimal depending on the
arrival time of vehicles. Different strategies were developed to define the
selection of a specific path from the options in the hyperpath. Recently,
Nassir et al., 2019 used smart-card database to develop a link based
recursive model that estimates the links’ attractiveness.
One of the aspects in traffic assignment is the prioritization of pas-
sengers. Bar-Gera et al. (2012) termed the “proportionality” property for
private car assignment, suggesting that for any link in the network, the
same proportion can be applied to all passengers, regardless of their
origin or destination. This assumption is plausible for private car
assignment. However, in transit assignment upstream passengers should
be prioritized over downstream passengers, according to the order in
* Corresponding author.
E-mail addresses: nurit.oliker@umontreal.ca (N. Oliker), sbekhor@technion.ac.il (S. Bekhor).
Contents lists available at ScienceDirect
EURO Journal on Transportation and Logistics
journal homepage: www.journals.elsevier.com/euro-journal-on-transportation-and-logistics
https://doi.org/10.1016/j.ejtl.2020.100005
2192-4376/© 2020 Published by Elsevier B.V. on behalf of Association of European Operational Research Societies (EURO). This is an open access article under the CC
BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
EURO Journal on Transportation and Logistics 9 (2020) 100005

2. which they board a vehicle. This property is mainly dealt in schedule
based assignment where passenger distribution is modeled over time,
while in frequency based assignment the assignment is averaged over
time and the order of the passengers is typically not handled.
The modelling of passenger distribution in a transit network requires
detailed graph representation. A link that is common to more than one
transit line, requires multiple arcs in the graph to distinguish between
passengers riding different lines. In a seminal work, Nguyen and Pallot-
tino (1988) addressed this aspect by forming the ’hypernetwork’ of the
transit service. A ’hypernetwork’ comprise explicit arcs for each segment
in the transit network: transit lines, walking, boarding, and aligning
links. This method is widely used in assignment models and enables the
consideration of complex routes. The first step of any assignment model
is to form a ‘hypernetwork’ that enable to distinguish passengers that are
walking, boarding, alighting and riding different transit line segments.
The journey of a transit traveler may comprise several components:
walking to and from a stop, waiting, riding transit lines and transferring
between them. Passengers assess these time components differently; it
has been shown that waiting and walking time are valued 2–3 times
longer than in-vehicle time (Wardman, 2004). These value perceptions
are expressed by different coefficients assigned to travel times; usually,
some penalty is given to the waiting, walking and transferring times
(Hamdouch et al., 2014).
The consideration of capacity limitations has received relatively little
attention in frequency based assignment models (Cepeda et al., 2006).
However, as many transit networks reach their maximal capacity, there is
an increasing need to limit the maximal flow when assessing passenger
distribution in the network (Fu et al., 2012). Frequency based transit
assignment models that consider capacity constraints commonly attri-
bute longer travel time to over-loaded transit lines (Cepeda et al., 2006).
Extending the travel time of an over-loaded path reduces its attractive-
ness, and thereby the number of passengers assigned to it in the next
iteration. This equilibrium problem is commonly solved using the
Method of Successive Averages (MSA) procedure, which has slow
convergence properties. Another drawback is that some of the equilib-
rium models do not ensure the solution meets capacity constraints.
De Cea and Fernandez (1993) modified congestion effects in bus stops
and abroad vehicle by extending the waiting and boarding times of
crowded bus stops and over-loaded transit line segments. They developed
an equilibrium model and used Jacoby method to solve it. The model did
not apply strict capacity limitations, and the computational complexity
resulted in flows that often exceed the service capacity. Wu et al. (1994)
developed a heuristic model in which the waiting time for a vehicle in-
creases with passenger flow, but the split between the different paths
remain constant. Bouzaïene-Ayari et al. (1995) extended this model to
update the split according to the updated times. However, their MSA
solution scheme required long running times.
Lam et al. (1999) developed a stochastic user equilibrium model
(SUE). They applied a linear function to penalize in-vehicle travel time of
crowded vehicles and a multinomial logit model to describe path selec-
tion. Using Lagrange multipliers, they insured capacity constraints are
met. In a following study, they elaborated their model to include delays
in boarding and alighting at stops (Lam et al. (2002)). Both models are
exemplified only for toy networks. Cominetti and Correa (2001)
considered the limitation of vehicle capacity with more realistic
waiting-time functions at stops. Their model address changes in fre-
quencies due to over-loaded vehicles and consider queuing processes at
transit stops to reflect changes in the set of attractive routes. Cepeda et al.
(2006) further developed their model to a gap minimization problem that
was solved heuristically by MSA with an improved efficiency. Codina and
Rosell (2017) tested different solution scheme for the problem formu-
lated in Cepeda et al. (2006). They suggest an MSA-based heuristic that
can solve large-scale problems while insuring feasibility.
Kurauchi et al. (2003) assigned cost to the probability of failing to
board a transit vehicle. They considered separately passengers on board a
crowded vehicle and passengers at stops, and used Markov chains, in
which the boarding probability is dependent on the residual capacity of
the transit vehicle. Schm€
ocker et al. (2008) developed a combined fre-
quency and dynamic approaches, solving a frequency based model in
subsequent time intervals to consider non-boarding passengers. In a
following study, Schm€
ocker et al. (2011) considered seat availability,
dealing with the probability of a failure to sit and the change in route
choice due to the discomfort of standing in a vehicle. Szeto et al. (2013)
developed a user equilibrium model and used linear programming
formulation to evaluate link use under uncertain travel times caused by
congestion. Sun and Szeto (2018) elaborated this approach to a
logit-based model with a stochastic user equilibrium (SUE) and presented
real-size network applications. Cheung and Shalaby (2017) suggested a
heuristic model that minimize the total congestion in the network, by
finding system optimum in a user-constrained model. Their model was
demonstrated on the Toronto Transit Commission (TTC) network, and
showed convincing results, reducing the ratio of over-crowded transit
lines in the network. However, the model does not ensure solution that
meets capacity constraints.
The objective of this study is to develop an efficient heuristic that
integrates capacity constraints in a frequency based transit assignment
model that considers online information. The main idea of the proposed
model is to first apply unconstrained assignment procedure, and then
handle only the over-loaded transit line segments, without updating the
travel times.
It should be emphasized that this is not an equilibrium model and the
discomfort of crowding in vehicles is not reflected in the utility function
of the suggested model. Our approach suggests an efficient algorithm that
impose strict capacity constraints by handling only the excess passengers.
Surplus passengers are re-assigned to feasible alternatives without
updating the travel times. Upstream passengers are prioritized over
downstream passengers in the re-assignment procedure. The avoidance
of an equilibrium model does not enable to reflect crowding discomfort
but implies significant reduction in running times. As the frequency
based assignment model is the lower-level problem of planning proced-
ures, its iterative solution scheme gives importance to its efficiency. In
practice, over-crowded lines become less attractive due to the discomfort
of travel, but still attracts enough passengers to remain full. We believe
that satisfying approximation of passenger distribution, especially for
planning purposes, can be achieved without updating the travel times.
A frequency based transit assignment model that considers online
information of predicted arrival times is available to passengers, was
developed by the authors (Oliker and Bekhor, 2018) and used in this
study. This is a path based assignment model that includes the finding of
attractive path set, setting of route choice decision rules for different
cases of predicted arrival times, and calculating the probability for these
different cases. The presented study extends the developed framework of
online information to account for occupancy information and capacity
constraints. The model is developed for two cases of occupancy infor-
mation: (1) passengers are informed of the vehicle occupancy in real-time
and may change their route selection accordingly, (2) passengers do not
have prior knowledge of the vehicle occupancy and will necessarily
choose an inferior alternative, which departs later than the desired
over-loaded vehicle.
Online information of predicted arrival times is becoming widely
available to passengers. The consideration of estimated arrival times
online information in a frequency based assignment model was per-
formed in several studies [Gentile et al. (2005), N€
okel and Wekeck
(2009), Barabino et al. (2015), Estrada et al. (2015), Chen and Nie (2015)
and Oliker and Bekhor (2018)]; all found that information has a signif-
icant impact on assignment results. Shimamoto et al. (2005) have made
first steps in investigating the effects of estimated arrival times available
information within a capacity constrained assignment model. They
modified the capacity constrained model of Kurauchi et al. (2003) to
include partial online information of predicted arrival times at the
boarding stop. Their model is well defined but developed only for a toy
network of 4 nodes and 2 lines. The suggested model elaborates the scope
N. Oliker, S. Bekhor EURO Journal on Transportation and Logistics 9 (2020) 100005
2

3. of Shimamoto et al. (2005) to include the information of occupancy
conditions as well, in addition to a vaster information of arrival times.
The estimated arrival times are assumed to be available not only for the
boarding stop, but also for the potential transfer stops. In addition to the
vaster scope, the model is developed for a large-scale network.
This paper is first to consider both capacity constraints and online
information of occupancy conditions in a frequency based transit
assignment model. The online information includes both estimated
arrival times (at boarding stops and transferring stops) and vehicles’
occupancy conditions in the available information. Strict capacity con-
straints are imposed in the assignment modelling. The scope of the sug-
gested research, within frequency based transit assignment models, is
presented in Table 1, it features the integration of capacity limitations
together with different online information conditions.
The contributions of this paper are as follows. The proposed model
includes the implementation of strict capacity constraints by a heuristic
that re-assign the surplus passengers without updating the travel times.
The heuristic allows for short running times, in comparison to common
MSA-type equilibrium models. In addition, the proposed model allows
for evaluating the potential effects of available occupancy information on
passenger distribution.
The remainder of the paper is organized as follows. The following
section describes the proposed methodology for the two cases of occu-
pancy information. The subsequent section illustrates the model appli-
cation for the Winnipeg network. Section 4 presents the results of the two
models and compares them with the unconstrained application. The last
section summarizes the results and provides directions for further
research.
2. Methodology
The developed model implements capacity constraints in a frequency
based transit assignment model. The main idea is to first perform
assignment without capacity constraints, find over-loaded transit line
segments and re-assign the surplus passengers. Since the transit assign-
ment algorithm is path based, it is possible to identify the O-D demand for
each over-loaded link. The model considers the order of boarding to the
over-loaded transit link, prioritize early boarding and re-assign the later
ones. Time dimension is considered only in the arrival times average
estimations, by calculating the probability of a certain line to precedes
other lines and vice versa. The model is static, features a frequency based
assignment that evaluates the average passenger loading over time.
The starting point of the proposed methodology is a non-capacitated
frequency based transit assignment model that considers online infor-
mation (Oliker and Bekhor, 2018). For completeness, this model is briefly
described in section 2.1. The subsequent sections describe the proposed
method that handles the over-loaded transit line segments.
The model prioritizes up-stream passengers, assuming the passengers
at the first stop of the line are the first to board it, enjoying a full capacity.
However, in the case of a circular line there will be some passengers on-
board at any segment of the route. One way to tackle this case is by pre-
processing the assignment of a circular line, to evaluate the average flow
in all segments, included the first listed one. This pre-processing simu-
lates the first run of the line and yields its average flow as the starting
point for the capacity constrained assignment.
2.1. Non-capacitated transit assignment
This model assumes that information of estimated arrival times is
available to the passenger at her origin and is used for the minimization
of her generalized cost, which can be expressed by:
Generalized cost ¼ in vehicle time þ α waiting time
þ β walking time þ γ transfer No: (1)
The generalized cost includes the different components of travel time:
in-vehicle, waiting, walking and transferring, and the penalty weights α;
β and γ. Waiting and transfers significantly reduce the comfort of travel
and most passengers will prefer to extend their travel in some time if a
direct line is available (Wardman, 2004). These preferences are
expressed by the selected penalty weights (α;β and γ).
The strategy of the passenger comprises an a-priori specification of
the ‘hyperpath’. Where the actual path is selected according to the pre-
dicted arrival times; for example: “If line 1 arrives 5 min or more before
line 2 take it, otherwise take line 2”. Based on this behavioral assump-
tion, the model finds the attractive path set, determines decision rules for
different cases of arrival times, calculates the probabilities for these cases
and assigns the demand accordingly.
For each O-D demand pair, the model generates a ‘hyperpath’ that
comprise a set of attractive paths. Each of these paths may be optimal,
under different conditions of arrival times. Therefore, the passenger is
assumed to select her actual path among the ‘hyperpath’ possibilities,
considering the online information of predicted arrival times.
The generation of the ‘hyperpath’ is conducted by a variation of Yen’s
k-shortest path algorithm (Yen, 1970); instead of finding the k shortest
path, the algorithm is adjusted to find all paths that are shorter than a
certain threshold, which is the worst case waiting time(s) scenario of the
shortest possible path. To confine the path set for each O-D pair to a
reasonable size, we exclude paths that highly overlap with another path
in the set. For example, a path is excluded if it consists riding the exact
same lines as another path in the set but transfer between them in an
adjacent stop.
The finding of the attractive paths for each O-D is followed by the
computation of their selection probabilities. Under the assumption that
waiting times in the network are independent, the sum of waiting times
along a route has the probability of the sum of independent random
variables. This sum is computed analytically by convolution.
After computing the probability of waiting times’ sum along a single
path, the joint probability that a candidate path has the minimal gener-
alized cost in the ‘hyperpath’ is computed. In the case there are only two
attractive paths, this probability is analytically computed. The difference
between two independent random variables is computed by convolution
again, using the difference function. For more than two attractive paths
the computation is more complex; probability relations between all paths
Table 1
The scope of the model within frequency based transit assignment models.
Online information consideration
No Arrival times Arrival times þ
Occupancy
conditions
Congestion
consideration
No Dial (1967), Spiess and Florian (1989), Nguyen and Pallottino (1988) Gentile et al. (2005), N€
okel and Wekeck
(2009), Chen and Nie (2015), Oliker and
Bekhor (2018)
Yes De Cea and Fernandez (1993), Wu et al. (1994), Bouzaïene-Ayari et al. (1995),
Cominetti and Correa (2001), Kurauchi et al. (2003), Cepeda et al. (2006),
Schm€
ocker et al. (2008, 2011), Szeto et al. (2013), Cheung and Shalaby (2017),
Codina and Rosell (2017)
Shimamoto et al. (2005) This paper
N. Oliker, S. Bekhor EURO Journal on Transportation and Logistics 9 (2020) 100005
3

4. are conditional. Therefore, for the case of more than two paths in the set
the probability is calculated by a Monte Carlo method. A random large
sample (5000 in this paper) of waiting time occurrences is drawn for all
candidate paths, using the calculated PDF’s. The drawn waiting times are
added to the other components of the travel time (i.e., in-vehicle, walking
and transfer times) to simulate the generalized cost. The proportion of
samples in which a path had the minimal general cost is used as its
probability of selection. After the probabilities for all paths in the
‘hyperpath’ are computed, the demand is assigned accordingly.
A detailed explanation of the non-capacitated transit assignment
model can be found in (Oliker and Bekhor, 2018).
2.2. Capacitated transit assignment
The proposed model scheme is presented in Fig. 1 and comprises the
following steps.
The initial step is composed of 3 procedures: first, the assignment
model is run without capacity constraints (section 2.1 above). The transit
assignment algorithm is path based, and thus allows for storing the O-D
flows for each transit link. The O-D identities of all assigned passengers,
together with their attractive path set, are saved to enable their re-
assignment.
Next, the given capacities of the transit line segments are compared to
the assigned loads, and line segments that exceeds their maximal ca-
pacity (or any given threshold) are traced. Note that the capacity is
calculated for each transit line, as the sum of vehicle capacities serving
that line in the solved time period.
The last initialization step is the formation of the surplus list. This list
indicates the surplus quantity in boarding links that leads to over-loaded
transit line segments. To form the surplus list, the environment of the
over-loaded transit link is analyzed. An over-loaded transit link envi-
ronment is schematically shown in Fig. 2. The trip components are rep-
resented by transit links a 2 A of the transit hypernetwork G ¼ (I, A). Let
aþ
ða Þ denote the alighting (boarding) links to transit link a, and ap
represent the transit link that precedes a.
It should be noted that the boarding link a denotes the boarding that
occurs just before the over-loaded transit link a. The O-D pair(s) with
path(s) that pass through a are added to the surplus list. The quantity of
the surplus is determined according to the over-loading condition and the
total assignment in a . The free capacity of a transit link a, denoted as
FCa, is given by:
FCa ¼ Ca VaP þ Vaþ (2)
where Ca is the link capacity and Vp and Vaþ denoted the flows in ap
and
aþ
respectively. After finding the free capacity of the transit link, the
surplus flow quantity of each O-D pair, for each boarding link (a ), is
found by:
Sa ;O D ¼ Va ;O D FCa
Va ;O D
Va
(3)
where Va denotes the total flow in the boarding link. If the free capacity
equals zero, then the demand that was originally assigned to a , is added
fully to the surplus list. Otherwise, the proportional ratio of the denied
boarding is added to the surplus list.
It should be emphasized here that the “proportionality” property
(Bar-Gera et al., 2012) is assumed only for boarding links, i.e., passengers
waiting together in a specific stop and boarding the same vehicle. Among
these passengers, we assume that the successful boarding ratio of an O-D
group is proportional to its share among the passengers intending to
board the vehicle at the stop. We inclusively prioritize the upstream
passengers throughout the assignment. Only at a given stop, when pas-
sengers aim on boarding the same vehicle, we assume proportionality.
2.3. Outer loop – select O-D related surplus
The path-based transit assignment allows to find the O-D pairs that
cause the surplus on the transit line segments. The surplus list is sorted
with respect to the quantity of each O-D total surplus. The elements in the
list are dealt iteratively until the surplus list is empty and the assignment
is completely feasible. In each iteration, the O-D with the maximal sur-
plus is handled.
The attractive path set of the O-D is analyzed. Paths that pass through
an over-loaded boarding link are traced. If a path passes through several
over-loaded line segments, the maximal surplus along the route is
selected and removed from the assignment.
2.4. Inner loop – re-assign selected O-D surplus
This step constitutes the key part of the algorithm. Here we consider
two different cases of information: (1) passengers are informed of the
vehicles occupancy condition, and (2) passengers have no information on
the occupancy condition.
In case passengers are informed of the vehicles’ occupancy condition,
they can change their route selection a-priori, and thus avoid from
waiting for a fully loaded vehicle. In this case, the surplus passengers are
routed to all other feasible (i.e., not fully loaded) alternatives in their
Fig. 1. Model scheme. Fig. 2. Over-loaded transit link environment.
N. Oliker, S. Bekhor EURO Journal on Transportation and Logistics 9 (2020) 100005
4

5. attractive path set. In case passengers have no information on the occu-
pancy condition, they can only select routes with later departing time
than their denied boarding. This case, passengers are routed only to al-
ternatives that are inferior to their (previously) desired but fully loaded
vehicle.
Note that in some cases, the initial path set may be insufficient for the
capacity constrained assignment. In those cases, additional paths are
found and added to the set.
The re-assignment procedures for the two information cases are
specified below. Any re-assignment is conducted only if it is feasible, i.e.,
meets the capacity constraints. In many cases only some fraction of the
desired assignment is feasible. In such cases the feasible fraction is
assigned, and the remainder is directed to other feasible alternatives in
the path set.
2.4.1. Available occupancy info
In this case passengers are re-routed to all other feasible alternatives
in their path set. This is an iterative procedure, repeated until the surplus
passengers were fully re-assigned.
First, the feasible path set is checked to be non-empty. If the list is
empty, new paths are found. In the uncommon case that no feasible path
exists, the maximal walking time threshold, is relaxed to enable the path
set generation. Then, the probabilities of the paths to be selected are
calculated. The attractive path set formation and the probability calcu-
lation are described in section 2.1.
The desired loading in path p, denoted by Vp, is given by:
VP ¼ SO D πP (4)
where SO-D is the O-D total surplus and πP is the computed path proba-
bility. Vp is checked for feasibility prior to its assignment; any transit link
along the path is checked to comply with:
Ca Va þ r Vp (5)
where the assigned fraction is denoted by r and initialized to 1. If the
capacity constraint is met for all transit line segments in the path, Vp (eq.
(4)) is assigned fully. If the desired assignment over-loads a transit link
(i.e., the condition in eq. (5) is violated), the previous link in the path
(i.e., prior to the transit link) is checked for two cases:
(1) It is also a transit link – in this case, if the assignment was feasible
for the previous transit link, the over-loading necessarily com-
prises passengers that board right before the over-loaded transit
link (i.e., a in Fig. 2). These passengers board later than the
desired re-assigned passengers and thus, their boarding should be
denied. The desired loading is preferred and the passengers that
were previously assigned to a , should be re-assigned. The new
created excess is added to the surplus list.
(2) It is a boarding link – in this case the passengers of the desired
assignment cause the potential over-load as they are boarding
right before it. Therefore, this assignment is only partially con-
ducted. The feasible fraction (r) of the load is computed by:
r ¼
ðCa FCaÞ
Vp
(6)
Note that the free capacity in the link (FCa) is computed by eq. (2).
The transit line segments of a path are checked serially, so that if r is
updated, the desired path load (Vp) is updated respectively and the
following transit line segment check (i.e., eq. (5)) is performed with the
updated (i.e., reduced) path load (Vp).
2.4.2. No occupancy info
The assumption here is that passengers receive no prior information
on the vehicle occupancy condition. A passenger may intend to board a
vehicle but miss the boarding due to congestion. In this case this
passenger is compelled to choose an inferior path, which necessarily
departs later than the denied boarding.
In this model we distinguish the surplus passengers by their denied
boarding time, which is the waiting time for their missed vehicle, and
enable their re-assignment only to later departing alternatives. The Sur-
plus by Time (ST) list, conducted for each O-D, has the following struc-
ture:
STO D ¼
pax time
::: :::
(7)
where each entry in the list specify the number of surplus passengers in a
group and their corresponding minimal departing time for re-assignment.
The O-D attractive paths are checked for feasibility for each ST entry
separately. A path is considered feasible if it is not fully loaded, and the
waiting time for the first vehicle is longer than the indicated minimal
time, or if the next link in the path is a walking link. Walking is always a
feasible option. For each entry in ST, the probabilities of the feasible
paths are then computed, and the surplus passengers are re-assigned
accordingly.
This procedure is longer than the assignment under available infor-
mation assumption (section 2.4.1) as some paths are considered invalid
due to their early departure times, even when they do not include fully
loaded transit line segments. Therefore, more iterations are required for
each O-D surplus assignment.
2.5. Stopping criteria
There are two stopping criteria, one for the inner loop that handles
the specific O-D surplus, and the second for the outer loop that handles
the entire surplus re-assignment procedure.
In the inner loop, the path set of the handled O-D is assigned with the
O-D surplus, according to the calculated probabilities and the paths oc-
cupancy and validity. After assigning the feasible load to the network, it
is reduced from the handled O-D surplus. If all the desired loadings were
feasible, meaning r was set to 1 (eq. (6)), for all paths in the set, the
updated O-D surplus equals zero, and the internal iteration stopping
criterion is met. Otherwise, the path set is updated; paths with fully
loaded transit line segments are eliminated from the set and if the set is
empty, new valid paths are found. New probabilities are calculated for
the updated ‘hyperpath’ and the remainder O-D surplus is re-assigned
accordingly.
When the inner loop stopping criterion is met, the algorithm returns
to step 2.3 and the next O-D surplus with the maximal quantity is dealt.
The outer loop stopping criterion is met when all O-D surplus were re-
assigned, and the surplus list is empty. Then, the assignment is fully
feasible.
2.6. Small example
Fig. 3 exemplifies the model, presenting the re-assignment process
and prioritization of upstream passengers for the simplest case. Fig. 3a
shows a small example of 3 transit lines and 2 O-D demand pairs. Two
lines connect A to C, where the first line is a direct line and the second
line stops at B. The travel times of these lines are 5 and 10 min respec-
tively. The third line connects B to C with a long travel time of 20 min. It
is an inferior alternative and therefore denoted in a dashed line. The
demand from A to C is 80 passengers, and the demand from B to C is 30
passengers. For simplicity, the headways and capacities of the lines are
identical and equal to 10 min and 50 passengers respectively. For this
example, no penalties were given (α; β γ in Eq. (1) equals 1,1,0). Fig. 3b
shows the initial assignment without capacity constraints. Recall that the
behavioral assumption is that a passenger boards the shortest path, given
available online information of predicted arrival times. The direct line
provides the shortest path from A to C in case the second line does not
precedes it in 5 min or more. Assuming uniform distribution of the arrival
N. Oliker, S. Bekhor EURO Journal on Transportation and Logistics 9 (2020) 100005
5

6. times, the probability that the direct line is the shortest equals 0.875
(Oliker and Bekhor, 2018). Therefore, 70 of the 80 A-C passengers are
assigned to the direct line, and 10 are assigned to the line that stops in B.
This assignment is infeasible, considering capacity constraints, as the
direct line was assigned with 30 surplus passengers, as indicated in the
surplus list. Fig. 3c presents the re-assignment of this surplus, the 30
passengers are moved to the indirect line. This re-assignment creates a
new surplus, of 10 passengers, in the second section of the indirect line,
from B to C. The A-C passengers are prioritized over the B–C passengers,
since they board the line earlier. Therefore, 10 B–C passengers are added
to the surplus list. Fig. 3d shows the re-assignment of the secondary
surplus passengers. As the ‘hyperpath’ of B–C included only 1 path. The
alternative path was found and added to the attractive path set and the
B–C surplus was re-directed to it. Note that the alternative path is inferior
to their origin path as it was not included in the initial attractive path set.
At this step, the surplus list is empty, and the assignment is feasible.
2.7. Differentiation from equilibrium modeling
This section aims to clarify the conceptual difference between the
proposed model and the equilibrium modeling approach.
Equilibrium models are defined according to Wardrop’s first principle
(Wardrop, 1952). Since travel times are formulated as flow dependent,
the problem is solved by iteratively update some component of the
perceived travel times according to the estimated flows. Crowded lines
are assigned with longer travel times and thereby the discomfort of
crowding is expressed and their selection probability in the next iteration
is reduced. In some equilibrium models, strict capacity constraints are not
enforced, and the general solution algorithm does not ensure feasibility
(e.g., De Cea and Fern
andez, 1993; Cominetti and Correa, 2001).
Refraining from applying an equilibrium model, the suggested model
does not express the crowding discomfort in the utility function. Our
approach imposes strict capacity constraints by allocating over-loaded
line segments and re-assign surplus passengers to feasible paths,
without updating the travel times in the network and without re-
assigning the total demand.
The solution of the suggested model is not in equilibrium state; for the
same O-D, some of the passengers will enjoy shorter travel times than
others, determined by the waiting time and occupancy condition they
encounter. In the scope of the suggested method, the model could be
adjusted to an equilibrium state by an outer third loop. This loop would
update the perceived travel that are used in the assignment model.
However, the main strength of the model is its efficiency, its re-
assignment method handles violation of capacity constraints as an
alternative method for an equilibrium approach. Thus, we find the reach
of an equilibrium within the suggested re-assignment method as redun-
dant and illogical.
As discussed in the introduction section, transit assignment models
are used for planning purposes. As the frequency based assignment model
is the lower-level problem in the transit network planning procedure, its
functionality in the solution scheme gives importance to its efficiency.
In practice, over-crowded lines become less attractive due to the
discomfort of travel, but still attracts enough passengers to remain full
(Kurauchi et al., 2003). In an urban transit network path selection is often
opportunistic, passengers may not pre-determine their route but decide
“on the move” in accordance with the available alternatives. If a bus
arrives full to a stop, the passenger will have other alternatives and may
not refrain from arriving to that stop in the following day. In the same
way that passengers do not check the schedule when the service is
frequent.
Assuming available information of estimated arrival times and oc-
cupancy conditions, some passengers are expected to use it and change
Fig. 3. Capacity constrained assignment example.
N. Oliker, S. Bekhor EURO Journal on Transportation and Logistics 9 (2020) 100005
6

7. their route selection accordingly. The role of information in route choice
was widely studied and found as significant (e.g., Hickman and Wilson
(1995); Gentile et al., 2005; N€
okel and Wekeck (2009); Barabino et al.
(2015); Estrada et al. (2015)). Passengers tend to use the available in-
formation as it may reduce their average travel time in up to 20%
(Gentile et al., 2005; Oliker and Bekhor, 2018). Using online information
requires flexibility of the passenger in her route choice; the passenger
may alternate the route according to the presented information, and her
travel time will vary accordingly. Such route choice behavior has an
un-equilibrized characteristics.
Re-assigning only the surplus passengers enables to fully exploit the
attractive lines, and thereby achieve significantly shorter travel times in
comparison to the equilibrium solution. The small example of Cepeda
et al. (2006), shown in Fig. 4, illustrates this property well. The example
presents two lines connecting A and C, an express line and a local line
making a stop at B. The demands are: 10 trips from A to B, 10 trips from B
to C, and 100 trips from A to B. Capacities are 120 and 320 passengers per
hour in the local and express lines respectively. Travel times are 24 min
in the express line and 20 min in each segment of the local line. Dwell
time at stops is 0.01 min. The passengers travelling from A to B and from
B to C has only one alternative, riding the local line. The passengers from
A to C may choose between the local and the express lines.
As the express line offers a significantly shorter travel time and its
capacity enables to assign 100% of the A-C demand to it, the suggested
model would apply it and results in an average travel time of 23.34 min
per passenger. The equilibrium solution assigns only 70% of the A-C
demand to the express line, results in an average travel time of 25.44 min
per passenger. The average travel time in the equilibrium solution is
higher in 9% compared to the suggested model. Therefore, we believe
that the re-assignment of the surplus passenger, without reaching an
equilibrium, may provide a plausible estimation of passenger distribution
in the network.
3. Model application
The developed methodology is tested on the well-known Winnipeg
network (INRO Consultants, 1999), a common benchmark in trans-
portation network modeling (e.g., Cepeda et al., 2006). The network
comprises 154 zones, 2975 links, 903 nodes and 5394 transit O-D pairs,
with a total demand of 18210 passengers (see Fig. 5). The transit network
includes 67 two-way routes comprising 4234 transit line segments. The
network topology and transit service were processed to form the ’hyper-
network’, composed of explicit walking, boarding, alighting and transit
links. The unfolded network comprises 15719 links and 5383 nodes. The
penalty values were set according to the default parameters of the network
(INRO Consultants, 1999); α; β γ in Eq. (1) equals 2,2 and 5 respectively.
The following assumptions were made in the application: (1) fixed
dwell times at each stop; (2) walking is possible in all links; (3) a path
does not include more than a certain number of transfers (3 in this paper);
(4) passengers are indifferent to minor travel time differences of paths
within a certain threshold (2 min in this paper); (5) passenger do not
walk more than a certain threshold (15minuters in this paper). Note that
in case no feasible path is found for a certain O-D, assumption 5 is
relaxed.
The running times of the models: without capacity constraints (Oliker
and Bekhor, 2018), with constraints and occupancy info (section 2.4.1)
and with constraints but no occupancy info (section 2.4.2) were
approximately 30, 33 and 35 min respectively on an Intel Core i7-860
processor (8 M Cache, 2.80 GHz). The addition of the capacity consid-
eration procedure to the total assignment run is relatively small.
In the assignment model, most of the running time is consumed by the
path finding procedures. The probabilities calculations are performed by
a Monte Carlo sampling method and consume negligible running time.
The inclusion of capacity constraints required relatively short running
times as most of the paths were found in the initial unconstrained
assignment.
4. Results
4.1. Overall travel times
After applying the initial unconstrained assignment, 769 of the 4234-
transit line segments (18%) were found to be over-loaded. Approximately
10300 of the 18210 passengers were re-assigned in the capacity con-
strained models.
The results of the 3 models are summarized in Table 2: without ca-
pacity constraints (section 2.1), capacity constrained without occupancy
information (section 2.4.2) and capacity constrained with available oc-
cupancy information (section 2.4.1). Vehicle capacity in the base run was
assumed to be 50 passengers per vehicle for the whole fleet.
As expected, the generalized cost per passenger increased in both
models that account for congestion: 76.4% when the passengers have no
information regarding occupancy condition, and 71% when passengers
are informed of the occupancy condition. This result demonstrates the
importance of occupancy information. When passengers are informed of
fully loaded vehicles, they can re-choose their path immediately; other-
wise, they wait for the next vehicle who might be full, which will result in
a later departing alternative. For the Winnipeg transit network, with a
fixed set of parameters, the occupancy information was found to save
about 5% of the average generalized cost.
Overall, both capacity constrained models showed similar trend of
changes in the travel time components, where the changes for the model
with no occupancy information were more substantial. The average in-
vehicle time decreased in both models, as the surplus passengers of the
over-loaded vehicles were re-assigned to other paths that included longer
waiting and walking times. The decrease is greater for the model without
information due to paths with shorter waiting times that are not fully
loaded but invalid in this model (i.e., depart prior to the denied boarding
occurrence). The decrease in the in-vehicle time is compensated by a
significant increase in the waiting and walking times. The average
number of transfers increased too, as the fully loaded transit line seg-
ments enforce passengers to perform additional transfers between
vehicles.
Fig. 6 presents a sensitivity analysis that examines the generalized
cost as a function of the vehicle capacity for the two cases of occupancy
information. The model shows consistency with a monotonic decrease in
the generalized cost for higher capacities. As expected, the solution of the
model with occupancy information outperform the solution without in-
formation for all capacity values. The gap between the results is the
highest in the intermediate range of 55–60 passengers per vehicle.
Possibly this range enables the best use of the information, where lower
capacities limits alternative paths and higher capacities reduces the
congestion and thus by the potential benefit of the information.
4.2. Specific O-D example
The assignment of O-D pair 3214, from node 81 to node 4, is exem-
plified in Table 3 for the model that assumes occupancy information.
Initially, without considering capacity constraints, the O-D demand was
assigned to 19 paths. When capacity constraints were applied, 17 of the
Fig. 4. Small example (Cepeda et al. (2006)).
N. Oliker, S. Bekhor EURO Journal on Transportation and Logistics 9 (2020) 100005
7

8. 19 paths were found to include over-loaded transit line segments. Node
81 is situated downstream for all the lines that pass nearby and therefore
most passengers that try to board the over-loaded vehicles are denied.
63.8 of the 79.54 passengers of this O-D demand were classified as sur-
plus and added to the surplus list. The selection probabilities of the 2
feasible paths, denoted as paths 12 and 19 in Table 3, were re-calculated.
The re-assignment was checked for feasibility before conducted (i.e.,
using eq. (5)). The desired assignment in path 12 was not fully possible,
and only the feasible fraction (eq. (6)) was assigned. The desired
assignment in path 19 was fully feasible and the path had additional free
space. In the next iteration, the feasible path set included only path 19,
the probability of selection equaled 1, and the remainder of the surplus
was assigned to it. This assignment was completely feasible. Note, that if
only some fraction of the desired assignment was feasible, the path set
was empty after the last iteration and new paths were found and added to
the ‘hyperpath’ in the next iteration. In the given example, the 63.8
surplus passengers were divided between paths 12 and 19.
4.3. Specific transit link example
Examination of the change in transit link loads shows that the variety
of O-D pairs that are assigned to a transit link often decrease when ca-
pacity constraints are considered. When the initial load exceeds the
maximal capacity, the surplus O-D demand is redirected to alternative
paths. Passengers boarding the transit line earlier, are prioritize over the
later boarding passengers. Therefore, the variety of origins in the con-
strained assignment tend to decrease, representing less downstream
passengers. For example, Table 4 presents the split of different origins
assigned to transit link 56, before and after applying capacity constraints,
for the model that assumes occupancy information. The line capacity is
120 passengers per hour, and the initial unconstrained load in the link
was 171 passengers; 30% of the demand that was initially assigned to the
link, was re-assigned to other paths. The initial unconstrained assignment
counted 219 different O-D pairs assigned to the link, compared to only
168 O-D pairs after the surplus re-assignment procedure. After the re-
assignment the dominance of the main origins has increased.
Fig. 5. Winnipeg transit network (INRO Consultants, 1999)).
Table 2
Assignment results for the Winnipeg network.
Average
results per
passenger
No capacity
constraints
(section 2.1)
Capacity constraints
No info
(section
2.4.2)
deviation With info
(section
2.4.1)
deviation
Generalized
cost (min)
38.08 67.19 76.42% 65.13 71.0%
Number of
transfers
0.85 1.06 24.68% 0.97 14.4%
Waiting time
(min)
14.84 26.86 80.93% 26.61 79.2%
In-vehicle
time (min)
11.88 10.97 7.70% 10.61 10.7%
Walking time
(min)
11.36 29.16 156.72% 27.71 144.0%
Fig. 6. Generalized cost as a function of vehicle capacity.
N. Oliker, S. Bekhor EURO Journal on Transportation and Logistics 9 (2020) 100005
8

9. 4.4. Comparison to an equilibrium model
This section applies an equilibrium model to enables its comparison
with the suggested model. We applied the commonly used MSA method
with the ’optimal strategies’ model (Spiess and Florian, 1989) as the
iterative procedure. In each iteration the model calculates an ‘effective
frequency’ for each boarding link according to the estimated flow and
updates the assignment accordingly.
The effective frequency of a boarding link is calculated as specified in
Cepeda et al. (2006) and following studies, and can be expressed by:
faðvÞ ¼
8
:
μ
1
Va
μc Va’ þ Va
β
if Va’ μc
0 otherwise
(8)
where μ is the nominal frequency, c is the line capacity, Va is the flow of
the passengers boarding at the stop and Va’ is the on-board flow right
after the stop, β is a parameter that determines the degree of update of the
effective frequency. Note that Va’ μc where the capacity constraint is
met. If the capacity in the link is exceeded the frequency is nullified, so
that the auxiliary solution will disable the boarding to the over-loaded
link.
The parameter β was set to 0.02, this value was found suitable by trial
and error calibration. The convergence of the model was fast and smooth,
completed after 18 iterations in average. The results of the equilibrium
model, compared to the unconstrained ‘optimal strategies’ model and the
suggested model (with info case, section 2.4.1) are presented in Table 5,
for two cases of vehicle capacity, 50 and 75.
The equilibrium model reduced the infeasibility of the solution,
resulting in less over-loaded links. Yet, its solution includes a significant
share of over-loaded transit line segments. Along iterations, crowded
segments were assigned with less passengers, but apparently the initial
solution was too far from feasibility to receive a feasible solution. When a
network is over-saturated, this equilibrium model cannot reach feasi-
bility (Cepeda et al., 2006). The over-loading was “pushed” up-stream in
the assignment and the limited MSA update of the solution in late iter-
ations kept the solution infeasible.
The root mean square error (RMSE) indicates the solutions are very
different. As the initial solution of the equilibrium model (i.e., the
optimal strategies) is a very saturated network, the ‘effective frequencies’
were significantly updated, and consequently the assignment result was
different. The equilibrium solution was also very different from the
suggested model solution. Generally, the equilibrium model reduced the
use of initially crowded lines. Conversely, the suggested model only
limits the loading to the vehicle capacity, assuming the vehicles can be
fully assigned. In the equilibrium model, a larger share of the demand is
assigned to inferior paths, and the attractive lines are far from being fully-
loaded, this property is well exemplified in section 2.7 (Fig. 4), where the
equilibrium model assigns only 70% of the demand to the attractive line,
and the suggested model assigns 100% to it. In the higher capacity case
(75 passengers per vehicle) the over-crowding was lower and therefore
the infeasibility of the equilibrium solution was lower and its update was
less substantial. Accordingly, the RMSE between the equilibrium model
to the other models was lower, but still significant.
Overall, this implementation presents the difficulty in acheving a
feasibile solution using the classic equilibrium model. Feasibility can be
reached using additional heuristics, adjustments and calibrations, but is
not straightforward.
5. Summary and outlook
This paper describes the implementation of capacity constraints in a
frequency based transit assignment model that considers online infor-
mation of both estimated arrival times and occupancy conditions.
Table 3
The assignment of O-D pair 3214 (from 81 to 4).
Path Probability Minimal travel time (excluding the wait time) Initial assignment Surplus After re-assignment Surplus
1 0.003 13.72 0.24 0.24 0.00 0
2 0.002 14.32 0.16 0.09 0.07 0
3 0.003 14.32 0.24 0.24 0.00 0
4 0.0008 14.32 0.06 0.06 0.00 0
5 0.0022 14.32 0.17 0.10 0.07 0
6 0.0006 14.32 0.05 0.05 0.00 0
7 0.0012 14.32 0.10 0.10 0.00 0
8 0.0026 14.32 0.21 0.21 0.00 0
9 0.0018 14.34 0.14 0.14 0.00 0
10 0.0022 14.34 0.17 0.18 0.00 0
11 0.1316 14.37 10.47 9.83 0.63 0
12 0.1372 14.37 10.91 0.00 47.79 0
13 0.1144 14.37 9.10 8.36 0.74 0
14 0.1164 14.37 9.26 8.05 1.21 0
15 0.1228 14.37 9.77 9.33 0.44 0
16 0.1242 14.37 9.88 8.27 1.61 0
17 0.1214 14.90 9.66 9.66 0.00 0
18 0.1124 14.90 8.94 8.91 0.03 0
19 0.0002 21.88 0.02 0.00 26.94 0
Sum 79.54 63.81 79.54 0
Table 4
O-D split in transit link 56.
No capacity constraints Capacity constraints
Number of O-D assigned to the link Number of O-D assigned to the link
219 168
Origins Percentage of the link
assignment
Origins Percentage of the link
assignment
34 0.03% 34 0.03%
16 0.03% 33 0.35%
31 0.04% 27 1.08%
27 0.55% 26 3.07%
33 1.23% 23 3.33%
23 5.69% 24 3.72%
24 8.77% 22 6.56%
26 11.00% 35 21.40%
45 14.17% 45 25.92%
22 15.21% 38 34.54%
35 15.92%
38 27.36%
N. Oliker, S. Bekhor EURO Journal on Transportation and Logistics 9 (2020) 100005
9

10. The main idea is to first apply an unconstrained assignment proced-
ure, and then handle only the over-loaded transit line segments, without
iteratively update the travel times. The proposed model applies capacity
constraints by allocating over-loaded transit line segments and re-assign
the surplus passengers to feasible alternatives. The assignment model
applied in this paper is a path based model. As most of the paths are found
in the initial unconstrained assignment procedure, the re-assignment
process only slightly extends the running time. This finding is highly
encouraging when comparing to common time-consuming equilibrium
models. The path based property of the model allows for prioritization of
passengers within a frequency based assignment framework; storing O-D
identities in the assignment procedure enables to prioritize upstream
passengers over later boarding passengers without model the assignment
over time.
The suggested model considered two types of information: (1) pas-
sengers are informed of the vehicle occupancy in real-time, and (2)
passengers have no prior knowledge of the vehicle occupancy. The two
cases were independently modeled and compared. It was found that
available information of the occupancy condition may reduce the travel
time significantly. Passenger that receive this information may change
their route choice accordingly, while uninformed passengers are forced
to choose inferior alternatives which departs later than their denied
boarding. This characteristic highlights the potential benefit of providing
occupancy information to the passenger.
Current research examines the role of online information and capacity
constrained assignment in the planning of transit networks. The impact
of considering these elements on the optimal design is investigated.
The model was applied using parameter values from the literature.
Further research will incorporate calibration of the model parameters.
Acknowledgements
This work was supported in part by the Israeli Ministry of Science and
Technology (3-12547) and in part by the Israel Science Foundation
(1532/16).
References
Bar-Gera, H., Boyce, D., Nie, Y.M., 2012. User-equilibrium route flows and the condition
of proportionality. Transp. Res. Part B Methodol. 46 (3), 440–462.
Barabino, B., Di Francesco, M., Mozzoni, S., 2015. Rethinking bus punctuality by
integrating Automatic Vehicle Location data and passenger patterns. Transport. Res.
Pol. Pract. 75, 84–95.
Bouzaïene-Ayari, B., Nguyen, S., Gendreau, M., 1995. Equilibrium-Fixed Point Model for
Passenger Assignment in Congested Transit Networks. U. de Montr
eal. Technical
Report CRT-95-57.
De Cea, J., Fern
andez, E., 1993. Transit assignment for congested public transport
systems: an equilibrium model. Transport. Sci. 27 (2), 133–147.
Cepeda, M., Cominetti, R., Florian, M., 2006. A frequency-based assignment model for
congested transit networks with strict capacity constraints: characterization and
computation of equilibria. Transp. Res. Part B Methodol. 40 (6), 437–459.
Chen, P.W., Nie, Y.M., 2015. Optimal transit routing with partial online information.
Transp. Res. Part B Methodol. 72, 40–58.
Cheung, L.L., Shalaby, A.S., 2017. System optimal re-routing transit assignment heuristic:
a theoretical framework and large-scale case study. Int. J. Transport. Sci. Technol. 6
(4), 287–300.
Chriqui, C., Robillard, P., 1975. Common bus lines. Transport. Sci. 9 (2), 115–121.
Codina, E., Rosell, F., 2017. A heuristic method for a congested capacitated transit
assignment model with strategies. Transp. Res. Part B Methodol. 106, 293–320.
Cominetti, R., Correa, J., 2001. Common-lines and passenger assignment in congested
transit networks. Transport. Sci. 35 (3), 250–267.
Dial, R.B., 1967. Transit Pathfinder Algorithm. Highway Research Record, pp. 67–85.
Issue Number: 205.
Estrada, M., Giesen, R., Mauttone, A., Nacelle, E., Segura, L., 2015. Experimental
evaluation of real-time information services in transit systems from the perspective of
users. In: CASPT 2015 - Conference on Advanced Systems in Public Transport.
(Rotterdam, The Netherlands ,19-23 July 2015).
Fu, Q., Liu, R., Hess, S., 2012. A review on transit assignment modelling approaches to
congested networks: a new perspective. Procedia-Soc. Behav. Sci. 54, 1145–1155.
Gentile, G., Nguyen, S., Pallottino, S., 2005. Route choice on transit networks with online
information at stops. Transport. Sci. 39 (3), 289–297.
Hamdouch, Y., Szeto, W.Y., Jiang, Y., 2014. A new schedule-based transit assignment
model with travel strategies and supply uncertainties. Transp. Res. Part B Methodol.
67, 35–67.
Hickman, M.D., Wilson, N.H., 1995. Passenger travel time and path choice implications of
real-time transit information. Transport. Res. C Emerg. Technol. 3 (4), 211–226.
INRO Consultants, 1999. Emme/2 User’s Manual: Release 9.2. Montr
eal, Canada.
Kepaptsoglou, K., Karlaftis, M., 2009. Transit route network design problem. J. Transport.
Eng. 135 (8), 491–505.
Kurauchi, F., Bell, M.G., Schm€
ocker, J.D., 2003. Capacity constrained transit assignment
with common lines. J. Math. Model. Algorithm. 2 (4), 309–327.
Lam, W.H.K., Gao, Z.Y., Chan, K.S., Yang, H., 1999. A stochastic user equilibrium
assignment model for congested transit networks. Transp. Res. Part B Methodol. 33
(5), 351–368.
Lam, W.H., Zhou, J., Sheng, Z.H., 2002. A capacity restraint transit assignment with
elastic line frequency. Transp. Res. Part B Methodol. 36 (10), 919–938.
Nassir, N., Hickman, M., Ma, Z.L., 2019. A strategy-based recursive path choice model for
public transit smart card data. Transp. Res. Part B: Methodol. 126, 528–548.
Nguyen, S., Pallottino, S., 1988. Equilibrium traffic assignment for large scale transit
networks. Eur. J. Oper. Res. 37 (2), 176–186.
N€
okel, K., Wekeck, S., 2009. Boarding and alighting in frequency-based transit
assignment. Transport. Res. Rec.: J. Transport. Res. Board (2111), 60–67.
Oliker, N., Bekhor, S., 2018. A frequency based transit assignment model that considers
online information. Transport. Res. C Emerg. Technol. 88, 17–30.
Schm€
ocker, J.D., Bell, M.G., Kurauchi, F., 2008. A quasi-dynamic capacity constrained
frequency-based transit assignment model. Transp. Res. Part B Methodol. 42 (10),
925–945.
Schm€
ocker, J.D., Fonzone, A., Shimamoto, H., Kurauchi, F., Bell, M.G., 2011. Frequency-
based transit assignment considering seat capacities. Transp. Res. Part B Methodol.
45 (2), 392–408.
Schm€
ocker, J.D., Shimamoto, H., Kurauchi, F., 2013. Generation and calibration of transit
hyperpaths. Transport. Res. Part C 36, 406–418.
Shimamoto, H., Kurauchi, F., Iida, Y., 2005. Evaluation of effect of arrival time
information provision using transit assignment model. Int. J. ITS Res. 11–18. In press.
Spiess, H., Florian, M., 1989. Optimal strategies: a new assignment model for transit
networks. Transp. Res. Part B Methodol. 23 (2), 83–102.
Sun, S., Szeto, W.Y., 2018. Logit-based transit assignment: approach-based formulation
and paradox revisit. Transp. Res. Part B Methodol. 112, 191–215.
Szeto, W.Y., Jiang, Y., Wong, K.I., Solayappan, M., 2013. Reliability-based stochastic
transit assignment with capacity constraints: formulation and solution method.
Transport. Res. C Emerg. Technol. 35, 286–304.
Wardman, M., 2004. Public transport values of time. Transport Pol. 11 (4), 363–377.
Wardrop, J.G., 1952. Road paper. Some theoretical aspects of road traffic research. Proc.
Inst. Civ. Eng. 1 (3), 325–362.
Wu, J.H., Florian, M., Marcotte, P., 1994. Transit equilibrium assignment: a model and
solution algorithms. Transport. Sci. 28 (3), 193–203.
Yen Y, Jin, 1970. An algorithm for finding shortest routes from all source nodes to a given
destination in general networks. Q. Appl. Math. 27 (4), 526–530. In press.
Table 5
Comparison to an equilibrium model.
Optimal
strategies
Equilibrium
model
Suggested
model
Vehicle capacity 50
Over-loading (over-load/
total load)
19.70% 18.57% 0.00%
Average results per passenger
Travel time (min) 39.32 41.92 65.13
Number of transfers 0.67 0.50 0.97
Waiting time (min) 14.79 13.04 26.61
In-vehicle time (min) 11.94 12.37 10.61
Walking time (min) 12.58 16.51 27.71
RMSE from equilibrium
solution
53.36 54.56
Vehicle capacity 75
Over-loading (over-load/
total load)
9.52% 8.15% 0.00%
Average results per passenger
Travel time (min) 39.32 40.41 45.35
Number of transfers 0.67 0.69 0.95
Waiting time (min) 14.79 14.79 19.73
In-vehicle time (min) 11.94 12.32 12.20
Walking time (min) 12.58 13.30 13.21
RMSE from equilibrium
solution
41.34 31.66
N. Oliker, S. Bekhor EURO Journal on Transportation and Logistics 9 (2020) 100005
10