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Transportmetrica A: Transport Science, 2015
Vol. 11, No. 2, 158–185, http://dx.doi.org/10.1080/23249935.2014.944241
Elastic demand with weibit stochastic user equilibrium flows and
application in a motorised and non-motorised network
Songyot Kitthamkesorna, Anthony Chena∗and Xiangdong Xub
aDepartment of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110,
USA; bKey Laboratory of Road andTraffic Engineering, Tongji University, Shanghai 201804, People’s
Republic of China
(Received 26 September 2013; accepted 8 July 2014)
In this paper, we propose a new elastic demand (ED) stochastic user equilibrium (SUE) model
with an application to the combined modal split and traffic assignment (CMSTA) problem.
This new model, called the path-size weibit (PSW) SUE model with ED, is derived based on
the Weibull distribution, which does not require the identically distributed assumption typically
imposed in the multinomial logit (MNL) model with the Gumbel distribution. In addition, a
path-sizefactorisincludedtocorrectthechoiceprobabilitiesofroutesthatarenottrulyindepen-
dent (i.e. another assumption typically required in the MNL model). Equivalent mathematical
programming (MP) formulation of the PSW-SUE-ED model is developed to simultaneously
consider both travel choice and route choice. The travel choice is determined based on the
ED function that explicitly considers the network level of service based on the logarithmic
expected perceived cost of the Weibull distribution to determine the travel demand, while the
route choice accounts for both route overlapping and non-identical perception variance with
respect to different trip lengths. Qualitative properties of the proposed MP formulation are
rigorously proved.A path-based partial linearisation algorithm combined with a self-regulated
averaginglinesearchstrategyisdevelopedforsolvingthe PSW-SUE-EDmodelandits applica-
tion to the CMSTA problem. Numerical examples are also provided to demonstrate the features
of the proposed PSW-SUE-ED model as well as a real-case study in a bi-modal network with
motorised and non-motorised mode choices.
Keywords: logit; weibit; stochastic user equilibrium; elastic demand; combined modal split
and traffic assignment problem
1. Introduction
Modelling the elasticity of travel demand in network equilibrium analysis was introduced by
Beckmann, McGuire, and Winsten (1956) to explicitly consider the equilibrium between supply
and demand. The supply (or performance) functions are determined by the link travel costs under
congestion, and the travel demand functions are determined by the user benefits (Florian and
Nguyen 1974), generally derived based on the level of service (LOS) of the network (Sheffi
1985). For example, as congestion increases, the network LOS decreases. Travellers may exercise
theiravailablechoices by consideringa different mode of travel (mode choice),going to adifferent
destination (destination choice), foregoing some trips altogether (travel choice), and in addition
to choosing a different route (route choice). These choices will have an effect on the traffic flow
patterns. In addition to the multi-dimensional travel choice applications (e.g. combined travel and
∗Corresponding author. Email: anthony.chen@usu.edu
© 2014 Hong Kong Society for Transportation Studies Limited
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Transportmetrica A: Transport Science 159
route choice problem, combined mode and route choice problem, combined destination and route
choiceproblem,andcombinedtravel-destination-mode-routechoiceproblem),modellingdemand
elasticityhas importanttransportation applicationsin predictingfuture traveldemand patterns and
assessing network improvement. Using the network analysis with fixed demand (FD) to assess
such behaviour may cause a biased future travel demand pattern prediction, misevaluation of
network performance, and result in an inefficient budget allocation.
Beckmann, McGuire, and Winsten (1956) provided the pioneer work of formulating the deter-
ministicuser equilibrium (DUE) model with elastic demand (ED) (orDUE-ED) asa mathematical
programming (MP) formulation, where the ED is a function of the equilibrium route travel cost
between each origin–destination (O–D) pair. For a historical review of Beckmann’s DUE-ED
model, readers are referred to Boyce (2013). Since the seminal work by Beckmann, McGuire,
and Winsten (1956), many researchers have further developed different ideas as shown in Table 1
to enhance the modelling realism and applications of the DUE-ED model.
In terms of the methodology used in formulation and analysis, Beckmann, McGuire, and
Winsten (1956) provided the first MP formulation, or more specifically a convex programming
(CP) formulation, for the DUE-ED model. Carey (1985) provided two dual formulations of the
DUE-ED problem using node-link and link-path variables, and explored the relationship between
the primal and dual formulations. Aashtiani (1979) gave the first nonlinear complementarity prob-
lem (NCP) formulation for modelling the interactions in a multimodal network, while Gabriel
and Bernstein (1997) introduced the nonadditive user equilibrium (NaUE) problem as an NCP
formulation in which the cost incurred on each path is not simply the sum of the link costs that
constitute that path. Dafermos (1982) offered a variational inequality (VI) formulation for the
multimodal traffic equilibrium model with ED, where the link travel costs depend on the entire
link flow vector and the travel demands depend on the entire mode-specific O–D cost vector. Fisk
and Boyce (1983) provided alternativeVI formulations for the network equilibrium travel choice
problem, which does not require invertibility of the travel demand function. Cantarella (1997)
provided a fixed point (FP) formulation for the multi-mode multi-user equilibrium assignment
with ED, where users have different behavioural characteristics as well as different choice sets.
The major drawback of the above models is the assumption that all travellers have perfect
knowledge of network conditions (i.e. know all available routes and have perfect perception of
all route costs). In reality, travellers rarely know all available routes, and certainly do not always
select the minimum cost route. To overcome such drawback of the DUE-ED model, several
researchers extended the stochastic user equilibrium (SUE) principle suggested by Daganzo and
Sheffi (1977) from a FD to an ED version, or the SUE-ED model for short. In the SUE principle,
a random error term is introduced into the route cost function to mimic the perception error of
network travel times due to the travellers’ imperfect knowledge of network conditions. At the
SUE state, no travellers can improve his or her perceived travel time by unilaterally changing
routes (Sheffi 1985). In the transportation literature, Gumbel and normal distributions are the two
commonly used random error terms to develop the probabilistic route choice models, which result
in the multinomial logit (MNL) and multinomial probit (MNP) route choice models (Dial 1971;
Daganzo and Sheffi 1977), respectively.
Yang and Bell (1998) extended Fisk’s (1980) MP formulation for the MNL-SUE model with
FD to the ED case (or MNL-SUE-ED), Xu and Chen (2013) extended Zhou, Chen, and Behkor’s
(2012) MP formulation for the C-logit-SUE model with FD to the ED case (or C-logit-SUE-ED),
while Ryu et al. (2014a) extended Bekhor and Prashker’s (1999) MP formulation for the paired
combinatorial logit (PCL) SUE model with FD to the ED case (or PCL-SUE-ED). Maher (2001),
Meng and Liu (2012) and Meng, Liu, and Wang (2014), on the other hand, provided MP and VI
formulations for the MNP-SUE-ED model without and with link interactions (LI), respectively.
As is well known, the MNL model needs the independently and identically distributed (IID)
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160 S. Kitthamkesorn et al.
Table 1. Summary of some key traffic equilibrium models with ED.
Approach Reference Model Remark
MP Beckmann, McGuire,
and Winsten (1956) DUE-ED Provide the first DUE model with ED as a CP formulation for modelling travellers’
trip-making behaviour
Carey (1985) DUE-ED Provide two dual formulations of the DUE-ED problem using node-link and path-link
variables, and explore the relationship between the primal and dual formulations
Yang and Bell (1998) MNL-SUE-ED Provide an equivalent MP formulation for the MNL-SUE-ED problem
Xu and Chen (2013) C-logit-SUE-ED Provide an equivalent MP formulation for the C-logit-SUE-ED problem, and develop a
path-based partial linearisation algorithm for solving it
Maher (2001) General SUE-ED Provide a new objective function for the SUE assignment with ED, and develop a
balanced demand algorithm for solving it
Kitthamkesorn et al.
(2013) CMSTA Provide a CMSTA problem based on the excess demand formulation with nested logit
for mode choice and CNL for route choice
Ryu et al. (2014a) PCL-SUE-ED Provide an equivalent MP formulation for the PCL-SUE-ED problem, and develop a
path-based partial linearisation algorithm for solving it
NCP Aashtiani (1979) Multimodal traffic
assignment problem Provide the first NCP formulation for the multimodal traffic assignment problem, and
develop one of the early path-based linearisation algorithms for solving the traffic
assignment problem
Gabriel and Bernstein
(1997) NaUE Introduce the nonadditive traffic equilibrium problem in which the cost incurred on
each path is not simply the sum of the link costs that constitute that path, and propose
the nonsmooth equations/sequential quadratic programming method for solving the
nonadditive traffic equilibrium problem
(Continued)
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Transportmetrica A: Transport Science 161
Table 1. Continued.
Approach Reference Model Remark
Variational
inequality (VI) Dafermos (1982) Multimodal traffic
assignment problem Provide a VI formulation for modelling the general multimodal traffic equilibrium
model with ED, where the link travel costs depend on the entire link flow vector and
the travel demands depend on the entire mode-specific O–D cost vector, and develop
an iterative relaxation algorithm for computing the equilibrium flow pattern
Fisk and Boyce (1983) DUE-ED Provide alternativeVI formulations for the network equilibrium travel choice problem,
which does not require invertibility of the travel demand function
Wu and Lam (2003a,
2003b)CMSTA-ED Provide a CMSTA-ED model based on the MNL-SUE flows for modelling combined
mode and route choice decisions in a bimodal network, and develop a MSA with
cost approximation algorithm and block Gauss-Seidel decomposition method
Meng and Liu (2012)
and Meng, Liu, and
Wang (2014)
MNP-SUE-ED-LI Develop twoVI models and two hybrid prediction–correction cost averaging algorithms
combined with a two-stage Monte Carlo simulation-based stochastic network loading
method for the MNP-SUE-ED problem with link interactions
FP Cantarella (1997) FP Develop a FP formulation for the multi-mode multi-user equilibrium assignment with
ED, where users have different behavioural characteristics as well as different choice
sets
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162 S. Kitthamkesorn et al.
assumption with the Gumbel variates in order to derive a closed-form probability expression. The
IID assumption comes with two known limitations: (1) inability to handle route overlapping and
(2) inability to handle perception variance with respect to (w.r.t.) different trip lengths. These
drawbacks may cause a biased result in estimating the expected perceived cost (EPC) (i.e. the
well-known MNL’s log-sum term) used in the ED function to determine the travel demands.
On the other hand, the MNP-SUE-ED model does not have the two known limitations of the
MNL model since it does not require the IID Gumbel variate assumption. However, the MNP-
SUE-ED model does not have a closed-form probability expression, and hence would require
significant computational efforts using either Monte Carlo simulation (Sheffi and Powell 1982),
analytic approximations (Maher 1992;Maher and Hughes 1997;Connors, Hess, and Daly 2014)
or numerical method (Rosa and Maher 2002).
In terms of applications, (Wu and Lam 2003a,2003b) adopted the VI formulation to develop a
combined modal split and traffic assignment (CMSTA) model with ED based on the MNL-SUE
flows for modelling mode choice and route choice decisions in a bimodal (i.e. motorised and non-
motorised) network. Since the CMSTA model adopts the MNL-SUE-ED model, it also inherits
the same two drawbacks identified above with the same bias in estimating the travel demands
for the two modes. To partially address the drawbacks, Kitthamkesorn et al. (2013) developed
an equivalent MP for the CMSTA problem that explicitly considers mode and route similarities
under congested networks. The mode choice is modelled using the nested logit model (Ben-Akiva
and Lerman 1985) and the route choice is modelled through the cross-nested logit (CNL) model
(Bekhor and Prashker 1999). Although the model captures the similarities of both mode and route
choices in the CMSTA problem, the identically distributed assumption still remains (i.e. inability
to account for the route-specific perception variance) due to the classical logit assumption of
homogeneous perception variance.
Inthis paper, we providean alternative to relaxthe IID assumption embedded inthe MNL-SUE-
ED model by using the Weibull distribution. The path-size weibit (PSW) model (Kitthamkesorn
and Chen 2013) is used to develop an equivalent MP formulation for modelling demand elasticity
in the SUE framework. The proposed PSW-SUE model with ED (or PSW-SUE-ED) has the
following two significant features that are distinct from the literature shown inTable 1.
•The network LOS is captured through the logarithmic expected perceived cost (log EPC) of
the Weibull distribution, which is used to develop the PSW-SUE-ED model, to determine
the travel demands and SUE flows. The advantage is that the log EPC explicitly considers
the route overlapping and non-identical perception variance problems in the SUE assign-
ment, and avoids the bias caused by the two known limitations of the MNL-SUE model in
estimating the travel demands.
•An application of the PSW-SUE-ED model is developed to consider both mode choice and
routechoice as a CMSTAproblem. It is demonstrated witha casestudy ina bimodal network
with motorised and non-motorised modes using the Winnipeg network.
This paper not only develops a new PSW-SUE-ED model, but also provides qualitative properties,
as well as a solution algorithm, accompanied by convergence results, numerical examples, and
application to a bimodal network with motorised and non-motorised modes.
The remainder of this paper is organised as follows. The next section gives some background of
the weibit route choice models. In Section 3, the equivalent MP formulations for the PSW-SUE-
ED model and its application as a CMSTA problem are provided along with some qualitative
properties. Section 4describes a path-based partial linearisation method combined with a self-
regulated averaging (SRA) stepsize scheme for solving the PSW-SUE-ED model. Numerical
results are presented in Section 5, and some concluding remarks are provided in Section 6.
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Transportmetrica A: Transport Science 163
2. Weibit route choice models
In this section, we provide some background of the weibit route choice models and their EPC. The
section begins with the multinomial weibit (MNW) model and is followed by the PSW model.
2.1. MNW model
2.1.1. Model formulation
Castillo et al. (2008) developed the MNW model to resolve the identical variance issue. Unlike
the MNL model which uses the conventional additive random utility model (RUM), the MNW
model adopts the multiplicative RUM with the Weibull random error (Fosgerau and Bierlaire
2009;Kitthamkesorn and Chen 2013), i.e.
Uij
r=gij
r−ζijβij
εij
r∀r∈Rij,ij ∈IJ, (1)
where IJ is the set of O–D pairs, Rij is the set of routes between O–D pair ij,gij
ris the travel cost
on route rbetween O–D pair ij,εij
ris the independently Weibull distributed random error term
on route rbetween O–D pair ij, and ζij ∈[0,gij
r)and βij ∈(0,∞)are the location parameter and
shape parameter of the Weibull distribution.According to the MNW disutility in Equation (1), we
have the MNW route choice probability (see Kitthamkesorn and Chen 2013,2014 for details):
Pij
r=gij
r−ζij−βij
k∈Rij gij
k−ζij−βij ∀r∈Rij ,ij ∈IJ.(2)
To show how this MNW model handles the perception variances of different trip lengths,
consider a two-route network configuration as shown in Figure 1. For both networks, the upper
route cost is larger than the lower route cost by 5 units. However, the upper route cost is two times
larger than the lower route cost in the short network, while it is only less than 5% larger in the
long network. As expected, the MNL model produces the same route choice probability for both
short and long networks. This is because the MNL model assumes that each route has the same
and fixed perception variance (i.e. π2/6θ2)as shown in Figure 2(a). In other words, the solution
Figure 1. MNL and MNW probabilities under the two-route networks. (a) Short network and (b) long
network.
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164 S. Kitthamkesorn et al.
Figure 2. Perception variance of the MNL and MNW models. (a) MNL model and (b) MNW model.
is solely based on the absolute cost difference irrespective of the overall trip length (Sheffi 1985).
The MNW model, in contrast, produces different route choice probabilities for the two networks.
Ituses a relativecost difference to handle different trip lengths and has the followingroute-specific
perception variance as a function of route travel cost (Kitthamkesorn and Chen 2014), i.e.
σij
r2=⎡
⎣gij
r−ζij
1+1/βij ⎤
⎦
2
1+2
βij −21+1
βij ∀r∈Rij,ij ∈IJ, (3)
where () is the gamma function. With this, a larger-cost route will have a higher perception
variance as shown in Figure 2(b), such that the probability of choosing each route becomes more
similar for a longer network as shown in Figure 1(b).
Note that scaling the dispersion parameter can partially relax the identical variance issue in
the MNL model in the traffic assignment context (Chen et al. 2012). By scaling the dispersion
parameter according to the O–D trip length and maintaining a coefficient of variation (CV) of 0.5
as shown in Figure 3(i.e. a larger θfor the short network or a smaller θfor the long network), we
can obtain similar results as that of the MNW model, where the probability of choosing the lower
route is higher for the short network according to a smaller perception variance. Nonetheless,
Figure 3. Probability and perception variance of MNL with scaling.
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Transportmetrica A: Transport Science 165
this scaling technique cannot obtain the route-specific perception variance like the MNW model
in Equation (3). All routes between an O–D pair must still have the same and fixed perception
variance of π2/6θ2in the lower portion of Figure 3due to the classical logit assumption of
homogeneous perception variance.
2.1.2. EPC of the MNW model
The EPC can be used to represent the LOS of a transportation network. Yang and Bell’s (1998)
MP formulation for the MNL SUE model with ED has the ED as a function of the MNL EPC –
the log-sum term (Sheffi 1985). Further, Oppenheim’s (1995) MP formulation for the MNL-based
combined travel demand model also has the multi-dimensional travel choices as a function of the
MNL EPCs. However, the MNW model does not have a closed-form EPC in general (Fosgerau
and Bierlaire 2009). To overcome this drawback, we consider the logarithmic EPC of the MNW
model (Kitthamkesorn and Chen 2014) to represent the LOS of a transportation network, i.e.
¯μij =− 1
βij ln
r∈Rij gij
r−ζij−βij ∀ij ∈IJ.(4)
This logarithmic MNW EPC satisfies some important properties as follows: (1) its partial deriva-
tive w.r.t. the logarithmic route travel cost gives back the MNW choice probability; (2) it is
monotonically decreasing w.r.t. the number of routes and (3) it is concave w.r.t. the vector of
logarithmic route travel costs (Kitthamkesorn and Chen 2014).
2.2. PSW model
2.2.1. Model formulation
Even though the MNW model can successfully address the identical perception variance prob-
lem, it still inherits the independently distributed assumption. To overcome this shortcoming,
Kitthamkesorn and Chen (2013) adopted the path-size factor ij
r(Ben-Akiva and Bierlaire 1999)
to handle the route overlapping problem. This path-size factor accounts for different route sizes
determined by the length of links within a route and the relative lengths of routes that share a link,
i.e.
ij
r=
a∈ϒr
la
Lij
r
1
k∈Rij δij
ak
∀r∈Rij,ij ∈IJ, (5)
where lais the length of link a∈A,Lij
ris the length of route rconnecting O–D pair ij,ϒris the
set of links in route rbetween O–D pair ij, and δij
ar is equal to 1 for link aon route rbetween O–D
pair ij and 0 otherwise. Routes with a heavy overlapping with other routes have a smaller value
of ij
r. Note that other functional forms of ij
rcan be found in Bovy, Bekhor, and Prato (2008)
and Prato (2009). The path-size factor is used to modify the MNW RUM model as follows:
Uij
r=gij
r−ζijβij
ij
r
εij
r∀r∈Rij,ij ∈IJ, (6)
which gives the PSW probability
Pij
r=
ij
rgij
r−ζij−βij
k∈Rij ij
kgij
k−ζij−βij ∀r∈Rij ,ij ∈IJ. (7)
For the detailed derivation of the choice probability, see Kitthamkesorn and Chen (2013).
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166 S. Kitthamkesorn et al.
2.2.2. EPC of the PSW model
Since the PSW model modifies the deterministic term of the MNW model, its logarithmic EPC
can be expressed as (Kitthamkesorn and Chen 2014):
¯μij =− 1
βij ln
r∈Rij
ij
rgij
r−ζij−βij ∀ij ∈IJ. (8)
Note that this logarithmic PSW EPC also has the same property as the logarithmic MNW EPC.
The partial derivative of this logarithmic PSW EPC w.r.t. the logarithmic route travel cost gives
back the PSW choice probability. It is monotonically decreasing w.r.t. the number of routes, and
it is concave w.r.t. the vector of logarithmic route travel costs.
3. MP formulation
This section presents equivalent MP formulations for the PSW-SUE-ED model and its application
to the CMSTA problem. Specifically, we present these MP formulations with some qualitative
properties. The section starts with some necessary assumptions, followed by the MP formulations
for the PSW-SUE-ED model and the CMSTA problem.
3.1. Assumptions
We start with some necessary assumptions. First, we make a general assumption on the link travel
cost:
Assumption 1The link travel cost τa, which can be a function of link travel time, is a
monotonically increasing function of its own flow.
Since the weibit model falls within the category of multiplicative random utility maximisation
model, the deterministic part of the disutility function is simply a set of multiplicative explanatory
variables (Cooper and Nakanishi 1988). Then, we make an assumption of the route travel cost:
Assumption 2The route travel cost is a function of multiplicative link travel costs, i.e.
gij
r=
a∈ϒr
τa∀r∈Rij ∀ij ∈IJ. (9)
Assumption 2is necessary to maintain the weibit choice probability expression based on the
relative cost difference (in contrast to the absolute cost difference of the logit choice probability
expression). In essence, the travel costs under this assumption are investigated using the logarith-
mic scale and the underlying model is multiplicative rather than the typical additive model (see
Fosgerauand Bierlaire 2009;Li 2011). Correspondingly,travellersare concerned with the relative
difference rather than the absolute difference of the travel costs when making their route choice
decisions. In fact, Fosgerau and Bierlaire (2009) found that the multiplicative model outperforms
the additive model in several cases leading to a better fit with empirical data sets. Castillo et al.
(2008) also commented that the ability of the MNW-SUE model to handle unequal path perceived
variances is suitable for heterogeneous travellers with different knowledge levels of network con-
ditions as well as different trip lengths. In addition,Assumption 2allows the MP formulation as a
convex optimisation problem to incorporate the ED (travel choice) and mode choice as a function
of the logarithmic PSW EPC.
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Transportmetrica A: Transport Science 167
Following the path-size logit (PSL) SUE formulation provided by Chen et al. (2012), the
lengths used in the path-size factor for the MP formulation is flow-independent. We make another
assumption on the path-size factor attributes:
Assumption 3The lengths laand Lij
rused in ij
rare flow-independent.
This assumption is also used by most of the existing extended logit-SUE models to account
for route overlaps (see Prashker and Bekhor 2004; Chen et al. 2012). In the MNP-SUE model,
the random error term is assumed to follow the normal distribution with zero mean and flow-
independentvariance(Sheffi1985;MengandLiu2012).Notethatwecanalsoadoptthevariational
inequality (VI) formulation to incorporate the flow-dependent path-size attributes similar to the
congestion-based C-logit-SUE model developed by Zhou, Chen, and Behkor (2012).
Since ζij cannot be decomposed into the link level easily, we make another assumption:
Assumption 4ζij is equal to zero.
This assumption indicates that each route between an O–D pair is assumed to have the same CV.
From Equation (3), the route-specific CV can be expressed as
ϑij
r=σij
r
gij
r
=gij
r−ζij
gij
r
1+2/βij
1+1/βij 2−1∀r∈Rij,ij ∈IJ, (10)
with ζij =0, ϑij
rof each route is equal. One limitation of assuming the location parameter equal
to zero (or the CV of each routes is the same within an O–D pair) is that we lose one degree of
freedom (or flexibility) of fully making use of the Weibull distribution as the random error term.
Without the location parameter, it may not be able to account for any arbitrary multiplier on the
route cost. In essence, the MNW-SUE model without the location parameter, which determines
the choice probability based on the relative cost difference, is insensitive to an arbitrary scale
(Kitthamkesorn and Chen 2014). Note that we can adopt the VI formulation of the congestion-
based C-logit-SUE model developed by Zhou, Chen, and Behkor (2012) to incorporate a non-zero
ζij into the PSW-SUE-ED model.
Finally, we make an assumption on the ED function Dij():
Assumption 5The ED function is a monotonically decreasing function of the logarithmic PSW
EPC.
3.2. MP formulation for the PSW-SUE-ED model
Inthis section, we provide the PSW-SUE-EDmodel,where theED is a function of the logarithmic
PSW EPC. Consider the following MP formulation:
minZ(f,q)=Z1+Z2+Z3+Z4+Z5
=
a∈Ava=ij∈IJ r∈Rij fij
rδij
ar
0lnτa(ω) dω+
ij∈IJ
r∈Rij
1
βij fij
rlnfij
r−1
−
ij∈IJ qij
0D−1
ij (ω)dω−
ij∈IJ
1
βij qij lnqij −1−
ij∈IJ
r∈Rij
1
βij fij
rlnij
r(11)
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168 S. Kitthamkesorn et al.
s.t.
r∈Rij
fij
r=qij ∀ij ∈IJ, (12)
qij ≥0, fij
r≥0∀r∈Rij,ij ∈IJ, (13)
where fij
ris the flow on route rbetween O–D pair ij,qij is the travel demand between O–D pair
ij from the ED function Dij() and vais the flow on link a. Equation (12) is the flow conservation
constraint, and Equation (13) is the non-negativity condition. The main differences between this
PSW-SUE-ED andYang and Bell’s (1998) MNL-SUE-ED are Z1and Z5.Z1is the multiplicative
Beckmann’stransformation.Itusesthelogtransformationto facilitate the route costcomputations.
With this, the PSW-SUE-ED has the ED Dij() as a function of the logarithmic PSW EPC at the
equilibrium. On the other hand, Z5incorporates ij
rto handle the route overlapping problem.
When there is no route overlapping (i.e. ij
r=1), the PSW-SUE-ED model collapses to the
MNW-SUE-ED model.
Proposition 1The solution of the MP formulation given in Equations (11)through (13)satisfies
the PSW route choice probability and the ED function.
Proof Note that the logarithmic terms in Equation (11) implicitly require both fij
rand qij to be
positive. By constructing the Lagrangian and then setting its partial derivative to 0, we obtain
lngij
r+1
βij lnfij
r−1
βij lnij
r−λij =0∀r∈Rij,ij ∈IJ, (14)
λij −D−1
ij qij−1
βij lnqij =0∀ij ∈IJ, (15)
r∈Rij
fij
r=qij ∀ij ∈IJ, (16)
where λij is the Lagrangian multiplier for the flow conservation constraint in Equation (12).
Rearranging Equation (14)gives
fij
r=exp βij λijij
rgij
r−βij ∀r∈Rij,ij ∈IJ. (17)
From Equation (16), we have
r∈Rij
fij
r=qij =exp βij λij
r∈Rij
ij
rgij
r−βij ∀ij ∈IJ. (18)
Dividing Equation (17) by Equation (18) gives the PSW route choice probability:
Pij
r=fij
r
qij =
ij
rgij
r−βij
k∈Rij ij
kgij
k−βij ∀r∈Rij,ij ∈IJ. (19)
On the other hand, by rearranging Equation (18), we obtain
λij =1
βij lnqij −1
βij ln
r∈Rij
ij
rgij
r−βij ∀ij ∈IJ. (20)
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Transportmetrica A: Transport Science 169
Substituting Equation (20) into Equation (15)gives
D−1
ij qij=− 1
βij ln
r∈Rij
ij
rgij
r−βij ∀ij ∈IJ. (21)
By rearranging Equation (21), we have the following ED as a function of the logarithmic PSW
EPC, i.e.
qij =Dij ¯μij=Dij ⎛
⎝−1
βij ln
r∈Rij
ij
rgij
r−βij ⎞
⎠∀ij ∈IJ. (22)
This completes the proof.
Proposition 2The route flow and the O–D demand solutions of the PSW-SUE-ED model are
unique.
Proof It is sufficient to prove that the objective function in Equation (11) is strictly convex
and that the feasible region is convex. The convexity of the feasible region is assured for linear
equality constraint in Equation (12). The nonnegative constraint in Equation (13) does not alter
this characteristic. From Proposition 1and Assumption 3, the second-order derivative w.r.t. route
flow is
∂2Z(f)
∂fij
r∂frs
k
=∂lngij
r
∂frs
k
−∂
∂frs
kD−1
ij ⎛
⎝
r∈Rij
fij
r⎞
⎠+1
βij
∂lnfij
r
∂frs
k
−1
βij
∂lnr∈Rij fij
r
∂frs
k. (23)
From Assumptions 1and 5, both ∂ln gij
r/∂frs
kand −(∂/∂ frs
k)D−1
ij (r∈Rij fij
r)are positive semi-
definite. Clearly (1/βij )(∂ ln fij
r/∂frs
k)is positive definite since
1
βij
∂lnfij
r
∂frs
k
=⎧
⎨
⎩
1
βijfij
r>0if(ij,r)=(rs,k),
0 otherwise. (24)
Finally, we can observe that −(1/βij)(∂ ln r∈Rij fij
r/∂frs
k)is positive semi-definite since all ele-
ments of the block matrix w.r.t. O–D pair ij are equal to −1/βij r∈Rij fij
r. Thus, the objective
function in Equation (11) is strictly convex w.r.t. route flows. Therefore, the equilibrium route
flow is unique. According to the flow conservation condition, the equilibrium travel demand is
also unique. This completes the proof.
3.3. Application of the PSW-SUE-ED model
In this section, we provide an application of the PSW-SUE-ED model to consider both mode
choice and route choice as a CMSTA problem. It will be demonstrated in the numerical result
section with a case study in a bimodal network with motorised and non-motorised modes using
the Winnipeg network.
To begin with, a model with a binary logit mode choice is presented. The ED term Z3in
Equation (11) is modified using the argument-complementing function (Sheffi 1985), i.e. (see
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170 S. Kitthamkesorn et al.
Figure 4. Network representation and excess demand as a binary logit mode choice.
also Figure 4)
ij∈IJ qij
0D−1
ij (ω) dω=−
ij∈IJ
1
γij qij
m=2
0ln ω
qij
m=1−ω
−ijdω, (25)
where qij
mis the demand of mode m=1, 2 between O–D pair ij,γij is the binary logit model
parameter between O–D pair ij and ij is the exogenous modal attractiveness difference between
the two modes connecting O–D pair ij. By incorporating Equation (25) into Equation (11), we
haveacombined binarylogit mode choice and PSW-SUEroute choice model. Note that the binary
logitmode choice is also a function of the logarithmic PSW EPC propagated fromthe route choice
level under congestion.
To extend the binary logit mode choice to the MNL mode choice, we restate Equation (25)as
ij∈IJ
1
γij qij
m=2
0ln ω
qij
m=1−ω
−ijdω=
ij∈IJ
2
m=1
1
γij qij
mlnqij
m−1−
ij∈IJ
2
m=1
1
γij qij
mijm,
(26)
and then change the mode choice index from binary (2) to multinomial (Mij) as follows:
ij∈IJ
2
m=1
1
γij qij
mlnqij
m−1−
ij∈IJ
2
m=1
1
γij qij
mijm →
ij∈IJ
m∈Mij
1
γij qij
mlnqij
m−1
−
ij∈IJ
m∈Mij
1
γij qij
mijm, (27)
whereqij
misnowthe traveldemand of modem∈Mij between O–D pairij and ijm is theexogenous
modal attractiveness of mode mbetween O–D pair ij. Note that γij is now the MNL model
parameter for the mode choice. By incorporating Equation (27) into Equation (11), we have a
combined MNL mode choice and PSW-SUE route choice (or MNL-PSW-SUE) model, i.e.
minZ=Z1+Z2+Z3+Z4+Z5
=
a∈Ava=ij∈IJ m∈Mij r∈Rijm fij
mrδij
amr
0lnτa(ω) dω+
ij∈IJ
m∈Mij
r∈Rijm
1
βijm fij
mr lnfij
mr −1
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Transportmetrica A: Transport Science 171
−
ij∈IJ
m∈Mij
qij
mijm +
ij∈IJ
m∈Mij 1
γij −1
βijm qij
mlnqij
m−1
−
ij∈IJ
m∈Mij
r∈Rijm
1
βijm fij
mr lnij
mr (28)
s.t.
r∈Rijm
fij
mr =qij
m∀m∈Mij,ij ∈IJ, (29)
m∈Mij
qij
m=qij ∀ij ∈IJ, (30)
fij
mr ≥0, qij
m≥0∀r∈Rijm,m∈Mij,ij ∈IJ, (31)
where fij
mr is the flow on route rof mode mbetween O–D pair ij,βijm is the weibit parameter of
mode mbetween O–D pair ij,ij
mr is the path-size factor on route rof mode mbetween O–D pair
ij and δij
amr is equal to 1 if link ais on route rof mode mbetween O–D pair ij and 0 otherwise.
Equations (29) and (30) are the flow/travel demand conservation constraints, and Equation (31)
is the non-negativity constraint on the decision variables (i.e. route flows and mode-specific O–D
flows).
Proposition 3The MP formulation in Equations (28)through (31)has the MNL mode choice
solution and the PSW route choice solution.
Proof The Lagrangian of this problem can be expressed as
L=Z+
ij∈IJ
m∈Mij
λijm ⎛
⎝qij
m−
r∈Rijm
fij
mr⎞
⎠+
ij∈IJ
φij ⎛
⎝qij −
m∈Mij
qij
m⎞
⎠, (32)
where λijm and φij are Lagrangian multipliers corresponding to the constraints in Equations (29)
and (30), respectively. Following the same principle of Proposition 1, we have the PSW route
choice solution, and λijm can be expressed as a function of the mode-specific O–D demand of
mode mand the logarithmic PSW EPC of mode m(¯μijm), i.e.
λijm =1
βijm lnqij
m−1
βijm ln
r∈Rijm
ij
mr gij
mr−βijm =1
βijm lnqij
m+¯μijm ∀m∈Mij ,ij ∈IJ. (33)
By setting the partial derivative of the Lagrangian w.r.t. qij
mto zero, we have
−ijm +1
γij −1
βijm lnqij
m+λijm −φij =0∀m∈Mij,ij ∈IJ. (34)
According to Equation (33), Equation (34) can be restated as
qij
m=exp γij φij +γij ijm −¯μijm ∀m∈Mij,ij ∈IJ. (35)
Then, qij can be expressed as
qij =
m∈Mij
qij
m=exp γij φij
m∈Mij
exp γij ijm −¯μijm ∀ij ∈IJ. (36)
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172 S. Kitthamkesorn et al.
Incorporating Equations (35) and (36) gives the MNL mode choice probability, i.e.
Pij
m=qij
m
qij =exp γij ijm −¯μijm
n∈Mij exp γij ijn −¯μijn ∀m∈Mij,ij ∈IJ. (37)
Therefore, the MP formulation in Equations (28) through (31) has the MNL mode choice solution
and PSW route choice solution. This completes the proof.
Proposition 4The solution of MNL-PSW-SUE model is unique.
Proof Following the same principle of Proposition 2,Z1and Z2are positive semi-definite and
positive definite, respectively. Since ijm and ij
mr are flow-independent, the second derivatives
of Z3and Z5are zero. Assuming that βijm >γ
ij, the second derivative of Z4can be expressed as
∂2Z4
∂qij
m∂qrs
n
=⎧
⎨
⎩ 1
γij −1
βijm !1
qij
m>0if(ij,m)=(rs,n),
0 otherwise. (38)
Thus, Z4is positive definite, and the objective function in Equation (28) is strictly convex w.r.t.
the route flows and mode-specific O–D demands. Therefore, the equilibrium route flows and
mode-specific O–D demands are unique. This completes the proof.
4. Solution algorithm
In this section, we adopt the path-based partial linearisation algorithm for solving the PSW-
SUE-ED and MNL-PSW-SUE models. The partial linearisation method belongs to the descent
direction algorithm for solving continuous optimisation problems (Patriksson 1994). A search
direction and a stepsize determination are iteratively performed to obtain a new iterative solution
untilsome convergencecriterionis satisfied. The partial linearisationmethod hasbeen adoptedfor
solvingvariousSUE models includingthe MNL-SUE model(Damberg,Lundgren, and Patriksson
1996;Chen et al. 2013), C-logit-SUE model (Chen, Kasikitwiwat, and Ji 2003), PSL SUE model
(Chen et al. 2012), CNL SUE model (Bekhor, Toledo, and Prashker 2008;Bekhor, Toledo, and
Reznikova 2008), paired combinatorial logit (PCL) SUE model (Chen et al. 2014) and PSW
SUE model (Kitthamkesorn and Chen 2013). All these are for the FD, except Xu and Chen
(2013) applied to the C-logit-SUE model with ED. In the partial linearisation method, the search
direction is obtained by solving a partial linearised subproblem. This can be done by updating the
link costs and route costs, computing the O–D demands and PSW route choice probabilities, and
assigning the auxiliary O–D demands (or the mode-specific O–D demands) and auxiliary route
flows according to the ED function for travel choice (or the mode-specific O–D demands for mode
choice) and the PSW probabilities for route choice, respectively. Note that the objective function
includes two integral terms and two entropy terms for the PSW-SUE-ED model (or one integral
term, two entropy terms for the MNL-PSW-SUE model). Thus, it is computationally expensive to
evaluate the complex objective function as well as its derivatives. In this study, we adopt the SRA
scheme recently proposed by Liu, He, and He (2009) to determine a stepsize without the need
to evaluate the complex objective and/or its derivatives. Compared to the method of successive
averages (MSA) stepsize scheme, the SRA stepsize scheme makes use of the residual error and
the stepsize in the current iteration to smartly determine a stepsize for the next iteration. The SRA
scheme can be shown to satisfy the convergence condition (see Robbins and Monro 1951;Blum
1954;Liu, He, and He 2009 for details). Detailed implementation steps of the partial linearisation
method for solving the PSW-SUE-ED and MNL-PSW-SUE models are provided below:
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Transportmetrica A: Transport Science 173
Step 0: Initialisation.
•Set iteration counter n=0;
•Calculate the path-size factor: ij
r=a∈ϒr(la/Lij
r)(1/k∈Rij δij
ak);
•Update link travel costs τ(0)
a=τa(0)and route travel costs gij(0)
r="a∈ϒrτ(0)
a;
•Calculate initial O–D demands (Dij() for the PSW-SUE-ED model and MNL mode choice
for the MNL-PSW-SUE model);
q(0)
ij =Dij ¯μij=Dij ⎛
⎝−1
βij ln
r∈Rij
ij
rgij(0)
r−βij ⎞
⎠
Perform the PSW loading to obtain initial route flows and link flows;
fij(0)
r=q(0)
ij
ij
rgij(0)
r−βij
k∈Rij ij
kgij(0)
k−βij ,v(0)
a=
ij∈IJ
r∈Rij
fij(0)
rδij
ar;
Step 1: Direction Finding.
•Increment iteration counter n:=n+1;
•Update link and route travel costs: τ(n)
a=τav(n−1)
a,gij(n)
r="a∈ϒrτ(n)
a;
•Calculate auxiliary O–D demands:
˜q(n)
ij =Dij(¯μij )=Dij ⎛
⎝−1
βij ln
r∈Rij
ij
rgij(n)
r−βij ⎞
⎠;
Perform the PSW loading to obtain auxiliary route flows:
˜
fij(n)
r=˜q(n)
ij
ij
rgij(n)
r−βij
k∈Rij ij
kgij(n)
k−βij ,˜v(n)
a=
ij∈IJ
r∈Rij
˜
fij(n)
rδij
ar;
•Search direction ˜
v(n)−v(n−1),˜
f(n)−f(n−1), and ˜
q(n)−q(n−1).
Step 2: Line Search.
•Use the SRA scheme to obtain α(n);
α(n)=1
η(n)
η(n)=⎧
⎨
⎩
η(n−1)+λ1if #
#
#
˜
f(n)−f(n−1)#
#
#≥#
#
#
˜
f(n−1)−f(n−2)#
#
#
η(n−1)+λ2otherwise
where λ1>1 and λ2<1.
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174 S. Kitthamkesorn et al.
Step 3: Move.
•f(n)=f(n−1)+α(n)˜
f(n)−f(n−1);v(n)=v(n−1)+α(n)˜
v(n)−v(n−1);
q(n)=q(n−1)+α(n)˜
q(n)−q(n−1).
Step 4: Convergence Test.
•If RMSE =$#
#
#
˜
f(n)−f(n−1)#
#
#2/|R|≤εholds, terminate, where |R|is the number of routes;
otherwise, go to Step 1.
Remark In Step 2, depending on the residual error relationship between the previous and the
current iterations, the decreasing speed of stepsize is different which is controlled by either λ1or
λ2. Also note that the stepsize sequence from the SRA scheme is strictly decreasing, satisfying
the convergence conditions:
α(n)>0,
∞
n=1
α(n)=∞ and lim
n→∞ α(n)=0or
∞
n=1α(n)2<∞
(Robbins and Monro 1951;Blum 1954;Liu, He, and He 2009).
5. Numerical example
In this section, we provide three examples to present the features of the proposed models. Exam-
ple 1 is the two-route network used to investigate the effect of different trip lengths. Example 2
is the loop–hole network adopted to investigate the effect of both route overlapping and route-
specific perception variance problems. Lastly, Example 3 uses the Winnipeg network to show
the algorithmic performance along with an application of bi-modal network considering both
motorised mode (i.e. auto) and non-motorised mode (i.e. bicycle) as a case study. Without loss
of generality, all routes are assumed to have the same CV ϑij
r=0.3 (i.e. βij =3.7 for all O–D
pairs, see Equation (10)) unless specified otherwise. A CV of 0.3 is typically used in testing the
statistical significance of parameter estimates, corresponding to the t-statistics of 3.33 (Zhao and
Kockelman 2002). laand Lij
rused in the path-size factor ij
rare set to the link free-flow travel
cost and route free-flow travel cost, respectively. The ED function (in vehicle per hour; vph) is
assumed to be a function of the logarithmic EPC, i.e.
qij =100exp −0.05 ׯμij∀ij ∈IJ. (39)
5.1. Example 1: two-route networks
This example modifies the two-route networks in Figure 1to incorporate the congestion effect.
The travel cost of each route includes a flow-dependent cost fij
r/100 as shown in Table 2. The
equilibrium solutions produced by the PSW-SUE-ED model are compared with the MNL-SUE-
ED and MNL-SUE-ED with scaling (or MNLs-SUE-ED) models as presented in Table 3. The
MNL-SUE-ED model has the dispersion parameter θof 0.1, and the MNLs-SUE-ED model has
a scaling θwith ϑij
r=0.3 using the lowest uncongested travel cost route (see Chen et al. 2012).
Theresults show that all models giveasmaller travel demand asthe overalltriplength increases.
The MNL-SUE-ED model produces the same route choice probabilities for both short and long
networks. This is because it cannot handle the heterogeneous perception variance w.r.t. different
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Transportmetrica A: Transport Science 175
Table 2. Flow-dependent route travel cost
for the two-route networks.
Network Upper route Lower route
Short 10 +fij
u/100 5 +fij
l/100
Long 125 +fij
u/100 120 +fij
l/100
Table 3. Travel demand and flow allocation for the two-route networks.
Probability
Model Network Demand Upper route Lower route
MNL-SUE-ED Short (θ=0.1) 99.79 0.38 0.62
Long (θ=0.1) 78.86 0.38 0.62
MNLs-SUE-ED Short (θ=0.86) 91.94 0.03 0.97
Long (θ=0.04) 79.40 0.46 0.54
PSW-SUE-ED Short 91.72 0.11 0.88
Long 79.36 0.47 0.53
Note: The italic values are used to show that the MNL model gives the same results for both
short and long networks.
trip lengths. Meanwhile, the MNLs-SUE-ED and PSW-SUE-ED models assign a smaller amount
of flows on the lower route as the overall trip length increases. The MNLs-SUE-ED model gives
a larger O–D demand. It further gives a higher probability of choosing the lower route, especially
on the short network. This is because the scaling technique still assumes the same and fixed
perception variance of each route within the same O–D pair. Instead, the PSW-SUE-ED model
provides flexibility in capturing the route-specific perception variances.
5.2. Example 2: loop–hole network
Inthis example,a loop–holenetwork inFigure 5is adopted to consider both route overlappingand
route-specific perception variance problems. This loop–hole network has three routes, and each
route has the same capacity of 100 vph. The two upper routes have a free-flow travel time (FFTT)
of 30min with an overlapping section of x. The lower route is truly independent with a FFTT of
y. The standard Bureau of Public Road function is adopted to represent the flow-dependent link
travel time, i.e.
ta=t0
a%1+0.15va
ca4&, (40)
where t0
ais the FFTT on link a, and cais the capacity on link a. Without loss of generality, the
travel cost is assumed to be an exponential function (Hensher and Truong 1985;Mirchandani and
Soroush 1987;Polak 1987), i.e.
τa=exp (0.05ta). (41)
5.2.1. Effect of route overlapping
The effect of route overlapping is investigated first where all routes have the same trip length
(i.e. y=30). The results produced by the PSW-SUE-ED model would be compared with the
MNLs-SUE-ED model under the same cost configuration in Equation (41). As expected, the
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176 S. Kitthamkesorn et al.
Figure 5. Loop–hole network and its characteristics.
Figure 6. Probability of choosing the lower route for the loop–hole network (y=30).
(a) (b)
Figure 7. EPC and travel demand patterns for the loop–hole network (y=30). (a) Logarithmic EPC and
(b) O–D demand.
MNLs-SUE-ED model has difficulty in handling the overlapping problem as shown in Figure 6.
As xincreases, it assigns more flows on the two upper routes (hence, a smaller amount of flows
on the lower route), while the PSW-SUE-ED model assigns a relatively larger amount of flows
on the lower route as xincreases. At x=30, only two routes with equal trip length exist. The
PSW-SUE-ED model assigns an equal amount of flows on both routes, while the MNLs-SUE-ED
model only assigns 36% of the total flows to the lower route as it still considers all three routes
as feasible routes in the loop–hole network.
However, the logarithmic PSW EPC is larger as shown in Figure 7(a). This is because the
logarithmic PSW EPC incorporates the route overlapping problem through the path-size factor
ij
r. As the overlapping section increases, ij
rof the upper routes decreases. As a result, the
PSW-SUE-ED model produces a lower O–D demand level, especially for a longer overlapping
section x, as shown in Figure 7(b).
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Transportmetrica A: Transport Science 177
Figure 8. Traffic flow patterns for the loop–hole network (y=32).
Figure 9. EPC and travel demand patterns for the loop–hole network (y=32). (a) O–D demand and (b)
logarithmic EPC.
5.2.2. Effect of both route overlapping and heterogeneous perception variance problems
To consider both route overlapping and heterogeneous perception variance problems, yis set at
32min. As such, the FFTT of the lower route is longer than that of the upper routes by 2min.
The results show that the PSW-SUE-ED model can produce a comparable flow and O–D demand
patterns to the MNP-SUE model with ED (or MNP-SUE-ED) as shown in Figure 8. The MNP-
SUE-ED and PSW-SUE-ED models give a smaller O–D demand, especially for a larger xvalue
as shown in Figure 9(a). This is because both models account for both route overlapping and
route-specific perception variance problems.As such, their logarithmic EPC is larger than that of
the MNLs-SUE-ED model as shown in Figure 9(b).
5.3. Example 3: Winnipeg network
This example uses the Winnipeg network shown in Figure 10 as a real-case study. This network
consists of 154 zones, 2535 links and 4345 O–D pairs. The network topology, link characteristics,
and O–D demands can be found in Emme/2 software (INRO Consultants 1999). For comparison
purposes, a behavioural working route set generated by Bekhor,Toledo, and Prashker (2008) was
adopted. For the Winnipeg network, the routes were generated using a combination of the link
elimination method of Azevedo et al. (1993) and the penalty method of De La Barra, Perez, and
Anez (1993), with a penalty of 5% on the travel times on the shortest path links. Only acyclic
pathswere considered in these methods.Inspection on the generated routesfor different O–D pairs
reveals that this route set includes both completely disjointed routes and very similar routes due
to the methods (i.e. link elimination and link penalty) used to generate the routes. In this working
route set, there are 174,491 routes with an average of 40.1 routes for each O–D pair, and the max-
imum number of routes generated for any O–D pair is 50. The total travel demand is 54,459 trips,
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178 S. Kitthamkesorn et al.
Figure 10. Winnipeg network with bike lanes.
and deterministic user equilibrium (UE) assignment results show that this is a moderately con-
gested network (the average volume to capacity, V/C, ratio less than 0.7). This route set was used
by Bekhor, Toledo, and Prashker (2008) to examine the effects of choice set size and route choice
models on path-based traffic assignment algorithms. It should be noted that some routes are quite
circuitous due to the link elimination method used to find disjoint routes (because of the removal
of all links belonging to the shortest path). For the O–D pairs with the maximum 50 routes, many
are not direct routes that meander from the origins to the destinations (see Bekhor, Toledo, and
Prashker 2006 for an example of the 50 routes generated for a given O–D pair plotted in geo-
graphical information systems). For the UE solution with FD, only a subset of routes are used
according to the Wardrop’s first principle (i.e. all used routes have route costs equal to the
minimum O–D cost, while unused routes have route costs higher than or equal to the mini-
mum O–D cost). As for the SUE models (e.g. MNL, C-logit, CNL, generalized nested logit,
PCL, MNW, PSW and MNP), all routes in the route set are used under the SUE conditions.
However, lower cost routes are preferred. In general, the SUE models would predict a more dis-
persed traffic pattern with a higher utilisation of network capacity due to more used routes and
more dissimilar routes with potentially longer route lengths. In addition, the SUE models that
can handle both route overlapping and non-identical perception variance problems (e.g. PSW
and MNP) typically would produce an even more dispersed traffic pattern and a larger num-
ber of links with higher V/Cratios due to the need to account for routes with overlapping and
larger perception variances for longer trip lengths. Note that this working route set has been
adopted in many studies including the length-based and congestion-based C-logit-SUE models
in Zhou, Chen, and Behkor (2012) and Xu et al. (2012), the CNL-SUE model in Bekhor, Toledo,
and Prashker (2008) and Bekhor, Toledo, and Reznikova (2008), the PCL-SUE model in Chen
et al. (2014), the scaling effect and route overlapping problem in logit-based SUE models in Chen
et al. (2012), and both additive and nonadditive traffic equilibrium model in Chen et al. (2013) and
Chen, Zhou, and Xu (2012). Comparisons of the UE and various SUE models with FD have been
reportedin theabovestudies. The focus of this example is to examinethe algorithmic performance
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Transportmetrica A: Transport Science 179
Table 4. Sensitivity analysis of λ1and λ2in solving the PSW-SUE-ED model.
λ1
No. of iterations 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00
λ20.01 562 574 572 572 548 551 553 583 594 536 531
0.05 538 547 562 594 623 627 633 574 587 590 592
0.10 529 514 548 565 579 598 606 597 623 675 677
0.15 1146 1147 1251 1273 1362 1421 1455 1427 1414 1578 1596
0.20 2015 2121 2129 2245 2387 2451 2477 2564 2679 2796 2948
Note: The bold/italic value indicated the best value in terms of the minimum number of iterations.
Figure 11. Convergence characteristics of the path-based partial linearisation algorithm with two line
search schemes.
Table 5. Computational efforts of the path-based partial linearisation
algorithm with two line search schemes.
Algorithm No. of iterations CPU time (s) CPU time/iteration (s)
MSA 3217 569 0.18
SRA 514 102 0.20
of the two parameters used in SRA scheme and to develop an application of the MNL-PSW-SUE
model in a bi-modal network with motorised and non-motorised modes.
5.3.1. Algorithmic performance
The algorithm is coded in Intel Visual FORTRAN 6.6 and runs on a 3.80GHz processor with
2.00GB of RAM. The stopping criterion εis set to 10−8. We first analyse the sensitivity of the
parameters λ1and λ2in the algorithmic performance in Table 4. In this case, it seems that λ2has
a larger impact on the number of required iterations. Without loss of generality, we will use λ1
and λ2(1.55, 0.10) which gives the best performance for further analysis.
The convergence characteristics of the path-based partial linearisation algorithm with two line
search schemes are shown in Figure 11. It appears that the SRA scheme outperforms the MSA
scheme with a stepsize of 1/n. The SRA scheme requires 514 iterations to reach the desired level
of accuracy, while the MSA scheme requires more than 3000 iterations as presented in Table 5.
In terms of average computational effort required in each iteration, the SRA scheme needs only
0.02s per iteration more than the MSA scheme.
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180 S. Kitthamkesorn et al.
5.3.2. Application in a bi-modal network with motorised and non-motorised modes
To construct a bi-modal network case, we incorporate the cycling routes obtained from the Win-
nipeg city’s website (www.winnipeg.ca) to the Winnipeg network. The dedicated bike lanes are
assumedto be locatedalong some streets as shown in Figure 10.The results producedby the MNL-
PSW-SUE model will be compared with the MNL-MNLs-SUE model. For a fair comparison, we
continue to use the same cost configuration in Equation (41). Both models use the logarithmic
EPC for the mode choice level with γij =0.5 and ijm =0 for the auto and bike modes. For the
bike routes, we use the behavioural route set from Bekhor, Toledo, and Prashker (2008) with a
travel distance less than 8 miles, which is the longest distance for the majority of cyclists in the
USA (US Census Bureau 2000). The bike travel time is assumed to be flow-independent deter-
mined by its trip length and an average speed of 10mph on a dedicated bike lane (Jensen et al.
2010). With this setting, we have 421 O–D pairs (out of 4345 O–D pairs) with both auto and bike
modes available.
We compare the flow allocations at the disaggregate and aggregate levels.At the disaggregate
level, we examine the route choice and mode choice probabilities produced by the MNL-MNLs-
SUE and MNL-PSW-SUE models. For demonstration purposes, we use O–D pairs (3, 147) and
(74, 60) to respectively represent a short O–D pair and a long O–D pair that include both auto
and bike modes. The route choice and mode choice probabilities shown in Figure 12 are under
the equilibrium solutions. Five routes with relatively large auto flow proportions in each O–D
pair are selected, where the first three routes (i.e. routes 1, 2 and 3) also include the bike routes.
Recall that the MNL-MNLs-SUE model can partially handle the different trip lengths through the
scaled dispersion parameter whereas the MNL-PSW-SUE model can simultaneously handle both
Figure 12. Comparison of route choice and mode choice probabilities of O–D pairs (3, 147) and (74, 60).
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Transportmetrica A: Transport Science 181
route overlapping and non-identical perception variances of different trip lengths. Therefore, the
differentroutechoice probabilitiescan be expected.The MNL-MNLs-SUE model seems toassign
more flows to the shortest routes. This is according to the same and fixed perception variance for
all routes between an O–D pair. Meanwhile, the MNL-PSW-SUE model seems to produce a more
dispersed flow pattern. It gives a lower route choice probability to routes that have couplings with
other routes. At the same time, it assigns a larger amount of flows to the longer routes compared
to the MNL-MNLs-SUE model. As a result, by combining the effect of route overlapping and
non-identical perception variances of different trip lengths, the MNL-PSW-SUE model produces
a higher share for the bike mode, especially for the shorter O–D pair (i.e. O–D pair (3147)).
Atthe aggregate level, we examinetheeffectof route overlappingandheterogeneous perception
variance problems on the link flow patterns and mode shares.As expected, the link flow difference
between the MNL-MNLs-SUE and MNL-PSW-SUE models can be found mostly in the central
business district (CBD) area as shown in Figure 13. This is because this area consists of more
overlapping between routes and different trip lengths.As such, the MNL-PSW-SUE model gives
a higher bike mode share than that produced by the MNL-MNLs-SUE model in the CBD area as
shown in Table 6.
Figure 13. Link flow difference between MNL-MNLs-SUE and MNL-PSW-SUE models.
Table 6. Mode share comparison between MNL-MNLs-SUE and MNL-PSW-SUE models.
MNL-MNLs-SUE MNL-PSW-SUE
Mode All areas CBD Non-CBD All areas CBD Non-CBD
Auto 99.63% 99.48% 99.80% 99.26% 98.82% 99.75%
(54,258) (28,756) (25,501) (54,056) (28,568) (25,488)
Bike 0.37% 0.52% 0.20% 0.74% 1.18% 0.25%
(201) (151) (50) (403) (340) (63)
Note: (), number of trips.
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182 S. Kitthamkesorn et al.
6. Conclusions
This study provided a new SUE model with ED that explicitly considers the heterogeneous
perception variance and route overlapping under congested conditions. This new PSW SUE-
ED (or PSW-SUE-ED) model was derived based on the Weibull distribution and a path-size
factor, similar to the PSW-SUE model for the FD case developed by Kitthamkesorn and Chen
(2013) to handle the two drawbacks of the MNL derived from the IID Gumbel variates (i.e. the
route overlapping problem and the identical perception variance problem). The core lies in the
derivation of the logarithmic EPC of theWeibull distribution (Kitthamkesorn and Chen 2014)as
the network LOS used in the ED function to determine the travel demands and SUE flows.The log
EPC explicitly captures the route overlapping and non-identical perception variance problems in
the SUE assignment, and avoids the bias caused by the two known limitations of the MNL-SUE
model in estimating the travel demands. In addition, an application of the PSW-SUE-ED model
was developed to consider both mode choice and route choice as a CMSTA problem. It was
demonstrated with a case study in a bimodal network with motorised and non-motorised modes
using the Winnipeg network. Through the numerical examples, the results revealed that
•the PSW-SUE-ED model can handle the route-specific perception variance better than the
MNL-SUE with scaling technique (Chen et al. 2012), which still assumes the same and fixed
perception variance for all routes connecting an O–D pair,
•the proposed PSW-SUE-ED model can capture both the route-specific perception variance
and route overlapping as well as travel demands under congested conditions in the similar
way as the MNP-SUE-ED model and
•the application of MNL-PSW-SUE as a CMSTA problem in the bimodalWinnipeg network
gave a larger non-motorised mode share than the ordinary logit-based CMSTA model due to
the logarithmic EPC which accounts for both route overlapping and heterogeneous percep-
tion variance. The result difference was more pronounced in the downtown area with more
route overlaps and different trip lengths.
For future research, it would be of interest to: (1) further develop the PSW-SUE-ED model to
include a non-zero location parameter, (2) consider flow-dependent attributes to account for route
overlapping (Zhou, Chen, and Behkor 2012), (3) extend to other travel choice dimensions to
include trip generation, trip distribution, modal split and traffic assignment (Zhou, Chen, and
Wong 2009;Yang, Chen, and Xu 2013), (4) incorporate various side constraints (e.g. physical
and environmental constraints) to improve the realism of the network equilibrium models (Chen,
Zhou, and Ryu 2011;Liu, Meng, and Wang 2014;Xu, Chen, and Cheng 2014;Ryu et al. 2014b)
and (5) explore the projection methods for solving the PSW-SUE-ED model (Chen, Lo, and Yang
2001;Chen, Lee, and Jayakrishnan 2002;Chen et al. 2010;Chen, Zhou, and Xu 2012;Ryu, Chen,
and Choi 2014;Xu et al. 2012).
Acknowledgements
The authors are grateful to three anonymous referees and Professor William Lam for their constructive
comments and suggestions to improve the quality and clarity of the paper.
Funding
The authors would like to acknowledge the financial support from the Royal Thai Government Scholarship
from Thailand, the China Scholarship Council Fellowship from China, the Oriental Scholar Professorship
Program sponsored by the Shanghai Ministry of Education in China to Tongji University, Shanghai, China,
and the Utah Transportation Center at Utah State University. These supports are gratefully acknowledged.
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Transportmetrica A: Transport Science 183
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