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A reliability-based traffic assignment model for multi-modal
transport network under demand uncertainty
Xiao Fu*, William H. K. Lam and Bi Yu Chen
Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
SUMMARY
In densely populated and congested urban areas, the travel times in congested multi-modal transport net-
works are generally varied and stochastic in practice. These stochastic travel times may be raised from
day-to-day demand fluctuations and would affect travelers’route and mode choice behaviors according to
their different expectations of on-time arrival. In view of these, this paper presents a reliability-based user
equilibrium traffic assignment model for congested multi-modal transport networks under demand uncer-
tainty. The stochastic bus frequency due to the unstable travel time of bus route is explicitly considered.
By the proposed model, travelers’route and mode choice behaviors are intensively explored. In addition,
a stochastic state-augmented multi-modal transport network is adopted in this paper to effectively model
probable transfers and non-linear fare structures. A numerical example is given to illustrate the merits of
the proposed model. Copyright © 2012 John Wiley & Sons, Ltd.
KEY WORDS: multi-modal transport network; travel time reliability; demand uncertainty; traffic assignment
model; route and mode choice behaviors
1. INTRODUCTION
In metropolitan areas, travel times in multi-modal transport networks are generally varied from day to
day because of random demand fluctuations and supply degradations. Many empirical studies have
found that travel time uncertainty has significant impacts on travelers’route and mode choice
behaviors [1–4]. These empirical studies revealed that travelers indeed consider the travel time
uncertainty as a risk for their travels. To reduce the risk of late arrival, travelers may have more
concerns on the probability that a trip can be successfully fulfilled within a given travel time, referred
as travel time reliability in the literature. Therefore, travelers’concerns on travel time reliability should
be explicitly considered in the travel behavior modeling.
In view of this, Lo et al. [5] extended the well-known user equilibrium (UE) model [6] to reliability-
based user equilibrium (RUE) model by using a concept of travel time budget. The travel time budget
is the summation of mean route travel time and a safety margin. The safety margin is an extra time
added by a traveler to achieve his or her desired probability of on-time arrival. Under RUE status,
travelers choose the optimal route with minimum travel time budget instead of expected travel time
in the UE model.
Following this RUE framework, many researchers have been focusing their attentions to travel
behavior modeling in either road or transit networks. In road networks, Shao et al. [7] proposed a
RUE model to investigate the effects of demand uncertainty. Siu and Lo [8] developed a RUE model
considering both demand and supply uncertainties. Zhou and Chen [9] compared three RUE models
under demand uncertainty. Reliability-based stochastic user equilibrium (RSUE) models were further
developed to take account of travelers’perception errors [10,11].
*Correspondence to: Xiao Fu, Department of Civil and Structural Engineering, The Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong. E-mail: xiao.fu@polyu.edu.hk
Copyright © 2012 John Wiley & Sons, Ltd.
JOURNAL OF ADVANCED TRANSPORTATION
J. Adv. Transp. 2014; 48:66–85
Published online 13 March 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/atr.202
In transit networks, Yang and Lam [12] presented a RSUE model in the congested network with
unreliable transit services. Zhang et al. [13] developed a schedule-based RSUE model to investigate
travel choice behaviors, in terms of departure time and route choices, in transit networks with demand
and supply uncertainties.
However, little effort has been found in modeling travelers’choice behaviors in multi-modal trans-
port networks under uncertainties. In reality, there is a practical need for providing a reliability-based
traffic assignment model in the multi-modal networks. There are two main reasons. First, public transit
networks and road networks interact with each other, especially during rush hours. A regular frequency
of bus service may be disrupted by traffic congestion that occurs in road networks. Using either transit
or road networks in modeling cannot demonstrate the interactions between public transit and road traffic.
Thus, a multi-modal network model is practically required. Second, combine-mode trips have increased
in magnitude in recent years. Travelers may fulfill trips by autos, or by public transit, or by park and ride
for their daily travels. Therefore, exploring travelers’route and mode choice behaviors in multi-modal
networks considering travel time reliability has become an important issue.
In view of these, this paper presents a RUE model in multi-modal networks. To capture the effects of
demand uncertainty, we formulated passenger flows and generalized travel times of different transport
modes as random variables. Stochastic bus frequency derived from the variability of road travel time is
explicitly considered, and the derivations of mean and standard deviation (SD) of link and route travel
times are provided. Additionally, unrealistic transfers are avoided, and the difficulty of non-linear fare
structures is tackled by using a state-augmented multi-modal (SAM) network developed by Lo et al. [14].
The outline of this paper is as follows. The model assumptions and the network representation are
described in Section 2. The distributions of passenger flow and travel time are derived in Section 3.
The RUE model formulation and the solution algorithm are presented in Section 4. A numerical
example illustrating the proposed model is provided in Section 5. Finally, conclusions and recommen-
dations for further studies are given in Section 6.
2. MODEL ASSUMPTIONS AND NETWORK REPRESENTATION
2.1. Model assumptions
To facilitate the presentation of the essential ideas without the loss of generality, we made the following
basic assumptions in this paper.
A1 Origin–destination (OD) demands are assumed to follow independent normal distributions similar
to the assumptions made in previous studies [15–17].
A2 Route flows are assumed to be mutually independent and follow the same type of statistical
distribution as that in the OD demand distribution. The coefficient of variation (CV) of route
flow is assumed to be equal to that of OD demand distribution as the works of Shao et al. [7,10].
A3 Link and route travel times are assumed to be mutually independent and follow normal
distributions [7,10].
A4 The multi-modal network model investigated in this study falls within the category of static model
for long-term planning at the strategic level. Therefore, it is assumed that all travelers in the multi-
modal network can have perfect knowledge toward traffic condition on the basis of their past
experiences.
A5 All travelers can get on the buses or trains, that is, no vehicle capacity constraint.
2.2. Multi-modal transport network
In this study, the SAM network, proposed by Lo et al. [14], is adopted to avoid unrealistic transfers and
represent non-linear fare structures involved in multi-modal networks.
Consider a multi-modal transport network M=(U,V), where U={i} and V={v} are, respectively,
the set of physical nodes and the set of physical links. The multi-modal network Mcan be divided into
wsub-networks M
b
=(U
b
,V
b
), b2B,U
b
⊆U,V
b
⊆V, and w=|B|, where b2Bis a specified transport
mode, and U
b
and V
b
, respectively, are the set of nodes and the set of links associated with the
67
RELIABILITY-BASED MULTI-MODAL ASSIGNMENT MODEL
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
sub-network M
b
. In this study, three transport modes (subway, auto, and bus), respectively denoted by
b
1
,b
2
, and b
3
, are considered. These three sub-networks are combined and represented by a strongly
connected graph G=(N,A) through a state-augmentation approach [18], where Nis a set of nodes
and Ais a set of links. The resultant network Gis termed the SAM network.
In the SAM network, each node is described as (i,s,n, and l), where iindicates the physical
location of the node, sis the transfer state used to model probable transfers, nis the number of
transfers that has been made by a traveler, and lis the alight or aboard indicator. The value of
lequals to 1 (0), indicating that a traveler is at the beginning (end) of a direct in-vehicle link. Specifically,
each transfer state s2Sassociates with a transport modal usage (s)2Band a set of probable transfers
x(s)⊆S. If travelers are at state s, it indicates that these travelers are using mode (s), and they can only
transfer to any state in x(s).
Links in the SAM network are divided into two categories, that is, A=A
t
∪A
d
,whereA
t
is the set of
transfer links between modes and A
d
is the set of direct in-vehicle links that are made up of physical links.
Each transfer link a
t
2A
t
is constructed according to the probable transfer states. Each in-vehicle link
aij
sn 2Adrepresents a direct in-vehicle movement from location ito location jwith transfer state sas
its nth mode in the trip. It should be noted that a direct in-vehicle link may consist of more than one
consecutive physical links. Inthis way, non-linear fares can be directly represented by node-to-node basis.
3. FORMULATION OF PASSENGER FLOW AND TRAVEL TIME DISTRIBUTIONS
3.1. Passenger flow distribution
For notation consistency, the capital letters used throughout this paper represent random variables, and
the lowercase letters represent deterministic variables. Following the model assumption A1, the travel
demand between OD pair rs (denoted as Q
rs
) is a random variable following a normal distribution,
Qrs ¼qrs þe;(1)
where q
rs
is the mean demand, E[Q
rs
]=q
rs
;eis the random term, E[e] = 0. Let srs
qbe the SD of the
OD demand:
srs
q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
var Qrs
½
p¼ffiffiffiffiffiffiffiffiffiffiffi
var e½
p(2)
The CV of the travel demand between OD pair rs (denoted as cv
rs
) can be expressed as
cvrs ¼srs
q
qrs (3)
Denote the passenger flow along a route p2P
rs
as F
p
. Following the model assumption A2, the flow
conservation can then be expressed by following equations:
Qrs ¼X
p2Prs
Fp(4)
qrs ¼X
p2Prs
fp(5)
sp
f¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Var Fp
q¼fpcvrs 8p2Prs (6)
where f
p
and sp
f, respectively, are the mean and the SD of passenger flow along route p.
Denote the passenger flow on the direct in-vehicle link aij
sn as Fij
sn . It can be expressed by the
summation of passenger flows on all routes using this in-vehicle link:
68 X. FU ET AL.
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
Fij
sn ¼X
p2Prs
dp;aij
sn
Fp8aij
sn 2Ad(7)
fij
sn ¼X
p2Prs
dp;aij
sn
fp8aij
sn 2Ad(8)
sij
fsn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Var Fij
sn
q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
p2Prs
Var Fp
dp;aij
sn
s8aij
sn 2Ad(9)
where fij
sn and sij
fsn, respectively, are the mean and the SD of passenger flow on link aij
sn;dp;aij
sn
is the
incidence relationship between in-vehicle link and route; dp;aij
sn
¼1 indicates that the in-vehicle link
aij
sn is on route p, and dp;aij
sn
¼0 otherwise.
The passenger flow on each transfer link a
t
2A
t
is denoted as Fat. It can be calculated by summing
the passenger flows of all routes using this transfer link as
Fat¼X
p2Prs
dp;at
ðÞFp8at2At(10)
fat¼X
p2Prs
dp;at
ðÞfp8at2At(11)
sat
f¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Var Fat
½
p¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
p2Prs
Var Fp
dp;at
ðÞ
s8at2At(12)
where fatand sat
fare the mean and the SD of passenger flow on link a
t
, respectively; d(p,a
t
) is the in-
cidence relationship between transfer link and route; d(p,a
t
) = 1 means that transfer link a
t
is on route
p, and d(p,a
t
) = 0 otherwise.
Let F
v
be the passenger flow of mode bon physical link v2V
b
. It can be expressed as the summation
of passenger flows on all direct in-vehicle links consisting of this physical link:
Fv¼X
aij
sn 2Ad
daij
sn ;v
Fij
sn ¼X
aij
sn 2AdX
p2Prs
daij
sn ;v
dp;aij
sn
Fp8v2Vb; sðÞ¼b(13)
fv¼X
aij
sn 2Ad
daij
sn ;v
fij
sn ¼X
aij
sn 2AdX
p2Prs
daij
sn ;v
dp;aij
sn
fp8v2Vb; s
ðÞ¼b(14)
sv
f¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Var Fv
½
p¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
aij
sn2Ad
Var Fij
sn
daij
sn;v
s
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
aij
sn 2AdX
p2Prs
daij
sn ;v
dp;aij
sn
Var Fp
s8v2Vb; sðÞ¼b(15)
where f
v
and sv
fare the mean and the SD of passenger flow on link v, respectively; daij
sn;v
is the
incidence relationship between in-vehicle link and physical link; daij
sn;v
¼1 indicates that physical
link vis in the in-vehicle link aij
sn and daij
sn;v
¼0 otherwise.
According to the model assumptions A1 and A2, passenger flows of links and routes all follow
normal distributions:
Fij
sn Nf
ij
sn;sij
fsn
2
8aij
sn 2Ad(16)
FatNf
at;sat
f
2
8at2At(17)
FvNf
v;sv
f
2
8v2V(18)
FpNf
p;sp
f
2
8p2Prs (19)
69
RELIABILITY-BASED MULTI-MODAL ASSIGNMENT MODEL
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
3.2. Link travel time distribution
In this section, travel time distributions for physical links, direct in-vehicle links, and transfer links are
derived.
3.2.1. Physical links
In this paper, the concept of generalized travel time is adopted to model crowding discomfort in
vehicles and congestion in road traffic. The generalized travel time of the physical link vis assumed
to be strictly increasing with respect to its link flow F
v
:
Tv¼tvFv
ðÞ (20)
tv¼ET
v
½¼
Zþ1
1
tvxðÞ’vxðÞdx 8v2V(21)
sv
t
2¼Zþ1
1
tvxðÞðÞ
2’vxðÞdx Zþ1
1
tvxðÞ’vxðÞdx
2
8v2V(22)
where T
v
is the generalized physical link travel time; t
v
(.) is the physical link travel time function; ’
v
(.)
is the probability density function of link flow; t
v
and sv
t, respectively, are the mean and the SD of travel
time on physical link v.
3.2.1.1. Subway (mode b
1
). The generalized travel time considering in-vehicle crowding discomfort
[19,20] on physical link vfor mode b
1
can be expressed as
Tv¼Tvt0
v;Fv;hb1;gb1
¼t0
v1þb1
Fv
hb1gb1
k1
!
8v2Vb1(23)
where hb1is the subway vehicle capacity (passengers per vehicle) and gb1is the subway deterministic
frequency (vehicles per hour); t0
vis the free flow travel time of link v;b
1
and k
1
are model parameters.
As indicated earlier, link flows follow normal distributions, so the probability density function of
link flow distributions can be expressed as
’vxðÞ¼ 1
ffiffiffiffiffiffi
2p
psv
f
exp xfv
ðÞ
2
2sv
f
2
0
B
@1
C
A8v2V(24)
Substituting Equations (23) and (24) into Equations (21) and (22), we have re-written the mean and
the SD of physical link travel time in mode b
1
as
tv¼t0
vþt0
v
b1
hb1gb1
ðÞ
k1X
k1
i¼0;i¼even
k1
i
sv
f
i
fv
ðÞ
k1ii1ðÞ!! 8v2Vb1(25)
sv
t¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t0
v
b1
hb1gb1
ðÞ
k1
!
2X
2k1
i¼0;i¼even
2k1
i
sv
f
i
fv
ðÞ
2k1ii1ðÞ!!
X
k1
i¼0;i¼even
k1
i
sv
f
i
fv
ðÞ
k1ii1
ðÞ
!!
!
2
0
B
B
B
B
B
@
1
C
C
C
C
C
A
v
u
u
u
u
u
u
u
u
t8v2Vb1(26)
The detailed derivations of Equations (25) and (26) can be found in Shao et al. [10].
70 X. FU ET AL.
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
3.2.1.2. Auto (mode b
2
). The link travel time considering congestion in road traffic (mode b
2
) can be
modeled by the most widely used Bureau of Public Roads function [21] as
Tv¼Tvt0
v;Xv;kv
¼t0
v1þg1
Xv
kv
k2
!
8v2Vb2(27)
where X
v
is the total traffic volume on road link v;k
v
is the capacity of the road link, and g
1
and k
2
are
parameters. It should be noted that X
v
consists of two components, that is, traffic volume of mode b
2
(auto) and traffic volume of mode b
3
(bus), because they both belong to road traffic. Therefore,
Xv¼Fvb2
e0
eb2þGb3eb38v2V⧹Vb1(28)
where Gb3is the bus frequency; Fvb2is the passenger flow of mode b
2
on road link v;eb2and eb3are
passenger car equivalents for modes b
2
and b
3
;e
0
is the average vehicle occupancy parameter
representing the number of passengers per auto. The first term in Equation (28) represents the traffic
volume of auto, and the second term represents the traffic volume of bus.
In the congested multi-modal transport network, the bus frequency may be not fixed because of the
variability of road travel time. Therefore, in this paper, bus frequency is considered to be a random
variable to model the interactions between bus usage and auto usage. For simplicity, the bus fleet size
s
0
is assumed to be fixed, and Gb3is determined by s
0
and the cycle time of bus route. It is also assumed
that cycle time can be represented by 2T
p
, where T
p
is the one-way travel time of the bus route that
contains the physical link v, and TpeNt
p;sp
t
ðÞ
2
(following model assumption A3 and will be
discussed in Section 3.3). Thus, the stochastic bus frequency Gb3can be calculated as
Gb3¼s0
2Tp
(29)
The mean and the variance of the second term in Equation (28) can be obtained according to Li et al. [22]:
EG
b3eb3
½
¼Es0eb3
2Tp
¼s0eb3
2tp
1þsp
t
ðÞ
2
tp2
! (30)
Var Gb3eb3
½¼Var s0eb3
2Tp
¼s02eb3
2sp
t
ðÞ
2
4tp4(31)
As mentioned in Section 3.1, Fvb2eNf
vb2;svb2
f
2
where fvb2is the mean passenger flow of mode
b
2
on road link vand svb2
fis the SD of passenger flow of mode b
2
on road link v. Assuming that the
traffic volume of bus and auto on the road are mutually independent, the mean and the SD of X
v
(denoted as x
v
and sxv, respectively) can be expressed as follows:
xv¼eb2
e0
fvb2þs0eb3
2tp
1þsp
t
ðÞ
2
tp2
!
8v2V⧹Vb1(32)
sxv¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e2
b2
e2
0
svb2
f
2þs02eb3
2sp
t
ðÞ
2
4tp4
s8v2V⧹Vb1(33)
Then, the mean and the SD of physical link travel time in mode b
2
can be expressed as follows.
tv¼t0
vþt0
v
g1
kv
ðÞ
k2X
k2
i¼0;i¼even
k2
i
sxv
ðÞ
ixv
ðÞ
k2ii1ðÞ!! 8v2Vb2(34)
sv
t¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t0
v
g1
kv
ðÞ
k2
!
2X
2k2
i¼0;i¼even
2k2
i
sxv
ðÞ
ixv
ðÞ
2k2ii1ðÞ!!
X
k2
i¼0;i¼even
k2
i
sxv
ðÞ
ixv
ðÞ
k2ii1ðÞ!!
!
2
0
B
B
B
B
B
@
1
C
C
C
C
C
A
v
u
u
u
u
u
u
u
u
t8v2Vb2(35)
71
RELIABILITY-BASED MULTI-MODAL ASSIGNMENT MODEL
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
3.2.1.3. Bus (mode b
3
). The generalized travel time on physical link vfor mode b
3
can be expressed as
Tv¼Tvt0
v;Fv;hb3;Gb3;Xv;kv
¼t0
v1þb2
Fv
hb3Gb3
k1
þg2
Xv
kv
k2
!
8v2Vb3(36)
where hb3denotes the bus capacity, Gb3is the stochastic bus frequency (referring to Section 3.2.1.2), and
b
2
and g
2
are model parameters. The last two terms in Equation (36) represent the crowding discomfort in
vehicle and the congestion in road traffic.
The mean and the SD of physical link travel time in mode b
3
are as follows:
tv¼t0
vþ2k1b2t0
v
hb3
ðÞ
k1s0
ðÞ
k1
!
X
k1
i¼0;i¼even
k1
i
sv
f
i
fvk1ii1ðÞ!! X
k1
j¼0;j¼even
k1
j
sp
t
jtpk1jj1ðÞ!!
þg2t0
v
kv
ðÞ
k2X
k2
i¼0;i¼even
k2
i
sxv
ðÞ
ixvk2ii1ðÞ!! 8v2Vb3
(37)
sv
t¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2k1b2t0
v
hb3
ðÞ
k1s0
ðÞ
k1
!
2X
2k1
i¼0;i¼even
2k1
i
sv
f
i
fv2k1ii1ðÞ!! X
2k1
j¼0;j¼even
2k1
j
sp
t
jtp2k1jj1ðÞ!!
2k1b2t0
v
hb3
ðÞ
k1s0
ðÞ
k1
!
X
k1
i¼0;i¼even
k1
i
sv
f
i
fvk1ii1ðÞ!! X
k1
j¼0;j¼even
k1
j
sp
t
jtpk1jj1ðÞ!!
" #2
þg2
ðÞ
2t0
v
2
kv
ðÞ
2k2X
2k2
i¼0;i¼even
2k2
i
sxv
ðÞ
ixv2k2ii1ðÞ!! X
k2
i¼0;i¼even
k2
i
sxv
ðÞ
ixvk2ii1ðÞ!!
!
2
2
43
5
v
u
u
u
u
u
u
u
u
u
u
u
u
u
u
t
8v2Vb3(38)
The detailed manipulations on deducing Equations (37) and (38) are given in Appendix A.
3.2.2. In-vehicle links
The in-vehicle link travel time is the summation of relevant physical link travel times.
As the means and the SDs of the physical link travel time of the three modes are given in Section
3.2.1, the in-vehicle link travel time (denoted as Tij
sn) of each mode can be obtained as
Tij
sn ¼X
v2Vb
Tvdaij
sn;v
8aij
sn 2Ad;sðÞ¼b(39)
Following model assumption A3, the mean and the SD of in-vehicle link travel time (denoted as tij
sn
and sij
tsn, respectively) can be expressed as
tij
sn ¼X
v2Vb
tvdaij
sn;v
8aij
sn 2Ad(40)
sij
tsn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
v2Vb
sv
t
2daij
sn;v
s8aij
sn 2Ad(41)
3.2.3. Transfer links
There is no transfer waiting time if travelers transfer to mode b
2
(auto). When travelers transfer to b
1
(subway) or b
3
(bus), the waiting time for transfer link a
t
(denoted as Tat) can be expressed as
Tat¼l
Gbþ1
Gb
Fatþ
Fbi
Gbhb
m
8at2At;b2b1;b3
fg (42)
where Fatis the passenger volume at transfer link a
t
and
Fbi is the prior passenger volume already in
mode bprior to picking up passengers at location i;G
b
and h
b
, respectively, are the frequency and
72 X. FU ET AL.
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
vehicle capacity of the boarding transport mode b;mand lare model parameters. The first term in
Equation (42) expresses the waiting time for the next arriving vehicle, and the second term is related
to the boarding congestion effect.
Specifically, the prior passenger volume can be obtained by summing the passenger flows of relevant
direct in-vehicle links that do not start from location i[14]:
Fbi ¼X
ayz
sn2Ad;8y6¼i; s
ðÞ¼
b
Fyz
sndayz
sn;vi;iþ1
¼X
ayz
sn2Ad;8y6¼i; s
ðÞ¼
bX
p2Prs
Fpdayz
sn;vi;iþ1
dp;ayz
sn
(43)
where vi;iþ1is the physical link from ito its next station i+ 1 in mode b
1
or b
3
;Fyz
sn is the passenger flow on
in-vehicle link ayz
sn from yto zthat does not start from i. For simplicity, Fatand
Fbi are assumed to be in-
dependent from each other, then the mean and the SD of Fatþ
Fbi (denoted as f
1
and s
1
)canbeexpressed
as follows: f1¼X
p2Prs
fpdp;at
ðÞþ X
ayz
sn2Ad;8y6¼i; sðÞ¼bX
p2Prs
fpdayz
sn;vi;iþ1
dp;ayz
sn
(44)
s1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
p2Prs
sp
f
2dp;at
ðÞþ X
ayz
sn2Ad;8y6¼j; sðÞ¼bX
p2Prs
sp
f
2dayz
sn;vi;iþ1
dp;ayz
sn
ðÞ
s(45)
Following model assumption A2, Fatþ
Fbi
ðÞNf
1;s2
1
.
As indicated earlier, the subway frequency Gb¼gb1is a constant. Thus, the mean and the SD of
waiting time for each transfer link to b
1
(subway) can be expressed as
tat¼l
gb1þ1
gb1
ðÞ
mþ1hb1
ðÞ
mX
m
i¼0;i¼even
m
i
s1
ðÞ
if1
ðÞ
mii1ðÞ!! 8at2At(46)
sat
t¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
gb1
ðÞ
mþ1hb1
ðÞ
m
!
2X
2m
i¼0;i¼even
2m
i
s1
ðÞ
if1
ðÞ
2mii1ðÞ!!
X
m
i¼0;i¼even
m
i
s1
ðÞ
if1
ðÞ
mii1
ðÞ
!!
!
2
0
B
B
B
B
B
@
1
C
C
C
C
C
A
v
u
u
u
u
u
u
u
u
t8at2At(47)
However, when travelers transfer to b
3
(bus), G
b
(i.e., Gb3) is stochastic as formulated in Section
3.2.1.2. Thus, the mean and the SD of waiting time for each transfer link to b
3
(bus) can be expressed as
tat¼2ltp
s0þ2mþ1
s0mþ1hb3
m
X
m
i¼0;i¼even
m
i
s1
ðÞ
if1mii1ðÞ!!
X
j¼0;j¼even
mþ1mþ1
j
sp
t
jtpmþ1jj1ðÞ!!
(48)
sat
t¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4l2
s02sp
t
2þ22mþ2
s02mþ2hb3
2m
X
2m
i¼0;i¼even
2m
i
s1
if12mii1ðÞ!! X
2mþ2
j¼0;j¼even
2mþ2
j
sp
t
jtp2mþ2jj1ðÞ!!
2mþ1
s0mþ1hb3
m
X
m
i¼0;i¼even
m
i
s1
if1mii1ðÞ!! X
mþ1
j¼0;j¼even
mþ1
j
sp
t
jtpmþ1jj1ðÞ!!
" #2
þ22mþ2l
s0mþ2hb3
m
X
m
i¼0;i¼even
m
i
s1
if1mii1ðÞ!! X
mþ2
j¼0;j¼even
mþ2
j
sp
t
jtpmþ2jj1ðÞ!!
22mþ2ltp
s0mþ2hb3
m
X
m
i¼0;i¼even
m
i
s1
if1mii1ðÞ!! X
mþ1
j¼0;j¼even
mþ1
j
sp
t
jtpmþ1jj1ðÞ!!
v
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
t
(49)
73
RELIABILITY-BASED MULTI-MODAL ASSIGNMENT MODEL
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
The detailed manipulations on deducing Equations (48) and (49) can be found in Appendix B.
3.3. Route travel time distribution
The route travel time T
p
can be obtained by summing the travel times on direct in-vehicle links and
travel times on transfer links:
Tp¼X
aij
sn 2Ad
Tij
sndp;aij
sn
þX
at2At
Tatdp;at
ðÞ8p2Prs (50)
Following model assumption A3, the mean and the SD of route travel time can be expressed as
tp¼X
aij
sn 2Ad
tij
sndp;aij
sn
þX
at2At
tatdp;at
ðÞ8p2Prs (51)
sp
t¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
aij
sn2Ad
sij
tsn
2dp;aij
sn
þX
at2At
sat
t
ðÞ
2dp;at
ðÞ
s8p2Prs (52)
and the route travel time follows a normal distribution: TpNt
p;sp
t
ðÞ
2
4. MODEL FORMULATION AND SOLUTION ALGORITHM
4.1. Model formulation
As mentioned earlier, demand uncertainty leads to travel time variability. Under travel time uncer-
tainty, travelers would assign an extra time to ensure a high probability of on-time (referring to gener-
alized travel time in this paper) arrival. The concept of travel time budget proposed by Lo et al. [5] is
adopted. The travel time budget in this paper is defined as the summation of mean generalized route
travel time and a safety margin of generalized route travel time.
Let abe the probability of arriving at destination within the travel time budget and c
p
be the travel
time budget for a given reliability threshold a. The value of aexpresses travelers’risk attitude toward
on-time arrival. A larger aindicates a higher expectation of on-time arrival. This value of acan be
pre-determined according to travelers’socio-economic characteristics and trip purposes [23].
Because TpeNt
p;sp
t
ðÞ
2
, the travel time budget can be expressed as
cp¼tpþΦ1aðÞsp
t8p2Prs (53)
where Φ
1
(a) is the inverse of standard normal cumulative distribution function at the probability of a.
If a= 50%, Φ
1
(a) = 0, and the safety margin is equal to zero. This indicates that travelers are risk-
neural and only concern mean travel time for their travels. Under this circumstance, the travel time
budget c
p
is equal to the mean travel time t
p
. Therefore, the RUE results should be close to that of
UE model when a= 0.5. However, it should be noted that in the RUE model when a= 0.5, the mean
travel times are still related to the SD of trafficflow [see Equations (25, 34, 37, 46, and 48)]. As such,
the RUE model is not exactly equivalent to the traditional UE model when a= 0.5.
In this paper, the travel time budget and the route fare both contribute to the route dis-utility
(denoted as ’
p
). The route dis-utility function can be represented as
’p¼o1cpo2rp8p2Prs (54)
where r
p
is the fare of a specific route p, and o
1
and o
2
are parameters. Specifically, r
p
can be calcu-
lated by summing the fares of in-vehicle links as
rp¼X
aij
sn2Ad
rij
sndp;aij
sn
8p2Prs (55)
where rij
sn is the fare with respect to the in-vehicle link aij
sn. The fare of each in-vehicle link is repre-
sented node to node. Thus, it is non-linear, not the summation of individual physical link fares.
Because this study falls in the category of static model for long-term planning at the strategic level, it
is postulated that all travelers in multi-modal networks would have a RUE route choice pattern: for
each OD pair, the dis-utilities of all used routes are smallest and equal, and all unused routes have
74 X. FU ET AL.
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
larger dis-utilities. Denote p2P
rs
as the most reliable route that has the smallest route dis-utility. The
RUE condition can be formally expressed as
fp’p’p
¼08p2Prs (56)
’p’p≥08p2Prs (57)
The aforementioned RUE problem can be further expressed as the following gap function formulation:
min GAP ¼X
p2Prs
fp’p’p
(58)
’p’p≥0 (59)
fp≥0 (60)
The gap function refers to the overall gap capturing the complementary slackness conditions of the
RUE model.
4.2. Solution algorithm
Most traditional solution algorithms cannot be used to solve the proposed RUE model, because it is
difficult to determine the decent direction for solving the problem concerned. The widely used method
of successive average (MSA) is a heuristic method with a forced convergence property. Therefore, in
this paper, a solution algorithm based on MSA is proposed for solving the aforementioned RUE prob-
lem. The detailed steps for the solution algorithm are presented as follows.
Step 0 Transform the traditional multi-modal network to a SAM network. List all the feasible routes
in the SAM network according to the pre-defined probable transfer states.
Step 1 Calculate free-flow route travel times {t
p
}. Set sp
t
fg
¼0
fg
. Then get free-flow route dis-
utilities {’
p
} on the basis of {t
p
}, sp
t
fg
and fares. Perform all-or-nothing assignment on the
basis of {’
p
} to obtain route flows and link flows f1¼fv;8v2V
.Setn=1.
Step 2 Get the SDs of link flow sv
f
no
and SDs of route travel time sp
t
fg
.Usefn,sv
f
no
,{t
p
}, and sp
t
fg
to update link travel times. Then get new {t
p
}and sp
t
fg
. After that, get new route dis-utilities {’
p
}.
Step 3 Perform all-or-nothing assignment on the basis of route dis-utilities {’
p
}, yielding auxiliary
link flows f
n.
Step 4 Calculate new link flows using an MSA scheme fnþ1¼fnþ1=nðÞ
fnfn
.
Step 5 For an acceptable convergence level t,if max
v
fnþ1fn
≤t, stop; otherwise, set n=n+1,
go to Step 2.
5. NUMERICAL EXAMPLE
To demonstrate the aforementioned RUE formulation, we adopted in this study the network used in
Lo et al. [14]. As shown in Figure 1, the network consisted of 9 nodes and 20 links. Only one OD
pair was considered in this example. The origin was set as Node 1, and the destination was set as
Node 9. Following model assumption A1, OD demand followed a normal distribution. The CV of
the OD demand is set as 0.3. Modal transfers follow the probable transfer states defined in Lo et al.
[14]. The resulting feasible routes are listed in Table I.
In this numerical example, the model parameters were set as follows: b
1
= 0.01, k
1
=2, g
1
= 0.3,
k
2
=2, e
0
= 1.2, eb2¼1;eb3¼3;b
2
= 0.007, g
2
= 0.01, l= 0.5, m=2, o
1
= 0.123, and o
2
= 0.09. The
link capacity of each road link was 800 vehicles per hour. Free-flow travel time of each roadway link
was set as 20 minutes, and travel time of each subway link was set as 19 minutes. The fare of traveling
by auto was nine units per link (fuel cost), whereas the non-linear fares of subway and bus are shown
in Tables II and III, respectively. The capacity of subway was 3200 passengers per vehicle, and the
frequency was eight vehicles per hour. The capacity of bus was 180 passengers per vehicle, and the
fleet size was 20 vehicles.
75
RELIABILITY-BASED MULTI-MODAL ASSIGNMENT MODEL
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
Figure 1. The multi-modal network.
Table I. Feasible routes according to probable transfer states.
Route number Modal usage (1—subway; 2—auto; 3—bus) Transfer segment (node to node)
11 1!9
23 1!9
32 1!9
43–11!6!9
52–11!6!9
61–31!6!9
72–11!5!9
82–31!6!9
92–31!3!9
10 2–31!2!9
11 2–11!4!9
12 2–3–11!3!6!9
13 2–1–31!5!6!9
14 2–3–11!2!6!9
15 2–1–31!4!6!9
Table III. Non-linear fares of the bus.
From node
To node
2369
1 8888
2 888
344
6 4
Table II. Non-linear fares of the subway.
From node
To node
4569
1 10254545
4 102545
5 10 25
6 10
76 X. FU ET AL.
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
The convergence characteristics of the proposed RUE solution algorithm are illustrated in Figure 2.
It can be seen that the RUE condition at relative gap of 10
4
has been achieved after 9864 iterations
(when q= 3000, t= 0.1). This result indicates that the proposed MSA solution algorithm can solve
the RUE problem for this typical network with an acceptable accuracy level.
Travelers’travel behaviors, in terms of mode and route choices, were investigated under different
levels of OD demand and on-time arrival probability. Note that the on-time arrival in this paper refers
to generalized travel time. Figure 3 shows the variation of modal split when the mean of the OD
demand (q) increases from 3000 passengers per hour to 30 000 passengers per hour under different
probabilities of on-time arrival (a). Figure 3a–c, respectively, depicts the percentages of travelers using
different transport modes (subway, bus, and auto).
It can be seen from Figure 3 that as the mean of the OD demand increases from 3000 to 30 000, the
percentage of modal split for subway increases dramatically. For example, with a 90% probability of
on-time arrival ( a= 0.9), the percentage of subway travelers increases from 36.74% to 84.51%. On the
contrary, the percentages of modal split for bus and auto both decrease (from 39.67% to 6.52% and
from 23.59% to 8.97%, respectively). This may be due to that large OD demand results in severe traffic
congestion on the road. In view of this, travelers tend to choose more reliable subway that has fixed
frequency and no congestion interactions with bus and auto.
From the earlier discussions, it can be found that the OD demand level influences travelers’route
and mode choice behaviors. In addition, the on-time arrival probability also has significant effects
on travelers’route and mode choice behaviors.
For a certain level of OD demand, with the increase of a, the number of people traveling by subway
rises, whereas the numbers of people traveling by auto and bus decrease. For example, when q= 3000,
with a= 50%, 25.65% of travelers use subway, and this percentage increases to 36.74% when a
reaches 90%. However, for bus and auto usage, the percentages decrease from 46.45% to 39.67%
and from 27.90% to 23.59%, respectively. This result may be because the demand variation has a
slighter impact on the generalized travel time of subway than on road traffic.
However, when qbecomes larger, this phenomenon is less prominent. For instance, when
q= 30 000, with the increase of a, variations of modal split are all within 1% (84.37% to 84.51%
for subway, 6.81% to 6.52% for bus, and 8.82% to 8.97% for auto). For auto, there is even a slight
increase (8.82% to 8.97%) as compared with the downtrend in a smaller demand. This shows that
when OD demand is very large, most travelers will not change their mode choices to improve the prob-
ability of on-time arrival. The reason may be that the large OD demand makes the crowding discomfort
in a subway considerable. In this situation, none of the available transport modes are reliable; and
4.88E-04
9.77E-04
1.95E-03
3.91E-03
7.81E-03
1.56E-02
3.13E-02
6.25E-02
1.25E-01
2.50E-01
5.00E-01
1.00E+00
1
3001 6001 9001
Relative Gap*
Number of iterations
7.80E-04
8.00E-04
8.20E-04
8.40E-04
8.60E-04
8.80E-04
8864 9364 9864
*Relative Gap = /
rs
pp
pP
GAPf
∈
∑
Figure 2. Convergence characteristics of the solution algorithm.
77RELIABILITY-BASED MULTI-MODAL ASSIGNMENT MODEL
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
therefore, some people prefer to use auto to avoid the discomfort in transit vehicles. Thus, in the
congested transport networks at metropolitan areas, the subway is normally more reliable than other
transport modes if a traveler has a higher expectation of on-time arrival, but when the network
(a) (b)
(c)
3000
12000
21000
30000 0.5
0.6
0.7
0.8
0.9
20
30
40
50
60
70
80
90
Mean of OD demand (q)
% Percentage of modal split (subway)
3000
12000
21000
30000
0.5
0.6
0.7
0.8
0.9
5
10
15
20
25
30
35
40
45
50
Probability of on-time arrival ( )
Mean of OD demand (
q
)
% Percentage of modal split (auto)
23.59%
27.90%
8.82%
8.97%
25.65%
36.74%
84.51%
84.37%
3000
12000
21000
30000
0.5
0.6
0.7
0.8
0.9
5
10
15
20
25
30
35
40
45
50
Probability of on-time arrival ( )Probability of o n-time arrival ( )
Mean of OD demand (q)
% Percentage of modal split (bus)
39.67%
6.81%
6.52%
46.45%
Figure 3. Modal splits under different levels of OD demand and on-time arrival probabilities: (a) subway, (b) bus,
and (c) auto.
506
404
2494
2596
2440
2480
2520
2560
2600
2640
360
400
440
480
520
560
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Route passenger flow
(without modal transfers)
Route passenger flow
(with modal transfers)
Probability of on-time arrival (α)
Transfer No Transfer
Figure 4. Travelers’attitudes toward modal transfer under different on-time arrival probabilities.
78 X. FU ET AL.
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
becomes extremely overcrowded, the subway will no longer be attractive because of the considerable
crowding discomfort on the trains.
Additionally, travelers’attitudes toward modal transfers under different probabilities of on-time arrival
were demonstrated. The results are illustrated in Figure 4. As discussed in Section 3, each modal transfer
needs a transfer waiting time and thus brings a penalty in the route utility. Transfer waiting times are
generally varied and stochastic because of the demand uncertainty, so modal transfers may bring some
uncertainties for people’s on-time arrivals. As aincreases, travelers tend to choose the routes without
any modal transfer. For example, when aincreases from 50% to 90% (q= 3000), the number of travelers
using single transport mode increases from 2494 to 2596, whereas the number of travelers who use modal
transfers decreases from 506 to 404.
6. CONCLUSIONS
This paper presented a new travel time reliability-based traffic assignment model in multi-modal
transport networks (including auto, bus, and subway modes) with demand variations. To capture the
effects of demand uncertainty, we formulated passenger flows and generalized travel times of different
transport modes as random variables. In this paper, the bus fleet size was assumed to be fixed, and the
stochastic bus frequency derived from unstable travel time of bus route was explicitly considered. The
probable transfers and the non-linear fare structures, involved in the multi-modal transport networks,
were explicitly modeled by using the SAM network.
Travelers’route and mode choice behaviors under stochastic multi-modal networks were also incor-
porated in the proposed model. To capture travelers’route and mode choice behaviors, we adopted the
travel time budget, which is defined as the summation of the mean and the safety margin of generalized
route travel time, in this new model. On the basis of this travel choice criterion, a RUE condition was
then proposed. The multi-modal RUE problem was solved by a path-based solution algorithm using
the MSA. The model and the solution algorithm were tested using a hypothetical network. The results
of the numerical example indicated that with a high expectation of on-time arrival, travelers tend to use
the subway and avoid modal transfers.
In this paper, some model assumptions are adopted. These assumptions may cause some potential
biases. It is assumed that OD demands are mutually independent, which may overestimate or underes-
timate the variance and the covariance of link/route flows. The ignorance of link travel time correla-
tions may underestimate the route travel time variances [24]. The incorporation of such correlations
in the proposed multi-modal traffic assignment model is a significant extension of this paper. For the
assumption of no vehicle capacity constraint, in some Asian cities, not all travelers can get on the first-
arrival transit vehicles during peak periods. Capacity constraints do exist and may result in crowding
effects on transit systems, so the proposed model can be extended by the incorporation of the capacity
constraints. For all normal distribution assumptions, it should be noted that normal distribution allows
negative values in principle, which may not be plausible in the real world. However, if the model fits well
to the observed data, this disadvantage can be negligible [25]. Other types of probability distribution can
also be adopted, such as log-normal distribution [26], Poisson distribution [27], and truncated normal dis-
tribution. If different distributions are adopted, further investigations on the properties of these distri-
butions and different analysis approaches should be carried out. In addition, in this paper, a path
enumeration technique was adopted to solve the proposed multi-modal RUE problem, because of the
non-additive property of generalized travel time budget. This path numeration technique, however, is only
appropriate for small multi-modal networks [23]. How to develop efficient path-based solution algorithm
based on column generation technique without requirement of path enumeration needs to be further
investigated.
7. LISTS OF SYMBOLS AND ABBREVIATIONS
7.1. Abbreviations
UE user equilibrium
RUE reliability-based user equilibrium
79
RELIABILITY-BASED MULTI-MODAL ASSIGNMENT MODEL
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
RSUE reliability-based stochastic user equilibrium
SD standard deviation
SAM state-augmented multi-modal
OD origin-destination
CV coefficient of variation
MSA method of successive average
7.2. Parameters
b
1
,k
1
model parameters in Equation (23)
g
1
,k
2
model parameters in Equation (27)
eb2;eb3passenger car equivalents for mode b
2
and b
3
e
0
average vehicle occupancy parameter representing the number of passengers per auto
b
2
,g
2
model parameters in Equation (36)
m,lmodel parameters in Equation (42)
o
1
,o
2
model parameters in Equation (54)
7.3. Variables
Mmulti-modal transport network; M=(U,V)
Uset of physical nodes; U={i}
Vset of physical links; V={v}
Bset of transport modes
bindividual transport mode; b2B;b
1
(subway), b
2
(auto), b
3
(bus)
U
b
set of physical nodes associated to mode b;U
b
⊆U
V
b
set of physical links associated to mode b;V
b
⊆V
wnumber of transport modes; w=|B|
M
b
sub-network of mode b;M
b
=(U
b
,V
b
)
GSAM network; G=(N,A)
Nset of SAM nodes
iphysical location of SAM node
stransfer state of SAM node; s2S
nnumber of prior transfers to current SAM node
lalight or aboard indicator
Sset of probable transfer states
Aset of SAM links; A=A
t
∪A
d
A
t
set of transfer links in SAM network; A
t
={a
t
}
A
d
set of direct in-vehicle links in SAM network; Ad¼aij
sn
(s) associated transport mode of state s;(s)2B
x(s) set of probable transfers from state s;x(s)⊆S
Q
rs
travel demand between OD pair rs
q
rs
mean of OD demand
srs
qSD of OD demand
cv
rs
CV of travel demand between OD pair rs
P
rs
set of routes between OD pair rs
F
p
passenger flow along route p2P
rs
f
p
mean of passenger flow along route p
sp
fSD of passenger flow along route p
Fij
sn passenger flow on direct in-vehicle link aij
sn
fij
sn mean of passenger flow on in-vehicle link aij
sn
sij
fsn SD of passenger flow on in-vehicle link aij
sn
dp;aij
sn
incidence relationship between in-vehicle link and route
dp;aij
sn
¼1 if in-vehicle link aij
sn is on route p; 0 otherwise
80 X. FU ET AL.
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
Fatpassenger flow on transfer link a
t
fatmean of passenger flow on transfer link a
t
sat
fSD of passenger flow on transfer link a
t
d(p,a
t
) incidence relationship between transfer link and route
d(p,a
t
) = 1 if transfer link a
t
is on route p; 0 otherwise
F
v
passenger flow of mode von physical link v2V
b
f
v
mean of passenger flow on link v
sv
fSD of passenger flow on link v
daij
sn;v
incidence relationship between in-vehicle link and physical link
daij
sn;v
¼1 if physical link vis in in-vehicle link aij
sn; 0 otherwise
T
v
generalized travel time of physical link v
t
v
mean travel time of physical link v
sv
tSD of travel time of physical link v
hb1subway vehicle capacity (passengers per vehicle)
gb1deterministic frequency of subway (vehicles per hour)
t0
vfree flow travel time of link v
X
v
total traffic volume on road link v
x
v
mean of total traffic volume on road link v
sxvSD of total traffic volume on road link v
k
v
capacity of the road link v
Gb3stochastic bus frequency
Fvb2passenger flow of mode b
2
on road link v
fvb2mean passenger flow of mode b
2
on road link v
svb2
fSD of passenger flow of mode b
2
on road link v
hb3bus capacity
Tij
sn travel time of in-vehicle link aij
sn
tij
sn mean of in-vehicle link travel time
sij
tsn SD of in-vehicle link travel time
Tatwaiting time of transfer link a
t
tatmean of transfer link waiting time
sat
tSD of transfer link waiting time
Fatpassenger volume at transfer link a
t
Fbi prior passenger volume already in mode bprior to picking up passengers at location i
G
b
frequency of the boarding transport mode b
h
b
vehicle capacity of the boarding transport mode b
vi;iþ1physical link from ito its next station i+ 1 in mode b
1
or b
3
f
1
mean of Fatþ
Fbi
s
1
SD of Fatþ
Fbi
T
p
travel time of route p
t
p
mean route travel time
sp
tSD of route travel time
aprobability of on-time arrival
c
p
travel time budget
Φ
1
(a) inverse of standard normal cumulative distribution function at the probability of a
’
p
route dis-utility
r
p
fare of route p
rij
sn fare of in-vehicle link aij
sn
pthe most reliable route with the smallest route dis-utility; p2P
rs
ACKNOWLEDGEMENTS
This work described in this paper was jointly supported by a Postgraduate Studentship and a research
grant from the Research Grant Council of the Hong Kong Special Administrative Region to the Hong
Kong Polytechnic University (Project No. PolyU 5215/09E).
81
RELIABILITY-BASED MULTI-MODAL ASSIGNMENT MODEL
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
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82 X. FU ET AL.
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
APPENDIX A.
Substituting Equation (29) in Equation (36) gives the following equation
Tv¼t0
vþb2t0
v
2FvTp
hb3s0
k1
þg2t0
v
Xv
kv
k2
¼A1þA2þA3
As discussed before, TpNt
p;sp
t
ðÞ
2
,FvNf
v;sv
f
2
,XvNx
v;sxv
ðÞ
2
Then, the following equations can be obtained.
EA
2
½¼ 2k1b2t0
v
hb3
ðÞ
k1s0
ðÞ
k1
!
X
k1
i¼0;i¼even
k1
i
sv
f
i
fvk1ii1ðÞ!! X
k1
j¼0;j¼even
k1
j
sp
t
jtpk1jj1ðÞ!!
EA
3
½¼
g2t0
v
kv
ðÞ
k2X
k2
i¼0;i¼even
k2
i
sxv
ðÞ
ixvk2ii1ðÞ!!
var A2
½¼ 2k1b2t0
v
hb3
ðÞ
k1s0
ðÞ
k1
!
2X
2k1
i¼0;i¼even
2k1
i
sv
f
i
fv2k1ii1ðÞ!! X
2k1
j¼0;j¼even
2k1
j
sp
t
jtp2k1jj1ðÞ!!
2k1b2t0
v
hb3
ðÞ
k1s0
ðÞ
k1
!
X
k1
i¼0;i¼even
k1
i
sv
f
i
fvk1ii1
ðÞ
!! X
k1
j¼0;j¼even
k1
j
sp
t
jtpk1jj1
ðÞ
!!
" #2
var A3
½¼
g2
ðÞ
2t0
v
2
kv
ðÞ
2k2X
2k2
i¼0;i¼even
2k2
i
sxv
ðÞ
ixv2k2ii1ðÞ!! X
k2
i¼0;i¼even
k2
i
sxv
ðÞ
ixvk2ii1ðÞ!!
!
2
2
43
5
Particularly, the manipulations of E[A
2
] and var[A
2
] are similar to the detailed manipulations of E
[I
2
] and var[I
2
] in Appendix B.
For simplicity, it is assumed that A
2
and A
3
are mutually independent. Then, Equations (37) and
(38) can be obtained as follows.
tv¼EA
1
½þEA
2
½þEA
3
½
¼t0
vþ2k1b2t0
v
hb3
ðÞ
k1s0
ðÞ
k1
!
X
k1
i¼0;i¼even
k1
i
sv
f
i
fvk1ii1ðÞ!! X
k1
j¼0;j¼even
k1
j
sp
t
jtpk1jj1ðÞ!!
þg2t0
v
kv
ðÞ
k2X
k2
i¼0;i¼even
k2
i
sxv
ðÞ
ixvk2ii1ðÞ!!
sv
t¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
var A2
½þvar A3
½
p
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2k1b2t0
v
hb3
ðÞ
k1s0
ðÞ
k1
!
2X
2k1
i¼0;i¼even
2k1
i
sv
f
i
fv2k1ii1ðÞ!! X
2k1
j¼0;j¼even
2k1
j
sp
t
jtp2k1jj1ðÞ!!
2k1b2t0
v
hb3
ðÞ
k1s0
ðÞ
k1
!
X
k1
i¼0;i¼even
k1
i
sv
f
i
fvk1ii1ðÞ!! X
k1
j¼0;j¼even
k1
j
sp
t
jtpk1jj1ðÞ!!
" #2
þg2
ðÞ
2t0
v
2
kv
ðÞ
2k2X
2k2
i¼0;i¼even
2k2
i
sxv
ðÞ
ixv2k2ii1ðÞ!! X
k2
i¼0;i¼even
k2
i
sxv
ðÞ
ixvk2ii1ðÞ!!
!
2
2
43
5
v
u
u
u
u
u
u
u
u
u
u
u
u
u
u
t
83
RELIABILITY-BASED MULTI-MODAL ASSIGNMENT MODEL
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
APPENDIX B.
The detailed manipulations on deducing Equations (48) and (49) are given as follows.
Substituting Equation (29) in Equation (42) gives the following equation.
Tat¼2lTp
s0þ2mþ1
s0mþ1hb3
m
Fatþ
Fbi
ðÞ
mTpmþ1¼I1þI2
As discussed before, TpNt
p;sp
t
ðÞ
2
,Fatþ
Fbi
ðÞNf
1;s2
1
. For simplicity, we transfer the
notations in the following equations, that is, t=t
p
,s2¼sp
t,F¼Fatþ
Fbi , and f=f
1
. Then,
EI
1
½¼
2lt
s0
var I1
½¼
4l2
s02s22
EI
2
½¼ 2mþ1
s0mþ1hb3
m
Rþ1
1 Rþ1
1 xmymþ11
ffiffiffiffiffiffi
2p
ps1
exp xfðÞ
2
2s12
!
1
ffiffiffiffiffiffi
2p
ps2
exp ytðÞ
2
2s22
!
dxdy
Substituting z1¼xf
s1;z2¼yt
s2in the aforementioned equation gives
EI
2
½¼ 2m
ps0mþ1hb3
m
Rþ1
1 z2s2þtðÞ
mþ1exp z22
2
Rþ1
1 z1s1þfðÞ
mexp z12
2
dz1
dz2
¼2m
ps0mþ1hb3
m
Rþ1
1 X
mþ1
j¼0
mþ1
j
z2s2
ðÞ
jtmþ1jexp z22
2
Zþ1
1 X
m
i¼0
m
i
z1s1
ðÞ
ifmiexp z12
2
dz1
" #dz2
¼2m
ps0mþ1hb3
m
Rþ1
1 X
mþ1
j¼0
mþ1
j
z2s2
ðÞ
jtmþ1jexp z22
2
X
m
i¼0
m
i
s1
ðÞ
ifmiZþ1
1
z1iexp z12
2
dz1
" #dz2
According to the MathWorld website [28], it follows that
Zþ1
1
z1iexp z12
2
dz1¼0 if i is odd
ffiffiffiffiffiffi
2p
pi1ðÞ!! if i is even
Therefore,
EI
2
½¼ 2m
ps0mþ1hb3
m
Rþ1
1 X
mþ1
j¼0
mþ1
j
z2s2
ðÞ
jtmþ1jexp z22
2
X
m
i¼0;i¼even ffiffiffiffiffiffi
2p
pm
i
s1
ðÞ
ifmii1ðÞ!!
"#
dz2
¼
2m
ps0mþ1hb3
m
X
m
i¼0;i¼even ffiffiffiffiffiffi
2p
pm
i
s1
ðÞ
ifmii1ðÞ!! X
mþ1
j¼0
mþ1
j
s2
ðÞ
jtmþ1jZþ1
1
z2jexp z22
2
dz2
"#
¼2mþ1
s0mþ1hb3
m
X
m
i¼0;i¼even
m
i
s1
ðÞ
ifmii1ðÞ!! X
mþ1
j¼0;j¼even
mþ1
j
s2
ðÞ
jtmþ1jj1ðÞ!!
84 X. FU ET AL.
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
With similar manipulations, the following equations can be obtained.
EI
22
½¼ 22mþ2
s02mþ2hb3
2m
X
2m
i¼0;i¼even
2m
i
s1
ðÞ
if2mii1ðÞ!! X
2mþ2
j¼0;j¼even
2mþ2
j
s2
ðÞ
jt2mþ2jj1ðÞ!!
var I2
½¼EI
22
½EI
2
½ðÞ
2
¼22mþ2
s02mþ2hb3
2m
X
2m
i¼0;i¼even
2m
i
s1
ðÞ
if2mii1ðÞ!! X
2mþ2
j¼0;j¼even
2mþ2
j
s2
ðÞ
jt2mþ2jj1ðÞ!!
2mþ1
s0mþ1hb3
m
X
m
i¼0;i¼even
m
i
s1
ðÞ
ifmii1ðÞ!! X
mþ1
j¼0;j¼even
mþ1
j
s2
ðÞ
jtmþ1jj1ðÞ!!
" #2
EI
1I2
½¼ 2mþ2l
s0mþ2hb3
m
X
m
i¼0;i¼even
m
i
s1
ðÞ
ifmii1ðÞ!! X
mþ2
j¼0;j¼even
mþ2
j
s2
ðÞ
jtmþ2jj1ðÞ!!
cov I1;I2
ðÞ¼EI
1I2
½EI
1
½EI
2
½
¼2mþ2l
s0mþ2hb3
m
X
m
i¼0;i¼even
m
i
s1
ðÞ
ifmii1ðÞ!! X
mþ2
j¼0;j¼even
mþ2
j
s2
ðÞ
jtmþ2jj1ðÞ!!
2mþ2lt
s0mþ2hb3
m
X
m
i¼0;i¼even
m
i
s1
ðÞ
ifmii1ðÞ!! X
mþ1
j¼0;j¼even
mþ1
j
s2
ðÞ
jtmþ1jj1ðÞ!!
Therefore, Equations (48) and (49) can be obtained by the following equation.
tat¼ET
at
½¼EI
1
½þEI
2
½
¼2ltp
s0þ2mþ1
s0mþ1hb3
m
X
m
i¼0;i¼even
m
i
s1
ðÞ
if1mii1ðÞ!! X
mþ1
j¼0;j¼even
mþ1
j
sp
t
jtpmþ1jj1ðÞ!!
sat
t¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
var I1
½þvar I2
½þ2cov I1;I2
ðÞ
p
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4l2
s02sp
t
2þ22mþ2
s02mþ2hb3
2m
X
2m
i¼0;i¼even
2m
i
s1if12mii1ðÞ!! X
2mþ2
j¼0;j¼even
2mþ2
j
sp
t
jtp2mþ2jj1ðÞ!!
2mþ1
s0mþ1hb3
m
X
m
i¼0;i¼even
m
i
s1if1mii1ðÞ!! X
mþ1
j¼0;j¼even
mþ1
j
sp
t
jtpmþ1jj1ðÞ!!
" #2
þ22mþ2l
s0mþ2hb3
m
X
m
i¼0;i¼even
m
i
s1if1mii1ðÞ!! X
mþ2
j¼0;j¼even
mþ2
j
sp
t
jtpmþ2jj1ðÞ!!
22mþ2ltp
s0mþ2hb3
m
X
m
i¼0;i¼even
m
i
s1if1mii1ðÞ!! X
mþ1
j¼0;j¼even
mþ1
j
sp
t
jtpmþ1jj1ðÞ!!
v
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
t
85RELIABILITY-BASED MULTI-MODAL ASSIGNMENT MODEL
Copyright © 2012 John Wiley & Sons, Ltd. J. Adv. Transp. 2014; 48:66–85
DOI: 10.1002/atr
