Mixed SUE behaviour under traveller information provision services with
heterogeneous multi-class multi-criteria decision making
Xiaoqing Jaber
Centre for Transport Research and Innovation for People (TRIP)
Department of Civil, Structural and Environmental Engineering
Trinity College Dublin, Dublin 2, Ireland
Email: lixq@tcd.ie
Margaret O’Mahony
Centre for Transport Research and Innovation for People (TRIP)
Department of Civil, Structural and Environmental Engineering
Trinity College Dublin, Dublin 2, Ireland
Email: margaret.omahony@tcd.ie
Abstract- This paper studies travellers’ mixed stochastic user equilibrium (SUE) behaviour
under traveller information provision services with heterogeneous multi-class multi-criteria
decision making. A multi-class, multi-criteria mixed SUE assignment model under traveller
information provision services is proposed and modelled as a nonlinear complementary
problem (NCP). The model considers heterogeneous road travellers who are assumed to be
multi-criteria decision makers. The route choice behaviour of equipped and unequipped
travellers under traveller information provision services is then formulated as an optimisation
program, in which the net economic benefit is maximised and the total generated emissions
are constrained. The optimisation model has an interpretation from economic, behavioural
1
and environmental viewpoints. The solution of this program satisfies the logit-based SUE
assignment. By deriving first-order optimality conditions from this program, the marginal cost
pricing policy is obtained. This pricing is not only class-dependent but also link-dependent.
Index Terms—traveller information provision services, mixed SUE assignment, nonlinear
complementarity problem, multi-class multi-criteria decision making
INTRODUCTION
Over the last two decades, there has been a considerable interest in the transportation area on
the analysis of the effects of providing traveller information. Much research work has been
undertaken to evaluate the impact of traveller information provision in terms of welfare
economic considerations (e.g. Arnott et al., 1991; Verhoef et al., 1995; Emmerink et al.,
1998), potential travel time savings (e.g. Abdel-Aty and Abdalla, 2004; Adler, 2001; Adler et
al., 2005; Jou et al., 2005; Mahmassani and Liu, 1999; Toledo and Beinhaker, 2006), driver
behaviour (e.g. Dell’Orco and Teodorovic, 2009; Emmerink et al., 1996; Iida et al., 1992;
Khattak et al., 1995; Mannering et al., 1994; Tsirimpa et al., 2007; Yang et al., 1993), safety
implications (e.g. Srinivasan et al., 1995; Al-Deek et al., 1998), and the efficiency of road
usage (e.g. Al-Deek and Kanafani, 1993; Emmerink et al., 1995a,b; Mahmassani and
Jayakrishnan, 1991), and so on. Only very few studies were focused on studying the
environmental impact of traveller information provision, particularly from increased vehicular
emissions. For example, Al-Deek et al. (1995) proposed an analytical method for evaluating
the impact of Advanced Traveller Information System (ATIS) on air quality in a simple
network where traffic experienced incident congestion. Kanninen (1996) discussed congestion
2
relief and the environmental impact expected of Intelligent Transportation Systems (ITS). It
was mentioned that ITS might induce latent travel demand which would probably increase
vehicular emissions.
The impact of traveller information on vehicular emissions requires more evaluation, because
any underestimation or overestimation of the environmental impact of traveller information
provision could lead to an unexpected outcome. Conventionally, it is intended that providing
travellers with the dissemination of traffic information would allow equipped vehicles to
spread from the congested to the less congested areas. Hence the travel times of both
equipped drivers and the total system travel time (TSTT) would be reduced. However,
diverted traffic may impose extra environmental externalities, such as air pollution, noise
pollution, and accidents, on existing drivers and residents living in the neighbourhood. The
increased pollution is hazardous to human health, so in this case, the aim of implementing
traveller information provision is not only to enhance the mobility of a transportation system
but also to maintain the system which is sustainable in an environmentally-friendly way.
Szeto et al. (2008) found out paradoxical phenomena present with the provision of traveller
information services, in which the provision of traveller information cannot reduce TSTT and
vehicular emissions simultaneously. With the above considerations, more analysis is required
to evaluate the impact of traveller information provision services on these two issues
simultaneously.
Traditionally, the travellers consider two criteria when making route choices, which are travel
times and travel costs. For those who choose to use information provision services, they make
their decisions based on the obtained information of travel times and costs on a network. For
those who do not use the services, they make route choices based on their experience or their
3
knowledge of the network. Nagurney et al. (2002) pointed out that it is not unreasonable to
assume that certain classes of travellers factor an environmental criterion into their decision
making process with the increasing concern of the degradation of the environment. Also with
the growing interests in studying Intelligent Transportation Systems, it is expected that the
emissions can be broadcast to the travellers as well as travel times and costs (Nagurney et al.,
2002). The consideration of the environmental criterion regarding emissions may cause
travellers to change their previous choices which they would have made when emissions were
not taken into account. Additionally, the weights of travellers’ multiple criteria of travel times,
costs and emissions will have direct impacts on their route choices as well, and consequently
TSTT and emissions. Therefore, it would be very meaningful to consider incorporating the
environmental criterion into the travellers’ decision making process when analysing the
impact of traveller information provision services. However, no one has attempted this in the
transportation literature to date.
In this paper, we proposed a multi-class multi-criteria mixed stochastic user equilibrium (SUE)
assignment model under the traveller information provision services with heterogeneous road
travellers. Traveller heterogeneity is considered by assuming a discrete set of value of times
(VOTs) for several traveller classes. The travellers of each class having the same VOT are
further divided into different groups with different travel cost perception variations. It is
assumed that equipped travellers have lower perception variation due to provided traveller
information and unequipped travellers have higher perception variation due to the lack of
current traffic information. The route choice behaviour of the equipped and unequipped
travellers is modelled to follow a mixed SUE assignment. In addition, the route choice
behaviour of equipped and unequipped travellers is formulated as an optimisation model, in
which the net economic benefit is maximised and the total generated emissions are
4
constrained. The proposed model has an interpretation from economic, behavioural and
environmental viewpoints. The solution of this model satisfies the logit-based SUE
assignment. The marginal cost pricing was then derived from the proposed model. This
marginal cost pricing is not only class-dependent but also link-dependent. The rest of the
paper is organised as follows: The next section describes NCP formulation and performance
measures. A system optimised pricing policy in a mix SUE assignment comes after that
followed by illustrative examples, concluding remarks, acknowledgement, and references.
NCP FORMULATION AND PERFORMANCE MEASURES
In this paper, a multi-class multi-criteria mixed SUE assignment model is developed under
traveller information provision services based on Szeto (2007), where user heterogeneity and
multi-criteria decision making are not considered. The travellers’ multi-criteria decision
making process is incorporated into the proposed model in which there is an explicit
environmental criterion. The multi-class multi-criteria mixed SUE assignment model is
different from those proposed in Yang and Zhang (2002), Huang and Li (2007), Yang and
Zhang (2008), and Zhang et al. (2008), as the latter models do not consider the environmental
criterion. This multi-class multi-criteria mixed SUE assignment problem with driver
heterogeneity is modelled as a nonlinear complementary problem (NCP). The model then can
be solved by any existing optimisation program. In this study, the model is solved using the
Generalised Reduced Gradient (GRG) method (Abadie and Carpentier, 1969).
5
NCP Formulation of the Heterogeneous Multi-class Mixed SUE Assignment Problem
with an Environmental Criterion
The multi-class route choice behaviour of equipped and unequipped travellers is modelled to
follow the principle of SUE by only varying travel cost perception variation as in Lo and
Szeto (2002), which implies that all the equipped travellers have a lower travel cost
perception variation and all the unequipped travellers have a higher travel cost perception
variation. It is assumed there are N information service providers (ISP) who provide traffic
information service over the entire road network for the corresponding N equipped driver
groups, who only pay for the information from one service provider. There is an unequipped
driver group who does not pay for any information service. Therefore, this problem has N  1
driver groups in each class having their own VOT. Each driver is assumed to be a multicriteria decision maker, who considers travel times, travel costs and generated emissions
when making route choices. Let M denote the number of driver classes in the network and
let m be a typical driver class, m  1, 2,..., M . The problem has M classes of drivers. For the
drivers in any class m , they all follow N  1 sets of SUE conditions with all groups of
equipped drivers having lower perception variations than those of unequipped drivers. Using
the logit model, the SUE conditions are formulated as follows:
f
rs
p ,i , m

exp  i  prs,i ,m 
 exp    
rs
k ,i , m
i
 qirs,m  0, rs, p, i, m ,
(1)
k
w
rs
p ,i , m

exp  i  prs,i ,m 
 exp    
i
rs
k ,i , m
, rs, p, i, m ,
(2)
k
where f prs,i ,m , wrsp,i ,m , and  prs,i ,m represent respectively the route flows, the proportion, and the
generalised travel cost or the disutility of group i drivers in class m on route p between
6
origin-destination (OD) pair rs ;  i represents the travel cost perception variation of group i
drivers; qirs,m stands for the demand of group i drivers in class m between OD pair rs .
In (1), the route flows are calculated according to the proportion defined by the logit model
(2). The generalised route travel cost or the disutility  prs,i ,m in (2) is a weighted average of
three criteria: travel times, travel costs and emission costs, which in turn are functions of the
link flow va . The link flow va can be determined by summing up all the route flows on that
link:
va 
 
mM pP rs iN 1
f prs,i ,m   ap , a ,
(3)
where  ap is the link-route incidence indicator -  ap  1 if link a is on route p ;  ap  0
otherwise, and P rs is the set of paths between OD pair rs .
The Bureau of Public Roads (BPR) type performance function is applied in this study and
shown as follows to calculate the link travel time t a :


 va  
ta  t 1    0   , a ,

 ca  
0
a
(4)
where t a0 , va and ca0 are the link free flow travel time, link flow and capacity, and
  0.15,   4 .
The route travel cost can be computed as follows:
 prs,i ,m  CISP ,i    Bmta  a    ap , rs, p, i, m,
(5)
a
where C ISP ,i is the information service charge for group i drivers; Bm is the value of time of
7
class m drivers;  a is the toll on link a . The sum of the travel time cost Bm ta and its toll  a
is the travel cost on link a . The route travel cost is composed of the service charge and the
sum of the travel costs of the links on that route.
According to Nagurney et al. (2002), the environmental costs associated with travelling on
link a can be expressed as follows:
ea  ea (va ), a.
(6)
If a single pollutant is simply considered, ea (va ) can be assumed to be the average emissions
of this single pollutant generated by the travellers on link a , for example, carbon monoxide.
Although speed and grade variations are essential considerations in estimating vehicular
emissions, they are not modelled in macroscopic traffic assignment models (Benedek and
Rilett, 1998).
Once the travel time, travel cost and emission costs are clearly defined, one can obtain the
following generalised route cost:
 prs,i ,m   tat ,a ,m ap   prs,i ,m  (c ,a ,m ap )  eae,a ,m ap , rs, p, i, m ,
a
a
(7)
a
where  prs,i ,m is the generalised travel cost or the disutility of group i drivers in class m on
route p between origin-destination (OD) pair rs ; t ,a,m , c,a,m and e,a,m denote the nonnegative weights associated with traveller’s travel time, cost and emissions on link a
respectively. The weights t ,a,m , c,a,m and e,a,m are not only class-dependent but also linkdependent. Equation (7) implies that each group i drivers in class m has their own perception
of the trade-offs among travel time, travel cost, and emissions generated when they travel on
route p between origin-destination (OD) pair rs . The trade-offs are represented by these
8
weights associated with each criterion. The link-dependent weights allow us to incorporate
some link-dependent factors as safety, comfort, view, sociability factors, as well as sensitivity
to pollution (Nagurney et al., 2002).
The total travel demand in the network q rs can be calculated as follows:
 q
rs
i ,m
m
 q rs , rs, i, m,
(8)
i
where qirs,m is the demand of group i drivers in class m between OD pair rs , which is fixed in
this study. It can also be modelled through the newly proposed multinomial logit elastic
market penetration model shown in Szeto et al. (2008), which captures the elasticity of the
demand for the services measured by market penetration.
The NCP formulation is obtained by adding the non-negativity conditions and multiplying
route flows by the SUE conditions (1) as follows:
 f prs,i ,m  f prs,i ,m  wrsp ,i ,m  qirs,m   0

f prs,i ,m  0
, rs, p, i, m .

rs
rs
rs

f p ,i ,m  wp ,i ,m  qi ,m  0

If f prs,i ,m  0 in (9), the term
f
rs
p ,i , m
(9)
 wrsp ,i ,m  qirs,m  must be zero to satisfy NCP. It means (1)
must be satisfied. If f prs,i ,m  0 , the last constraint in (9) is satisfied, i.e. f prs,i ,m  wrsp,i ,m  qirs,m  0 .
Put everything together and let y represents a strategic interaction of decisions makers as
follows:
9
i , i  1,..., N

C , i  1,..., N

 ISP ,i

  a , a  1,..., A


,
y
t ,a ,m , a  1,..., A, m  1,..., M 


c ,a ,m , a  1,..., A, m  1,..., M 


e ,a ,m , a  1,..., A, m  1,..., M 
(10)
x  y    f prs,i ,m , rs, p, i, m  , and
(11)
F  x    f prs,i ,m  wrsp ,i ,m  qirs,m , rs, p, i, m  ,
(12)
the NCP (9) can then be expressed as finding x*  0 such that:
F  x*   0, x*  y   F  x*   0 ,
T
(13)
where  i and C ISP ,i are the strategy of information providers;  a is the strategy of a toll
operator; and the weights t ,a,m , c,a,m and e,a,m are respectively travellers’ own perception
of the trade-offs among travel time, travel cost, and emissions; and wrsp,i ,m is defined by (2).
Performance Measures of the Total System Travel Time (TSTT) and Vehicular
Emissions
The total system travel time (TSTT) is calculated to measure system performance regarding
congestion and it is the sum of the travel times of all drivers on all links, expressed as:
TSTT   vata .
(14)
a
According to (14), TSTT is a function of link flows, and hence is a function of route flows
based on (3).
10
In terms of vehicular emissions, there are two types of vehicular emissions: link and network
(or overall). The link vehicular emission is defined through the link emission factor approach.
The key of estimating vehicular emissions is the relationship that volume of emissions is
equal to the product of an emission factor and link load (DeCorla-Souza et al., 1995). This
emission factor obtained using MOBILE model proposed by the Environmental Protection
Agency (EPA) is based on the federal test procedure (FTP), typical driving conditions for an
urban vehicle trip (DeCorla-Souza et al., 1995). This link emission factor approach adopted
by Nagurney et al. (1998) and others can be expressed as follows:
Qa  ha va , a,
(15)
where Qa is the vehicular emissions on link a ; va represents the hourly traffic flow on link
a ; ha is the emission factor on link a , which is assumed to be given for all links. The factors
affecting the value of ha are discussed in Nagurney (2000). The overall vehicular emissions
can be calculated as follows:
Q   Qa .
(16)
a
According to (15), the vehicular emissions on a particular link are the product of the link
flows and the corresponding emission factor. And the overall vehicular emissions can be
obtained as in (16) by summing of the vehicular emissions on each link.
A SYSTEM OPTIMISED PRICING POLICY IN A MIXED SUE ASSIGNMENT
PROBLEM
Traveller Information Services with Net Economic Benefit Maximisation
11
According to the trip consumer approach in Oppenheim (1995), an individual traveller
considered as a consumer of urban trips maximises direct utility through an optimal choice of
the aggregate demand. The direct utility of a representative group i traveller f prs,i ,m with a
level of travel cost perception variation  i in class m corresponding to the aggregate demand
qirs,m can be defined as follows:
U i ,m  
1
i

rs pP
f prs,i ,m ln f prs,i ,m 
rs
1
i
q
rs
i ,m
ln qirs,m .
(17)
rs
Then the gross direct utility of representative consumers in class m with various levels of
travel cost perception variations (various  i ) can be expressed as follows:
U
i ,m
i
  (
i
1
i

rs pP
rs
f prs,i ,m ln f prs,i ,m )   (
i
1
i
q
rs
i ,m
ln qirs,m ).
(18)
rs
According to Yang and Huang (2005), net economic benefit of all the travellers is the
difference between their gross direct utility and total travel cost. Therefore, the net economic
benefit (NEB) of class m consumers is the difference between the gross direct utility of
travellers in class m ,
U
i ,m
, and the total travel cost in class m ,
i
 
i
f prs,i ,m prs,i ,m , and
rs pP rs
can be expressed as follows:
NEBm   (
i
1
i

 
rs pP

f
rs
f prs,i ,m ln f prs,i ,m )   (
i

rs
p ,i , m
rs
p ,i , m
1
i
q
rs
i ,m
ln qirs,m )
rs
(19)
.
rs pP rs
i
One then can obtain the net economic benefit of all consumers from all classes as follows:

1
NEB     (
m 
 i i
 
i


rs pP rs
rs pP rs
f

rs
p ,i , m
f prs,i ,m ln f prs,i ,m )   (
i
rs
p ,i , m

.

12
1
i
q
rs
i ,m
rs
ln qirs,m )
(20)
The maximisation of the net economic benefit is equivalent to the minimisation of the
negative net economic benefit. In this model, the negative net economic benefit is minimised
so that a convex minimisation program can be formulated and the strict convexity of F (f ) in
the following Model 1 with respect to flow variable, f prs,i ,m over the feasible region can be
proved.
minF (f )   (
f
Model 1:
m
i
i
 
i
subject to
1
i
rs pP

f
pP rs
i

rs
p ,i , m
1
i
q
rs
i ,m
rs
ln qirs,m
(21)
)
f prs,i ,m   qirs,m  q rs , rs, p, i, m,
m
a
m
rs
rs
p ,i , m
 h  
a
f prs,i ,m ln f prs,i ,m  
rs pP rs
 
m

i
rs pP
(22)
i
f prs,i ,m ap  Q, a, rs, p, i, m
(23)
rs
f prs,i ,m  0, rs, p, i, m,
(24)
where  i is the parameter representing the travel cost perception variation of group i drivers;
f prs,i ,m and  prs,i ,m are respectively the route flows and the route travel cost of group i drivers in
class m on route p between origin-destination (OD) pair rs ; qirs,m is the demand of group i
drivers in class m between OD pair rs ; q rs is the total travel demand between OD pair rs ;
ha is the emission factor on link a ; Q is the maximum allowable vehicular emissions in the
transportation system. qirs,m , q rs and Q are exogenously decided. Equation (22) guarantees that
the path flow pattern satisfies the travel demands. Constraint (23) guarantees that the path
flow pattern does not exceed the maximum allowable vehicular emissions. Constraint (24) is
to ensure route flows are non-negative. It is observed that conditions (22)-(24) correspond to
Linear System 4.1 in Nagurney (2000), the existence of a solution to this linear system of
13
equations and inequalities guarantees that the demand associated with OD pairs can be
satisfied by a path flow pattern, which also satisfies the maximum allowable vehicular
emissions simultaneously. This means that the solution guarantees viability of a transportation
system with given OD pairs and travel demands. Nagurney (2000) has pointed out that there
may be more than one such flow pattern to Linear System 4.1. It is proved in Jaber (2009) that
the solution f prs,i ,m to the minimisation problem is a strict local minimum of F (f ) and then
that is the unique minimum of F (f ) over the feasible region defined by conditions (22)-(24).
The Equivalent Logit-Based SUE Assignment Conditions and a System Optimised Toll
The solution of Model 1 can be derived to have the form of the logit-based Stochastic User
Equilibrium assignment conditions. This is demonstrated by proving that the first-order
conditions of Model 1 are equivalent to the logit-based SUE assignment conditions. The
Lagrangian for the minimisation Model 1 can be formulated as:
L(f , λ , μ)  F (f )    rs ( qirs,m  
rs
m
  ( ha 
a
i
i

m
f
i

pP rs
f prs,i ,m )
(25)
  Q),
rs
p
p ,i , m a
rs pP rs
where  rs and  are Lagrange multipliers associated with constraints (22) and (23)
respectively. Equating the partial derivatives of L(f , λ , μ) with respect to the flow variables
f prs,i ,m to zero will lead to the conditions that a stationary point of F (f ) subject to constraints
(22)-(24) must satisfy. It is important to note that
L
does not exist when f prs,i*,m  0 ,
rs
f p ,i ,m
because of the singularity at f prs,i*,m  0 in the logarithmic function in the objective function
F (f ) , a solution to the stationary point conditions will only be valid if all components of
14
f prs,i ,m are strictly positive. The first-order conditions obtained by calculating the partial
derivatives of L(f , λ , μ) with respect to the flow variables f prs,i ,m are expressed as follows:

f
rs
p ,i , m

where
f prs,i ,m

L(f , λ, μ) 
F (f ) 
1
i
F (f )   rs    ha  ap  0, rs, p, i, m,
f prs,i ,m
(ln f
rs*
p ,i , m
(26)
a
 1)  
rs
p ,i , m
  f
m
rs*
p ,i , m
i
d prs,i ,m
df prs,i ,m
, rs, p, i, m .
Therefore

f prs,i ,m
L(f , λ , μ) 
1
i
rs*
p ,i , m
(ln f
 1)  
rs
p ,i , m
  f
m
i
rs*
p ,i , m
d prs,i ,m
df prs,i ,m
  rs
(27)
   ha  ap  0, rs, p, i, m.
a
Equation (27) can be modified as follows:
1
i
(ln f
rs*
p ,i , m
 1)  
rs
p ,i , m
  f
m
i
rs*
p ,i , m
d prs,i ,m
df
rs
p ,i , m
   ha  ap   rs , rs, p, i, m. (28)
a
It is observed that the left-hand side of (28) consists of the sum of four items. The first of
which is the derivative of the first item in the objective function F (f ) , which has the form of
an entropy function defined on the path flow. The second is route travel cost, which is defined
in (5). The third is congestion externality, which is the additional travel time burden that an
additional traveller inflicts on each one of the travellers already using path p (Sheffi, 1985).
The fourth is the marginal contribution of the total cost on path p due to vehicular emissions
generated by travelling on path p (Nagurney, 2000). The Lagrange multiplier  associated
with vehicular emission constraint (23) is interpreted as being the marginal cost of emission
abatement. The sum of the second, third and fourth items on the left-hand side of (28) can be
interpreted as the marginal cost of group i drivers in class m travelling on path p between
OD pair rs ,  prs,i ,m , associated with the congestion and emission externalities that the
additional traveller imposes on others:
15

rs
p ,i , m

rs
p ,i , m
  f
m
i
rs*
p ,i , m
d prs,i ,m
df
rs
p ,i , m
   ha  ap , rs, p, i, m.
(29)
a
Then (28) becomes:
1
i
(ln f prs,i*,m  1)   prs,i ,m   rs , rs, p, i, m.
(30)
Combine (22) and (30), carrying out some manipulations, there is:
f prs,i*,m  qirs,m exp(i prs,i ,m )
 exp( 
kK
rs
i k ,i , m
) , p.
(31)
Observing that (31) obtained from the first-order optimality conditions of Model 1 has the
form of the logit-based SUE assignment conditions. The differences between the above SUE
condition and the ones derived in Sheffi (1985) and Yang (1999) are the usage of the route
cost function. Here, the route cost function is (29), where  prs,i ,m is the marginal cost of group
i drivers in class m travelling on path p between OD pair rs associated with the congestion
and emission externalities. However, the one in Sheffi (1985) uses the usual travel cost
function without considering the congestion and emission externalities. The one derived in
Yang (1999) only incorporates the congestion externality in the travel cost function not
considering the emission externality. The solution of Model 1 satisfying the logit-based SUE
assignment not only maximises the net economic benefit of all the travellers but also ensures
that total emissions does not violate its constraint. This means, at equilibrium, the flow pattern
is system optimised, but travellers still behave in a manner following the mixed SUE. To
achieve such an optimal equilibrium point in any network where equipped and unequipped
travellers follow the mixed SUE assignment, it is required that the congestion and emission
externalities should be simultaneously incorporated into the travel cost function. This can be
obtained by charging travellers a toll equal to the congestion and emission externalities, which
is the difference between the marginal travel cost and generalised travel cost as follows:
16

rs
p ,i , m

rs
p ,i , m

  f
rs
p ,i , m
m
where
 f
m
i
rs*
p ,i , m
d prs,i ,m
df
df
i
and   ha  ap
rs
p ,i , m
d prs,i ,m
rs*
p ,i , m
rs
p ,i , m
   ha  ap , rs, p, i, m,
(32)
a
are respectively the congestion and emission
a
externalities.
Karush-Kuhn-Tucker (K-K-T) Optimality Conditions
Since the objective function (21) is to maximise the net economic benefit, Karush-KuhnTucker (K-K-T) optimality conditions can be derived for the system-optimised problem given
by (21)-(24). K-K-T conditions for Model 1 can be stated as follows:
1
i
(ln f
rs*
p ,i , m
 1)  
rs
p ,i , m
  f
m
i
rs*
p ,i , m
d prs,i ,m
df prs,i*,m
(33)
  rs*   *  ha  ap  0, rs, p, i, m,
a
1
d prs,i ,m
rs*
rs
rs*
 (ln f p ,i ,m  1)   p ,i ,m   f p ,i ,m rs*
df p ,i ,m
m
i
i
T

 rs*   *  ha  ap  f prs,i*,m  0, rs, p, i, m,
a

 * ( ha 
a
m
i

f prs,i*,m ap  Q)  0, rs, p, i, m,
m
i
pP
rs
(35)
rs pP rs
 *  0,
 
(34)
f prs,i*,m   qirs,m  0, rs, p, i, m,
m
(36)
(37)
i
f prs,i*,m  0.
17
(38)
Because of the singularity at f prs,i*  0 in the logarithmic function in the objective function
F (f ) , it is required that all f prs,i* are strictly positive. Therefore, (33) must hold as equality,
which has been proved to be the conditions equivalent to the logit-based SUE conditions. This
implies that at equilibrium the system optimised flow pattern follows the logit-based SUE. In
conditions (35) and (36), the positive marginal cost of emission abatement 
*
is to ensure the
vehicular emissions not to exceed the maximum allowable vehicular emissions. When the
vehicular emissions are smaller than its maximum allowable value, the marginal cost of
emission abatement is zero. Conditions (37) and (38) are respectively the flow conservation
and non-negativity constraints.
Weights in a Bicriteria Model: Relation to the Optimal Toll
If it is assumed that the travellers in the network are environmentally conscious and consider
travel costs and generated emissions in their route choice decision making, there is the
following generalised route cost function, which incorporates two criteria that travellers
consider:
 prs,i ,m   prs,i ,m  e, p ,m  ea ap , rs, p, i.m,
(39)
a
where e , p , m is the weight associated with emissions generated on path p for class m drivers.
Obviously, the weight associated with travel cost is assumed to be 1. It is assumed as in
Nagurney et al. (2002) that the emissions function ea on link a is equal to the emission
factor on the link, which represents emissions generated by a single traveller travelling on that
link. Then the above generalised route cost function can be rewritten as follows:
18
 prs,i ,m   prs,i ,m  e, p ,m  ha  ap , rs, p, i.m.
(40)
a
To achieve the optimal equilibrium solution in Model 1, a toll should be charged and equal to:
 prs,i ,m   prs,i ,m   prs,i ,m
  f
rs*
p ,i , m
  f
rs*
p ,i , m
m
m
i
i
d prs,i ,m
df
rs
p ,i , m
d prs,i ,m
df
rs
p ,i , m
   ha  ap  e, p ,m  ha  ap
a
(41)
a
 (   e, p ,m ) ha  ap .
a
The above equation represents the optimal toll which guarantees a SUE flow pattern is system
optimised. When the weight associated with generated emissions is equal to lagrange
multiplier  , traveller only needs to pay a toll equal to the congestion externality. When the
traveller is not aware of the generated emissions, he needs to pay a toll equal to the congestion
and emission externalities. When the traveller has very high awareness of generated emissions
(i.e., e , p ,m   ), he pays a toll less than the congestion externality. This implies that the
weight that the traveller place on emissions is directly related to the charged optimal toll
under the assumption mentioned before. The higher the awareness of generated emissions is,
the lower the optimal toll is.
ILLUSTRATIVE EXAMPLES
Here, two examples are given for an illustrative purpose to analyse the effectiveness of
derived marginal cost pricing. The proposed NCP formulation is used as an analysis
methodology, route flow patterns of equipped travellers and unequipped travellers can be
solved and next system performance can be evaluated. It is assumed that there are two classes
of travellers with different VOTs. Each class is further divided into two groups, equipped and
19
unequipped travellers. The scenario example used here is broadly based upon an existing
section of the road network in Ireland between North of Balbriggan and Dundalk in Co. Louth,
which consists of two links, two nodes and one OD pair shown in Figure 1 below.
1
3
1
2
Figure 1 The example network.
13
The parameters adopted in the first example are as follows: a) demand parameters: q1,1
 150
13
13
vph; q13
2,1  700 vph; q1,2  800 vph; q2,2  2600 vph; b) route choice parameters: value of
time B1 = € 15/ hr; B2 = € 10/ hr; travel cost perception variation parameter: unequipped
drivers 1 = 0.15 €-1; equipped drivers  2 =2 €-1; c) toll operation parameter: toll 1   2  € 0
d) information service parameter: service charge CISP =
€ 0.5; e) vehicular emission
parameter: link emission factors h1 =0.9 litre/veh; h2 =1.3 litre/veh; maximum allowable
vehicular emissions: Q =4310 litre; f) Network parameters: c10  2500 vph; c20  2000 vph;
t10  21 mins; t20  29 mins.
Table 1 demonstrates system performance before and after implementation of marginal cost
pricing. Before setting a maximum allowable vehicular emission constraint in the network, no
marginal toll is needed. Travellers are bircriteria decision makers but only concerned with
travel costs in their decision making process, which means c =1, e =0. The generalised route
cost function in (39) substituting into (2) in NCP formulation is reduced to (5). The calculated
TSTT and total emissions are respectively 118363.66 minutes and 4351.86 litres. To limit the
total emissions not to exceed 4310 litres, a marginal toll as expressed in (32) is implemented,
20
which reduces the total emissions to be 4310 litres but increases TSTT to 120513.73 minutes
whether or not travellers are concerned with generated emissions in their decision making. In
this case, the marginal cost pricing under the logit-based SUE does not necessarily diminish
TSTT. This point has been indicated in Yang (1999), although Yang’s marginal cost pricing
only considers congestion externality.
Table 1 System performance before and after the marginal cost pricing under SUE.
Total Emissions
Maximum Allowable
(litre)
Vehicular Emissions (litre)
TSTT (min)
Before Marginal
c =1, e =0
118363.66
4351.86
None
c =1, e =0
120513.73
4310.00
4310
c =1, e =5
120513.73
4310.00
4310
Cost Pricing
After Marginal
Cost Pricing
Table 2 shows the optimal tolls under various emission weights when the same maximum
allowable vehicular emissions are required. It is revealed that congestion tolls are classdependent. Class 1 drivers face higher congestion tolls than class 2 drivers when they traverse
the same link. It is observed that emission tolls are not class-dependent like congestion tolls,
but are link-dependent. Emission tolls are related to the emission factors on links. The higher
the emission factor, the higher the emission toll, and vice versa. When travellers are more
environmentally conscious (higher e ), the marginal cost of emission abatement is less.
Therefore, emission tolls are less. However, congestion tolls remain. It is observed that the
difference between optimal marginal costs of abatement is 5, which is exactly the same as the
21
weight travellers place on generated emissions in their decision making. This coincides with
the relationship derived in (41).
Table 2 The optimal tolls under different emission weights with the same total emission
constraint.
Class 1 Drivers
Class 2 Drivers
the Marginal
Maximum
(B1=15 €/hr)
(B2=10 €/hr)
Cost of Emission
Allowable
Abatement μ
Vehicular
Link 1
Link 2
Link 1
Link 2
Emissions (litre)
Congestion
6.86
c =1,
Toll (€)
e =0
Emission
0.59
4.58
0.39
11.54
16.66
11.54
16.66
6.86
0.59
4.58
0.39
12.82
4310
7.82
4310
Toll (€)
Congestion
c =1,
Toll (€)
e =5
Emission
7.04
10.16
7.04
10.16
Toll (€)
In the second example, the same network and parameters are adopted as in the first example
with the exception of the following: link emission factors h1 =1.3 litre/veh; h2 =0.9 litre/veh;
maximum allowable vehicular emissions: Q =4950 litre and 4840 litre. Table 3 illustrates how
the system performance changes before and after implementing the marginal cost pricing with
various levels of total emission constraints. It shows that the marginal cost pricing decreases
both TSTT and total emissions, which are 118363.66 minutes and 4998.14 litres originally.
To lower total emissions such that they are less than 4950 litres, the marginal cost pricing
used in the network is where the traveller has no environmental consciousness. Surprisingly,
22
total emissions, which are 4851.37 litres, are less than the maximum permitted level, which is
4950 litres.
Table 3 System performance before and after the marginal cost pricing under SUE.
TSTT (min)
Total Emissions
Maximum Allowable
(litre)
Vehicular Emissions
(litre)
Before Marginal
c =1, e =0
118363.66
4998.14
None
c =1, e =0
115374.94
4851.37
4950
c =1, e =0
115437.87
4840.00
4840
c =1, e =0.3
115437.87
4840.00
4840
Cost Pricing
After Marginal
Cost Pricing
It is shown in Table 4 that with this emission constraint there are only congestion tolls. If the
maximum allowable vehicular emissions are further lowered to be 4840 litres, it can be seen
in Table 3 that total emissions are further reduced to the maximum allowable level. However,
in this case, congestion tolls alone are not enough. As shown in Table 4, there are emission
tolls on all links, which is 0.96 euro on link 1 and 0.67 euro on link 2. As discussed in the last
example, emission tolls are link-dependent. Link 1 has a higher emission toll since link 1 has
a higher emission factor. If the travellers are assumed to be environmentally conscious
( e =0.3) in the same network with the same maximum allowable emissions, which are 4840
litres, emission tolls are reduced but congestion tolls remain the same. Observing that, before
and after travellers place some weights on emissions, the difference between optimal marginal
costs of emission abatement is 0.3, which is exactly equal to the weight the traveller puts on
23
generated emissions. This implies that there is a strong relationship between weights
associated with emissions and the marginal cost pricing, more specifically the marginal cost
of emission pricing. It is found that an increased weight associated with emissions results in
less optimal tolls.
It is also observed in Table 3 that TSTT is pushed up from 115374.94 minutes to 115437.87
minutes when a tighter emission constraint is adopted. This means that there is a trade-off
between TSTT and total emissions. When one tries to lower total emissions, TSTT is
worsened. As appeared in the last example, the marginal cost pricing cannot lower TSTT and
total emission simultaneously. This does not mean that the system is not optimised. The
phenomenon that the marginal cost pricing under SUE cannot necessarily decrease TSTT is
not a paradox but a pseudo paradox. This is rooted in the stochastic nature of the mixed
equilibrium model, which is the randomness of the perceived travel times (Sheffi, 1985). With
traveller information provision services, equipped travellers still have perception variation of
travel times, which is only less than that of unequipped travellers. Therefore, in the mixed
stochastic network loading model, travellers’ perceived travel times are minimised. An
improved system does not necessarily mean a reduction in TSTT. An optimised system here
is referred to as an optimal situation with a given emission bound. Whether the marginal cost
pricing under the logit-based SUE can reduce TSTT and total emissions simultaneously or not
depends on the topology, characteristics and the emission bound of a network.
Table 4 The optimal tolls under different emission constraints with various emission
weights.
Class 1 drivers
Class 2 drivers
the Marginal
Maximum
(B1=15 €/hr)
(B2=10 €/hr)
Cost of
Allowable
24
Link 1
c =1,
e =0
Link 2
Link 1
Emission
Vehicular
Abatement μ
Emissions (litre)
0
4950
0.74
4840
0.44
4840
Link 2
Congestion
3.50
2.19
2.33
1.46
Toll (€)
Emission
0.00
0.00
0.00
0.00
3.34
2.34
2.23
1.56
Toll (€)
c =1,
e =0
Congestion
Toll (€)
Emission
0.96
0.67
0.96
0.67
3.34
2.34
2.23
1.56
Toll (€)
c =1,
e =0.3
Congestion
Toll (€)
Emission
0.57
0.40
0.57
0.40
Toll (€)
CONCLUDING REMARKS
In this paper, a multi-class multi-criteria mixed SUE assignment model is presented under the
traveller information provision services incorporating traveller heterogeneity. The novelty of
this study is to incorporate the travellers’ multi-criteria decision making process into the
proposed mixed SUE model in which there is an explicit environmental criterion, which has
not been attempted in the transportation literature to date. An optimisation model is proposed
to derive the marginal cost pricing, which was demonstrated to be applicable in the network
under the logit-based mixed SUE.
25
It is noteworthy that the marginal cost pricing derived is not only class-dependent but also
link-dependent. Gomez-Ibanez and Small (1994) provided a brief review of congestion
pricing schemes and technologies available for implementing them. They discussed that the
practical possibilities of implementing complex pricing schemes are changing rapidly with
new developments in automated vehicle identification and charging. The implementation of
more sophisticated congestion pricing schemes more depend on automated charging
technologies. Three tasks involved in every such system are: to recognize the identities of
valid users; to detect and classify vehicles for the purpose of enforcement; and to manage the
financial transactions. With the application of such technologies such as in-vehicle units, it is
possible to implement the proposed class-dependent and link-dependent marginal cost pricing
in practice. Niskanen and Nash (2008) provided a comprehensive review of experiences of
road pricing in Europe and elsewhere, both in research and practice. They mentioned an
approach, which is now often referred to as marginal cost-based pricing, in which prices
remain based on marginal cost and various second-best rules may be derived to indentify the
optimal adjustments from marginal cost pricing. Verhoef et al. (2008) discussed the
implementation paths (IPs) for marginal cost-based pricing in urban transport, proposed a
structural economic approach to the design and evaluation of such IPs and applied such
approach to analyse IPs in the context of urban transport. Even though the practical
possibilities of implementing marginal cost pricing are increasing, Niskanen and Nash (2008)
pointed out that the importance of institutional implementation issues including political
acceptability have been given too little attention both in research and in actual attempts to
implement road pricing measures. Both the analysis of theoretical approaches and the
feasibility of practical applications of marginal cost pricing, therefore, require more
investigation.
26
This paper opens up a number of research directions. The total travel demand is assumed to
be fixed. Kanninen (1996) pointed out that ITS may induce potential travel demand. Ignoring
latent travel demand may overestimate the benefits of information provision services. It is
thus interesting to study the effect of information provision in a network with an elastic travel
demand. The proposed models are static. It would be meaningful to model mixed SUE
behaviour under endogenous market penetration of traveller information services in a
dynamic traffic assignment framework. This type of dynamic model considering traveller
information services was previously proposed by Lo and Szeto (2004), which allowed the
study of the impact of traveller information provision services to capture the changes in
departure times and the impact of traveller information provision services under non-recurrent
network congestion. As it is not the focus of this study, we leave the dynamic analysis of
marginal cost pricing for future studies. Also, it would be very meaningful to calibrate the
parameters of perception variations  i , which represent travel uncertainty, once the survey
data is available. The method proposed by Huang (1995) can be employed for this purpose.
Moreover, our proposed NCP formulation allows the analysis of strategic interactions among
information providers, toll road operators, and travellers’ multicriteria decision making. It
would be interesting to analyse how each strategy of these decision makers affects the
marginal cost pricing and the concerns of other parties. In addition, extending the existing
model to incorporate travel time, cost and network uncertainties would be another challenging
research direction.
ACKNOWLEDGEMENT
27
This research is funded under the Programme for Research in Third-Level Institutions
(PRTLI), administered by the Irish Higher Education Authority.
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