Road pricing is currently considered one of the most powerful tools for managing
transport demand in urban areas. In recent years several European cities (such
as London, Stockholm and Milan) have introduced road pricing as a travel demand
management strategy. Many reasons, such as congestion, noise and air pollution
reduction, lead public administrations to use pricing policies in urban contexts.
These measures allow transportation systems to be managed more efficiently and
sustainably since they can generally yield a temporal, spatial and modal redistribution
of travel and particularly rebalance the modal split between private vehicles
and mass-transit systems. In a transportation network, the natural interaction
between mobility demand and transportation supply leads the transportation system
to a condition defined as User Equilibrium (UE). In a simple case of one origin-destination
pair connected by one link, user equilibrium can be graphically represented
by the intersection between the demand function and the supply function, as
shown in Fig. 1. The UE condition is known to be non-efficient
in economic terms, where efficiency entails total cost minimization, in the
demand case (Beckmann et al., 1956).
||User Equilibrium (UE) and System Equilibrium (SE) conditions
This condition is known as System Optimum (SO) when the deterministic approach
is used to solve the network assignment problem and System Equilibrium (SE)
(Gentile et al., 2005) or Stochastic System Optimum
(SSO) (Stewart, 2007) when the user choice model is
a random utility model. As shown in Fig. 1, the SE condition
can be graphically represented by the intersection between the demand function
and marginal cost function (indicated in the literature also as social cost
or marginal social cost function). This function is equal to the individual
cost plus the additional cost incurred by all other vehicles when one extra
vehicle is on the road. The discrepancy between the User Equilibrium and the
System Equilibrium comes from user behaviour in making mobility choices: an
additional user, entering a traffic flow, considers a travel cost that does
not include the cost increase imposed on the other travellers in the network.
In other words, travellers try to maximise their own utility or private benefits
instead of considering social welfare. It is shown that efficient transportation
system use can be achieved by charging efficient tolls on network links. The
optimal situation can be reached by the imposition of a tax (or toll) that will
reconcile the private cost and the social cost (Prudhomme
and Bocarejo, 2005). These tolls, called Marginal Social Cost Pricing (MSCP)
tolls, are equal to negative externalities (such as congestion cost, travel
delays, air pollution, accidents) imposed on other travellers by an additional
user and are one of the most popular tools for road pricing applications (Yildirim
and Hearn, 2005). According to Marginal Social Cost (MSC) theory, many advanced
models are formulated to design pricing strategies (Yang
and Huang, 2004; Hamdouch et al., 2007; Kuwahara,
2007). Generally in the economic literature a condition where all tolls
are equal to marginal external costs on each link is referred to as the first-best
condition. However, it is not always possible to determine the optimal fare
for one good because of the distortions in the market for related goods (Rouwendal
and Veroef, 2006).
Hence in this case we consider sub-optimal solutions (second-best tolls) in
which not all links are tolled. The second-best tolls issue has been extensively
studied (Verhoef et al., 1996; Lawphongpanich
and Hearn, 2004; Zhang and Ge, 2004; Ekstrom
et al., 2009). First best tolls are achieved only in a theoretical
condition for mobility demand segmentation: for instance, users can differ from
each other with respect to the Value Of Time (VOT). Several authors have studied
the anonymous tolls problems (Arnott and Krauss, 1998;
Yang and Zhang, 2008). Furthermore, there are several
practical reasons why transport regulators often consider second-best solutions:
capital and monitoring costs may be prohibitive or they may prefer to start
with a demonstration project before implementing road pricing on a system-wide
scale (Verhoef, 2002). When pricing strategies are applied
in a real network, a more important issue for policy makers is what to do with
road pricing revenues. Investment in public transport is generally one of the
preferred options for the allocation of revenues (Farrel
and Salesh, 2005). This option is chosen when the aim of pricing strategies
is congestion reduction. Further, investment in public transport can reduce
acceptance problems. Moreover, as stated by Kottenhoff and
Freij (2009), the use of pricing revenue for improving public transport
is required in order to increase the elasticity of road system users (who generally
tend to keep their modal choices unchanged) and compensate for a decrease in
service quality due to the increase in crowding of public transport vehicles.
However, it is worth noting that the improvement in public transport should
be implemented before or simultaneously with pricing policy application in order
to avoid the above-mentioned problems. Indeed, four months before the beginning
of the Stockholm congestion charging trial the public transport services were
extended by 7% (Eliasson et al., 2009). Likewise,
when the Ecopass System was implemented in Milan, the bus service improved by
16% (Municipality of Milan, 2009).
In Norway, tolls are used to finance both urban and inter-urban road projects.
In the three largest cities of Oslo, Bergen and Trondheim cordon tolls make
up the main funding of road investments and, to a certain extent, public transport
investment programmes (Odeck and Brathen, 2008). As
part of the revenues are used for the pricing system operating costs, the revenue
rate that can be allocated to public transport improvement depends on the toll
collection system adopted. In other words, if we consider the Oslo Toll Ring,
an electronic cordon pricing scheme, operational costs are only 10% of the total
revenue (Ieromonachou et al., 2007). Instead,
if we consider London Congestion Charging, an area pricing scheme with no electronic
system for toll collection, for the financial year 2007/2008 total costs (scheme
operation, publicity, enforcement, Transport for London staff, traffic management,
Transport for London central costs) were £131 million and total revenues
(daily vehicle charges, enforcement income received) were £268 million
(TFL, 2008), making operational costs for London Congestion
Charging 49% of total revenue. Obviously, it is impossible to compare the Oslo
Toll Ring and London Congestion Charge because their aims are very different
(to provide investments for transport programmes, in the first case and to reduce
congestion, in the second). It is important to note that the revenue rate available
can vary greatly according to the toll collection system.
In this study we provide an analysis of the fare definition problem when part
of pricing revenues are used for improving public transport. Indeed, a multimodal
approach in the toll computation model is generally adopted, neglecting revenue
use (Gentile et al., 2005) or calculating fares
in the case of homogeneous networks (Huang, 2002; Ferrari,
2005). Therefore, we formulate a toll computation model through a multidimensional
constrained optimisation problem in multimodal and multiuser context with elastic
demand. This model is then applied on a heterogeneous (i.e., any origin-destination
pair has different accessibility) trial network to analyse several second-best
strategies (such as cordon and parking pricing) and estimate possible effects
of road pricing revenue use on social welfare and fare levels.
TOLL COMPUTATION MODEL AND ALGORITHM
In the literature (Cascetta, 2001), a transportation
network can be modelled by means of graph theory: a graph can be defined as
an ordered pair of sets N and L, where N is the set of elements indicated as
nodes and L is a set of pairs of nodes belonging to N indicated as links.
Links of a graph in a transportation network represent trip phases, such as physical movements on a road or train waiting at a station. Likewise, nodes of a graph describe the transition among trip phases (i.e., links), such as vehicles that pass through an intersection. Moreover, it is necessary to associate to each link: an impedance function and a flow value. Impedance functions, known as cost functions represent the resource consumption associated to the link crossing; flow values indicate the average number of users or vehicles which, in a unit of time, cross the considered links.
In a transportation network, there is a peculiar class of nodes associated to each traffic zone (a portion of territory in which the analysed area is divided) indicated as centroid nodes that represent the origin and/or destination of all trips related to the considered zone. Finally, a path can be defined as a sequence of successive links connecting an origin centroid node with a destination centroid node, where each origin-destination pair can be connected by means of more than one single path.
In order to model effects of the transportation system on people choices, it is necessary to divide the population into user classes, where each class represents a homogeneous category with respect to socio-economic features, such as car availability or income levels.
The User Equilibrium (UE) condition, that is the network configuration resulting
from the interaction between people preferences and transportation system performance,
can be formulated mathematically through the interaction between two kinds of
models: supply models, that simulate the performance of transportation systems
depending on user (or vehicle) flows and demand models, that imitate user choices
influenced by transportation system performance. A supply model can be summarised
by the following analytical relation:
where: Cmi,h is the vector of path generalised costs
for user class i and mode m (such as car or bus) at time period h, of dimensions
(nm-Paths x1), whose generic element expresses
the generalised cost of path k for user class i on mode m at time period h;
Am is the link-path incidence matrix for mode m, of dimensions (nm-Links
x nm-Paths), whose generic element al,km =
1 if link l belongs to path k, 0 otherwise; cmi,h is the
vector of link generalised costs for user class i and mode m at time period
h, of dimensions (nm-Links x1), whose generic element clm,i,h
expresses the average cost perceived by users of class i when they pass through
link l of mode m at time period h (in a congested network this vector depends
on total link flows fhm of mode m at time period h); is
the vector of link flows of mode m at time period h, of dimensions (nm-Links
x1), whose generic element flm,h expresses the average
number of travellers that pass through link l on mode m in a time unit during
time period h, which is equal to the sum of flow vectors fmi,h
related to each user class i, of dimensions (nm-Links x1), whose
generic element flm,i,h expresses the average number of
class i users that pass through link l on mode m in a time unit during time
is the vector of non-additive path costs for user class I and mode m at time
period h, of dimensions (nm-Paths x1), whose generic element expresses
costs that depend only on path k of mode m for user class i at time period h
(such as road tolls at motorway entrance/exit points).
The demand model can be defined as a mathematical relationship that associates
average demand flows to a given activity and transportation supply systems.
Travel demand models are usually derived from random utility theory based on
the hypothesis that every user is a rational decision-maker maximising utility
relative to his/her choice (Cascetta, 2001). Generally
the utility associated to each travel choice is a linear function of generalised
travel cost. The demand model can be described by the following mathematical
where, Fmi,h is the vector of path flows for user class
i and mode m at time period h, of dimensions (nm-Paths x1), whose
generic element Fkm,i,h expresses the average number of
class i users that choose path k of mode m in a time unit during time period
h; Pmi,h is the path choice probability matrix for user
class i and mode m at time period h, known as the path choice map, of dimensions
(nm-Paths xnOD-Pairs), whose generic element expresses
the probability that class i users travelling between origin-destination pair
od choose path k of mode m at time period h, which generally depends on path
generalised cost matrix Cmi,h; dmi,h
is the travel demand vector for user class i and mode m at time period h, of
dimensions (nOD-Pairs x1), whose generic element expresses
the average number of class i users travelling between origin-destination pair
od on mode m in a time unit during time period h. If travel demand is assumed
elastic, vector dmi,h depends on path cost vectors Cmi,h;
otherwise it is fixed.
Combining Eq. 1 with 2, we obtain the well-known assignment
model formulated as a fixed-point model (Cantarella, 1997):
the vector of link flows for mode m in the UE condition at time period h; is
the vector of link flows of user class i for mode m in the UE condition at time
Because the UE condition is not efficient from an economic point of view, tolls need to be charged to affect user behaviour in order to achieve the best utilisation of the transportation system. Since there are two kinds of user costs, additive and non-additive (as shown in Eq. 1), we need to introduce two kinds of fares: additive (indicated as y) and non-additive (indicated as Y). Formally, non-additive fares have to be considered as path fares. Such tolls are applied to groups of paths that satisfy several conditions (such as all paths that join the same origin-destination pair). Thus, the real number of variables to be optimised is lower (indeed, we need a variable for each group) and it is not necessary to explicitly enumerate all considered paths (path enumeration is generally performed implicitly by the assignment algorithm). An example of an additive fare is the toll that depends on the length of the link (if a car crosses two links it will pay a fare equal to the sum of the two corresponding tolls), while an example of a non-additive fare is parking pricing because it depends only on the place where users park and parking duration and does not depend on the number (or length) of crossed links to reach ones destination.
Moreover, if we consider a fixed policy (i.e., the fare does not depend on
the time spent in the system), variables y and Y represent the fare value, while
if we consider hourly policies (i.e., fares depend on the time spent in the
system, such as parking fares), variables y and Y represent the cost per unit
time and the corresponding value for each user can be obtained by multiplying
these values by the time spent in the system.
Toll determination can be formulated as a multidimensional constrained optimisation problem which, according to economic theory, consists in finding out the fare values í and ì which maximise the social surplus (the opposite of objective function Z), that is:
where: y is the vector of additive fares, of dimensions (nRoad-Links x1), whose generic element yl represents the value of pricing applied to link l; Y is the vector of non-additive fares, of dimensions (nRoad-Paths x1), whose generic element Yk represents the value of pricing applied to path k; í [ì] is the optimal value of vector y [vector Y]; Sy [SY] is the feasibility set of vector y [vector Y], which expresses minimum and maximum value for each additive [non-additive] fare; Z is the objective function to be minimised, which is equal to the opposite of social surplus; Λ is the assignment function; Sfm is the feasibility set of link flows of mode m, which expresses flow consistency (for instance, the sum of all incoming flows in a node has to be equal to the sum of all outgoing flows if the node is not a centroid).
The first constraint (Eq. 4) represents the multimodal assignment
constraint that provides user flows on all transportation systems as
a function of design variables (y and Y) and user flows on all transportation
However, it is worth noting that with respect to the multimodal assignment model
proposed in the literature in this case we also take into account the crossed
congestion phenomenon, that is an increase in flows on the road system provides
an increase in travel times also for public transport if operating on shared
links. This assumption introduces some theoretical complications because it
does not satisfy some assumptions proposed by Cantarella
(1997) for stating the uniqueness of the equilibrium solution in the case
of multimodal approaches. Instead, the second constraint (Eq.
5) indicates that flows have to satisfy consistency conditions (as indicated
in the definition of set Sfm). The objective function Z is assumed
equal to the opposite of social surplus and can be expressed as the sum of User
Surplus (US), Traffic Revenues (TR), System Costs (SC) and Environmental Costs
The US can be used as an indicator of transportation user welfare. It thus represents the net utility that users obtain by making travel choices and equals the average maximum perceived utility in a stochastic approach, that is the EMPU (Expected Maximum Perceived Utility) variable. The Logit model used for modal split allows the EMPU variable to be expressed in closed form as indicated by the following relation:
where: is the travel demand on pair o-d related to user class i at time period
h; is the EMPU variable, calculable with the well-known formula:
is the systematic utility that users of class i associate to mode m on pair
o-d at time period h and is
the Logit parameter.
The second and third terms in the social surplus expression are the amount of traffic revenues, provided by public and private systems and the amount of system costs, respectively. The latter term is the sum of the operating costs of the transit system, that are a function of performance parameters and the operating costs of the pricing system, assumed constant with respect to network flows:
where is the path flow vector in equilibrium condition for each user class
i and each mode m at time period h, of dimensions (nm-Paths x1),
whose generic element Fkm,h expresses the average number
of travellers that choose path k on mode m in a time unit during time period
h; this vector is equal to
where: OCMT are the operating costs of the mass transit system, depending on performance parameters of the mass transit system (PPTS); OCPS are the operating costs of the pricing system, that are constant with respect to network flows and variable according to the toll collection system.
The last term is the amount of external costs, which are expressed in the literature as a function of equilibrium flows, that is:
In order to solve the optimisation problem (3) we have adopted the meta-heuristic
algorithm proposed by DAcierno et al. (2006).
This algorithm is based on three steps: an Exhaustive Mono-dimensional Optimisation
(EMO) phase; a Starting Solution Definition (SDD) phase and a Neighbourhood
Search Optimisation (NSO) phase.
In the first phase, each variable is optimised exhaustively, assuming the values of other variables as constant. Phase 2 is obtained by setting each variable equal to the optimal value calculated in the previous phase. Finally, the last phase evaluates the solutions that can be obtained from the current solution by an elementary move (i.e., only one variable is increased or decreased while the other variables are maintained constant) and identifies a new solution to be analysed.
This meta-heuristic algorithm was adapted in order to manage the peculiar design
variable framework (variables y and Y) that, is different from the optimisation
model proposed by DAcierno et al. (2006).
Characterisation of the variables is given in the following section.
TRIAL NETWORK APPLICATION
The model formulated in the previous section was applied on a trial network
to verify the efficiency of several pricing policies. The trial network is shown
in Fig. 2. Network users can choose among three transport
modes: car, transit and pedestrian. However, the accessibility variable is the
main network feature, according to a real network of a large metropolitan area:
for instance, between node 4 and node 2 there is no transit system that joins
this pair directly. For the demand, we considered three user classes that differ
in modal choice set and parking time: users who have no car availability (10%
of total demand for each O-D pair), users who have car availability and have
to stay at their destination for 2 h (20% of total demand for each O-D pair)
and users who have car availability and have to stay at their destination for
6 hours (70% of total demand for each O-D pair). The demand for each O-D pair
is such as to produce high congestion levels in the network when no pricing
policy is applied.
Only second-best policies were tested. We applied cordon pricing and some parking
pricing policies: a parking policy with undifferentiated fares (in each zone
the same fare is applied), a destination-based parking policy (the traditional
strategy where road users pay for parking and the fare is a function of where
they park) and an Origin-Destination (OD) parking policy, that is a strategy
proposed by DAcierno et al. (2006) where
road users pay a fare based on both trip origin and destination. All proposed
parking strategies have been applied in two cases: the case of fixed fares (in
this case variables y and Y express the fare value) and the case of hourly fares
(in this case variables y and Y express the fare per unit time and the corresponding
value for each user can be obtained by multiplying these values by the parking
time). Obviously, in the case of cordon pricing policy the fare can be only
fixed (indeed, the application of an hourly cordon pricing policy would require
the implementation of advanced technologies able to calculate the time spent
by each vehicle in the charging area). Furthermore, for each policy we considered
different applications related to the use of pricing revenue for improving public
transport. In particular, we considered five use schemes: pricing revenues not
used, 25% of pricing revenues used, 50% of pricing revenues used, 75% of pricing
revenues used and 100% of pricing revenues used.
However, it is worth noting that, in real contexts, the percentage of pricing
revenues which can be used to improve other transportation systems (such as
public transport) depends on operating costs of the pricing system. For instance,
when a cordon pricing strategy with electronic toll collection system is implemented,
operating costs can be very low such as in the Oslo Toll Rings system (Ieromonachou
et al., 2007), such that there is a considerable availability of
money to finance other projects or systems.
On the contrary, in parking strategies, it could be difficult to use a fully
electronic collection system (except in restricted areas such as controlled
parking lots) because ticket inspections would require physical inspection by
traffic policeman or other operators with the consequent increase in operating
costs. Therefore, for parking strategies the percentage of revenues which can
be used for improving public transport is seldom higher than 50%.
Although the trial network is the same as that reported by DAcierno
et al. (2006), in this case some modifications to the algorithm framework
were applied. In particular, the optimisation model proposed by DAcierno
et al. (2006) designed only hourly parking fares. In our application
we considered both fixed parking fares (in this case design variables represent
the real value of the fare and not the fare per time unit) and cordon pricing
fares (in this case the elements of variable Y are equal to zero if the destination
of the path is a non-cordoned area and equal to the pricing value if the destination
is the cordoned area). Moreover, the solution algorithm was modified to take
systems due to the use of a share of pricing revenues for improving public transport.
Figure 3-7 show optimisation results for
each pricing policy with different percentages of revenue used for improving
public transport. In particular in each figure, the value of the objective function
is provided, expressed in its Euro equivalent (i.e., travel times are expressed
in euros by means of value of time coefficients) per hour.
||Objective function value when pricing revenues are not utilised
||Objective function value when 25% of pricing revenues are
utilised for improving public transport
||Objective function value when 50% of pricing revenues are
utilised for improving public transport
It is worth noting that in any case pricing policies with hourly fares (and with any percentage of revenue used to improve public transport) are always better than cordon pricing. Moreover, the best policy seems to be the parking one based on origin-destination fares.
Figure 8 provides a comparison among the analysed strategies
(both in terms of fixed and hourly fares) in the case of different percentages
of pricing revenues used to improve public transport. The main result is that
the best value of the objective function can be found for any strategy in the
case of between 25 and 50% revenue used.
||Objective function value when 75% of pricing revenues are
utilised for improving public transport
||Objective function value when 100% of pricing revenues are
utilised for improving public transport
The reason for this result is related to the fact that in the case of values
lower than 25% the increase in public transport performance can be neglected,
while in the case of values higher than 50% the great attractiveness of public
transport over the private system provides a reduction in pricing revenues and
therefore the amount of resources available for public transport. Hence the
system reaches an optimal configuration that represents a compromise between
the need to reduce the number of car users and the fact that they themselves
are implicitly the main backers of public transport.
Analysis of the modal split (Fig. 9-13)
shows that hourly fares always provide higher travel demand for public transport
and the strategy that provides the highest value of public transport use is
always parking pricing based on both trip origin and destination. With respect
to revenue use, for values greater than 25% the modal split values are constant
because the system achieves an equilibrium condition related to the reduction
in public transport backers.
Generally, the adopted objective function does not provide a maximisation of
public transport demand but the joint maximisation of revenues and accessibility.
Indeed, modal splits indicated in Fig. 9-13
represent the average values on the network without taking into account different
levels of accessibility of each origin destination pair. Therefore, the adopted
objective function (described in Eq. 6), which allows to take
into account the maximisation of relative accessibility of each origin destination
pair, can be considered a good tool for design transportation systems.
Finally, it may be shown that revenue use for increasing service frequencies
allows fare levels to be decreased, where this reduction depends on performance
parameters of bus system. Figure 14 describes these features
comparing the cordon pricing strategy in two cases: the use of 50% of revenues
for improving public transport and the lack of revenue use.
||Objective function value for different pricing policies and
different percentage of pricing revenue use
To give details, we have provided the fare variation with respect to initial
service frequency of bus lines in the cordon pricing case. The service frequency
is supposed equal for three bus lines of the trial network. The fare reduction
due to pricing revenues increases as the initial service frequency increases.
From an economic standpoint, the revenue use is positive, because the social
surplus improves and the fares can be reduced, thereby also gaining public opinion
acceptability. Thus we can say that a well-designed parking policy can be more
effective as a travel demand management instrument than a policy which charges
users a toll to reach the city centre.
||Public transport demand when pricing revenues are not used
||Public transport demand when 25% of pricing revenues are utilised
for improving public transport
||Public transport demand when 50% of pricing revenues are utilised
for improving public transport
||Public transport demand when 75% of pricing revenues are utilised
for improving public transport
Revenue use is to be preferred because the social surplus increases and fares
can be reduced.
According to economic theory (Pigou, 1920; Beckmann
et al., 1956) the best utilisation of a transport system can be achieved
by charging each user with an additional cost equal to the difference between
the marginal cost and the individual average cost. Since users generally have
different socio-economic features, their individual average cost could be different.
Hence, the best strategy to be adopted is the one that allows different (socio-economic)
classes of users to be charged different fares.
||Public transport demand when 100% of pricing revenues are
utilised for improving public transport
||Fare reduction obtained when 50% of pricing revenues are utilised
for improving public transport
In our trial network the main differences between users is car availability
and parking time at destination. Therefore, in this case, the best strategy
(i.e., the strategy that allows different fare levels to be applied according
to class features) is hourly parking pricing based on the origin-destination
of trips since it only charges car users a fare depending on their parking times.
CONCLUSIONS AND RESEARCH PROSPECTS
In this study we formulated a toll computation model according to economic
theory of efficient tolls that can be charged on road links to achieve the best
use of a transportation network. Since for several reasons theoretical tolls
are impossible to apply, we analysed the effectiveness of some second-best policies,
such as cordon and parking pricing, through the application of the formulated
model on a trial network. In line with a real network in a large metropolitan
area, we used a multimodal network with variable accessibility among Origin-Destination
pairs and mobility demand segmented according to socio-economic features. In
particular, if we consider users with different parking times, road pricing
may not be the best strategy, while a parking pricing policy, such as an Origin-Destination
(OD) parking policy, may prove more effective as a mobility management instrument.
Furthermore, a parking pricing policy involves fewer unacceptable problems than
a road pricing policy, which hardly ever meets the approval of both public opinion
and political decision-makers. As many authors suggest, the revenue use for
public transport improvement is positive, because not only does the social surplus
value improve but fares can also be reduced, thus also gaining public opinion
acceptability. Obviously, these considerations on the effectiveness of the examined
pricing policies are related to the features of the analysed trial network.
In a future project, we propose to analyse these conditions on a real network
to explore the problem of the revenue percentage to be utilised, since current
indications show that the percentage is variable and depends on the toll collection