Fares for mass-transit services are generally set below the profitability threshold:
Ticket revenues cover only a (minor) part of operational costs while the other
part is subsidised by public funds (details on methodologies for calculating
public transport subsidies can be found, for instance, in the study of Hao
et al. (2009), Yang et al. (2010) and
Tscharaktschiew and Hirte (2012). Such fare policies
are adopted for social and environmental reasons. Social aspects of the problem
have been explored by Chapleau (1995), Hodge
(1998), Obeng (2000), Parry
and Bento (2002), Paulley et al. (2006),
Abrate et al. (2009), Nuworsoo
et al. (2009), Attard (2012) and Kallbekken
et al. (2013). More generally, the effects of mass-transit pricing
policies have been studied by Ballou and Mohan (1981),
Ferrari (1999), Karlaftis and
McCarthy (2002), Zhang et al. (2006), Jansson
(2008), Proost and van Dender (2008) and Basso
and Jara-Diaz (2012).
In this study, an optimisation model for establishing mass-transit fares is
proposed. In accordance with the problem characteristics, the model satisfies
the following requirements:
||It assumes transportation demand as elastic, for simulating
changes in modal split
||It simulates road and mass-transit transportation systems jointly, to
take into account effects of modal split changes on congestion
||It considers different user classes with different socio-economic attributes
||It assumes an objective function that considers all costs: Operational
costs, user costs of both road and mass-transit systems and external costs
Therefore, this problem can be seen as a multimodal network design problem
(Montella et al., 2000; DAcierno
et al., 2011a; Cipriani et al., 2012)
in which mass-transit fares assume the role of decision variables. A general
formulation of the pricing problem was provided by Cascetta
(2009) and the multimodal nature of the analysed problem was highlighted
by Huang (2000), Casello (2006),
Chien and Tsai (2007), Gallo
et al. (2011a, b), Gkritza
et al. (2011) and Basso and Jara-Diaz (2012).
Finally, Osula (1998) showed how changes in mass-transit
fares can also modify trip generation.
GENERAL SOLUTION APPROACH
Generally, establishing public transport fares may requires five phases (Fig.
1) that can differ according to the concerned study area. In the area identification
phase the area covered by the mass-transit services under study is identified;
it can be a city, or part of it, a province or a region. The fare zone definition
phase consists of subdividing the territorial area into zones if it is so large
that fares have to be differentiated by trip length. The zones should take account
of administrative divisions, land uses, pre-existing fare regulations and user
perception of fared zones. The mass-transit fare optimisation phase searches
for the optimal configuration of mass-transit fares and is the focus of this
study. Evaluation of results can suggest modifications to area zoning, by comparing
different results the best zoning-fare combination can be identified.
FARE ZONE DEFINITION
If the area is so large that it is not fair to fix the same fare for all trips,
the area has to be subdivided into fare zones. This is common practice in metropolitan
areas, provinces and regions.
|| General approach to mass-transit fare optimisation
Several methods have been proposed in this respect, based on distances (Cervero,
1982; Daskin et al., 1988; Clark
et al., 2011; Borndorfer et al., 2012),
time intervals (Cervero, 1981), average travel times
(Phillips and Sanders, 1999) or number of zones crossed
(Schobel, 2006; Sharaby and Shiftan,
2012). However, three main methods of fare zoning can be identified (Fig.
2): Concentric zones, circular rings and sector zones and alveolar zones.
Concentric zones are mainly adopted where there is a major centre that attracts
and/or generates most trips (e.g., a capital city). In this case, also fares
for tangential trips are provided. If such trips are considerable, circular
rings and sector zones are more commonly used. In this case fares depend on
the number of zones crossed. Finally, when there is no major centre, the area
can be subdivided into alveolar zones: Fares depend on the number of crossed
zones as well.
Although, theoretically speaking, each fare can be independent of the others,
in general each fare can be defined according to the minimum fare, that is the
fare of a trip within a single zone and the number of crossed zones. This relation
can be expressed as:
where, Tb is the mass-transit fare matrix of dimensions (nTicketsxnMaxCrossedZones),
whose generic element tin is the fare of the i-th ticket
type that allows up to n zones to be crossed; tb0 is the
vector of basic mass-transit fare of dimensions (nTicketsx1), whose
elements ti0 are the first row elements of matrix Tb
and represent fares that allow travel only within the same zone (intra-zonal
trips); 1 is a vector of dimensions (nTicketsx1), whose elements
are all equal to 1; tb* is the vector of basic variation in fares
of dimensions (nTicketsx1), whose elements represent the unit variation
in fares without considering any corrective coefficient; A is the corrective
coefficient matrix of dimensions (nTicketsxnMaxCrossedZones),
whose generic element ain is a corrective coefficient
related to the i-th ticket type that allows up to n zones to be crossed; N is
a vector of dimensions (nMaxCrossedZonesx1), whose generic element
nj is equal to (j-1).
|| Types of fare zone
Equation 1 is equivalent to the following:
where, Γ is a matrix of dimensions (nTicketsxnMaxCrossedZones),
whose generic element γin is a coefficient related
to the i-th ticket type that allows up to n zones to be crossed.
Indeed, since both previous equations are linear, assuming:
where, A* is a vector whose generic element ai* is equal to the
ratio between ti* and ti0, Eq.
1 may be transformed into Eq. 2 via the following relation:
is a matrix of dimensions (nTicketsxnMaxCrossedZones),
whose elements are all equal to 1.
Thus, we proposed to adopt Eq. 2 for defining fare zone criteria
using the matrix Γ as the only parameter.
MASS-TRANSIT FARE OPTIMISATION MODEL
The proposed mass-transit fare optimisation model can be formulated as follows:
is the optimal value for Tb,
is the feasibility set of matrices Tb, Z(·) is the objective
function to be minimised, fc* is the equilibrium link flow matrix
for the road system of dimensions (nRoadLinksxnUserCategories),
fb* is the equilibrium link flow matrix for the mass-transit system
of dimensions (nTransitLinksxnUserCategories), Λ(·)
is the multimode assignment function which provides equilibrium flows as a solution
of a fixed-point problem, TR is the monetary value obtained by selling public
transport tickets, generally indicated as mass-transit revenues which obviously
depends on fare values (Tb) and number of mass-transit system users
(fb*), α is a term which expresses the part of operational costs
not covered by subsidies, B is the budget used for implementing the mass-transit
system, generally equal to the total operational cost,
is the feasibility set for fc* and
is the feasibility set for fb*.
Constraint (Eq. 5) represents the multimodal equilibrium
assignment; it constrains road and mass-transit flows to be in multimodal equilibrium
for each configuration of mass-transit fares, Tb. Equilibrium flows
can be calculated by means of three kinds of models: Supply, demand and flow
Supply models describe the performance of transportation systems in relation
to user flows (e.g., an increase in vehicles on a road generally produces an
increase in travel times). Such models (Cascetta, 2009)
can be formulated by means of the following equation:
where, Cim is the vector of generalised path costs associated
to mode m (where m = c in the case of road systems and m = b in the case of
mass-transit systems) for user category i, of dimensions (nRoadPathsx1)
in the case of road systems and (nTransitHyperpathsx1) in the case
of mass-transit systems. Indeed, in the case of mass-transit systems, we adopt
the hyperpath approach proposed by Nguyen et al.
(1998) in order to simulate preventive-adaptive choice behaviours since
users are unable to set their itineraries preventively. Indeed, since the physical
path depends on the arrival of vehicles at boarding stops, users choose the
set of attractive lines beforehand (preventive stage) and the line used according
to arrival events (adaptive stage), Δm,i is the link-path (or
link-hyperpath) incidence matrix associated to mode m for user category i, whose
is equal to 1 if link l belongs to path (or hyperpath) k, 0 otherwise, of dimensions
(nRoadLinksxnRoadPaths) in the case of the road system
and (nTransitLinksxnTransitHyperaths) in the case of the
mass-transit system; cim is the vector of generalised
link costs associated to mode m for user category i, of dimensions (nRoadLinksx1)
in the case of the road system and (nTransitLinksx1) for the mass-transit
system, fc is the vector of generic link flow associated to the road
system, of dimensions (nRoadLinksx1), fb is the vector
of generic link flow associated to the mass-transit system, of dimensions (nTransitLinksx1),
is the vector of non-additive path costs associated to mode m for user category
i, whose generic element
expresses costs that depend only on path k (such as road tolls at motorway entrance/exit
points in the case of the road system and waiting time at bus stops/train stations
or fares for the mass-transit system), of dimension (nRoadPathsx1)
for the road system and (nTransitHyperpathsx1) for the mass-transit
system. Therefore, path cost vectors for the road system can be assumed as constants
while in the case of the mass-transit system they have to be assumed at least
as a function of vector Tb, that is:
Likewise, demand models imitate user choices influenced by transportation system
performance. In particular, these models are based on the assumption that users
are rational decision makers maximising utility (or equivalently, minimising
generalized costs) through their choices. These models can be formulated by
means of the following equation:
is the vector of path (or hyperpath) flows associated to mode m for user category
i, of dimensions (nRoadPathsx1) for the road system and (nTransitHyperpathsx1)
for mass-transit; Pim is the matrix of path (or hyperpath)
choice probabilities associated to mode m for user category i, whose generic
expresses the probability of people travelling between origin o and destination
d on path (or hyperpath) k, of dimensions (nRoadPathsxnOriginDestinationPairs)
for the road system and (nTransitHyperpathsxnOriginDestinationPairs)
for mass-transit system; dim is the vector of travel demand
associated to mode m for user category i, whose generic element
expresses the average number of users travelling between origin o and destination
d in a time unit, of dimensions (nOriginDestinationPairsx1).
It is worth noting that matrix Pim depends only on path
costs of the same transportation system while vector dim
depends on path costs of all transportation systems.
Flow propagation models describe the relation between path costs and link flows.
In particular, with the assumption of intra-period stationarity, such models
can be expressed as:
Since the flow on a link can be expressed as the sum of all flows on the same
link belonging to different user categories, that is:
The interaction between supply, demand and flow propagation models, in order
to obtain equilibrium flows, provides the following equation:
By splitting Eq. 12 for each transportation system and explicitly
expressing the dependence of mass-transit non-additive hyperpath costs on mass-transit
fares, we obtain the following fixed-point formulation (Cantarella,
1997; D'Acierno et al., 2011b):
which can be reduced to:
that is constraint (Eq. 5). The solution of the fixed-point
problem (Eq. 13) or equivalently (Eq. 5),
consists in obtaining road and mass-transit flows which generate road and mass-transit
generalised costs that produce a modal split and path choices such that the
same flows are reproduced. For solving the multimodal equilibrium assignment
problem, in this study we adopt the fixed-point model and the solution algorithms
proposed by DAcierno et al. (2011b).
Under constraint (Eq. 6) ticket revenues have to cover at
least the share of operational costs which is not subsidised by public authorities.
According to constraint (Eq. 7) flows have to belong to feasibility
sets that express consistency of flows (for instance, the sum of all ingoing
flows in a node has to be equal to the sum of all outgoing flows if the node
is not a centroid).
Since the calculation and check of Eq. 5 requires implementation
of suitable algorithms, the proposed model shows a bi-level formulation: The
upper level is the optimisation model (Eq. 4) and the lower
is the assignment problem (Eq. 5) subject to Eq.
7. Therefore, if matrix Γ is fixed, through Eq. 2,
the optimisation model (Eq. 4) can be simplified as:
is the optimal value and
the feasibility set of tb0. This second model (Eq.
14-17) has less complexity since vector tb0
has only nTickets elements, while matrix Tb has nTicketsxnMaxCrossedZones
elements to be optimised.
In this study, two objective functions are tested; the first is the objective
function adopted in several mass-transit network design problems that considers
only system costs and user costs, adapted to our multimodal problem:
where, NOTC(·) is the Net Operational Transit Cost and UGC(·)
is the User Generalised Cost of all transportation systems.
The NOTC term can be expressed as the difference between the Total Operational
Transit Cost (TOTC) and Ticket Revenue (TR), that is:
In particular, we proposed to adopt the following formulations:
is the standard cost per kilometre of line λ, expressed in euro/km; Lλ
is the length of line λ, expressed in km; φλ is the
service frequency of line λ, expressed in buses/h or trains/h;
is the average number of users of the mass-transit system, belonging to user
category i, travelling from origin o to destination d, expressed in users/h;
is the value of the ticket for user category i entitling him/her to travel from
origin o to destination d, expressed in Euros.
The UGC term can be expressed as the sum of the Road System User Cost (RSUC)
and Transit System User Cost (TSUC), that is:
is the vector of waiting times of mass-transit system users belonging to category
i, whose generic element Twib,k expresses the average
total waiting time at bus stops/train stations spent by users travelling along
hyperpath k, of dimensions (nTransitHyperpathsx1).
We expressed the non-additive hyperpath cost for the mass-transit system by
splitting waiting time and ticket costs in order to show that in the objective
function (Eq. 18) the term TR has both a positive (namely
mass-transit system revenues) and negative (ticket costs) sign. Hence, it can
be omitted. Therefore, mass-transit fares in the proposed optimisation model
do not appear explicitly in the cost function formulation but implicitly by
means of assignment (Eq. 5) and budget (Eq.
Moreover, we propose to adopt a second objective function which also considers
external costs, EC(·), produced by road traffic, that is:
Also in this second case mass-transit revenues and ticket costs for users are
not present since they cancel each other out. Details on external cost formulation
adopted in this model can be found in a Gallo et al.
(2011a), where the same numerical parameters are applied in a real scale
The proposed model was tested on a trial network and on a real-scale network.
The descriptions of networks and results of the tests are reported as follows.
|| Simple network framework
Trial network: The proposed optimisation model was tested on the simple
network of Fig. 3, where, neglecting Eq. 6
(or equivalently Eq. 15) the problem may be solved in a closed
form; this network has one link, a, shared by road and mass-transit systems
and there is a single bus line l. Moreover, it is assumed that there is a single
fare zone, a single ticket type, a single user category and hence a single fare,
The Total Operational Transit Cost (TOTC) is calculated as:
where, cloc is the unit operational cost of line l, Ll
is the length of line l and φl is the service frequency of line
Road System User Cost (RSUC) on the link is equal to the sum of user travel
times, indicated as UTCc,a and expressed by means of cost function
ca (fc,a) and the Road Monetary Costs (road/parking pricing),
RMCa, applied on the link.
Likewise the User Transit System User Cost is equal to the sum of user travel
times, indicated as UTCb,a and the mass-transit ticket cost tb0.
In particular, UTCb,a is equal to the sum of the on-board time, assumed
equal to the road travel time, ca(fc, a), the waiting
time, equal to the ratio between a regularity term ηl and the
frequency φl of the line and the access-egress time, cp,
The mode choice model is expressed by means of a Multinomial Logit model, where
the systematic utility of road and mass-transit users, indicated as Vc
and Vb, are as follows:
where, MSEc and MSEb are the socio-economic variables
of modal choice model, respectively for the road and mass-transit system. Moreover,
the β terms are the parameters of the model. Therefore the road travel
demand can be estimated as:
and the mass-transit travel demand is:
Assuming that capacity constraints are not present on the mass-transit system,
the mass-transit (user) flow, fb,a, is equal to the mass-transit
demand, db; moreover, the road (vehicle) flow, fc,a, is
equal to the ratio between the road demand, dc and the occupancy
index, δc, assumed constant. Therefore, objective function (Eq.
18) can be written as:
Since objective function (Eq. 32) is continuous with continuous
first and second partial derivatives, the road travel time function is continuous
with continuous first and second partial derivatives and the feasibility set
of mass-transit fares is a closed interval [0, tb0,max],
of Eq. 14 is one of the points among endpoints of feasibility
interval (i.e., 0 and tb0,max) and values
(of the above feasibility set) that satisfy conditions:
If road travel time is estimated by means of a BPR function, that is:
Equation 33 may be stated to be satisfied with:
|| Objective function chart by TWT value
where, ca0 is the free-flow road travel time on link
a, that is equal to the ratio between the length of link a and the free-flow
average speed on the same link; αaBPR and βaBPR
are parameters of the BPR function; Capa is the capacity of link
a. Therefore, if condition of Eq. 36 is not satisfied, the
solution of Eq. 14 is one of the endpoints of interval [0,
Adopting the parameters in Table 1, the solution of Eq.
4 may be analysed. In particular, Fig. 4 shows objective
function (Eq. 32) by different TWT values, with:
where, the TWT variation is obtained by φl variation.
|| Parameter values
A major result is that with TWT values lower than 8 min and higher than 10
min the solution is an endpoint of feasibility interval (in this case tb0,max
is equal to _10.00). Besides when the solution is a value that puts the first
derivative equal to zero, the mass-transit fare increases when the TWT value
increases. This means that, when the mass-transit travel time increases with
respect to the road travel time, the system optimum (hence the minimum value
of the objective function) can be obtained by increasing mass-transit fares
and hence moving users from the mass-transit system to the road system. Indeed,
although the increase in the number of road users increases both road and mass-transit
travel times and hence the objective function value, the decrease in the number
of mass-transit users reduces the number of users that incur the Transit Waiting
Time (TWT) and yields a decrease in the objective function that counterbalances
the increase due to travel time increases.
Figure 5 shows that Eq. 35 has an asymptotic
trend to endpoints of its feasibility interval (Eq. 36).
In this case the TWT variation yields a translation of
and endpoints of Eq. 35, where the difference is equal to
the product of the value δc, TWT variation and the ratio between
βaTIME and βaCOST.
|| Mass-transit fares by RMCa and TWT values
|| Mass-transit fares by RMCa and values of time
Another important result is that, in fixing TWT values, an increase in road
monetary costs yields a decrease in mass-transit fares. Indeed, an increase
in road monetary costs increases the objective function value while a decrease
in mass-transit fares yields a decrease in the number of road users that entails
a decrease in both road and mass-transit travel times, a reduction in users
that have to support road monetary costs and hence a reduction in the objective
function value that counterbalances the increase due to the rise in road monetary
costs. Finally, the dotted line corresponding to the road monetary cost of _1.50
intercepts curves at solution points in Fig. 4.
An increase in the ratio between βaTIME and βaCOST
(for instance by fixing βaTIME and reducing βaCOST),
yields an increase in both the width of the feasibility interval and the mass-transit
fare (Fig. 6).
|| Objective function chart by βTUV values
|| Real-scale features
Introducing the external costs in the objective function, assuming that:
where, βTUV is the Transit User Value that expresses the value
that society associates to a user travelling on the mass-transit system, the
objective function (Eq. 32) can be written as:
Also in this case solution
of Eq. 14 is one of the endpoints of the feasibility interval
(i.e., 0 and tb0,max) and values
(of the above feasibility set) that satisfy the Eq. 33 for
Finally, Fig. 7 shows that an increase in term βTUV
yields a decrease in mass-transit fares.
Real-scale network: The proposed approach was tested also on a real-scale
network (Fig. 8, with features reported in Table
2) in order to ascertain the applicability of the proposed model. In particular,
we sought the optimal value of multiplier μ which allows us to:
||Modify all existing fares proportionally
||Minimise both the proposed objective functions
||Jointly satisfy assignment and budget constraints
We developed and implemented a solution algorithm on a PC Intel Core2 Quad
Q6600 2.40 Ghz which provided results in 26.3 min (i.e., about 0.8 min per solution
examined). Table 3 synthesises cost function terms for each
considered multiplier value. Figure 9 shows objective function
values depending on the multiplier value, while Figure 10-16
highlight trends in single terms.
Generally, an increase in mass-transit fares will produce a reduction in mass-transit
travel demand and an increase in road travel demand. This in turn means an increase
in road congestion and related road travel times, fuel consumption and external
costs. Mass-transit user costs generally increase with respect to the zero-fare
condition because in the case of shared lanes (i.e., buses in a rural context)
road congestion will also produce an increase in mass-transit travel times.
Mass-transit user costs could decrease with a reduction in travel demand. However,
as regards the average travel cost per user, there could be a decrease since
users who chose to modify their mode choice were initially using high cost hyperpaths.
|| Real-scale network, (a) Road network, (b) Bus lines, (c)
Rail network and (d) Rail lines
|| Real-dimension network application
|| Objective function values
|| Total operational costs
|| Road user costs
|| Mass-transit user costs (except ticket costs)
|| Ticket revenues
|| External costs
|| Public transport share
|| Required subsidy
Finally, the trend in ticket revenues shows that there is initially an increase
in revenues because the reduction in consumers is compensated by the increase
in fares. Compensation effects then tend to diminish until a reduction in revenues
In this study we proposed a multimodal and multiuser model for determining
optimal mass-transit fares in real contexts. The problem was formulated with
a multidimensional constrained minimisation model where constraints are strongly
interdependent (e.g., the assignment constraint influences the values of other
constraints). The use of a multimodal and multiuser assignment model allows
the problem to be broken down into a bi-level problem: The upper level is the
optimisation problem with budget constraints and the lower is the assignment
problem with consistency constraints.
Application on a trial network highlighted the properties of the problem and
testing on a real-scale network showed that the proposed model provides results
in reasonable calculation times. However, initial results indicate that the
budget constraint greatly affects the optimal solution irrespective of the explicit
consideration of external costs. Future research could be directed at testing
other real-scale networks in order to ascertain whether budget constraints are,
after all, the only parameter to be considered.
This study is partially supported under research project PON-DIGITAL PATTERN
grant No. PON01_01268.