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Beyond the Mohring effect: Scale economies induced by transit lines structures design

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Abstract

In this paper we study how the spatial arrangement of transit lines (lines structure) influences scale economies in public transport. First we show that the degree of scale economies (DSE) increases discretely whenever passenger volume induces a change in lines structure. The technical elements behind this are explained by using a new three-dimensional concept called directness, encompassing number of transfers, number of stops and passenger route lengths. This is first exemplified in a simple ad-hoc network, and then applied to examine the structural changes that occur in the design of transit lines in a fairly general representation of a city. We show that directness increases whenever lines structure changes as a response to larger demand volumes - increasing DSE at the particular value of flow where this change occurs - because systems with more direct lines for each OD pair diminish in-vehicle times while increasing waiting times mildly, such that users are benefited by lower travel times and operators are benefited by lower idle capacity. After the change, however, DSE decreases within the demand range where the new line structure is maintained, just as in the one line model. The possibility of deciding the line structure introduces directness as a new source of economies of scale which are finally exhausted after full directness is achieved.
Beyond the Mohring effect: scale economies induced by
transit lines structures design.
Andrés Fielbaum, Sergio Jara-Diaz and Antonio Gschwender
Universidad de Chile and Instituto Sistemas Complejos de Ingeniería (ISCI)
Abstract
In this paper we study how the spatial arrangement of transit lines (lines structure)
influences scale economies in public transport. First we show that the degree of scale
economies (DSE) increases discretely whenever passenger volume induces a change
in lines structure. The technical elements behind this are explained by using a new
three-dimensional concept called directness, encompassing number of transfers,
number of stops and passenger route lengths. This is first exemplified in a simple ad-
hoc network, and then applied to examine the structural changes that occur in the
design of transit lines in a fairly general representation of a city. We show that
directness increases whenever lines structure changes as a response to larger demand
volumes - increasing DSE at the particular value of flow where this change occurs -
because systems with more direct lines for each OD pair diminish in-vehicle times while
increasing waiting times mildly, such that users are benefited by lower travel times and
operators are benefited by lower idle capacity. After the change, however, DSE
decreases within the demand range where the new line structure is maintained, just as
in the one line model. The possibility of deciding the line structure introduces directness
as a new source of economies of scale which are finally exhausted after full directness
is achieved.
Keywords: public transport; scale economies; lines structures; directness
1. Introduction
Cost functions and economies of scale are economic concepts that are quite relevant
for the normative analysis within production theory, including industry structure and
optimal pricing policies. Behind cost functions lies the technical process of conversion of
inputs into outputs, such that cost functions can capture scale economies (a technical
property indeed). In transport, the main technical elements are frequencies, vehicle
sizes and the organization of lines in space. In this paper we aim at understanding the
relations behind the third design element and scale economies in public transport.
The provision of public transport services exhibits various technical characteristics that
have been shown to affect its degree of scale economies. First of all, the so-called
Mohring effect, where an increase in patronage makes optimal frequency larger and
waiting times lower. Mohring (1972) found the frequency of the service to be
proportional to the square root of the demand when only this effect is modeled in an
isolated public transport line. In addition to this waiting time effect, as demand increases
the system can also be adapted by incorporating new lines, thus reducing another
component of users’ cost, namely the walking time. This has been modeled for a bus
feeder system by Hurdle (1973), in a rectangular area by Kocur and Hendrickson (1982)
1
for a single period, by Chang and Schonfeld (1991) for multiple periods, and by Small
(2004), who analyzed the impact of road pricing on public transport. All of them obtain a
cube root formula for both the optimal frequency of each line and for the optimal number
of routes.
A third variable that can be adapted according to the demand level is the size of the
vehicle, which also increases with patronage. As operators’ cost per passenger diminish
with vehicle size (due to fixed costs per vehicle), this is also a source of scale
economies. However, when vehicles size increases the time spent at each stop also
increases because more passengers board to, and alight from, each single vehicle,
thereby increasing cycle time - which affects operators’ cost as a larger fleet is needed -
and users’ in-vehicle time. Both effects reduce the degree of scale economies. Including
these effects in his model of an isolated public transport line, Jansson (1980) obtained a
modified square root formula for optimal frequency. In all models the adjustment of
frequency and vehicle size generates scale economies that, nevertheless, diminish as
flow increases.
An important element of design that responds in a discrete way to increases in flow is
lines structure, i.e. the way in which vehicles serve a number of routes in order to move
a given set of flows (product). Such a structure can be optimized together with fleet and
vehicle sizes, admitting many possible arrangements in space, with public transport
lines organized as, for example, cyclical, hub-and-spoke, feeder-trunk or direct services.
As flows grow these arrangements might evolve in a way that should be studied
specifically; understanding the evolution of design including lines structure and
analyzing its impact on total costs and scale economies is the main objective of the
paper. Considering operators’ costs only, Basso and Jara-Díaz (2006b) study the
difference in the analysis of scale economies when lines structures are fixed or a
variable to be optimized. Kraus (2008) formulates the problem including users and
operators’ costs over a cost minimizing network, which in public transport would imply
that users choose system optimal routes rather than individually optimal ones. In this
paper we analyze scale economies looking at the evolution of lines structure design as
total flow grows considering total costs and recognizing that users choose individually
optimal routes. The main conclusions are that changes in lines structure induce scale
economies at the particular value of total flow where this change occurs; that the
technical elements behind this are the reduction of stops, transfers and route lengths;
and that vehicle sizes and frequencies grow as well, as in single line models.
The paper is organized as follows. In the remainder of this Section we summarize the
various ways in which scale has been studied in transport. Section 2 contains a
discussion of what it means to introduce lines structure in scale analysis, showing that
the degree of scale economies (DSE) - the ratio between average and marginal cost -
increases discretely whenever lines structure changes as a response to a continuous
proportional increase in flows. In Section 3 we use a simple network to illustrate this
property and to introduce the multi-dimensional concept of directness that helps
describing the evolution of lines structure as flows grow. This concept is used to
present, in Section 4, a more general case that rests on a parametric description of a
2
city; most importantly, the technical elements that help explaining the change in lines
structure as patronage grows are presented in detail. Section 5 concludes, emphasizing
the role of directness in scale economies analysis of public transport systems.1
Although transport processes usually involve many inputs and outputs, the engineering
technology has been usually formulated using aggregates, where product was
described using single scalar measures as ton or passenger-miles until mid-eighties,
and by means of a vector of a very small dimension thereafter, including flows related
variables, service quality variables and network description variables. The compact
description of output prompted two definitions in the literature around the analysis of
scale economies, both referring to proportional expansions of output: returns to density
(called RTD) and returns to scale with variable network size (called RTS). The former
considered a proportional expansion of outputs keeping network size fixed, while the
latter considered a simultaneous expansion of both flows and the network by the same
proportion (Caves et al., 1984; Keaton, 1990). However, using aggregate output
descriptions blurs the technical relations with inputs and has some unpleasant
consequences in the analysis of economies of scale in transport activities.
Behind any compact description of transport output lays the true output of any transport
firm: a vector of origin-destination (OD) flows of different things during different periods
(Jara-Díaz, 1982a). In very simple transport systems the analytical derivation of the
technical relations between inputs and flows - the production function can be done,
such that the corresponding cost functions can be obtained analytically as well.2 This
approach proved very useful to show that the use of aggregates introduced ambiguity in
the economic analysis in transport because, for example, the same amount of
passenger-miles could require very different types and amounts of inputs depending on
how this passenger-miles are distributed in space. Most importantly, scale economies
should be studied holding the origin-destination system constant, as introducing new
OD pairs means introducing new products, which would require the analysis of
economies of scope; this means that “economies of scale with variable network size” is
actually an ill-defined concept, as shown by Basso and Jara-Díaz (2006a) while
“economies of density” is better suited to the definition of economies of scale. 3 A
corollary from this story is that more attention has to be paid to the transport production
process itself in order to fully understand scale economies. This is the main objective of
this paper.
1 Scale economies in public transport have also been reported in other dimensions. Tirachini et al. (2010a), for
example, show that when crowding discomfort is considered diseconomies of scale are found for high levels of
patronage, a result that vanishes when more than one line is considered (Tirachini et al., 2010b). Tirachini and
Hensher (2011) and Jara-Díaz and Tirachini (2013) have studied the impact of the boarding-alighting-paying
methods, finding yet another source of economies of scale. Considering different modes also impacts the analysis, as
shown by Tirachini and Hensher (2012) or Basso and Jara-Díaz (2010, 2012).
2 See for example the analysis of the backhaul transport system involving two flows only (Jara-Díaz, 1982b) or the
three-nodes system studied by Jara-Díaz and Basso (2003) involving a discrete decision regarding the spatial
arrangement of the vehicles (service structures).
3 Sometimes RTD has been defined adding the condition that route structure is unchanged after an increase in flows
(Basso and Jara-Díaz, 2006b).
3
2. The impact of the discrete nature of lines structure choice on DSE.
In this section we analyze the general relation between the adjustment of lines
structures and scale economies in transit networks. Let us formally define a “lines
structure” as a set of spatially organized transit routes that operates on a given network
serving all flows. A simple example is shown in Figure 1, where a three nodes network
(a) - with potentially six OD pairs - can be served in different ways, such as a single line
running counterclockwise (b), or with two lines each one circulating between two nodes
(c). How to decide which lines structure is best for a given origin-destination (OD) flow
matrix
Y
? In transport production this is part of the search for the optimal input
combinations that yield the minimum total cost, so choosing the best structure has to be
done together with other design variables like frequencies and vehicle sizes in order to
find the smallest value of the resources consumed (
VRC
) provided those design
variables are technically able to produce
Y
; for short, a cost function has to be
obtained, which requires finding the optimal input demand functions depending on
product and input prices, noting that the
VRC
includes all resources, i.e. operators’
and users’. This requires a certain procedure which we now summarize for a general
case.
a) b) c)
Figure 1. Network (a), and two alternative lines structures.
Consider a physical network (e.g. streets of a city) and a given OD matrix of flows
Y
.
For this setting, each candidate lines structures
Ei
is composed of a series of transit
lines that, altogether, are capable to serve all trips4. In turn, each of the lines
l
that
form a lines structure has to be assigned a frequency
fl
and a vehicle size
Kl
. In
order to find the optimal values of these variables for a given lines structure
i
composed of
lines, one has to minimize the total value of the resources consumed
VRCi
, that depends on the set of frequencies
(
f1, … , f mi
)
=f
and vehicle sizes
K
(¿¿1, … ., K mi)=K
¿
of all lines in structure
i
provided
f
and
K
can carry flows
Y
.
VRCi
can be expressed as the sum of the resources consumed by operators
VRCiO
and users
VRCiU
, i.e.
VRCi(f , K , Y )=VR CiO (f , K )+VR CiU (f , K , Y )
(1)
4 In practice, the total number of possible lines structures is huge and cannot be obtained. Nevertheless, this is not
needed for the analysis in this section. In sections 3 and 4, where specific networks are analyzed, we will work with a
set of pre-conceived lines structures.
4
VRCi
is a function of
Y
directly because users’ costs increase with
Y
. The
optimal values of
(f , K )
for a given
Y
, denoted
(f¿, K ¿)
, are those that minimize
VRCi
subject to technical feasibility constraints, as explained in Appendix A. Then
solving (1) we get
f¿=f¿
(
Y
)
and
K¿=K¿(Y)
. When these optimal values are
plugged back into
VRCi
one obtains the conditional cost function
Ci
(
Y
)
VR Ci(f¿
(
Y
)
, K ¿
(
Y
)
,Y )
as defined in Jara-Diaz (2007), i.e. the minimum
VRC
to serve flow
Y
for a given lines structure
i
. Finally, the best lines structure for
each
Y
is given by the
argmi niCi(Y)
. This way the optimal lines structure for a
given
Y
is found together with the optimal frequencies and vehicle sizes that were
found in the previous step. Note that
fl
¿=fl
¿(Y)
and that, for some
Y
values, some
frequencies can become nil and some can become positive.
In equation (1) it is assumed that operators’ costs increase with frequencies and bus
sizes, while users’ costs decrease. For the proposition below, these expressions can be
general; we only need them to be differentiable and to include at least one component
inversely related with frequency. For the sake of clarity, in what follows we will use
waiting cost as representative of this component5.
The question we want to address is whether adjusting line structures contributes to
scale economies in transport networks. First, we have to recall that the degree of scale
economies
DSE
is defined as the ratio between average and marginal costs such
that there are scale economies iff
DSE>1
. The proposition formulated and proved
below states that the
DSE
increases when the lines structures changes (unless
common lines exist everywhere6). The proof will be based on the discrete change in
lines structure when passengers choose their routes minimizing their individual costs
(an example using the network in Figure 1 is offered and analyzed in detail in the next
Section). If we are using a set of predefined lines structures, this change is obviously
discrete (and the first part of the proof below is not necessary); if all the lines are always
candidates to appear, the crucial fact is that nobody will wait for a line with a frequency
that is extremely low when there are no common lines.
Let us define a vector of OD flows
q
as a threshold point if there exists at least one
line
l
such that optimal frequency
fl
¿(q)=0
and
fl
¿(q(1+ε))>0ε>0
with no
common lines for some of its passengers (see footnote 6). This means that when
Y
just exceeds
q
at least one new line appears because it minimizes
VRC
and
5 There are more components of
VRC
inversely related with frequency, as in-vehicle users’ cost (because of
time at stops and crowding) or bus size-related operators’ cost (Jara-Díaz and Gschwender, 2009; Hörcher and
Graham, 2018).
6 In the literature the case known as “common lines” appears when for some portions of the route, the passenger is
indifferent to choose within a certain set of lines because they all make almost the same trip. Using Figure 1 as an
example, if both line structures (b and c) coexist, passengers travelling from the bottom node to the upper-right one
could use the line of structure (b) or the right line of structure (c). For them, both are common lines. Passengers
travelling from the bottom node to the upper-left one could face common lines or not, depending on the relative
frequencies and travel times of the line of structure (b) and the left line of structure (c). For a precise definition, see
Chriqui and Robillard (1975) or Cominetti and Correa (2001).
5
becomes best for some users7. Note that a vector
Y
at which a change in lines
structure occurs is, by definition, a threshold point, in which new routes emerge for
some users (and other inferior ones can disappear).
The main idea behind the proposition that follows is that the emergence of a new line
(i.e. a line whose optimal frequency goes from zero to a positive value) triggers an
upward jump in the DSE.
Proposition: Consider a network served by a public transport system. Then at every
threshold point the
DSE
increases discretely, i.e.
ε → o¿DSE (qε)
ε → o+¿DSE (qε)>lim
¿¿
lim
¿¿
, with
qε=q(1+ε)
.
Proof: the proof has two parts.
1) First, let us show that when some
fl
¿
becomes strictly positive due to a growth of
Y
at
q
, it increases in a discontinuous way from zero. Define
f0
such that if
fl
<
f0
, then the waiting cost for passengers using line
l
will induce a travel time
cost that is larger than the current total user cost of any other route in the system. In
that case no passenger chooses
l
and the optimal frequency is zero; in other words,
fl
will never be in the interval
(0, f0)
, but jumps in a discontinuous way from
0
to some positive value
fl
¿≥ f 0
.
As an example, suppose that the “south” node is the only origin in Figure 1. If we only
had the red long line from Figure 1b, and the blue line from Figure 1c was added
(starting from a different bus stop, such that users have to decide which line to take in
advance), this blue line will be used by those passengers whose destination is the
upper-left node only if its frequency is large enough to compensate for waiting plus the
longer in-vehicle time if using the red line.
What we have just shown is that when a change in line structure occurs, there is a
discrete jump in the value of at least one frequency: the choice of an optimal line
structure is essentially discrete. Then the production of flows can be represented by a
function that involves the choice of a discrete variable (the lines structure
i
, indicating
which lines are present in the system) and several variables that are continuous for a
given discrete variable (such as the frequencies and vehicle capacities of each of the
lines in
i
).
The second part of the proof applies to any kind of cost function that results from the
optimal choice of a discrete variable
Z
(in our case the lines structure
i
) and
several variables
X
(in our case the frequencies and vehicle capacities of each of the
lines in
i
) that are continuous for a given
Z
. As scale analysis deals with
7 The potential adaptation of route structure following a growth in flows is at “the kernel of transport production;
changes in the flow vector
Y
potentially induce changes in input usage as well in route structures and operating
rules in general” (Jara-Diaz (2007).
6
expansions of output vector
Y
along a ray
µY
, with
µ ≥ 1
(Baumol et al, 1982),
it is equivalent to a single product like analysis. Then, in the rest of the paper,
Y
will
be treated as a scalar given by the total sum of the flows, i.e.
Y=∑ yi
; note that this
means that
µY =µyi=µY
.
2) Consider
X¿, Z¿
optimal to produce
Y
. Consider
Y0
such that any increase
leads to a change from
Z¿=Z1
to
Z¿=Z2
(equivalent to a threshold point above).
Then
ε → o¿DSE(Y0+ε)
ε → o+¿DSE (Y0+ε)>lim
¿¿
lim
¿¿
.
To prove this, consider the conditional cost functions
C1
and
C2
associated to
Z1
and
Z2
respectively obtained by optimizing
X
only given
Zi
(in our case
this means optimizing frequencies and vehicle sizes for a given lines structure). Let us
look at the average and marginal costs for
C1
and
C2
at
Y0
. As
C1
and
C2
are continuous functions, then
C1(Y0)/Y0=C2(Y0)/ Y0
. Regarding the marginal costs,
the derivative of the cost with respect to
Y
verifies
C2
∂ Y <∂ C1
Y
, because
C2
becomes lower than
C1
when
Y
grows. As average costs are equal and marginal
costs are lower for
C2
, it is direct to conclude that the ratio between average and
marginal cost, i.e
DSE
, increases.
ε → o¿DSE(Y0+ε)
ε → o¿DS E1(Y0+ε)=lim
¿¿
ε → o+¿DS E2(Y0+ε)= DS E 2(Y0)>DS E1(Y0)=lim
¿¿
ε → o+¿DSE (Y0+ε)=lim
¿¿
lim
¿¿
Q.E.D.
The Proposition is represented in Figures 2, where average and marginal costs for
C1
and
C2
are shown. At the exact point where the two average costs coincide
(i.e. where the optimization process induces a change from
Z1
to
Z2
), a black
arrow shows that a) the marginal cost is lower for
Z2
and b) the global
DSE
increases discretely.
As a conclusion, the structural design of public transport systems involves variables as
frequency
f
(and the associated fleet
B
), density
D
and vehicle capacity
K
,
that can be treated (or approximated) as continuous8, and lines structure, which has a
discrete nature and introduces technical novelties that are worth studying. In the
following Section we will introduce a multidimensional concept that helps analyzing the
technical relations between lines structure and scale economies.
8 Frequency and capacity are discrete variables, although the former can be fractional when looked at on
a per hour basis. The latter, though, is constrained by commercially available vehicles.
7
a)
b)
Figure 2. Change in DSE due to (discrete) change in lines structure
3. Directness in lines structures: a source of scale economies.
3.1 Introducing directness.
We have proved that changes in line structures always lead to a discrete (local)
increase in scale economies. This general result, however, says nothing about what
exactly are the transport-related technical elements that help understanding what lies
behind this. We do know from the literature that increasing demand induces higher
frequencies, larger vehicles and an increase in the density of lines. As a result, waiting,
access and egress times diminish (scale economies) while in-vehicle and cycle times
increase (scale diseconomies). What is the equivalent technical effect that links overall
demand with lines structure and scale economies? And how do scale economies
behave once a change in lines structure has occurred?
These are quite complex questions because frequency, vehicle size or lines density can
be represented each by a single, well-defined variable, whereas a lines structure, e.g.
feeder-trunk or hub-and-spoke, can be conceptually described with some precision by a
generic description but cannot be represented by a single variable. Further, changes in
line structures are not continuous but discrete, occurring at some specific levels of total
patronage. Both elements not only increase the mathematical complexity of the
associated optimization problem, but also add new challenges to scale economies
analyses.
Generally speaking the literature on lines structures in the last fifteen years shows that,
for low levels of overall demand distributed in space, those structures involving transfers
tend to be appropriate, e.g., hub-and-spoke or feeder-trunk systems9. As patronage
increases, lines get organized along the idea of routes that follow more closely the
origin-destination pattern avoiding transfers, increasing what can be called “directness”,
such that each new passenger generates positive externalities on the rest of the
9 Gschwender et al. (2016), for example, study a Y-shaped city. They show that as the patronage increases, the
optimal structure changes in one of the following ways (depending on trip distribution): from No transfers to No stops,
from Feeder-trunk to No stops, or - the only odd case - from No transfers to Feeder-trunk. Daganzo (2010) studies a
grid city served with direct lines within an internal region and with hub and spoke from the external region, optimizing
the size of the internal region; he shows that the larger the patronage, the larger the zone served with direct lines
(internal region). Badia et al. (2014) extend the paper by Daganzo (2010) and this conclusion remains valid; also, the
set of lines becomes denser when the number of passengers increases.
8
passengers because a) transfers diminish, b) distances travelled diminish, and c)
number of stops diminish10. Element a) has a clear positive impact on users, b)
diminishes in-vehicle-time for all, and c) diminishes in-vehicle-time for users and cycle
time for operators. So these elements seem to contribute to increase the
DSE
through the reduction of average users’ costs, but all effects should be analyzed. In
order to represent directness in a more precise way, we propose the following three
(continuous) indices: average transfers required per trip, average stops required per trip
(including the extremes) and the average across all passengers of the ratio between
their traveled distance and the length of the shortest path that link their origin and
destination (relative distance). Note that these flow-related indices can also be defined
as averages across OD pairs, such that these new “network indices” can be calculated
irrespective of the assignment of flows.
The concept of directness has an extreme case in non-stop services (which have been
called exclusive in previous papers), where each OD pair is served by one line only,
providing a service similar to a private car but with lower operating costs per passenger
and larger waiting times. From this viewpoint, as directness increases the number of
passengers with different origins and destinations sharing the same vehicle diminishes.
It is worth noting that a connection between patronage and directness has emerged in
the transit network design literature. For example, Fielbaum et al. (2018) studied the
heuristics proposed by Dubois et al. (1979) and Ceder and Wilson (1986), that are built
around direct services: both create spatial arrangements of lines depending on a
parameter
σ
that controls the maximum admissible deviations from the shortest
paths, i.e., represents exactly the trade-off between more directness (
σ=0
) and bus-
sharing;
σ
is inversely related with directness. When searching for the best
σ
,
Fielbaum et al. (2018) found systematically that small values for this parameter were
optimal when patronage was large, i.e. increasing directness was the optimal response
to demand increases.
3.2 An illustrative model
In order to illustrate in a simple way what has been discussed above, let us consider the
network we introduced in Section 2 (Figure 1), with network and flow characteristics as
represented in Figure 3, where two destinations are located at the same distance
L0
from a single origin, forming an isosceles triangle; the distance between the destinations
is
Q
(Figure 3a). The total number of passengers in the system is
Y
half on
each OD pair as represented in Figure 3b – and the question is whether it is better to
have only one line carrying all the passengers (full bus sharing, Figure 3c), or two lines,
one for each destination (full direct, Figure 3d);
λ
represents the load of the lines on
each directed arc. The directness indices are shown in Table 1 (note that in this case
the flow indices and the network indices coincide, as there is only one flow assignment
option).
10 This is an extension of the concept of OD-directness originally defined by Laporte et al. (2011) on the lines
network as the fraction of the OD-pairs that can be joined without transfers.
9
(a) (b) (c) (d)
Figure 3. Network (a), transport demand (b) and alternative service structures:
one shared line (c) and two direct lines (d).
SERVICE STRUCTURE
DIRECTNESS INDICES
Bus-Sharing Direct
Number of transfers 0 0
Number of stops 2.5 2
Distance traveled/Minimum distance
1+Q
2L0
1
Table 1. Indices of directness for the alternative service structures.
Let us represent the value of the resources consumed necessary to serve total flow
Y
by each system as
VRC=B(c0+c1K)+ pwY tw+pvY tv
, (2)
where
pv
and
pw
are the values of in-vehicle and waiting times respectively, and
the parameters
c0
and
c1
define the operators’ cost per bus.
Model (2) follows Jansson (1980), Jara-Díaz and Gschwender (2009) and Fielbaum et
al (2016), among others; it has been shown to be enough to capture the most relevant
aspects of public transport costs in a system without transfers. In this approach, fleets,
capacities, waiting times and in-vehicle times for each of the two systems can be
expressed as functions of the corresponding frequency
f
- that becomes the (only)
design variable to be optimized -, the vehicle speed
V
, boarding-alighting time
t
,
Y
,
Q
and
L0
. A simple analysis (in Appendix B) yields the expressions shown
in Table 2. Note that in the two-lines case lines are symmetric and exhibit the same
frequency.
One line (Bus-sharing) Two lines (Direct)
10
Bus capacity
K
Y/f
Y/2f
Fleet
B
f(2L0+Q)
V+2tY
4f L0
V+2tY
Waiting time
tw
Y/2f
Y/2f
In-vehicle time
tv
1
2
(
L0
V+1
4ftY
)
+1
2
(
L0+Q
V+3
4ftY
)
L0
V+tY
4f
Table 2. Elements of the alternative service structures as a function of frequency.
Replacing the respective functions from Table 2 into equation (2), the optimal
frequencies are obtained from the first order conditions (as shown in Appendix B). Both
optimal frequencies and capacities are shown to increase with
Y
(as in Jansson,
1984), such that the scale effects (explained in section 1) are preserved. By plugging
optimal frequencies back into
VRC
we obtain the cost function
Ci
for each system:
C1=2
c0(2L0+Q)
VY(2c1tY +pw+pvtY
2)+2c0tY +c1Y(2L0+Q)
V+pvY
2
(2L0+Q)
V
(3)
C2=2
c04L0
VY(c1tY +pw+pvtY /2
2)+2c0tY +2c1Y L0
V+pvYL0
V
(4)
Note that
C1
and
C2
can be written as
Ci(Y)=
αiY2+βiY+εiY ,
with
α1>α2, ε1>ε2
and
β1<β2
. For high values of patronage,
α
and
ε
dominate,
such that
C2
is smaller, i.e. the two-lines structure (full directness) is better; shorter
routes are good for both users (through
pv
) and operators (through
c1
). On the
other hand, when
Y
is small,
β
dominates, such that the system with only one line
(full bus sharing) is better due to the lower waiting times (through
pw
). The average
costs resulting from
C1
and
C2
are shown in Figure 4a using the parameters
shown in Appendix B.
DSE
is represented in Figure 4b for each system, with the
solid lines representing
DSE
for the optimal structure.
a)
b)
Figure 4. Average costs (a) and DSE (b) for Bus-sharing and Direct services.
11
The general property advanced in Section 2 and Figure 2 emerges very clear: the
DSE
“jumps” when
Y
reaches a certain volume that makes the direct lines
superior, which is explained because of more direct routes and fewer stops. What about
DSE
after the lines structure changes? Using the short notation introduced above
DSE
can be expressed as
DS Ei=1+βi
2αiY+βi+2εiY
αi+βi/Y
(5)
This expression shows that economies of scale are always present, but
lim
Y → ∞
DSE=1
,
suggesting that the positive externalities induced by each of the elements that
constitutes “directness” in this model get exhausted in spite of the upward jump in
DSE
induced by the change in lines structure: eventually everybody travels along the
shortest possible route and with no intermediate stops11.
4. Directness and scale economies in a representative urban setting.
Transit systems can be spatially organized in many (and complex) ways. In order to
visualize the technical elements that intervene in the relation between lines structure
and scale economies, a better representation of the underlying spatial setting is needed
such that lines could be structured following many possible arrangements. To do this we
will apply the lines structure analysis by Fielbaum et al. (2016) over the simplified
parametric urban model introduced by Fielbaum et al (2017) shown in Figure 5, where
trips go from
n
peripheries P (that only generate trips) to both the CBD and the
n
subcenters SC. There are also trips from the subcenters to other subcenters and to the
CBD. In other words, peripheries only generate trips and the CBD only attracts trips,
representing a simplified morning peak situation. The proportions of total trips
Y
departing from the peripheries and from the sub-centers, and the proportions going from
the peripheries to the CBD, own sub-center and other sub-centers are treated
parametrically, such that all types of cities can be represented (monocentric, polycentric
and dispersed).
Figure 5. A simplified parametric urban model (Fielbaum et al., 2017).
11 This refers to scale economies induced by directness. If the number of passengers gets too large, new sources of
economies (or diseconomies) of scale might emerge, such as congestion or a change in technology (e.g. metro).
12
Under this setting, one can search for the best lines structure as total trips
Y
grow
keeping trip distribution constant12, in order to visualize a relation between lines
structure, demand and scale economies. Finding an optimal set of routes is an NP-Hard
problem, which is the reason why this evolution can only be analyzed over a reasonable
set of strategic line structures13. We will consider four traditional generic schemes with
different directness indices: Feeder-Trunk (FT), Hub and spoke (HS), No Transfers (NT,
or “direct”), No Stops (NS, or “exclusive”). The lines that belong to each structure are
represented in Figure 6; as all structures are radially symmetric, only lines emerging
from one zone are shown. Each type of line (e.g. radial, circular) is represented by a
different color. Lines of the same type that share one link are grouped (such as the
three black lines in 6a).
a.FT b. HS
c. NT d. NS
12 In Fielbaum et al. (2017) the distribution of total flow in trips from the peripheries and subcenters to the CBD and
(other) subcenters is represented by three parameters. These proportions (i.e. the parameters) are held constant in
the analysis of scale, looking only at the effect of
Y
(ray analysis).
13 Informally, a problem is NP-Hard when any algorithm that seeks the exact solution would take absurdly long
times. Quak (2003), Schöbel and Scholl (2005), and Borndörfer et al. (2007) have shown that finding an optimal set
of routes is an NP-Hard problem for various specifications.
13
Figure 6. Four strategic line structures
A brief description of each of these four generic line structures establishes a connection
with the three indices that describe directness as defined above:
- FT: each periphery is connected with its subcenter, and subcenters are linked by direct
trips that follow shortest paths, that could follow the circular line or one of the lines
connecting each subcenter with the 3 “opposite” subcenters. The number of transfers is
always 1 for trips from a periphery unless the destination is the own subcenter. Trips are
the shortest possible. Buses stop at each node.
- HS: all peripheries are connected to the CBD that acts as a hub, and there are two
additional circular lines (clockwise and counterclockwise; only one is shown in Figure 6)
connecting the ring of subcenters. The number of transfers is 1 for most trips that do not
finish at the CBD. Trips may be longer than the shortest path but only for a small
fraction of the trips. Buses stop at each node.
- NT: nobody needs to transfer. Trips may be longer than the shortest path but only for a
small fraction of the trips. There are specific lines connecting each OD-pair (some of
them vanish because their optimal frequency is nil). Buses stop at each node.
- NS: nobody needs to transfer. As buses are OD specific, their routes are as short as
possible and each bus travels non-stop from start to end (only 2 “stops” per trip).
The trips paths followed by the passengers are not known a priori because they depend
on optimal frequencies (some of which could be zero) that in turn depend on
Y
. In
order to characterize the structures in terms of directness independent of
Y
, Table 3
shows the network indices of the four structures calculated as averages across OD
pairs – instead of passenger trips – in a city with eight zones (
n
=8, 136 OD pairs)14.
Directness increases from FT to HS, then to NT and finally to NS.
Structure FT HS NT NS
Number of transfers 0.47 0.35 0 0
Number of stops 3.06 3.06 3.06 2
Distance traveled/Minimum distance 1 1 1 1
Table 3. Network indices describing directness for each lines structure.
In Fielbaum et al. (2016), frequencies of lines within each structure are optimized,
minimizing a
VRC
function similar to equation (2) studied in section 3.2, but now
including a penalty
pT
for each of the
T
transfers in the system, as shown in
equation (6). As in the illustrative model, users are assumed to be homogeneous
regarding time valuation, crowding and congestion are not considered, and the number
of users is exogenous (i.e min
VRC
for a given
Y
, which yields a cost function).
VRC=B
(
c0+c1K
)
+pwY tw+pvY tv+pTT
(6)
14 Note that whenever some lines vanish as a result of the optimization process (zero frequency) the flow directness
indices may increase.
14
Because of the complexity of the network, users now may have more than one route to
choose from. All passenger routes are assumed to have the same fare such that
assignment of passenger to routes are commanded only by the operational
characteristics of the system; their choices depend on frequencies and frequencies
depend on choices (as formulated in Appendix A), which prevents analytical solutions.
Therefore, an iterative procedure is needed to find the optimal frequency and vehicle
size for each line within a given lines structure, where each iteration rests on finding a
relation between (
B , K , t w, t v, T
) and the vector of frequencies.Using the parameters
shown in Appendix C, the optimal vector of frequencies is obtained for each structure15;
again, both frequencies and bus sizes increase with patronage. Plugging these back
into
VRC
the cost function
Ci
for each lines structure is obtained.
Figure 7a shows the results of Fielbaum et al. (2016) regarding the average cost of
each line structure; as
Y
increases the optimal structure changes from hub and
spoke, to no transfers and finally to no stops, i.e., directness increases (and feeder-
trunk is never optimal)16. In Figure 7b this evolution is shown by means of the
corresponding
DSE
of the optimal structure for each level of the total flow: scale
economies indeed increase after each change (including a change within HS when the
circular line emerges), and decrease thereafter. For synthesis, the possibility of deciding
the line structure introduces directness as a new source of economies of scale which
are finally exhausted after full directness is achieved.
a) b)
Figure 7. Average costs and overall DSE as directness increases.
Let us now address the central question: which design elements lie behind these results
regarding scale economies? Having found the superior structures, an analysis of
directness can be made taking into account the passengers’ trips. Figures 8 show the
evolution of each of the three flow indices that define directness as a function of the
number of passengers whose growth induces lines structure changes from HS to NT to
NS.
As represented in Figure 8a, transfers occur only for low values of
Y
where the hub
and spoke structure dominates; the emergence of a new line (whose frequency jumps
15 Parameters were chosen from Fielbaum et al. (2016), including meaningful values for trip distribution. For
example, 80% of the trips depart from the peripheries and half of them go to the CBD.
16 We use a logarithmic scale to be able to represent both low and high volumes of passengers and the
corresponding dominant structures (this scale will also be used for all subsequent figures). Flow is shown in
passengers per minute.
15
discretely from zero) within the HS design that connects directly some OD pairs (a new
lines structure rigorously speaking) generates a reduction in the number of transfers
and also in the number of stops and distance traveled, which shows up in Figures 8b
and 8c. The average stops per trip decreases down to 2 when the no-stops structure
dominates for high values of
Y
(Figure 8b). The ratio between the distance traveled
and the minimum distance possibly required (called “detour” in Figure 8c) generally
decreases except when changing from hub and spoke to the no-transfers structure, as
some passengers experience longer trips because some short lines disappear in favor
of longer ones that collect more passengers; note that this is counterbalanced by the
reduction in transfers, showing that sometimes there is a trade-off between the different
components of directness.
a) b)
c)
Figure 8. The three flow indices of directness as a function of patronage.
The physical measures of directness translate into users’ time and users’ costs, which
are shown in Figures 9. Figure 9a summarizes the “equivalent time” associated to each
of the directness indices: length of the routes translates into time-in-motion, the number
of stops (together with vehicle load) translates into time at stops, and each transfer is
valuated as 24 minutes in motion (as in Fielbaum et al., 2016). Their sum is the total
equivalent time (TET) presented at the top of Figure 9a, and it synthesizes the total
effect of directness on users; the fact that TET diminishes when lines structure changes
clearly shows that increasing directness as patronage increases, contributes to scale
economies. The slight increase of TET within each structure is caused by the larger time
at stops induced by larger vehicles, an effect that is almost irrelevant when compared
with the rest including the reduction in the number of stops each time the structure
changes. Note that the more than 10 minutes reduction of TET is mostly explained by
the reduction in time-in-motion and transfers (some 4 minutes each) against the 2
minutes reduction in time at stops.
16
Figure 9b shows the average costs per passenger due to in-vehicle time, waiting time
and transfers, which are the three components of the users’ cost function. Looking at
the points where lines structure changes, it becomes evident that increasing directness
makes in-vehicle time and transfer cost decrease, but there is a local increase in waiting
time because directness diminishes bus-sharing and each passenger now has less
lines to choose from. This local increase in waiting times, however, is more than
compensated by the frequency growth as patronage increases within each structure
(Mohring effect).
a) b)
Figure 9. Effects of directness on equivalent users’ times and users’ costs.
So far, we have interpreted scale economies in terms of users’ costs; what about
operators’ costs? Which are the effects of directness? To tackle these questions, let us
recall that total operators’ costs are given by
c0B+c1Σ
where
B
is total fleet and
Σ=BK
is total number of seats. Let us analyze both variables.
In Figures 10 we show (a) number of seats per passenger and (b) number of buses per
passenger as a function of patronage. Seats per passenger drop significantly when
lines structure changes. This effect occurs because bus-sharing diminishes (when
directness increase) reducing the idle capacity of buses as we now explain in detail: the
size of the buses for a line is given by its most loaded segment, such that idle capacity
is present in the rest of the arcs used by the line; only in the NS structure buses are
always full. On the other hand, within a given structure increasing
Y
increases cycle
time through boarding-alighting time, which makes
Σ/Y
an increasing function of
Y
17.
Figure 10b reveals that the number of vehicles per passenger decreases nearly in a
continuous way, which shows that the effect of the change in lines structure over total
fleet as
Y
grows is less important than the increase in bus size. In other words, when
Y
increases, optimal frequencies and vehicle capacities increase as well, but
frequency grows at a decreasing rate precisely because the capacity grows making fleet
per capita decrease.
17 This (novel) result can be obtained analytically in the one-line case using the expressions for optimal frequency
and capacity in Jansson (1984) or Jara-Díaz and Gschwender (2009).
17
a) b)
Figure 10. Effect of directness on the components of operators’ costs.
In summary, including lines structures as part of the (optimal) design of public transport
services in an urban space introduces yet another source of scale economies which has
been defined here as directness, a concept that encompasses many elements
summarized by three indices that capture transfers, routes length, and stops; as
directness increases the total equivalent time for users decreases, approaching the
(time related) characteristics of a private car trip. All in all, searching for the optimal lines
structure is both affected by scale effects (such as the Mohring effect) and triggers new
scale sources.
5. Conclusions.
In this paper we have introduced a basic structural design element - the spatial
arrangement of transit lines - in the analysis of scale economies in public transport
systems. We have shown that the discrete change from one structure to another as
patronage increases is a source of scale economies. This change occurs because at
the threshold point average costs are equal but the marginal cost of the new structure is
lower (which justifies the change), such that the degree of scale economies increases at
that point. The difficulty in the detailed analysis of what lies behind the effect on scale
economies emerges due to the lack of a single variable that captures the evolution of
lines structures as flows grow, which makes a substantial difference with the analysis of
frequency and vehicle capacity in a single line.
In order to understand the engineering aspects behind the relation between lines
structure and scale economies, we have proposed a three-dimensional concept called
directness encompassing number of transfers, number of stops and passenger route
lengths. We have shown in a very simple network as well as in a fairly general
representation of a city, that all these indices improve (diminish) when a change in lines
structure takes place due to an increase in passenger volume. Grossly speaking, as
more passengers use public transport, it is possible to evolve towards systems with
more direct lines for each OD pair, diminishing in-vehicle times while keeping
reasonable waiting times, such that users are benefited by lower travel times and
operators are benefited by lower idle capacity.
The change in lines structure occurs at specific levels of patronage, such that there are
segments of demand where the same lines structure remains as the best. Within those
segments scale economies analysis replicates the case of the single-line models, i.e.
18
frequencies and bus capacities increase with patronage, such that waiting times for
users diminish (Mohring effect), and average cost for operators diminish, which
outbalances the diseconomies of scale induced by larger times at bus stops. For
synthesis, the degree of scale economies increases locally when lines structure
changes and diminishes afterwards until the next change occurs. And this happens until
full directness is achieved; from then on frequencies and vehicle sizes increase until
scale economies are exhausted.
Next steps in the analysis should take into account that, besides frequency and vehicle
size, another relevant source of scale economies has also emerged from simple
models: lines density, which has been represented as parallel lines that provide the
same service (i.e. same frequency and bus size) affecting access time for a single OD
pair. When dealing with lines structures, the introduction of density would require the
inclusion of yet another variable in the model, namely the number of actual streets
represented by each arc in the city network. As patronage continues increasing it is very
likely that the density of lines running between each pair of nodes in the parametric
representation of the city should increase as well. Future research should consider the
joint evolution of density, frequencies, bus sizes, and lines structures as patronage
grows.
Finally, the analysis presented here involves only the variation of total patronage
keeping trip distribution constant; in the terminology created by Baumol et al. (1982)
within a multi-output framework, this is a ray analysis, where flows in every OD pair
grow by the same proportion. When a city exhibits an evolution of flows that involves a
change in its basic urban structure, e.g. from monocentric to polycentric, the scale
effects recognized and analyzed in this paper should be complemented with the study
of economies of scope.
Acknowledgements
This research was partially funded by Fondecyt, Chile, Grant 1160410, and CONICYT
PIA/BASAL AFB180003.
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Appendix A: Procedure to optimize
(f , K )
for a given lines structure and a given
Y
.
The problem is
Mi nf , K VR C i(f , K ,Y )=VR CiO(f , K )+VR CiU (f , K ,Y )
(A.1)
21
where the combinations of frequencies and vehicles sizes can generate flows
Y
and
the minimization must respect capacity constraints: if we denote the number of
passengers that use each line
l
at each segment
e
as
λ¿
, then
Kl≥ λ¿e , l
(A.5). In order to find these
λ¿
, for each OD-pair
w
all the possible routes
rRw
need to be identified18 with their corresponding users costs
cr
(note that
VRCiU
is
then defined by the costs of the selected routes), such that the
Yw
passengers use
the less costly one19 (A.3), which will be identified by
xr=1
(A.2 and A.6); note that in
(A.3), when
xr ' =1
the corresponding cost must be the minimum, and when
xr ' =0
the inequality is trivially fulfilled. The load of each line at each arc is obtained as the sum
of the passengers over the set of routes that use line
l
in arc
e
defined as
S¿
(or a portion
θlew (f , K )
of them, if common lines are present) as written in (A.4).
Formally, the program being solved for each lines structure is the following:
min
f , K
VRC
(
f , K
)
subject to
rRw
xr=1w
(A.2)
xr ' cr '
(
f , K , xr '
)
≤ cr
(
f , K , xr
)
r , r ' Rw
(A.3)
λ¿=
w
Yw
rRwS¿
xrθlew (f , K )e , l
(A.4)
Kl≥ λ ¿l , e
(A.5)
f 0, x{0,1}
(A.6)
Appendix B. Optimal frequencies and capacities in the simple model (Section 3.2)
Let us solve rst the one-line system. Each of its components can be expressed
as a function of frequency:
Bus capacity
(K)
: total passengers per unit time
Y
use
f
buses per unit
time, such that the load of each bus is
K=Y/f
.
Cycle time
(tc)
: regarding vehicle in motion, each bus needs to travel across a
path whose length is
2L0+Q
, taking a time of
(2L0+Q)/V
; regarding time at
stops, each passenger needs
2t
to board and alight a bus whose load is
Y/f
passengers, which makes a total of
2tY /f
. Total cycle time is the sum of these
two terms:
2L0+Q
V+2tY
f
.
Fleet
(B)
: recalling that
f=B
tc
,
it becomes apparent that
B=f2L0+Q
V+2tY
.
18 By definition, when there are common lines in a segment, they are recognized as part of a single route.
19 Other criteria for passenger assignment to routes can be used without affecting the analysis, provided high cost
routes are discarded; one possible such criteria could be some distribution across routes with similar low costs.
22
Waiting time
(tw)
: passengers arrive at an homogeneous rate to the bus stop, and
buses exhibit a constant headway such that on average each passenger will wait
half the headway
(1/2f)
.
In-vehicle time
(tv)
: it needs to be calculated as the average between two types of
OD-passengers. Passengers that alight from the bus at the first stop travel a
distance
L0
such that time in-motion is
L0/V
. At the first stop the bus stays
Y
2ft
, and users that alight there spend on average half of that time. Passengers
that alight at the second stop travel a distance
L0+Q
; they stay in the vehicle
Y
2ft
at the first stop, and - on average - half that time at the second stop. The
average in-vehicle time for passengers is then
1
2
[
(
L0
V+Y
4ft
)
+
(
L0+Q
V+Y
2ft+Y
4ft
)
]
.
Replacing these expressions in
VRC=B(c0+c1K)+ pwY tw+pvY tv
yields
VRC=(f2L0+Q
V+2tY )(c0+c1
Y
f)+ pwY1
2f+pvY1
2
[
(
L0
V+Y
4ft
)
+
(
L0+Q
V+Y
2ft+Y
4ft
)
]
(A.7)
Making the derivative with respect to
f
equal to zero yields:
c
Y[¿¿ 1+pv
4]
pw
2+2t¿
¿
YV ¿
¿
f¿=
¿
,
c
Y[¿¿ 1+pv
4]
pw
2+2t¿
¿
V¿
Y2c0(L0+Q)
¿
K¿=
¿
(A.8)
Both expressions increase with
Y ,
with
f¿
tending to a linear function, and
K¿
tending to some constant when
Y → ∞
.
The solution for the two-lines system is the following:
Bus capacity
(K)
: total passengers per unit time per line are now
Y/2
, and use
f
buses per unit time, such that the load of each bus is
K=Y/2f
.
Cycle time
(tc)
: each bus travels across a path whose length is
2L0
, so time in
motion is
2L0/V
; regarding time at stops, each passenger needs
2t
to board
and alight a bus whose load is
Y/2f
passengers, which makes a total of
tY /f
.
Cycle time is the sum of these two terms:
tc=2L0
V+tY
f
.
Fleet
(B)
: there are two identical lines so
B=2f tc=4f L0
V+2tY
.
Average waiting time
(tw)
: as in the one line system
tw=1/2f
.
23
Average in-vehicle time
(tv)
: Passengers spend in motion
L0
V
. At stops, each
bus spends
1
2ftY
, such that passengers spend on average half that time, which
yields
tv=L0
V+tY
4f
.
Replacing these expressions in
VRC=B(c0+c1K)+ pwY tw+pvY tv
yields
VRC=
(
4f L0
V+2tY
)
(
c0+c1
Y
2f
)
+pwY
2f+pvY
(
L0
V+tY
4f
)
(A.9)
Making the derivative with respect to
f
equal to zero yields:
f¿=
YV
(
pw
2+tY
[
c1+pv
4
]
)
4L0c0
,
K¿=
Y L0c0
V
(
pw
2+tY
[
c1+pv
4
]
)
(A.10)
Again, both expressions increase with
Y ,
with
f¿
tending to a linear function, and
K¿
tending to some constant when
Y → ∞
.
24
Appendix C. Definitions and values of the parameters for simulations.
Symbol Meaning Value
α
Fraction of trips starting at the peripheries that go to the CBD. 0.5
β
Fraction of trips starting at the peripheries that go to the own subcenter. 0.25
a
Fraction of trips that start at the peripheries. 0.8
~
α
Fraction of trips starting at the sub-centers that go to the CBD. 0.67
c0
Unitary cost per bus per period of time. 0.17 [US$/min]
c1
Unitary cost per seat per period of time. 0.0034 [US$/min]
g
Distance periphery-subcenter/distance subcenter-CBD. 0.33
n
Number of zones in the city. 8
pT
Users’ cost of a transfer. 0.59 [US$]
pv
Value of in-vehicle time. 1.48 [US$/h]
pw
Value of waiting time. 2.96 [US$/h]
T0
Vehicle in-motion time between a subcenter and the CBD. 30 [min]
L0
Distance from origin to each destination in triangle city 30 [km]
Q
Distance between destinations in triangle city 2 [km]
V
Commercial speed of the buses 13 [km/h]
25
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