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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

mi

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))>0∀ε>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

r∈Rw

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

∑

r∈Rw

xr=1∀w

(A.2)

xr ' cr '

(

f , K , xr '

)

≤ cr

(

f , K , xr

)

∀r , r ' ∈Rw

(A.3)

λ¿=∑

w

Yw∑

r∈Rw∩S¿

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