University of California Transportation Center
UCTC- FR- 2010- 19
Cost- Saving Properties of Schedule Coordination in a
Simple Trunk- and- Feeder Transit System
Karthikgeyan Sivakumarana, Yuwei Li,
Michael J. Cassidy, and Samer Madanat
University of California, Berkeley
August 2010
1
Cost- Saving Properties of Schedule Coordination
in a Simple Trunk- and- Feeder Transit System
Karthikgeyan Sivakumaran a, *, Yuwei Li a , Michael J. Cassidy a , Samer Madanat b
a
Department of Civil and Environmental Engineering
University of California
416 McLaughlin Hall
Berkeley, CA 94720, United States
b
Department of Civil and Environmental Engineering
University of California
110 McLaughlin Hall
Berkeley, CA 94720, United States
Abstract
The paper explores how the coordination of vehicle schedules in a public transit system affects
generalized costs. We consider an idealized system that delivers its users to a common
destination by requiring each to transfer from a feeder- to a trunk- line vehicle. Continuum
models are used first to analyze cases in which the trunk- line vehicle schedule is given
exogenously. We find that when feeder vehicles are dispatched in coordination with this
exogenous trunk- line schedule, the reduction in user cost often outweighs the added cost to the
feeder operation. In cases when the frequencies of trunk and feeder services can be established
jointly, the models show that coordination can be Pareto improving, meaning that operator and
user costs both diminish. Conditions that give rise to these cost savings are specified. Practical
implications are discussed.
Keywords: schedule coordination; feeder transit
1. Introduction
By reducing the times spent in transferring between vehicles, schedule coordination can diminish
the costs that a transit system imparts to its users. In the long run, this can bring broader benefits
to both the transit agency and society at large by inducing greater transit ridership. These
matters have been studied extensively in the literature ( Chien and Schonfeld, 1998; Chowdhury
and Chien, 2002; Li et al., 2009).
What seem to have garnered less attention are questions on how coordination can affect
aspects of transit cost beyond just the user cost. Yet, a decision on whether to deploy some
proposed coordination scheme will often depend upon these other costs; e. g. a scheme that saves
user transfer time is more likely to be adopted if it also reduces, or at least does not significantly
increase, transit operating costs ( Hickey, 1992; Schumann, 1997).
* Corresponding author. Telephone: + 1( 304) 517- 0961; Fax: + 1( 510) 643- 8919; E- Mail: ksivakum@ berkeley. edu
2
The present paper therefore explores how schedule coordination affects some of the costs
that are imparted to transit operators as well as its users. This is done by applying continuum
approximations of generalized cost to a simple transit network of many parallel feeder lines that
connect to a single trunk.
Background is furnished in the following section. We describe: existing models for
designing public transit systems, the scope of the present study, and the hypothetical system
upon which the study is based. A continuum model is used in section 3 to explore impacts of
coordination when the schedule of trunk- line service is given exogenously. We find that by
dispatching feeder vehicles in coordination with the given trunk schedule, total user cost can
significantly diminish while little or no extra cost is imparted to the operator of the feeder
service. The continuum model is expanded and used in section 4 to explore cases in which the
trunk and feeder service frequencies can be optimized jointly. Here we find that coordination
can be Pareto improving, such that costs diminish for all parties. Practical implications are
discussed in section 5.
2. Background
This section: reviews relevant literature ( sec. 2.1); describes our general approach to the present
analyses ( sec 2.2); and presents the trunk- feeder network to be used in these analyses ( sec. 2.3).
2.1 Literature Review
Numerous models have been developed for designing transit systems. Most furnish values for
decision variables ( e. g. the distances between stops and between routes, vehicle headways, etc.)
that minimize some generalized cost. Roughly speaking, these models fall into two categories,
as described below.
In the first category of models, input parameters and decision variables take values that
are discrete in space. Thus for example, travel demands are specified as origin- destination
matrices, and the locations available for stops and routes may be constrained by details of the
local topography. Models of this discrete type can generate design solutions that incorporate
many realistic details. This feature has made these models popular in the literature: they have
been used to design systems that deliver users from many origins to many destinations ( Chien
and Schonfeld, 1998; Kuah and Perl, 1989; Kuan et al., 2006; Martins and Pato, 1998; Shrivastav
and Dhingra, 2001; Shrivastava and O’Mahony, 2006; Verma and Dhingra, 2006); as well as
systems in which the distribution of trips is many- to- one ( Chien and Yang, 2000; Chowdhury
and Chien, 2001).
The realism imbedded in these discrete models adds complexity, however. As a result,
solutions are typically obtained through heuristic methods that do not always guarantee global
optimality. And these solutions often do not unveil relations between the input parameters and
the optimal values of the decision variables. Thus it can be difficult to glean general insights
from these models to inform high- level system design.
3
In the second category of models, input parameters and decision variables are
approximated as smooth, continuous functions, such that travel demands, routes and stops are
expressed per unit length or unit area ( Daganzo, 1999; Newell, 1973). Continuum
Approximation ( CA) models of this type have primarily been used to design many- to- one
systems ( Byrne and Vuchic, 1972; Chien, et al., 2002; Clarens and Hurdle, 1976; Hurdle, 1973;
Kuah and Perl, 1988; Wirasinghe, 1977; Wirasinghe, 1980). In essence, CAs omit much of the
real- world detail of discrete models in favor of simplicity, and therefore tend to generate
solutions that take relatively simple, closed forms. These can often unveil relations between the
input parameters and the optimal values of the decision variables. The resulting insights
therefore tend to be more general than those obtained from a discrete model.
2.2 Study Scope
In the pursuit of general insights, we will analyze the costs of a simple trunk- feeder transit
network using CAs. Previous research involving models of this kind typically ignored trunk- line
costs, and assumed that trunk- vehicle headways were sufficiently small as to render user transfer
costs negligible as well. Our models will differ from these earlier CAs in that we will initially
add a term to describe a user’s cost in transferring from a feeder- to a trunk- line vehicle, and will
later add another term to describe trunk- line operating cost ( Clarens and Hurdle, 1975; Hurdle,
1973; Kuah and Perl, 1988; Wirasinghe, 1980). We will thus consider a user’s costs incurred as
she: accesses a feeder line by travelling toward it; waits for a feeder vehicle; and eventually
transfers to a trunk vehicle. These will be estimated for the so- called average- case user during
her one- way trip. The costs of accessing a feeder- line stop while travelling parallel to the line,
and of travelling aboard vehicles are both ignored, since these are invariant to schedule
coordination.
The costs for the system operator( s) will depend upon the vehicle- hours of service that
are to be provided. These will depend on factors that include: service frequencies, the density of
feeder lines, the physical lengths of those lines and vehicle travel speeds. We will assume that
any cost of controlling vehicles to maintain a schedule is the same, whether or not the schedule is
aimed at coordinating trunk and feeder services; and that all transit vehicles have sufficient
capacities to accommodate boarding demands. 1
2.3 Case Study
The idealized network on which we will base our study is shown in Figure 1. It consists of a
trunk line, operating at a headway , that runs in the y- direction to a Central Business District
( CBD) at location ; and parallel feeder lines. The latter are each of length L, run
perpendicular and connect to the trunk, and collectively span distance along the trunk. The
service region is thus a rectangle of dimension x , as shown in the figure.
1 The presumed invariant cost of control is therefore ignored in our analyses. Moreover, new control methods for
maintaining a transit schedule can be deployed quite inexpensively; see Daganzo and Pilachowski, 2009 and
Pilachowski, 2009. Further discussion on the costs of control, and on issues regarding vehicle capacity, are offered
in section 5.
4
We will assume that is sufficiently large that the service region holds many feeder
lines; and that these lines can be located anywhere throughout the region, as might occur when
the feeder service is provided by buses on a dense network of streets. Trunk- line stops will not
necessarily be placed at every junction with the feeder lines, meaning that a feeder vehicle may
be required to travel in the - direction to reach a trunk- line stop. However, this travel distance
will be assumed negligible as compared with , a feeder vehicle’s one- way travel distance in the
- direction.
Travel demand in the service region is expressed as a continuous, time- independent
density function. We will assume that this density varies gradually along and is independent of
, as might occur, for example, if development arose along the trunk line and gradually
diminished at greater distances from it. We therefore denote this density function . By
assuming that demand is independent of , both the spacing between neighboring feeder lines, ,
and the feeder- vehicle headway, , will be fixed throughout the service region. This will
simplify our analysis for the case when trunk and feeder schedules are established jointly ( in
Section 4). This assumption of uniform demand also means that all feeder lines will have the
same number of stops, since we will take stop locations to be unconstrained by topography and
would be selected instead to minimize costs ( see Kuah and Perl, 1988). We can therefore
assume that feeder- vehicle speeds are the same on all lines. We select a feeder- vehicle
commercial speed, , that is slower than the cruise speed to roughly account for the time lost in
serving passengers. ( Crude estimates for commercial speed suffice, as we shall see in sec. 3.)
Finally, we will assume that the distribution of trips is many- to- one, with all users
bound for the CBD. A case like this might arise ( approximately) during the morning rush in a
mono- centric city.
3. Exogenous Trunk- Line Schedules
We first consider the case in which trunk- line vehicles operate with a headway that is given and
that cannot be altered to accommodate feeder operation. This case can arise when, for example,
the trunk line is long and connects both to feeder lines inside our service region, and to additional
lines that reside outside it. We present CAs for estimating generalized costs when trunk and
feeder services are uncoordinated ( sec. 3.1); reformulate the models to estimate costs when
feeder service is operated in coordination with the exogenously- specified trunk- line schedule
( sec. 3.2); and make comparisons ( sec 3.3).
3.1 Cost Models for Uncoordinated Service
The CAs presented below are comparable to those derived in earlier work ( Hurdle, 1973;
Wirasinghe, 1980; Kuah and Perl, 1988). We will merely add a term to estimate the user cost of
transfers. All parameters and decision variables to be used in our models are defined in Table 1.
We define to be the generalized cost per unit time for the service region. It is the sum
of four cost components: the users’ access to feeder lines ( along the direction), their wait times
at feeder stops, their wait ( i. e. transfer) at the trunk stop, and the feeder operating cost. Thus,
5
( 1)
Where
.
We minimize , taking and as decision variables. The optimal spacing for the feeder
lines, as a function of , is therefore
. ( 2)
Substituting ( 2) into ( 1), we minimize with respect to and find the optimal feeder- vehicle
headway to be
, ( 3)
which leads to an optimal feeder line spacing of
. ( 4)
Substituting ( 3) and ( 4) into ( 1), we find that the minimum generalized cost for the service region
is
. ( 5)
Note from ( 5) how this minimum cost ( excluding the waiting cost at the trunk stop) is
robust to variations in the input parameters, meaning that coarse estimates for their values will
yield near- optimal designs for the feeder system. 2
Note too that ( 5) furnishes a cost in the absence of coordination between the trunk and
feeder schedules; this is evident from the third term in ( 1) which takes the average transit time to
be half the trunk- vehicle headway. Schedule coordination is examined next.
3.2 Coordinating Feeder Service with the Trunk
Consider a simple scheme in which we coordinate schedules within the service region by
choosing a common feeder headway, , that is some integer multiple of the trunk line’s
exogenous headway; i. e.
where is any positive integer.
One could select the value of that makes closest to the optimal feeder line headway,
, obtained from ( 3). Referencing ( 2), one would then obtain:
( 6)
2 Methods for transforming idealized CA design variables to real- world environments can be found in Kuah and Perl
( 1988) and Wirasinghe ( 1980).
6
and
. ( 7)
3.3 Cost Comparisons
Selecting a line spacing as in ( 6) can, to a large degree, compensate for the added cost of
choosing an  , such that the user and operating cost incurred within the feeder subsystem
changes very little. To illustrate this, we note first that can be rewritten as ,
where is some non- negative constant that should be close to 1 ( by virtue of having chosen
suitable k). In the uncoordinated case, the costs within the service region for user access, for the
wait for feeder vehicles and for feeder operation are each . In the coordinated case, the
access and feeder operating costs are each , and the user wait cost for feeder
vehicles is . The percent difference in the sum of access, waiting, and feeder
operating cost between coordinated and uncoordinated operation is therefore . This
difference is small for close to 1, as shown in Figure 2.
Since schedule coordination eliminates the user transfer cost at the trunk stop, the
difference between and is . And since ,
coordination yields a savings in total cost only when is small. This makes sense: when H is
small compared to , there is no need for schedule coordination.
4. Endogenous Trunk- Line Schedules
Suppose now that the trunk- line component of the system serves only the feeder lines that reside
within the service region of Figure 1. For this case, we will compare the costs of uncoordinated
services against those that occur when the trunk and feeder lines operate with the same headway
and in coordinated fashion. We will also demonstrate how this simple coordination scheme can
be Pareto improving. CAs are presented for the uncoordinated and coordinated cases ( secs. 4.1
and 4.2, respectively). Estimated costs are compared to identify the conditions needed to achieve
Pareto improvements ( sec. 4.3); and the cost savings are quantified for a range of operating
environments on our hypothetical network ( sec. 4.4).
4.1 Uncoordinated Trunk- Feeder Service
The trunk and feeder headways, and respectively, are treated for now as separate decision
variables, such that , the sum of relevant generalized costs for the entire service region is given
by
( 8)
Where
.
7
The first term in ( 8) is the trunk operating cost, obtained by multiplying the operating
cost rate, , with the needed number of trunk- line vehicles ( ). The second term is the
average transfer cost, where in the absence of coordination, the average wait time is again
assumed to be . The third term describes the remaining two user- cost components and the
feeder operating cost, given in ( 5).
We find that
and that
( 9)
where is the service region’s total hourly travel demand.
Note that the minimum cost, , is expressed purely as a function of its input parameters.
4.2 Coordinated Service
When trunk and feeder services are coordinated and operate at a common headway, , the sum
of relevant generalized costs for the service region, , is given by
( 10)
where
.
The second term in ( 10) is the user waiting cost for a feeder vehicle ( only), since we assume zero
user cost in transferring to the trunk. The third term, consisting of the access to feeder lines and
the feeder operating cost, comes from ( 7). Note that ( 10) is convex in , since the sum of
convex functions is itself convex. Thus, the minimum cost, , can be obtained numerically
from ( 10).
4.3 Conditions for Pareto Improvement
We now compare the cost models of secs. 4.1 and 4.2 to determine the bounds for which
schedule coordination is Pareto improving. The user costs will be examined first. These
diminish with coordination if the sum of the user- cost components of ( 9) is less than that of ( 8);
i. e., if
. ( 11)
Next we determine the conditions in which coordination reduces the operating cost for
trunk and feeder services combined. To this end, we compare the operating- cost components of
( 9) and ( 8). If
, ( 12)
the trunk and feeder operating cost is reduced.
8
If both the trunk and feeder services are furnished by a single entity, coordination will be
Pareto improving whenever ( 11) and ( 12) both hold. Of course, trunk and feeder services are
often provided separately by distinct agencies. In this latter case, coordination can still reduce
the costs for every party involved.
To see why this is true, note first from ( 12) that the operating cost for feeder service
( alone) is reduced if , which can also be expressed as
. ( 13)
Referring to ( 3), we can see that ( 13) reduces to the inequality .
As regards the operating cost of trunk service ( alone), inspection of ( 7) and ( 9) reveals
that the optimal trunk- vehicle headway is guaranteed to be larger with coordination. This is
because i) the coordinated model, given by ( 9), shifts the waiting cost ( ) at trunk stops to
feeder stops, and ii) this waiting cost is now weighed against both the trunk and feeder operating
costs ( rather than solely against the trunk operating cost). These two factors combined assure
that coordination results in a larger trunk headway. The larger headway leads to a lower trunk
operating cost.
4.4 Illustrations of Pareto Improvement
To illustrate the benefits of coordination, we present two scenarios. The first corresponds to a
region where the average wage rate is low, as typically occurs in a developing country; and the
second to a region with a high average wage rate, as in a more industrialized country. We
assume in both scenarios that the user value of time, for both access and waiting, is equivalent to
the user wage rate. Additionally, while some components of the hourly cost rates for both the
feeder and trunk systems will remain roughly the same across the two scenarios ( fuel,
depreciation, etc.), the overall hourly operating cost will differ due to differences in labor costs.
Accordingly, the input cost parameters for the developing country (“ low”) and the industrialized
country (“ high”) are shown in Table 2.
We assume that both scenarios are governed by the same system characteristics: = 5
km/ hr, = 10 km, = 5 km, and = 20 km ( refer again to Table 1 for the definitions of these
parameters). Feeder- and trunk- vehicle speeds are chosen to reflect typical urban bus and light
rail speeds: 15 and 30 km/ hr, respectively.
Both scenarios are analyzed under a range of demand densities for the trip origins. For
simplicity, uniform densities are used. ( Similar results were found for demand densities that
varied with y and caused uncoordinated feeder- vehicle headways to vary across feeder lines.)
These uniform densities ranged from 10- to 200 pax/ km 2 - hr, in increments of 10 pax/ km 2 - hr,
such that the total demand in the service region ranged from 500- to 10,000 pax/ hr.
Costs saved through coordination are shown as functions of these demands in Figures 3
and 4. Figure 3 displays savings for the low- cost scenario, and Figure 4 for the high- cost one.
Both figures present curves for the savings in total user cost, total ( trunk and feeder) operating
cost and feeder operating cost alone.
9
5. Conclusions
By applying continuum approximations to a simple trunk- feeder transit network, we find that
schedule coordination can save more than just user costs. In cases when the vehicle headway on
the trunk is given exogenously, service can be coordinated by operating the feeder vehicles at
headways that are integer multiples of those on the trunk. By suitably adjusting the feeder- line
spacing, this simple coordination scheme can eliminate user waiting cost at the trunk station,
while often adding little or no cost elsewhere in the system. Better still, if the headways for
trunk and feeder vehicles are both decision variables, schedule coordination can often be Pareto
improving and benefit all parties.
We acknowledge that these findings came by analyzing an idealized network, and by
adopting a number of ( often simplifying) assumptions. And we concede that there are limits on
the extent to which these findings can be exploited. For example, when a trunk line’s schedule is
exogenous, an operator does not always have free reign to locate the feeder lines in response to
that schedule. Even when this freedom exists, moreover, feeder- line spacing would usually
have to be optimized with respect to a limited portion of the day ( e. g. the morning rush), since
the headways scheduled for a trunk tend to change over the day. Furthermore, we did not
account for any real- world “ control” costs of coordination, which might include transit signal
priority, vehicle tracking, etc. If this is the case, the cost of control may override coordination
benefits, particularly when the optimal coordinated headway takes a low value. Finally, the
passenger- carrying capacity of trunk- line vehicles can also limit possibilities, since the joint
selection of a feeder- and trunk- vehicle headway tends to expand the latter. This limitation can
be remedied at relatively low cost when trunk vehicles can be enlarged; e. g. by using articulated
buses or by adding cars to trains. Or, an operator might increase capacity by dispatching trunk
vehicles in small platoons, though the resulting increase in trunk operating cost might then
become large ( Sivakumaran et al., 2010).
All this notwithstanding, the present findings can inform transit system design. In those
instances when trunk and feeder services are provided by distinct agencies, the findings speak to
the benefits that might come via institutional cooperation. They may also motivate transit
agencies to explore alternative schemes for delivering service. Consider, for example, a case in
which many- to- one service is provided on a network with a long trunk line that spans a long
service region. Pareto improvements might come by partitioning the network into narrower sub-regions
and assigning trunk- vehicles to serve sub- regions in dedicated fashion. A better
understanding of the cost- saving potential of schedule control might ultimately give rise to any
number of innovations in transit service. The present paper represents a step forward in this
regard.
Acknowledgement
The present work was funded by the University of California Transportation Center.
10
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Fig. 1. Hypothetical trunk- and- feeder operating environment.
CBD ( at
Demand density
Adjacent feeder lines
Trunk line
( at )
)
Figure 1
Fig. 2. Percent change in the sum of access, waiting, and feeder operating costs through coordination.
0%
2%
4%
6%
8%
10%
0.6 0.8 1 1.2 1.4
Percent Difference in Cost
m
Figure 2
Fig. 3. Cost Savings from Coordination, for “ Low” Cost Parameter Values
0
200
400
600
800
1000
1200
1400
0 2000 4000 6000 8000 10000
Cost Savings Rate [$/ hr]
Hourly Demand [ pax/ hr]
Total User Cost Savings
Total Operating Cost
Savings
Feeder Operating Cost
Savings
Figure 3
Fig. 4. Cost Savings from Coordination, for “ High” Cost Parameter Values
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0 2000 4000 6000 8000 10000
Cost Savings Rate [$/ hr]
Hourly Demand[ pax/ hr]
Total User Cost Savings
Total Operating Cost
Savings
Feeder Operating Cost
Savings
Figure 4
Table 1
Description of input parameters and decision variables
Term Description Units
Demand density pax/ km 2 - hr
User speed in accessing the transit system km/ hr
Feeder vehicle speed km/ hr
Trunk vehicle speed km/ hr
User value of access time $/ hr
User value of waiting time $/ hr
Feeder vehicle operating cost rate $/ hr
Trunk vehicle operating cost rate $/ hr
Width of the service region km
Length of the service region km
Distance to CBD from y = 0 km
Feeder line spacing km
Feeder headway hrs
Trunk headway hrs
Table 1
Table 2
Cost parameter values for two scenarios
Term Low Value [$/ hr] High Value [$/ hr]
3 15
3 15
40 100
180 300
Table 2