INSTITUTE OF TRANSPORTATION STUDIES
UNIVERSITY OF CALIFORNIA, BERKELEY
Public Transportation Systems:
Basic Principles of System Design,
Operations Planning and Real- Time
Control
Carlos F. Daganzo
Course Notes
UCB- ITS- CN- 2010- 1
October 2010
i
Institute of Transportation Studies
University of California at Berkeley
Public Transportation Systems:
Basic Principles of System Design,
Operations Planning and Real- Time Control
Carlos F. Daganzo
COURSE NOTES
UCB- ITS- CN- 2010- 1
October 2010
ii
Preface
This document is based on a set of lecture notes prepared in 2007- 2010 for the U. C. Berkeley
graduate course “ CE259- Public Transportation Systems”-- a course targeted to first year graduate
students with diverse academic backgrounds.
The document is different from other books on public transportation systems because it is
informal, has a narrower focus and looks at things in a different way. Its focus is the planning,
management and operation of public transportation systems. Important topics such as financing,
governance strategies and urban transportation policy are not covered because they are not
specific to transit systems, and because other books and courses already treat them in depth. The
document is also different because it deemphasizes facts in favor of ideas. Facts that constantly
change and can be found elsewhere, such as transit usage statistics and transit system
characteristics, are not covered.
The document’s way of looking at things, and its structure, is similar to the author’s previous
book “ Logistics systems analysis” ( Springer, 4th edition, 2005) from which many basic ideas are
borrowed. ( Transit systems, after all, are logistics systems for the movement of people.) Both
documents espouse a two- step planning approach that uses idealized models to explore the
largest possible solution space of potential plans. The logical organization is also similar: in both
documents systems are examined in order of increased complexity so that generic insights
evident in simple systems can be put to use as knowledge “ building blocks” for the study of
more complex systems.
The document is organized in 8 modules: 5 on planning ( general; shuttle systems; corridors; two-dimensional
systems; and unconventional transit); 2 on management ( vehicles; and employees);
and 1 on operations ( how to keep buses on schedule). The planning modules examine those
aspects of the system that are usually visible to the public, such as routing and scheduling. The
management and operations modules analyze the more mundane aspects required for the system
to work as designed. Two more modules are in the works: management of special events ( e. g.,
evacuations; Olympics); and operations in traffic.
Although the document includes new ideas, which could be of use to academics and
professionals, its main aim is as a teaching aid. Thus, a companion document including 7
homework exercises and 3 mini- laboratory projects directly related to the lectures is also made
available. It can be obtained by visiting the Institute of Transportation Studies web site and
looking for a publication entitled: “ Public Transportation Systems: Mini- Projects and Homework
Exercises”. Versions of these exercises and mini- projects were used in the 2009 and 2010
installments of CE259: a 14- week course with two 1- hour lectures and one 1- hr discussion
session per week. Sample solutions to the mini- projects and exercises can be obtained by
university professors by writing to the ITS publications office and requesting a third document
entitled: “ Public Transportation Systems: Solution Sets”.
The various modules were originally compiled by PhD students Eric Gonzales, Josh Pilachowski
and Vikash Gayah, directly from the lectures. Subsequently, my colleague Prof. Mike Cassidy
used them in an installment of CE259 and offered many comments. This published version has
been edited and reflects the input of all these individuals. Their help is gratefully acknowledged.
The errors, of course, are mine. The financial support of the Volvo Research and Educational
Foundations is also gratefully acknowledged.
Carlos F. Daganzo
September, 2010
Berkeley, California
iii
iv
CONTENTS
Preface .…………………………………………………………………….…………………… i
Module 1: Planning— General Ideas ……………...………………………..……………… 1- 1
• Course substance and organization ………………………...………………….……… 1- 1
• Transit Planning …………………………………………………………………….… 1- 2
o Definitions ……………………………………………………….…………… 1- 2
o How to account for politics ………………………………………...………… 1- 3
o How to account for demand ……………………………………………..…… 1- 6
o The shortsightedness tragedy …………………………………….…………… 1- 6
o Planning and design approaches ……………………………………………… 1- 7
• Appendix: Class Syllabus ……………………………………………………...…… 1- 10
Module 2: Planning— Shuttle Systems ………………………………………….………… 2- 1
• Overview ……………………………………………………………………..……….. 2- 1
• Shuttle Systems ……………………………………………………………………….. 2- 2
o Individual Transportation ………………………………………………….….. 2- 2
􀂃 Time- independent Demand …………………………………..……….. 2- 2
􀂃 Time- Dependent Demand – Evening ( Queuing) ……………………… 2- 3
􀂃 Time- Dependent Demand – Morning ( Vickrey) ……………………… 2- 4
o Collective Transportation ………………………………………………..…… 2- 7
􀂃 Time- Independent Demand ………………………………………..… 2- 7
􀂃 Time- Dependent Demand …………………………………….……… 2- 8
o Comparison between Individual and Collective Transportation ……….…… 2- 10
• Appendix A: Vickrey’s Model of the Morning Commute ………………..…………. 2- 12
Module 3: Planning— Corridors ……………………………………………………..…… 3- 1
• Idealized Analysis …………………………………………………………………… 3- 2
o Limits to The Door- to- Door Speed of Transit ……………………………… 3- 2
o The Effect of Access Speed: Usefulness of Hierarchies ………..…………… 3- 5
• Realistic Analysis ( spatio- temporal) ………………………………………………… 3- 8
o Assumptions and Qualitative Issues ………………………………………… 3- 8
o Quantitative formulation …………………………………………………… 3- 11
o Graphical Interpretation ……………………………………………..……… 3- 12
o Dealing with Multiple Standards …………………………………………… 3- 13
o No transfers ………………………………………………………………… 3- 14
o Transfers and Hierarchies …………………………………………..……… 3- 17
o Insights ……………………………………………………………...……… 3- 22
o Standards- Revisited ……………………………………………...………… 3- 24
o Space- and Time- Dependent Services ………………………………...…… 3- 26
􀂃 Average Rate Analysis ………………………………………..…… 3- 26
􀂃 Service Guarantee Analysis ……………………………..….……… 3- 28
v
Module 4: Planning— Two Dimensional Systems ……………………...………………… 4- 1
• Idealized Case ( New 2- D Issues) ………………………………..……………….. … 4- 1
o Systems without Transfers ………………………………..…….…………… 4- 2
o The Role of Transfers in 2- D Systems …….……………………………. 4- 4
• Realistic Case ( No Hierarchy) ……………………..……………………………….… 4- 9
o Logistic Cost Function ( LCF) Components ……………………………..…… 4- 9
o Solution for Generic Insights …………………………………………..…… 4- 10
o Modifications in Practical Applications ……………………………….…… 4- 12
o General Ideas for Design …………………………………………………… 4- 14
• Realistic Case ( Hierarchies-- Qualitative Discussion) ………………….…………… 4- 16
• Time Dependence and Adaptation ………………………………………………… 4- 17
• Capacity Constraints ………………………………………………………………… 4- 19
• Comparing Collective and Individual Transportation ……………………………… 4- 20
Module 5: Planning— Flexible Transit ………………………….………………………….. 5- 1
• Ways of delivering flexibility ………………………………..…………………..……. 5- 1
o Individual Public Transportation ……………………………………………… 5- 1
o Collective Transportation …………………………………………………...… 5- 2
• Taxis ……………………………………………………………………………..…… 5- 2
• Dial- a- Ride ( DAR) …………………………………………………………….……… 5- 6
• Public Car- Sharing …………………………………………………………..………. 5- 10
• Appendix: Determination of Expected Distance to a Taxi …………………….……. 5- 13
Module 6: Management— Vehicle Fleets …………………………………………………… 6- 1
• Introduction ………………………………………………………………………..….. 6- 2
• Schedule Covering 1 Bus Route …………………………………...………………….. 6- 3
o Fleet Size: Graphical Analysis ………………………………….…………….. 6- 4
o Fleet Size: Numerical Analysis ……………………………………………….. 6- 6
o Terminus Location …………………………………………………………….. 6- 7
o Bus Run Determination ……………………………………………………….. 6- 8
• Schedule Covering N Bus Routes ……………………………………….…………….. 6- 9
o Single Terminus Close to a Depot …………………………………………….. 6- 9
o Dispersed Termini and Deadheading Heuristics …………………………….. 6- 10
• Discussion: Effect of Deadheading ………………………………………………….. 6- 12
• Appendix: The Vehicle Routing Problem and Meta- Heuristic Solution Methods …... 6- 13
vi
Module 7: Management— Staffing …………………………………………………..…… 7- 1
• Recap ……………………………………………………………………………….…. 7- 1
• Staffing a Single Run ……………………………………………………………..…… 7- 2
o Effect of Overtime ………………………………………………………..…… 7- 3
o Effect of Multiple Worker Types ………………………………………...…… 7- 4
• Staffing Multiple Runs …………………………………………………………...…… 7- 5
o Run- Cutting …………………………………………………………………… 7- 5
o Covering …………………………………………………………….………… 7- 6
o Simplified estimation of cost ……………………………………………...…… 7- 6
• Choosing Worker- Types …………………………………………………………….… 7- 8
• Dealing with Absenteeism ………………………………………………………..…… 7- 9
• What is Still Left to be Done ………………………………………………………… 7- 11
Module 8: Reliable Transit Operations ……………………………………………..……… 8- 1
• Reliability …………………………………………………………………………..… 8- 1
• Systems of Systems …………………………………………………………………… 8- 1
o Example 1: a stable single agent ……………………..……………………….. 8- 2
o Example 2: an unstable single agent ……………….………………………….. 8- 4
o Example 3: two agents …………………………………………..…………….. 8- 5
• Uncontrolled Bus Motion ………………………………………………….………….. 8- 6
• Conventional Schedule Control ……………………………………………………….. 8- 8
o Optimizing the Slack ………………………………………………………….. 8- 9
• Dynamic ( Adaptive) Control …………………………………………..…………….. 8- 11
o Forward looking Method …………………………………………………….. 8- 11
o Two Way Looking Method ( Cooperative) ………………………….……….. 8- 14
Public Transportation Systems: Planning— General Ideas
1- 1
Module 1: Planning— General Ideas
( Originally compiled by Eric Gonzales and Josh Pilachowski, January 2008)
( Last updated 9- 22- 2010)
Outline
• General course info ( admin)
• Course substance and organization
• Transit Planning
o Definitions
o How to account for politics
o How to account for demand
o The shortsightedness tragedy
o Planning and design approaches
Course Substance and Organization
Goal of the Course
• What transit can and can’t do realistically
• How to do it ( large/ small scale)
• How to make it happen practically ( focus on engineering)
Brief Explanation of Syllabus ( see Appendix)
• The planning part of the course explores what is possible and how to do it with building
blocks of increasing realism and complexity; it shows the limits of transit systems and
gives you the tools to develop systematic plans.
• The management and operations part explores the “ plumbing” of transit systems. This
includes management items that are hidden from the user’s view such as fleet
sizing/ deployment and staffing plans, as well as more visible operational items such as
adaptive schedule control and traffic management.
• Planning ideas will be reinforced with two lab projects and five homework exercises.
Management/ operations ideas will be reinforced with one lab project and two exercises.
Imagine public transit in a linear city. Many people travel between different origins and
destinations at different times ( thin arrows in the time- space diagram below). Note how people
have to adapt their travel in space to the location of stops and in time to the scheduled service in
order to use transit ( thick arrow), and how this adaptation could be reduced by providing more
transit services ( more thick arrows). Unfortunately, the thick arrows cost money; and this
Public Transportation Systems: Planning— General Ideas
1- 2
competition between supply costs versus demand adaptation turns out always to be at the heart of
transit planning. It will be a central theme in this course.
city
transit
veh trip
User desired
x
t
stop
stop
adaptation
Transit Planning
Definitions
• Guideway – fixed parts of a transportation system, modeled as links and nodes
( infrastructure)
• Network – set of links and nodes, uni- or multi- modal
• Path – a sequence of links and nodes
• Origin/ Destination – beginning and end of a path through a network
• Terminal – node where users can change modes
• Planning – art of developing long term/ large scale schemes for the future
• Mobility – the distance people can reach in a given time ( e. g. VKT/ VHT)
• Accessibility – the opportunities people can reach in a given time ( depends on land use)
We can improve accessibility by improving mobility and/ or by changing the distribution of
opportunities. But if the opportunities are fixed in space, then a change in mobility is equivalent to
a change in accessibility.
As shown in the previous figure, there is a trade- off inherent in public transportation because users
give up flexibility ( suffering a “ level of service” penalty) for economy. To strike this balance
between level of service ( LOS) and supply cost in networks for individual modes ( e. g. highway,
bike- lanes, and sidewalks), planners can only change the infrastructure. But in collective
transportation, planners also have control over the vehicles’ routes and schedules.
Public Transportation Systems: Planning— General Ideas
1- 3
The goal of planning is to achieve efficiency, measured as a combination of LOS and supply costs.
Costs come in different forms, such as time, T, comfort, safety, and money, $, and should be
reduced to some common units. The result is called a generalized cost or disutility, which can be
defined both for individuals and groups, and is usually expressed as a linear combination of
component costs; e. g. for one individual experiencing time T and cost $ it could be:
Generalized Cost = βTT + β $
$
How to Take into Account Politics
Note that βT and β $
will vary between individuals, so even though an individual may have a well-defined
generalized cost, the choice of appropriate weights to represent a diverse group is always a
political decision that cannot be resolved objectively.
Note too that transit systems involve costs to non- users— energy, pollution, noise, etc.— and that
since people also disagree about how these should be valued, they further complicate the decision-making
picture.
Clearly, we need to simplify things! ( but without ignoring the effects of politics).
To this end, we will assume in this course that there is a political process that has converged to the
establishment of some standards, which would apply to all the non- monetary outputs of the transit
system; e. g.,
T – Door- to- door time ( no more than a standard, T0)
E – Energy consumed ( no more than E0)
M – Mobility ( at least M0)
A – Accessibility ( at least A0)
And our goal will be minimizing the cost, $, of meeting the standards; i. e.,
Mathematical Program ( MP): min{ $: T ≤ T0; E ≤ E0; M ≥ M0; A ≥ A0 … }
Note how each standard is associated with an inequality constraining the value of the performance
output in question. Since these outputs are usually directly connected to 4 key measures of
aggregate motion: VHT, VKT, PHT, PKT, we can often reformulate the standards in terms of
passenger time ( distance) and vehicle time ( distance).
Alternatively, since all variables in this MP ( both monetary and non- monetary), which we
collectively call y = ($, T, E, M, A), are functions of the system design, x, ( i. e., the routes and
schedules used for the whole system) and the demand, α ( which we assume to be given), we can
express the MP in terms of x and α.
Public Transportation Systems: Planning— General Ideas
1- 4
To make this formulation more concrete, let us define these relations by means of a vector- valued
function Fm:
y = Fm( x, α)
where,
y – performance outputs for the entire system ( both monetary and non- monetary)
m – mode
x – design variables for the entire system
α – demand
We then look for the value of x that minimizes the $- component of y while the other components
satisfy the standards constraints. The result is as a best design, x*( α), which if implemented would
yield y*( α) = Fm( x*( α), α) = Gm( α). This function represents the best performance possible from
mode m with given demand α. We will, in this course, compare the Gm( α) for different modes.
To see all this more concretely, consider a simple transit system where all users are concentrated at
two points.
In this case we have:
x – frequency of service ( a single design variable: buses/ hr)
α – demand ( a single demand variable: pax/ hr)
Define now the components of Fm. We assume that each vehicle dispatch costs cf monetary units.
Thus we have:
$ = Fm
$( x, α) = cf x/ α [$/ pax]
Note: we have defined $ as an average cost per passenger. We could instead have defined $ as the
total system cost per hour. Both definitions lead to the same result since they differ by a constant
factor: the demand, α. If we now assume that headways are constant but the schedule is not
advertised, we have:
T = Fm
T( x, α) = 1/ x [ hrs] ( out of vehicle delay assumes ½ headway at origin and ½ headway
at the destination)
And finally, if each vehicle trip consumes ce joules of energy we also have:
E = ce x/ α [ joules/ pax]
Public Transportation Systems: Planning— General Ideas
1- 5
If the political process had ignored energy and simply yielded a standard T0 for T, and if we choose
the monetary units so cf = 1, the MP would then be:
min{ x/ α: 1/ x ≤ T0 }.
Note that the OF is minimized by the smallest x possible. Thus, the constraint must be binding,
and we have:
x* = 1/ T0
Therefore the “ optimum” monetary cost per passenger would be:
$* ≡ Gm
$( α) = 1/( αT0)
We call the above the “ standards approach” to finding efficient plans.
There is another approach, which we call the “ Lagrangian approach.” It involves choosing some
shadow prices, β, and minimizing a generalized cost with these “ prices” without any constraints.
Although the selection of prices cannot be made objectively, one can always find prices that will
meet a set of standards ( see your CE 252 notes). So the Lagrangian approach is equivalent to the
standards approach. For example, we can formulate:
minx { $+ βT ≡ x/ α + β( 1/ x) }
The solution is:
x* = αβ
You can verify that the “ standards” solution ( x* = 1/ T0 and $* = x*/ α = 1/( αT0) is achieved for
( 1/ 2 )( 1/ ) . So no matter what standard you choose, there is a price that achieves it.
0 β = T α
In summary, there are 2 approaches to obtain low cost designs that satisfy policy aims:
1. Standards: min { $ s. t. T ≤ T0, E ≤ E0… }
This minimizes the dollar cost subject to policy constraints, e. g. for trip time, energy
consumption and possibly other outputs. Usually, as shown in the example, constraints
become binding when solved → T = T0, E = E0
2. Lagrangian: min { $( x, α) + βT( T( x, α)) + βE( E( x, α)) }
Public Transportation Systems: Planning— General Ideas
1- 6
This minimizes the generalized cost, and gives the same solution as the standards method
when suitable shadow prices, βT and βE, are chosen. The shadow prices can be found by
solving the Lagrangian problem for some prices, finding the optimum T and E and then
adjusting the prices until T and E meet the standards. In simple cases, such as the above
example, this can be done analytically in closed form.
How to Account for Demand: Some Comments about Demand Uncertainty and Endogeneity
So far, we have assumed that the demand, α, is given, and critics could say that this is not realistic.
However, if we are lucky and the design one provides happens to be optimum for the demand that
materializes, then the issue is moot. Suppose we design x for a chosen level of demand, α, that is
expected to materialize at some point in the future. Normally, we expect realized demand to
change with time, and for a well- designed system that provides improved service this demand
should be increasing. Then, the question of whether the system design is optimal in reality ( given
that we assumed a demand α0) is less a question of if, but of when, since the demand α0 will
eventually be realized. Furthermore, we will learn later in the course that the cost associated with
a design, x*, that is optimal for α0 is also near- optimal for a broad range of values of α ( within a
factor of 2 of α0). Thus, if the realized demand does not change quickly with time, the system
design is likely to produce near optimal costs for a long period of time.
Furthermore, we should remember that demand is difficult to predict in the long run. So, building
complicated models that endogenize α in order to predict precise values is not a worthwhile
activity in my opinion. Rough estimates of future demand are sufficient for design purposes. This
is not to say that a vision for the future is not important; only that it does not need to be anticipated
precisely. The following example illustrates what happens if one ignores the vision.
The Shortsightedness Tragedy
This example shows that when demand changes with time, then incrementally chasing optimality
with short- term gain objectives in mind can lead us to a much worse state than if we design from
the start with foresight and long term objectives.
Now, consider the investment decisions for a system with potential for 2 modes:
automobile – divisible capacity with cost per unit capacity, cg
subway – indivisible and very large capacity with cost for a very large capacity, c0
Politicians, who make decisions about how much money to invest in transportation infrastructure,
tend to focus on short- run returns because of the relatively short political cycle. If elections for
city leaders occur every couple of years, then politicians have incentives to look at costs only in
the near future. This can be “ tragic.”
Suppose that demand for transportation in a city is growing over time and is expected to continue
growing long into the future ( this tends to be the case in nearly all cities around the developing
Public Transportation Systems: Planning— General Ideas
1- 7
world). Suppose too that the goal is supplying ( at all times) enough capacity to meet demand. The
politicians must decide whether to invest a large amount of money, c0, in digging tunnels and
laying track for a subway that will have enormous capacity to handle demand for decades into the
future or to incrementally expand road infrastructure to handle the demand αi expected over the
next political cycle, i. This would cost ci = cgαi monetary units and will be the decision made if ci
< c0 ( assuming cost is the main political issue.) The result of this “ periodic review” decision
making is shown by this figure:
$
t
$ auto( t)
periodic review based on
political cycles
now
c0
ci = cgαi
$ subway( t)
t’
If the decision rule for investing in infrastructure is to chose the lowest cost over the next political
cycle and demand increases gradually, “ automobile” will always win because with gradual
increases in demand: ci < c0. In the long run, however, the cost of investment in automobile
infrastructure is unbounded. Had decisions been made with a view to the long run
( t > t’), the subway ( i. e. the less costly investment) would have been chosen.
Another point pertaining to “ the future demand vision” is that systems often create their own
demand; and this should be recognized ( even exploited) when developing design targets. Planning
actions that have long- term consequences should be made with a long- term horizon and long term
vision.
Planning and Design Approaches
Comparative Analyses – This is planning by looking at what similar cities have done and trying to
copy it. Although this is useful, “ safe” and often done, it can exclude opportunities to come up
with innovative solutions that may only be appropriate for the case of concern. ( We will not do
this in this course; we will instead create designs from scratch, systematically.)
Public Transportation Systems: Planning— General Ideas
1- 8
Step- wise Approach – This is how systematic planning must be done -- problems are too big to be
explored in one shot. We first plan generally for the big picture; then fill in the design/ engineering
step.
In order to conduct broad planning for the large scale, it is useful to simplify the analyses.
Decision variables, such as number of buses, number of stops, and number of bus routes are
integer values in reality, but we will treat them as divisible ( continuous) variables. This greatly
simplifies matters, for example turning integer programming problems into linear programs, so
that complex problems can be solved much more easily. This will work if the simplification does
not introduce large errors.
Decision Methods
1. Planning Large/ Long scale Simplified/ Broad
2. Design Detailed/ Specific
Example
Consider a simple mathematical ( integer) program, e. g. for maximizing personal mobility subject
to a budget constraint:
max { z = 22x + 18y }
s. t. 2.1x + 1.9y ≤ 2
x, y ∈ Z ( integer valued)
This is so simple that the solution can be obtained graphically ( try it); the solution is:
x* = 0, y* = 1, z* = 18.
Now, if we start with the planning approach and simplify the problem by treating x and y as
continuous variables. We are now solving a linear program which has the ( optimistic) solution:
x* = 0.952, y* = 0, z* = 20.95,
( The solution is optimistic because it is the optimum for a problem with fewer constraints.) To
obtain a feasible solution the LP solution can be rounded up or down. Because of the constraint,
we must round down and we obtain:
x* = 0, y* = 0, z* = 0.
Public Transportation Systems: Planning— General Ideas
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This solution will be pessimistic since it is feasible, but not necessarily optimal. In fact, this is
much worse than the optimum solution! So, the simplifying assumptions of the step- wise approach
do not work so well for this small scale problem.
Now, if we do the same problem on a much larger scale ( e. g. for a budget that would cover a
whole city) we would solve instead the mathematical program,
max { z = 22x + 18y }
s. t. 2.1x + 1.9y ≤ 200
x, y ∈ Z ( integer valued)
Starting with a planning step, assuming the variables can take non- integer values ( linear program),
the ( optimistic) solution is
x* = 95.2, y* = 0, z* = 2095.
Rounding to the nearest integer value ( the design step) gives a pessimistic final objective function
value:
x* = 95, y* = 0, z* = 2090
Now the pessimistic value associated with the integer solution we obtained with the step- wise
approach is very close to the optimistic value, and therefore should be even closer to the real
optimum that could have been obtained. So, simplifying the problem for large- scale planning
purposes, as we will do in this course, is not detrimental to the results of the analysis.
Public Transportation Systems: Planning— General Ideas
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Appendix: Class Syllabus ( spring 2010)
The schedule below lists the topics covered in 1- hr lecture periods in the spring semester ( 2010)
and how they were coordinated with the homework exercises and the mini- project activities. Not
listed, a 1- hr weekly discussion session was also scheduled to cover the homework exercises and
the mini- projects.
─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─ ─
Period Date Lecture subject Problems Mini- project
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1 1/ 19 Introduction: general ideas, politics
2 1/ 21 Introduction: standards, demand uncertainty
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3 1/ 26 Planning: shuttle systems, fixed demand 1 ( EOQ)
4 1/ 28 Planning: shuttle systems, adaptive demand 1
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5 2/ 5 Planning: modal comparisons, idealized corridors 2 ( Vickrey)
6 2/ 4 Planning: idealized corridor hierarchies 2
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7 2/ 9 Planning: corridors ( detailed analysis, standards)
8 2/ 11 Planning: corridors ( standards vs. generalized costs)
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9 2/ 16 Planning: inhomogeneous corridors 3 ( spacing only CA) 1
10 2/ 18 Planning: idealized grid systems ( issues) 3 1
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11 2/ 23 Planning: realistic grid systems ( no hierarchy) 1
12 2/ 25 Planning: grid systems ( practical issues) 1
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13 3/ 2 Planning: hybrid systems ( modal comparisons) 4 ( modal competition) 2
14 3/ 4 Planning: hierarchical systems, adaptation 4 2
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15 3/ 9 Planning: paratransit ( general concepts; taxis) 5 ( hierarchy design) 2
16 3/ 11 Planning: paratransit ( dial- a- ride) 5 2
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17 3/ 16 Planning: paratransit ( car- sharing) 2
18 3/ 18 Management: vehicle fleets ( 1 route) 2
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SPRING BREAK
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Public Transportation Systems: Planning— General Ideas
1- 11
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Period Date Lecture subject Problems Mini- project
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19 3/ 30 Management: vehicle fleets ( n routes) 6 ( feeder DAR)
20 4/ 1 Management: methodology ( meta- heuristics) 6
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21 4/ 6 Management: staffing ( 1 run) 3
22 4/ 8 Management: staffing ( n runs) 3
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23 4/ 13 Operations: vehicle movement ( theory, systems of systems) 3
24 4/ 15 Operations: vehicle movement ( pairing) 3
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25 4/ 20 Operations: vehicle movement ( pairing avoidance) 7 ( bus pairing)
26 4/ 22 Operations: right- of- way ( issues, nodes) 7
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27 4/ 27 Operations: right- of- way ( links, systems)
28 4/ 29 Operations: special events ( capacity management)
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Public Transportation Systems: Planning— Shuttle Systems
2- 1
Module 2: Planning-- Shuttle Systems
( Originally compiled by Eric Gonzales and Josh Pilachowski, February, 2008)
( Last updated 9- 22- 2010)
Outline
• Overview
• Shuttle Systems
o Individual Transportation
􀂃 Time- independent Demand
􀂃 Time- Dependent – Evening ( Queuing), Morning ( Vickrey)
o Collective Transportation
􀂃 Time- Independent
􀂃 Time- Dependent
o Comparisons and Competition
Overview
Recall from Module 1 that public transportation can be thought of as a system that consolidates
individual trips in time and space to exploit economies of scale that result from collective travel.
Since this course is about developing insights as well as recipes, we will analyze simple systems
starting with point- to- point shuttles, then expand to transit in corridors, and finally build up to
the more realistic case of organizing public transportation in 2 dimensions.
1. Shuttle Systems – Assume the population is already consolidated at two points ( an origin
and destination) so that there is no spatial consolidation of trips. Collective transportation,
in this case, will involve temporal consolidation as individuals adjust their departure
times to match the scheduled departure of transit vehicles from the shared origin to the
shared destination.
2. Corridors – Assume now that the population is spread along a corridor so that all travel is
made in 1 dimension along which transit service is provided. Here, collective
transportation must involve spatio- temporal consolidation as individuals must travel to
discrete stations where they can board transit vehicles which depart at discrete times.
Public Transportation Systems: Planning— Shuttle Systems
2- 2
3. Cities – Finally we consider the more realistic case of a population spread across 2
dimensions. Now transit services must be aligned in a route structure to cover the 2- D
space, and this routing adds circuity to travel as transit systems carry individuals out of
the way of their shortest path in order to consolidate trips spatially.
Shuttle Systems
We start by analyzing point- to- point shuttle systems. For comparison purposes we will do this
for both, individual and collective transportation modes. In both cases we look first at the time-independent
case where we assume steady state conditions ( supply and demand are constant over
time). This is the way many economic models treat transportation. We then look at the ( more
interesting) time- dependent case. Individual modes, like private automobiles, incur significant
guideway costs in proportion to the capacity provided, which cannot be easily adapted to a time-dependent
demand. Public transit modes without extensive guideways will be shown to be more
flexible, because a significant part of their costs come from vehicle operations.
Individual Transportation Modes
Time- Independent Demand
In order for individuals to travel in private vehicles ( such as automobiles) without much delay,
some amount of capacity, μ ( pax/ hr), must be provided to serve the demand, λ ( pax/ hr). For
private modes, there is a roughly constant infrastructure cost, cg, per unit of capacity provided.
There is also a cost per vehicle trip, cf, that each driver perceives as a fixed cost of making a trip
by private car. Assuming as an approximation that there is no delay whatsoever when the
capacity exceeds demand ( μ ≥ λ), the cost per passenger of a private vehicle system is
f
= g + c
λ
c μ
$ , for μ ≥ λ.
Public Transportation Systems: Planning— Shuttle Systems
2- 3
c c
In order to minimize this cost, we would always choose to provide the least possible capacity,
which means μ = λ. Therefore the minimum cost per passenger is given by
= g f $ +
which is independent of demand, so there are no economies of scale in our idealization of private
transportation; i. e., the total cost accrues at rate λ$. Doubling the number of drivers on the road
would double the total cost of transportation when just enough capacity is provided to serve
demand. We now look at the time- dependent case, both for the evening and morning rush hours,
which are different.
Time- Dependent Demand – The Evening Commute with Known Demand ( Queuing Analysis)
Until now, we have assumed that demand is time- independent so that as long as capacity
matches demand there is no delay, but in reality travel demand rises and falls over the course of a
day. Below is a cumulative plot of demand showing the difference between the daily average
demand, λ , and the maximum demand in the peak of rush hour, λm. We assume that the demand
curve is given and ( for simplicity only) that the day has a single rush instead of two. Note that
λm ≥ λ , and that in a time- independent system where the demand rate does not fluctuate over
the course of the day, λm would equal λ .
t
#
TD = 24 hours
λ
λm μ
V( t)
D( t)
Figure 1.
The minimum monetary cost of providing service subject to a travel delay standard, T0, can take
a range of values depending on the standard and the capacity it requires. This range can be
Public Transportation Systems: Planning— Shuttle Systems
2- 4
easily identified. A lower bound for the cost is obtained by relaxing the standard and simply
assuming, T < ∞. This relaxed standard is achieved by providing just enough capacity to meet
the average daily demand ( μ = λ ) such that there are no unserved vehicles carrying over from
day to day. This yields a lower bound equal to the monetary cost of the time- independent case:
cg + cf. An upper bound for the cost is obtained by tightening the standard to T0 = 0. This
standard is achieved by providing sufficient capacity so that there is never congestion: μ = λm.
The upper bound is therefore as shown below:
f
m
g f g c c T T c c + ⎟⎠
⎞
⎜⎝
+ ≤ ≤ ≤ ⎛
λ
λ
min{$ : } 0
Note that these bounds apply whether we interpret T as the average delay experienced by drivers,
or as the maximum delay experienced in the worst case. The choice of which standard to use is a
political decision. But these bounds show that a rush hour can only make costs worse than in the
time- dependent case because the cost of serving uniform demand is the lower bound of this
expression. So, we still do not see economies of scale.
Aside ( showing how to calculate the actual values T* and $*): If desired, one can also
estimate T* and $* ( not just the bounds) by using a cumulative plot diagram and/ or a
spreadsheet. For example, if T and T0 are averages across drivers, we would evaluate the
total time delay, TT( μ), for a given capacity, μ, as the area between the arrival curve described
by V( t) and the departure curve, D( t), determined by the capacity, μ. The average time delay
per driver, T( μ), is thus given by
λ
μ T( ) = T ( )
T μ .
Note from the picture that the area between V( t) and D( t), and therefore T( μ) declines with μ;
and since the monetary cost of private transportation always increases with capacity, $( μ) ≡
cg μ/ λ , the constraint of our mathematical program must be binding. Thus,
0 T( μ *) = T
which yields μ* ( and $*).
Time- Dependent Demand – The Morning Commute ( Vickrey Model with Endogenous Demand)
In our idealization of the morning commute the times at which people leave their homes and
would arrive at our mythical bottleneck are not given. Instead, the demand is driven by work
appointments characterized by a cumulative curve of desired departure times through the
bottleneck, which we call the wish curve, W( t). If the slope of the wish curve, s, is less than the
capacity of the bottleneck, μ, all drivers can pass through the bottleneck exactly when they would
Public Transportation Systems: Planning— Shuttle Systems
2- 5
e L
like; then there would be no delay. Curves V( t), D( t) and W( t) would match. However, if the s
exceeds capacity, some drivers would have to depart the bottleneck earlier or later than their
wished time and the three curves could not match.
To see what could happen as drivers adjust their home departure times ( over days) in response to
their delays, we suppose that each driver values time in queue at a rate β ($/ hr), time arriving
early at rate eβ and time late at a rate Lβ. The constants e and L are dimensionless and such that:
≤ 1 ≤
N N N
According to Vickrey ( 1969), if s exceeds μ and drivers minimize their generalized costs
including delay, earliness, and lateness, an equilibrium curve of arrival times to the bottleneck
arises in which the order of arrivals to the bottleneck is the same as the order of wished
departures.
The equilibrium principle is that no driver should be able to decrease its generalized cost by
changing their arrival time. In Vickrey’s equilibrium, shown in Fig. 2, there is a critical driver,
numbered Nc in the sequence of arrivals and departures, who experiences no earliness or lateness
and whose entire cost is time in queue. ( Note how the departure curve D( t) crosses W( t) for the
ordinate of this driver.) All drivers who arrive before Nc will depart the bottleneck before their
wished departure time. We will define Ne as the count of such drivers. All drivers who arrive
after Nc will depart the bottleneck after their desired departure time. We will define NL as the
count of such drivers. If there are a total of NR drivers then the following is true:
e L R + =
You can convince yourselves that the queuing diagram for the equilibrium is uniquely defined if
you are given T, Ne and NL. It can be shown ( see Appendix) that:
( L e)
N Le
T R
+
=
μ
;
L e
N LNR
e +
= ; and
L e
N eNR
L +
= .
It also turns out that if s >> μ, the generalized level of service cost ( including both queuing delay
and unpunctuality cost) is nearly the same for all commuters, approximately βT. When L >> e,
this generalized cost is βNR/ μ.
Public Transportation Systems: Planning— Shuttle Systems
2- 6
t
#
s
NR
μ
D( t)
W( t)
V( t)
T
NL/ μ
TD = 24 hours
Nc
Ne
NL
Ne/ μ
Figure 2.
The total cost of congestion in this morning commute is the sum of total queuing delay ( the area
between V( t) and D( t)), the total earliness penalty ( e times the area between D( t) and W( t) where
D( t) > W( t)), and the total lateness penalty ( L times the area between W( t) and D( t) where D( t) <
W( t)). This calculation can be most easily done based on the geometry of the figure.
A little reflection shows that if we choose a bottleneck capacity that minimizes the out- of- pocket
cost per person $ required to cover the cost of said capacity subject to a time standard ( say for
the critical commuter), we obtain the same bounds as in the evening rush: 1
g f g f c s c T T c c + ⎟⎠
⎞
⎜⎝
+ ≤ ≤ ≤ ⎛
λ
min{$ : } 0 ,
where R D λ = N / T
.
So, in the morning rush we continue to be worse- off than in the time- independent case; and
economies of scale still do not appear.
1 This is true because the practical range of μ is [ λ , s] and$ μ / λ f g = c + c .
Public Transportation Systems: Planning— Shuttle Systems
2- 7
Collective Transportation
We now repeat this analysis for public transit and find that the results are quite different ( and
encouraging).
Time- independent Demand
Consider now a shuttle service provided on an existing guideway from a common origin to a
common destination, where the frequency of service is the decision variable that the transit
agency can determine.
We assume that shuttle vehicles ( e. g., trains) are large enough to carry any number of passengers
that may show up and define:
H – headway between vehicle dispatches [ hours]
x – frequency of vehicle dispatch [ number of vehicles per hour]
H
= 1
cf – cost per vehicle dispatch of providing shuttle service [ dollars per vehicle]
λ – demand [ number of passengers per hour]
So, the monetary cost per passenger, $, of providing shuttle service is given by the cost per hour
of dispatching the transit vehicles divided by the total number of passengers using the system.
λ
$= f
c x
The out- of- vehicle delay experienced by passengers in the system ( ignoring the time in motion
between the origin and destination, which is the same for every traveler) is always proportional
to the headway of service. For example, if people know the headways but not the schedule and
they have specific appointments at the destination ( as in the morning commute), they will leave
home with at least one headway of slack, which they will spend either at the origin or at the
destination. Combined, their total delay would be H. If people do not have specific appointments
( as happens for many people in the evening commute) their delay would be ½ H on average.
Thus, for the worst- case situation ( with appointments) the average delay T is:
x
T = 1
So if we apply a standard T0 ( as we did for individual modes) we have to solve:
Public Transportation Systems: Planning— Shuttle Systems
2- 8
⎭ ⎬ ⎫
⎩ ⎨ ⎧
≡ ≤ 0
min : 1 T
x
c x
$ f
λ
and since the constraint is binding, we find:
0
$*
T
c f
λ
=
Note: There are economies of scale in providing collective transportation because the monetary
cost, $*, decreases with the demand! This is the promise of public transportation vis a vis
individual transportation. In reality the contrast is not so pronounced because as we shall see
there exist compensating complications, but the promise is real. The reason is that with more
demand more individuals can consolidate their travel onto each vehicle without changing the
number of vehicle runs; and this lowers the cost of providing transportation per person. We now
show that economies still arise if we allow the demand to vary with time.
Time- Dependent Demand
The analysis above assumes that the demand is uniformly spread throughout the course of the
day, but in reality the demand for travel is concentrated into rush hours. Let us now evaluate the
cost of providing collective transportation for this case, assuming that the passenger arrivals are
given. 2
Consider now a simplified case of a day with two demand periods: a peak demand, λ p
, for a
period of Tp hours of the day, and an off- peak demand, λ o , for the remaining TD – Tp hours. The
cumulative plot of Fig. 3 shows this demand profile and that Np passengers travel in the peak,
leaving ND – Np passengers for the off- peak hours.
2 This assumption can now be used for both the evening and morning commutes ( with and without appointments)
because with our large- vehicles, passengers do not have to compete for limited system capacity.
Public Transportation Systems: Planning— Shuttle Systems
2- 9
t
#
TD = 24 hours
λ p
λ o
ND
Np
Tp
Figure 3.
To design a transit system for this demand, we can break up the day into two regimes and choose
a peak period headway, Hp, and an off- peak headway, Ho, to minimize the cost in providing
transit service over the course of the whole day. This can be done by minimizing the total
generalized cost by the Lagrangian approach with the two decision variables, Hp and Ho:
min{ Z ( Total amount of waiting time) c ( Number of bus dispatches)} f = β +
⎪⎭
⎪⎬ ⎫
⎪⎩
⎪⎨ ⎧
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛ −
+ + ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
= + −
o
D p
p
p
p p o D p f H
T T
H
T
min Z β H N H ( N N ) c
The headways that minimize the generalized cost are
p
f
p
f p
p
c
N
c T
H
β βλ
* = =
o
f
D p
f D p
o
c
N N
c T T
H
β βλ
=
−
−
=
( )
( )
* .
Using these optimal headways gives a minimum total generalized cost of
* 2 ( ( )( ) ) f p p D p D p Z = βc T N + T − T N − N .
Public Transportation Systems: Planning— Shuttle Systems
2- 10
Note that for a given ratio Np/ ND this total generalized cost is proportional to D N , so the
generalized cost of collective transportation per person is proportional to 1/ D N ; i. e., it
decreases with increasing ridership, ND, and therefore with the average daily demand λ = ND/ TD.
So even with time- dependent demand, public transit displays economies of scale.
Technical aside: Note that the optimum cost does not change much if the demand is spread
evenly across the whole day. Suppose, for example, that the coefficient 2 = 1 f βc and 30%
of the trips are made in 4 of the 24 hours in a day ( i. e., there is quite a bit of peaking). If we
use a dummy value ND = 10 in the formula, we find that the total generalized cost for this
time- dependent case is
1( 4 × 3 + ( 24 − 4)( 10 − 3) )= 15.30 .
Using the same logic we see that if the ND = 10 trips had been spread uniformly across the
entire 24 hrs, the generalized cost would have been: ( 24×10) ½ = 15.49.
Note the very small difference, and that peaking actually reduces the cost to society, which
was not the case for individual modes! You can also convince yourself that the relative
difference between these two costs is independent of ND. The relative difference is so small
because we can adapt the provision of transit service to match demand. The small and
favorable relative error suggests that to plan collective transportation systems with dominant
vehicle costs ( as in our examples) one can assume a time- independent demand as a
simplification. Infrastructure costs, on the other hand, must be provided in a time- invariant
( non- adaptable) way, so the same cannot be said when guideway costs are important, as
happens for transportation by individual modes and some collective kinds ( e. g., subways).
Comparison between Individual and Collective Transportation Modes
In many cases, individual modes are used in parallel with public transit lines, and an equilibrium
is reached in which some trips are made by individual modes and the rest by transit. If a
traveler’s decision of which mode to take is based only on the level of service ( LOS) cost ( i. e.
the delay time), the equilibrium will be reached when the level of service costs are the same for
both choices.
We have seen from Vickrey’s model that the generalized cost of delay for automobile commuters
is approximately βNR/ μ, when L >> e and s >> μ. Note that this cost increases proportionally
with the number of individuals using the roadway, NR, and decreases as capacity, μ, is expanded.
For collective transportation, by contrast, the level of service cost is always proportional to the
service headway, H, and is independent of the number of individuals using the transit system. It
is βH if everyone has appointments. Assuming the vehicles are sufficiently large, this makes
Public Transportation Systems: Planning— Shuttle Systems
2- 11
sense because the time cost of riding a transit shuttle depends only on how long a rider must wait
for the vehicle, not on how many other people are sharing the vehicle.
So the following diagram plotting general cost vs. number of users helps explain what happens
when the two modes provide competing shuttle services for a population of NR travelers and we
have to decide where to allocate funds for increased capacity. The increasing lines correspond to
“ automobile” and the horizontal lines to “ public transit”.
NCar
Generalized
Cost
cf
( car) + βN/ μ , low μ
cf
( car) + βN/ μ , medium μ ( initial value)
cf
( car) + βN/ μ , high μ
cf
( transit) + βH, high H
cf
( transit) + βH, medium H
( initial value)
cf
( transit) + βH, low H
NR
Initial
Equilibrium
1
2
NCar NTransit
Improvement in
generalized cost
Figure 4.
Assume now that the automobile and public transit systems are initially described by the two
curves labeled “ medium” in the figure. If people choose shuttle service based on generalized
cost, then the intersection of these two curves is the initial equilibrium. The total generalized cost
is then the sum of the total cost for all modes ( which is the same for all trips, regardless of
mode), depicted by the shaded area: NR( cf
( transit) + βH).
Now, suppose some public funds become available and we can choose whether to invest in
public transit or individual modes. We can choose to improve the headway for transit service, H,
( option 2 in the figure) or the roadway capacity, μ, ( option 1); so… where should we spend the
money?
An investment in automobile infrastructure lowers the cost of driving which will cause a shift in
mode share to more drivers ( point 1). The user cost ( shaded area), however, remains unchanged
because drivers fill the new road capacity until the time delay is equivalent to the time cost of
taking transit.
Public Transportation Systems: Planning— Shuttle Systems
2- 12
Investing in public transit, however, lowers the user cost for transit riders by reducing the
headway, and this creates a mode share shift towards transit ( point 2). In this case the
improvement benefits both transit riders and drivers ( by taking drivers off the road). Therefore,
in this idealized example everyone benefits from investing more funds in collective
transportation, even those people who never set foot on a transit vehicle.
Related Reading
Vickrey, W. S. ( 1969). “ Congestion theory and transportation investment.” The American
Economic Review, 59( 2) 251– 260.
Appendix A: Vickrey Model of the Morning Commute
We look for an equilibrium where the critical driver is indifferent to any arrival time, and the
first and last drivers to the bottleneck experience no delay. Thus, given a fixed slope, μ, of D( t),
we can find this equilibrium ( see Figure 2) by setting the delay experienced by the critical driver,
T, equal to the earliness cost experienced by arriving first or the lateness cost experienced by
arriving last:
μ
T = Nee and
μ
T = NLL .
With these two equalities and the relation Ne + NL = NR we can solve for T, Ne + NL, with the
result of the text:
1 1 ( )
1
L e
N Le
L e
T N R
R +
=
⎟ ⎟ ⎟ ⎟
⎠
⎞
⎜ ⎜ ⎜ ⎜
⎝
⎛
+
=
μ
μ ;
L e
N LNR
e +
= and
L e
N eNR
L +
=
So this shows that the critical driver would not have an incentive to change its arrival position.
But for the curves of Figure 2 to be in equilibrium, other drivers— whether their wished times are
before or after the critical time— would also have to lack an incentive to change their arrival
positions. A good way to verify this is in two steps:
( a) Draw an “ indifference curve” for a generic non- critical driver ( with a given wish time)
showing for each possible arrival position from 0 to NR the time at which the driver would have
to join the virtual queue when arriving in this position to achieve the generalized cost currently
experienced. ( Note that each arrival position has a given earliness or lateness for this driver.)
Public Transportation Systems: Planning— Shuttle Systems
2- 13
( b) Noting that the latest time at which the queue can be joined for any position is given by
V( t); and that V( t) is never to the right of the indifference curve; i. e., the indifference times are
not feasible and the driver cannot improve his or her position.
Step ( a) requires some care. The following references can perhaps help. They are not required
reading, but they contain more detail and additional applications.
Related Reading
Daganzo, C. F. ( 1985). “ The uniqueness of a time- dependent equilibrium distribution of arrivals
at a single bottleneck.” Transportation Science. 19( 1) 29– 37.
Daganzo C. F. and Garcia, R. C. ( 2000). “ A Pareto improving strategy for the time- dependent
morning commute problem.” Transportation Science. 34( 3) 1– 9.
Public Transportation Systems: Planning— Corridors
3- 1
Module 3: Planning— Corridors
( Originally compiled by Eric Gonzales and Josh Pilachowski, February, 2008)
( Last updated 9- 22- 2010)
Outline
• Idealized Analysis
o Limits to The Door- to- Door Speed of Transit
o The Effect of Access Speed: Usefulness of Hierarchies
• Realistic Analysis ( spatio- temporal)
o Assumptions and Qualitative Issues
o Quantitative formulation
o Graphical Interpretation
o Dealing with Multiple Standards
o No transfers
o Transfers and Hierarchies
o Insights
o Standards- Revisited
o Space- and Time- Dependent Services
􀂃 Average Rate Analysis
􀂃 Service Guarantee Analysis
In the previous module we looked at the special case where all trips originate at one point and
end at another point. Now, we consider demand spread along a corridor, so trips must be
consolidated both in time and in space. The design of transit service in a corridor requires
choosing a stop spacing, S, and service headway, H.
We will first focus exclusively on S in order to isolate the effect of spatially distributed demand
from that of its temporal distribution, which we saw in Module 2. Whereas temporal
consolidation involved a trade- off between out- of- vehicle ( waiting) time and vehicle operating
cost, which had huge economies of scale as demand increased, we will now see that in the spatial
case the trade- off is between out- of- vehicle ( access) time and in- vehicle time, and that this trade-off
is less favorable to public transit: it imposes a severe limit on door- to- door speed even if we
make the most favorable assumptions possible for collective transportation.
Public Transportation Systems: Planning— Corridors
3- 2
Idealized Analysis
Limits to Door- to- Door Speed
Consider a very long transit corridor serving customers that travel from left to right. Customer
origins are continuously distributed anywhere along the corridor and their trips can take any
length up to a maximum ℓ . The stops are separated by distances, s ≤ ℓ . We are interested in the
tightest door- to- door travel time guarantee that can be extended to all customers.
s
ℓ
Now we will make a number of optimistic ( although unrealistic) assumptions in order to identify
this guarantee while accounting for the fact that passengers must access the transit stop and then
ride vehicles which make periodic stops to pick up and drop of passengers. This bound will be
independent of demand and many other parameters, so it is very general.
• Assume vehicles are dispatched so frequently that once a passenger arrives at a stop, he
or she does not wait at all for the next vehicle; i. e., H = 0.
• Assume the doors of the vehicle open and close instantly, and passengers take no time to
get in or out of the vehicles.
• Finally assume that there is no upper bound to the speed that can be achieved by a transit
vehicle while traveling between stops, so that vmax = ∞.
Although we would agree that these conditions would favor operation extremely, the transit
system will still be limited by:
• A maximum acceleration above which passengers will feel physical discomfort from the
force ( a0 ≈ 1 m/ s2).
• The average walking speed at which passengers travel to access their nearest transit stop
( va ≈ 1 m/ s).
There are two components of travel time in this case: access time, ta, and riding time, tr. In the
worst case, the access time results from a passenger walking half of a stop spacing from the
origin and another half stop spacing to the destination. So:
a
a v
t = s
Public Transportation Systems: Planning— Corridors
3- 3
Riding time is the consequence of the commercial speed of transit ( the average speed of the
vehicle vv) which is affected by the stop spacing. If there is no maximum speed, then the transit
vehicle will accelerate as it departs a stop until it is half way between stops. Then the vehicle
will decelerate to make the next stop ( see figure below). Under these conditions, the riding time
ts for a trip between stops can be decomposed into two equal parts of length: s/ 2 = ½ a0( ts/ 2) 2.
From this we find:
0
2
a
t s s = ,
and the riding time tr for a trip of length ℓ >> s will be approximately ℓ / s times longer; i. e.:
0 sa
t r
l ≈ 2 .
Note that the commercial speed is therefore:
2
0 sa
t r
≈ l .
t
x
vv = ℓ / tr
s
x( t)
Figure 5.
We assume that people walk to the nearest station. Then, you can verify that for any spacing s
you choose, there always is an unlucky passenger who would have to walk a distance s and then
Public Transportation Systems: Planning— Corridors
3- 4
ride for a distance s⎡ ℓ / s⎤. 1 As a result, the total door- to- door time for this worst- case passenger
is: t = ta + tr = s/ va + 2s⎡ ℓ / s⎤/( sa0) ½ . This function increases with s except and declines only
when s is a sub- multiple of ℓ . At these points it takes on the form:
v sa0
t
a
l = s + 2 .
So we look for the minimum of this expression, and as ( a very good) approximation we ignore
the fact that s should be a sub- multiple of ℓ . There is a trade- off here for choosing the stop
spacing s. On the one hand, a longer stop spacing increases the distance passengers must walk to
access the mode, so the access time increases with s. However, a greater space between stops
allows vehicles to accelerate to higher speeds so that riding time decreases with s. Therefore, an
optimal stop spacing, s*, can be chosen to minimize the door- to- door travel time. The result of
this optimization is:
3
1
0
2 2
* ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
=
a
v
s a l ;
3
1
0
2
0 3 ) , , ( * ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
=
v a
t a v
a
a
l
l
Of course, this result is valid only if s* ≤ ℓ , as we assumed; i. e., only if ℓ ≥ va
2/ a0. Fortunately,
since realistic values of va
2/ a0 are comparable with 1 m, this requirement is comfortably satisfied
for the trip lengths that interest us. Since the unluckiest passenger has a trip length close to ℓ we
can approximate the speed of this passenger by:
( ) 3
1
0 3
1
*
ˆ v a
t
v l a
l ≈ = ,
This expression can also be interpreted as the door- to- door speed that can be guaranteed to all
passengers with trips of length close to ℓ .
Let us plug in some numbers to see how this upper bound of door- to- door speed changes with
the length of trips made. If passengers walk with speed va = 1 m/ s and the maximum allowable
1 To see this, draw a picture with an unlucky trip as follows: ( i) an origin displaced by an infinitesimal amount ε
toward the left of a mid- point between stations, and ( ii) a trip length, y = ℓ if s = ℓ ; or else, y = s⎣ ℓ / s⎦+ 2ε if s < ℓ .
( This is an admissible choice, since for sufficiently small ε the trip length is valid: y < ℓ .) Now note that in both
cases the trip length is a multiple of s, so both the origin and the destination are near a mid- point and access distance
is s. Note too that both cases involve severe backtracking with total in- vehicle distance s⎡ ℓ / s⎤ ≥ ℓ . You can also
convince yourselves that s⎡ ℓ / s⎤ is also an upper bound to the in- vehicle distance traveled by any passenger; and that
therefore, our unlucky passenger is actually the unluckiest.
Public Transportation Systems: Planning— Corridors
3- 5
acceleration is a0 = 1 m/ s2, the figure below shows the fastest door- to- door speeds that can be
guaranteed.
ℓ 2 km
v ˆ
8 km 50 km
4.2 m/ s
6.7 m/ s
12.28 m/ s
~ 1 mi ~ 5 mi ~ 30 mi
7.5 mph
15 mph
27.5 mph
This result is very slow, even with all the favorable assumptions we have made for transit
( including vmax = ∞). Why? We are minimizing total travel time including the access time ( i. e.
maximizing door- to- door travel speed) which relies on passengers walking to the stops. Since
people walk very slowly, the stops must be spaced closely enough to limit the time passengers
spend accessing transit. This spacing, along with the limit of acceleration, prevents the vehicles
from achieving high speeds. With individual transport modes the results are better. 2 Is there a
way of improving collective transportation so it can be more competitive? The answer, as we
shall see next day, is yes.
( Hint: the door- to- door speed of public transit depends on the access speed; and if we could
increase this speed by some means, the door- to- door speed would increase.) We will explore this
issue next, and how to exploit it. We will also study how to plan real corridor systems without
the simplifying assumptions we have made – fully recognizing spatiotemporal effects.
The Effect of Access Speed: Usefulness of Hierarchies
For the moment we continue with our idealized and favorable scenario for public transit service.
So far, our goal has been to understand how transit door- to- door service speed depends on ℓ . We
2 If we made similar favorable assumptions for individual transportation modes on uncongested guideways, their
commercial speed would be close to the mode’s maximum speed for all l; i. e., much better than for public transit.
The reason is that by being individual these modes do not require much of an access displacement: a great virtue.
Public Transportation Systems: Planning— Corridors
3- 6
made a couple of assumptions, shown below, in order to obtain an optimistic but very simple
upper bound of door- to- door time. The demand, λ, does not matter for this bound.
= ∞
=
≅
max
0
0
v
t
H
s
Recall that the door- to- door travel time for the unluckiest passenger was shown to be:
0
2
v sa
t s
a
l = + .
By minimizing this expression with respect to s we obtained the following approximate formulae
for the door- to- door travel time and speed of the unluckiest passenger with trip length ℓ :
3
1
0
2
3 ) ( ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
=
v a
t
a
l
l and ( ) 3
1
0 3
v ˆ 1 v a = l a .
Note how if we could increase the speed of access the situation would improve. We can do this
by using another transit service to provide access!
ℓ
s0
s1
Let’s reexamine our logic assuming this is done. By providing a local transit service with stop
spacing, s0, to access an express service with stop spacing, s1, the access speed would now be:
3
1
0
1 1
3 2
1
2
ˆ ⎟⎠
⎞
⎜⎝
⎛ = ⎟⎠
⎞
⎜⎝
v = v⎛ s s v a a w
where vw is the speed of walking. The derivation of this would actually be slightly different so
we do not double- count access time, so for simplicity we will assume some small transfer time
Public Transportation Systems: Planning— Corridors
3- 7
equal to
w v
s
2
0 . This will allow us to continue using the same equation. The improved door- to-door
travel time is then:
3
2
1
3
1
0
3
1
2
1
1
0
( 1 ) 2 s 3 2 ( v a ) s
a
t s w l ⎥⎦
⎤
⎢⎣
⎡
= + ×
l − −
You can verify that:
*
1
* *
0 s < s < s .
( ) 1 t s l will be the best travel time for a fixed s1, assuming that you have optimized s0 already.
Note: you can notice that this equation is in the form:
z = Axn + Bx− m ; with n, m > 0
which we will be analyzing in more detail in Homework # 2. You will find that the optimum
solution m n
An
x Bm +
⎟⎠
⎞
⎜⎝
= ⎛
1
* is insensitive to different values of A and B.
After we optimize tl with respect to s1 we find the result to be:
7
4
7
1
7
3
0
7
1
3
0
4
5.3 5.3 l
l − − = ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
≈ w
w
l a v
a v
t
This equation shows that tl is of order 7
4
l and
l t
v l ˆ ∝ is of order 7
3
l and of order 7
1
w v . By
plotting v with respect to ℓ with and without a hierarchy we can see for which trip lengths it is
optimum to provide a local service.
ˆ
Public Transportation Systems: Planning— Corridors
3- 8
ℓ
v ˆ
hierarchy
no hierarchy
ℓ *
below this you do not
need any hierarchy
Realistic Analysis with Spatio- Temporal Effects
We have so far made a number of favorable and unrealistic assumptions about our transit system
in order to derive generic insights about the effects of the spatial dispersion of passengers along a
corridor. So with these insights in mind we now turn our attention to the development of specific
plans introducing more realism. The analysis will include both, the spatial and temporal effects
of dispersed demand; combining the ideas we have so far seen with those of Module 2. We shall
see that in addition to ℓ , two other important variables affect a corridor system’s structure: the
trip generation rate, λ, and the “ user’s value of time” β.
Assumptions and Qualitative Issues
Here are the improvements to realism we now consider:
1) Remove the assumption that vmax = ∞; for example define vmax = vauto ( for buses)
Public Transportation Systems: Planning— Corridors
3- 9
) where vmax
x x( t) where vmax = ∞
x(
vmax
t
t
= vauto
s
t a/ 2 s/ vmax ta/ 2
2) Remove the assumption that ts = 0. If we approximate the trajectory of the bus with
piecewise linear segments of vmax and stop time then we can define ts as the dwell time at
a stop plus the loss time due to acceleration and deceleration. The total travel time will
then be:
s stops t
v
t dist (# )
max
= +
3) Remove the assumption that H = 0
Before starting quantitative analysis, let us compare the spatio- temporal accessibility provided by
different modes with a plot showing the area that a person can reach in a given time depending
on their mode of transportation.
Public Transportation Systems: Planning— Corridors
3- 10
t
x
s
auto transit
pedestrian
vauto
vtransit
vwalk
H
We can look at the area covered by a single stop spacing and headway. Notice how a person,
depending on their origin in space and time, will choose a bus stop based on their accessibility:
t
x
s
H
vwalk
vtransit
Public Transportation Systems: Planning— Corridors
3- 11
Quantitative Formulation
Let’s try to design a realistic corridor without any hierarchy. We propose choosing the H* and s*
that minimize the cost of service given some door- to- door travel time standard. For example:
min { cost of service}
s. t. t( ℓ ) ≤ T0
We assume for now that we focus on a single “ ℓ “; e. g. the longest trips people make. To do this,
we need formulae for the cost of service and the constraint in terms of our decision variables:
Cost of service =
sH
c s
sH
c s d
λ λ
+
H
v
s
s
t
v
t
a
s = + + + l l
l
max
( )
Note: λ is the average demand density in the corridor ( trips/ time · dist) and λsH is the number of
customers associated with one stop and one vehicle. The constants cs and cd are unit costs for a
bus stop and a bus- mile. How would you derive these?
To solve the problem we can write the Lagrangian as below. Can you associate the four terms
with specific passenger activities?
max
$ $
v v
s
s
t
sH
c
H
H
c
z
WD IVD AD LH
a
s
d s
moving stop
β
β
λ
β
λ
l l + ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
+ + + ⎟⎠
⎞
⎜⎝
= ⎛ +
+ + + + +
which ( ignoring the “ cs“ term) has the solution:
( ) 2
1
*
2
1
* ; a s l
d t v s c H ≅ ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
≅
λβ
giving us:
( ) ⎪ ⎪
⎩
⎪ ⎪
⎨
⎧
⎪ ⎪
⎭
⎪ ⎪
⎬
⎫
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
+ ⎟⎠
⎞
⎜⎝
= ⎛
− upper bound
lower bound
v t
c
c
c
a s
d
s
d
2
1
2
1
2
1
*
0
$
l
λ
λ β
β
Public Transportation Systems: Planning— Corridors
3- 12
max
2
1
2
1
* 2
v v
T c t
a
d s l l + ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
+ ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
=
λβ
Note: the UB solution is obtained by sticking H* and s* into the neglected term and adding the
result to $*.
Graphical interpretation:
This picture shows how the solution depends on λ, ℓ , and β.
Where WD represents waiting delay, AIVD represents access and in- vehicle- delay, and LH
represents line haul time.
Note: “ β” is a proxy for the wealth of a city and the diagram illustrates the kind of system that
cities of wealth might use to satisfy a demand characterized by λ and ℓ .
Public Transportation Systems: Planning— Corridors
3- 13
β
s* H*
s*
H*
Dealing with Multiple Standards
A more realistic situation would require adherence to level of service for more than a single trip
length. Let’s examine the situation where we our constraint is:
( l) ≤ ( l);∀ l 0 t T
ℓ
T0
( ) T0 l given
Public Transportation Systems: Planning— Corridors
3- 14
}
We end up with a minimization problem that looks like:
min{ cos ( , )
,
agency t s H
s H
( 1)
s. t. ( , , l) ≤ ( l);∀ l 0 T s H T ( 2)
Note: There will always be at least one binding constraint when the problem is minimized. We
will call this ( unknown) binding trip length ℓ c. If we knew it and we knew this length provided
the only binding constraint ( a reasonable assumption), we could formulate the problem as a
single- constraint problem and solve it:
min{$( , )}
,
s H
s H
s. t. ( ) ( , , ) 0 0 = = c c T l T s H l
This would be an easy task because it can be done with the Lagrangian method we have just
seen. Note that the remaining constraints would be satisfied as strict inequalities. If we don’t
know the critical length, this property of the optimal solution of the single- constraint problem
can be used to see if a test value for ℓ is the correct one. So to solve the problem we can solve the
single- constraint Lagrangian problem for different ℓ until we find one that exhibits this property.
No Transfers
For our specific corridor formulae and assuming no transfers, this procedure can be simplified
even more and the result is intuitive. This is now explained.
ℓ
s
Given our assumptions, the mathematical program corresponding to ( 1) and ( 2) is:
⎭ ⎬ ⎫
⎩ ⎨ ⎧
=
H
c d
s H λ
min $
,
s. t. ( l) l
l l + + + ≤ ,∀ 0
max
T
v
t
v s
H s s
a
Public Transportation Systems: Planning— Corridors
3- 15
Notice that the cost function omits the component related to making a stop ( c sH s λ ) because
this value is small, and the objective function here gives a lower bound. Notice too that the
constraint separates into a part that depends only on the choice of headway, H; we call this the
waiting delay ( WD). The rest of the constraint depends only on the choice of stop spacing, s;
this will be called the access and in- vehicle time ( AIVT ≡ T( ℓ | s)).
When we plot the expression T( ℓ | s) for a fixed s the result is a straight line: the vertical intercept
is the fixed maximum access time ( s/ va), and the slope the vehicle’s average pace ( 1/ vmax + ts/ s).
The minimum vertical distance between the travel time standard, T0( ℓ ), and an AIVT line for a
given s, T( ℓ | s), represents the fixed amount of waiting delay that can be added to every trip and
still keep the travel time with the constraint. Note that this minimum vertical distance is the
maximum vertical displacement of our AIVT line until it becomes tangent from below to the
T0( ℓ ) curve. This vertical displacement is the maximum headway, H, that can be chosen for a
given s and still meet the standard, thus minimizing the cost of providing transit service. Now,
the AIVT line can be changed by our choice of s, so let’s choose the s that gives us the maximum
displacement so we can choose the greatest possible H and therefore achieve the lowest possible
operating cost. This is the sought result.
ℓ
t ( ) T0 l
( | ) 2 T l s
target LOS
convex hull
LE of =
msin ( l | ) L ( l)
T s = T
T( l | s)
( | ) 1 T l s
( | ) 3 T l s
H*( s)
ℓ *
Public Transportation Systems: Planning— Corridors
3- 16
This optimization can be done in one shot by considering the lower envelope ( LE) of travel time
across all choices of s.
Lower Envelope of ( l ) s { ( l )} L ( l)
T| s = min T | s = T
To this end, note that when an AIVT line is displaced it cannot possibly touch T0( ℓ ) in an upward
bulge; so we only need to look for points of tangency on the convex hull ( CH) of T0( ℓ ). 3 So, we
propose the following: slide TL( ℓ ) up until it touches ( and is tangent to) the convex hull of the
time standard T0( ℓ ). 4 Then, the displacement is the optimum headway H*, and the tangent to the
envelope at the point of contact ( ℓ = ℓ *) is the optimum AIVT line ( with s = s*). 5
Applying this result,
( )
a
s
L v
t
v
T l l
l 2
max
= +
s a s* = l * t v
To summarize, we have split the optimization into two parts: ( i) a spatial step to find a stop
spacing, s, that minimizes the access and in- vehicle time and ( ii) a temporal step to find the
headway, H, to minimize the cost of meeting the service constraint.
This is approximate and works neatly because we left out the cost of the stopping. So the
analysis above gives us a lower bound of cost. If the stopping cost were left in the analysis, the
mathematical program can still be solved with brute force in a spreadsheet, but this gives us very
little insight. If we solve the simplified formulation and then plug the resulting TL( ℓ ) and s* into
the cost function, we will get an upper bound for the cost. No further analysis is necessary when
the lower bound and upper bound are close.
What if buses run in both directions along a corridor?
3 The CH is the highest convex curve that can be drawn without exceeding T0( ℓ ).)
4 Note that this point of tangency does not have to be on T0( ℓ ), as occurs on the figure.
5 Why is this true? ( i) You see from the geometry of the picture that the displacement of the optimum AIVT line
( which is straight) to first contact with T0( ℓ ), i. e. the optimum headway H( s*) for the s = s*, is always equal to the
displacement of the LE to first contact with the CH; thus, the displacement we propose is the optimum headway for
s*. And ( ii) s* is the optimum spacing because no other AIVT line can be displaced by a greater amount.
Public Transportation Systems: Planning— Corridors
3- 17
ℓ
s
The stop spacing will remain unchanged, because s is chosen only to minimize travel time, and
the demand plays no role in the travel time expression. The cost of operating service will
double, however, because twice as many buses are needed to serve the same demand per unit
length.
Exercise: Consider transit service in a loop demand uniformly distributed between all points.
Would we want to serve trips with bi- directional transit routes or is it better to reduce
headways by putting all vehicles in service in the same direction? You should be able to
convince yourself that if the route has 4 buses or more, it is always better to operate bi-directional
service. ( Hint: If you had only one bus, it should be obvious that it is most time
efficient to operate service in one direction. Likewise, if you had an infinite number of
buses, it should be obvious that buses should be deployed in both directions to serve the
demand. Where is the tipping point where it becomes more efficient to operate buses in the
both directions?)
Transfers and Hierarchies
Now, what if we introduce transfers to an express service operating in parallel to the local
service with frequent stops. There are couple ways this service could be structured. So far, we
have been looking at translationally symmetric route patterns, but this need not be the case. We
could run offset local- express services as shown below.
Public Transportation Systems: Planning— Corridors
3- 18
A B
local express
The disadvantages of such a network design outweigh the benefits for cases where the demand is
spread out because for trips between points such as A and B we would require multiple transfers.
But if all the trips have a common destination ( e. g., for feeder systems that collect passengers
from many destinations and deliver them to a single hub) the strategy has merit. For spread- out
( many- to- many) service it makes sense to consider a local bus service that is paralleled by an
express service where passengers can transfer from one service to the other at designated transfer
stops.
ℓ
s0
s1
Assume that the headways are synchronized with the same H for local and express services, but
the local buses stop with spacing, s0, and the express buses make less frequent stops with spacing
s1. Even this structure of service can be operated in different ways.
Strategy 1: Express buses are scheduled at consistent headways, and the local feeders are
dispatched in to depart in both directions along the corridor every time an express bus reaches a
transfer station. At some point between transfer stations, the local buses wait and then begin a
return trip, bringing passengers to the transfer station just in time for the arrival of the next
express bus.
Public Transportation Systems: Planning— Corridors
3- 19
t
x
s1
H
e1 e2
s0
2
distribute Δ
distribute
collect
collect
dead time
Strategy 2: Express buses are again dispatched at a scheduled headway. Instead of running
feeder buses in both directions, a bus is dispatched from the transfer station after the arrival of an
express bus, and a second feeder is dispatched in the same direction to collect passengers and
drop them off at the downstream transfer station in time to catch the next arriving express bus. 6
6 If service is not synchronized there is no need for “ dead- times” and buses can both collect and deliver passengers.
The two bus systems can even have different headways, H0 and H1. Could you draw a picture such as those above?
Public Transportation Systems: Planning— Corridors
3- 20
t
x
s1
H
e1 e2
s0
2
Δ
distribute
collect
dead time
Both of these operational strategies tessellate across time and space and require two local bus
dispatches for each express bus dispatch. Therefore they require the same number of vehicle
kilometers of service, and a lower bound to the cost of providing service based on vehicle- km is
H
c d
λ
3
$ =
for both timed- transfer strategies. ( Convince yourselves that the coefficient would be “ 2” for un-synchronized
service with H0 = H1 = H). To be complete we must account for bus- hrs while
stopping. Then, the cost in a system with timed transfers is
( ) ( )
⎭ ⎬ ⎫
⎩ ⎨ ⎧ + ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
= + + d s t t t c c
s
c s s
s H
$ s , s , H 1 3 1 2 2 dead time
0
1
1
1
0 1 λ
The unsynchronized case with H0 = H1 = H would have a very similar form except for some of
the coefficients: “ 3” would be “ 2”, the next ‘ 2” would be “ 1” and the final “ 2” would be “ 0”.
Test yourselves and see if you can derive the unsynchronized expression for H0 ≠ H1.
The door- to- door travel time T is composed of the following components:
H = waiting delay
Public Transportation Systems: Planning— Corridors
3- 21
w v
s 0 = access time
v0 = average speed of local vehicle including stops but not dead time
0
1
v
s = local in- vehicle travel time
v1 = average speed of express vehicle including stops but not dead time ( v1 > v0)
1 v
l = express in- vehicle travel time
Δ = transfer time
where the vehicle pace ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
= +
i
s
i s
t
v v max
1 1
So the door- to- door travel time is given by7
( ) ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
+ + ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
= + Δ + + +
0 max 1 max
1
0
0 1
, , 1 1
s v
t
s v
t
s
v
s
T s s H H s s
w
l ; ℓ > s1 > s0
and we can optimize the system with a mathematical program of the familiar form:
( s s H)
s s H
min $ , , 0 , 1 , 0 1
=
s. t. ( , , ) ≤ ( l),∀ l 0 1 0 T s s H T
The lower bound of the cost is now 3cd/ λH, and the door- to- door time,
T( s s H) H T( s) v
, , l | 0 1 = + . The maximum possible H can be determined by the same method
described for a system with only local service, although here we determine a lower envelope of
travel time in 2 parameters, s0 and s1.
T ( ) { T( s )}
L s
v
l l v = min |
Example: Considering s1 for the time being as a constant, find the optimal s0*.
s w s s t v 0 1 * =
7 The only changes for the unsynchronized cases involve the coefficient of H ( or of H0 and H1 , if H0 ≠ H1).
Public Transportation Systems: Planning— Corridors
3- 22
( )
max 1
1 1
max
1 * | 2
s
t
v
s
v
s t
v
T s s
w
l s l
l = Δ + + + +
The s1/ vmax term is typically much less than s w 2 t v so we can ignore s1/ vmax and get an
approximate solution.
( ) 3
1
2
1 * s w s ≅ t l v
( ) ( ) 3
1
2
max
3
1
2
max
3 1 s w
w
s
L t v
v v
t
v
T l
l l
l + ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
= Δ + +
Insights ( Comparisons across Countries)
Imagine combining the cost and time into a Lagrangian expression of generalized cost when we
value time at a rate of β dollars per unit time. If we neglect the ( small) effects of dead times and
transfer times, the result is:
⎥⎦ ⎤
⎢⎣
⎡
= + + + + + +
max 1 max
1
0
3 0 1 1
s v
t
v
t s
s
s
v
s
H
H
c
z s s
w
d l
β β l
λ
Three of the parameters that appear in this expression ( λ, β and ℓ ) can vary by orders of
magnitude across cities and countries, and the others vary much less. Therefore, ( λ, β and ℓ ) can
be thought of as the main drivers of system structure or design. Now, if we divide through the
above expression by β so that the generalized cost ( GC) is always expressed in units of time,
then λβ always appear together so z*( λ, ℓ , β)/ β is really a function of only two drivers of design:
( λβ and ℓ ). This generalized cost in units of time is the total time required to make a trip
including the time people must spend working to afford system. We can think of the λβ driver as
the “ wage generation rate per unit time and distance” because λ is the trip generation rate and β
the value of time associated with each trip generated, which should be similar to the wage rate.
It is nice to use intrinsic units that are independent of a currency or country. We can express
wages β in any equivalent units we want. For example we could use units of ct ( where ct is the
operating cost per unit time of running a bus), using β/ ct as our wage metric. Note that this ratio
is the number of buses that can be continuously operated with the wages of one person. ( In rich
countries the ratio can be close to 1 and in poor countries much, much less.) Thus, we can think
of λβ/ ct as the “ bus generation rate”.
Whether one uses intrinsic units or not, the fact that demand and wealth can be combined into a
single driver means that low- density wealthy neighborhoods in developed countries and poor
Public Transportation Systems: Planning— Corridors
3- 23
dense neighborhoods in developing countries ( with the same bus- generation rates) should have
approximately the same system structure. And they should also share the time- based GC. ( This
happens because as we have seen the time- based GC depends only on the combined value of
λβ.) Isn’t it nice that we can say this even before optimizing the system?
Example: Plugging some numbers into this model helps illustrate the difference between transit
competitiveness in wealthy versus poor countries. Using extrinsic units of hrs, km, $:
vw ≅ 3 km/ hr
vmax ≅ 36 km/ hr
ts ≅ 5 x 10- 3 hr
• β ~ 1 → 20 $/ hr
cd ≅ 1 $/ km
cs ≅ 10- 1 $/ stop
ct ≅ 20 $/ hr
• ℓ ~ 2 → 40 km
• λ ~ 1, 2.5, 10, 20, 50, 200 trips/ km2
The values with the greatest range of values ( marked with • ) are our drivers of design. The
figure below shows how the generalized cost ( in units of time) relates to the length of a trip for
transit serving neighborhoods of different values of λβ and the cost of making the trip by car in a
wealthy or poor country. More accessibility is associated with greater trip length for a
generalized cost.
ℓ
Time, z*/ β
λβ = 200 λβ = 50 λβ = 20
50 km
15 km
100 min
car ( wealthy)
car ( poor)
140 min
Public Transportation Systems: Planning— Corridors
3- 24
Standards– Revisited ( Two Additional Points)
The first point is that every length- based standard can be reduced to a “ simple standard”. Recall
from the earlier discussion how, for a defined “ political” standard T0( ℓ ) for door- to- door trip
time, we were able to find the critical length of trip and critical headway to satisfy that standard
with the graphical construction below.
ℓ
t
( ) T0 l
convex hull of
H*
( ) T0 l
*
0 T
l*
AIVD line s*
LE
simple
standard
Note that if we replace T0( ℓ ) with the simple standard shown with its corner at point ( ℓ *, T0
*) we
arrive at the same solution! This simple standard can be interpreted such that all trips shorter than
a certain length ( ℓ *) must be completed within a certain time ( T0
*) and longer trips can be
ignored.
The simplification is useful because it involves just two parameters ( ℓ * and T0
*). Therefore, by
exploring the structure of optimum transit systems for all possible values of these two parameters
one would have explored all possible optimum solutions. Note too from the figure that ℓ * must
be the binding length and therefore we can treat it as the only ( equality) constraint. As a result,
there is a 1: 1 relationship between ( ℓ *, T0
*) and ( ℓ *, β), and we see that we can alternatively
explore the space of all solutions by plotting the Lagrangian solution for all values of ( ℓ *, β), as
we had suggested earlier.
Public Transportation Systems: Planning— Corridors
3- 25
min$
The second point is that there is a neater way of eliminating the socioeconomic drivers ( λ and β)
from the formulation of the problem, simply by working with the total system costs per day,
rather than the unit cost per passenger carried. In the standards formulation we wrote formulas
for $( s, H) and T( s, H) with units per passenger. But if instead we had ( equivalently) used
$ T( s, H) ≡ λ$( s, H), with units of cost per unit time and length, then you can see from the earlier
notes that the parameter λ would not appear in any of our formulas for $ T( s, H). In fact, the
mathematical program:
. . ≤ ( l);∀ l 0 s t T T
T
would not include either of our socioeconomic drivers ( λ or β) in its formulation! This allows
you to find the optimum yearly cost and the system structure by defining a standard and nothing
else. The socioeconomic variables enter the picture only when a city chooses the standards it can
afford. The average cost per passenger carried expressed in units of local wages, which is
$ / β $ /( ) T ≡ λβ
( ) = T + P
, should be an important factor in any such decision.
Example: ( optional problem for students to solve to understand these two ideas)
Show that the equivalent simple standard to the linear standard T for the
lower bound formulation of the case with no transfers is:
0 l 0 0l
2
max
0
*
1
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
−
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
=
v
P
v
t
a
s
l if
max
0
1
v
P >
and that:
Public Transportation Systems: Planning— Corridors
3- 26
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎬
⎫
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
−
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
−
=
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
−
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
= +
max
0
0
*
0
2
max
0
0
0
*
0
1
$
1
v
P
v
t
T
c
v
P
v
t
P
T T
a
s
d
a
s
if
max
0
0 1
v
P
v
t
T a
s
−
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
>
Note how the solution does not involve λ or β. Then, use the Lagrangian approach to
show that the shadow price that would achieve the above is:
2
max
0
0
*
1
( )
⎟ ⎟ ⎟ ⎟ ⎟
⎠
⎞
⎜ ⎜ ⎜ ⎜ ⎜
⎝
⎛
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
−
−
=
v
v P
t
T
c
a
s
λβ d
To repeat: The importance of this is that standards are connected to total costs, and you don’t
need anything else to determine this cost.
Space- and Time- Dependent Service
Assuming we have a corridor, we want to see how performance is affected by changing the
design variables in space and time. Of our two decision variables, s and H, spacing is a physical
aspect of the route, and so is only a function of space: s( x), while headway will remain constant
as buses travel the route, and so is only a function of time: H( t). The ( t, x) area of concern can be
partitioned into space ( i) and time ( j) slices as shown below and we can find the cost of
delivering service for si and Hj. We will do this first for average- case analysis ( which you should
know) and then for the service guarantee ( standards) approach.
Average Case Analysis
For average case analysis, demand plays an important role, so we start by defining an OD matrix
of trip selection rates. The OD matrix can be represented as λ i
i' j., where i is the origin, i’ the
destination and j the time ( the units of λ would be pax/ time · dist2). We shall find that it is not
necessary to use the entire OD matrix, only the relevant parts for which we want standards.
Public Transportation Systems: Planning— Corridors
3- 27
t
x
i
j
If we ignore the cost of stops, the total cost of service is:
= Σ
j
j
j
d
T T
H
$
c L
Note: it does not depend on the OD matrix.
The generalized cost of waiting delay, where λ · j is the total number of trips generated per unit
time along the complete corridor during time slice j ( units of pax/ time), is:
Σ ⋅
j
j j j β H ( λ T )
Similarly, the generalized cost of inbound access is:
Σ ⋅
i
i i
a
i L
v
1 ( )
4
β λ s
where λ i
is the total number of trips generated per unit distance with destinations for the whole
corridor during the course of a day ( units = pax/ dist). Since the cost of egress should be the
Public Transportation Systems: Planning— Corridors
3- 28
same, we can multiply this equation by 2 to account for the total access cost. Finally, if we let Λ i
be the number of people crossing a screen- line in region i during the course of a day
( units= pax/ hr), we can express the generalized cost of stops as:
Σ Λ ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
i
s i
i
i t
s
L β
As you can see, we don’t need to know the whole OD matrix, only the summary information
embodied in { λi · , λ · j, and Λ i
}.
Also note that the optimization is very simple. The first two equations are functions of Hj and
not si and can be optimized alone and separately for each time period. Likewise, the last two
equations are functions of si and not Hj and can be optimized alone and separately for each
location.
Service Guarantee Analysis
Instead of optimizing for the average case with a choice of β, we can choose a set of time
standards T0( i, i’, j) for selected origin and destination pairs and times of day. Then, there is no
need to know the demand to estimate the optimum cost. It would be the job of policy- makers to
decide on a reasonable standard. The objective function is the same as above, and the standards
would simply introduce constraints of the form:
i i ii j T( i, i', j) ≥ AT + AT + IVTT + H 0 ' '
for relevant sets of ( i, i’, j). Note that the four terms of the RHS have simple subscripts. This
MP can often be solved by introducing shadow prices and decomposing the Lagrangian into
parts that can be optimized separately. If this does not work we can resort to a numerical
solution.
Further Readings
The following readings may be useful to reinforce the concepts you have learned in this module.
Clarens, G. and Hurdle, V. ( 1975) “ An operating strategy for a commuter bus system”,
Transportation Science 9, 1- 20. ( Average- case analysis of non- hierarchical many- to- one 2- D systems
with inhomogeneous demand.)
Wirasinghe, C. S., Hurdle, V. F. and Newell, G. F. ( 1977) “ Optimal parameters for a coordinated rail
and bus transit system” Transportation Science 11, 359- 74. ( Average- case analysis of a 2- mode
hierarchy serving 1- D, many- to- one demand.)
Public Transportation Systems: Planning— Two Dimensional Systems
4- 1
Module 4: Planning— Two- Dimensional Systems
( Originally compiled by Eric Gonzales and Josh Pilachowski, March, 2008)
( Last updated 9- 22- 2010)
Outline
• Idealized Case ( New 2- D Issues)
o Systems without Transfers
o The Role of Transfers in 2- D Systems
• Realistic Case ( No Hierarchy)
o Logistic Cost Function ( LCF) Components
o Solution for Generic Insights
o Modifications in Practical Applications
o General Ideas for Design
• Realistic Case ( Hierarchies-- Qualitative Discussion)
• Time Dependence and Adaptation
• Capacity Constraints
• Comparing Collective and Individual Transportation
Remember from previous modules the types of systems we have analyzed. Shuttle systems had
one decision variable, H, and could only be optimized temporally. Corridors had two decision
variables, H and s, and could be optimized temporally and spatially. These design decisions
defined all the passengers travel choices; i. e., when and where to board a transit vehicle. Think
now about a two- dimensional system and the new travel choices available to passengers. This
should illuminate the extra issues that must now enter into the analysis. They include
considerations of total route length and layout, the role of transfers and travel circuity. As before
we start with an idealized analysis that isolates the new issues and then proceed with a more
realistic treatment that combines them all.
Idealized Case
We will perform the idealized analysis in a similar manner as the corridor analysis. We consider
a system with a single line with no transfers allowed and bi- directional service. We assume H= 0
and ts= 0. For the two- dimensional system we will also assume that a0=∞, which removes all
penalty for stopping meaning that v= vmax at all times. We make this assumption because if we
had allowed a0=∞ in the shuttle and corridor analysis then the door- to- door speed would be vmax.
Yet, this turns out not to be true in the two- dimensional case. So, this set of assumptions allows
us to isolate the new effect introduced by the second spatial dimension. Let us see…
Public Transportation Systems: Planning— Two Dimensional Systems
4- 2
Systems without Transfers
Consider a square city with sides φ, area R= φ2 and an infinitely dense grid of streets; see figure
below. No matter how long a transit line is, it cannot cover all points. Therefore, we anticipate
that coverage and access become important issues in 2- D, and that our new decision variable will
be route length and placement. To minimize worst- case access time in 2- D we should place
stops on a ( square) grid, with spacing s to be determined. The worst- case access time would then
be 2s/ vw since there is an access distance of s at both the origin and destination. What then about
travel time? Note that since stops don’t matter it will be the maximum distance a person spends
in a vehicle, divided by vmax. And since service is bi- directional, the maximum distance is ½ of
the length of the line, which we denote L. Thus, IVTT= L/ 2vmax, and we minimize IVTT by
choosing the shortest route to cover our lattice of stops. The problem of shortest- path routing for
pre- existing points is a famous and complex problem known as the Traveling Salesman Problem.
Fortunately for us, the solution for a two- dimensional lattice structure with an even number of
points, such as the one shown above, is easy and efficient since there always is a path where the
distance between any two consecutive stops along the route is s ( you can convince yourself of
this.)
s
s
φ
φ
R
Public Transportation Systems: Planning— Two Dimensional Systems
4- 3
If you now imagine cutting the grid between parallel route lines then the area can be imagined as
a corridor with length L and width s
s
L
where L is the total length of the route. Thus, the area can be expressed as:
Ls = R ,
and the in- vehicle travel time for the worst case person would be:
max max 2 2sv
R
v
IVTT = L = .
The door- to- door travel time guarantee is then:
w max 2
2
sv
R
v
t = s +
Note: This is an EOQ expression with respect to the lattice spacing. When optimized the
solutions is:
max
2
1
max
* 2 2
v v v v
t R
w w
φ
= ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
=
This gives a door- to- door travel speed for the worst- case person:
max
1 v v w ≈ = * 2
ˆ
t
v
φ
If we assume values of vw= 3kph and vmax= 36kph, then the resulting door- to- door speed is
5.5kph, which is not much faster than walking speed. The underperformance arises because
to achieve low access time the route needs to be very winding. And buses in a windy route
entrap passengers unfortunate to go a long distance. So, how can we improve the system? If we
allow for transfers then passengers are no longer entrapped, and all we have to do is look for
routings that give good coverage while providing good travel options to passengers that can
transfer. So what are these routings? To get an understanding of this issue, we look at some
idealized systems with one transfer.
v ˆ ≈
Public Transportation Systems: Planning— Two Dimensional Systems
4- 4
The Role of Transfers in 2- D Systems
Two extreme possibilities are considered here. A hub and spoke system ( H) with only one
transfer point; and a grid system that allows for transfers at every stop. See the illustrations
below. Note that for the same route spacing, the grid system requires more route- kilometers; so
it should be more expensive to cover with vehicles.
Another disadvantage of the grid system relative to the hub is that coordination is more difficult.
An advantage is that users can always choose a direct route without backtracking. We now
compare the performance of these two systems ( and of the no- transfer, single- line system ( O))
for different values of L. This is reasonable because if one holds H and the commercial speed of
vehicles invariant across scenarios, then L is the most important driver of cost.
s
φ
φ
A Hub and Spoke System ( H)
Public Transportation Systems: Planning— Two Dimensional Systems
s
s
φ
φ
A square grid system ( G)
We now change notation and use L to denote the kilometers of undirected service provided. A
little bit of reflection shows that the total lengths of service for the three cases are: 1
s
L
s
G
H
4φ 2
=
L
s
L
2
2
0
3
2
φ
φ
=
=
For the same L, the three services provide different coverage, as represented by s:
4- 5
L
s
L
s
L
s O H G
2φ 2 ; 3φ 2 ; 4φ 2
= = =
1 To get these simple expressions, it is assumed that it takes 1 spacing to turn the buses at the end of each route.
Public Transportation Systems: Planning— Two Dimensional Systems
4- 6
These values represent the sideways spacing between lines achieved by the three system types.
Thus, the worst- case sideways access times are: , , and . If we
ignore the longitudinal access times ( which should be the same for the three systems) and focus
on cross- town trips ( of length
2φ 2 / Lvw w 3φ 2 / Lv w 4φ 2 / Lv
l ≈ φ ), the worst- case door- to- door travel times are then:
w max
G
w max
H
w max
Lv v
T
Lv v
T
v
L
Lv
T
φ φ
φ φ
φ
= +
= +
= +
2
2
2
0
4
3 2
4
2
If we now choose max ~ 10
v
v
w
then we can compare the three cases based on the dimensionless
variable
φ
L . The formulae become:
⎪ ⎪ ⎪
⎩
⎪ ⎪
⎨
+ ⎟⎠⎞
⎜⎝
⎛
+ ⎟⎠
⎞
⎜⎝
⎛
⎝ ⎠
= ×
( G )
L
( H )
v L
T
max
40 1
30 2
φ
φ φ
⎪ ⎧
⎟ ⎟ ⎞
⎜ ⎜ ⎛
+ ⎟⎠
⎞
⎜⎝
⎛ L ( O )
L 4
20 1
φ
φ
These expressions can be expressed graphically as follows:
Public Transportation Systems: Planning— Two Dimensional Systems
4- 7
2
20 30 40
4
6
8
φ
T ⋅ vmax
φ
L
10
O
H
G
This illustrates that “ short systems” with few stops, whose total length is not much greater than
the perimeter of their service region do not require transfers. The figure also illustrates that long
systems with many stops do benefit, and that in these cases the longer the system the greater the
benefit.
Just so you get a feel for the meaning of
φ
L we look at four common routing examples and their
φ
L values:
Public Transportation Systems: Planning— Two Dimensional Systems
4- 8
Campus periphery
≈ 4
φ
L
Small metro/ small town bus ≈ 12
φ
L Large system ≈ 20
φ
L
We find that the optimal routing layout depends on the value of
φ
L and, if we add a 25% access
penalty to the grid system to reflect the added cost of an uncoordinated transfer, we find that the
critical points are as follows:
L/ φ
O H G
≈ 10 ≈ 20
This explains why systems in real- life often have the structures shown in the above figures.
Also, note that when we allow one transfer, then max v ˆ = v as L → ∞ for the grid system. So,
transfers really do help with performance in 2- D.
Public Transportation Systems: Planning— Two Dimensional Systems
4- 9
Realistic Case – No Hierarchies
We could do a realistic analysis for each case we introduced earlier, however in the interest of
time we will be concentrating on the grid case since it is the most useful for larger networks.
Since we are dealing with worst case analysis we will also only concentrate on square grids. A
rectangular grid would introduce directionality and add unneeded complexity.
First we will introduce stop spacing, s, within route spacing, S, such that s< S. We will also use S
as a decision variable instead of L. We need to make assumptions about how people travel. In
this case, we will assume that people only make one transfer and they choose their origin and
destination stops in order to minimize their access distance.
We will then develop formulas for agency cost and passenger time ( access + waiting + in- vehicle
travel time)
Logistic Cost Function ( LCF) Components
Recall that the transit service in 2 dimensions can be described by 3 decision variables: stop
spacing s, line spacing S, and service headway H.
s
S
The total cost for such a system is cost of driving and stopping a bus multiplied by the number of
buses operating per unit area.
⎟⎠
⎞
⎜⎝
= ⎛ +
s
c c
SH
s
T d
$ 4 in units of time dist 2
cost
⋅
⎟⎠
⎞
⎜⎝
= ⎛ +
s
c c
SH
s
λ d
$ 4 in units of
pax
cost
Public Transportation Systems: Planning— Two Dimensional Systems
4- 10
Notice this is very similar to the case for a corridor, the only difference being a factor 4/ S,
expressing the fact that cost depends on the number of lines.
The travel time is composed of access time ( AT), waiting time ( WT), and in- vehicle travel time
( IVTT) just as we saw for corridors. For the worst case passenger whose trip starts and ends as
far as possible from transit service ( the middle of the square),
w w v v
AT ⎥⎦
⎢⎣
⎟⎠
⎜⎝
⎟⎠
⎜⎝
2 2 2 2
s S s S s + S
=
= 1 ⎡⎛ + ⎞ + ⎛ + ⎞⎤
WT = H + Δ or 2H + Δ or3H + Δ
where Δ represents time required to make a transfer, such as walking time from one stop to
another. The number of headways included in WT depends on the assumptions we make about
the synchronization of schedules ( H if services are perfectly synchronized so that passengers
only wait at the first stop where they board; 2H if services are not coordinated and passengers
have to wait when they transfer, or else if service is coordinated but passengers have
appointments at the destination; etc…).
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
= +
s
t
v
IVTT s
max
0
1
l
where the longest possible trip length is ℓ 0 ≈ 2φ. So, the worst case time for a 2- D system is
given by the sum,
⎟ ⎟⎠
⎞
= +
s
t
T s 2 ⎜ ⎜⎝
⎛
+
+
+ Δ +
v v
H S s
w max
0
1
l
Notice again that this is very similar to the time associated with transit service in a corridor. The
difference is the waiting time, 2H + Δ, and an additional component of access time, S/ vw.
Solution for Generic Insights
If we consider the lower bound of cost, assuming that the cost of stopping is small, the standards
approach is described by the following mathematical program:
SH
c d 4
min ( 6.1)
s. t. ( ) ( 2 T + ⎟ ⎟
⎜ ⎜
+ l 0 l) ( 6.2) | s T
v
H S
w
≤
⎠
⎞
⎝
⎛
Public Transportation Systems: Planning— Two Dimensional Systems
4- 11
where ( ) ⎟ ⎟ ⎞
⎜ ⎜ ⎛
= Δ + + + T l | s s l 1 ts
v ⎝ v s ⎠ w max
.
The constraint will be an equality at optimality because for any T( ℓ | s) the cost is minimized by
choosing the highest values of S and H. Therefore, the lower envelope method ( explained in
Module 3) can be used to solve for s*, and with TL( ℓ ) we can determine ℓ *. The mathematical
program can thus be obtained with pencil and paper.
Alternatively, we can use the Lagrangian approach, expressing the generalized cost in dollars per
person as
⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
+ +
+
= + + Δ +
s
t
v v
H S s
SH
c
z s
w
d
L
l l
max
2
4 β
λ
( 6.3)
which decomposes so that the stop spacing, s, is isolated. Solving for s* and substituting,
s w s* = lt v ( 6.4)
⎟ ⎟ ⎠
⎞
⎜ ⎜
⎝
⎛
= + + Δ + + +
max
2 2
4
v v
t
v
H S
SH
z c
w
s
w
d
L
l l β
λ
The optimal headway, H*, and line spacing, S*, can be solved in closed form.
λSβ
c
H d 2
* = ( 6.5)
( ) ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
= + + Δ + +
w
s
w
d
L v
t
v v
S
S
c
z S l l 2
2
* 4
max
β β
λ
β
3
1
S
8 2
* ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
=
λβ
d w c v
( 6.6)
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
+ + Δ + ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
=
w
s
w
d
L v
t
v v
c
z l l * 6 2
max
3
1
2
β
λ
β
We now compare this cost to the generalized cost for corridors, assuming the same values as in
module 3:
vw ≅ 3 km/ hr
vmax ≅ 36 km/ hr
Public Transportation Systems: Planning— Two Dimensional Systems
4- 12
ts ≅ 5 x 10- 3 hr
cd ≅ 1 $/ km
In 2- D, the generalized cost is
⎟⎠
⎞
⎜⎝
⎛ + + ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
=
36
* 4.2 0.08
3
1
2 l
β l
λ
β
L z
and in universal units of time:
36
4.2 0.08 l
l + + ⎟ ⎟⎠
⎜ ⎜⎝
=
β λβ
L * 1 3
1
z ⎛ ⎞
compared to a generalized cost in a corridor of
⎟⎠
⎞
⎜⎝
⎛ + + ⎟⎠
⎞
⎜⎝
= ⎛
36
* 2 0.08 2
1
l
β l
λ
β
L z
36
* 2 1 2 0.08
1
l
l + + ⎟ ⎟⎠
⎞
⎜ ⎜⎝
⎛
=
β λβ
L z
Note that the universal generalized cost per person declines with demand, λ, and wealth, β, more
slowly in the 2D case than in the 1D case. In other words, the second dimension somewhat
dilutes the economies of scale in collective transportation. Note too that the effect of distance is
the same in both cases.
Remember, however, that λ is expressed in demand per area in the 2- D case, and demand per
distance in the corridor case, so these expressions cannot be compared for the “ same” λ.
For the hypothetical case of long trips in a relatively poor city ( ℓ 0 = 40 km, β = $ 1 / hour, and λ =
103 pax / hr⋅ km2), the generalized cost zL/ β = 2.1 hours which decomposes to 0.17 hours of work,
0.9 hours of delay ( access, waiting, and in- vehicle stopping), and 1.11 hours of travel time ( like
in a car).
Modifications for Practical Applications
1) Some lines may require fixed infrastructure ( BRT, rail, etc.), so the cost of construction,
bond finance, etc. should be amortized over the life of the infrastructure. Convince
yourselves that for an infrastructure cost rs $/ hr⋅ stop, this contributes
Public Transportation Systems: Planning— Two Dimensional Systems
4- 13
sS
s
λ
HsS
r
to the objective function.
2) Stops may be skipped if the demand is low. In this case we work with expectations.
E( time stopped per unit length) = E(# pax boarding/ alighting moves per distance) tm
+ E(# stops per distance) ts
where tm is the marginal time for one passenger move and ts is the marginal time for a
vehicle stop. The expectations are now given. First, note that
E(# of pax moves per stop) = 2λ
Therefore, since there are 1/ s stops per km;
E(# pax moves per distance) =
s
1 E(# pax moves per stop) = 2λHS ,
and
s
E(# stops) = 1 Pr{ stopping},
where
Pr{ stopping} = ( 1− e− 2λHsS )
if the demand for stops follows a Poisson process with the given mean.
3) Cities have centers, so we may want to orient our grid towards the center. Notice that if
we zoom in on a part of a ring- radial network it looks like a grid. Nothing prevents us
from making a constant density of service in a ring- radial network by adding radial lines
as we move out from the city center.
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4- 14
We can also use this strategy if we want to have the flexibility to have different densities
of service and headways in different parts of the city as shown in the figure below. To do
this systematically, we can set different standards for trips in different parts of the city.
For example,
T0
( A)( ℓ ) for all trips in A
T0
( B)( ℓ ) for all trips in B ( or, even better, B ∪ A)
T0
( AB)( ℓ ) for all trips between A and B
B
A
SB, sB, HB
SA, sA, HA
General Ideas for Design
1) Think of a family of design concepts, qualitatively – e. g. grid system, ring- radial
network, etc.
2) Identify members of the family by list of decision variables – e. g. stop spacing s, line
spacing S, and headway H.
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3) Estimate the cost and translate the specific concept into a detailed plan – e. g.
OR
Considering all regions ( r = A,…) and time periods of the day ( j = 1, 2,…) solve the
following mathematical program for the decision variables: {…, ( sr, Sr), …} and {…, Hrj,
…}:
Σ ( )
=
=
=
1,2,...
,...
min$ $ ( ) ,
j
r A
r rj
r
T T S H
( ) A Aj Aj S H T 0 T , A A s. t. s , ≤ , j = rush, off- peak, night
( ) B Bj Bj S H T 0 , , B A A B T s , S , s ≤ , j = rush, off- peak, night
( ) AB A ABj T s 0 , A B B Aj Bj S , s , S , H , H ≤ T , j = rush, off- peak, night
You may want to include separate constraints for access time or waiting time, depending
on what the city wants, but you should always use your judgement. Anything is possible,
but the more complicated the problem, the more difficult it is to solve the problem
exactly.
Lagrangian decomposition can help us solve this mathematical program. It may be
possible to simplify the problem and eliminate many of the decision variables. So, we
can use shadow prices to simplify these complicated mathematical problems by assigning
a different β to each of the constraints. Increasing the values of β will reduce the left side
of the constraint when the Lagrangian is optimized, so we start with an estimated value of
β and then increase it until the constraints are met. If we have a closed form for the
optimal decision variable values in terms of β, it is easy to adjust the solution by
changing the shadow price.
Note: This approach can be used to solve ( 6.1,2) by working instead with ( 6.3). All you
should have to do is plug in ( 6.4,5,6) into ( 6.2) and find the β that solves ( 6.2) as an
equality.
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2- D Systems: Realistic Case ( Hierarchies)
Until now, we have looked only at local systems in 2- D. However, we could introduce a
hierarchy with the same method as for corridors ( see module 3). There are now decision
variables for the stop spacing and line spacing of both the local and express services.
Express Line
Local Line
The introduction of hierarchies also gives us the flexibility to design non- isotropic systems. For
example, a grid may serve as a basis for local buses guaranteeing a length- based but uniform
standard for short- medium trips. A radial express service may be overlaid to provide better
service for inter- zonal travel ( e. g. Chicago). Such an express network may be described in as
few as 3 additional decision variables: # of radial lines, # of ring lines, service headway.
Express Line
Local Line
Perhaps one system can be designed to act a radial network outside of the city center and look
more like a grid in the city center ( e. g. Washington DC, London). The possibilities are many,
but in all cases the goal is to reduce these concepts to as few descriptors as possible which will
describe the shape and design that the system should have once the variables are chosen.
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2- D Systems: Time- Dependence and Adaptation
Over time, demand for transportation in a city changes. Some of the decision variables are easier
to change over time than others. The headway, H, can be varied very easily even within the
course of a day. The stop spacing, s, can be changed with a little more effort, and line spacing, S,
is relatively fixed.
Suppose we have a linear city of length one, and we place a station to minimize access distance.
A single station divides the city into two halves and should be placed in the center to minimize
worse case and average access distance; S* = 1/ 2.
S = 1/ 2 S = 1/ 2
As demand grows over time, we may want to add stations incrementally to the city, one at a
time. If we can pull up old stations and always re- optimize, the placement should make the
spacing follow the progression, S* = 1/ 2, 1/ 3, 1/ 4, 1/ 5, etc. However, if the stations are fixed
once they are placed, subsequent placement of stations will not always give us the minimum
access cost.
For a worst case analysis, imagine that we have the city above with n = 2 spacings ( S* = 1/ 2) and
we add one more station. Only half of the city benefits, so the worst case access is unchanged.
The worst case access cost is only improved when symmetry is established at n = 2, 4, 8, … The
incremental addition of station