-
Notifications
You must be signed in to change notification settings - Fork 0
/
intro-talk.tex
650 lines (544 loc) · 17.6 KB
/
intro-talk.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
\documentclass[handout]{beamer}
% \documentclass[mathserif,color=option]{beamer}
% \usepackage{lucida} % Lucida Bright (SO Version)
\usepackage[small]{eulervm} % Euler VM
% This file is a solution template for:
% - Giving a talk on some subject.
% - The talk is between 15min and 45min long.
% - Style is ornate.
% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice.
\mode<presentation>
{
\usetheme{Antibes}
% or ...
\setbeamercovered{transparent}
% or whatever (possibly just delete it)
}
\usefonttheme[onlylarge]{structurebold}
\setbeamerfont*{frametitle}{size=\normalsize,series=\bfseries}
\setbeamertemplate{navigation symbols}{}
\usepackage[english]{babel}
% or whatever
\usepackage[latin1]{inputenc}
% or whatever
\usepackage{times}
% \usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
% Change the footnote style
\renewcommand{\thefootnote}{\arabic{footnote}}
\renewcommand{\footnotesize}{\tiny}
\setlength{\skip\footins}{.5cm}
% Use subfigure package
\usepackage{subfigure}
% To display columns dynamically
\usepackage{colortbl}
% drawing trees
% \usepackage{pstricks, pst-node, pst-tree}
% Put several slides on a single page
% \usepackage{pgfpages}
% \pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
\usepackage{tikz}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{verbatim}
\usetikzlibrary{arrows,shapes}
\usepackage{slashbox, multirow}
\usepackage{array}
\usepackage{url}
% The DeclareGraphicsRule statement causes all file extensions that are not
% associated with a well-known graphic format to be treated as Encapsulated
% PostScript files.
% \usepackage{graphicx}
% \DeclareGraphicsRule{*}{mps}{*}{}
% set caption font style
\usepackage{caption}
\renewcommand{\captionfont}{\scriptsize}
\renewcommand{\captionlabelfont}{\scriptsize}
\setlength{\abovecaptionskip}{3pt}
% For every picture that defines or uses external nodes, you'll have to
% apply the 'remember picture' style. To avoid some typing, we'll apply
% the style to all pictures.
\tikzstyle{every picture}+=[remember picture]
% By default all math in TikZ nodes are set in inline mode. Change this to
% displaystyle so that we don't get small fractions.
\everymath{\displaystyle}
\title[Which Route to Follow?] % (optional, use only with long paper titles)
{Which Route to Follow in Urban Transportation Network?}
\subtitle
{An Introduction to Traveler's Route Choice Behavior} % (optional)
\author[Xiong Y.L.] % (optional, use only with lots of authors)
{Xiong Yiliang}
% - Use the \inst{?} command only if the authors have different
% affiliation.
\institute[HKPolyU] % (optional, but mostly needed)
{
Department of Civil and Structural Engineering\\
Hong Kong Polytechnic University
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.
}
\date[July 12, 2010] % (optional)
{July 12, 2010}
\subject{Wardrop Equilibrium, Network Design}
% This is only inserted into the PDF information catalog. Can be left
% out.
% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:
% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
% \logo{\pgfuseimage{university-logo}}
% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
\AtBeginSection[]
{
\begin{frame}<beamer>{Outline}
\tableofcontents[currentsection,currentsubsection]
\end{frame}
}
%\AtBeginSubsection[]
%{
% \begin{frame}<beamer>{Outline}
% \tableofcontents[currentsection,currentsubsection]
% \end{frame}
%}
% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command:
%\beamerdefaultoverlayspecification{<+->}
\begin{document}
\begin{frame}
\titlepage
\note{
Thank all of you for coming to this seminar. I really appreciate this opportunity
to talk a little about traveler's route choice problem.
As not everyone in this room work in the field of transportation planning or
travel demand modeling, so I try to make this talk as simple as possible.
}
\end{frame}
\begin{frame}{Outline}
\tableofcontents
% You might wish to add the option [pausesections]
\end{frame}
% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:
% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
% 15 and 30 frames, all told.
\section{Introduction}
% - A title should summarize the slide in an understandable fashion
% for anyone how does not follow everything on the slide itself.
\subsection{Traffic Congestion}
\begin{frame}{Traffic congestion}
\begin{figure}
\centering
\subfigure[Cross-Harbor Tunnel in Hong Kong]{
\includegraphics[height=.45\textheight]{fig/HK_Cross_Harbour_Tunnel.jpg}
}%
\subfigure[Heavy traffic in Beijing]{
\includegraphics[height=.45\textheight]{fig/Daily-Life-in-Beijing_10.jpg}
}
\caption{
Traffic congestion in modern cities
\footnote{
The images were downloaded respectively from the Wikipedia
(\url{http://en.wikipedia.org/wiki/File:HK_Cross_Harbour_Tunnel.jpg},
visited on July 9, 2010)
and the United Press International
(\url{http://www.upi.com/News_Photos/gallery/Daily-Life-in-Beijing/2012/2},
visited on July 9, 2010).
}
}
\end{figure}
\note{
We are all familiar with congested roads, and perhaps also with congestion in
other networks such as the Internet, so it is obviously important to have a general
understanding of how and why congestion occurs in networks. The Cross-Harbor
Tunnel is seriously congested, and the situation is even worse in mainland China.
}
\end{frame}
\begin{frame}{Non-decreasing travel time}
\uncover<1->{
The travel time of a road depends on the traffic flows on this road:
\begin{quote}
The \alert{more vehicles} on a road, the \alert{more time} needed to
pass through it.
\end{quote}
}
\begin{columns}
\column{0.6\linewidth}
\uncover<2->{
Road travel time, $t(x)$, is defined as a \alert{non-decreasing} function of flows $x$.
\begin{block}{What does \emph{non-decreasing} mean here? }
Mathematically, for any $x_1 > x_2$, we have
$$t(x_1) \geq t(x_2)$$
\end{block}
}
\column{0.4\linewidth}
\invisible<1>{
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig/latency-000}
\caption{Link travel time}
\end{figure}
}
\end{columns}
\note{
The more vehicles on a link, the more time needed to pass through it.
Put it mathematically, we get this definition.
}
\end{frame}
\subsection{A Simple Example}
\begin{frame}{Which route to follow?}
\begin{example}
\begin{figure}
\centering
\includegraphics[width=.5\linewidth]{fig/2-node-000}
\caption{Two roads between $rs$}
\end{figure}
\end{example}
\pause
\begin{block}{Travel Time}
\begin{itemize}
\item Free travel time on minor road is 20 minutes.
\begin{itemize}
\item The travel time \alert{increases} 5 minutes for each additional vehicle.
\end{itemize}
\item Travel time on highway is \alert{always} 50 minutes.
\end{itemize}
\end{block}
\end{frame}
\begin{frame}{Flows on a 2-node network}
\begin{columns}
\column{.6\linewidth}
\begin{itemize}
\item The travel time functions:
\begin{itemize}
\item Minor road: $ c_{min}(x) = 20 + 5 \cdot x$
\item Highway: $c_{high}(x) = 50 $
\end{itemize}
\item There are 10 persons traveling between $rs$.
\uncover<2->{
\item The total travel time for Pattern 1 is
$(3 \times 35 + 7 \times 50) \div 10 = 45.5$
}
\uncover<3->{
\item The total travel time for Pattern 2 is
$(5 \times 50 + 4 \times 50) \div 10 = 50$
}
\end{itemize}
\column{.4\linewidth}
\begin{example}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig/2-node-001}
\caption{Traffic Flow Pattern 1}
\end{figure}%
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig/2-node-002}
\caption{Traffic Flow Pattern 2}
\end{figure}
\end{example}
\end{columns}
\end{frame}
\begin{frame}{All possible traffic flow patterns}
\begin{table}
\small
\centering
\caption{All possible traffic flow patterns}
% \rowcolors[]{1}{blue!20}{blue!10}
\begin{tabular}{%
r%
c<{\onslide<1->}!{\vrule}%
c<{\onslide<1->}%
c<{\onslide<1->}!{\vrule}%
l<{\onslide<2->}c%
}
\hline
$x_{min}$ & $x_{high}$ & $t_{min}$ & $t_{high}$ & $\bar{t}$\\
\hline
0 & 10 & 20 & 50 & 50.0 \\
1 & 9 & 25 & 50 & 47.5 \\
2 & 8 & 30 & 50 & 46.0 \\
3 & 7 & 35 & 50 & \textbf{45.5}$^*$ \\
4 & 6 & 40 & 50 & 46.0 \\
5 & 5 & 45 & 50 & 47.5 \\
6 & 4 & 50 & 50 & \textbf{50.0} \\
7 & 3 & 55 & 50 & 53.5 \\
8 & 2 & 60 & 50 & 58.0 \\
9 & 1 & 65 & 50 & 63.5 \\
10 & 0 & 70 & 50 & 70.0 \\
\hline
\end{tabular}\\
\vspace{3pt}
{\scriptsize
Note: Average travel time $\bar{t} = (x_{min} \cdot t_{min} + x_{high} \cdot t_{high})
\div (x_{min} + x_{high})$
}
\end{table}
\end{frame}
\begin{frame}{Some insights}
In larger networks, the number of possible traffic flow patterns can be
\alert{astronomical}. And different patterns result in different network
performances.
\pause
\begin{block}{Two questions}
\begin{enumerate}
\item Which traffic flow pattern is \alert{most likely} to occur in the real world?
\item What is the \alert{source} of traffic congestion?
\end{enumerate}
\end{block}
\end{frame}
\section{Traffic Equilibrium}
\subsection{Individual's Route Choice Behavior}
\begin{frame}{The shortest path problem}
Every traveler tries to find the \alert{quickest route} between $r$ and $s$.\\
\pause
\begin{minipage}{.4\linewidth}
\begin{block}{Day 1}
\begin{figure}
\centering
\invisible<1>{ \includegraphics[width=\linewidth]{fig/2-node-003} }
\end{figure}
\begin{itemize}
\item Initially, there is no traveler on the network.
\item The \alert{minor road} is the quickest route between $r$ and $s$.
\end{itemize}
\end{block}
\end{minipage}%
\pause
~$\Rightarrow$~%
\begin{minipage}{.4\linewidth}
\begin{block}{Day 2}
\begin{figure}
\centering
\invisible<1-2>{ \includegraphics[width=\linewidth]{fig/2-node-004} }
\end{figure}
\begin{itemize}
\item All the travelers crowd into the minor road.
\item The \alert{highway} is the quickest route between $r$ and $s$.
\end{itemize}
\end{block}
\end{minipage}%
\pause
~$\Rightarrow$~%
\begin{minipage}{.1\linewidth}
\begin{block}{Day n}
\begin{center}
$\cdots$
\end{center}
\end{block}
\end{minipage}
\end{frame}
\begin{frame}{Complex interaction}
Route Choices \& Route Travel Time:
\begin{itemize}
\item The travelers attempt to \alert{choose} the quickest route, and this choice
depends on the \alert{travel time of each route}.
\item However, the \alert{route travel time} in turn depends on the
\alert{route choices} made by all the travelers in this network.
\end{itemize}
\pause
\begin{block}{Question}
If this process goes to infinity, is there a \alert{stationary} state or an equilibrium?
\end{block}
\note{
However, the pattern of the flow of traffic through a network is the consequence of
a subtle and complex interaction between different users. For example, in a road
network we would normally expect each driver to attempt to choose the most
convenient route, and this choice will depend upon the delays the driver expects to
encounter on different roads; but these delays will in turn depend upon the choices of
routes made by others. This mutual interdependence makes it difficult to predict
the effects of changes to the system, such as the construction of a new road or
the introduction of tolls in certain places.
If this process goes to infinity, is there a stationary state?
Maybe, this process can reach a state that no one wants to change his route choice
any more.
}
\end{frame}
\subsection{Equilibrium Traffic Flows}
\begin{frame}{Wardrop's Principle}
Each traveler \alert{non-cooperatively} seeks to \alert{minimize} his travel time.
An equilibrium is reached when no traveler may lower his travel time through
\alert{unilateral action}. \\
\pause
\begin{block}{Wardrop's Principle}
\begin{quote}
"The journey times on all the routes actually used are \alert{equal}, and \alert{less than}
those which would be experienced by a single vehicle on \alert{any unused route}."~
\footnote{
Wardrop, J. G., 1952. Some theoretical aspects of road traffic research,
Proceedings, Institute of Civil Engineers, Part II, Vol.1, pp. 325-378.
}
\end{quote}
\end{block}
\note{
An equilibrium is reached when no traveler may lower his travel time by his
own effect. If he cooperate with others, he may further lower his travel time,
but he does not or cannot cooperate with others, the equilibrium is reached.
}
\end{frame}
\begin{frame}{Find the equilibrium traffic flows}
Wardrop's Principle has two-fold meaning.
\begin{columns}
\pause
\column{.45\linewidth}
\begin{block}{"The journey times on all the routes actually used are equal."}
At equilibrium, the travel time on minor road is equal to that of highway,
\begin{align*}
t_{min} &= t_{high}\\
20 + 5 \cdot x_{min} &= 50 \\
\therefore x_{min} = 6 &\text{ and } x_{high} = 4
\end{align*}
\end{block}
\pause
\column{.45\linewidth}
\begin{block}{"\dots and less than those on any unused route."}
If the number of trips between $r$ and $s$ is 5, all the travelers will choose
the minor road, because of
\begin{align*}
t_{min}(5) &= 45\\
t_{high}(0) &= 50\\
\therefore t_{min}(5) &< t_{high}(0)
\end{align*}
\end{block}
\end{columns}
\end{frame}
\begin{frame}{Implication of the equilibrium}
\begin{minipage}{.35\linewidth}
\begin{block}{Day 1}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig/2-node-003}
\caption{Initial traffic flows. }
\end{figure}
\end{block}
\end{minipage}%
~$\Rightarrow$~%
$\cdots$
~$\Rightarrow$~%
\begin{minipage}{.35\linewidth}
\begin{block}{Day n}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig/2-node-002}
\caption{Equilibrium traffic flows.}
\end{figure}
\end{block}
\end{minipage}
\pause
\begin{block}{Implication of the equilibrium}
If we assume that every traveler chooses the \alert{quickest route}, the traffic flows
that \alert{satisfy Wardrop's Principle} is most likely to occur in real world.
\end{block}
\note{
This principle helps us understand the underlying travel behavior in urban
transportation network. And it also provides a method to predict the traffic
flows in the network.
}
\end{frame}
\subsection{Source of Traffic Congestion}
\begin{frame}{Equilibrium flows vs. Optimal flows}
\begin{table}
\centering
\caption{Equilibrium traffic flows are different form optimal flows. }
\small
\begin{tabular}{
l|p{.1\linewidth}|p{.1\linewidth}|p{.1\linewidth}|p{.1\linewidth}|p{.1\linewidth}
}
\hline
Flows patterns & $x_{min}$ & $x_{high}$ & $t_{min}$ & $t_{high}$ & $\bar{t}$\\
\hline
Optimal flows & 3 & 7 & 35 & 50 & 45.5$^*$ \\
Equilibrium flows & 6 & 4 & 50 & 50 & 50.0 \\
\hline
\end{tabular}\\
\vspace{3pt}
{\scriptsize
Note: $\bar{t} = (x_{min} \cdot t_{min} + x_{high} \cdot t_{high})
\div (x_{min} + x_{high})$
}
\begin{itemize}
\item Compared to optimal flows, \alert{too many} travelers crowd into
the minor road in equilibrium flows.
\item The average travel time of equilibrium flows is 4.5 minutes
\alert{larger than} that of optimal equilibrium.
\end{itemize}
\pause
\begin{block}{Source of traffic congestion}
\begin{center}
Choosing quickest route $\Rightarrow$%
Crowding effects $\Rightarrow$%
Traffic congestion
\end{center}
\end{block}
\end{table}
\end{frame}
\section{Summary}
\begin{frame}{Summary}
% Keep the summary *very short*.
Investigate the traffic congestion in light of travelers' route choice behaviors:
\begin{description}
\item[Individual's behavior] The quickest path problem.
\item[Aggregate quantity] Equilibrium traffic flow.
\item[Macroscopic phenomenon] Traffic congestion.
\end{description}
\note{
The main contributions of this research are that in addition to accounting for traveler's
heterogeneity in preference and perceptions, it also
investigates probabilistic activity/travel behaviors in uncertain environment. This can help us
to understand the source of uncertainty in the demand side of urban transportation system.
}
\end{frame}
\begin{frame}{State-of-the-Art Research}
Considerable number of studies have emerged in recent years:
\begin{itemize}
\item Road congestion pricing
\item Random errors in travelers' choices
\item Uncertainties in travel time
\item $\dots$
\end{itemize}
\end{frame}
\appendix
%
%\section{Transportation Network Design}
%
%\begin{frame}{Transportation Network Design}
%$\dots$
%\end{frame}
\section*{Appendix}
\begin{frame}{Network Structure}
\begin{columns}
\column{.6\linewidth}
An abstract model for urban transportation network:
\begin{itemize}
\item A set of nodes $N$, and a set of links $A$.
\item Origin-destination (O-D) pair $rs$, where $r,s \in N$.
\item Number of trips made between each O-D pair $f^{rs}$.
\item For each link $a \in A$, a travel time function $t_a(x_a)$ is given.
\end{itemize}
\column{.4\linewidth}
\begin{example}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig/braess-000}
\caption{Braess's network}
\end{figure}
\end{example}
\end{columns}
\end{frame}
\end{document}