NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR
TRANSPORT AND THE EVOLUTION OF URBAN SPATIAL STRUCTURE*
Alan G. Wilson**
October 1984 CP-84-41
Contribution to the MetropoZitan Study:
1 0*Presented at the A.I.R.0 Conference in Naples, September 1983.
**School of Geography The University of Leeds Leeds LS2 9JT
UNLTED KINGDOM Telephone 431751
C o Z Z a b o r a t i v e P a p e r s report work which has not been performed solely at the International Institute for Applied Systems Analysis and which has received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organi- zations supporting the work.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
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CONTRIBUTIONS TO T E E METROPOLITAN STUDY:
Anas, A . a n d L.S. D u a n n ( 1 9 8 3 ) D y n a m i c F o r e c a s t i n g o f T r a v e l Demand. C o l l a b o r a t i v e P a p e r , C P - 8 3 - 4 5 .
I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d S y s t e m s A n a l y s i s ( I I A S A ) , A - 2 3 6 1 L a x e n b u r g , A u s t r i a .
C a s t i , J . ( 1 9 8 3 ) E m e r g e n t N o v e l t y a n d t h e M o d e l i n g o f S p a t i a l P r o c e s s e s . R e s e a r c h R e p o r t , R R - 8 3 - 2 7 . I IASA, L a x e n b u r g , A u s t r i a .
L e s s e , P.F. ( 1 9 8 3 ) The S t a t i s t i c a l D y n a m i c s o f
S o c i o - E c o n o m i c S y s t e m s . C o l l a b o r a t i v e P a p e r , C P - 8 3 - 5 1 . I I A S A , L a x e n b u r g , A u s t r i a .
Haag,
G.a n d W. W e i d l i c h ( 1 9 8 3 ) An E v a l u a b l e T h e o r y o f a C l a s s o f M i g r a t i o n P r o b l e m s . C o l l a b o r a t i v e P a p e r , CP-83-58. I I A S A , L a x e n b u r g , A u s t r i a .
N i j k a m p , P. a n d U. S c h u b e r t ( 1 9 8 3 ) S t r u c t u r a l Change i n U r b a n S y s t e m s . C o l l a b o r a t i v e P a p e r , C P - 8 3 - 5 7 .
I I A S A , L a x e n b u r g , A u s t r i a .
L e o n a r d i ,
G.( 1 9 8 3 ) T r a n s i e n t a n d A s y m p t o t i c B e h a v i o r o f a R a n d o m - U t i l
i t yB a s e d S t o c h a s t i c S e a r c h P r o c e s s i n C o n t i n o u s S p a c e a n d T i m e . W o r k i n g P a p e r , WP-83-108.
I I A S A , L a x e n b u r g , A u s t r i a .
F u j i t a , M. ( 1 9 8 4 ) The S p a t i a l G r o w t h o f T o k y o M e t r o p o l i t a n A r e a . C o l l a b o r a t i v e P a p e r , C P - 8 4 - 0 3 .
I I A S A , L a x e n b u r g , A u s t r i a .
A n d e r s s o n , A.E. a n d 0 . J o h a n s s o n ( 1 9 8 4 ) K n o w l e d g e I n t e n s i t y a n d P r o d u c t C y c l e s i n M e t r o p o l i t a n R e g i o n s . W o r k i n g P a p e r , U P - 8 4 - 1 3 . I I A S A , L a x e n b u r g , A u s t r i a . J o h a n s s o n , B . a n d P. N i j k a m p ( 1 9 8 4 ) A n a l y s i s o f
E p i s o d e s i n U r b a n E v e n t H i s t o r i e s . W o r k i n g P a p e r , WP-84-75. I I A S A , L a x e n b u r g , A u s t r i a .
W i l s o n , A.G. ( 1 9 8 4 ) T r a n s p o r t a n d t h e E v o l u t i o n o f U r b a n S p a t i a l S t r u c t u r e . C o l l a b o r a t i v e P a p e r ,
C P - 8 4 - 4 1 . I I A S A , L a x e n b u r g , A u s t r i a .
FOREWORD
Contribution t o
t h eMetropolitan Study:
1 0The p r o j e c t " N e s t e d D y n a m i c s o f M e t r o p o l i t a n P r o c e s s e s a n d P o l i c i e s " s t a r t e d a s a c o l l a b o r a t i v e s t u d y i n 1 9 8 3 . The S e r i e s o f c o n t r i b u t i o n s i s a means o f c o n v e y i n g i n f o r m a t i o n b e t w e e n t h e c o l l a b o r a t o r s i n t h e n e t w o r k o f t h e p r o j e c t .
T h f s p a p e r d e v e l o p s a s k e l e t o n f o r a n a l y z i n g how t h e e v o l u t i o n o f t r a n s p o r t s y s t e m s i s i n f l u e n c e d b y t h e d y n a m i -
c a l l y c h a n g i n g l o c a t i o n s o f r e g i o n a l a c t i v i t i e s . A t t e m p t s t o a s s e s s d y n a m i c i m p a c t s o f t h i s k i n d a r e o f s p e c i a l
i n t e r e s t i n t h e M e t r o p o l i t a n S t u d y i n t h e s t u d i e s o n t h e e f f e c t o f a c t i v i t y c h a n g e s o n t h e much s l o w p r o c e s s o f
c h a n g e i n t r a n s p o r t n e t w o r k s .
A k e E . A n d e r s s o n L e a d e r
R e g i o n a l I s s u e s P r o j e c t
S e p t e m b e r
1 9 8 4SUMMARY
Methods o f dynamical a n a l y s i s d e v e l o p e d i n l o c a t i o n t h e o r y
a r e a p p l i e d t o t h e problem o f t h e e v o l u t i o n o f t r a n s p o r t s y s t e m s .
I ti s shown t h a t t h e i n f l u e n c e of t r a n s p o r t v a r i a b l e s on u r b a n
s t r u c t u r e c a n be modelled; h e r e , t h e r e v e r s e i s a t t e m p t e d . The
a n a l y s i s p r o v e s t o be d i f f i c u l t b e c a u s e of t h e c o m b i n a t o r i a l
problems a s s o c i a t e d w i t h l a r g e n e t w o r k s and a s u g g e s t i o n i s
e x p l o r e d f o r making p r o g r e s s u s i n g t h e c o n c e p t o f ' s p i d e r ' n e t -
works.
CONTENTS
1. INTRODUCTION: DEVELOPMENTS I N LOCATION THEORY AND T H E I R
I M P L I C A T I O N S FOR TRANSPORT MODELLING
1
2.
URBAN S P A T I A L STRUCTURE AND EVOLUTION A S A FUNCTION O FTRANSPORT SYSTEM V A R I A B L E S 2
3 . MAKING TRANSPORT-SYSTEM SUPPLY VARIABLES E X P L I C I T 3
4 .
SUMMARY O F A FORMAL MODEL7
5.
THE EVOLUTION O F URBAN S P A T I A L STRUCTURE, INCLUDINGTRANSPORT SYSTEMS
9
6 .
CONCLUDING COMMENTS: AN ONGOING RESEARCH PROGRAMME15
F I G U R E S
1.
R e t a i l P a t t e r n sf o r
V a r i o u s A l p h aand
B e t a V a l u e s1 7 2.
E f f e c tof
D e c r e a s i n g T r a v e l C o s tt o
C i t y C e n t e r18 3.
E l e m e n t sof a
G e n e r a l M o d e l , E m p h a s i z i n g T r a n s p o r t19
4 .
A' s p i d e r '
N e t w o r k20
5.
( a ) T i m e - F l o w R e l a t i o n s h i p s ; ( b ) C a p t i a lc o s t s v s . Scale 2 1 of
I n v e s t m e n t6 . Sources of
D e m a n df o r
New T r a n s p o r tI n f r a s t r u c t u r e 22
REFERENCES
23
- v i i -
1. I n t r o d u c t i o n : developments i n l o c a t i o n theory and t h e i r imPlioations f o r t r a n s p o r t m o d e l l i n g
The f i r s t f l u s h o f urban modelling i n t h e
1960s
and e a r l y 1970s was concerned w i t h population a c t i v i t y and only t o a l e s s e r e x t e n t w i t h economic a c t i v i t y and t h e supply of houses, s e r v i c e s and s o on. This r e s e a r c h l e a n e d h e a v i l y on e a r l i e r developments i n t r a n s p o r t modelling, mainly again concerned w i t h demand r a t h e r t h a n supply. Since t h e l a t e 1970s, t h e r e have been u s e f u l developments i n modelling t h e supply subsystems o f a comprehensive urban model andit i s
now a p p r o p r i a t e t o t u r n t h e wheel f u l l c i r c l e and t o c o n s i d e r t h e i m p l i c a t i o n s o f t h e methodologyunderpinning t h e s e developments f o r modelling t h e supply o f t r a n s p o r t i n f r a s t r u c t u r e . The new developments i n urban modelling t o be d e s c r i b e d h e r e stem from t h e work of H a r r i s and Wilson (1978) a p p l i e d t o r e t a i l i n g systems (but r e l e v a n t t o a v a r i e t y of s i m i l a r l y - s t r u c t u r e d s e r v i c e s y s t e m s ) . The developments a r e r e p o r t e d i n more d e t a i l i n Wilson (1981). It has l a t e r been shorn t h a t t h e i d e a s can be extended t o r e s i d e n t i a l l o c a t i o n and housing supply and even i n d u s t r i a l l o c a t i o n and a g r i c u l t u r a l l o c a t i o n
-
s e e Wilson (1983) f o r a review. As we w i l l s e e , t r a n s p o r t - r e l a t e d v a r i a b l e s p l a y an important r o l e a s exogenous v a r i a b l e s i n all. .
t h e s e models and t h e problem t o be posed i s , i n e f f e c t , how t o make them endogenous.
There i s , of c o u r s e , a s u b s t a n t i a l l i t e r a t u r e i n t h e t r a n s p o r t j o u r n a l s which i s r e l e v a n t t o t h e t o p i c i n hand. Key r e f e r e n c e s a r e t h e s p e c i a l i s s u e s o f
T r a n s p o r t a t i o n Research,
B
on T r a n s p o r t a t i o n network design ( ~ o y c e , 1979) and on T r a n s p o r t a t i o n s u p p l y models ( ~ l o r i a n and Caudry, 1 9 8 0 ) . The f i r s t h a s a d i r e c t r e l e v a n c e t o t h e modelling o f network e v o l u t i o n-
through s e e k i n g optimum a d d i t i o n a l l i n k s , and s o on; t h e second h a s a b r o a d e r r e l e v a n c e , p a r t i c u l a r l y i n t h es p e c i f i c a t i o n o f supply-cost f u n c t i o n s , t h e importance o f which we
w i l l
s e e l a t e r . There i s a l s o a u s e f u l g e n e r a l paper on t r a n s p o r t supply by Manheim (1980) and a l a t e r e x t e n s i o n of t h e i r own framework by F l o r i a n and Gaudry (1983).The o b j e c t i v e o f t h i s paper i s t o t a c k l e some o f t h e problems which a r e common t o t h i s l i t e r a t u r e w i t h a method which
i s
t r a n s l a t e d from l o c a t i o n a l a n a l y s i s . A t t h i s s t a g e , it i s p o s s i b l e t o develop t h e s k e l e t o n o f t h e i d e a o n l y , and a t a l a t e r s t a g eit
i s hoped t h a t some more e f f e c t i v e i n t e g r a t i o n w i l l be p o s s i b l e .I n s e c t i o n 2 below, we s k e t c h i n b a r e s t o u t l i n e t h e e s s e n c e o f r e c e n t r e s e a r c h t o d a t e . It t u r n s o u t t h a t supply-side developmentsturn on t h e d i f f e r e n c e between c o s t s and revenue ( o r b e n e f i t s ) . I n s e c t i o n 3, t h e r e f o r e , we s e t up t h e
-
a p p r o p r i a t e v a r i a b l e f o r d e s c r i b i n g t r a n s p o r t system supply w i t h p a r t i c u l a r r e f e r e n c e t o t h e n a t u r e o f c o s t s and b e n e f i t s . The model developed i s summarised f o r m a l l y i n s e c t i o n4.
We t h e n c o n s i d e r t h e fundamental problem o f t h e e v o l u t i o n o f networkstructures (and other supply-side variables) in section 5. This allows us to tackle the problem of transport-land-use interaction and we make some concluding comments in section 6.
2. Urban spatial structure and evolution as a function of transport system variables
A
typical urban model which represents both supply-side and demand side behaviour can be put together in a general way as follows (from Wilson, 1983).
Consider m-type organisations(or people) in zone i demanding goods or services of type g from organisationsof type n in zone j. Then the interaction array -
the intersection of supply and demand - can be taken as
where x . is demand by m for g at i, w.'png is the attractiveness of (n,j ,g) ~ ~
1 1 J
supply for (m,i ,g) demand and cg is the cost of travel from i to j for
g.This ij
would be a model of consumers' behaviour and it can take a variety of explicit f o m . Let h? be a vector of characteristics measuring supply of
gby n at j and
-J
a?
a set of corresponding parameters. Then, formally, clntracting wiYg to P g ,
-J J
W ng = W ng (hng ang)
j j
-J 9 - j( 2 )
If is measured in money units, then
is the total revenue attracted to the (n,jYg) combination.
Let z:~ be the amount of g produced by n at j. Then at least one of the
J
elements of h?g will be a function of z?. Let q?g be the
Zth input needed to
-J J J t
produce zng and let be the unit price of that input. Then the cost of j
production is
with
and
to allow for economies or diseconomies of scale. Then a typical assumption
about supply-side dynamics is
with possible equilibrium states as the solutions of
There are enormous complexities beneath the apparent simplicity of this formal presentation. When all the appropriate substitutions have been made the systems
(7) or (8) - for disequilibrium or equilibrium modelling respectively - are
coupled non-linear simultaneous equations in the supply-side variables
( 2 " ) .J
Solutions can disappear or change their nature at critical values of parameters and the system can jump between alternative equilibi-ium states as a result of pertubations. 'Historical accidentsf could have a crucial impact on the form of system development.
There has been much exploration of the kinds of system state which can arise, and the nature of transformations between states, mainly using numerical
experiments in relation to idealised systems. Examples of possible states for retail supply as a function of two parameters (a associated with attractiveness,
0with ease of travel - the larger a, the more important are consumer scale economies; the larger B, the more difficult in general travel is) are shown in Figure 1 which is taken from Clarke and Wilson (1983).
The transport system has an obvious influence on these models through the arrays {cij). Indeed, in Figure 2, we show modifications to the pivotal case of Figure 1 (a
=1.3, B
=3.5) obtained by factoring city centre costs by 0.95, 0.85 and 0.75 respectively. The scale and nature of the influence is obvious. However, these c .-variables are exogenous. The next step in the argument is to make them
i~
endogenous, first by relating interaction to congest'ion; and secondly, by making assumptions about the development of transport .supply. We define variables and tackle the first (and most traditional) of these issues in the next section. Firs;, however, we make a remark which generalises the fonnal model presented in equations
(1)-(8) above.
Equations (1) and (2) contain a hypothesis about consumers behaviour, (3)-(6) represent the 'production functionf and the way it is perceived, while (7) and (8) are alternative hypotheses about supply-side behaviour. The remark is this:
any of these components can be modified without changing the essence of the main idea - interdependence and non-linearities will produce bifurcations. In
particular, the supply-side equations could be modified in a public sector case to maximise benefits subject to a budget constraint.
3. Making transport-system supply variables explicit
Each element of the (cij) array which appears in the models above should,
i d e a l l y , though
it
i s not always p r a c t i c a b l e , be t a k e n a s a ' g e n e r a l i s e d ' c o s t . Suppose we d i s t i n g u i s h mode by a s u p e r s c r i p t , k . Then t a k e , f o r examplewhere rn i s t h e out-of-pocket money c o s t o f t h e journey,
i
j t h e t r a v e l l i n g t i m e ,t i j
( I l k e;f forms o f ' e x c e s s t time
-
such a s w a i t i n g time f o r p u b l i c t r a n s p o r t , and p lA J
and t h e t e r m i n a l c o s t s a t i and j r e s p e c t i v e l y . For a c a r d r i v e r ,
m . k l w i l l
1 1 j
be marginal c o s t s , such a s p e t r o l and p ( ; 2 ) k
w i l l
be a combination o f parking chargesJ
a t j and any time ( a p p r o p r i a t e l y weighted) spent walking from c a r park t o f i n a l d e s t i n a t i o n . For t h e p u b l i c t r a n s p o r t t r a v e l l e r ,
m w i l l
r e p r e s e n t f a r e s , e ki j
i
jt h e time s p e n t w a i t i n g
-
and s o w i l l be i n v e r s e l y r e l a t e d t o frequency o f s e r v i c e , ( 2 ) k('Ik
w i l l be time s p e n t between home and t h e p u b l i c t r a n s p o r t f a c i l i t y and p .Pi 1
t h e time t o r e a c h t h e f i n a l d e s t i n a t i o n (and each of t h e s e may i n c l u d e all t h e c o s t s o f journeys by s u b s i d i a r y modes). The c o e f f i c i e n t s a l and a: k r e p r e s e n t d i f f e r e n t v a l u e s of time. Corresponding d e f i n i t i o n s could be produced f o r f r e i g h t t r a n s p o r t c o s t s , say cgk f o r type o f goods g by mode k , though i n t h e r e s t of t h e d i s c u s s i o n below we r e s t r i c t o u r s e l v e s t o person t r i p s f o r convenience.
ki
This r e p r e s e n t a t i o n i s o f t h e averaae consumerrs p e r c e p t i o n o f d i f f e r e n t components o f d i s u t i l i t y . This
w i l l
s u f f i c e f o r p r e s e n t purposes, and c l e a r l y b e g i n s t o make t h e t r a n s p o r t supply v a r i a b l e s e x p l i c i t . We can summarise t h e p o s i t i o n reached i n t h i s r e s p e c t a s follows:( i )
c h a r a c t e r i s t i c s o f t h e l i n k s o f t h e road networkw i l l
determine c a r (and o t h e r road v e h i c l e ) t r a v e l t i m e s-
s a yt
i j i f we t a k e k = 1 t o b e t h e c a r mode-
through a r a t h e r complicated procedure which we d e s c r i b e s h o r t l y ;
( i i ) a combination o f network p r o v i s i o n and v e h i c l e p r o v i s i o n and o p e r a t i n g procedures
w i l l
determine p u b l i c t r a n s p o r t t r a v e l t i m e s ,t i j ;
2( i i i )
p e t r o l c o s t s and t a x e s ( s a y )w i l l
determinem
i j'
( i v ) f a r e s p o l i c y w i l l determine
m
2 - i j'
( v ) frequency of s e r v i c e d e c i s i o n s w i l l determine e 2 - i j
'
( v i ) network d e s i g n i n t h e form of spacing o f t h e r o u t e s i n r e l a t i o n t o housing (and t h e d e s i g n of any s u b s i d i a r y modes) w i l l determine
!'I2
and p ( 1 ) 2 .j
'
( v i i ) t h e p r o v i s i o n o f p a r k i n g spaces and t h e p o l i c y f o r c h a r g i n g f o r
it w i l l
(211determine p j
.
The next s t a g e i n t h e argument i s t o show how t h e t r a v e l time element o f g e n e r a l i s e d c o s t r e l a t e s t o network p r o v i s i o n . Consider f i r s t c a r t r a v e l t i m e s ,
ti: .
Consider a s i n g l e (m,n,g) category i n e q u a t i o n ( 1 ) f o r convenience and r e w r i t e t h i s a sY i j
- -
Y i j (ti:, o t h e r v a r i a b l e s ) (10:f o r c a r t r i p s . The d i f f i c u l t y i s t h a t w h i l e Y
i s
obviously a h n c t i o n o f1 i j
i s e q u a l l y obviously a f u n c t i o n o f c o n g e s t i o n l e v e l s which a r e determined t i j , t i j
by t h e Y . . s . The next s t e p , t h e r e f o r e , i s t o a s s i g n t h e flows, c a l c u l a t e d on a
1 J
guessed s e t of t ' I S , t o t h e network. ~ , t
(r,s)
be a l i n k o f t h e network andi
jl e t xrs b e a measure of i t s ' s i z e ' ; l e t QrS be t h e flow on l i n k
( r , s ) .
L e t R?" be t h e s e t of l i n k s which make up t h e b e s t r o u t e fromi
t o j (measured i n1 J
terms o f g e n e r a l i s e d c o s t ) . Then
I J
where y l i s t h e t r a v e l time f o r c a r on l i n k ( r , s ) and rs
and, t o complete t h e c i r c l e ,
where V min
r s
i s
t h e s e t of t r i p i n t e r c h a n g e s ( i , j ) f o r which l i n k ( r , s ) i s on t h e b e s t r o u t e .The key exogenously given t r a n s p o r t supply v a r i a b l e i s now x
.
Given t h i s , I rst h e e q u a t i o n s ( 1 0 ) - ( 1 3 ) can be s o l v e d i t e r a t i v e l y : guess
t i j ,
f i n d Y i j from( l o ) ,
f i n d Qrs from ( 1 3 ) , f i n d yr: from ( 1 2 ) and t h e n ti: from ( 1 1 ) . R e c a l c u l a t e Y
1 i j
from ( 1 0 ) w i t h new t i j , and r e p e a t u n t i l convergence i s achieved. Equation ( 1 2 )
1' i s t h e key e q u a t i o n connecting t h e x - v a r i a b l e s t o t h i s system:
it
i s a time-(or r ssometimes p r e s e n t e d a s speed-) f l o w r e l a t i o n s h i p which obviously depends on t h e p h y s i c a l n a t u r e of t h e l i n k a s r e p r e s e n t e d by x
.
An example o f t h e use of t h e.rs
U . S . speed-flow r e l a t i o n s h i p i n t h e network d e s i g n problem i s provided by Dantzig, Harvey, Landsdowne, Robinson and Maier (1979)
.
A corresponding a n a l y s i s can be c a r r i e d through f o r p u b l i c t r a n s p o r t . I n t h e c a s e o f a r a i l network, t h e c o n g e s t i o n e f f e c t s i n t h e form above can be n e g l e c t e d and r o u t e s and t i m e t a b l e s can be planned d i r e c t l y . This i s a l s o t r u e t o some e x t e n t f o r bus networks, b u t i n c o n s t r u c t i n g t i m e t a b l e s
it w i l l
b e n e c e s s a r y t o t a k e account of t h e impact o f c a r - t r a f f i c c o n g e s t i o n i n i t s i n t e r a c t i o n w i t h t h e b u s e s . There i s a complication a r i s i n g from t h e f a c t t h a t t h e p u b l i c t r a n s p o r t agency p l a n s r o u t e s . Suppose t h e agency o p e r a t e s a s e t of r o u t e s R = 1 , 2 ,....
Then we need t o d e f i n e s e t s analogous t o R~!" and
vmin,
say R ( 2)min1 J rs i j ('Imin and Vrs 9
a n t i c i p a t i n g a l a t e r development i n n o t a t i o n . - It might a l s o be u s e f u l t o d e f i n e ,
-
s a y , (3'') a s t h e s e t of r o u t e s t o be used i n g e t t i n g from i t o j and Iik2)as t h ei j
s e t o f ( i , j ) p a i r s which use r o u t e R a t some s t a g e . A f u r t h e r c o m p l i c a t i o n , o f c o u r s e , i s t h a t a passengers1 r o u t e (PR) from
i
t o j may i n v o l v e more t h a n oneagency r o u t e ( A R ) . The assignment problem i s t h e n analogous t o t h a t f o r c a r u s e r s : t r a v e l l e r s should b e a l l o c a t e d t o t h e most advantageous r o u t e i n g e n e r a l i s e d c o s t t e r m s . Then, n o t o n l y can t h e t r a v e l time be c a l c u l a t e d i n a r e l a t i o n s h i p analogous t o ( 1 1 ) (though t h i s should now i n c l u d e any i n t e r - r o u t e t r a n s f e r t i m e s ) , b u t a l s o t h e a p p r o p r i a t e f a r e c a l c u l a t i o n can be made.
I n t h e d i s c u s s i o n of assignment above,
it
has been assumed t h a t t r a v e l l e r s t a k e t h e l e a s t ( g e n e r a l i s e d ) c o s t r o u t e i n each c a s e . I n p r a c t i c e , t h e r e i s more l i k e l y t o b e some d i s p e r s i o n w i t h second-best, t h i r d - b e s t , and s o on,r o u t e s b e i n g used t o a n e x t e n t determined by t h e g e n e r a l i s e d c o s t d i f f e r e n c e s
between them. For t h e purposes of t h i s paper, we simply n o t e t h a t such procedures can b e i n c o r p o r a t e d w i t h o u t undue d i f f i c u l t y and would n o t b e expected t o change o u r r e s u l t s h e r e i n any s i g n i f i c a n t way.
We can now summarise t h e d i s c u s s i o n s o f a r by n o t i n g t h a t t h e t e c h n i c a l supply- s i d e v a r i a b l e s t o b e s p e c i f i e d a r e {xrS, e
ig,
p11)2, p j 2 ) 2 } . The p r i c i n gv a r i a b l e s t o be determined a r e {m
1
( t h e p e t r o ltax
p a r t ) ,mi:
( f a r e s ) , and p (2.11-i~
jp a r k i n g c h a r g e s ) . The p r i c i n g v a r i a b l e s a r e obviously r e l a t e d t o t h e t e c h n i c a l ones and
w i l l
depend on b r o a d e r a s p e c t s o f p o l i c y , such a s t h e requirement o f t h e p u b l i c t r a n s p o r t agency t o b r e a k even o r n o t . Then, b e a r i n g t h i s summary i n mind, we c a n proceed t o a d i s c u s s i o n o f t h e costs:of supply.A t
l e a s t t h r e e time s c a l e s c a n b e i d e n t i f i e d o v e r whichit
i s r e l e v a n t t o c o n s i d e r c o s t s : t h e v e r y l o n g - l a s t i n g c a p i t a l investment i n networks; t h e s h o r t e r - t e r m c a p i t a l investment i n , f o r example, p u b l i c t r a n s p o r t v e h i c l e s ; and t h er e c u r r e n t running c o s t s o f p a r t i c u l a r systems. Decisions on t h e s e d i f f e r e n t
s c a l e s a r e o f t e n r e l a t i v e l y i n d e p e n d e n t . We need, t h e r e f o r e , t h e c a p i t a l c o s t s f c r b u i l d i n g ( o r e x t e n d i n g ) a l i n k ( r , s ) f o r mode k a t ' s i z e ' x r s
'
s a yFrs
k ( x r S ) . Le:( x ) be t h e r e c u r r e n t c o s t s o f m a i n t a i n i n g and running such a l i n k . I n t h e
'rs rs
p u b l i c t r a n s p o r t c a s e , we have a l r e a d y s e e n t h e s i g n i f i c a n c e o f supply r o u t e s and
it w i l l
b e b e t t e r t o r e l a t e s h o r t e r - t e r m c a p i t a l c o s t s and non-network running Costs t o t h e s e . Let t h e s e begR
( y R ) and gR(yR) r e s p e c t i v e l y f o r p r o v i d i n g ' c a p a c i t y ' y R on t h e Rth r o u t e (though n o t e t h a t t h e s ew i l l
be dependent on network supply a l s o ).
We pursue t h e a n a l y s i s f u r t h e r i n t h e n e x t s u b s e c t i o n w i t h more e x p l i c i t assumptions about t h e form of t h e s e c o s t f u n c t i o n s .F i n a l l y , we need t o d e f i n e , a t l e a s t f o r m a l l y , measures o f b e n e f i t a s s o c i a t e d w i t h a p a r t i c u l a r system s t a t e . For p r i v a t e c a r u s e r s , t h e u s u a l measure i s consumers s u r p l u s (though it c o u l d b e something s i m p l e r , l i k e g e n e r a l i s e d c o s t
s a v i n g s ) . For a r e c e n t s u r v e y o f t h e problems o f b e n e f i t measurement, p a r t i c u l a r l y b u i l d i n g on t h e concept o f consumers s u r p l u s , s e e Jara-Diaz and F r i e s z ( 1 9 8 2 ) .
T h i s c a n a l s o b e a p p l i e d t o p u b l i c t r a n s p o r t u s e r s , o r a l t e r n a t i v e l y i n t h i s c a s e ,
some market mechanism c a n be used i f t h i s i s r e f l e c t e d i n t h e o p e r a t i n g p o l i c y of t h e p u b l i c t r a n s p o r t agency. For t h e p r e s e n t , we c a n l e t B k ( (xrs}, (y
1 ,
1 2
R
(mijl, (mij
1,
( p ! 2 ) 2 ~ ) be t h e b e n e f i t t o u s e r s o f mode k a r i s i n g from supply-side Jd e c i s i o n s and p o l i c i e s and we
w i l l
pursue t h e consequences of more e x p l i c i t s p e c i f i c a t i o n s i n s e c t i o n5
below.4.
Summarv of a formal modelThe d i s c u s s i o n i n s e c t i o n 3 h a s been informal w i t h t h e main i d e a s of each r e l e v a n t submodel b e i n g e x p l a i n e d i n t u r n . It i s now u s e f u l t o draw t h e s e i d e a s t o g e t h e r more f o r m a l l y , t o e x t e n d t h e n o t a t i o n and t o make
it
more c o n s i s t e n t where a p p r o p r i a t e . A s a f i r s t s t e p we show t h e main submodels and t h e i rr e l a t i o n s h i p s i n F i g u r e 3 . T h i s
i s ,
i n e f f e c t , a diagram f o r a comprehensive urban model re-arranged fromi t s
u s u a l form t o a c c e n t u a t e t h e r e l a t i o n s h i p s which a r e r e l e v a n t t o t h e t r a n s p o r t subsystem. m e r e a r e s o many obvious feedback l o o p s i n t h e system t h a t t h e r e i s no c l e a r c y c l e o f c a u s a t i o n . One of t h e i n t e r e s t i n g i s s u e s i n t h e d i s c u s s i o n below i s how t o t a c k l e t h i s q u e s t i o n . The main o u t p u t v a r i a b l e s from each s t a g e a r e shown on t h e diagram and any amendments t o n o t a t i o n w i l l be explained i n t h e summary below.mg i n e q u a t i o n C 1 ) needs t o be broken i n t o Cm,nYg) The i n t e r a c t i o n v a r i a b l e Y i j
c a t e g o r i e s f o r t h e purposes of modelling urban s t r u c t u r e . From a t r a n s p o r t view- p o i n t , however, we a r e more i n t e r e s t e d i n s p l i t by mode, k and w e would a g g r e g a t e o v e r t h e o t h e r i n d i c e s p r o v i d e d . t h e flows can a l l b e t r a n s l a t e d i n t o a p p r o p r i a t e u n i t s
-
s a y passenger c a r u n i t s ( p . c . u . ' s ) . Here, and i n subsequent e q u a t i o n s , w e use a n a s t e r i s k t o denote summation and a l s o , where necessary, a conversion t o a p p r o p r i a t e u n i t s . We assume, t h e r e f o r e , t h a t e q u a t i o n ( 1 ) c a n be w r i t t e n(where we now r e p l a c e c i j by ci;). g
A s a shorthand, we can t h e n make W mg i j a f u n c t i o n o f p r o v i s i o n a t j
, zng
j and t r a v e l c o s t s , c s o t h a t t h e urban s t r u c t u r e submodel c a n b e w r i t t e ni j '
(making a n o t h e r approximation f o r s i m p l i c i t y , t a k i n g c o s t s a s p r o p o r t i o n a l t o s c a l e of p r o v i s i o n , and eimg i s t h e e x p e n d i t u r e p e r t r i p by ( i s m ) o r g a n i s a t i o n s o r people f o r g ) I f we t h e n c o n c e n t r a t e on e q u i l i b r i u m s t a t e s f o r i l l u s t r a t i v e purposes, we
can s a y t h a t
zng
i s t h e s o l u t i o n of jand t h i s i s a s e t of l i n k e d n o n l i n e a r simultaneous equations when a l l t h e s u b s t i t u t i o n s a r e made from ( 1 4 ) - ( 1 7 ) i n t o
(18).
The next s t e p i s t r i p assignment t o t r a n s p o r t networks. Generalised c o s t can be t a k e n a s i n
( l o ) ,
which i s r e p e a t e d h e r e f o r convenience:***k
This makes Y i n
( 1 4 )
dependent ont i j .
k We t h e n proceed w i t h a common i jn o t a t i o n f o r each of t h e two main modes, b u t l a t e r a d a p t t h i s t o recognise t h a t
it
i s f e a s i b l e f o r p u b l i c t r a n s p o r t a g e n c i e s t o p l a n r o u t e s b u t t h a t t h i s i s n o t t h e c a s e f o r road p l a n n e r s . We a l s o have t o c o n f r o n t t h e problem t h a t buses use t h e highway network and u s u a l l y s h a r e c o n g e s t i o n w i t h c a r s .Let Ri;in(k) be t h e s e t o f l i n k s ( r , s ) which form t h e l e a s t g e n e r a l i s e d c o s t r o u t e from
i
t o j f o r mode k . I n t h e p u b l i c t r a n s p o r t c a s e , t h e a l g o r i t h m w i l l have t o be s o designed t h a t t h e s u c c e s s i v e l i n k s a r e on agency r o u t e s , R , and t h a t any t r a n s f e r s a r e f e a s i b l e . Let Vmin(k) be t h e s e t o f i n t e r c h a n g e s( i
, j ) f o rr
swhich ( r , s ) i s on t h e b e s t r o u t e . Then e q u a t i o n s (11)-(13) c a n be r e w r i t t e n :
We have subdivided t h e c a r and p u b l i c t r a n s p o r t l i n k t r a v e l time e q u a t i o n (21) t o show a c o n v e n t i o n a l speed-flow r e l a t i o n s h i p f o r k = 1 ( b u t i n c l u d i n g t o t a l p u b l i c t r a n s p o r t useage a s exogenous where t h i s i s l o a d e d on t o buses s h a r i n g t h e l i n k ) and t o show p u b l i c t r a n s p o r t times depending on l o a d and r o u t e p l a n n i n g ( w i t h
f o r m a l l y p r e s e n t f o r t h e c o n g e s t i o n l i n k ) , though we a l s o show yr: t o be a Qrs
f u n c t i o n of x t h e r e l e v a n t l i n k c a p a c i t y . r s
'
The next a p p r o p r i a t e s t e p i s t o r e c a p on c o s t s and b e n e f i t s . Link c a p i t a l c o s t s a r e
I
w i t h r e c u r r e n t c o s t s
I n t h e o r y ,
it
would b e p o s s i b l e t o combine t h e s e i n t o , s a y , a n n u a l c o s t s by t h e u s e o f a n a p p r o p r i a t e d i s c o u n t i n g r a t e , b u t network l i n k s t y p i c a l l y have such a l o n g l i f e t h a tit
seems b e s t t o keep t h e s e s e p a r a t e . It i s c o n s i d e r e d t h a t a l l o t h e r c a r o p e r a t i n g c o s t s a r e b o r n e by t h e u s e r ; and t h a t p u b l i c t r a n s p o r t r o u t e - r u n n i n g c o s t s c a n , i n t h i s c a s e , be combined i n t o a simple r e c u r r e n t f i g u r ewhere y i s t h e l e v e l of a c t i v i t y on r o u t e R . We assume t h a t w a i t i n g t i m e i s
R
a l s o a f u n c t i o n o f t h e y R t s :
Another t a s k which we n e g l e c t f o r t h e t i m e b e i n g , b u t which can e a s i l y b e r e i n t r o d u c e d , i s t h e f i x i n g o f t h e a r r a y o f p u b l i c t r a n s p o r t f a r e s [mi:}.
F o r s i m p l i c i t y , we n e g l e c t p e t r o l t a x e s and a l l o r i g i n c o s t s , and assume t h a t p a r k i n g c h a r g e s ,
,
a r e p o l i c y v a r i a b l e s . The v a r i a b l e s t o b e determined i n t h e t r a n s p o r t s u p p l y model, y e t t o b e s p e c i f i e d , a r e t h e nA The s p e c i f i c a t i o n o f t h i s model
w i l l
b e t h e main t a s k o f s e c t i o n5.
5 .
The e v o l u t i o n o f urban s p a t i a l s t r u c t u r e , i n c l u d i n g t r a n s p o r t s y s t e m s We have a l r e a d y s e e n , i n F i g u r e 2 i n r e l a t i o n t o F i g u r e 1, t h a t t r a n s p o r t s y s t e m s have a major impact on u r b a n s t r u c t u r e . It i s p o s s i b l e , and i n t e r e s t i n g , t o e x p l o r e such impacts more s y s t e m a t i c a l l y . ' I n F i g u r e 2, we e x p l o r e d , i n e f f e c t , t h e impact o f a l t e r n a t i v e u n d e r l y i n g networks i n v o l v i n g d i f f e r e n t d e g r e e s o fc e n t r a l o r i e n t a t i o n . Formally, t h i s c o u l d b e c o n s i d e r e d t o come a b o u t from xrs c h a n g e s which g e n e r a t e t h e kinds of c i j change which were i n v e s t i g a t e d . I n t h a t k i n d o f s i n g l e mode s i t u a t i o n , we c o u l d a l s o c o n s i d e r t h e i m p a c t s on s t r u c t u r e o f o t h e r network c o n f i g u r a t i o n s combined w i t h a l t e r n a t i v e p o l i c i e s on p a r k i n g c h a r g e s . I t would b e more i n t e r e s t i n g , o f c o u r s e , t o develop a two-mode model and t o e x p l o r e t h e e f f e c t s of a l t e r n a t i v e p o l i c i e s f o r t h e p u b l i c t r a n s p o r t mode, b o t h i n terms of networks and p r i c i n g . These e x p l o r a t i o n s c o u l d b e c a r r i e d o u t f o r a s t r u c t u r e i n a s i n g l e system
-
s u c h a s a r e t a i l s y s t e m , a s used f o r F i g u r e s 1 and 2 , o r f o r a more comprehensive b a s e , such as a m o d i f i e d Lowry model. T h i s a n a l y s i s c o u l d b e s h a r p e n e d f u r t h e r by t h e s e a r c h f o r c r i t i c a l v a l u e s o f t r a n s p o r t s y s t e m p a r a m e t e r s which demarcate d i f f e r e n t forms o f urban s t r u c t u r e . These -e x p l o r a t i o n s w i l l a l l b e r e p o r t e d i n a l a t e r p a p e r . For p r e s e n t p u r p o s e s ,
it
i s more i m p o r t a n t t o t u r n t o new t h e o r e t i c a l q u e s t i o n s and f i r s t l o o k a t t h e q u e s t i o n i n r e v e r s e-
how do t r a n s p o r t s y s t e m s e v o l v e ?-
and t h e n t o l o o k a t t h e j o i n te v o l u t i o n o f u r b a n s t r u c t u r e and t r a n s p o r t systems.
To make p r o g r e s s , t h e model g i v e n i n s e c t i o n
4
has t o be extended by t h e d e t a i l e d s p e c i f i c a t i o n of c o s t and b e n e f i t f u n c t i o n s t o g e t h e r w i t h a model- g e n e r a t i n g h y p o t h e s i s about how t h e t r a n s p o r t supply agency works. To f i x i d e a s , c o n s i d e r t h e s i n g l e ( c a r ) mode c a s e and suppose t h a t r e c u r r e n t c o s t s can b e n e g l e c t e d . I t a l s o makes s e n s e t o assume t h a t l i n k s a r e b e i n g added t o t h e network i n c r e m e n t a l l y from a g i v e n s i t u a t i o n .We have t o imagine t h e e x i s t e n c e o f a c o s t f u n c t i o n f & x r s ) f o r each p o s s i b l e l i n k ( r , s ) . T h i s w i l l v a r y most o b v i o u s l y w i t h s i z e , xrs, b u t a l s o w i t h l o c a l topography and l a n d u s e . I t w i l l b e lower i n f l a t t e r rural c o u n t r y , a v o i d i n g t h e need f o r b r i d g e s , a v o i d i n g d e m o l i t i o n o f b u i l d i n g s and s o o n . The s e t o f l' ' s r e p r e s e n t s a d e t a i l e d s t a t e m e n t o f network b u i l d i n g p o s s i b i l i t i e s f o r
r s
a n a r e a , and of c o u r s e
it
i s c o n t i n u a l l y changing as t h e a r e a d e v e l o p s . To make t h i s e x p l i c i t , l e t us u s e t h e l a b e lt
t o r e p r e s e n t a t i m e p e r i o d , s a y t t o t + 1, and l e trr;
( x r S ) b e t h e c o s t o f b u i l d i n g a l i n k xrz i n t h a t p e r i o d .t
L e t4.
b e t h e s e t o f l i n k s a l r e a d y e x i s t i n g up t o t h a t p e r i o d and
C~
a r e p r e s e n t a t i o nof t h e s e t o f l a n d u s e s . Then, f o r m a l l y , l'r s c o u l d b e w r i t t e n
rr;
( X rs'-
N~ 3-
L ~ )t o show i t s dependence on t h e s e q u a n t i t i e s .
The a g g r e g a t e b e n e f i t f u n c t i o n , B ~ , w i l l b e t h e b e n e f i t s d e r i v i n g from t h e b u i l d i n g o f a s e t o f l i n k s {xrS1 i n t i m e p e r i o d
t t
and t h i s w i l l a l s o depend on t h e e x i s t i n g c o n f i g u r a t i o n :t t t t
B ( {xrS1,.N
, t ,
{ Y i j l ) . We a l s o showit
as dependent on t h e f l o w s , {Y; ;1.
T h i s c o u l d b e measured, f o r example, as t h eA d
a d d i t i o n a l consumers s u r p l u s a c c r u i n g t o t r a v e l l e r s from t h e b u i l d i n g of {xrz}.
We now need t o s p e c i f y a n a p p r o p r i a t e p o l i c y f o r t h e network b u i l d i n g agency.
Suppose
it
w i s h e s t o maximise b e n e f i t s , s u i t a b l y d i s c o u n t e d , o v e r a s e r i e s o f p e r i o d s s u b j e c t t o a t o t a l budget c o n s t r a i n t o p e r a t i n g i n each p e r i o d . T h i s c a n b e r e p r e s e n t e d asI t
Max B = C B ( { x r z ~ , g , L t {yi:}) / ( 1 + ~ ) ~
{ X t = O
r
ss u b j e c t t o
C t
t
rrs
( x r s )5
r s twhere R
i s
a s u i t a b l e d i s c o u n t r a t e a n d Kt
i s t h e t o t a l budget f o r p e r i o dt .
S i n c e t h e b e n e f i t f u n c t i o n would i n v o l v e a knowledge o f a l l t h e f l o w s , Y i j ,t
t h e n t h e problem r e p r e s e n t e d by ( 2 8 ) and ( 2 9 ) i m p l i c i t l y i n v o l v e s t h e f u l l network-c o n s t r a i n e d t r a n s p o r t model a s a s e t o f c o n s t r a i n t s .
I t i s i n t e r e s t i n g t o compare t h i s f o r m u l a t i o n w i t h t h a t f o r t h e c a l c u l a t i o n of {z?} d e s c r i b e d i n s e c t i o n 2 . I n t h e l a t t e r , we f o c u s on
zng
f o r each zoneJ
j
j , and t h e number of p o s s i b l e c o n f i g u r a t i o n s , a l t h o u g h v e r y g r e a t , a r e much l e s s t h a n involved i n t h e network problem as formulated above. I n p r i n c i p l e , t h e r e i s a v e r y l a r g e number o f x c o m b i n a t i o n s . I n t h e problem a s formulated i n a n
rs
i n c r e m e n t a l model, x can b e t a k e n a s v a r y i n g c o n t i n u o u s l y o r d i s c r e t e l y . A s rs
we noted e a r l i e r , b o t h a p p r o a c h e s a r e used i n t h e l i t e r a t u r e . I n t h e o t h e r
problem t h e s i z e a t a l o c a t i o n c a n more e a s i l y b e t r e a t e d as a continuous v a r i a b l e over t i m e . There a r e a l s o , of c o u r s e , fewer v a r i a b l e s . I t i s i n t e r e s t i n g t o a t t e m p t t o r e d e f i n e t h e network problem t o make it more l i k e t h e f a c i l i t y - c e n t r e problem, and t h e n a l t e r n a t i v e m o d e l l i n g f o r m u l a t i o n s c o u l d b e used t o r e p l a c e
( 2 8 ) and ( 2 9 ) which may b e more amenable t o a n a l y s i s . T h i s i n v o l v e s s e e k i n g new ways o f making a p p r o x i m a t i o n s .
One p o s s i b i l i t y i s t o f i x t h e main nodes o f t h e network (perhaps t h e
c e n t r o i d s of o r i g i n and d e s t i n a t i o n s ) , t o connect each node t o a number o f n e a r neighbours and t h u s t o d e f i n e what used t o be c a l l e d a ' s p i d e r network' i n t h e e a r l y days of t r a n s p o r t p l a n n i n g when it was d i f f i c u l t t o h a n d l e l a r g e networks.
An example i s shown i n F i g u r e
4 .
The r ' s and s ' s , t h e n become i t s and j ' s andm i n t .
Ri
j i s t h e s e t o fi
I s and j l ' s which form t h e b e s t r o u t e betweeni
and j , and s o on. I n F i g u r e4 ,
minR1
,14
might b e ( 1 , 2 , 7 , 8 , 9 , 1 4 ) f o r example. We c o u l d a l s o l e t S b e t h e s e t o f ( i , j ) p a i r s f o r which n o t i o n a l s p i d e r l i n k s have been d e f i n e d The problem c o u l d t h e n become one o f d e t e r m i n i n g t h e c a p a c i t y o f e a c h such l i n k , xi
j-
S e a r i n g i n mind t h a tit
c o u l d , o f c o u r s e , b e z e r o . We c o u l d t h e n u s e a d i f f e r e n c e e q u a t i o n f o r m u l a t i o n ( c h o o s i n g t h i s r a t h e r t h a n a d c f f e r e n t i a le q u a t i o n because
it
seems more s e n s i b l e t o work i n terms o f a n n u a l , s a y , b e n e f i t s and c o s t s ) :where Ax t i s t h e increment i n c a p a c i t y o v e r t h e p e r i o d t t o
t + l ,
A B t h e g a i n ~ ~i
ji n b e n e f i t s from t h i s increment and
r i
jits
c o s t , R is a s u i t a b l e discount r a t e . I f t h e t o t a l budget was exceeded, t h e problem c o u l d b e r e r u n w i t h R s e t h i g h e r . The e q u i l i b r i u m c o n d i t i o n would b et o b e s o l v e d f o r
Axij
a f t e r a l l t h e r e l e v a n t s u b s t i t u t i o n s have been made.Let u s c a l l t h e models we have developed so f a r SML ( s u p p l y model 1) and SM2 r e s p e c t i v e l y . I n t h e f i r s t c a s e , probably t h e o n l y r e a s o n a b l e way t o proceed i s t o d e f i n e a p o o l o f p o s s i b l e p r o j e c t s , ( x
.
But even t h e n ,r s
t h e r e w i l l b e a l a r g e number o f p o s s i b l e combinations of t h e s e which
w i l l
c o m p l i c a t e t h e c a l c u l a t i o n o f t h e b e n e f i t f u n c t i o n i n each c a s e . I n t h e second c a s e , it i s i m p o r t a n t t o n o t e t h a t a h i e r a r c h i c a l a n a l y s i s i s i m p l i e d . The increment Axij, o f c a p a c i t y t o b e added t o a n o t i o n a l l i n k betweeni
and j h a s , a t a lower and s u b s e q u e n t s t a g e , t o b e transformed i n t o ' s e n s i b l e ' a d d i t i o n s t o t h e a c t u a l network. T h i s i n t u r n i m p l i e s a f e e d b a c k between t h e two l e v e l s on c o s t s : r e a l c o s t s a t t h e lower, network, s c a l e , have t o b e t r a n s l a t e d i n t o ( approximate) f u n c t i o n s a t t h e h i g h e r ( i - j ) s c a l e .Numerical e x p e r i m e n t s would b e p o s s i b l e w i t h e i t h e r SMl o r SM2. However, a b e t t e r chance o f a n a l y t i c a l i n s i g h t s seem t o b e o f f e r e d by SM2 and s o we proceed w i t h t h a t . We s p e l l o u t t h e s e c t i o n
4
model f o r t h e single-mode assumptions we have been working w i t h and e x p l o r e t h e consequences. Wes i m p l i f y f u r t h e r by assuming a g i v e n demand s e c t o r which i s p r o p o r t i o n a l t o t h e p o p u l a t i o n d i s t r i b u t i o n P i a n d a s i n g l e s e r v i c e s e c t o r (W.1 which
i s
d e t e r m i n e dJ
w i t h i n t h e model. T h i s c a n b e s e e n as t h e f i r s t s t e p t o w a r d s t h e a n a l y s i s o f t h e j o i n t e v o l u t i o n of t r a n s p o r t systems and urban s t r u c t u r e .
The demand f o r t r a n s p o r t i s Y s a y , given by i j
'
e . is t h e p e r c a p i t a demand f o r s e r v i c e s measured i n u n i t s o f t r i p s p e r head o f
1
p o p u l a t i o n . We a l s o assume t h a t W:
i s
t h e s i z e o f f a c i l i t i e s a t j a n d t h a t t h i sJ
can r e p r e s e n t a t t r a c t i v e n e s s . Then revenue a t j
i s
g i v e n b ywhere f i s a c o n s t a n t t r a n s l a t i n g t r i p u n i t s i n t o money u n i t s a n d t h e c o s t s a r e
f o r s u i t a b l e c o n s t a n t s ,
k j
.
W . i s t h e s o l u t i o n o f t h e e q u i l i b r i u m c o n d i t i o n Jc i j i n t h e s e e q u a t i o n s i s t a k e n as a g e n e r a l i s e d c o s t :
i j = i j +
(?-I
where a i s t h e v a l u e o f t i m e . It
i s
determined t h r o u g h t h e a s s i g n m e n t model a sf o l l o w s : l e t R i j min b e t h e s e t o f spider-modes which form t h e s h o r t e s t p a t h
I t
from
i
t o j ; l e tVIf?
b e t h e s e t o f( i
, j ) t r i p b u n d l e s which u s e t h e l i n k1 J
i , .
Let S b e t h e s e t o f l i n k s ( i , j ).
Let y b e t h e t r a v e l t i m e on ai
jl i n k . Then
C a p i t a l c o s t s a r e g i v e n by
r i j - - r i j
( b x i j )The b e n e f i t f u n c t i o n i s
The {Ax..) t o b e chosen a r e t h e n t h e s o l u t i o n s o f
1 J
Three f u n c t i o n s remain t o b e s p e c i f i e d t o complete t h e model:
y i j i n e q u a t i o n
(39) , ri
i n( 4 1 )
and dBij i n ( 4 2 ) . I n t h e c a s e o f t h e f i r s t two, we c a n show what we would e x p e c t i n g r a p h i c a l form, and t h i s i s done i n F i g u r e5.
I n t h e c a s e o f AB we can u s e a r e s u l t f i r s t d e r i v e d i n t h e c o n t e x t o f random i j '
u t i l i t y t h e o r y t h a t change i n consumers s u r p l u s i s : ( 1 )
.-Beij AB =
-
1 l o g i jB
( 0 ).-Beij i j
where t h e s u p e r s c r i p t s ( 0 ) and ( 1 ) on t h e c o s t t e r m s d e n o t e ' b e f o r e ' and ' a f t e r ' r e s p e c t i v e l y . There i s a d i f f i c u l t y t h a t t h i s r e l a t e s t o t h e whole system. We would have t o d e f i n e ABij a s t h e v a l u e o f AB r e s u l t i n g from a change Axij w i t h a l l o t h e r A i j f s z e r o . T h i s , as we w i l l s e e l a t e r , exposes a v e r s i o n o f t h e
' b a c k c l o t h f problem
-
a s i n Wilson and C l a r k e ( 1 9 7 9 ) . ThusL e t us now c y c l e t h r o u g h a number o f imaginary r u n s o f t h e model. Suppose t h e p o p u l a t i o n i s growing and s p r e a d i n g and t h i s i s r e f l e c t e d i n {P.); l e t
1
{ e i ) b e growing even more r a p i d l y t o s i g n i f y a p e r i o d o f i n c r e a s i n g c a r ownership;
suppose t h e
m
a r e f i x e d ; l e t a b e i n c r e a s i n g and 6 d e c r e a s i n g ; assume f i x e d i jv a l u e s o f any o t h e r c o n s t a n t s and known f u n c t i o n a l forms where a p p r o p r i a t e . Assume a n i n i t i a l v a l u e f o r t h e t r a v e l c o s t a r r a y {c
.
The c r u c i a l d e c i s i o ni
jt o t a k e i s t h e n how t o ' b r e a k i n t o ' t h e model and t o c y c l e t h r o u g h
it
( c f . F i g u r e 3 ) . We a d o p t o n e i l l u s t r a t i o n h e r e b u t r e c o g n i s e t h a t t h e r e i s a d i f f i c u l t r e s e a r c h q u e s t i o n i n v o l v e d . We i d e n t i f y t h e major s t e p s i n s t u r n : (1) S o l v e e q u a t i o n s ( 3 2 ) - ( 3 6 ) f o r
{Y.
. ) and {W.),
p r o b a b l y u s i n g a n i n c r e m e n t a l1J J
p r o c e d u r e f o r {W.)
-
meaning t h a tit
cannot d e c r e a s e ( o r c a n o n l y d e c r e a s e by Ja p r o p o r t i o n ) from one t i m e p e r i o d t o t h e n e x t .
( 2 ) S o l v e e q u a t i o n s ( 3 7 ) - ( 4 0 ) w i t h Axij = 0 i n i t i a l l y . C a l c u l a t e Qij from
( 4 0 1 , y i j from ( 3 9 ) and
t i j
from ( 3 8 ) . Then r e t u r n t o s t e p ( 1 ) w i t h new c f r o n i j ( 3 7 ) and r e c y c l e between ( 1 ) a n d ( 2 ) u n t i l e q u i l i b r i u m i s a c h i e v e d .( 3 ) Take a t r i a l s e t {Ax. . ) , r e r u n s t e p s ( 1 ) and ( 2 ) t o o b t a i n a new { c i j
1,
s a y1J
{ c i i l ) ) . C a l c u l a t e AB from ( 4 4 ) , and c a r r y o u t s e p a r a t e r u n s t o e s t i m a t e AB i j
'
t h e c o n t r i b u t i o n t o AB from Ax
his
c o u l d b e done, a p p r o x i m a t e l y by r u n n i n g i j 't h e model system f o r t h e Ax. change o n l y w i t h a l l o t h e r Axilj, = 0 ) .
1 j Use ( 4 3 )
t o o b t a i n a n R. from
1 j
and t h e n s c a l e Ax s o t h a t
i
jnew
-
Axi o l d R
-
A x i j. a
Recycle w i t h t h e s e new Axij from s t e p ( 2 ) . T h i s i s analogous t o t h e b a l a n c i n g p r o c e d u r e f o r c a l c u l a t i n g {W.). 1
We can now s e e w h e t h e r t h e model would do what we e x p e c t
it
t o do by assuming t h e s c e n a r i o s k e t c h e d a t t h e b e g i n n i n g o f t h i s d i s c u s s i o n . Let u s add t h a t i n one p a r t i c u l a r suburban zone t h e r e i s a s u b s t a n t i a l a d d i t i o n a l i n c r e a s e i n p o p u l a t i o n . S t e p s ( 1 ) and ( 2 ) o f t h e model would produce new c e n t r e s and measures o f c o n g e s t i o n . A p o s s i b l e b e f o r e and a f t e r s i t u a t i o n i s s k e t c h e d i n F i g u r e6.
I n s t e p ( 3 ) , t h e p r o c e d u r e s h o u l d t h e n g e n e r a t e h i g h r a t e s o f r e t u r n on l i n k s which r e l i e v e t h i s c o n g e s t i o n . So t h e model s h o u l d d e a l c o r r e c t l y w i t h b a s i c development.T h i s s k e t c h does l e a d t o some new n o t i o n s , however. F i r s t , t h e r e s h o u l d be z minimum a v a i l a b l e t r a n s p o r t c a p a c i t y t o anywhere i n t h e r e g i o n ( r e p r e s e n t i n g
1
<U c o u n t r y l a n e s o r w h a t e v e r ) t o a l l o w f o r t h e p o s s i b i l i t y o f new {W.) devel'opment.
1
S e c o n d l y ,
it
i s c l e a r t h a t a highway agency may b e o t h e r t h a n s i m p l y r e s p o n s i v e i n t h e manner assumed h e r e . P a r t i c u l a r A x i j l s may b e implemented t o f a c i l i t a t e development, f o r example; and t h i s would have a n a p p r o p r i a t e impact on W ' sj
t h r o u g h s t e p (1) above. T h i r d l y ,
it
i s c l e a r t h a t i n any r e a l p a r t i c u l a r c a s e , much work would have t o b e done on t h e d e t a i l e d s p e c i f i c a t i o n o f p a r a m e t e r s and f u n c t i o n a l forms. F i n a l l y ,it
i s c l e a r t h a t t h e model, which i s a l r e a d yc o m p l i c a t e d t h r o u g h s t e p
( 3 ) , w i l l
b e immensely more c o m p l i c a t e d when r e a l i s t i c d e t a i l i s added back i n t oi t .
6.
Concluding comments : an ongoing r e s e a r c h programmeI t
i s
a p p r o p r i a t e t o conclude w i t h comments under t h r e e main h e a d i n g s about t h e ways i n which t h e s e i d e a s can b e t a k e n forward. F i r s t , we c o n s i d e r t h e f u r t h e r t h e o r e t i c a l advances which a r e n e c e s s a r y ; s e c o n d l y , we l o o k a t how t o make t h e models more r e a l i s t i c ; and t h i r d l y , we examine t h e p o t e n t i a l u t i l i t y of t h e s e models.There a r e two main a s p e c t s t o t h e f i r s t h e a d i n g . F i r s t ,
it
would b e v a l u a b l e i f t h e models could b e made s u f f i c i e n t l y e x p l i c i t t h a tit
w a s p o s s i b l e t o c a r r y o u t t h e k i n d of a n a l y s i s p r e s e n t e d i n H a r r i s and Wilson (1978) f o r r e t a i l s y s t e m s . That i s , t o e x p l o r e a n a l y t i c a l l y ' t h e n a t u r e o f ABijand r i j as f u n c t i o n s of Ax
-
and t h e e q u i v a l e n t o f t h e s e i n more complex f o r m u l a t i o n s-
a721i
jt o u s e t h i s as t h e b a s i s o f g a i n i n g i n s i g h t i n t o t h e n a t u r e o f t h e t r a n s i t i o n s
fr-ln
one k i n d o f e q u i l i b r i u m s t a t e t o a n o t h e r . We can be c e r t a i n t h a t t h e d e g r e e o f i n t e r d e p e n d e n c e and n o n - l i n e a r i t i e s i n v o l v e dw i l l
l e a d t o jumps, f o r example.S e c o n d l y ,
it
w i l l be n e c e s s a r y t o be more e x p l i c i t a b o u t t h e l i n k i n g o f t h e two l e v e l s i n t h e h i e r a r c h y-
t h e c a p a c i t i e s on t h e s p i d e r network a s a morerealistic
network a t a f i n e r l e v e l o f r e s o l u t i o n . As u s u a l ,it
may w e l l b e t h a t s i g n i f i c ~ t p r o g r e s s would be p o s s i b l e t h r o u g h t h e c a r r y i n g o u t o f n u m e r i c a l e x p e r i m e n t s oni d e a l i s e d systems.
The second main heading r e f e r r e d t o t h e t a s k o f b u i l d i n g more r e a l i s t i c m o 6 d s . The c e n t r e of t h i s i s t h e s p e c i f i c a t i o n o f t h e t h r e e main f u n c t i o n s which w i l l d e t e r m i n e much o f t h e outcome: on c o s t s , on b e n e f i t s and on speed-flow r e l a t i o r r ( o r , i n t h e l a s t c a s e , t r a v e l t i m e
-
investment r e l a t i o n s ) . A f u r t h e r s t e p i n v o l v e s d e v e l o p i n g t h e model i n r e a l i s t i c complexity-
f o r example, t o h a n d l e multi-modal s i t u a t i o n s . I n t h i s c a s e ,it w i l l
b e p o s s i b l e t o f o r m u l a t e t h e mo&l i n s u c h a way t h a t t h e f o c u s c a n b e on t h e e f f e c t o f p a r t i c u l a r v a r i a b l e s-
s a y p u b l i c t r a n s p o r t f a r e s o r p a r k i n g p r o v i s i o n and c h a r g i n g-
and t h i s would t h e nc o n n e c t more d i r e c t l y t o some o f t h e p r e v i o u s l y c i t e d l i t e r a t u r e on t r a n s p o r t s q ? l y . It may a l s o b e p o s s i b l e t o develop t h e c o s t f u n c t i o n s i n such a way t h a t it i s
p o s s i b l e t o r e p r e s e n t l i n k s o f r o a d i n d i f f e r e n t l e v e l s o f a h i e r a r c h y .
T h i r d l y , we need b r i e f l y t o review t h e u t i l i t y of t h e approach. There a r e two a s p e c t s t o t h i s . F i r s t , it would b e i n t e r e s t i n g t o l o o k a t t h e l o n g r u n h i s t o r y o f t h e e v o l u t i o n o f network s t r u c t u r e s i n p a r t i c u l a r p l a c e s and t o a t t e m p t t o i n t e r p r e t t h i s u s i n g t h e models and c o n c e p t s developed. Secondly, a s w i t h t h e more t r a d i t i o n a l l i t e r a t u r e on t r a n s p o r t s u p p l y , it s h o u l d be p o s s i b l e t o f i n d ways of u s i n g t h e models i n a p o l i c y c o n t e x t . The f i r s t o f t h e s e t a s k s may be i n a t h e o r e t i c a l s e n s e e a s i e r t h a n t h e second, because
f o r e c a s t i n g , f o r example, i s d i f f i c u l t b e c a u s e of t h e i n f l u e n c e o f p e r t u r b a t i o n s and h i s t o r i c a l a c c i d e n t s . But it s h o u l d be p o s s i b l e t o s e e k ' b e s t a d d i t i o n s ' a n d a l s o t o i n v e s t i g a t e t h e s t a b i l i t y of s t r u c t u r e s .
FIGURE 1 : Retail Patterns for. various Alpha and Beta Values
FIGURE
2 : Effect of Decreasing Travel Cost to City CenterSpatial interaction:
the demand for transport
\I / \
\I
Land use structureAssignment to networks
I
I I
T r a n s ~ o r t user costsW
P I K\
Transport supplier costs \
and revenues /
I I
Operating policies
. Investment decisions
\/
FIGURE 3 : Elements o f a general model, emphasising transport
FIGURE b : A ' s p i d e r ' n e t w o r k
FIGURE
5
: (a) Time-flow r e l a t i o n s h i p s (b) c a p i t a l c o s t s v s s c a l e of investmentX New Centres
Congestion Levels
FIGURE
6
: Sources o f demand f o r new t r a n s p o r t i n f r a s t r u c t u r eREFERENCES
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