Optimization Models of Transportation Network Improvement: Review and Future Prospects

OPTIMIZATIOEI MODELS OF TRANSPORTATION NETWORK IMPROVEMENT: REVIEW AND FUTURE PROPSECTS

Ross

D.

llacKinnon

April 1976

Research Memoranda are interim reports on research being con- ducted by the International Institute for Applied Systems Analysis, and as such receive only limited scientific review. Views or opin- ions contained herein do not necessarily represent those o f the Institute or o f the National Member Organizations supporting the Institute.

T h i s p a p e r i s t h e s e c o n d i n a s e r i e s on " R e g i o n a l

Development and Land-Use Models". The p u r p o s e o f t h i s s e r i e s i s t o c o n s i d e r t h e a p p l i c a t i o n o f o p t i m i z i n g and b e h a v i o u r a l l a n d - u s e m o d e l s a s t o o l s i n t h e s t u d y o f r e g i o n a l d e v e l o p m e n t . The p r e s e n t p a p e r c o n s i d e r s t h e p r o b l e m o f d e s i g n i n g a n

o p t i m a l r e g i o n a l t r a n s p o r t a t i o n n e t w o r k t a k i n g i n t o a c c o u n t t h e s p a t i a l l a n d - u s e p a t t e r n . The p a p e r r e v i e w s c u r r e n t m o d e l s and i d e n t i f i e s w e a k n e s s e s i n them. I n t h e c o n t e x t o f t h i s

s e r i e s , t h e main v a l u e o f t h i s p a p e r , a s a complement t o ( I ) , i s i n i d e n t i f y i n g models w h i c h m i g h t b e combined i n t o i n t e g r a t e d l a n d - u s e and t r a n s p o r t a t i o n d e s i g n models.

J . R .

I.!iron March, 1976

Pawers i n t h e R e a i o n a l Development a n d Land-Use Models S e r i e s

1. John

R .

M i r o n , " R e g i o n a l Development and Land-Use Models:

An

Overview o f O p t i m i z a t i o n M e t h o d o l o g y " , PA-76-27, A p r i l , 1 9 7 6 .

2. Ross

D .

MacKinnon, " O p t i m i z a t i o n Models o f T r a n s p o r t a t i o n

Network Improvement: Review and' F u t u r e P r o s p e c t s " ,

RJl-76-23, A p r i l , 1976.

O p t i m i z a t i o n Models o f T r a n s p o r t a t i o n Network Improvements: Review and F u t u r e

P r o s p e c t s

( A b s t r a c t )

The paper b r i e f l y reviews t h e a l t e r n a t i v e approaches t o s p a t i a l improvements i n t r a n s p o r t a t i o n networks from t h e e a r l y l i n e a r programming a t t e m p t s t o t h e more r e c e n t d i s c r e t e programming a p p r o a c h e s ; t h e more a n a l y t i c a l g e o m e t r i c a l and o p t i m a l c o n t r o l methods; narrow c o s t mini- m i z a t i o n models and t h e more comprehensive a t t e m p t s t o i n c o r p o r a t e a broad r a n g e of economicand s o c i a l i m p a c t s . F i n a l l y some p e r s o n a l remarks a r e made c o n c e r n i n g t h e most promising a r e a s o f f u t u r e r e s e a r c h w i t h r e s p e c t t o p r a c t i c a l r e l e v a n c e , c o m p u t a t i o n a l f e a s i b i l i t y and t h e o r e t i c a l i n t e r e s t .

Ross D. MacKinnon U n i v e r s i t y of Toronto

OPTIMIZATION MODELS OF TRANSPORTATION NETWORK IMPROVEMENTS : REVIEW AND FLJTURE PROSPECTS*

I n v e s t m e n t s i n t h e t r a n s p o r t a t i o n s y s t e m r e p r e s e n t a m a j o r component ( f r e q u e n t l y t h e l a r g e s t s h a r e ) o f a c o u n t r y ' s p u b l i c c a p i t a l e x p e n d i t u r e s . These i n v e s t m e n t d e c i s i o n s c a n , o f c o u r s e , s t r o n g l y i n f l u e n c e how t h e t r a n s p o r t a t i o n s y s t e m w i l l b e u s e d f o r d e c a d e s t o come. I n . p a r t i c u l a r , t r a n s p o r t a t i o n i n v e s t m e n t s may a f f e c t s u b s e q u e n t d e c i s i o n s o f i n d i v i d u a l s and f i r m s r e g a r d i n g t h e t y p e and volume o f t h e i r a c t i v i t i e s and u l t i m a t e l y even t h e i r l o c a t i o n s . Thus, n o t o n l y t h e m a g n i t u d e o f c a p i t a l e x p e n d i t u r e s i n t r a n s p o r t a t i o n , b u t a l s o t h e i r s t r a t e g i c economic, s o c i a l and e n v i r o n - m e n t a l r a m i f i c a t i o n s would seem t o make t h e s t u d y o f them a p r a c t i c a l l y

i m p o r t a n t and i n t e l l e c t u a l l y c h a l l e n g i n g a r e a o f r e s e a r c h . It i s t h e p u r p o s e o f t h i s p a p e r t o r e v i e w t h e a p p l i c a t i o n s o f o p t i m i z a t i o n methods t o t h i s r e s e a r c h q u e s t i o n and make some t e n t a t i v e p r e s c r i p t i o n s r e g a r d i n g t h e most p r o m i s i n g a n d r e l e v a n t a r e a s o f f u t u r e r e s e a r c h . (A r e v i e w o f r e s e a r c h i s p a r t i c u l a r l y a p p r o p r i a t e i n view o f t h e many d i s c i p l i n e s and i n t e r e s t g r o u p s making c o n t r i b u t i o n s t o t h i s a r e a o f r e s e a r c h . They i n c l u d e e n g i n e e r s

( p r i m a r i l y c i v i l , e l e c t r i c a l and i n d u s t r i a l ) , o p e r a t i o n s r e s e a r c h e r s ,

e c o n o m i s t s , m a t h e m a t i c i a n s , g e o g r a p h e r s , r e g i o n a l s c i e n t i s t s , u r b a n p l a n n e r s , and p e r h a p s o t h e r s . Although t h e volume o f r e s e a r c h i s managable, i t s f a r - f l u n g d i s t r i b u t i o n makes t h e t a s k o f k e e p i n g informed o f c u r r e n t d e v e l o p - m e n t s a d i f f i c u l t o n e . )

*

The p r e s e n t a t i o n o f t h i s p a p e r a t t h e IFAC Workshop on O p t i m i z a t i o n A p p l i e d t o T r a n s p o r t a t i o n was made p o s s i b l e by g r a n t s from t h e 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 Systems A n a l y s i s and t h e Canadian Committee f o r IIASA ( I n s t i t u t e f o r P u b l i c P o l i c y R e s e a r c h i n M o n t r e a l ) . Some o f t h e r e v i e w s e c t i o n s h a v e b e e n a d a p t e d f r o m a t e x t i n t r a n s p o r t a t i o n s y s t e m s a n a l y s i s b e i n g w r i t t e n by my- s e l f and P r o f e s s o r G. B a r b e r of N o r t h w e s t e r n U n i v e r s i t y . H i s c o n t r i b u t i o n s t o t h i s p a p e r , b o t h d i r e c t and i n d i r e c t , a r e g r a t e f u l l y acknowledged.

+

Department o f Geography, U n i v e r s i t y of T o r o n t o

The s t u d y o f t r a n s p o r t a t i o n i s p r e - e r n i n a n t l y a s p a t i a l p r o b l e m . The u s e o f a t r a n s p o r t a t i o n s y s t e m m a n i f e s t s i t s e l f as a set o f s p a t i a l f l o w s ; t h e most i m p o r t a n t i n v e s t m e n t and c o n t r o l d e c i s i o n s are o f a s p a t i a l l y s p e c i f i c n a t u r e ; t h e c o n s e q u e n c e s o f t h e s e i n v e s t m e n t s c a n b e h i g h l y d i f f e r e n t i a t e d s p a t i a l l y ; t h e t r a n s p o r t a t i o n s y s t e m i s o n e of t h e c r u c i a l mechanisms which g i v e s rise t o l o c a t i o n a l v a r i a t i o n s and s p e c i a l i - z a t i o n i n economic a n d s o c i a l a c t i v i t i e s . I t i s t h e s e s p a t i a l o r l o c a t i o n a l a s p e c t s which a r e emphasized i n t h i s r e v i e w .

Approaches t o o p t i m i z e s u c h i n v e s t m e n t s c a n b e c a t e g o r i z e d r o u g h l y

and somewhat a r b i t r a r i l y i n t o two g r o u p s . The f i r s t and l a r g e s t i s u l t i m a t e l y c o n c e r n e d w i t h n u m e r i c a l o p t i m i z a t i o n p r o c e d u r e s . T h e s e a p p r o a c h e s f r e q u e n t l y a t t e m p t t o i n c o r p o r a t e s i g n i f i c a n t t h e o r e t i c a l r e l a t i o n s h i p s , b u t t h e y a r e d e v i s e d t o p r o v i d e c o m p u t a t i o n a l l y f e a s i b l e s o l u t i o n methods f o r r e a l

p l a n n i n g p r o b l e m s . Most o f t h e s e methods a r i s e d i r e c t l y from t h e m a t h e m a t i c a l programming l i t e r a t u r e . F i n a l l y , t h e y d e a l e x c l u s i v e l y w i t h d i s c r e t e s p a c e problem f o r m u l a t i o n s . T h a t i s , l o c a t i o n s are r e p r e s e n t e d b y d i s c r e t e n o d e s a n d t h e n e t w o r k improvement d e c i s i o n i s w h e t h e r two n o d e s s h o u l d b e d i r e c t l y c o n n e c t e d , a n d , i f s o , what t h e c a p a c i t y o f t h e l i n k s h o u l d b e .

The s e c o n d c l a s s o f o p t i m i z a t i o n methods i s more a n a l y t i c a l , t h e o r e t i c a l and i s f r e q u e n t l y c o n c e r n e d w i t h s p a c e as a c o n t i n u o u s v a r i a b l e . I t h a s

i t s r o o t s i n a n a l y t i c a l geometry a n d d i f f e r e n t i a l c a l c u l u s . R a t h e r t h a n a t t e m p t i n g t o s o l v e r e a l p l a n n i n g p r o b l e m s , t h i s a p p r o a c h a i m s t o p r o v i d e t h e o r e t i c a l , g e n e r a l , a n d e s s e n t i a l l y q u a l i t a t i v e i n s i g h t s o n l i m i t e d

a s p e c t s o f t h e t r a n s p o r t a t i o n i n v e s t m e n t problem. These two a p p r o a c h e s , w h i l e i n p r i n c i p l e complementing o n e a n o t h e r h a v e d e v e l o p e d l a r g e l y i n d e p e n d e n t l y w i t h minimal i n f l u e n c e s on o n e a n o t h e r .

1. Numerical Models o f O p t i m a l Network Improvement 1.1 L i n e a r Programming Models

T h e r e a r e , o f c o u r s e , many f o r m u l a t i o n s o f o p t i m a l n e t w o r k improvement p r o b l e m s , r e f l e c t i n g d i f f e r i n g i n i t i a l c o n d i t i o n s , s y s t e m b e h a v i o u r , p r i m a r y o b j e c t i v e s and c o n s t r a i n t s . P e r h a p s t h e most o b v i o u s f o r m u l a t i o n i s a s i m p l e e x t e n s i o n o f t h e c a p a c i t a t e d H i t c h c o c k

-

Koopmans T r a n s p o r t a t i o n Problem a s p r e s e n t e d by

Quandt (1960) :

I J

Minimize

g

C C

X..

i=1 j=1 i j l j J

s u b j e c t to.: C X..

. 2

Si i=l,

...

^I ( s u p p l y )

j = l 'J ⁽²⁾

C X = D 1

. . . ,

J (demand)

i j j ( 3 )

i = 1

i = l , . . . , 1 X..

-

AKij

2

Kij j = l , . . . , J

l j ( 4 )

( r o u t e c a p a c i t y )

( b u d g e t ) (5)

The o b j e c t i v e i s t o m i n i m i z e t h e t o t a l c o s t o f s h i p m e n t s o f a homogenous p r o d u c t s u b j e c t t o s p e c i f i e d s u p p l y c o n s t r a i n t s and demand r e q u i r e m e n t s w i t h o u t v i o l a t i n g t h e c a p a c i t y of any l i n k ,[old c a p a c i t y ( K , , )

+

added c a p a c i t y (AK..)] and n o t e x c e e d i n g t h e b u d g e t

1 J 1J

a v a i l a b l e f o r t r a n s p o r t a t i o n i n v e s t m e n t e x p e n d i t u r e s (B).

One o f t h e most i n t e r e s t i n g a s p e c t s o f t h i s f o r m u l a t i o n , - a s w i t h a l l l i n e a r programming p r o b l e m s , i s t h e d u a l problem:

J I I J

Maximize

C

V . D

- C

U i S i - C

C

W K - t B j=1

J

j i=l i=1 j=1 ij ij

-

_{< C}

subject to: Vj - ^Ui

Wij

- ij

All of the dual variables have the standard interpretation of the change in total costs associated with a unit change in the right hand side of the associated constraint in the primal. Thus, in particular, the dual variable

t

is the decrease in costs arising from an additional dollar expended on increasing the capacity of the transportation network. This imputed rate of return can be compared to the rate of return for expenditures from other investments in order to assess the wisdom of making more (or less) money available for transportation system improvements.

Quandt presents two alternative formulations of the same problem.

One of them minimizes construction cost expenditures, completely ignoring transfer costs. A tolerable level of total transportation costs could be incorporated into this version of the model as a constraint.) The other formulation minimizes the joint costs of operations and investments with the latter costs being amortized on an annual basis using appropriate interest rates.

In a paper which was published earlier but which derives from ~uandt's analysis, Garrison and Marble (1958) fsnnulate perhaps the first network generation problem which attempts to incorporate the economic consequences sf transportation investment. The objective is to minimize the sum of

operating and investment expenditures subject to supply, demand and

transportation capacity constraints:

I J I J

Minimize C

+

^{C C r AKij}

"ij *ij

i=l j=l

i=1 j=1

subject to:

C X > C C

b

AKkq

+ ^D

^(demand)

ij -

k q i j.q

i= 1 j

(link capacity) (12)

In inequalities (10) and (11) a . and b are empirically determined

~ l c q jkq

coefficients which measure the effects of capacity increments on route kq on the supply capacities and demand requirements at i and j respec- tively. The difficulties of measuring such coefficients at any one time are rather formidable. More important, perhaps, it should not be expected that they would remain constant as the transportation system, the way it is used, and the distribution of economic activity change over time. It would seem clear that the sensitivity of demands and production to changes

in system structure cannot be assumed to remain constant and linear as the system itself undergoes changes. Another apparent flaw in the model is that the "optimal" solution may result in less than optimal transportation investments since low capacities will discourage demands and supplies from increasing and thus total flows will be less than they would be with

higher investment levels. Thus, a cost minimizing objective may result in a stifling of economic development, which would of course, be contrary to the objective of most national governments. Having said all this, it must also be emphasized that as one of the first applications of optimization methods to the network generation problem, the Garrison

-

Marble paper

r e p r e s e n t s a n a m b i t i o u s i n i t i a l e f f o r t t o go beyond t h e more f a m i l i a r narrowly d e f i n e d f o r m u l a t i o n s .

Kalaba and Juncosa (1956) have p r e s e n t e d a n o t h e r e a r l y p a p e r i n t h i s r e s e a r c h a r e a . Although nominally concerned w i t h a t e l e p h o n e r a t h e r t h a n a p h y s i c a l t r a n s p o r t a t i o n network, many of t h e p r i n c i p l e s a r e s i m i l a r . Tn t h i s f o r m u l a t i o n b o t h nodes ( s w i t c h i n g c e n t r e s ) and l i n k s have c a p a c i t i e s which can be i n c r e a s e d . I n t e r s t a t i o n demands a r e g i v e n . The o p t i m a l

r o u t i n g problem i s d e f i n e d t o be one of maximizing a performance c r i t e r i o n f u n c t i o n which i s t h e r a t i o o f s a t i s f i e d demands t o t h o s e p r e d i c t e d

exogenously. For t h e d e s i g n problem, t h e a u t h o r s choose t o minimize t h e t o t a l c o s t of adding t o e x i s t i n g l i n k and s w i t c h i n g c a p a c i t i e s s u b j e c t t o c o n s t r a i n t s on l i n k and s w i t c h i n g c a p a c i t i e s a s w e l l a s a c o n s t r a i n t on t h e performance c r i t e r i o n r a t i o index j u s t mentioned. Thus a g a i n we s e e t h e f l e x i b i l i t y of mathematical programming methods

-

o b j e c t i v e s may appear i n t h e o b j e c t i v e f u n c t i o n o r i n t h e c o n s t r a i n t s . To a l a r g e d e g r e e t h e c h o i c e i s t h e m o d e l l e r ' s a l t h o u g h t h e o r e t i c a l p r i n c i p l e s and p r a c t i c a l r e q u i r e m e n t s may p r o v i d e some guidance.

A l l of t h e s e l i n e a r p r o g r a m i n g f o r m u l a t i o n s can be, modified s o a s t o i n c l u d e t h e p o s s i b i l i t y of t r a n s h i p m e n t o r can be p u t i n t o a n a r c - c h a i n format t o accommodate any g e n e r a l i z s d t r a n s p o r t a t i o n network. An a r c - c h a i n f o r m u l a t i o n o f a network s y n t h e s i s problem can be e x p r e s s e d i n t h e

f o l l o w i n g way :

I J Q 4 4

Minimize C C C

i=1 j = 1 q = l ^Cij Xij s u b j e c t t o :

(1

^J 4

c e xij

- <

si

q = l j = l

4

The d e c i s i o n v a r i a b l e X. i s t h e f l o w from node i t o node j o v e r t h e i j

qth c h a i n (qth s h o r t e s t p a t h ) from i t o j . The d e c i s i o n v a r i a b l e Akk i s l i n k - o r a r c - s p e c i f i c - i . e . t h e i n c r e a s e i n c a p a c i t y of t h e k t h l i n k . The c o e f f i c i e n t a e q i s d e f i n e d s o t h a t i t e q u a l s u n i t y i f l i n k k

i j

i s on t h e qth s h o r t e s t p a t h between nodes i and j and e q u a l s z e r o o t h e r - w i s e .

1 . 2 D i s c r e t e Programming Models

One of t h e major shortcomings o f l i n e a r programming r e p r e s e n t a t i o n s of t h e network improvement problem i s t h a t i n c r e a s e s t o t h e c a p a c i t y of any network l i n k a r e assumed t o b e c o n t i n u o u s l y v a l u e d v a r i a b l e s . More r e a l i s t i c a l l y , t h e s e c a p a c i t y improvements might b e l i m i t e d t o a s e t of i n d i v i s i b l e e n t i t i e s , s u c h a s t h e a d d i t i o n of e n t i r e l y new l a n e s of

t r a f f i c on t h e l i n k . A l t e r n a t i v e l y , t h e p o t e n t i a l changes t o a network a r c might b e r e s t r i c t e d t o a s e t of m u t u a l l y e x c l u s i v e d e s i g n p o s s i b i l i t i e s ;

i n t h i s c a s e t h e v a r i a b l e s may b e l i m i t e d t o t h e v a l u e s z e r o and one.

I n t e g e r v a l u e d v a r i a b l e s , t h e n , emerge v e r y n a t u r a l l y from t h e t y p i c a l l y r a t h e r lumpy i n v e s t m e n t d e c i s i o n s a r i s i n g i n many t r a n s p o r t a t i o n p l a n n i n g s i t u a t i o n s . I n p r i n c i p l e , e i t h e r c u t t i n g - p l a n e methods o r t r e e - s e a r c h i n g p r o c e d u r e s may b e u s e d , b u t f o r most problems of i n t e r e s t i n g s i z e o n l y t h e l a t t e r a r e f e a s i b l e .

A t y p i c a l mixed i n t e g e r p r o g r a m i n g model of network improvement could c l o s e l y r e s e m b l e t h e p r e v i o u s l i n e a r programming f o r m u l a t i o n :

I J

Q

4 9

Minimize C C 1

cij xij

i-1 j = l q = l S u b j e c t t o :

Q J

4

The f l o w

-

b l o c k i n g c o n s t r a i n t s (21) e n s u r e t h a t t h e t o t a l f l o w o n any l i n k L i s less t h a n o r e q u a l t o t h e c a p a c i t y o f t h e l i n k ( t h e p r o d u c t o f t h e c a p a c i t y k o f o n e u n i t e . g . a l a n e and t h e number o f u n i t s

L

(Ae) e . g . t h e number o f l a n e s ) . The c o e f f i c i e n t r

II

i s t h e c o s t of con-

s t r u c t i n g o n e u n i t o f c a p a c i t y on l i n k L . The i n v e s t m e n t d e c i s i o n v a r i a b l e s

XL

may b e r e s t r i c t e d t o t h e b i n a r y s e t , z e r o and o n e , and b e i n t e r p r e t e d as t h e d e c i s i o n t o add o r n o t t o add l i n k L t o t h e s y s t e m . Mixed i n t e g e r programming models o f t h i s t y p e h a v e b e e n a p p l i e d i n a l a r g e number o f d i f f e r e n t s i t u a t i o n s i n c l u d i n g i n t e r c i t y highway n e t w o r k s ( B e r g e n d a h l

(1969) a n d , Morlok

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e t a 1 ( 1 9 6 9 ) ) , i n t r a - u r b a n t r a f f i c s y s t e m s (Ochoa-Rosso ( 1 9 6 8 ) , Ochoa-Rosso a n d S i l v a ( 1 9 6 8 ) , and H e r s h d o r f e r ( 1 9 6 5 ) ) , a n d n a t i o n a l t r a n s p o r t s y s t e m s (Taborga (1968) and K r e s g e and R o b e r t s ( 1 9 7 1 ) ) .

One form o f n e t w o r k g e n e r a t i o n problem h a s r e c e i v e d c o n s i d e r a b l e a t t e n t i o n

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s o much t h a t i t i s f r e q u e n t l y r e f e r r e d t o as " t h e o p t i m a l n e t - work problem". The " o p t i m a l network" i s t h a t s e t o f a r c s l i n k i n g t o g e t h e r a g i v e n s e t o f n o d e s s u c h t h a t t h e sum o f t h e s h o r t e s t p a t h s o v e r t h e n e t - work between e v e r y p a i r o f n o d e s i s minimized w i t h t h e r e s t r i c t i o n t h a t

t h e t o t a l l e n g t h of t h e network d o e s n o t exceed some upper bound.

The s o l u t i o n t o t h i s problem c o n s i s t s of t h e b i n a r y i n c i d e n c e m a t r i x s p e c i f y i n g which p a i r s o f network l i n k s a r e t o b e c o n s t r u c t e d . The number of d i f f e r e n t p o t e n t i a l s o l u t i o n s q u i c k l y becomes enormous a s My t h e number of p o t e n t i a l l i n k s , i n c r e a s e s . I n g e n e r a l , t h e r e a r e 2 M p o t e n t i a l s o l u t i o n networks. T h i s problem can b e s o l v e d by any method which i s c a p a b l e o f r e s t r i c t i n g t h e t o t a l number of l i n k c o m b i n a t i o n s

t h a t must be examined t o some manageable s i z e . Many a l g o r i t h m s have been developed f o r t h i s p u r p o s e

-

b a c k t r a c k programming, b r a n c h and bound methods and h e u r i s t i c p r o g r a m i n g . (See S c o t t (1971) f o r a r e v i e w o f t h e s e a l g o r i t h m s and some a p p l i c a t i o n s . )

A l l t r e e s e a r c h i n g a l g o r i t h m s a r e e s p e c i a l l y f l e x i b l e s o l u t i o n

p r o c e d u r e s i n t h a t t h e y c a n b e used t o s o l v e a v a r i e t y o f mixed i n t e g e r

-

r e a l l i n e a r o r even n o n - l i n e a r programming problems s o l o n g a s t h e y r e t a i n t h e p r o p e r t y of m o n o t o n i c i t y i . e . t h e o b j e c t i v e f u n c t i o n may b e n o n - l i n e a r provided t h e d e l e t i o n of a l i n k c o n s i s t e n t l y d e c r e a s e s ( o r i n - c r e a s e s ) i t s c u r r e n t v a l u e .

I n s p i t e of t h i s f l e x i b i l i t y , improvements i n e x c l u s i o n o p e r a t o r s and bounding p r o c e d u r e s and t h e i n c r e a s i n g c a p a b i l i t i e s of computers, none of t h e s e a l g o r i t h m s can b e used t o o b t a i n s o l u t i o n s t o v e r y l a r g e problems.

*

I n t h i s e v e n t , h e u r i s t i c methods v a r y i n g from s i m p l e random sampling and t r i a l and e r r o r t o e l a b o r a t e s e a r c h p r o c e d u r e s may have t o b e used.

*

I n o r d e r t o a v o i d t h e c o m p u t a t i o n a l l y cumbersome c o m b i n a t i o n a l programming methods, Hodgson (1972) d e v e l o p s a l i n e a r programming f o r m u l a t i o n by assuming a n e x t e n s i v e two l a n e highway network a l r e a d y e x i s t s ; t h e i n v e s t m e n t problem c o n s i s t s o f d e t e r m i n i n g where t h e network should b e upgraded t o a f o u r l a n e f a c i l i t y . The d e c i s i o n v a r i a b l e i n t h i s c a s e i s t h e number of miles of f o u r l a n e roadway between two a d j a c e n t nodes. P a r t i a l l y upgraded l i n k s a r e t h u s p e r m i s s i b l e and m e a n i n g f u l . T h i s f o r m u l a t i o n e n a b l e s t h e r e s e a r c h e r t o t a k e a d v a n t a g e of t h e e x t e n s i v e s o f t w a r e and powerful c o m p u t a t i o n a l methods o f m a t h e m a t i c a l programming s y s t e m t o b e brought t o b e a r on l a r g e network improvement problems w i t h l a r g e numbers of c o n s t r a i n t s .

The models o f f e r t h e p o t e n t i a l o f p r o v i d i n g f e a s i b l e s o l u t i o n s w i t h

o n l y a minimum o f c o m p u t a t i o n a l e f f o r t . Although many o f t h e s e a l g o r i t h m s c o n v e r g e toward t h e g l o b a l l y o p t i m a l s o l u t i o n , t h e y c a n t e r m i n a t e w i t h a l o c a l l y o p t i m a l s o l u t i o n which d i f f e r s from t h e g l o b a l optimum by some unknown margin o f e r r o r . An example of s u c h h e u r i s t i c a l g o r i t h m 'is presented i n S c o t t (1969) and a p p l i e d t o a s y s t e m of 32 c i t i e s i n MacKinnon and

Hodgson (1970). V a r i o u s s p e c i a l h e u r i s t i c a l g o r i t h m s have been d e v e l o p e d by S t e e n b r i n k (1974) and B a r b i e r (1966) and a p p l i e d t o o p t i m a l n e t w o r k p r o b l e m s .

Another c a t e g o r y o f h e u r i s t i c p r o c e d u r e c o n s i s t s of methods which t a k e i n t o a c c o u n t t h e h i e r a r c h i c a l s t r u c t u r e of t r a n s p o r t a t i o n n e t w o r k s s o l v i n g p r o b l e m s f i r s t a t t h e most a g g r e g a t e l e v e l , s u b s t i t u t i n g a s i n g l e l i n k f o r many which p e r f o r m s i m i l a r f u n c t i o n s . A s t h e a l g o r i t h m p r o c e e d s a n e t w o r k d i s a g g r e g a t i o n p r o c e d u r e i s implemented g i v i n g more d e t a i l e d i n v e s t m e n t

p r e s c r i p t i o n s . ( S e e , f o r example, Manheim ( 1 9 6 6 ) , Chan (1969) and Chan

--

e t a 1 ( 1 9 6 8 ) ) . More s i m p l y , l a r g e r e g i o n s c a n b e decomposed i n t o s m a l l e r sub-

r e g i o n s w i t h i n which n e t w o r k o p t i m i z a t i o n i s i n d e p e n d e n t l y u n d e r t a k e n . The o v e r a l l s o l u t i o n network i s g e n e r a t e d a s a n a g g r e g a t i o n of t h e o p t i m a l s o l u t i o n s o f t h e s m a l l e r r e g i o n s .

F i n a l l y , i n t e r a c t i v e programming c a n b e u s e d . With t h i s p r o c e d u r e , t h e m o d e l l e r i n t e r a c t s d i r e c t l y w i t h a computer t e r m i n a l , r e s p o n d i n g t o t h e i n t e r i m s o l u t i o n s g e n e r a t e d by t h e program. L i n k s may be added o r d e l e t e d by t h e m o d e l l e r a s t h e computer s o l v e s and e v a l u a t e s e a c h of t h e p r o p o s e d n e t w o r k s . S t a i r s (1968) s u g g e s t s t h i s method f o r t h e n e t w o r k i m - provement problem and S c h n e i d e r (1971) u s e s i t t o p r o v i d e q u i t e r e a s o n a b l e r e s u l t s t o r e l a t e d p u b l i c f a c i l i t y l o c a t i o n a l l o c a t i o n problems.

Interactive programming in its many guises represents the least rigorous of all approaches to network optimization. At its best, it can effectively harness the subjective intuitive understanding that experienced planners can bring to bear on complex, ill-structured problems. At its

worst, heuristic programming generates plausible guesses at solutions to specific problems and offers few if any general insights into the problem of transportation network improvement.

1.3 Optimal Control Methods

Radically different are the attempts to apply control theoretic methods to the network generation problem.

Wang, Snell, and Funk (1968) assume a given rectangular network geometry, and a given generation, distribution and direction of traffic

(all converging on a single CBD). They formulate a problem which jointly minimizes the time costs of flow and investment costs; simultaneously

assigning flows and improving the network. Flows can be generated re-

cursively because of the assumed directionality and the well defined simple geometric structure. Travel times are assumed to be a non-linear funcf ion of traffic volumes and investment on a link. State variables are defined in terms of flows, investments, and travel time costs. The Hamiltonian function for a typical interior node is generated. Initial conditions for the state variables are of course zero (no flows or antecedent costs beyond the limits of the urban area.) The boundary conditions for the adjoint variables are thus readily determined. An application of a discrete version of the maximum principle then generates a solution of some generality to this network improvement problem. It is suggested that

t h e s p e c i a l c a s e where upper and lower bounds ( i . e . i n e q u a l i t y c o n d i t i o n s ) on i n v e s t m e n t l e v e l s i n d i f f e r e n t p a r t s of t h e network b e h a n d l e d by a

s e a r c h p r o c e d u r e guided by c u r r e n t v a l u e s o f t h e a d j o i n t and s t a t e v a r i a b l e s and t h e p a r t i a l d e r i v a t i v e s o f t h e Hamiltonian f u n c t i o n . Two h y p o t h e t i c a l n u m e r i c a l examples a r e p r e s e n t e d

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t h e f i r s t a p u r e network g e n e r a t i o n problem, t h e second a network improvement problem. The a b i l i t y of t h i s t y p e of approach t o d e a l w i t h n o n - l i n e a r f u n c t i o n s c e r t a i n l y recommends i t . Whether i t c a n b e extended t o model more r e a l i s t i c problems i s s t i l l un- known.

2. T h e o r e t i c a l Models of Optimal Network Improvement

Q u i t e d i f f e r e n t i n s t y l e from t h e models o u t l i n e d i n t h e p r e v i o u s s e c t i o n a r e t h e a p p r o a c h e s which a t t e m p t t o i s o l a t e and r i g o r o u s l y a n a l y z e a s m a l l e r s e t of f a c t o r s and where t h e u l t i m a t e a i m s , s t a t e d o r n o t , a r e t h e o r e t i c a l p r i n c i p l e s and a n a l y t i c a l e l e g a n c e r a t h e r t h a n s p e c i f i c n u m e r i c a l s o l u t i o n s t o p r a c t i c a l p l a n n i n g problems. Although t h i s l i t e r a t u r e i s r a t h e r small, a d e t a i l e d r e v i e w o f a l l t h e models i s n e v e r t h e l e s s i m p r a c t i c a l . I n t h i s s e c t i o n , t h e problems a r e enumerated and t h e g e n e r a l f l a v o u r of t h e a p p r o a c h e s a r e i n d i c a t e d .

The " S t e i n e r problem'' i s t h e most w i d e l y s t u d i e d of a l l t h e s e problems.

The c o n t i n u o u s s p a c e a n a l o g t o t h e minimal spanning t r e e problem (Prim, ( 1 9 5 7 ) ) , t h e S t e i n e r problem d e t e r m i n e s t h e network of minimal l e n g t h which c o n n e c t s a g i v e n s e t o f p o i n t s t o e a c h o t h e r . The s o l u t i o n t o t h i s problem i s of p a r t i c u l a r r e l e v a n c e i n c a s e s where flow c o s t s a r e i n s i g n i f i c a n t compared t o c o n s t r u c t i o n c o s t s (eg. communications networks and economies where c a p i t a l i s i n v e r y s h o r t s u p p l y ) and where t h e c o n s t r u c t i o n c o s t s u r f a c e i s u n i f o r m and t h u s E u c l i d e a n measures of d i s t a n c e a r e r e l e v a n t . The S t e i n e r

network p r o v i d e s a lower bound on network l e n g t h f o r systems which a r e t o connect a g i v e n s e t of r e g i o n s o r c i t i e s .

The s o l u t i o n t o t h e t h r e e p o i n t S t e i n e r problem h a s been known f o r many y e a r s - a s i m p l e "vee" network i f t h e t r i a n g l e j o i n i n g t h e t h r e e p o i n t s has a n a n g l e o f more t h a n 120'; o t h e r w i s e a "wye" network w i t h t h e t h r e e l i n k s m e e t i n g i n t h e i n t e r i o r of t h e t r i a n g l e , forming t h r e e 120 a n g l e s . The g e n e r a l n - p o i n t c a s e h a s proven t o b e more r e s i s t a n t t o 0

s o l u t i o n , b u t methods have been developed t o s o l v e n-point c a s e s f o r any s p e c i f i e d network topology. ( G i l b e r t and P o l l a k , 1968 and Werner, 1 9 6 9 ) . A s t h e number of t o p o l o g i e s i n c r e a s e s g e o m e t r i c a l l y w i t h n , i t may seem t h a t t h e s e methods a r e s e v e r e l y l i m i t e d ; however, i n most s i t u a t i o n s , many t o p o l o g i e s c a n b e d i s c a r d e d a s i m p l a u s i b l e c a n d i d a t e s . Thus, even f o r

l a r g e v a l u e s of n , networks which a r e c l o s e t o t h e o v e r a l l minimal l e n g t h can b e g e n e r a t e d .

I n s t e a d of i g n o r i n g f l o w c o s t s c o m p l e t e l y , i t would b e p o s s i b l e t o emphasize them t o t h e e x c l u s i o n of c o n s t r u c t i o n c o s t s . The s o l u t i o n t o t h i s problein i s t h e t r i v i a l network which c o n n e c t s e a c h p a i r of p o i n t s d i r e c t l y . C o n s i d e r a b l y more i n t e r e s t i n g a r e t h o s e network models which a t t e m p t t o i n c l u d e t h e t r a d e - o f f between c o n s t r u c t i o n and f l o w c o s t s . One v e r s i o n of t h i s problem i s t h e t h r e e p o i n t Weber problem. One raw m a t e r i a l s o u r c e P:

A i s t o b e connected by a t r a n s p o r t a t i o n network t o two m a r k e t s P and P s o

2 3

t h a t t h e j o i n t c o s t s of c o n s t r u c t i o n and f l o w a r e minimized. Working i n E u c l i d e a n s p a c e and assuming f l o w and c o n s t r u c t i o n c o s t s a r e l i n e a r , homogeneous f u n c t i o n s of d i s t a n c e , t h e problem r e d u c e s t o t h a t of f i n d i n g t h e l o c a t i o n c o o r d i n a t e s ( X , Y ) of an i n t e r i o r p o i n t such t h a t

i s minimized where (Xi, Y.) a r e t h e g i v e n l o c a t i o n c o o r d i n a t e s o f p o i n t

1

Pi, f i s t h e f l o w between i and j , and c and k a r e t h e known p e r m i l e i j

c o s t s o f c o n s t r u c t i o n and f l o w r e s p e c t i v e l y . Werner (1968a) s o l v e s t h i s f o r m u l a t i o n , r e - i n t e r p r e t i n g t h e p r o b l e m i n t r i g o n o m e t r i c t e r m s . With more t h a n t h r e e p o i n t s , t h e p r o b l e m o f s p e c i f y i n g t h e a p p r o p r i a t e n e t w o r k t o p o l o g y a g a i n a r i s e s . Werner d e v e l o p s a method by which a " f i r s t d e s i g n "

n e t w o r k i s decomposed and a s e q u e n c e o f c o s t m i n i m i z i n g a d j u s t m e n t s a r e made by a p p l y i n g t h e methodology d e v e l o p e d f o r t h e t h r e e p o i n t c a s e . Only l o c a l l y o p t i m a l s o l u t i o n s c a n b e d e r i v e d as t h e o r d e r o f t h e a d j u s t m e n t s a f f e c t s t h e n e t w o r k t o which t h e p r o c e d u r e c o n v e r g e s . The s p e c i f i c a t i o n o f o p t i m a l n e t w o r k t o p o l o g y c a n b e s o l v e d f o r a few s p e c i a l c a s e s , b u t Werner s t a t e s t h a t "...up t o now, p r a c t i c a l l y a l l e s s e n t i a l p r o b l e m s c o n c e r n i n g t o p o l o g i c a l n e t w o r k d e s i g n are u n s o l v e d . "

I t i s i n t e r e s t i n g t o n o t e t h a t t h e n e t w o r k which j o i n t l y m i n i m i z e s

c o n s t r u c t i o n and f l o w c o s t s i s a minimum l e n g t h n e t w o r k ( i . e . a S t e i n e r t r e e ) i f we assume t h a t f l o w s l i n e a r l y d e c l i n e w i t h d i s t a n c e . T h i s s h o u l d p r o v i d e -

a s t r o n g w a r n i n g a g a i n s t u s i n g s u c h models a s p l a n n i n g d e v i c e s . C l e a r l y f l o w s w i l l d e c l i n e w i t h i n c r e a s e d c h a r g e s , b u t t h e p r i m a r y p u r p o s e o f a t r a n s p o r t a t i o n s y s t e m i s - n o t t o m i n i m i z e c o s t s . One must b e c a u t i o u s i n d e f i n i n g t h e o b j e c t i v e f u n c t i o ? and c o n s t r a i n t s o f n e t w o r k improvement m o d e l s . T h i s comment i s c l o s e l y r e l a t e d t o t h e c r i t i c i s m o f r h e G a r r i s o n

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Marble model i n t h e p r e v i o u s s e c t i o n .

Another c a s e where demand i s s p e c i f i e d a t a f i n i t e number o f d i s c r e t e l o c a t i o n s i s a t r u n k l i n e problem s t u d i e d by Beckmann (1967) and Hauer ( 1 9 7 2 ) . Hauer c o n s i d e r s t h e c a s e o f a s i n g l e o r i g i n ( e g . p l a c e o f work) and m u l t i p l e d e s t i n a t i o n s l o c a t e d a t d i s c r e t e p o i n t s i n

E u c l i d e a n s p a c e ( e g . p l a c e s o f r e s i d e n c e s u c h as a p a r t m e n t b l o c k s ) . The problem i s t o d i s t r i b u t e t h e homeward bound t r a v e l l e r s t o p o i n t s c l o s e

t o t h e i r d e s t i n a t i o n t a k i n g i n t o a c c o u n t t h e i n c o n v e n i e n c e t h a t d i v e r s i o n s impose o n t r a v e l l e r s bound f o r d i f f e r e n t d e s t i n a t i o n s . More s p e c i f i c a l l y , t h e o b j e c t i v e i s t o m i n i m i z e some c o m p o s i t e o f w a l k i n g d i s t a n c e , t r a v e l t i m e f o r p a s s e n g e r s and t r a v e l t i m e o f t h e v e h i c l e . (Although n o m i n a l l y a r o u t i n g p r o b l e m , i t i s , w i t h s l i g h t m o d i f i c a t i o n s , a p p l i c a b l e t o t h e l o c a t i o n o f f i x e d f a c i l i t y s u c h a s a n i n t e r c i t y freeway o r a w a s t e d i s - p o s a l f a c i l i t y . ) A number o f g e n e r a l p r o p e r t i e s r e g a r d i n g t h e g e o m e t r i c s h a p e o f t h e o p t i m a l r o u t e are d e r i v e d and a g r a p h i c a l method o f con- s t r u c t i n g a r o u t e f o r any s e t o f p o i n t s i s d e v e l o p e d . As u s u a l , t h e o r d e r i n which t h e p o i n t s a r e t o b e s e r v e d , t h e n e t w o r k t o p o l o g y , must b e

s p e c i f i e d ; t h u s some e x p e r i m e n t a t i o n may b e n e c e s s a r y .

O t h e r a p p r o a c h e s r e l a x t h e a s s u m p t i o n r e g a r d i n g t h e d i s c r e t e n e s s o f t h e l o c a t i o n o f t r a n s p o r t a t i o n demand. For example, one c a n a t t e m p t t o d e r i v e t h e o p t i m a l t r a n s p o r t a t i o n n e t w o r k t o s e r v e a s p a t i a l l y c o n t i n u o u s p o p u l a t i o n d i s t r i b u t i o n . Tanner (1968) d e v e l o p s a methodology t o e v a l u a t e a l t e r n a t i v e n e t w o r k s o f t h i s t y p e b u t a v o i d s t h e more d i f f i c u l t p r o b l e m o f n e t w o r k s y n t h e s i s .

To make t h e n e t w o r k g e n e r a t i o n problem t r a c t a b l e , t h e s e f o r m u l a t i o n s f o c u s on a m a j o r t r i p d e s t i n a t i o n s u c h a s t h e CBD t o which a l l t r i p s

a r e assumed t o be d e s t i n e d . A homogenous t r a n s p o r t a t i o n s u r f a c e i s g e n e r a l l y assumed, superimposed on which i s t o b e a new a r t e r i a l

network on which t r a n s p o r t a t i o n c o s t s a r e t o b e some s p e c i f i e d f r a c t i o n o f "normal" c o s t s . F r i e d r i c h (1956) u s e s a c a l c u l u s of v a r i a t i o n s

a p p r o a c h t o d e t e r m i n e t h e optimum t r a j e c t o r y of a s i n g l e a r t e r i a l r o u t e s e r v i c i n g a v a r i a b l e t r i p g e n e r a t i n g d e n s i t y w i t h i n a r e c t a n g u l a r s e c t o r of a c i t y . Werner - - e t a 1 (1968) and Sen (1971) d e v e l o p e x t e n s i o n s and s p e c i a l c a s e s of t h i s approach. Sen g e n e r a t e s a n a r t e r i a l network w i t h m u l t i p l e b r a n c h e s ( a h e r r i n g - b o n e topology) t o s e r v e a u n i f o r m l y

d i s t r i b u t e d p o p u l a t i o n . For t h i s s p e c i f i e d network t o p o l o g y , i t i s

p o s s i b l e t o d e r i v e g e n e r a l s t a t e m e n t s a b o u t t h e r e l a t i o n s h i p between t h e dimensions of t h e r e g i o n , t h e t r a n s p o r t a t i o n c o s t d i f f e r e n t i a l , t h e a n g l e of i n c i d e n c e and t h e s p a c i n g o f t h e b r a n c h l i n e s w i t h t h e main stem of t h e network.

Another c o n t i n u o u s s p a c e problem i s t h e minimum c o s t r o u t e c o n n e c t i n g two p o i n t s where t h e c o s t s of c o n s t r u c t i o n , b e c a u s e of p h y s i c a l con-

d i t i o n s o r l a n d a c q u i s i t i o n c o s t s , v a r y w i t h l o c a t i o n . The g e n e r a l problem c a n be f o r m u l a t e d a s a c a l c u l u s o f v a r i a t i o n s problem (Werner and

B o u k i d i s , n.d.) b u t i n t h i s form can b e s o l v e d o n l y f o r s p z c i a l c a s e s .

*

Where f l o w c o s t s c a n b e i g n o r e d and t h e c o s t s u r f a c e c a n be approximated by a s e t of p o l y g o n a l homogeneous c o s t r e g i o n s , Werner (1968) shows t h a t t h e problem can b e s o l v e d a s a m u l t i v a r i a t e e x t e n s i o n of t h e law of r e f r a c t i o n .

*

M.J. Hodgson of t h e U n i v e r s i t y o f A l b e r t a i s c u r r e n t l y working on n u m e r i c a l a p p l i c a t i o n s of t h e s e methods i n c a s e s where t h e c o s t s can be d e s c r i b e d by a n a l y t i c a l s p a t i a l f u n c t i o n s .

Newel1 (1974) comments o n why t h e o p t i m a l network problem i s u n d e r - r e s e a r c h e d . Not o n l y i s i t t y p i c a l l y a t t h e l a s t s t a g e o f t r a n s - p o r t a t i o n p l a n n i n g p r o c e s s , b u t i t i s c h a r a c t e r i z e d by d i f f i c u l t i e s which n a t u r a l l y a r i s e o u t o f two c h a r a c t e r i s t i c s of t h e m a t h e m a t i c a l f u n c t i o n s r e l e v a n t t o s u c h p r o b l e m s . The f i r s t , t h e f a c t t h a t many o f t h e d e c i s i o n v a r i a b l e s are i n t e g e r v a l u e d h a s a l r e a d y been d i s c u s s e d a t some l e n g t h . The s e c o n d i s t h a t t h e r e l a t i o n s h i p between j o i n t f l o w and c o n s t r u c t i o n c o s t s on t h e one hand and f l o w s i s n o t i n g e n e r a l a convex f u n c t i o n . T h i s non- c o n v e x e t y a r i s e s from t h e economies o f s c a l e a c h i e v e d by l a r g e t r a n s p o r t a t i o n f a c i l i t i e s . Thus, t o t a l c o s t s on any l i n k r i s e i n a convex f a s h i o n f o r a n y f i x e d f a c i l i t y ( a s c o n g e s t i o n l e v e l s a r e e n c o u n t e r e d ) b u t as f l o w s r i s e , l a r g e r s c a l e f a c i l i t i e s c a n r e d u c e a v e r a g e f l o w c o s t s o n any r o u t e . The r e l e v a n t c o s t c u r v e t h e n i s t h e i n d i v i d u a l c o s t c u r v e s ' e n v e l o p e , which v e r y r o u g h l y , c a n b e s a i d t o b e concave. Concave programming p r o b l e m s a r e d i f f i c u l t t o s o l v e , b u t i n q u a l i t a t i v e t e r m s , i f t h e c o s t c u r v e i s c o n c a v e , t h e n f l o w s between a n y two d e s t i n a t i o n s

w i l l

b e r o u t e d o v e r a s i n g l e p a t h b e c a u s e of t h e economies o f s c a l e . That m u l t i p l e p a t h r o u t i n g a r i s e s i n r e a l i t y c a n b e a t t r i b u t e d t o t h e f a c t t h a t t h e c o s t c u r v e i s n o t s t r i c t l y c o n c a v e , b u t p i e c e w i s e convex.

Another o b s e r v a t i o n a r i s i n g from N e w e l l ' s d i s c u s s i o n i s t h a t p e r f e c t l y s y m m e t r i c a l n e t w o r k s ( e g . s q u a r e g r i d s , n o n - h i e r a r c h i c a l n e t w o r k s , e t c . ) a r e a l m o s t c e r t a i n t o b e s u b o p t i m a l e v e n i f t h e u n d e r l y i n g demands a r e p e r f e c t l y s y m m e t r i c a l . T h i s f o l l o w s d i r e c t l y from t h e a b o v e m e n t i o n e d economies of s c a l e .

Although Newel1 i s modest i n e v a l u a t i n g h i s c o n t r i b u t i o n , i t i s

p o s s i b l e t h a t t h e s e s o r t o f q u a l i t a t i v e f i n d i n g s a r e more s i g n i f i c a n t t h a n t h e more t e c h n i c a l and s u p p o s e d l y o p e r a t i o n a l r e s u l t s o f m a t h e m a t i c a l programming methods. More i m p o r t a n t t h a n e v a l u a t i n g t h e r e l a t i v e m e r i t s of t h e s e ' a p p r o a c h e s , i t i s a p p a r e n t t h a t m a t h e m a t i c a l programming f o r m u l a t i o n s s h o u l d a t t e m p t t o i n c o r p o r a t e t h e t h e o r e t i c a l f i n d i n g s of N e w e l 1 and o t h e r s . 3 . P e r s p e c t i v e s on F u t u r e R e s e a r c h P r i o r i t i e s

Having p r o v i d e d a b r i e f b u t h o p e f u l l y r e p r e s e n t a t i v e r e v i e w t o g e t h e r w i t h a n e x t e n s i v e i f n o t comprehensive b i b l i o g r a p h y on o p t i m i z a t i o n a p p r o a c h e s

t o t h e network improvement problem, i t i s a p p r o p r i a t e t o a s k what a r e t h e most i n t e r e s t i n g remaining r e s e a r c h t a s k s which c o u l d b e u n d e r t a k e n . The answer t o s u c h a q u e s t i o n i s n e c e s s a r i l y s u b j e c t i v e and s e l e c t i v e . I n t h i s s e c t i o n , no a t t e m p t i s made t o p r o v i d e a n o b j e c t i v e comprehensive l i s t of r e s e a r c h problems. I t i s n e v e r t h e l e s s hoped t h z t p a r t i c i p a n t s of t h e workshop w i l l make s u g g e s t i o n s r e g a r d i n g a d d i t i o n a l f o c a l p o i n t s f o r r e s e a r c h a s w e l l a s c o n c r e t e p r o p o s a l s r e g a r d i n g a p p r o p r i a t e r e s e a r c h s t r a t e g i e s t o r e s o l v e remaining problems.

The f i r s t and p e r h a p s o v e r r i d i n g r e s e a r c h p r i o r i t y i s a r a t h e r g e n e r a l one a n d , i n a s e n s e , a l l o t h e r s a r e s p e c i a l c a s e s o f i t

-

how t o imbed o p t i m a l network improvement models i n t o b r o a d e r t r a n s p o r t a t i o n p l a n n i n g and even b r o a d e r u r b a n and r e g i o n a l socio-economic p l a n n i n g c o n t e x t s ?

T r a n s p o r t a t i o n p l a n n i n g i s t y p i c a l l y decomposed i n t o t h e f o l l o w i n g sub- problems: ( a ) p r e d i c t i o n of t h e d i s t r i b u t i o n of p o p u l a t i o n , l a n d u s e and economic a c t i v i t i e s ; (b) t r i p g e n e r a t i o n ; ( c ) t r i p d i s t r i b u t i o n ; ( d ) modal

s p l i t ; ( e ) t r a f f i c a s s i g n m e n t ; ( f ) network a n a l y s i s , e v a l u a t i o n and m o d i f i c a t i o n . Although f e e d b a c k l o o p s a r e r e c o g n i z e d , i n p r a c t i c e t h e p l a n n i n g p r o c e s s t e n d s t o f o l l o w t h i s s e q u e n c e . For s i m p l i c i t y l e t u s i g n o r e t h e modal s p l i t component. Some a p p r o a c h e s t o n e t w o r k g e n e r a t i o n assume t r i p g e n e r a t i o n i s known a n d t h e model s i m u l t a n e o u s l y d i s t r i b u t e s t r i p s , a s s i g n s t r a f f i c , and g e n e r a t e s network improvements. The s i m p l e e x t e n s i o n s o f t h e c a p a c i t a t e d T r a n s p o r t a t i o n Problem i n e i t h e r t r a n s h i p - ment o r a r c - c h a i n form a r e models o f t h i s t y p e ( e g . Quandt ( 1 9 6 0 ) , C a r t e r

and S t o w e r s ( 1 9 6 3 ) .

More numerous a r e t h o s e which assume i n t e r n o d a l demands a r e s p e c i f i e d and i n d e p e n d e n t o f n e t w o r k s t r u c t u r e . These models a s s i g n t h e known t r i p s t o t h e t r a n s p o r t a t i o n n e t w o r k w h i l e a d d i n g l i n k s o r u p g r a d i n g t h e q u a l i t y of t h e c u r r e n t n e t w o r k ( e g . R i d l e y ( 1 9 6 8 ) , S c o t t ( 1 9 6 9 ) , Boyce, F a r h i and W e s c h i e d e l ( 1 9 7 3 ) , B u s h e l 1 ( 1 9 7 0 ) , Hodgson ( 1 9 7 2 ) , B e r g e n d a h l ( 1 9 6 9 b ) , and many o t h e r s . ) H u t c h i n s o n (1972) a r g u e s t h a t t h i s i s a r e a s o n a b l e s t r a t e g y

f o r improvements t o a n a l r e a d y w e l l - d e v e l o p e d t r a n s p o r t a t i o n s y s t e m . But of c o u r s e i n g e n e r a l t h e new s y s t e m c o n f i g u r a t i o n w i l l i n f l u e n c e f u t u r e

d i s t r i b u t i o n o f demands. Even f o r a g i v e n d i s t r i b u t i o n o f p o p u l a t i o n and economic a c t i v ' i t y , b e t t e r t r a n s p o r t a t i o n s e r v i c e w i l l i n d u c e i n d i v i d u a l s and f i r m s t o i n c r e a s e t h e i r u t i l i z a t i o n of t h e t r a n s p o r t a t i o n s y s t e m . Thus t h e i n t e g r a t i o n o f t h e n e t w o r k g e n e r a t i o n problem w i t h t r i p d i s t r i b u t i o n and g e n e r a t i o n models i s c e r t a i n l y d e s i r a b l e . T h i s h a s been done v e r y c r u d e l y by MacKinnon and Hodgson (1970) by i n c o r p o r a t i n g a n u n c o n s t r a i n e d g r a v i t y model i n t o t h e o b j e c t i v e f u n c t i o n . More r e c e n t l y Boyce and h i s a s s o c i a t e s h a v e d e v e l o p e d a method which m i n i m i z e s t h e c o s t s o f a t r i p s e t which i s

distributed by a doubly constrained interaction model. This approach,

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of course, assumes that trip generation is insensitive to transportation service characteristics. Even the same number of people with the same spatial distribution may make more trips as costs decrease and the quality of service improves in other ways. More importantly, perhaps, changes in the transportation system will in the longer run induce households,, firms, and other decision making units to change the location of their activities and therefore the spatial structure of their transportation requirements.

The relationships between transportation improvements and the location (and re-location) of other economic and social activities is not very

well understood. [See for example Holland (1972), Straszheim (1970) and Putnam (1975)l Models which identify the location of both transportation link improvements and production levels of economic activities such that the joint costs of transportation and production are minimized may be appropriate in some instances. Conceptually and operationally this is a rather simple integration of an interregional input-output model with location-allocation and network improvement models. Given a total

schedule of exogenously predicted "final demands", exports, and imports by sector, the cost minimizing location of industry and transportation investments can be determined. Barber (1973, 1975) has applied such a model to interregional systems in Indonesia and Colombia. (In the Indonesian context, see also ~ ' ~ u l l i v a n - - et a1 1975.) With less than rigidly and centrally planned economies, such formulations may have only heuristic value, setting a lower bound on one criterion of merit, re- cognizing that firms may make location decisions which do not result in

*

Source: personal communication with

D.

Boyce, University of Pennsylvania.

c o s t m i n i m i z a t i o n f o r t h e s y s t e m a s a whole. Even i n r i g i d l y p l a n n e d economies, o t h e r g o a l s may b e i m p o r t a n t y e t d i f f i c u l t t o i n c o r p o r a t e i n t h e model. A l s o f o r any t y p e o f economy, i n p u t - o u t p u t c o e f f i c i e n t s a r e n o t s t a b l e a n d more i m p o r t a n t l y t h e y w i l l be a f f e c t e d by c h a n g e s i n t h e t r a n s p o r t a t i o n c o s t s t r u c t u r e . A c o s t - s e n s i t i v e i n t e r r e g i o n a l i n p u t - o u t p u t submodel c o u l d b e i n c o r p o r a t e d i n t o t h e l a r g e r model, b u t t h i s would g i v e r i s e t o d i f f i c u l t n o n l i n e a r i t i e s .

L o c a t i o n f a c t o r s o t h e r t h a n t r a n s p o r t a t i o n c o s t s s h o u l d b e i n c l u d e d w i t h i n t h e model. S t r a s z h e i m (1970) and many o t h e r s would s u g g e s t t h i s i s p a r t i c u l a r l y t r u e f o r more advanced economies where t h e s e n s i t i v i t y o f l o c a t i o n d e c i s i o n s t o t r a n s p o r t a t i o n c o s t s i s r a t h e r low b e c a u s e of t h e r e l a t i v e homogeneity of t h e t r a n s p o r t a t i o n c o s t s u r f a c e and t h e

t y p i c a l l y s m a l l p r o p o r t i o n o f v a l u e added a c c o u n t e d f o r by t r a n s p o r t a t i o n c o s t s i n many of t h e f a s t e s t growing s e c t o r s i n s u c h economies. T r a n s - p o r t a t i o n e f f e c t s on s p a t i a l development would a p p e a r t o b e most s t r i k i n g a t two e x t r e m e s o f development

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on a n i n t e r r e g i o n a l s c a l e , t r a n s p o r t a t i o n c a n b e a n e f f e c t i v e t o o l t o s h a p e development i n e c o n o m i c a l l y l e s s advanced c o u n t r i e s where t r a n s p o r t a t i o n s e r v i c e i s a t a low l e v e l and t h e economy i s d e p e n d e n t o n t r a n s p o r t a t i o n - i n t e n s i v e a c t i v i t i e s ; on a l o c a l s c a l e , t r a n s p o r t a t i o n can s h a p e t h e f u t u r e s p a t i a l s t r u c t u r e of u r b a n a r e a s .

Dickey (1972) and Dickey and Azola (1972) h a v e p r o v i d e d a v a l u a b l e f i r s t s t e p i n b r o a d e n i n g t h e f o c u s o f n o r m a t i v e u r b a n t r a n s p o r t a t i o n p l a n n i n g by imbedding a n e c o n o m e t r i c model o f l a n d u s e development ( t h e

"Empiric" model) i n t o a n o p t i m i z a t i o n framework. The d e c i s i o n v a r i a b l e s a r e i n t e r z o n a l t r a v e l t i m e s and t h e g o a l s a r e s t a t e d i n t e r m s o f d e s i r e d

future social and economic land use patterns. The constraints are com- posed largely of the equations of the Empiric model. Thus, "future land use changes that take place are constrained to do so in the way the Empiric model says they will.

^{' I *}

Dickey rightly characterizes this approach as exploratory and preliminary Not only is the econometric model subject to some skepticism, but also the decision variables themselves (exponentially transformed travel times) are not in a form which is necessarily directly translatable into action by transportation planners. For one thing, it would not be difficult to generate combinations of "optimal" travel times which were inconceivable since changing times between one zonal pair will change the times between another pair if the first link is on the path between the second pair. Even if this difficulty is not encountered, the costs of changing travel times and the means by which this is to be done are not specified. Given the variety of means and costs (and our imperfect knowledge) this vagueness is perhaps a desirable characteristic. The model suggests directions and approximate magnitudes of actions implied by the goals, not the precise means of implementation.

Lundqvist (1973) presents an even more ambitious model which uses hierarchical decomposition to jointly determine the optimal expansion and use of transportation

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and other land uses within an urban area. The need for a "recursive dialogue between optimization and simulation techniques1' is stressed. The solution to the non-linear combinatorial problem is based on heuristic tree searching methods.

* One set of non-linearities which naturally arises in the constraints is

accommodated by using e-BCij (t+l)as a decision variable rather than the

travel times Cij (t+l) themselves

( 6

is a distance friction parameter

^'

estimated in the econometric model.) Multiplicative non-linearities are

handled by assuming one set of variables is constant and searching over

their known range of values for their best settings.

W i t h i n t h i s g e n e r a l p r e s c r i p t i o n o f b r o a d e n i n g t h e s c o p e of o p t i m a l n e t w o r k improvement models, two c a t e g o r i e s o f d i f f i c u l t i e s

t y p i c a l l y a r i s e

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f i r s t , d a t a o n s y s t e m i n p u t s , g o a l s , and p a s t b e h a v i o u r a r e s c a r c e , c o s t l y a n d / o r o f poor q u a l i t y ; s e c o n d , t h e o r e t i c a l d e f i c i e n c i e s a r e g e n e r a l l y j u s t a s s e v e r e . T h e r e i s a n a p p a r e n t need f o r models which a r e i m p l e m e n t a b l e , r e l e v a n t , and a b o u t which we h a v e some c o n f i d e n c e r e g a r d i n g t h e i r v a l i d i t y .

T h r e e c a t e g o r i e s o f s p e c i a l problems r e l a t i n g t o t h e s e models a r e t h o s e a s p e c t s r e l a t e d t o s y s t e m dynamics, r i s k and u n c e r t a i n t y , and s p e c i f i c a t i o n o f o b j e c t i v e s . Each of t h e s e w i l l b e c o n s i d e r e d i n t u r n .

For t h e most p a r t , o n l y r e l a t i v e l y c r u d e models h a v e t h u s f a r been f o r m u l a t e d t o o p t i m i z e improvements t o t r a n s p o r t a t i o n s y s t e m s o v e r t i m e . Most of t h e s e a p p r o a c h e s c o n s i d e r t h e network a d d i t i o n problem w i t h i n a dynamic programming framework. Funk and T i l l m a n (1968) and d e N e u f v i l l e and Mori ( 1 9 7 0 ) , f o r example, e x p l i c i t l y make t h e s i m p l i f y i n g a s s u m p t i o n t h a t t h e b e n e f i t s and c o s t s a r i s i n g from e a c h l i n k a d d i t i o n p r o j e c t a r e i n d e p e n d e n t o f e a c h o t h e r , and c a n t h u s be s p e c i f i e d n u m e r i c a l l y a t t h e o u t s e t . More p r e c i s e l y , t h e i n t e r d e p e n d e n c i e s a r e o n l y t h e r e l a t i v e l y t r i v i a l o n e s a r i s i n g from t h e f a c t t h a t b u i l d i n g a l i n k i n a n e a r l i e r p e r i o d may p r e c l u d e t h e c o n s t r u c t i o n o f o t h e r s i n s u b s e q u e n t p e r i o d s

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a s e q u e n c e w h i c h , t a k e n a s a w h o l e , c o u l d r e s u l t i n h i g h e r n e t b e n e f i t s t h a n o n e which s i m p l y maximizes t h e n e t b e n e f i t s f o r e a c h p e r i o d t a k e n s e p a r a t e l y .

O f c o n s i d e r a b l y more i n t e r e s t (and d i f f i c u l t y ) a r e models which

r e c o g n i z e t h e f l o w i n t e r d e p e n d e n c i e s between d i f f e r e n t l i n k s . That i s , t h e

opening of a new link will divert traffic from other links and may even change the flow of traffic between origin

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destination pairs in both a relative and absolute sense. Thus the value of adding a particular link will be dependent on what other links have already been added to the system. For example, the construction of a link in time period t may affect the value of adding a nearby link in period t+l

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positively if traffic generated and/or diverted as a result of the first link causes congestion on the network in this area, and negatively if one of the system objectives is to encourage regionally equitable balance of trans- portation services and investments.

Behavioural responses to network changes should be an important part of the network link sequencing problem. Changing the network will result in a different assignment distribution, and generation of traffic. Ultimately transportation investments can result in different geographical patterns of population, economic and recreational activities

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patterns which will be passed on to subsequent stages of the spatial development process.

Some of these behavioural responses are relatively easy to model.

Bergendahl (1969b) assumes that transportation demands between all pairs of points are predictable exogenously and that traffic need only be assigned to the network in each period. He assigns traffic using a linear programming multicommodity flow model. Incorporating link congestion costs,

u us hell

(1970)

uses a similar approach.) Other network assignment algorithms could be used, but this one provides a measure of the maximum savings in flow costs resulting from transportation investments. The optimal investment sequence is then

determined by minimizing a recursive function which is a composite of discounted operating and investment costs:

go

( s O ) = o

where

st

i s t h e s t a t e of t h e system a t t i m e t (a l i s t o f t h e

t r a n s p o r t a t i o n l i n k s and t h e i r c h a r a c t e r i s t i c s ) z t ( s t ) t h e minimum o p e r a t i n g c o s t s a s s o c i a t e d w i t h t h e s t a t e S t c ( S t

,

^st+') t h e i n v e s t m e n t c o s t s of changing t h e s y s t e m from

S t o S t t + l

V ( ST) t h e " s c r a p v a l u e n o f t h e system ( e g . t h e f u t u r e o p e r a t i n g c o s t s a s s o c i a t e d w i t h t h e "horizon" network S T

.

R and R' a p p r o p r i a t e d i s c o u n t f a c t o r s .

B e r g e n d a h l ' s model. i s o p e r a t i o n a l and has been a p p l i e d t o highway i n v e s t - ment d e c i s i o n s i n Sweden. The c o m p u t a t i o n a l t a s k h a s been reduced by e l i m i n a t i n g many of t h e i m p l a u s i b l e l i n k combinations and s e q u e n c e s . Although more a m b i t i o u s t h a n o t h e r e f f o r t s Bergendahl d o e s n o t

a t t e m p t t o i n c o r p o r a t e t h e t r a f f i c r e - d i s t r i b u t i o n and g e n e r a t i o n e f f e c t s o f network improvements. Even though o u r t h e o r e t i c a l knowledge i s r e l a t i v e - l y weak on t h i s s u b j e c t , i t i s of i n t e r e s t t o . n o t e how s u c h r e l a t i o n s h i p s could b e i n c o r p o r a t e d . Hodgson (1974) u s e s a s i m p l e g r a v i t y model t o re- d i s t r i b u t e and g e n e r a t e i n t e r c i t y t r a f f i c estimates o v e r t i m e i n r e s p o n s e t o network c h a n g e s . T r a f f i c i s a s s i g n e d by t h e s i m p l e s h o r t e s t p a t h

method. T r a n s p o r t a t i o n flow e s t i m a t e s a r e maximized o v e r t i m e s u b j e c t t o i n v e s t m e n t i n e a c h p e r i o d and t h e s p e c i f i c a t i o n o f a t e r m i n a l network which was g e n e r a t e d i n MacKinnon and Hodgson (1970) u s i n g a s i n g l e s t a g e

o p t i m i z a t i o n method. T h i s l a t t e r c o n d i t i o n i s c l e a r l y a r t i f i c i a l , b u t i t

is likely that considerable experimentation with crude approximations is going to be necessary before well-structured solutions to the temporal sequencing problem emerges.

In an important paper Frey and Nemhauser (1972) model the optimal timing of network expansion as a convex programming problem where flows are non-linear functions of service characteristis which in turn are functions in part of flows. The interdependencies between augmenting

capacity on different links is fully taken into account and the conditions under which a myopic decision strategy is optimal are delimited. Un-

fortunately the general nature of these findings are tempered somewllat by the statement that "...although these results may be extendable to large serial networks, it does not seem possible to generalize them to networks with more complex topologies."

Each of the four responses to network change

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assignment, distribution, generation, and spatial re-structuring

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takes time. The specification of these lags should ultimately be an integral part of a transportation in- vestment model. Although it would be tempting to claim that they are in ascending order of response lags, even this is not clear. Spatial re- structuring (e.g. plant re-location) may be initiated in response to anticipated transportation improvements whereas traffic generation and other "operational" responses must await the actual construction of the facility. The characteristics and timing of these responses are not well understood; much research needs to be done, particularly in terms of the spatial restructuring impacts as these often swamp the short term savings in transportation costs which are frequently used to provide the nominal justification for transportation investments.