COMPUTATIONAL RESULTS WITH "FABRICATED" DATA

( 1 )

m ( 2 ) ( 3 ( 4 ) (5 ( 6 )

aJ I d

d C I

P O n I (d al

r)e

ul ^(d d M

4 d C b (d

u LI 0 LI d (d u

r) (d

7 > ^a ^?

? a 3 ^m

a al al U-I al

o ~ 7 4 FI u 01 o a

LI 0 ~ a o I a LI

a ⁰_a⁰

₂

^w^a

Z

ⁿ^m

tu V ' w a J t u c a

2 3

^tu ^aJ

0 d ru Q

04'0 0 0 aJ ^C

I x ^ld ^{d d} U U al 0

LI LI al h ld h (d a r)

5 s g

^al ^al ^d^al

Problem

2

^P^m ^M ^al ^Q) ¹ ^u^m

% 3 2

^P^ul ^a^d

number

3 ³

^iZi4

^!2z

^P ^W

^$ 1 3

^z^B⁷^(d^:^al^, ³^PI^{0 -}^E:

TABLE 4.4

COMPUTATIONAL RESULTS WITH n ~ DATA: B ~

-

^B STAGE ALONE ~ ~ ~ ~ ~ ~ w

P r o b l e m N u m b e r of free N u m b e r of nodes CPU t i m e

n u m b e r 0-1 v a r i a b l e s evaluated (in seconds)

3 . 5 of Chapter 3, i f w e want t o s o l v e problems w i t h n o n t r i a n g u l a r input-output s t r u c t u r e , w e cannot count on a n e f f i c i e n t problem r e d u c t i o n s t a g e t o d e c r e a s e t h e s i z e o f t h e problem t o a more mana- g e a b l e s i z e . Because o f t h e c i r c u l a r n a t u r e o f t h e i n t e r d e p e n d e n c e i n such s i t u a t i o n s , any problem r e d u c t i o n s t a g e t h a t may be d e v i s e d w i l l n e c e s s a r i l y be e i t h e r more cumbersome, o r weaker ( a s t h e d a t a r e l a x a t i o n suggested i n s e c t i o n 3.5), and i n e i t h e r c a s e may p r o v e n o t t o b e worth t h e c o m p u t a t i o n a l e f f o r t r e q u i r e d . S i n c e it was shown i n Chapter 3 t h a t t h e s o l u t i o n t o t h e LP r e l a x a t i o n when A is n o t t r i a n g u l a r i s e s s e n t i a l l y no more d i f f i c u l t t h a n f o r t h e c a s e i n which A i s upper t r i a n g u l a r , t h e computational e x p e r i e n c e pro- vided in t h i s c h a p t e r w i t h t h e B-B s t a g e a l o n e ( T a b l e s 4.2 and 4.4) i n d i c a t e s t h a t o u r LP-based B-B approach would b e r a t h e r e f f i c i e n t in s o l v i n g problems w i t h n o n t r i a n g u l a r input-output s t r u c t u r e s w i t h o u t any problem-reduction a t t e m p t being made. Although improvements c o u l d p o s s i b l y b e made by such a t t e m p t s , our r e s u l t s i n d i c a t e in t h e l e a s t t h a t i t i s f e a s i b l e t o s o l v e v e r y l a r g e problems w i t h o u t depen- dence on t h e s u c c e s s of any form of problem r e d u c t i o n t e c h n i q u e s . T a b l e 4.5 g i v e s t h e s u m m a r y s t a t i s t i c s f r o m T a b l e s 4.2 and 4.4 on t h e performance o f t h e B-B s t a g e a l o n e f o r t h e two sets of problems.

Thus f a r n o t h i n g has been s a i d concerning t h e p a r t i c u l a r imple- m e n t a t i o n of t h e B-B a l g o r i t h m t h a t y i e l d e d t h e r e s u l t s j u s t d i s c u s s e d . W e n e x t a n a l y z e in t u r n : (1) t h e e f f e c t i v e n e s s of s t e p 7 of t h e B-B a l g o r i t h m ; (2) t h e a l t e r n a t i v e s e p a r a t i o n s t r a t e g i e s proposed in Chapter 3; (3) t h e q u a l i t y of t h e bounds o b t a i n e d from t h e LP r e l a x a - t i o n ; and (4) t h e t i g h t n e s s of t h e c o n d i t i o n a l upper bounds UB(*)

TABLE 4.5

SUMMARY STATISTICS FOR THE B

-

^B ^STAGE

Problems Average number of Average number of CPU t i m e

0-1 v a r i a b l e s * nodes e v a l u a t e d * ( i n seconds)

Note:

*

Rounded t o n e a r e s t i n t e g e r .

computed a t e a c h s e p a r a t i o n .

A t s t e p 7 of t h e B-B a l g o r i t h m a n a t t e m p t i s made t o peg a t t h e l e v e l 1 some of t h e Ak f r a c t i o n a l in t h e s o l u t i o n t o t h e r e l a x e d problem a t a g i v e n node. For t h i s p u r p o s e Ek i s computed f o r e a c h k c o r r e s p o n d i n g t o a f r a c t i o n a l A ~ . Computational tests performed w i t h a sample of t h e problems, b o t h w i t h and w i t h o u t S t a g e 1, showed t h a t a l t h o u g h it succeeded in many c a s e s in pegging some v a r i a - b l e s , i t d i d n o t p r o v e t o be worth t h e c o m p u t a t i o n a l e f f o r t expended

i n computing t h e

.

^Even in t h o s e c a s e s in which t h e s i z e ' of t h e Ek

tree d e c r e a s e d s i g n i f i c a n t l y a s a r e s u l t of u s i n g s t e p 7 , c o m p u t a t i o n a l t i m e s were g e n e r a l l y h i g h e r t h a n t h o s e in which t h e s t e p was bypassed.

T h i s w a s p a r t i c u l a r l y t r u e f o r problems w i t h a l a r g e number of f r e e v a r i a b l e s , r e g a r d l e s s of t h e s e p a r a t i o n s t r a t e g y u s e d . As a r e s u l t of t h e s e e x p e r i m e n t a l r u n s s t e p 7 was d i s c a r d e d , and a l o n g w i t h i t t h e s e p a r a t i o n r u l e of s e l e c t i n g t h e hk c o r r e s p o n d i n g t o t h e l a r g e s t Eke The s i m p l e s t implementation of t h e B-B a l g o r i t h m t h u s proved

t h e most e f f e c t i v e , and a l l t h e c o m p u t a t i o n a l r e s u l t s g i v e n in t h i s c h a p t e r p e r t a i n t o t h i s s i m p l e r form of t h e a l g o r i t h m .

The l a s t two i t e m s ( ( 3 ) and ( 4 ) ) c o n c e r n i n g t h e B-B a l g o r i t h m a r e b e s t d i s c u s s e d i n t h e c o n t e x t of t h e i n e f f e c t i v e n e s s of s t e p 7 , a s i t serves t o i l l u s t r a t e t h e s u c c e s s of t h e approach. The i n e f f e c t i v e n e s s of s t e p 7 may be e x p l a i n e d by s e v e r a l f a c t o r s . F i r s t , node evalua- t i o n s a r e c a r r i e d o u t e x t r e m e l y f a s t . I n d e p e n d e n t l y of any o t h e r f a c t o r t h i s i m p l i e s that an a l l - i n t e g e r s o l u t i o n ( i . e . , t h e f i r s t

f e a s i b l e s o l u t i o n of (P)) can be o b t a i n e d a t a v e r y s m a l l c o m p u t a t i o n a l c o s t

.u

Secondly, i f t h e f i r s t f a c t o r i s combined w i t h a n e f f i c i e n t s e p a r a t i o n s t r a t e g y one c a n conclude that t h e f i r s t a l l - i n t e g e r s o l u - t i o n o b t a i n e d should be r e a s o n a b l y good. I n o u r approach t h i s is e v i - denced by t h e f a c t t h a t in n e a r l y 50% of t h e test problems t h e f i r s t f e a s i b l e s o l u t i o n o b t a i n e d was in f a c t o p t i m a l . T h i r d l y , i f t h e two p r e v i o u s f a c t o r s a r e combined w i t h a r e l a x a t i o n t h a t y i e l d s t i g h t bounds, one c a n c o n f i d e n t l y e x p e c t t h a t t h e B-B tree should n o t grow v e r y l a r g e . The computat i o n a l e x p e r i e n c e provided h e r e shows t h a t t h e number of node e v a l u a t i o n s r e q u i r e d in any of t h e problems w a s never v e r y l a r g e i n r e l a t i o n t o t h e number o f 0-1 v a r i a b l e s . I n f a c t , i n no case d i d t h e number of node e v a l u a t i o n s exceed t h e number' of v a r i a b l e s ! Moreover, f o r most of t h e c a s e s in which t h e f i r s t a l l - i n t e g e r s o l u t i o n o b t a i n e d w a s o p t i m a l , every a c t i v e node of t h e t r e e s u b s e q u e n t l y examined was fathomed by bound w i t h o u t any f u r t h e r b r a n c h i n g t a k i n g p l a c e . F i n a l l y , t h e c o n d i t i o n a l upper bounds UB(-)

Even i f a l a r g e number of nodes must be e v a l u a t e d .

proved r e l a t i v e l y e f f e c t i v e in fathoming a t a l l l e v e l s of t h e B-B t r e e , c a u s i n g i n most c a s e s a s i g n i f i c a n t r e d u c t i o n in t h e number of node e v a l u a t i o n s r e q u i r e d .

It i s f a i r l y s a f e t o conclude from t h e above d i s c u s s i o n t h a t a l l t h e i n g r e d i e n t s f o r a s u c c e s s f u l B-B a r e p r e s e n t i n our s o l u t i o n approach t o such a n e x t e n t that a t t e m p t s a s t h o s e of s t e p 7 a r e rendered i n e f f e c t i v e o r "not worth t h e e f f o r t . " It i s important t o n o t e , in c o n c l u s i o n , t h a t t h e s i m p l e r B-B implementation seems t o p r o v i d e t h e b e s t of two worlds; n o t o n l y i t a p p e a r s t o be more e f f i c i e n t f o r t h e c a s e s i n which t h e input-output matrix A i s upper t r i a n g u l a r , b u t , most i m p o r t a n t l y , i t i s a l s o t h e v e r s i o n t h a t a p p l i e s when A i s n o t t r i a n g u l a r . - 1/

L/

A weaker form of s t e p 7 c o u l d a c t u a l l y be used, a s i n d i c a t e d in s e c t i o n 3.5 of Chapter 3 . I n view of t h e r e s u l t s of t h i s c h a p t e r , however, i t i s extremely u n l i k e l y that i t would be e f f e c t i v e .

CHAPTER 5

- 1

ajiWj

-

^Vi

-

^Gi, ^and Ci

2

x f o r a l l p o s s i b l e

j€I i

( P l ) is a n o - c a p a c i t y - s h a r i n g v e r s i o n of ( P ) . I t c a n b e viewed a l t e r n a t i v e l y as a L e o n t i e f s u b s t i t u t i o n problem w i t h economies-of- s c a l e , o r as a g e n e r a l i z a t i o n of break-even a n a l y s i s ' f o r inter- dependent p r o d u c t s .

S i n c e ( P l ) i s (P) w i t h B = I , i t is o b v i o u s that t h e s o l u t i o n a p p r o a c h t o (P) c a n b e d i r e c t l y a p p l i e d t o ( P I ) . For t h i s s i m p l e r s t r u c t u r e , however, we show that a s u f f i c i e n t c o n d i t i o n f o r i m p o r t s similar t o t h e c o n d i t i o n f o r d o m e s t i c p r o d u c t i o n u s e d in t h e problem r e d u c t i o n s t a g e ( S t a g e l ) , c a n b e o b t a i n e d . S t r o n g e r r e s u l t s can a l s o b e d e r i v e d f o r t h e B-B s t a g e based on

(c),

t h e LP r e l a x a t i o n o f ( P l )

,

g i v e n below:

/

-

If a i j = 0 ( P I ) decomposes i n t o I s i m p l e make-buy problems.

(x)

^hasp r e c i s e l y t h e s t r u c t u r e of

(F) ,

and t h e two-step s o l u t i o n approach a p p l i e s , w i t h t h e

Si

computed a t S t e p I ( w i t h Fi ) and t h e o b t a i n e d a t S t e p 11. Theorem 3 . 1

Si = Hi

-

i i

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

(5)

a l s o , and t h u s a t l e a s t one Si must be s t r i c t l y p o s i t i v e i f any domestic p r o d u c t i o n i s t o t a k e p l a c e .

Theorem 3.2, a s b e f o r e , c a n be used t o e l i m i n a t e a c t i v i t i e s noncompe- t i t i v e a t m a r g i n a l c o s t ; a s t h i s theorem a p p l i e s t o a l l t h e v e r s i o n s of ( P I ) s t u d i e d in t h i s c h a p t e r , w e assume h e r e a f t e r t h a t t h e set I c o n t a i n s o n l y t h e c o m p e t i t i v e a c t i v i t i e s ( i . e . ,

zi

^> ⁰ f o r each i E I ) .

The s u f f i c i e n t c o n d i t i o n s f o r domestic p r o d u c t i o n ( (3.5) and ( 3 . 6 ) ) , of S t a g e 1, become r e s p e c t i v e l y

and

w i t h t h e and

6

having t h e same i n t e r p r e t a t i o n a s i n Chapter 3.

I f I* i s t h e set of a c t i v i t i e s t h a t s a t i s f y ( 5 . 1 ) , t h e n any a c t i v i t y i

d

^I* t h a t s a t i s f i e s ( 5 . 2 ) i s added t o I* i n an i t e r a t i v e f a s h i o n a s d e s c r i b e d i n Chapter 3 . A t t h e end of t h i s s t a g e a set of a c t i v i - t i e s I*

c -

I i s i d e n t i f i e d which i s known t o b e o p t i m a l l y u n d e r t a k e n .

The problem r e d u c t i o n s t a g e f o r ( P I ) does n o t end h e r e , however. W e g i v e n e x t a s u f f i c i e n t c o n d i t i o n t h a t c a n be used i t e r a t i v e l y t o iden-

t i f y a c t i v i t i e s t h a t a r e o p t i m a l l y imported i n ( P I )

.

I n t h e a b s e n c e of c a p a c i t y s h a r i n g t h e t o t a l v a r i a b l e s a v i n g s a t t a b e d by u n d e r t a k i n g a g i v e n a c t i v i t y must c o v e r t h e f i x e d c o s t i n c u r r e d in u n d e r t a k i n g it.- I/ I n o t h e r words, t o t a l b e n e f i t s must c o v e r t o t a l c o s t s ; t h i s i s t r u e whether p r o d u c t i o n is f o r f i n a l consumption, f o r i n t e r m e d i a t e i n p u t in t h e production of o t h e r pro- d u c t s , o r both. I f x; i s t h e o p t i m a l l e v e l f o r a c t i v i t y i and H* i t s a s s o c i a t e d v a r i a b l e s a v i n g s ( i . e . , W i

-

H U s th e v a r i a b l e

1 1

c o s t of producing product i )

,

t h e n f o r e a c h i such t h a t x t > 0 t h e c o n d i t i o n H+x* > F must be s a t i s £ ie d . It i s e a s y t o see t h a t

1 1 i

i f t h i s c o n d i t i o n is n o t s a t i s f i e d t h e n t h e minimum a v e r a g e p r o d u c t i o n c o s t a t t a i n a b l e i s l a r g e r t h a n t h e import cost2' and t h u s no s a v i n g s c a n be o b t a i n e d by u n d e r t a k i n g domestic p r o d u c t i o n .

1/

Under c a p a c i t y s h a r i n g a n a c t i v i t y w i t h p o s i t i v e m a r g i n a l s a v i n g s o v e r import c o s t could o p t i m a l l y be undertaken which d i d n o t c o v e r i t s a s s o c i a t e d f i x e d c o s t s . It was o n l y r e q u i r e d t h a t a l l pro- c e s s i n t e r d e p e n d e n t a c t i v i t i e s j o i n t l y covered f i x e d c o s t s .

ZI

Let MACi = minimum a v e r a g e domestic p r o d u c t i o n c o s t a t t a i n a b l e f o r product i.

Then MACi = (Wi

-

^HZ)

+

-;i; I

Based on t h e c o n d i t i o n given above that must be s a t i s f i e d by t h e

- -

a c t i v i t i e s can be e a s i l y e s t a b l i s h e d wi'th a v e r y s i m p l e B-B s t a g e .

where Hi = Wi

- 1

^ajiWj

-

^Vkbki

-

^Gi, ^{and bki} ^isd e f i n e d a s in

j E 1

( P ) . Ck i s a n u p p e r bound on t h e t o t a l a c t i v i t y of i n d u s t r y k f o r any p o s s i b l e v a l u e of x i E Ik.

'

(P2) o b v i o u s l y i n c l u d e s ( P I ) a s a s p e c i a l c a s e , and t h e b a s i c d i s t i n c t i o n between (P2) and (P) i s t h a t in (P) Ik

n

0

f o r a t l e a s t one k

#

t . (P2) h a s t h e f o l l o w i n g LP r e l a x a t i o n :

(3)

Max k€K

1 [I

^&Ik[ H i - 7 b k i ] x i ] Fk

Theorem 3 . 1 o b v i o u s l y a p p l i e s t o

(x)

^; and c o n d i t i o n s (5.1) and ( 5 . 2 ) become r e s p e c t i v e l y

( 5 . l a ) and

I f t h e c o n d i t i o n of t h e Theorem 5 . 1 i s r e p l a c e d by

1 ^H.;.

^< ^F

1 1 - k ' i E I k

t h e same r e s u l t s o b t a i n e d f o r ( P I ) f o l l o w f o r (P2). A c h a r a c t e r i s t i c of t h e s o l u t i o n of ( ~ 2 ) c a n b e s e e n t o b e t h a t e a c h i n d u s t r y k w F l l e i t h e r p r o d u c e a l l p r o d u c t s i E I

'

o r none, where I k c o n t a i n s o n l y p r o d u c t s i s u c h t h a t

gi

^> 0 , a s w e assume t h a t a l l p r o d u c t s w i t h

H

^<0 have been e l i m i n a t e d from t h e set I .

-

5.2.2 A l t e r n a t i v e P r o d u c t i o n Techniques

I n t h i s s u b s e c t i o n w e e x t e n d t h e b a s i c model ( P I ) t o a l l o w f o r c h o i c e among a l t e r n a t i v e p r o d u c t i o n t e c h n i q u e s f o r e a c h p r o d u c t , and show how t h e r e s u l t s o b t a i n e d f o r ( P l ) a p p l y t o t h i s v e r s i o n o f t h e problem.

L e t T b e t h e set of a l t e r n a t i v e t e c h n i q u e s f o r p r o d u c t i, as i

w e l l as t h e c a r d i n a l i t y of t h e set. T f i x e d - c h a r g e c o s t f u n c t i o n s i

are t h u s s p e c i f i e d f o r e a c h i E I. T h i s v e r s i o n of t h e problem, l a b e l e d ( P 3 ) , h a s t h e f o l l o w i n g f o r m u l a t i o n ^:

t t t

where Hi = Wi

- 1

^ajiWj

-

^Vi ^and Ci is a n upper bound on j € 1

xi, t f o r any t ^ETi. S u p e r s c r i p t s t have been added t o t h e appro- p r i a t e p a r a m e t e r s t o d e n o t e t h e i r dependence on t h e p r o d u c t i o n t e c h - n i q u e s . A set of c o n s t r a i n t s of t h e form

i s n o t r e q u i r e d in (P3), as t h e s u b s t i t u t i o n theorem g u a r a n t e e s that o n l y one t e c h n i q u e t from e a c h T w i l l b e used i f p r o d u c t i i s

d o m e s t i c a l l y produced

.-

I n o t h e r words, a t most one x . , ^t

1 t E Ti, o r Yi ( t h e s l a c k v a r i a b l e ) w i l l b e a t a p o s i t i v e l e v e l f o r each

I t i s assumed in t h e f o r m u l a t i o n of (P3) that t h e i n p u t - o u t p u t s t r u c t u r e d o e s n o t v a r y w i t h t h e p r o d u c t i o n t e c h n i q u e . T h i s i s a r e a s o n a b l e assumption f o r c h o i c e of t e c h n i q u e problems w i t h i n p u t - o u t p u t r e l a t i o n s h i p s s p e c i f i e d a t t h e product l e v e l . It merely s a y s t h a t each t e c h n i q u e t E Ti under c o n s i d e r a t i o n f o r p r o d u c t i r e q u i r e s t h e same endogenous i n p u t s ( p a r t s , components, modules, sub- a s s e m b l i e s , e t c . ) i n i t s p r o d u c t i o n p r o c e s s , b u t may have c o m p l e t e l y

L/

See Chapter 3 .

d i f f e r e n t exogenous i n p u t r e q u i r e m e n t s , l a b o r s k i l l s , e t c . The c o s t

c o s t f u n c t i o n s . Thus, w e c a n c o n c l u d e that a f t e r t h e e l i m i n a t i o n of n o n c o m p e t i t i v e t e c h n i q u e s , t h e " a l t e r n a t i v e p r o d u c t i o n t e c h n i q u e s "

v e r s i o n of ( P I ) i s i d e n t i c a l t o t h e "multi-segment c o s t f u n c t i o n "

v e r s i o n .

The LP r e l a x a t i o n of (P3) i s

With A upper t r i a n g u l a r , t h e c o n s t r a i n t m a t r i x of

(x)

^isb l o c k t r i a n g u l a r ; row i now c o n t a i n s T.

+

1 p o s i t i v e e l e m e n t s , i n c l u d i n g

t h e c o e f f i c i e n t of t h e s l a c k v a r i a b l e y _i

.

I f w e s o l v e

(z)

^{w i t h}^{a l l} _Ci ₌ ( e q u i v a l e n t l y , a l l :F = 0) t h e n o b v i o u s l y any p r o d u c t i t h a t i s d o m e s t i c a l l y produced i n t h e s o l u t i o n t o s u c h problems w i l l be produced by t h e t e c h n i q u e w i t h t h e

t t

l o w e s t (Pi

+

^{G .)}

,

namely t e c h n i q u e Ti. Thus, r a t h e r t h a n working

w i t h t h e f u l l problem, t h e a p p r o a c h t a k e n h e r e c o n s i s t s o f working w i t h a sequence of problems s u c h t h a t e a c h h a s a s i n g l e p r o d u c t i o n t e c h n i q u e s p e c i f i e d f o r e a c h p r o d u c t . I n i t i a l l y w e c o n s i d e r t h e problem w i t h t = Ti o n l y , f o r e a c h i E I. To t h i s problem

m 1

-

< 0 c a n b e e l i m i - Theorem 3 . 2 a p p l i e s and a l l a c t i v i t i e s w i t h Hi -

n a t e d , a s t h e y a r e o p t i m a l l y imported i n (P3). Note t h a t n o t o n l y t e c h n i q u e

Ti i s e l i m i n a t e d in t h i s c a s e , b u t a l l Ti t e c h n i q u e s

a r e , s i n c e -Ti <

o

= -t H~ c

o

f o r any t E T ~ .

Proof. By c o n d i t i o n (5.4) we h o w t h a t Hi t > H ^{t L} f o r a l l i E I

r e q u i r e m e n t that

f

ct

^t ^t

b a s i c s o l u t i o n . Once a p r o d u c t i o n a c t i v i t y a t e n t e r s t h e b a s i s i t

i n t r o d u c e d i n t o t h e b a s i s , o r t e c h n i q u e s w i t c h i n g f o r t h e a c t i v i t i e s

5.2.3 General Concave Cost Function

Consider now t h e c a s e i n which t h e investment c o s t f u n c t i o n f o r c a p a c i t y i s given by a g e n e r a l concave c o s t f u n c t i o n Fi(xi), w i t h

dFi(xi) d2p (x )

> 0 _and

-

^c 0 f o r a l l p o s s i b l e v a l u e s of xi, f o r

dxi _dx2

-

each i E I . T h i s problem i s formulated a s

The Hi in t h i s c a s e a r e d i s t i n g u i s h e d from t h e Hi of t h e p r e v i o u s problems by t h e f a c t t h a t v a r i a b l e investment c o s t s a r e n o t included a s components of v a r i a b l e domestic p r o d u c t i o n c o s t s .

I n t h e p r e v i o u s s u b s e c t i o n was shown t h e e q u i v a l e n c e between (P3) and a v e r s i o n of ( P I ) w i t h multi-segment concave c o s t f u n c t i o n s . We show h e r e that (P4) can be solved by s o l v i n g a n "equivalent"

problem w i t h a p i e c e w i s e l i n e a r c o s t f u n c t i o n .

The " e q u i v a l e n t " problem w i t h p i e c e w i s e l i n e a r approximation of t h e c o s t c u r v e s F (x ) i s o b t a i n e d s e q u e n t i a l l y . The e q u i v a l e n c e

i i

i s in t h e s e n s e that a t t h e o p t i m a l s o l u t i o n t h e p i e c e w i s e l i n e a r approximation c o i n c i d e s w i t h t h e t r u e c o s t f u n c t i o n . Let Ci be a n

u p p e r bound on a l l p o s s i b l e v a l u e s of x

-

we lmve an o p p o r t u n l t y t o r e a p p l y Theorem 3 . 2 i f one o r more 1

CHAPTER 6 CONCLUSION

6 . 1 Summary

I n t h i s s t u d y a s o l u t i o n procedure w a s developed f o r a c l a s s of investment planning models which i n c o r p o r a t e s t h e f o l l o w i n g f e a - t u r e s : economies-of-scale i n p r o d u c t i o n , i n t e r m e d i a t e i n p u t - o u t p u t r e l a t i o n s h i p s among p r o d u c t i o n a c t i v i t i e s , and c a p a c i t y s h a r i n g . The c h o i c e i s between domestic p r o d u c t i o n and i m p o r t s t o s a t i s f y exogenously stateddemands f o r a g i v e n b i l l of goods. The model w a s p r e s e n t e d i n Chapter 2 , which a l s o p r o v i d e s a b r i e f d i s c u s s i o n of t h e complex i n t e r d e p e n d e n c i e s t h a t e x i s t among p r o d u c t i o n a c t i v i t i e s and t h e i r p o t e n t i a l e f f e c t s on investment d e c i s i o n s .

The t h e o r e t i c a l a n a l y s i s of t h e planning model was done i n

Chapter 3 . Simple s u f f i c i e n c y c o n d i t i o n s f o r import and f o r domestic p r o d u c t i o n of a g i v e n product were d i s c u s s e d and a problem r e d u c t i o n

s t a g e w a s developed which a p p l i e s t h e s e c o n d i t i o n s i n a n i t e r a t i v e f a s h i o n . A c t i v i t i e s which a r e n o t c o m p e t i t i v e w i t h i m p o r t s when maximum b e n e f i t s from interdependence a r e assumed a r e i d e n t i f i e d a t t h i s s t a g e , a s w e l l a s a c t i v i t i e s t h a t a r e p r o f i t a b l e even when no advantage i s t a k e n of interdependence. For t h e s o l u t i o n s t a g e an LP-based branch-and-bound (B-B) a l g o r i t h m w a s developed. It was shown t h a t t h e LP r e l a x a t i o n of t h e planning problem i s a s i m p l e maximiza- t i o n over a Leontief s u b s t i t u t i o n s t r u c t u r e f o r which a v e r y e f f i c i e n t s o l u t i o n approach e x i s t s . The development of t h i s c h a p t e r assumed a n

u p p e r - t r i a n g u l a r input-output s t r u c t u r e f o r t h e p l a n n i n g model. It was shown t h e n how "data r e l a x a t i o n 1 ' a l l o w s f o r t h e same problem r e d u c t i o n s t a g e t o b e used f o r n o n t r i a n g u l a r i n p u t - o u t p u t s t r u c t u r e s , and, most i m p o r t a n t l y , it w a s shown that t h e LP r e l a x a t i o n can a l s o be v e r y e f f i c i e n t l y solved in t h i s c a s e , and t h e B-B s t a g e i s t h u s

r e a d i l y a p p l i c a b l e t o t h i s more g e n e r a l problem.

Computational r e s u l t s o b t a i n e d from t h e implementation of t h e s o l u t i o n approach developed in Chapter 3 u s i n g t h e Korea s t u d y d a t a were given in Chapter 4. The r e s u l t s from 25 test problems g e n e r a t e d by making p a r a m e t r i c changes i n t h e K o r e a n d a t a p r o v i d e d s t r o n g evidence of t h e e f f i c i e n c y of o u r two-stage approach. The problem r e d u c t i o n

s t a g e w a s v e r y s u c c e s s f u l i n r e d u c i n g t h e s i z e of t h e problem. For s e v e r a l problems, i n f a c t , a l l p r o f i t a b l e p r o d u c t i o n a c t i v i t i e s w e r e i d e n t i f i e d a t t h i s s t a g e , a l t h o u g h t h i s c o u l d o n l y be laown a f t e r t h e B-B s t a g e . More important t h a n t h e f i r s t s t a g e , however, w a s t h e e f f i c i e n c y of t h e B-B s t a g e . Computational e x p e r i e n c e o b t a i n e d f o r t h e B-B s t a g e a l o n e showed t h a t i t s e f f i c i e n c y does n o t depend i n any s i g n i f i c a n t way on t h e s u c c e s s of a problem r e d u c t i o n s t a g e . T h i s i s important f o r two r e a s o n s : F i r s t , t h e e f f e c t i v e n e s s of t h e problem r e d u c t i o n s t a g e d e c r e a s e s w i t h t h e d e g r e e of i n t e r d e p e n d e n c e among t h e a c t i v i t i e s ; and secondly, f o r n o n t r i a n g u l a r i n p u t - o u t p u t s t r u c t u r e s it i s u n c e r t a i n whether i t "pays1' t o u s e a problem reduc- t i o n s t a g e and t h e s o l u t i o n t o such problems must t h e r e f o r e r e l y more h e a v i l y o r perhaps e n t i r e l y on t h e B-B s t a g e , o r some o t h e r

s o l u t i o n method.

The important c o n c l u s i o n from t h e computational e x p e r i e n c e of

Chapter 4 ^{i s}t h a t v e r y l a r g e problems c a n be e f f i c i e n t l y s o l v e d by o u r two-stage approach. Computational f e a s i b i l i t y s h o u l d t h u s no l o n g e r be an i s s u e in d e c i d i n g t h e l e v e l of a g g r e g a t i o n a t which such models should be e s t i m a t e d .

I n Chapter 5 v a r i o u s models of i n p u t - o u t p u t i n t e r d e p e n d e n c e (no c a p a c i t y s h a r i n g allowed) w e r e s t u d i e d , each i n c o r p o r a t i n g d i f f e r e n t f e a t u r e s . The two-stage a p p r a a c h was s p e c i a l i z e d t o e a c h . It w a s shown f o r t h e s e models that b e s i d e s t h e p o s s i b l e i d e n t i f i c a t i o n of a c t i v i t i e s known t o b e o p t i m a l l y u n d e r t a k e n , t h e problem r e d u c t i o n s t a g e y i e l d s a set of a c t i v i t i e s which c o n t a i n s a l l t h e a c t i v i t i e s i n t h e o p t i m a l s o l u t i o n t o t h e model and h a s t h e f o l l o w i n g p r o p e r t y : i f any one a c t i v i t y is dropped from t h e s e t , t o t a l s a v i n g s from u n d e r t a k i n g a l l t h e remaining a c t i v i t i e s d e c r e a s e s .

While i t i s u n l i k e l y that t h e simple models of Chapter 5 f i t any r e a l s i t u a t i o n , t h e y c o n s t i t u t e n e v e r t h e l e s s b a s i c s u b s t r u c t u r e s of many important p l a n n i n g problems, i n c l u d i n g t h e p r o c e s s a n a l y s i s models c i t e d in Chapter 1. These s u b s t r u c t u r e s can be e x p l o i t e d in many ways in decomposition schemes t o o b t a i n e f f i c i e n t s o l u t i o n

approaches t o more complex real problems, and t h i s is perhaps t h e m a i n v a l u e of t h e a n a l y s i s of Chapter 5.

6.2 Suggest i o n s f o r F u r t h e r Research

An obvious a r e a f o r f u r t h e r r e s e a r c h is t h e i n t r o d u c t i o n of s p a t i a l and dynamic e l e m e n t s i n t h e p l a n n i n g model ( P ) . Another important e x t e n s i o n of (P) would be t h e i n c o r p o r a t i o n of a l t e r n a t i v e p r o d u c t i o n t e c h n i q u e s f o r each product. It seems t h a t t h i s p a r t i c u l a r

e x t e n s i o n c o u l d be i n c o r p o r a t e d in our two-stage approach w i t h only minor changes a l o n g t h e l i n e s of t h e a l t e r n a t i v e t e c h n i q u e s e x t e n s i o n of ( P I ) .

It was p o i n t e d o u t in t h e p r e v i o u s s e c t i o n that t h e problems s t u d i e d in Chapter 5 c o n s t i t u t e i m p o r t a n t s u b s t r u c t u r e s o f many p l a n n i n g problems. ks most of t h e s u c c e s s f u l a p p r o a c h e s t o mixed i n t e g e r programming problems r e l y h e a v i l y o n e x p l o i t i n g s t r u c t u r e s , t h e r e s u l t s of Chapter 5 c a n b e u s e f u l i n s o l v i n g more complex problems.

A s e r i o u s d i f f i c u l t y , however, o c c u r s when c o n s t r a i n t s a r e added t o t h e s e s i m p l e models, o r even t o ( P ) , which e x p l i c i t l y p l a c e a l i m i t on some r e s o u r c e . In t h e s e c a s e s t h e dichotomy between d o m e s t i c pro- d u c t i o n and i m p o r t s w i l l n o t be in g e n e r a l a p r o p e r t y of t h e o p t i m a l s o l u t i o n . T h i s i s t h e c a s e , f o r example, i f a budget c o n s t r a i n t is added t o (P) o r t o any of t h e models of Chapter 5. It i s a n i m p o r t a n t e x t e n s i o n t h a t would a l l o w s e c t o r a l models l i k e (P) t o b e imbedded i n economy-wide models. T h i s c o n s t i t u t e s a v e r y d i f f i c u l t problem, a s no p r i c e c a n b e found in g e n e r a l f o r t h e l i m i t e d r e s o u r c e which would l e a d t o t h e o p t i m a l p r o d u c t i o n d e c i s i o n s . - I/

An example i s g i v e n n e x t of a n e x t e n s i o n of one of t h e s i m p l e models of Chapter 5 which c o n s t i t u t e s a v e r y i m p o r t a n t c l a s s of

problems and f o r which t h e d i f f i c u l t y d e s c r i b e d above d o e s n o t o c c u r . The g e n e r a l approach of this d i s s e r t a t i o n seems t o be promising f o r t h i s c l a s s of problems. Consider t h e f o l l o w i n g m u l t i - p e r i o d e x t e n s i o n of ( P I ) , t h e most b a s i c s t r u c t u r e s t u d i e d in Chapter 5 . W e u s e a

T h i s immediately r u l e s o u t an o t h e r w i s e promising approach t o t h i s c o n s t r a i n e d v e r s i o n of (P)

,

which i s Lagrangian r e l a x a t i o n ( d u a l decomposition) w i t h r e s p e c t t o t h e added c o n s t r a i n t ( s )

.

c o s t m i n i m i z a t i o n f o r m u l a t i o n of ( P I ) w i t h o u t s u b s t i t u t i n g f o r t h e import v a r i a b l e s s o that a l l t h e p a r a m e t e r s a p p e a r e x p l i c i t l y .

We assume that :V i n c l u d e s t h e v a r i a b l e p r o d u c t i o n c o s t component G:, and, a s b e f o r e , that t h e i n p u t - o u t p u t matrix A i s upper t r i - a n g u l a r . I f we g i v e t h e f o l l o w i n g i n t e r p r e t a t i o n t o t h e v a r i a b l e s and p a r a m e t e r s in (P5):

xt = q u a n t i t y of product i produced o r o r d e r e d a t t h e i

b e g i n n i n g o f p e r i o d t ,

yt = ending i n v e n t o r y of p r o d u c t i in p e r i o d t

,

v 1

⁼ v a r i a b l e c o s t of p r o d u c t i o n f o r product i i n p e r i o d t

'i ⁼ h e n t o r y h o l d i n g c o s t p e r u n i t of p r o d u c t i in t

,

: ₌ f i x e d c o s t component of p r o d u c t i o n c o s t , set-up c o s t , o r o r d e r i n g c o s t , f o r p r o d u c t i in p e r i o d t ,

t h e n (P5) r e p r e s e n t s a n important c l a s s of problems o c c u r r i n g in m a t e r i a l r e q u i r e m e n t s p l a n n i n g (MRP) systems. (P5) i s s i m i l a r t o V e i n o t t ' s [I9691 m u l t i - f a c i l i t y economic-lot-size model, and i t i n c l u d e s a s s p e c i a l c a s e s t h e f o r m u l a t i o n o f Crowston, Wagner and Williams [1973], and Crowston and Wagner [1973].

An o p t i m a l s o l u t i o n t o (P5) e x i s t s which is an extreme p o i n t of t h e m a t e r i a l b a l a n c e c o n s t r a i n t s , and t h e LP r e l a x a t i o n o f (P5), as in a l l t h e models s t u d i e d h e r e , y i e l d s a L e o n t i e f s u b s t i t u t i o n problem, I t can be s e e n immediately, by t h e s u b s t i t u t i o n theorem, t h a t a t most

t t t-1

i t-l w i l l be a t a p o s i t i v e l e v e l ( i . e . , xiyi ₌0) o n e o f x and yi

in t h e o p t d l s o l u + i o n o f (P5), and e x a c t l y o n e i f Di t > 0.

The r e s u l t s of t h i s s t u d y s u g g e s t t h a t t h e s p e c i a l i z a t i o n of o u r two-stage approach t o t h i s i m p o r t a n t c l a s s of problems might be a worthwhile r e s e a r c h e f f o r t .

REFERENCES

h n n e , A . S . , Investments f o r Capacity Expansion: S i z e , L o c a t i o n and Time-Phasing, George A l l e n and Unwin, L t d . , London, 1967.

Nam, J . W . , Y. W. Rhee and L. E. Westphal, "Data Development f o r a S t u d y of t h e Scope f o r C a p i t a l - L a b o r S u b s t i t u t i o n i n t h e

Mechanical E n g i n e e r i n g I n d u s t r i e s

,

^"Development R e s e a r c h C e n t e r , The World Bank, Washington, D . C . , and t h e Korean I n s t i t u t e o f S c i e n c e and Technology, S e o u l , 1973.

S t o u t j e s d i j k, A. and L. E. Westphal, e d s . , I n d u s t r i a l I n v e s t m e n t A n a l y s i s u n d e r I n c r e a s i n g R e t u r n s , M a n u s c r i p t , The World Bank, Washington, D. C . , September 1978.

V e i n o t t , A. F. J r . , "Minimum Concave-Cost S o l u t i o n of L e o n t i e f S u b s t i t u t i o n Models o f M u l t i - F a c i l i t y I n v e n t o r y S y s t e m s , "

O p e r a t i o n s R e s e a r c h , Vol. 1 7 , No. 2, March-April 1969.

V i e t o r i s z , T., "Programming of P r o d u c t i o n and E x p o r t s f o r Metalworking:

Models and P r o c e d u r e s , " i n P l a n n i n g and Programming of t h e Metalworking I n d u s t r i e s w i t h a S p e c i a l V i e w t o E x p o r t s ,

U n i t e d N a t i o n s I n d u s t r i a l Development O r g a n i z a t i o n , New York,

Westphal, L. E. and Y. W. Rhee, " P l a n n i n g i n t h e Mechanical

- ~ n ~ i n e e r i n ~ S e c t o r ^{, I 1} i n 1 n d u s t r i a l I n v e s t m e n t A n a l y s i s u n d e r I n c r e a s i n e R e t u r n s . A. S t o u t i e s d i i k and L. E. W e s t ~ h a l . e d s . . _" 4 4 - z

M a n u s c r i p t , The World Bank, Washington, D.C., September 1978.

Im Dokument Investment Programming for Interdependent Production Processes (Seite 67-100)