UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Algorithm for estimating the medians of a weighted graph subject to side constraints, and an application… Whitaker, Roy Alexander 1971

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ALGORITHM FOR ESTIMATING THE MEDIANS OF A WEIGHTED GRAPH SUBJECT TO S IDE C O N S T R A I N T S , AND AN A P P L I C A T I O N TO RURAL HOSPITAL LOCATIONS IN B R I T I S H COLUMBIA by ROY ALEXANDER WHITAKER B.A., U n i v e r s i t y of E x e t e r , 1962 MSc., P e n n s y l v a n i a S t a t e U n i v e r s i t y , 1965 A THES IS SUBMITTED IN P A R T I A L FULF ILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In the Department of G e o g r a p h y We accept this thesis as conforming to the required standard THE U N I V E R S I T Y OF B R I T I S H COLUMBIA M a r c h 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h C olumbia Vancouver 8, Canada Date ABSTRACT P l a n t l o c a t i o n as a c e n t r a l i z e d p l a n n i n g o b j e c t i v e i n w h i c h some a g e n c y has c o n t r o l o v e r m o s t o f t h e s y s t e m e l e m e n t s c a n be r e d u c e d , i n many c i r c u m s t a n c e s , t o t h e p r o b l e m o f f i n d i n g t h e m e d i a n s o f a w e i g h t e d g r a p h . T h i s c o n c e p t i s f e a s i b l e i f i t c a n be a s s u m e d t h a t e a c h l o c a t i o n s o u g h t i s c o n s t r a i n e d t o a s u b s e t o f p n o d e s on an n node n e t w o r k . T h i s c o m b i n a t o r i a l p r o g r a m m i n g p r o b l e m c a n be f o r m a l l y s t a t e d as f o l l o w s : i f G i s a w e i g h t e d g r a p h , w ( v . v . ) t h e w e i g h t e d d i s t a n c e o f node v . t o node v . , and Xp i s a n y s e t o f p n o d e s on G ( x l s x 2 , • • • , X p ) , t h e n t h e r e q u i r e d s e t o f p n o d e s Xp on G i s t h e p m e d i a n o f t h e g r a p h i f i t s a t i s f i e s t h e e x p r e s s i o n n n A l t h o u g h t h i s o b j e c t i v e c a n be e x p l i c i t l y o p t i m i z e d by b r a n c h b o u n d a l g o r i t h m s , t h o s e d e v e l o p e d t o d a t e c a n become c o m p u t a t i o n a l l y i n f e a s i b l e f o r some l a r g e s c a l e p r o b -l e m s . A f a s t m e t h o d f o r e s t i m a t i n g t h e m e d i a n s o f a w e i g h t e d J g r a p h i s g i v e n w h i c h w i l l p r o v i d e o p t i m a l o r n e a r o p t i m a l s o l u t i o n s on any t y p e o f n e t w o r k . The h e u r i s t i c p r o c e d u r e s a d o p t e d i n t h i s s t u d y c a n be g e n e r a l i z e d i n t e r m s o f t h r e e b a s i c s t e p s ; 1) p a r t i t i o n t h e g r a p h t o o b t a i n an i n i t i a l f e a s i b l e s o l u t i o n , 2) r e - i t e r a t e over ; s t e p 1 t o a c h i e v e a l o c a l m i n i m u m , and 3) p e r t u r b t h i s c o n v e r g e n c e t o t e s t f o r a l o w e r b o u n d . The d e s i g n o f s t e p s 1 and 3 a r e c r u c i a l t o t h e s u c c e s s o f t h e a l g o r i t h m i c m e t h o d . Two p r o c e d u r e s a r e g i v e n f o r t h e b a s i c p a r t i t i o n i n g o f t h e g r a p h , one o f w h i c h i s a m o d i f i c a t i o n o f a c r i t e r i o n o r i g i n a l l y d e v e l o p e d by S i n g e r ( 1 9 6 8 ) . The o t h e r m e t h o d i n t r o d u c e s a node e l i m i n a t i o n r e c u r s i o n w h i c h a p p e a r s , e x p e r i m e n t a l l y , t o be t h e more e f f i c i e n t p r o c e d u r e f o r c e r t a i n t y p e s o f w e i g h t e d n e t w o r k s . E f f i c i e n t p e r t u r b a t i o n m e t h o d s a r e d e v e l o p e d f o r t e s t i n g t h e l o w e r b o u n d s o b t a i n e d . The b a s i c mode l s t r u c t u r e i s m o d i f i e d by t h e i n t r o -d u c t i o n o f h e u r i s t i c s f o r t h e c o n s t r a i n e d p l a n t l o c a t i o n p r o b l e m u n d e r a w i d e v a r i e t y o f r e s t r i c t i o n s . N u m e r i c a l p r o c e d u r e s a r e s u g g e s t e d f o r r e s t r i c t i n g t h e s e a r c h t o a s u b s e t o f m p o t e n t i a l p l a n t s i t e s among a l l n n o d e s on t h e n e t w o r k . H e u r i s t i c s a r e d e v e l o p e d f o r f o r c i n g c e r t a i n l o c a t i o n s i n t o s o l u t i o n , f o r p l a c i n g u p p e r b o u n d c o n s t r a i n t s on p l a n t s i z e s , and f o r r e s t r i c t i n g t h e maximum l i n k d i s -t a n c e o v e r w h i c h a p a r t i c u l a r a l l o c a t i o n m i g h t be m a d e . A t t e n t i o n i s g i v e n t o t h e p r o b l e m o f e s t i m a t i n g t h e j o i n t m i n i m i z a t i o n o f p l a n t and t r a n s p o r t a t i o n c o s t f u n c t i o n s o v e r a n e t w o r k s u r f a c e . F o r d y n a m i c l o c a t i o n - a l l o c a t i o n s y s t e m s an e x p l i c i t d y n a m i c p r o g r a m m i n g f o r m u l a t i o n i s d e v e l o p e d f o r t h e o p t i m a l s e q u e n c i n g o f p l a n t l o c a t i o n s o v e r t i m e s u b j e c t , i f n e c e s s a r y , t o p e r i o d i c v a r i a t i o n s i n a l l c o s t f u n c t i o n s and node w e i g h t s . An a p p l i c a t i o n o f t h e b a s i c m e d i a n a l g o r i t h m t o t h e p r o b l e m o f r u r a l h o s p i t a l l o c a t i o n s i n S o u t h e a s t B r i t i s h C o l u m b i a i s d e m o n s t r a t e d , and c o m p u t e r c o d e s a r e l i s t e d f o r a l l t h e s p e c i f i e d m o d e l s . P a g e ABSTRACT '. i i T A B L E OF CONTENTS . v L I S T OF T A B L E S . . i x L I S T OF F IGURES x ACKNOWLEDGMENTS . . . x i i CHAPTER T - INTRODUCTION AND STUDY OUTLINE 1 O b j e c t i v e s o f S t u d y 2 S c o p e o f t h e S t u d y . . . . . . . . 4 S t u d y O u t l i n e 6 CHAPTER I I - REGIONAL HEALTH P L A N N I N G : SOME P E R S P E C T I V E S AND PROBLEMS 8 The R a t i o n a l e f o r R e g i o n a l H e a l t h P l a n n i n g . . . . . 8 H i s t o r i c a l A n t e c e d e n t f o r R e g i o n a l i z a t i o n . . . . 11 Some I n h e r e n t P r o b l e m s i n R e g i o n a l P l a n n i n g . . . 1 8 The R o l e o f t h e M o d e r n H o s p i t a l 19 The R o l e o f S o c i a l M e d i c i n e 22 I n s u r a n c e C o v e r a g e and t h e U t i l i z a t i o n o f M e d i c a l C a r e . 28 M e d i c a l C a r e i n t h e R u r a l S e t t i n g 30 Some P r e v i o u s M o d e l s o f H e a l t h F a c i 1 i t y L o c a t i o n 32 A G e n e r a l A p p r o a c h t o R u r a l H o s p i t a l L o c a t i o n S y s t e m s 38 CHAPTER I I I - AN ALGORITHM FOR ESTIMATING THE MEDIANS OF A WEIGHTED GRAPH SUBJECT TO S IDE CONSTRAINTS 41 P r o b l e m S t a t e m e n t 41 F o r m u l a t i o n o f t h e P r o b l e m i n Terms o f a W e i g h t e d G r a p h ' . . 46 Some P r e v i o u s A p p r o a c h e s t o t h e P r o b l e m 48 ( a ) E x a c t S o l u t i o n M e t h o d s . . . . 48 ( b ) H e u r i s t i c - S o l u t i o n s 52 The S i n g e r A l g o r i t h m 57 Some Comments on T h e s e A l g o r i t h m s 69 Some A l t e r n a t i v e P a r t i t i o n i n g and P e r t u r b a t i o n M e t h o d s 70 An A l g o r i t h m f o r E s t i m a t i n g t h e M e d i a n s o f a W e i g h t e d G r a p h 73 R e s t r i c t i o n s on P o t e n t i a l P l a n t S i t e s 76 P l a n t S i z e U p p e r B o u n d s 78 Maximum D i s t a n c e C o n s t r a i n t 92 P l a n t C o s t 93 D y n a m i c L o c a t i o n - A l l o c a t i o n S y s t e m s . . 97 A D y n a m i c P r o g r a m f o r t h e O p t i m a l S e q u e n c i n g o f P l a n t L o c a t i o n s O v e r T i m e 107 CHAPTER IV - AN A P P L I C A T I O N OF THE MEDIAN ALGORITHM TO RURAL HOSPITAL LOCATIONS IN A S U B - R E G I O N OF B R I T I S H COLUMBIA 119 C h o i c e o f R e g i o n a l S t u d y A r e a 120 G e n e r a t i o n o f a M e d i c a l Demand N e t w o r k f o r S o u t h e a s t e r n B r i t i s h C o l u m b i a . . . . 122 A M e a s u r e o f t h e L o c a t i o n a l E f f i c i e n c y o f t h e H o s p i t a l S y s t e m o f S o u t h e a s t B r i t i sh C o l u m b i a 1 30 P r e d i c t i o n o f A c t u a l F l o w P a t t e r n s W i t h i n t h e S u b - R e g i o n 132 A M e t h o d f o r E s t i m a t i n g O p t i m a l P l a n t S i z e O v e r t h e C u r r e n t S y s t e m o f H o s p i t a l s i n t h e R e g i o n 141 CHAPTER V - SUMMARY AND CONCLUSIONS 144 L I T E R A T U R E CITED 148 A P P E N D I C E S I. Node P o p u l a t i o n s and Demands by M e d i c a l C a t e g o r i e s 155 I I . S h o r t D i s t a n c e M a t r i x B e t w e e n Nodes and P o t e n t i a l H o s p i t a l S i t e s . . . . 156 I I I . A C o m p u t e r P r o g r a m f o r C a l c u l a t i n g S h o r t P a t h s T h r o u g h a N e t w o r k (The D a n t z i g A l g o r i t h m ) 160 I V . A C o m p u t e r P r o g r a m f o r W a r d ' s H i e r a r c h i c a l G r o u p i n g A l g o r i t h m W i t h D e n d o g r a m 167 V . An A l g o r i t h m f o r t h e G r a p h i c a l G e n e r a -t i o n o f a D e n d o g r a m From any S t e p w i s e H i e r a r c h i c a l G r o u p i n g M e t h o d 178 V I . A C o m p u t e r P r o g r a m f o r E s t i m a t i n g t h e M e d i a n s o f a W e i g h t e d G r a p h S u b j e c t , i f o p t i o n e d , t o s i d e C o n s t r a i n t s 184 V I I . A C o m p u t e r P r o g r a m f o r t h e O p t i m a l S e q u e n c i n g o f P l a n t L o c a t i o n s O v e r T i m e by D y n a m i c P r o g r a m m i n g . . 226 V I I I . P r o g r a m Deck S e t Up f o r a l l S a m p l e P r o b l e m s D i s c u s s e d i n C h a p t e r I I I . . 240 L I S T OF TABLES T a b l e r Page 1 . S h o r t D i s t a n c e M a t r i x f o r F i g u r e 4 63 2 . The A M a t r i x f o r F i g u r e 4 64 3 . P o s s i b l e S o l u t i o n s t o t h e 4 Med ium P r o b l e m o f F i g u r e 6 by C o o p e r ' s H e u r i s t i c s 81 4 . S o l u t i o n T a b l e u f o r F i g u r e 6 88 5 . O p t i m a l S e q u e n c i n g o f P l a n t L o c a t i o n s f o r a H y p o t h e t i c a l E x a m p l e 112 6 . A b s o l u t e Number o f S c h o o l D i s t r i c t S e p a r a t i o n s by M e d i c a l C a t e g o r i e s . . . . . 124 7 . C o m p a r i s o n o f D i s t a n c e M i n i m i z i n g R e s u l t s : C o n s t r a i n e d and U n c o n s t r a i n e d S o l u t i o n s 131 8 . C o m p a r i s o n o f C o n s t r a i n e d and U n c o n -s t r a i n e d L o c a t i o n S y s t e m s 133 9 . F l o w P a t t e r n P r e d i c t i o n : D i s t a n c e M i n i m i z i n g and I n t e r a c t a n c e M o d e l s 136 L I S T OF F IGURES F i g u r e Page 1 . R e l a t i o n s h i p B e t w e e n P l a n t and T r a n s p o r t a t i o n C o s t F u n c t i o n s 43 2 . P i e c e w i s e L i n e a r A p p r o x i m a t i o n o f a C o n c a v e P l a n t C o s t F u n c t i o n 43 3 . P r i n c i p l e s o f t h e A p p l i c a t i o n o f a B r a n c h - B o u n d A l g o r i t h m t o P l a n t L o c a t i o n 50 4 . A 10 Node S a m p l e N e t w o r k P r o b l e m . . 62 5 . An 11 Node S a m p l e N e t w o r k P r o b l e m 65 6 . A 10 Node S a m p l e N e t w o r k P r o b l e m f o r P l a n t S i z e C o n s t r a i n t s 65 7 . A 20 Node S a m p l e N e t w o r k P r o b l e m . . . . . . . . 66 8 . a ) . 2 M e d i a n S o l u t i o n f o r F i g u r e 4 67 b) . 3 M e d i a n S o l u t i o n f o r F i g u r e 4 67 c ) . 3 M e d i a n S o l u t i o n f o r F i g u r e 5 67 9 . 2 M e d i a n S o l u t i o n ( L i m i t e d P o t e n t i a l S i t e s ) f o r F i g u r e 4 70 1 0 . a ) . 4 M e d i a n S o l u t i o n ( P l a n t S i z e C o n s t r a i n t s ) f o r F i g u r e 6 89 b ) . 2 and 3 M e d i a n S o l u t i o n s ( P l a n t S i z e C o n s t r a i n t s ) f o r F i g u r e 4 . . . . . . . 89 F i g u r e Page c ) . 2 M e d i a n S o l u t i o n (Maximum D i s t a n c e C o n s t r a i n t ) f o r F i g u r e 4 89 1 1 . a ) . C o m b i n e d P l a n t - T r a n s p o r t a t i o n C o s t S o l u t i o n f o r F i g u r e 4 98 b ) . 3 M e d i a n S o l u t i o n ( P l a n t C o s t s and P l a n t S i z e C o n s t r a i n t s ) f o r F i g u r e 4 . . . 98 1 2 . F o r w a r d R e c u r s i v e D y n a m i c S o l u t i o n f o r F i g u r e 7 . . 101 1 3 . F o r w a r d R e c u r s i v e (Node 9 F o r c e d ) D y n a m i c S o l u t i o n f o r F i g u r e 7 102 1 4 . B a c k w a r d R e c u r s i v e D y n a m i c S o l u t i o n f o r F i g u r e 7 106 1 5 . S c h o o l D i s t r i c t L o c a t i o n s i n S o u t h e a s t B r i t i s h C o l u m b i a 123 1 6 . M e d i c a l Demand N e t w o r k : Node and P o t e n t i a l H o s p i t a l S i t e s 129 1 7 . I l l u s t r a t i o n o f D e n d o g r a m A l g o r i t h m f o r a S i x S u b j e c t E x a m p l e 182 1 8 . G r a p h i c a l D i s p l a y o f a 40 S u b j e c t D e n d o g r a m G r o u p e d by t h e Ward A l g o r i t h m 183 ACKNOWLEDGMENTS I s h o u l d l i k e t o e x t e n d my g r a t i t u d e t o t h e many p e o p l e w h o s e a s s i s t a n c e and e n c o u r a g e m e n t made p o s s i b l e t h e c o m p l e t i o n o f t h i s s t u d y . I am i n d e b t e d t o D r . D. 0. A n d e r s o n f o r h i s h e l p w i t h t h e e p i d e m i o l o g i c a l a s p e c t s o f t h e s t u d y , t o D r . G . G a t e s and K. G . D e n i k e f o r t h e i r comments on t h e l o g i c a l s t r u c t u r e o f t h e a l g o r i t h m s , and t o D r . M. C h u r c h and M. J . P a t t e r s o n f o r t h e i r v a l u a b l e s u g g e s t i o n s w h i c h l e d t o i m p r o v e m e n t s i n t h e c o m p u t e r c o d e s . To M r s . M. W e r n t z I e x p r e s s my t h a n k s f o r r e d u c -i n g t h e c o m p u t e r t a p e s o f h o s p i t a l s e p a r a t i o n d a t a t o m a n a g e a b l e p r o p o r t i o n s , and t o t h e B r i t i s h C o l u m b i a H o s p i t a l I n s u r a n c e Scheme f o r m a k i n g t h e s e t a p e s a v a i l -a b l e . F i n a l l y I s h o u l d l i k e t o a c k n o w l e d g e w i t h a p p r e c i -a t i o n t h e C e n t r a l M o r t g a g e and H o u s i n g C o r p o r a t i o n f o r t h e i r g e n e r o u s s u p p o r t o f t h i s s t u d y o v e r i t s l a s t two y e a r s . CHAPTER I INTRODUCTION AND STUDY OUTLINE S c i e n t i f i c and t e c h n o l o g i c a l a d v a n c e s i n m e d i c i n e h a v e c r e a t e d a n e e d f o r r e f i n e m e n t s i n t h e o r g a n i z a t i o n s t h r o u g h w h i c h h e a l t h c a r e i s g i v e n . As m e d i c a l t e c h n o l o g y g r o w s e v e r more c o m p l e x i t has become i n c r e a s i n g l y a p p a r e n t t h a t an i n h e r e n t t e n d e n c y e x i s t s t o w a r d s e v e r more s p e c i a l -i z a t i o n and f r a g m e n t a t i o n i n t h e p r o v i s i o n o f m e d i c a l c a r e . T h i s t r e n d , t o g e t h e r w i t h m o u n t i n g s o c i a l c r i t i c i s m a g a i n s t t h e c o n t i n u a t i o n o f i n e q u a l i t i e s i n t h e r e c e i p t o f h e a l t h c a r e among some u n d e r p r i v i l e g e d s e g m e n t s o f t h e p o p u l a t i o n , has s e t i n m o t i o n an i n f l a t i o n a r y s p i r a l whose r e s u l t s a r e " t o be s e e n i n m e d i c a l c o s t s r i s i n g more r a p i d l y t h a n t h e e c o n o m y c a n c o m f o r t a b l y s u s t a i n ( M c L a c h l a n , 1 9 6 7 ) . The i m p l i c a t i o n s o f s u c h a t r e n d a r e p a r t i c u l a r l y s e r i o u s i f c o n s i d e r e d a g a i n s t t h e r e a l i t y t h a t t h e n e e d s o f m e d i c i n e m u s t c o m p e t e w i t h o t h e r s e c t o r s o f t h e economy f o r t h e a l l o c a t i o n o f r e s o u r c e s a l l t o o s c a r c e f o r t h e number and v a r i e t y o f demands p l a c e d upon t h e m . The r a t i o n a l u s e o f s u c h r e s o u r c e s as a r e a l l o t t e d t o t h e h e a l t h c a r e s e c t o r has t h e r e f o r e become a d e s i r a b l e e n d , one who se n e e d has b e e n i n c r e a s i n g l y r e c o g n i z e d o v e r t h e l a s t f e w y e a r s . Y e t g i v e n t h e c o m p l e x i t y o f t h e i n t e r n a l r e l a t i o n s h i p s w h i c h e x i s t among t h e v a r i o u s c o m p o n e n t s o f t h e m e d i c a l s e c t o r , and g i v e n t h e m u l t i p l i c i t y o f e x o g e n -e o u s e n v i r o n m e n t a l v a r i a b l e s w h i c h b e a r upon i t f r o m w i t h o u t , s u c h a g o a l r e q u i r e s o r has r e q u i r e d new p l a n n i n g s t r a t e g i e s a t a l m o s t e v e r y l e v e l o f o p e r a t i o n . O b j e c t i v e s o f S t u d y I t i s t h e a i m o f t h i s s t u d y t o f o c u s upon one s u b s e t o f t h e s e d e c i s i o n m a k i n g p r o c e s s e s , s p e c i f i c a l l y t h o s e w h i c h a r e c o n c e r n e d w i t h some f a c e t s o f t h e l o c a t i o n o f h o s p i t a l s * i n r u r a l a r e a s . The c o n s t r u c t i o n and m a i n t e n a n c e o f h o s -p i t a l u n i t s r e p r e s e n t s t h e s i n g l e m o s t c o s t l y c o m p o n e n t o f t h e h e a l t h c a r e s y s t e m , a n d a c c o u n t s f o r n e a r l y f o u r t y p e r c e n t o f t o t a l e x p e n d i t u r e s on m e d i c a l h e a l t h ( K l a m a n , 1 9 6 9 ) . A t s u c h a p r i c e i t seems a p p a r e n t t h a t t h e c o m m u n i t y a t l a r g e c a n no l o n g e r a f f o r d t h e p r a c t i c e o f o p e r a t i n g a s e p a r a t e h o s p i t a l i n l o c a l i t i e s w h e r e n e i t h e r l o c a l n e e d s n o r t h e s u p p l y o f q u a l i f i e d p e r s o n n e l c a n j u s t i f y t h e c o n -s t r u c t i o n o f s u c h c o s t l y u n i t s ( M y e r s , 1 9 6 1 ) . The end r e s u l t o f s u c h p o l i c i e s , a l l t o o p r e v a l e n t i n t h e p a s t , i s t o be s e e n i n t h e u n d e r u t i 1 i z a t i o n o f f a c i l -i t i e s and c o n s e q u e n t h i g h o p e r a t i n g c o s t s whose b u r d e n m u s t e v e n t u a l l y be b o r n e by t h e t a x p a y e r a t l a r g e . In c o n n e c t i o n w i t h t h i s p o i n t M c G i b o n y and B l o c k c o u l d s t a t e as e a r l y a s 1949 t h a t w h i l e some . . . small hospitals ave needed . . . it is also obvious that by themselves they ave not able to pvovide a complete sevvice to the patient and * n o n - m e t r o p o l i t a n that unless in some way they ave tied in with larger hospitals they may provide 3 not service, but a disservice to their -patients ( M c G i b o n y a n d B l o c k , 1 9 4 9 ) . A t t h e o t h e r end o f t h e s c a l e i t i s e q u a l l y c l e a r t h a t t h e p r o v i s i o n o f t o o f e w f a c i l i t i e s , p a r t i c u l a r l y i n r u r a l a r e a s , may r e s u l t i n t h e d e n i a l t o p e o p l e o f r e a s o n a b l e a c c e s s t o a l e v e l o f c a r e n e c e s s a r y f o r t h e m a i n t e n a n c e o f t h e i r h e a l t h . The o b j e c t i v e s o f t h i s s t u d y r e p r e s e n t an a p p r o a c h t o t h e r e s o l u t i o n o f t h i s c o n f l i c t . In more f o r m a l t e r m s t h e y may be s t a t e d as f o l l o w s : 1. To i d e n t i f y and a s s e s s some o f the major com-ponents which c o n t r o l the movement o f p a t i e n t s i n r u r a l a r e a s , t h r o u g h the m e d i c a l care system t o some s u i t a b l e h o s p i t a l t r e a t m e n t . 2. To d e v e l o p an o p e r a t i o n a l model based upon the most s i g n i f i c a n t c r i t e r i a d e v e l o p e d i n 1., which can r e p r o d u c e the system o f f l o w s i n the e x i s t -i n g network w i t h some s u c c e s s , g i v e n the con-s t r a i n t o f p r e s e n t l o c a t i o n s a t f i x e d s i z e s (measured i n terms o f some f u n c t i o n o f the number o f beds now a v a i l a b l e ) . 3 . To d e v i s e a h y p o t h e t i c a l h o s p i t a l system by means o f such a model t h a t c o n s i s t s o f the number, s i z e s and l o c a t i o n s o f f a c i l i t i e s which can s a t i s f y t he p a r t i c u l a r needs o f p a t i e n t a r e a s s u b j e c t t o the c o n s t r a i n t s t h a t , i . no p a t i e n t , or maximum number o f p a t i e n t s , be more th a n a maximum p r e d e t e r m i n e d d i s t a n c e ( o r time) from h o s p i t a l c a r e and i i . t h a t some upper bound i n t o t a l c o s t , i . e . a system budget con-s t r a i n t , i s not exceeded. Comparison o f the c u r r e n t and h y p o t h e t i c a l l o c a -t i o n s e t s w i l l a l l o w the e f f i c i e n c y o f the p r e s e n t o p e r a t i n g system t o be a s s e s s e d w i t h r e s p e c t t o the p a r t i c u l a r s p a t i a l needs o f i t s dependent p a t i e n t a r e a s . Imbalances i n the p r e s e n t network, i f f o u n d , can suggest one b a s i s f o r p o l i c y d e c i s i o n s c o n c e r n i n g the f u t u r e l o c a t i o n o r r e l o c a t i o n o f h o s p i t a l f a c i l i t i e s , and f o r the d e t e r m i n a t i o n o r c o n t i n u a t i o n o f r e f e r r a l c e n t r e s f o r more s p e c i a l i z e d t y p e s o f c a r e . • 4. To make p r o v i s i o n f o r the d i r e c t i n c o r p o r a t i o n i n t o t h e model o f the p r e s e n t l o c a t i o n a l p a t t e r n a t i t s c u r r e n t s c a l e o f o p e r a t i o n s . Then, g i v e n t h i s , some f u t u r e o p t i m a l o r n e a r - o p t i m a l system o f h o s p i t a l l o c a t i o n s ( a t a p a r t i c u l a r p o i n t i n t i m e ) may be d e v i s e d i f based upon some f o r e c a s t o f p a t i e n t demand a t d i s c r e t e p o i n t s o r a r e a s a c r o s s t h a t demand s u r f a c e whose bounds a r e the bounds o f the network as a. whole. T h i s o b j e c t i v e s i m p l y r e c o g n i z e s t h a t h o s p i t a l s once b u i l t , and however w i s e l y l o c a t e d t h e y may or may not be, ar e l i a b l e t o be m a i n t a i n e d f o r as l o n g as i s p o s s i b l e . L o c a t i o n a l i n e r t i a o f t h i s k i n d i s d i r e c t l y a t t r i b u t a b l e t o the h i g h i n i t i a l i n -vestment c o s t s o f h o s p i t a l c o n s t r u c t i o n and t o the w i d e s p r e a d p r e s s u r e s w i t h i n communities t o r e s i s t t he c l o s u r e o f any m e d i c a l f a c i l i t y once opened, p a r t i c u l a r l y i f t h e r e a r e no p l a n s f o r th e r e p l a c e m e n t o f the f a c i l i t y i n t h a t l o c a l i t y . 5. To make a l l o w a n c e f o r the m u l t i p l e l o c a t i o n o f f a c i l i t i e s o v e r t i m e . I f h o s p i t a l s a r e t o be l o c a t e d i n space a t s u c c e s s i v e time i n t e r v a l s t h e n , f o r some g i v e n l o n g term time p e r i o d , which p a r t i c u l a r subset o f a l l p o t e n t i a l l o c a -t i o n s i s r e q u i r e d such t h a t some o v e r a l l c r i t e r -i o n i s m i n i m i z e d ? S c o p e o f t h e S t u d y T h i s s t u d y f o r m u l a t e s a n u m e r i c a l a l l o c a t i o n m o d e l t o f u l f i l l t h e o b j e c t i v e s a s s t a t e d a b o v e , and g i v e s a p a r t i a l a p p l i c a t i o n t o one r e g i o n a l l y d e f i n e d s u b n e t w o r k w i t h i n t h e p r o v i n c e o f B r i t i s h C o l u m b i a . F u n d a m e n t a l t o t h e q u e s t i o n o f h o s p i t a l c o s t s i s t h e p h e n o m e n o n o f r e t u r n s t o s c a l e w h i c h , i n t h i s f i e l d , a r e n o t o r i o u s l y d i f f i c u l t t o a s s e s s . A s i m p l e c o r r e l a t i o n o f t h e r e l a t i o n s h i p b e t w e e n t h e a v e r a g e c o s t o f c o n s t r u c t i n g and m a i n t a i n i n g a bed and t h e s i z e o f t h e h o s p i t a l p l a n t v / i l l a c t u a l l y s u g g e s t t h e o p p o s i t e t r e n d . The r e a l i t y o f t h e s i t u a t i o n i s more s u b t l e s i n c e i n p r a c t i c e t h e r e e x i s t s a h i e r a r c h y o f s e r v i c e s w h e r e b y l a r g e r h o s p i t a l s , a c t i n g a s r e f e r r a l c e n t r e s , m a i n -t a i n s p e c i a l i z e d f u n c t i o n s i n a d d i t i o n t o t h o s e l o w e r o r d e r s e r v i c e s w h i c h a l l f a c i l i t i e s o f a l o w e r r a n k p r o v i d e . T h i s c e n t r a l p l a c e c o n c e p t o f a n e s t e d h i e r a r c h i c a l s t r u c t u r e o f h o s p i t a l s e r v i c e s i s f u n d a m e n t a l t o an u n d e r s t a n d i n g o f s c a l e e c o n o m i e s f o r , a s a r u l e , t h e more s p e c i a l i z e d a f u n c t i o n i s , t h e more c o m p l e x i s t h e e q u i p m e n t r e q u i r e d , t h e more s k i l l e d i t s o p e r a t o r s m u s t b e , and t h e l e s s f r e -q u e n t l y i t i s i n u s e . C o n s e q u e n t l y s u c h s e r v i c e s a r e c o s t l y t o m a i n t a i n r e l a t i v e t o l o w e r o r d e r f u n c t i o n s . I f t h e s e e f f e c t s a r e h e l d c o n s t a n t h o w e v e r , t h e n , as B e r r y and o t h e r s h a v e d e m o n s t r a t e d , f o r a g i v e n l e v e l o f s e r v i c e s u n i t bed c o s t s w i l l t e n d t o d e c r e a s e w i t h i n c r e a s i n g s i z e o f h o s p i t a l ( B e r r y , 1 9 6 7 ) . In t e r m s o f t h e a p p l i c a t i o n o f t h e m o d e l i t h a s n o t p r o v e d p o s s i b l e t o d e r i v e a d e q u a t e c o s t c u r v e s s i n c e a s i n t i m a t e d , t h i s i s a m a j o r u n d e r t a k i n g i n i t s own r i g h t . F u r t h e r m o r e , a p p l i c a t i o n o f t h e a l g o r i t h m t o some d e s i r a b l e f u t u r e l o c a t i o n a l p a t t e r n i s d e p e n d e n t upon f o r e c a s t i n g f u t u r e demand f u n c t i o n s f o r d i s c r e t e p a t i e n t a r e a s a n d l i e s b e y o n d t h e s c o p e o f t h i s d i s s e r t a t i o n . The m a j o r c o m p o n e n t o f t h e s t u d y w i l l t h e n , be t h e d e v e l o p m e n t o f t h e m o d e l i t s e l f , a n d i t s t e s t i n g w i l l be c o n f i n e d t o 1) an a t t e m p t t o r e p r o d u c e t h e f l o w p a t t e r n s now e x t a n t on t h e e x i s t i n g n e t w o r k and 2) t h e d e r i v a t i o n o f a h y p o t h e t i c a l l o c a t i o n s y s t e m , w i t h t h e same number o f p l a n t s , W h i c h i s more e f f i c i e n t i n t e r m s o f c e r t a i n k e y c r i t e r i a t h a n i s t h e p r e s e n t o n e . T h i s l a t t e r a i m s h o u l d p r o v i d e some i n s i g h t s i n t o t h e p o t e n t i a l i m p r o v e m e n t s w h i c h m i g h t be made w i t h i n t h e c u r r e n t o p e r a t i n g s y s t e m . S t u d y O u t l i n e T h i s s t u d y c o n s i s t s o f t h r e e a d d i t i o n a l c h a p t e r s whose o u t l i n e i s g i v e n b e l o w . In C h a p t e r I I some c o n t e m -p o r a r y t r e n d s i n m e d i c a l h e a l t h p l a n n i n g a r e d i s c u s s e d t o g e t h e r w i t h some r e c e n t a n a l y s e s o f h o s p i t a l l o c a t i o n s y s t e m s . The r a t i o n a l e f o r h e a l t h p l a n n i n g i s e v a l u a t e d ; some a s p e c t s o f t h e movement o f t h e p a t i e n t t h r o u g h t h e m e d i c a l c a r e s y s t e m a r e e x a m i n e d , and some i m p l i c a t i o n s o f t h i s k n o w l e d g e i n t e r m s o f i m p r o v e d m e d i c a l p l a n n i n g a r e a s s e s s e d . The r o l e o f t h e h o s p i t a l i n t h e s y s t e m i s d i s c u s s e d a n d a c r i t i c a l r e v i e w o f some p r e v i o u s m e d i c a l l o c a t i o n m o d e l s i s a t t e m p t e d . C h a p t e r I I I p r o v i d e s , f i r s t o f a l l , a f o r m a l s t a t e -m e n t o f t h e g e n e r a l p l a n t l o c a t i o n p r o b l e m , o f w h i c h h o s -p i t a l l o c a t i o n , a t l e a s t i n r u r a l a r e a s o r t h e i n t e r u r b a n c o n t e x t , may be c o n s i d e r e d as a p a r t i c u l a r c a s e . C e r t a i n s p e c i f i c c o n s t r a i n t s w h i c h a r e o f s i g n i f i c a n c e i n t h e f o r m u l a t i o n o f t h e g e n e r a l p r o b l e m a r e i n c o r p o r a t e d i n t o t h i s s t a t e m e n t . S e c o n d l y , a r e v i e w o f p r e v i o u s a t t e m p t s t o s o l v e t h e p r o b l e m i s p r o v i d e d . F i n a l l y , a d e t a i l e d d e s c r i p t i o n o f t h e a l g o r i t h m i t s e l f i s g i v e n , and some s a m p l e p r o b l e m s on a v a r i e t y o f n e t w o r k s a r e i l l u s t r a t e d t o d e m o n s t r a t e t h e f l e x i b i l i t y o f t h e mode l d e s i g n f o r a f a i r l y w i d e r a n g e o f p o t e n t i a l p l a n t l o c a t i o n p r o b l e m s . C h a p t e r IV p a r t i t i o n s B r i t i s h C o l u m b i a i n t o s u b -r e g i o n s o f a s i z e w h i c h c a n be c o m p u t a t i o n a l l y h a n d l e d by t h e a l g o r i t h m . One o f t h e s e r e g i o n s i s t h e n s e l e c t e d and a n a l y s e d i n t e r m s o f t h e l i m i t a l o b j e c t i v e s p r e v i o u s l y s p e c i f i e d . The b a s i c a l g o r i t h m i s s u b j e c t e d t o some m o d i f -i c a t i o n s w h i c h seem a p p r o p r i a t e t o t h e p a r t i c u l a r c a s e u n d e r s t u d y , and t h e f u n d a m e n t a l s t r e n g t h s and w e a k n e s s e s o f t h e f o r m u l a t i o n a r e d i s c u s s e d t o g e t h e r w i t h some s u g g e s t i o n s f o r p o t e n t i a l i m p r o v e m e n t and d e v e l o p m e n t . CHAPTER I I REGIONAL HEALTH P L A N N I N G : SOME P E R S P E C T I V E S AND PROBLEMS The R a t i o n a l e f o r R e g i o n a l H e a l t h P l a n n i n g The l a s t d e c a d e and a h a l f i n N o r t h A m e r i c a has w i t n e s s e d a s u r g e o f i n t e r e s t i n t h e c o n c e p t o f r e g i o n a l p l a n n i n g as t h e a p p r o p r i a t e m i l i e u w i t h i n w h i c h t o i n s t i -t u t e much n e e d e d i m p r o v e m e n t s i n m e d i c a l h e a l t h . P r i o r t o t h e H i l l - B u r t o n A c t o f 1 9 4 6 , t h e r e s p o n s i b i l i t y f o r t h e p r o v i s i o n o f m e d i c a l c a r e had e s s e n t i a l l y b e e n i n l o c a l c o m m u n i t y h a n d s ( M c N e r n e y , 1 9 6 2 ) . As l o n g as t h e c o s t s o f m e d i c a l c a r e r e m a i n e d l o w r e l a t i v e t o t h e demands o f o t h e r s e c t o r s o f t h e economy t h i s r e s p o n s i b i l i t y d i d n o t p r o v e t o be a s e v e r e f i n a n c i a l b u r d e n t o m o s t c o m m u n i t i e s , a n d f o r t h e f e w f o r . w h i c h i t was ( f r e q u e n t l y r u r a l a r e a s ) , p u b l i c o p i n i o n a t l a r g e was n o t s u f f i c i e n t l y a g i t a t e d t o demand c o r r e c t i v e a c t i o n a t t h e n a t i o n a l l e v e l . As t h e s e c o s t s h a v e r i s e n s h a r p l y o v e r t h e l a s t t w e n t y y e a r s h o w e v e r , u n d e r t h e s t i m u l u s o f c h a n g i n g h e a l t h c a r e p a t t e r n s and i n -c r e a s i n g l y s o p h i s t i c a t e d m e d i c a l t e c h n o l o g y , t h e d i s p a r i t i e s b e t w e e n t h e t a x b a s e s o f r i c h and p o o r c o m m u n i t i e s , d e f i n e d a l l t o o o f t e n upon u r b a n - r u r a l l i n e s , and t h e a t t e n d a n t d i f f e r e n t i a l s i n t h e q u a l i t y o f a v a i l a b l e c a r e w h i c h t h i s h a s c a u s e d , h a v e become b o t h m e d i c a l l y and p o l i t i c a l l y i n t o l e r a b l e . In r e s p o n s e t o t h e s e n e e d s i n c r e a s i n g s u p p o r t f r o m b o t h p r o -v i n c i a l a n d n a t i o n a l s o u r c e s has now become t h e n o r m , and a l t h o u g h l o c a l r e v e n u e r e s o u r c e s w i l l c o n t i n u e t o be o f some i m p o r t a n c e i n t h e f i n a n c i n g o f l o c a l m e d i c a l s e r v i c e s f o r a c o n s i d e r a b l e t i m e t o c o m e , t h i s r o l e w i l l i n e v i t a b l y be d i m i n i s h e d i n t h e f u t u r e . As l o c a l f i n a n c i a l a u t o n o m y h a s d e c l i n e d so has t h e o p p o r t u n i t y a r i s e n f o r t h e r a t i o n a l p l a n n i n g o f h e a l t h f a c i l i t i e s upon a more r e a l i s t i c g e o g r a p h i c a l b a s i s t h a n was f o r m e r l y f e a s i b l e . The e v i l s o f u n c o - o r d i n a t e d p l a n n i n g h a v e b e e n w e l l d o c u m e n t e d i n t h e h e a l t h p l a n n i n g l i t e r a t u r e ( s e e f o r e x a m p l e ' L e t o u r n e a u , 1 9 6 5 ; C a r d w e l l and K u c k a , 1 9 6 1 ) . The d u p l i c a t i o n o f e x p e n s i v e e q u i p m e n t and p e r s o n n e l i n a d j a c e n t o r n e a r b y h o s p i t a l s ; t h e i n s t a l l a t i o n o f s p e c -i a l i z e d c a r e f a c i l i t i e s a t l o c a t i o n s w h e r e t h e demand i s n o t s u f f i c i e n t t o w a r r a n t t h e m ; and a v i r t u a l l y de f a c t o c o m p e t i t i v e s p i r i t among some h o s p i t a l s t o i n s t a l 1 f a c i 1 i -t i e s w i t h l i t t l e r e g a r d t o t h e o v e r a l l n e e d s o f t h e i r d e -p e n d e n t s e r v i c e a r e a s , a l l b e a r t e s t i m o n y t o t h e i n e f f i c i e n t u s e o f e x i s t i n g m e d i c a l r e s o u r c e s . C o n c o m i t a n t w i t h t h e s e a b u s e s a r e a s e r i e s o f p a r -a l l e l p r o b l e m s w h i c h a r e a r e f l e c t i o n o f t h e m a l d i s t r i b u t i o n o f r e s o u r c e s o v e r t h e c o n t i n e n t a t l a r g e . I n a d e q u a t e and l o w q u a l i t y f a c i l i t i e s a r e common i n many o u t l y i n g and u n d e r p r i v i l e g e d a r e a s o f b o t h C a n a d a and t h e U n i t e d ' S t a t e s . The d i s t r i b u t i o n o f f a c i l i t i e s and p e r s o n n e l a p p e a r s , i n many i n s t a n c e s , t o be more a r e f l e c t i o n o f t h e e c o n o m i c s t a t u s o f a c o m m u n i t y t h a n i t s a c t u a l h e a l t h n e e d s , and t h e o r g a n i z a t i o n o f t h a t h e a l t h c a r e s u f f e r s a l l t o o f r e q u e n t l y f r o m a c l o s e r c o r r e s p o n d e n c e w i t h l o c a l p o l i t i c a l j u r i s -d i c t i o n s , t h a n w i t h t h e n a t u r a l movement o f p e o p l e a l o n g w e l l d e f i n e d l i n k s o f c o m m u n i c a t i o n ( W e i n e r m a n , 1 9 5 1 ) . Few o r g a n i z a t i o n s t o o , a r e as f r a g m e n t e d as t h e v a r i e d c o m p o -n e n t s o f t h e h e a l t h c a r e s y s t e m . F o l l o w i n g S c h a e f e r , H i l l e b o e d e f i n e s f i v e l e v e l s o f o r g a n i z a t i o n a t w h i c h t h i s f r a g m e n t a t i o n o c c u r s . F i r s t t h e r e i s a h o r i z o n t a l f r a g -m e n t a t i o n , c a u s e d by a p a r t i t i o n i n g o f r e s p o n s i b i l i t y o f t e n i l l d e f i n e d , a t t h e f e d e r a l , s t a t e and l o c a l l e v e l s . S e c o n d l y , v e r t i c a l f r a g m e n t a t i o n i n t o a m u l t i p l i c i t y o f a g e n c i e s , c o m m i s s i o n s , d e p a r t m e n t s , d i v i s i o n s , b u r e a u s , and o f f i c e s , e a c h o f w h i c h o c c u r s a t t h e t h r e e h o r i z o n t a l l e v e l s o f o r g a n i z a t i o n . S e c t i o n a l a r r a n g e m e n t o f h e a l t h p r o g r a m s among p u b l i c , p r i v a t e and v o l u n t a r y o r g a n i z a t i o n s p r o v i d e s a t h i r d l e v e l o f f r a g m e n t a t i o n . N e x t i s t h e f r a g m e n t a t i o n o f h e a l t h a c t i v i t i e s a l o n g p r o f e s s i o n a l and t e c h n i c a l l i n e s , a n d f i n a l l y f r a g m e n t a t i o n e x i s t s i n t h e f i n a n c i n g o f h e a l t h s e r v i c e s t h r o u g h a v a r i e t y o f i n s u r a n c e and p u b l i c g r a n t s c h e m e s ( H i l l e b o e , 1 9 6 8 ) . The i n e v i t a b l e c o n s e q u e n c e o f s u c h d i v i s i o n s i s t o be s e e n i n i l l - d e f i n e d r o l e s , overr l a p p i n g r e s p o n s i b i l i t i e s and p o o r c o m m u n i c a t i o n among t h e v a r i e d l e v e l s o f t h e o r g a n i z a t i o n , a l l o f w h i c h c o n t r i b u t e t o a q u a l i t y o f m e d i c a l c a r e , b e l o w t h a t w h i c h i s now t e c h -n i c a l l y p o s s i b l e . w h i c h a r e e v o k e d by them has a r i s e n t h e c o n c e p t o f r e g i o n a l p l a n n i n g , o r t h e r e g i o n a l i z a t i o n o f m e d i c a l f a c i l i t i e s as i t has come t o be k n o w n . A s u c c i n c t d e f i n i t i o n o f t h i s c o n c e p t has b e e n g i v e n by D o n a b e d i o n and A x e l r o d ( 1 9 6 1 ) : . . . r e g i o n a l i z a t i o n e n v i s a g e s t h e u n i f i e d p l a n n i n g o f a f u n c t i o n a l l y d i f f e r e n t i a t e d a n d c a r e f u l l y c o - o r d i n a t e d s y s t e m o f h o s p i t a l s s e r v i n g a n e n t i r e g e o g r a p h i c r e g i o n d e m a r c a t e d , n o t by n a r r o w p o l i t i c a l b o u n d a r i e s , b u t a c -c o r d i n g to e s t a b l i s h e d p a t t e r n s o f s e e k i n g a n d p r o v i d i n g m e d i c a l c a r e i n a m a n n e r a n a l a g o u s to t r a d i n g a r e a s . U n d e r s u c h a p l a n , s t a n d a r d s o f m e d i c a l c a r e c o u l d b e c o n s i d e r a b l y i m p r o v e d t h r o u g h o u t t h e r e g i o n w i t h o u t c o s t l y d u p l i c a ^ - l t i o n o r i n e f f i c i e n t d e p l o y m e n t o f s c a r c e r e -s o u r c e s a n d s k i l l s . T h e r e w o u l d b e e s t a b l i s h e d a p p r o p r i a t e l y l o c a t e d c e n t r a l o r b a s e h o s p i t a l s to w h i c h o t h e r s m a l l h o s p i t a l s i n t h e r e g i o n w o u l d t u r n f o r h e l p a n d a d v i c e . T h e c e n t r a l h o s p i t a l , w o u l d , i n t u r n , a s s u m e c o n s i d e r a b l e r e s p o n s i b i l i t y f o r s u p p o r t i n g c o n t i n u i n g p r o -f e s s i o n a l e d u c a t i o n a n d m e d i c a l a n d a l l i e d s e r v i c e s i n t h e h o s p i t a l s a s s o c i a t e d w i t h i t . H i s t o r i c a l A n t e c e d e n t s f o r R e g i o n a l i z a t i o n The e a r l i e s t f o r m a l d o c u m e n t a t i o n o f t h e s e c o n c e p t s a p p e a r s t o h a v e b e e n c o n t a i n e d i n t h e Dawson R e p o r t , p u b -l i s h e d i n 1 9 2 0 , by t h e M i n i s t r y o f H e a l t h i n G r e a t B r i t a i n . In t h i s r e p o r t , s u b s e q u e n t l y t h e b a s i s f o r t h e o r g a n i z a t i o n o f t h e B r i t i s h H e a l t h S e r v i c e i n 1 9 4 7 , was e m b o d i e d t h e n o t i o n o f a n e t w o r k o f p r i m a r y and s e c o n d a r y h o s p i t a l s a f f i l i a t e d by s t r o n g l i n k s w i t h a l a r g e r t e a c h i n g h o s p i t a l ( M i n i s t r y o f H e a l t h o f G r e a t B r i t a i n , 1 9 2 0 ) . The t e a c h i n g h o s p i t a l was e n v i s a g e d as a c t i n g b o t h a s a r e f e r r a l c e n t r e f o r t h e s m a l l e r c o m m u n i t y h o s p i t a l and as a b a s e f r o m w h i c h m e d i c a l , t e c h n i c a l , a n d a d m i n i s t r a t i v e a s s i s t a n c e c o u l d be o f f e r e d w h e r e v e r n e c e s s a r y f o r t h e p r o v i s i o n o f h i g h e r q u a l i t y c a r e . The f i r s t e x p r e s s i o n o f t h e s e i d e a l s i n N o r t h A m e r i c a came e l e v e n y e a r s l a t e r i n 1931 when t h e B i n g h a m A s s o c i a t e s F u n d , p a r t i a l l y u n d e r ^ t h e s t i m u l u s o f w i d e s p r e a d p u b l i c d e m o r a l i z a t i o n c a u s e d by t h e s e v e r e e c o n o m i c d e p r e s s i o n o f 1 9 2 9 , o r g a n i z e d t h e e a r l i e s t s y s t e m o f r e g i o n a l h e a l t h c a r e i n t h e U n i t e d S t a t e s . T h i s p r o g r a m , b a s e d upon t h e New E n g l a n d M e d i c a l C e n t r e i n B o s t o n , s e t o u t t o o f f e r a c o m p l e t e d i a g n o s t i c and s u r g i c a l s e r v i c e t o t h e p a t i e n t s o f r e f e r r i n g p h y s i c i a n s , and t o p r o v i d e a c o n -t i n u i n g p o s t - g r a d u a t e e d u c a t i o n a l p r o g r a m f o r d o c t o r s i n t h e a f f i l i a t e d n e t w o r k o f s m a l l e r c o m m u n i t y h o s p i t a l s , w h i c h c o v e r e d m o s t o f M a i n e and much o f n o r t h e r n M a s s a c h u s e t t s ( L e m b o k e , 1 9 5 1 ) . B u t d e s p i t e t h e p r o m i s e o f t h i s e a r l y and s u c c e s s f u l b e g i n n i n g l i t t l e more was a c h i e v e d o v e r t h e n e x t f i f t e e n y e a r s . The p h i l o s o p h y o f i n d i v i d u a l r e s p o n s i b i l i t y i n a l l s p h e r e s o f l i f e , n o t l e a s t i n m e d i c a l c a r e , r u n s d e e p i n N o r t h A m e r i c a n l i f e , and t h e s t r e n g t h o f t h i s t r a d i t i o n t e n d e d t o f o s t e r an a t t i t u d e o f a p a t h y , i f n o t h o s t i l i t y , on t h e p a r t o f t h e g e n e r a l p u b l i c t o w a r d s w i d e - r a n g i n g r e g i o n a l p r o g r a m s ( S i g m o n d , 1 9 6 7 ) . I t was n o t u n t i l t h e p a s s i n g o f t h e H i l l - B u r t o n A c t i n 1 9 4 6 , t h a t t h e s t a g e became s e t f o r t h e s u b s e q u e n t w i d e s p r e a d g r o w t h o f r e g i o n -a l l y o r i e n t e d h e a l t h c a r e p r o g r a m s . The H i l l - B u r t o n A c t was i m p l e m e n t e d as a d i r e c t c o n -s e q u e n c e o f a s e v e r e s h o r t a g e o f d o c t o r s and f a c i l i t i e s i n r u r a l a r e a s , a n d was f o c u s e d p r i n c i p a l l y upon e f f o r t s t o a t t r a c t p h y s i c i a n s i n t o t h e s e r e m o t e l o c a t i o n s and t o p r o -v i d e them w i t h w e l l e q u i p p e d c e n t r e s f r o m w h i c h t o o p e r a t e . B u t t h e l e g i s l a t i o n has had a s i g n i f i c a n t i m p a c t upon r e g i o n a l m e d i c a l p l a n n i n g . O r g a n i z a t i o n o f t h e p r o g r a m a t t h e s t a t e r a t h e r t h a n t h e l o c a l l e v e l has s t r e n g t h e n e d t h e r o l e o f s t a t e h e a l t h d e p a r t m e n t s , a n e c e s s a r y p r e r e q u i s i t e f o r t h e f o u n d a t i o n o f a w i d e r and more f u n c t i o n a l p a t t e r n o f h e a l t h c a r e . A t t h e same t i m e t h e d e l e g a t i o n o f t h e r e s p o n s i b i l i t y f o r f u n d r a i s i n g t o b o t h s t a t e and c o m m u n i t y a g e n c i e s has h e l p e d t o s t i m u l a t e i n t e r e s t i n t h e h e a l t h p l a n n i n g g r o u p and has s e r v e d t o d i s s i p a t e some o f t h e e m o t i o n a l o p p o s i t i o n t o t h e r e g i o n a l p l a n n i n g c o n c e p t . F o r a l l i t s s t r e n g t h s h o w e v e r t h e r e a r e c e r t a i n i n -b u i l t r e s t r a i n t s i n t h e H i l l - B u r t o n l e g i s l a t i o n w h i c h h a v e p r e v e n t e d i t f r o m b e c o m i n g t h e b a s i s f o r a r e a l l y e f f e c t i v e s y s t e m o f m e d i c a l r e g i o n a l r e - o r g a n i z a t i o n . M o s t s i g n i f i -c a n t among t h e s e has b e e n t h e r e l i a n c e p l a c e d upon l o c a l p a r t i c i p a t i o n i n t h e f i n a n c i n g o f c o n s t r u c t i o n p r o j e c t s . D e p e n d i n g upon s t a t e and l o c a l p e r c a p i t a i n c o m e l e v e l s t h i s a m o u n t has come t o v a r y f r o m b e t w e e n one t o t w o - t h i r d s o f t h e t o t a l c o s t o f t h e p r o j e c t . U n f o r t u n a t e l y t h o s e l o c a t i o n s w i t h t h e g r e a t e s t n e e d f o r new f a c i l i t i e s h a v e p r o v e d a l l t o o o f t e n t o be t h o s e w h i c h a r e m o s t e c o n o m i c a l l y d e p r e s s e d and u n a b l e t o r a i s e e v e n t h e m i n i m u m r e v e n u e s n e c e s s a r y t o q u a l i f y f o r f e d e r a l a i d . As a c o n s e q u e n c e t h e r e has b e e n a t e n d e n c y t o w i t h h o l d f u n d s f r o m p r e c i s e l y t h o s e a r e a s i n m o s t u r g e n t n e e d o f them ( B a t t i s t e l 1 a , 1 9 6 6 ) . S e c o n d l y , t h e e m p h a s i s o f t h e p r o g r a m has b e e n upon t h e c o n s t r u c t i o n o f new b e d s t o t h e e x c l u s i o n o f t h e d e v e l o p m e n t o f b r o a d b a s e d c o - o r d i n a t e d s e r v i c e s a t d i f f e r e n t l e v e l s o f s t a t e and f e d e r a l o r g a n i z a t i o n . In M c N e r n e y ' s v i e w " t o o many h o s p i t a l s may h a v e b e e n c o n s t r u c t e d i n c o m m u n i t i e s w h e r e r e a l n e e d may n o t h a v e b e e n so much f o r i n - h o s p i t a l b e d s as f o r w e l 1 - e q u i p p e d d i a g n o s t i c o r e m e r g e n c y c l i n i c s . " ( M c N e r n e y , 1 9 6 2 ) A f i n a l f a i l u r e o f t h e H i l l - B u r t o n p r o g r a m has b e e n s e e n i n i t s v o l u n t a r y . t r i p a r t i t e s t r u c t u r e o f f e d e r a l , s t a t e a n d l o c a l g o v e r n m e n t s w h i c h has p r o v e d t o be an u n w i e l d y means f o r r e g i o n a l r e o r g a n i z a t i o n . P r o g r e s s has b e e n r e t a r d e d i n many s t a t e s by o u t d a t e d g o v e r n m e n t a l m a c h i n e r y w h i c h i s u n r e p r e s e n t a t i v e o f t h e n e e d s o f commu-n i t i e s , and by t h e l a c k o f v i a b l e h e a l t h o r g a n i z a t i o n s p r o f e s s i o n a l l y t r a i n e d a n d e m p o w e r e d by s u f f i c i e n t a u t h o r i t y t o c a r r y t h r o u g h r e f o r m s . The f a i l i n g s o f t h e H i l l - B u r t o n A c t p r o v i d e d l e s s o n s w h i c h w e r e n o t f o r g o t t e n i n t h e p r e p a r a t i o n o f t h e H e a r t D i s e a s e , C a n c e r and S t r o k e A c t o f 1 9 6 5 . T h i s p i e c e o f l e g i s l a t i o n , s o m e t i m e s known a s t h e R e g i o n a l M e d i c a l P r o g r a m s A c t , m a r k e d t h e f i r s t s i g n i f i c a n t a t t e m p t on a n a t i o n a l s c a l e i n N o r t h A m e r i c a t o o r g a n i z e h e a l t h s e r v i c e s i n t o a r e g i o n a l f r a m e w o r k . The a c t s p e c i f i c a l l y e n c o u r a g e d t h e c r e a t i o n o f r e g i o n s who se b o u n d a r i e s w e r e t o be d e f i n e d i n t e r m s o f m e d i c a l l y f u n c t i o n a l c r i t e r i a . As s u c h , t h e d e l i m i t a t i o n s o f t h e r e g i o n s h a v e , o f n e c e s s i t y , c u t a c r o s s l o c a l and s t a t e l i n e s i n some i n s t a n c e s , t h e r e b y o f f s e t t i n g some o f t h e m a j o r s h o r t c o m i n g s o f t h e H i l l - B u r t o n A c t . I n s o f a r as t h e p r o g r a m has b e e n s u c c e s s f u l l y i m p l e m e n t e d i s e r v i c e s have b e e n o r g a n i z e d i n t h e i n t e r d e p e n d e n t h i e r -a r c h i c a l s t r u c t u r e f u n d a m e n t a l t o t h e r e g i o n a l i z a t i o n c o n c e p t . D e c i s i o n s c o n c e r n i n g t h e p r e c i s e n a t u r e and r a n g e o f s e r v i c e s o f f e r e d w i t h i n e a c h r e g i o n h a v e b e e n d e c e n t r a l -i z e d a n d a r e b e i n g u n d e r t a k e n by l o c a l m e d i c a l p l a n n i n g g r o u p s w h i c h a r e , i n t h e o r y a t l e a s t , more s e n s i t i v e t o t h e n e e d s o f l o c a l a r e a s . F i n a l l y ; , and p e r h a p s more i m p o r t a n t o f a l l , t h e c o s t o f t h e p r o g r a m has t h u s f a r b e e n u n d e r -w r i t t e n by g r a n t s f r o m t h e N a t i o n a l I n s t i t u t e o f H e a l t h , and i s t h e r e f o r e i n d e p e n d e n t o f t h e f i n a n c i a l s t a t u s o f e a c h r e g i o n and i t s a b i l i t y o r o t h e r w i s e t o pay f o r t h e s e r v i c e s i t r e q u i r e s ( S a n a z a r o , 1 9 6 7 ; K i s s i c k , 1 9 6 7 ) . R e p o r t s on t h e c u r r e n t s t a t u s o f t h e R e g i o n a l M e d i c a l P r o g r a m s A c t i n d i c a t e t h a t a s u c c e s s f u l b e g i n n i n g has been m a d e , b u t a s w i t h any p r o g r a m on t h i s s c a l e o f m a g n i t u d e and i n n o v a t i o n , p r o b l e m s h a v e a r i s e n f r o m t i m e t o t i m e t o * i m p e d e i t s r a t e o f p r o g r e s s . C o m m i t m e n t f r o m many i n s t i -t u t i o n s has b e e n on a t e n t a t i v e b a s i s so f a r o w i n g t o t h e l i m i t e d d u r a t i o n o f c u r r e n t f i n a n c i a l s u p p o r t f o r t h e p r o ^ g r a m . A s s u r a n c e s o f l o n g e r s u p p o r t a r e s e e n as b e i n g , v i t a l t o t h e a c h i e v e m e n t o f t h e l o n g t e r m o b j e c t i v e s o f t h e p l a n ( S t e w a r t , 1 9 6 7 ) . The p r o g r a m has a l s o b e e n a d v e r s e l y a f f e c t e d by a s e r i o u s s h o r t a g e o f m e d i c a l manpower i n t h e U n i t e d S t a t e s . T h e s e s h o r t a g e s h a v e p l a c e d r e s t r a i n t s on t h e r a t e o f i m p l e m e n t a t i o n o f some o f t h e p r o g r a m ' s g o a l s . N e i t h e r i t i s known t o w h a t e x t e n t t h i s p r o g r a m ' s r e q u i r e -m e n t s f o r manpower h a v e d i m i n i s h e d t h e s u p p l y a v a i l a b l e t o o t h e r d e s e r v i n g m e d i c a l c a r e s c h e m e s . T h i s l i m i t a t i o n has r e s u l t e d n e c e s s a r i l y i n a r e l a t i v e l y g r e a t e r e m p h a s i s b e i n g g i v e n t o m e d i c a l t r a i n i n g a c t i v i t i e s i n t h e i n i t i a l o p e r -a t i n g p h a s e s o f t h e p l a n , and h a s c r e a t e d t h e f a l s e i m p r e s -s i o n i n some q u a r t e r s t h a t t h e R e g i o n a l M e d i c a l P r o g r a m s a r e p r i m a r i l y d e s i g n e d f o r t h i s t a s k . In t h e C a n a d i a n c o n t e x t t h e c o n c e p t o f r e g i o n a l p l a n n i n g has b e e n f i r m l y e n d o r s e d by t h e r e p o r t o f t h e R o y a l C o m m i s s i o n on H e a l t h S e r v i c e s and has been i m p l e m e n t e d t o some e x t e n t e v e r s i n c e t h e e s t a b l i s h m e n t o f t h e f e d e r a l g o v e r n m e n t s g r a n t - i n - a i d p r o g r a m f o r c o n s t r u c t i o n i n 1 9 4 8 . In t h e p a s t h o w e v e r , r e g i o h a l i z a t i o n , w h i c h i s o r g a n i z e d on t h e p r o v i n c i a l l e v e l i n C a n a d a , has s u f f e r e d f r o m t o o much c e n t r a l i z a t i o n a t t h e t o p and a l a c k o f s e n s i t i v i t y t o l o c a l n e e d s . In W a h n ' s v i e w an i n h e r e n t d e f e c t has b e e n t h a t " t h o s e who u l t i m a t e l y h a v e t h e o b l i g a t i o n t o i m p l e m e n t t h e p l a n a r e n o t a c t i v e l y i n v o l v e d i n t h e p r e p a r a t i o n o f t h e p l a n " ( W a n n , 1 9 6 7 ) . M o r e r e c e n t l y a movement t o w a r d s d e c e n t r a l i z a t i o n , i n t h e p l a n n i n g f u n c t i o n a t l e a s t , has t a k e n p l a c e , and b o t h B r i t i s h C o l u m b i a and O n t a r i o h a v e now e s t a b l i s h e d a number o f r e g i o n a l p l a n n i n g a r e a s . The B r i t i s h C o l u m b i a p l a n came i n t o e f f e c t i n 1967 w i t h t h e c r e a t i o n o f t w e n t y - s e v e n r e g i o n a l h o s p i t a l d i s t r i c t s t h r o u g h o u t t h e p r o v i n c e . In d e s i g n t h e s e d i s t r i c t s a r e s e e n t o h a v e s e v e r a l i n t e r d e -p e n d e n t f u n c t i o n s . E a c h a r e a has e s t a b l i s h e d an A d v i s o r y P l a n n i n g C o m m i t t e e whose t a s k i s t o r e v i e w t h e p r o j e c t s f o r c o n s t r u c t i o n a n d e x p a n s i o n s u b m i t t e d by t h e m a n a g e m e n t b o a r d s o f i n d i v i d u a l h o s p i t a l s i n t h e d i s t r i c t , and t o f i t them i n t o a p r i o r i t y p l a n ( w h i c h may i n c l u d e r e c o m m e n d a -t i o n s f o r t h e c r e a t i o n o f new f a c i l i t i e s ) w h i c h i s t h e n s u b m i t t e d t o t h e c e n t r a l p l a n n i n g a u t h o r i t y f o r a p p r o v a l . One m a j o r p u r p o s e o f t h e p l a n has b e e n t o p r o v i d e a g r a n t -i n - a i d s c h e m e w h i c h o f f e r s a more e q u i t a b l e way o f s h a r i n g c o n s t r u c t i o n c o s t s o v e r t h e e n t i r e a r e a w h i c h u s e s a h o s -p i t a l ' s f a c i l i t i es . As p r e s e n t l y c o n s t i t u t e d t h e p r o v i n c i a l g o v e r n m e n t p a y s s i x t y p e r c e n t o f c o n s t r u c t i o n c o s t s i n d i s t r i c t s w h e r e t h e l e v y i s up t o f o u r m i l l s and e i g h t y p e r c e n t i n r e g i o n s w h e r e t h e a s s e s s m e n t i s g r e a t e r t h a n t h i s . The r e m a i n d e r i s a c c o u n t e d f o r by d i s t r i c t and f e d e r a l c o n t r i b u t i o n s . O r g a n i z a t i o n o f s e r v i c e s on t h i s b a s i s t h u s a l l o w s t h e l o c a l s h a r e o f c o n s t r u c t i o n c o s t s t o be r a i s e d on a b r o a d e r t a x b a s e t h a n was f o r m e r l y p o s s i b l e . Some c o m m e n t a t o r s s e e p o s s i b l e l o n g t e r m w e a k n e s s e s ' i n t h e f i n a n c i n g a r r a n g e m e n t s . In p a r t i c u l a r t h e 6 0 : 4 0 p e r c e n t b a s i s f o r c o s t s h a r i n g i s n o t c o n s i d e r e d a d e q u a t e by some p l a n n e r s e s p e c i a l l y i n t h e c a s e o f t h e l a r g e r r e f e r r a l h o s p i t a l s , w h i c h t r e a t many p a t i e n t s f r o m b e y o n d t h e i r d i s -t r i c t b o u n d a r i e s , b u t r e c e i v e no e x t r a r e c o m p e n s e f o r t h i s s e r v i ce ( A n o n , 1 9 6 7 ) . I t i s t o o e a r l y a s y e t t o e v a l u a t e c r i t i c a l l y t h e s u c c e s s e s o f r e g i o n a l p l a n n i n g on t h i s c o n t i n e n t . C h a n g e s i n t h e b a s i c p h i l o s o p h y o f o r g a n i z a t i o n a l s t r u c t u r e on t h i s m a g n i t u d e i n e v i t a b l y g i v e r i s e t o p r o b l e m s u n f o r e s e e n and t o r o l e s a s y e t n o t f i r m l y d e f i n e d . To some d e g r e e t h i s has a l r e a d y become a p p a r e n t . A p a r t f r o m s u c h t e c h n i c a l c o n s i d e r a t i o n s as t h e p r i o r d e l i m i t a t i o n o f r e g i o n a l a r e a s ( s e e K l a r m a n , 1 9 6 4 ) e x p e r i e n c e w i t h a r e a w i d e p l a n n i n g a g e n c i e s h a s f r e q u e n t l y d e m o n s t r a t e d a number o f c o n f l i c t i n g i n t e r e s t s and p u r p o s e s . C o - o p e r a t i o n among a g e n c i e s a n d o t h e r p l a n n i n g b o d i e s i s c l e a r l y f u n d a m e n t a l t o s u c c e s s f u l p l a n n i n g on t h e r e -g i o n a l l e v e l b u t w h e r e , a s i s o f t e n t h e c a s e , d i f f e r e n t a g e n c i e s o r e v e n d e p a r t m e n t s w i t h i n t h e same p l a n n i n g b o a r d , h a v e d i f f e r e n t t e c h n i c a l b a c k g r o u n d s and o r i e n t a t i o n s , t h e r e has b e e n a t e n d e n c y t o d e f i n e and i d e n t i f y c o m m u n i t y h e a l t h s e r v i c e n e e d s i n d i f f e r e n t w a y s . T h i s i n t u r n has e n c o u r -a g e d s e l e c t i v e a t t e n t i o n upon t h e means t o s o l v e p a r t i c u l a r p r o b l e m s , r a t h e r t h a n t h e p r i o r f i r s t s t e p o f c r i t i c a l l y e v a l u a t i n g t h e u n d e r l y i n g p r o b l e m s t h e m s e l v e s ( A r n o l d and H i n k , 1 9 6 8 ) . A s e c o n d g r o u p o f p r o b l e m s r e f l e c t s a l a c k o f d e f i -n i t i o n i n some i n s t a n c e s o f t h e r o l e s w h i c h e a c h o f t h e v a r i o u s c o m p o n e n t s o f h e a l t h c a r e i s t o p l a y i n t h e o v e r a l l c o n t e x t o f h e a l t h p l a n n i n g . The f o c u s o f t h e p r e s e n t c o n -c e p t o f r e g i o n a l p l a n n i n g on a h i e r a r c h i c a l s t r u c t u r e o f h o s p i t a l r e f e r r a l s n e e d s t o be e x t e n d e d t o i n c l u d e a l l p u b l i c h e a l t h o r g a n i z a t i o n s and o t h e r r e l a t e d b o d i e s , whose i n t e r e s t s h a v e a b e a r i n g upon b o t h p r e v e n t i v e and c u r a t i v e m e d i c i n e . The t o t a l s y s t e m s a p p r o a c h t o h e a l t h p l a n n i n g has b e e n a d v o c a t e d by s e v e r a l a u t h o r s ( s e e f o r e x a m p l e W e i n e r m a n , 1 9 6 9 ) . H i l l e b o e h a s e x t e n d e d t h e i d e a f u r t h e r by u r g i n g t h e d e v e l o p m e n t o f s t r o n g l i n k s w h e r e p o s s i b l e w i t h o t h e r c o m m u n i t y p l a n n i n g o b j e c t i v e s . In h i s v i e w h e a l t h p l a n n i n g i s i n s e p a r a b l e f r o m t h e s e o t h e r g o a l s . The q u e s t i o n o f t h e t o t a l r e s o u r c e s a v a i l a b l e t o m e e t a l l p l a n n i n g demands i s one w h i c h i s o f t e n o v e r l o o k e d by h e a l t h a d m i n i s t r a t o r s . The c o n s t r u c t i o n o r m o d i f i c a t i o n o f h e a l t h f a c i l i t i e s n e e d s t o be m a t c h e d a g a i n s t t h e c a p a c i t y o f a v a i l a b l e b u i l d i n g r e s o u r c e s , and e s t i m a t e d manpower r e q u i r e -m e n t s s h o u l d be a r e f l e c t i o n o f a v a i l a b l e o r p o t e n t i a l l y a v a i l a b l e e d u c a t i o n a l f a c i l i t i e s i f r e a l i s t i c o b j e c t i v e s a r e t o be a t t a i n e d ( H i l l e b o e , 1 9 6 8 ) . The R o l e o f t h e M o d e r n H o s p i t a l In any d e f i n i t i o n o f r o l e s t h a t o f t h e m o d e r n h o s -p i t a l i n , t h e o r g a n i z a t i o n o f h e a l t h c a r e i s c l e a r l y one o f f u n d a m e n t a l s i g n i f i c a n c e . Many o f t h e p r o b l e m s w h i c h h a v e r e s u l t e d f r o m t h e p a s t c o n s t r u c t i o n o f f a c i l i t i e s d e s i g n e d f o r t h e t r e a t m e n t o f s p e c i f i c d i s e a s e s h a v e l o n g b e e n r e c o g -n i z e d , p a r t i c u l a r l y w h e r e t h e c o n t r o l o f i n f e c t i o u s a i l m e n t s has b e e n t h e m a j o r o b j e c t i v e . The t u b e r c u l o s i s s a n i t o r i u m , l o n g a f a m i l i a r s i g h t upon t h e l a n d s c a p e , has b e e n r e n d e r e d a l m o s t o b s o l e t e by m e d i c a l a d v a n c e s o v e r t h e l a s t t w e n t y y e a r s ( H o g e , 1 9 6 7 ) . B u t g e r i a t r i c and p s y c h i a t r i c n e e d s a r e on t h e i n c r e a s e i n m o d e r n s o c i e t y , and a c c o u n t f o r a p r o p o r t i o n a t e l y l a r g e r s h a r e o f m e d i c a l , r e s o u r c e s w i t h e v e r y p a s s i n g d e c a d e . The r o l e o f t h e c o m m u n i t y g e n e r a l h o s p i t a l w i t h r e s p e c t t o t h e s e n e e d s i s s t i l l as y e t , a m a t t e r o f some d e b a t e . The c o n t i n u e d u s e o f s e p a r a t e l o n g t e r m g e r i a t r i c and p s y c h i a t r i c c a r e i s s o m e t i m e s t h o u g h t d e s i r a b l e on t h e g r o u n d s t h a t l a r g e s c a l e i n c o r p o r a t i o n o f t h e s e s e r v i c e s i n t o t h e s t r u c t u r e o f t h e g e n e r a l h o s p i t a l w i l l d i m i n i s h i t s e f f i c i e n c y by v i r t u e o f d i s e c o n o m i e s o f s c a l e , and w i l l , a t t h e same t i m e , p r o v e t o be i n a c c e s s i b l e t o many o f t h e p a t i e n t s h o m e s , a f a c t o r o f some s i g n i f i c a n c e w h e r e l o n g t e r m c a r e i s i n v o l v e d . A g a i n s t t h i s i t may be s a i d t h a t t h e c o n s e q u e n c e s o f s u c h s e g r e g a t i o n a r e p o t e n t i a l l y more s e r i o u s , p a r t i c u l a r l y i n an a g e w h e r e m o d e r n t r a n s p o r t has i n c r e a s e d t h e r a n g e o f a c c e p t a b l e t r a v e l d i s t a n c e s . The m o s t i n t r a c t a b l e p r o b l e m a s s o c i a t e d w i t h t h e m a i n t e n a n c e o f s e p a r a t e i n s t i t u t i o n s f o r t h i s t y p e o f c a r e i s t h a t o f o b t a i n i n g s u i t a b l e q u a l i f i e d s t a f f . G e r i a t r i c and P s y c h i a t r i c h o s p i t a l s h a v e f o r l o n g b e e n r e g a r d e d as d e -p r e s s e d a r e a s by b o t h t h e p u b l i c and t h e m e d i c a l p r o f e s s i o n . A p p o i n t m e n t s t o t h e s e p o s i t i o n s a r e n o t , by and l a r g e , c o m -p e t i t i v e , and s t a f f i n g s h o r t a g e s r e m a i n a p e r e n n i a l f a c t o r , w i t h a l l t h e a t t e n d a n t e f f e c t s on t h e l e v e l and q u a l i t y o f m e d i c a l c a r e t h a t t h i s i m p l i e s . M o r e o v e r s e g r e g a t i o n f r o m t h e d i s t r i c t h o s p i t a l c e n t r e t e n d s t o d e p r e s s t h e i n t e r e s t s o f m e d i c a l r e s e a r c h i n t o t h e s e a i l m e n t s , and t h e i s o l a t i o n o f t h e p s y c h i a t r i c p a t i e n t s u f f e r s t h e f u r t h e r d i s a d v a n t a g e o f p e r p e t u a t i n g p u b l i c p r e j u d i c e s a b o u t t h e p a r t i c u l a r n a t u r e o f t h i s d i s o r d e r ( M c K e o w n , 1 9 6 6 ) . P a t i e n t s s e g r e -g a t e d upon t h e b a s i s o f s u c h d i s e a s e c h a r a c t e r i s t i c s s e l -d o m , i f e v e r , r e m a i n h o m o g e n e o u s w i t h r e s p e c t t o t h e i r m e d i c a l n e e d s . C h r o n i c a l l y i l l p a t i e n t s may d e v e l o p s y m p -toms o f a c u t e i l l n e s s w h i l e g e r i a t r i c p a t i e n t s w i l l f r e -q u e n t l y show s i g n s o f m e n t a l d i s o r d e r s . The n e t e f f e c t i s t h a t t h e i n s t i t u t i o n i n q u e s t i o n e i t h e r f i n d s i t s e l f r e s p o n s i b l e f o r c l a s s e s o f p a t i e n t s whose n e e d s a r e b e y o n d i t s r e s o u r c e s o r m u s t p r o v i d e t h o s e r e s o u r c e s i t s e l f j t h e r e b y e n c o u r a g i n g a f u r t h e r d u p l i c a t i o n o f f a c i l i t i e s and p e r -s o n n e l . The a l t e r n a t i v e a p p r o a c h , w h i c h i s now f a v o u r e d by many m e d i c a l a d m i n i s t r a t o r s , i s t o a b o l i s h s e p a r a t e i n s t i -t u t i o n s f o r t h e s e t y p e s o f c a r e and t o p r o v i d e t h e w h o l e s p e c t r u m o f c a r e i n t h e c o m m u n i t y g e n e r a l h o s p i t a l . P r o -p o n e n t s o f t h i s p o l i c y a r g u e t h a t l o n g s t a y u n i t s a t t h e d i s t r i c t h o s p i t a l n e e d n o t be i n o r d i n a t e l y e x p e n s i v e s i n c e c h r o n i c a l l y and p s y c h i a t r i c a l l y i l l p a t i e n t s n e e d n o t be a d m i t t e d t o and r e t a i n e d i n w a r d s d e s i g n e d f o r t h e more e x p e n s i v e t y p e s o f a c u t e c a r e . B e c a u s e o f t h e r e l a t i v e e a s e w i t h w h i c h p a t i e n t s c a n be t r a n s f e r r e d w i t h i n a s i n g l e h o s p i t a l c o m p l e x , t h e r e seems t o be no i n h e r e n t r e a s o n why. s p e c i a l l o n g s t a y u n i t s c a n n o t be c o n s t r u c t e d a t l o w e r j. c a p i t a l and m a i n t e n a n c e c o s t s f o r t h i s s p e c i f i c p u r p o s e . S u c h u n i t s m i g h t f u r t h e r s e r v e a d o u b l e f u n c t i o n by a c t i n g a s c o n v a l e s c e n t homes f o r p a t i e n t s who no l o n g e r n e e d a c u t e c a r e b u t who m u s t r e m a i n h o s p i t a l i z e d f o r a r e h a b i l i t a t i o n p e r i o d ( B r i d g m a n , 1 9 6 7 ) . The R o l e o f S o c i a l M e d i c i n e Thus f a r t h i s d i s c u s s i o n has c e n t r e d upon t h e p r o -v i s i o n a n d o r g a n i z a t i o n o f t h e c u r a t i v e a s p e c t s o f h o s p i t a l c a r e . The e m p h a s i s on c u r a t i v e m e d i c i n e , p a r t i c u l a r l y i n t h e h o s p i t a l s e t t i n g , made s e n s e i n t h e p a s t when a c u t e d i s e a s e o f t h e i n f e c t i o u s v a r i e t y was t h e o v e r r i d i n g m e d i c a l p r o b l e m , and c h r o n i c d i s e a s e was o f much l e s s s i g n i f i c a n c e o w i n g t o t h e s h o r t e r s p a n o f l i f e w h i c h t h e n p r e v a i l e d . B u t t h i s s i t u a t i o n i s no l o n g e r c o m m o n , a t l e a s t i n t e c h -n i c a l l y d e v e l o p e d p a r t s o f t h e w o r l d , and t h e r o l e o f p u b l i c h e a l t h p r o g r a m s and s o c i a l and p r e v e n t i v e m e d i c i n e w h i c h has r e c e i v e d l e s s a t t e n t i o n i n t h e p a s t , i s now s e e n a s b e i n g o f p o t e n t i a l l y g r e a t e r c o n s e q u e n c e i i i t h e f u t u r e . The n e e d f o r a l t e r n a t i v e a p p r o a c h e s t o i n - p a t i e n t h o s p i t a l c a r e t a k e s on an a d d e d s i g n i f i c a n c e i f t h e o p i n i o n o f t h o s e who b e l i e v e t h a t h o s p i t a l u s e i s d i r e c t l y r e l a t e d t o t h e s u p p l y o f b e d s c a n be g i v e n c r e d e n c e ( R o e m e r , 1 9 6 1 ) . A l t h o u g h R o s e n t h a l h a s t a k e n a c o n t r a r y v i e w p o i n t i n t h i s c o n t r o v e r s y , F e l d s t e i n s u p p o r t s R o e m e r ' s f i n d i n g s , and i n h i s a n a l y s i s o f h o s p i t a l u t i l i z a t i o n i n E n g l a n d , i s d r a w n i r r e s i s t a b l y t o t h e c o n c l u s i o n t h a t t h e e x i s t i n g s u p p l y o f b e d s i s t h e c o r r e c t o n e , w h a t e v e r i t s l e v e l may be ( R o s e n t h a l , 1 9 6 4 ; F e l d s t e i n , 1 9 6 7 ) . The s e a r c h f o r a l t e r n a t i v e s t o h o s p i t a l c a r e r e q u i r e s some p r i o r d i s t i n c t i o n o f t h e c o n c e p t s o f " n e e d " and " d e -m a n d . " I f demand c a n be l o o s e l y d e f i n e d as t h e a c t u a l r e -c e i p t o f m e d i c a l c a r e t h e c o n c e p t o f n e e d i s c l e a r l y more e p h e m e r a l i n n a t u r e . W h e t h e r o r n o t v a s t h e a l t h n e e d s a c t u -a l l y e x i s t as has s o m e t i m e s b e e n c l a i m e d i s a c t u a l l y a d e -b a t a b l e p o i n t . I n d e e d many p h y s i c i a n s who p r a c t i c e among i n s u r e d u r b a n p o p u l a t i o n s a r e o f t h e o p i n i o n t h a t t h e demands f o r h e a l t h s e r v i c e s o f t e n e x c e e d t h e t r u e n e e d s ( T a s k F o r c e R e p o r t , 1 9 7 0 ) . B u t w h a t e v e r t h e t r u t h o f t h e m a t t e r i s , i t seems c l e a r t h a t one o f t h e p r i m a r y a i m s o f p r e v e n t i v e m e d i c i n e i s t o k e e p p a t i e n t s , w h e r e p o s s i b l e , f r o m r e a c h i n g t h e l e v e l o f i n - p a t i e n t h o s p i t a l c a r e . P r e v e n t i v e m e d i c i n e has a l r e a d y a c h i e v e d some s t r i k -i n g s u c c e s s e s i n t h e c o n t r o l o f i n f e c t i o u s d i s e a s e s . V a c -c i n a t i o n p r o g r a m s , c o n d u c t e d on a mass b a s i s , h a v e v i r -t u a l l y e l i m i n a t e d many o f t h e c o n t a g i o u s i n f e c t i o n s w h i c h t o o k s u c h a h e a v y t o l l i n t h e p a s t . Mass s c r e e n i n g p r o g r a m s h a v e o f f e r e d a d d i t i o n a l o p p o r t u n i t i e s f o r p r e v e n t i v e m e d i -c i n e . B u t a l t h o u g h t h e y h a v e p r o v e d t h e m s e l v e s i n t h e c o n t r o l o f p r e l i m i n a r y t u b e r c u l o s i s , t h e r e i s much l e s s c o n c e n s u s on t h e i r c u r r e n t o v e r a l l w o r t h a s a w i d e s p r e a d t o o l f o r t h e e a r l y d e t e c t i o n o f many t y p e s o f d i s e a s e . Many o f t h e s e p r o g r a m s h a v e p r o v e d t o be e x p e n s i v e p r o p o s i t i o n s when c o n d u c t e d upon a l a r g e s c a l e , and t h e y h a v e n o t a l w a y s p r o v e d t o be as m e d i c a l l y r e l i a b l e as t h e i r p r o p o n e n t s h a v e c l a i m e d . I t seems c l e a r t h a t more c o s t e f f e c t i v e n e s s s t u d i e s a r e r e q u i r e d i n t h i s f i e l d a n d , a t t h e v e r y l e a s t , i t i s f e l t t h a t t h e y s h o u l d be d i r e c t e d more s e l e c t i v e l y t o w a r d s h i g h r i s k g r o u p s i n t h e p o p u l a t i o n ( T a s k F o r c e R e p o r t , 1 9 7 0 ) . One o f t h e r e c u r r i n g p r o b l e m s w h i c h f a c e s s o c i a l m e d i c i n e i s t h a t t h e p u b l i c p e r c e p t i o n o f w h a t c o n s t i t u t e s a s e r i o u s h e a l t h h a z a r d i s n o t a l w a y s c o i n c i d e n t w i t h t h a t o f t h e m e d i c a l p r o f e s s i o n . C i g a r e t t e s m o k i n g , o f t e n c i t e d as a c a s e i n p o i n t , o f f e r s an e x c e l l e n t e x a m p l e o f t h e t y p e o f c o n f l i c t w h i c h c a n o c c u r b e t w e e n , on t h e one h a n d , t h e f i n a n c i a l b e n e f i t s b e s t o w e d by a p a r t i c u l a r i n d u s t r y on t h e n a t i o n a l t r e a s u r y , and t h e a i m s and a s p i r a t i o n s o f p u b l i c h e a l t h p r o g r a m s on t h e o t h e r . J a m e s has a p t l y s u m m a r i z e d t h i s k i n d o f a m b i g u i t y i n a t t i t u d e s by r e m a r k i n g t h a t p u b l i c p r e s s u r e f o r t h e c o n t r o l o f many m e d i c a l p r o b l e m s i s a p t t o s e t q u i t e d i f f e r e n t p r i o r i t i e s f r o m t h o s e o f t h e h e a l t h p r o f e s s i o n a l . " D i s e a s e s , " he c o m m e n t s , " a p p e a r t o r u n an u n p o p u l a r i t y c o n t e s t b e f o r e t h e p u b l i c a n d i t i s r a r e l y s e t t l e d by i n c i d e n c e s t a t i s t i c s " ( J a m e s , 1 9 6 7 ) . One o f t h e k e y s t o g r e a t e r s u c c e s s i n p r e v e n t i v e m e d i c i n e w o u l d seem t o l i e i n an i n c r e a s e d u n d e r s t a n d i n g o f t h e c o m p l e x p r o c e s s e s by w h i c h i n d i v i d u a l s a r e m o t i v a t e d t o s e e k h e a l t h c a r e and by t h e way i n w h i c h p a t i e n t s move t h r o u g h t h e m e d i c a l c a r e s y s t e m . In c o n s i d e r i n g some o f t h e m e c h a n i c s o f t h i s p r o c e s s i t i s known t h a t an i n t r i c a t e n e t -w o r k o f f o r m a l and i n f o r m a l i n t e r a c t i o n s e x i s t s b e t w e e n p a t i e n t s o r p o t e n t i a l p a t i e n t s on t h e one hand and t h e p r o -v i d e r s o f ; h e a l t h c a r e , t h e p h y s i c i a n s and o t h e r m e d i c a l p e r s o n n e l ^ o n t h e o t h e r . The e v e n t s w h i c h c h a r a c t e r i z e t h e s e i n t e r a c t i o n s o c c u r i n p a r t i c u l a r s e t t i n g s i n w h i c h a t t i t u d e s w h i c h r e f l e c t s o c i a l , e c o n o m i c a n d c u l t u r a l a t t r i b u t e s h a v e a p r o f o u n d i n f l u e n c e on t h e o u t c o m e s ( D o n a b e d i o n , 1 9 6 8 ) . Many s t u d i e s h a v e b e e n made o f t h e u t i l i z a t i o n o f h e a l t h s e r v i c e s by p a t i e n t s o f w i d e l y v a r y i n g b a c k g r o u n d s . F rom t h e s e c e r t a i n g e n e r a l i z a t i o n s may be d e d u c e d w h i c h show i n e f f e c t t h a t f a c i l i t i e s a r e u t i l i z e d m o s t f r e q u e n t l y by t h e y o u n g m i d d l e - a g e d , by f e m a l e s r a t h e r t h a n m a l e s , and by t h o s e who h a v e a c h i e v e d a h i g h e r e d u c a t i o n a l and s o c i o -e c o n o m i c s t a t u s . By c o r o l l a r y t h o s e i n d i v i d u a l s o f l o w e r e d u c a t i o n s a t t a i n m e n t s and i n c r e a s e d a g e w o u l d a p p e a r t o r e a c t more s l o w l y t o t h e s e symptoms and t o d e l a y i n s e e k i n g h e l p ( O d o r o f f and A b b e , 1 9 5 7 ; F e l d s t e i n , 1 9 6 6 ) . T h e s e c o n -c l u s i o n s w o u l d seem t o be e q u a l l y v a l i d f o r t h o s e who a r e u s i n g p r e v e n t i v e and d e t e c t i o n s e r v i c e s and f o r t h o s e who a r e s e e k i n g a c t u a l d i a g n o s i s a n d t r e a t m e n t . F r i e d s o n , Suchman and o t h e r s h a v e u n d e r t a k e n more s o p h i s t i c a t e d a n a l y -s i s i n w h i c h l i n k a g e m e c h a n i s m s b e t w e e n s o c i a l c h a r a c t e r i s -t i c s and i n d i v i d u a l s p e r c e p t i o n s o f t h e i r s y m p t o m s and t h e i r r e s p o n s e s t o them a r e p r o p o s e d ( S u c h m a n , 1 9 6 5 ; F r i e d s o n , 1 9 6 1 ) . R o s e h s t o c k who h a s r e v i e w e d some o f t h e s e s t u d i e s comments upon t h e s p e c i f i c l i m i t a t i o n s o f t h e i r f o c u s w h i c h i s c o n f i n e d t o h e a l t h b e h a v i o u r u n d e r t a k e n i n r e s p o n s e t o s p e c i f i c s y m p t o m s . In h i s w o r d s : T h e f i n d i n g s a v e t h u s o f u n k n o w n v e l e v a n c e to t h e s i t u a t i o n a o n f v o n t i n g t h e p e v s o n w h o m u s t d e c i d e w h e t h e v to s e e k p v e v e n t i v e o r d e t e c t i o n s e v v i c e s b e f o v e t h e a p p e a v a n c e o f e v e n t s t h a t h e i n t e v p v e t s a s s y m p t o m s . ( R o s e n s t o c k , 1 9 6 6 ) The a u t h o r t h e n g o e s on t o p r o p o s e a m o d e l w h i c h a t t e m p t s t o e x p l a i n h e a l t h b e h a v i o u r i n i n d i v i d u a l s who b e l i e v e t h e m s e l v e s t o be p r e s e n t l y f r e e o f a n y d e f i n i t i v e s y m p t o m s . He h y p o t h e s i z e s t h a t t h e d e c i s i o n t o s e e k p r e -v e n t i v e m e d i c i n e i n s u c h a s i t u a t i o n i s d e p e n d e n t upon t h e e x t e n t t o w h i c h an i n d i v i d u a l p e r c e i v e s h i s s u s c e p t i b i l i t y t o some p a r t i c u l a r c o n d i t i o n ; t h e e x t e n t t o w h i c h he p e r -c e i v e s i t s p o s s i b l e o c c u r r e n c e as b e i n g s e r i o u s t o h i m ; and t h e d e g r e e t o w h i c h he f e e l s t h a t some p r e v e n t i v e t e s t i s a p p r o p r i a t e f o r h i m i n s o f a r a s i t w i l l d i m i n i s h h i s p e r -c e i v e d s u s c e p t i b i l i t y . Once t h e s e c o n d i t i o n s a r e o b t a i n e d , t h e n some cue o f s t i m u l u s , e i t h e r i n t e r n a l l y , o r e x t e r n a l l y s u c h as an i m p a c t f r o m some c o m m u n i c a t i o n m e d i a , i s r e q u i r e d t o t r i g g e r an a p p r o p r i a t e r e a c t i o n ( R o s e n s t o c k , 1 9 6 6 ) . The v a l i d i t y o f R o s e n s t o c k ' s m o d e l r e m a i n s l a r g e l y unknown a t t h e p r e s e n t t i m e , s i n c e t h e r e has b e e n l i t t l e e m p i r i c a l e v i d e n c e t h u s f a r t o s u b s t a n t i a t e t h e s e h y p o t h e s e s . S u c h e v i d e n c e a s i s a v a i l a b l e seems t o s u g g e s t , as R o s e n s t o c k h i m s e l f n o t e s , t h a t t h e b e h a v i o u r p a t t e r n s i m p l i c i t i n t h e h e a l t h b e l i e f m o d e l t e n d t o be e x h i b i t e d more among t h e u p p e r s o c i o e c o n o m i c g r o u p s t h a n t h e l o w e r . I t i s p r e c i s e l y among t h e s e l o w e r s t a t u s g r o u p s h o w e v e r , t h a t s o c i a l m e d i c i n e has b e e n l e a s t e f f e c t i v e t o d a t e , and w h e r e t h e n e e d i s c o n s e q u e n t l y g r e a t e r . The c l a s s i c a l r e f e r r a l p a t t e r n w h e r e -by p a t i e n t s s e e k t h e s e r v i c e s o f a p h y s i c i a n and a r e r e -f e r r e d by h i m w h e r e n e c e s s a r y t o a s p e c i a l i s t , o r t h e l a y r e f e r r a l s y s t e m i n w h i c h p a t i e n t s r e f e r t h e m s e l v e s d i r e c t l y t o a s p e c i a l i s t a r e l e s s f r e q u e n t l y s e e n i n t h e s e s o c i a l g r o u p s ( W h i t e , 1 9 6 7 ) . The p r o b l e m i s f u r t h e r c o m p o u n d e d by a r e l a t i v e i n a c c e s s i b i l i t y o f a l l t y p e s o f m e d i c a l c a r e t o t h e s e p e o p l e . The d e c l i n e o f t h e g e n e r a l p r a c t i t i o n e r i n N o r t h A m e r i c a n m e d i c a l p r a c t i c e i s a w e l l known p h e n o m e n o n . A r e c e n t s t u d y by D o r s e y o f t h e c h a n g e s i n t h e d i s t r i b u t i o n o f p h y s i c i a n s i n B o s t o n b e t w e e n 1 9 4 0 a n d 1961 r e v e a l e d a d e c l i n e i n t h e r a t i o o f p h y s i c i a n s p e r 1 0 0 , 0 0 0 p o p u l a t i o n f o r a l l s o c i o - e c o n o m i c a r e a s o f t h e c i t y . F o r s o c i a l c l a s s e s 3 - 5 , d e f i n e d i n t h e s t u d y as t h e l o w e r s t a t u s g r o u p s , t h i s t r e n d i s p a r t i c u l a r l y d i s t u r b i n g s i n c e t h e a r e a s o c c -u p i e d by t h e s e g r o u p s h a v e r e l a t i v e l y l o w p h y s i c i a n r a t i o s t o b e g i n w i t h ( D o r s e y , 1 9 6 8 ) . I f , as i n F r i e d s o n ' s v i e w , c o n v e n i e n t a c c e s s t o m e d i c a l c a r e i s i m p o r t a n t t o p a t i e n t s e s p e c i a l l y i n t e r m s o f r o u t i n e p r i m a r y c a r e , t h e n t h e i n f e r -e n c e i s t h a t t h e u r b a n p o o r w i l l , i n many c a s e s , e i t h e r i g n o r e t h e i r i n i t i a l s y m p t o m s , o r r e f e r t h e m s e l v e s t o t h e e m e r g e n c y d e p a r t m e n t o f t h e n e a r e s t h o s p i t a l , a m e d i c a l l y u n d e s i r a b l e h a b i t w h i c h a p p e a r s t o be v e r y much on t h e i n -c r e a s e i n n e a r l y e v e r y m a j o r c o m m u n i t y on t h e c o n t i n e n t ( F r i e d s o n , 1 9 6 1 ; C o l e m a n , 1 9 6 7 ) . The d e c l i n e o f t h e g e n e r a l p r a c t i b n e r a n d t h e f l i g h t o f o t h e r p h y s i c i a n s and s p e c i a l i s t s t o t h e s u b u r b s o r t o o f f i c e s l o c a t e d a d j a c e n t , i n many i n s t a n c e s , t o h o s p i t a l c o m p l e x e s , has l e f t a m e d i c a l v a c u u m f o r many p o t e n t i a l p a t i e n t s i n t h e s l u m s and g h e t t o a r e a s o f t h e l a r g e r c i t i e s . The c o n c e p t o f t h e n e i g h b o u r h o o d h e a l t h c e n t r e o r c l i n i c has b e e n p r o p o s e d as an a n s w e r t o t h i s m e d i c a l g a p . Y e r b y has d e s c r i b e d t h e e s s e n t i a l f e a t u r e s o f t h e s e c e n t r e s . They a r e c o m m u n i t y o r i e n t e d i n t h e s e n s e o f b e i n g l o c a t e d w i t h r e s p e c t t o s i g n i f i c a n t c o n c e n t r a t i o n s o f t h e l o w e r i n c o m e g r o u p s ; a r e o p e n d a y and n i g h t ; p r o v i d e b a s i c d i a g n o s t i c and t h e r a p u t i c s e r v i c e s ; a r e s t a f f e d n o t o n l y by d o c t o r s , l a b o r a t o r y t e c h n i c i a n s and f r e q u e n t l y r e q u i r e d s p e c i a l i s t s , b u t a l s o by p u b l i c h e a l t h n u r s e s and s o c i a l w o r k e r s ; a n d h a v e a d i r e c t a m b u l a n c e l i n k w i t h some h o s p i t a l f o r i m m e -d i a t e r e f e r r a l w h e r e n e c e s s a r y ( Y e r b y , 1 9 6 7 ) . I t i s h o p e d t h a t s u c h c e n t r e s w i l l u l t i m a t e l y d i m i n i s h u n n e c e s s a r y h o s p i t a l i n - p a t i e n t a d m i s s i o n , a n d w i l l do much t o r e d u c e t h e p r e s s u r e on h o s p i t a l e m e r g e n c y d e p a r t m e n t s w h i c h a r e n o t t h o u g h t t o be t h e a p p r o p r i a t e m i l i e u f o r p r i m a r y h e a l t h c a r e . I n s u r a n c e C o v e r a g e and t h e U t i l i z a t i o n o f M e d i c a l C a r e I n t e r e s t i n a l t e r n a t i v e a p p r o a c h e s t o h o s p i t a l c a r e has b e e n f u r t h e r e n c o u r a g e d by s p e c u l a t i o n as t o t h e p o t e n t t i a l l o n g t e r m e f f e c t s o f m e d i c a r e and g o v e r n m e n t s p o n s o r e d i n s u r a n c e p l a n s on i n - p a t i e n t h o s p i t a l u t i l i z a t i o n . T h e r e i s a c o n s i d e r a b l e amount o f e v i d e n c e t o show t h a t m e d i c a l s e r v i c e s a r e u s e d more f r e q u e n t l y by t h o s e who c a r r y h e a l t h i n s u r a n c e . I n d e e d t h e c o r o l l a r y o f t h i s — t h a t p e o p l e may be d e t e r r e d f r o m o b t a i n i n g n e c e s s a r y h e a l t h c a r e by v i r t u e o f t h e i r l a c k o f c o v e r a g e i s o b v i o u s l y t h e r e a s o n b e h i n d t h e r e c e n t e x p a n s i o n o f t h e r o l e o f g o v e r n m e n t s i n t h e m e d i c a l " i n s u r a n c e f i e l d ( T a s k F o r c e R e p o r t , 1 9 7 0 ) . S t u d i e s i n t h e U n i t e d S t a t e s s u g g e s t t h a t h o s p i t a l a d m i s s i o n r a t e s a r e much h i g h e r among t h e i n s u r e d t h a n t h e u n i n s u r e d ( S o m e r s and S o m e r s , 1 9 6 2 ) . W h e t h e r o r n o t t h i s a c t u a l l y l e a d s t o o v e r -u t i l i z a t i o n i s a d e b a t a b l e i s s u e among m e d i c a l a d m i n i s t r a t o r s s i n c e t h e h o s p i t a l l e n g t h o f s t a y i s g e n e r a l l y s h o r t e r f o r t h e i n s u r e d p e r s o n . In C a n a d a t h e r e l a t i o n s h i p b e t w e e n h o s p i t a l a d m i s s i o n r a t e s and i n s u r a n c e c o v e r a g e a p p e a r s t o be l e s s c l e a r . A n d e r s e n and H u l l s t a t e t h a t t h e i n t r o d u c -t i o n o f u n i v e r s a l h o s p i t a l i n s u r a n c e i n t o t h i s c o u n t r y has n o t p r o d u c e d a m a r k e d i n c r e a s e i n t h e r a t e s o f a d m i s s i o n ( A n d e r s e n and H u l l , 1 9 6 9 ) . P e t e r s o n on t h e o t h e r hand a s s e r t s t h a t u n n e c e s s a r y a d m i s s i o n s a r e o c c u r r i n g among some p a t i e n t s a d m i t t e d p u r e l y f o r d i a g n o s t i c p u r p o s e s ( P e t e r s o n , 1 9 6 7 ) . I t may be ( a l t h o u g h t h e r e i s no d i r e c t e v i d e n c e a s y e t ) t h a t t h i s a p p a r e n t c o n t r a d i c t i o n r e f l e c t s a s l i g h t b u t p e r c e p t i b l e c h a n g e i n t h e p h y s i c i a n c u l t u r e o f C a n a d a w h i c h t r a d i t i o n a l l y , a p p e a r s t o h a v e b e e n h o s p i t a l o r i e n t e d . Thus w h e r e u n n e c e s s a r y h o s p i t a l a d m i s s i o n s f o r d i a g n o s t i c p u r -p o s e s o c c u r , t h e y m a y , t o some e x t e n t , be c o u n t e r b a l a n c e d by t h e movement t o w a r d s g r o u p p r a c t i c e w h i c h , as t h e T a s k F o r c e has t e n t a t i v e l y i n d i c a t e d , r e s u l t s i n a r e d u c t i o n o f h o s p i t a l u s a g e ; M e d i c a l C a r e i n t h e R u r a l S e t t i n g Many o f t h e d i f f i c u l t i e s w h i c h b e s e t h e a l t h a d m i n i s -t r a t o r s i n t h e p r o v i s i o n o f m e d i c a l c a r e i n m e t r o p o l i t a n a r e a s a r e m a g n i f i e d i n t h e c o n t e x t o f r u r a l l i f e . R u r a l h e a l t h c a r e i s b u r d e n e d by some p a r t i c u l a r p r o b l e m s w h i c h a r e p e c u l i a r t o i t s s e t t i n g , A l t h o u g h t h e m o r t a l i t y r a t e s f o r r u r a l i n h a b i t a n t s a r e l e s s t h a n f o r t h o s e o f t h e i r u r b a n c o u n t e r p a r t s m o r b i d i t y d u r i n g l i f e a p p e a r s t o be g r e a t e r , e s p e c i a l l y f o r c h r o n i c d i s e a s e s ( R o e m e r and A n z e l , 1 9 6 8 ) . . The p r o v i s i o n o f m e d i c a l c a r e f o r m i g r a n t a g r i c u l t u r a l w o r k e r s and f o r I n d i a n s , who do n o t f i t e a s i l y i n t o t h e m a i n s t r e a m o f N o r t h A m e r i c a n l i f e , p r e s e n t s a l m o s t u n i q u e d i f f i c u l t i e s . The b e n e f i t s o f good o c c u p a t i o n a l h e a l t h s e r v i c e s a r e g e n e r a l l y d e n i e d t o r u r a l w o r k e r s . F u n d a m e n t a l t o r u r a l a r e a s t o o , i s t h e l o w p o p u l a t i o n d e n s i t y w h i c h i s r e f l e c t e d i n g r e a t e r d i s t a n c e s b e t w e e n p a t i e n t s a n d t h e s e r v i c e s t h e y m i g h t e x p e c t t o r e c e i v e . The p r o v i s i o n o f t h e s e s e r v i c e s i s f u r t h e r c o m p o u n d e d by t h e v e r y l o w d e n s i t y o f h e a l t h p e r s o n n e l p e r u n i t a r e a . The l a c k o f h e a l t h m a n -p o w e r i n r u r a l a r e a s i s a p e r e n n i a l p r o b l e m . The r a t i o o f m e d i c a l p r o f e s s i o n a l s t o p o p u l a t i o n i s l e s s i n t h e r u r a l s e t t i n g f o r e v e r y c a t e g o r y o f h e a l t h p e r s o n n e l ( M o t t and R o e m e r , 1 9 6 8 ) . F o r t h e m o s t p a r t s m a l l r u r a l c o m m u n i t i e s h a v e f e w e r a d v a n t a g e s t o o f f e r t h e y o u n g p h y s i c i a n when c o m p e t i n g f o r h i s s e r v i c e s i n t h e o p e n m a r k e t . In p a r t i c u -l a r t h e a m e n i t i e s a v a i l a b l e , b o t h t e c h n i c a l and s o c i a l , a r e f e w e r ( M c N e r n e y and R i e d e l , 1 9 6 2 ) . M o r e o v e r , e v e n i f i t w e r e p o s s i b l e t o a t t r a c t p e r s o n n e l i n t o t h e s e a r e a s i n l a r g e n u m b e r s , t h e r e i s c o n s i d e r a b l e d o u b t as t o w h e t h e r i t w o u l d be w o r t h t h e c o s t i n t e r m s o f t h e d e g r e e t o w h i c h t h e y c o u l d be f u l l y e m p l o y e d . G r e a t e r f i n a n c i n g i s t h e n , c l e a r l y n o t t h e a n s w e r i n i t s e l f . What i s r e q u i r e d f o r t h e s e a r e a s i s a f l e x i b l e and m o b i l e s y s t e m o f m e d i c a l c a r e w h i c h c a n be a d e q u a t e l y i n c o r p o r a t e d i n t o t h e c o n c e p t o f t h e r e g i o n a l i z a -t i o n o f f i x e d f a c i l i t i e s as p r e v i o u s l y d e s c r i b e d . To some e x t e n t t h e s e g o a l s h a v e a l r e a d y b e e n a c h i e v e d . A number o f s c h e m e s h a v e been p r o p o s e d and i m p l e m e n t e d . F l y i n g d o c t o r s e r v i c e s and t h e u s e o f h e l i c o p t e r s i n e m e r -g e n c y ' s i t u a t i o n s h a v e b e e n c o m p r e h e n s i v e l y d o c u m e n t e d i n t h e h e a l t h p l a n n i n g l i t e r a t u r e ( D u n c a n , 1 9 6 6 ; J a c o b s a n d M c L a u g h l i n , 1 9 6 7 ) . The t r a i n i n g o f p a r a - m e d i c a l p e r s o n n e l , who p o s s e s s a l i m i t e d number o f t h e s k i l l s o f t h e q u a l i f i e d p h y s i c i a n , h a v e b e e n s u g g e s t e d f o r u s e i n e x t r e m e l y r e m o t e l o c a t i o n s ( W o r l d H e a l t h O r g a n i z a t i o n , 1 9 5 4 ) . In more a c c e s s -i b l e a r e a s m o b i l e h e a l t h u n i t s h a v e b e e n a d v o c a t e d as o f f e r -i n g an e x c e l l e n t m e t h o d o f p r o v i d i n g c a r e t o c o m m u n i t i e s t o o s m a l l t o m a i n t a i n a p e r m a n e n t h e a l t h f a c i l i t y ( B o d e n h e i m e r , 1969 ; K i 1 s d o n k , 1 9 6 6 ) . S u c h u n i t s h a v e been v a l u a b l e i n t h e t r e a t m e n t o f e n d e m i c i n f e c t i o u s d i s e a s e s i n A f r i c a , and as a p r e v e n t i v e m e d i c a l a i d , p a r t i c u l a r l y f o r t u b e r c u l o s i s s c r e e n i n g , i n t h e more d e v e l o p e d n a t i o n s . P h y s i c i a n s i n r u r a l a r e a s h a v e b e e n e n c o u r a g e d t o p o o l t h e i r r e s o u r c e s , r e c o r d s and e q u i p m e n t i n p a r t i c u l a r c e n t r a l l o c a t i o n s , a n d t o m a i n t a i n a s e r i e s o f s i m p l e s a t e l l i t e o f f i c e s i n s u r r o u n d i n g c o m m u n i t i e s a t w h i c h t h e y s p e n d a f e w h o u r s p e r week a t s p e c i f i e d d a y s and t i m e s ( C a n a d i a n M e d i c a l A s s o c i a t i on , 1 9 6 7 ) . A l t h o u g h i n a l l t h e s e ways a m e a s u r e o f much n e e d e d f l e x i b i l i t y has b e e n i n t r o d u c e d i n t o t h e r u r a l h e a l t h s y s t e m t h e r e s t i l l r e m a i n s t h e f u n d a m e n t a l p r o b l e m o f t h e p r o v i s i o n o f a number o f f i x e d f a c i l i t i e s , s u c h as c l i n i c s o r h o s p i -t a l s , w h i c h c a n s e r v e as a b a s i s f o r t h e o r g a n i z a t i o n o f r u r a l m e d i c a l c a r e , and t o w h i c h p a t i e n t s c a n be s e n t when t h e s e o t h e r m e a s u r e s p r o v e i n a d e q u a t e t o t h e i r n e e d s . B e a r -i n g i n m i n d t h e c o s t l y f i n a n c i a l b u r d e n o f m a i n t a i n i n g h o s p i t a l s i n t h e r u r a l c o n t e x t i t i s p e r h a p s s u r p r i s i n g t h a t so l i t t l e o b j e c t i v e a n a l y s i s has b e e n u n d e r t a k e n on t h e m u l t i p l e l o c a t i o n o f m e d i c a l f a c i l i t i e s i n t h i s s e t t i n g . As r e c e n t l y as 1 9 6 6 L o r d L I e w e w y n - D a v i e s c o u l d s a y , Very l i t t l e s t u d y h a s b e e n m a d e w i t h r e g a r d to t h e l o c a t i o n o f h e a l t h b u i l d i n g s . . . t h e l a c k o f s t u d y o f t h i s t o p i c i s s t r i k i n g w h e n t h e l i t e r a t u r e o f u r b a n p l a n n i n g a n d d e v e l o p m e n t i s e x a m i n e d . ( L o r d L I e w e l y n - D a v i e s , 1 9 6 6 ) S i n c e t h a t s t a t e m e n t was made a f e w r e s e a r c h e r s h a v e t u r n e d t h e i r a t t e n t i o n t o t h i s p r o b l e m a n d a b r i e f r e v i e w o f t h e i r m o d e l s now f o 1 1 o w s . Some P r e v i o u s M o d e l s o f H e a l t h F a c i l i t y L o c a t i o n A t l e a s t t h r e e s t u d i e s i n r e c e n t y e a r s h a v e c o n s i d e r e d m e t h o d s by w h i c h t h e l o c a t i o n a l e f f i c i e n c y o f e x i s t i n g o r p o t e n t i a l h o s p i t a l n e t w o r k s m i g h t be m e a s u r e d . In t h e f i r s t o f h i s two s t u d i e s on t h i s p r o b l e m S c h n e i d e r c o n s i d e r s t h e c a s e o f u r b a n h o s p i t a l s w i t h p a r t i c u l a r r e f e r e n c e t o C i n c i n n a t t i ( S c h n e i d e r , 1 9 6 7 ) . He p o s t u l a t e s t h a t t h e l o c a t i o n a l i m b a l a n c e o f a n y one h o s p i t a l c a n b e s t be a s s e s s e d by a n a l y z i n g t h e n e t p u l l i n g f o r c e o f h o s p i t a l i n - p a t i e n t f l o w s . G i v e n d i s c r e t e r e s i d e n t i a l l o c a t i o n s t o w h i c h a w e i g h t , r e p r e s e n t i n g n u m b e r s o f p a t i e n t s i s a t t a c h e d , he s e e k s t o f i n d among them t h e p o i n t o f m i n i m u m a g g r e g a t e t r a v e l by means o f t h e K u h n - K u e n n e a l g o r i t h m ( K u h n a n d K u e n n e , 1 9 6 2 ) . The d i f f e r e n c e b e t w e e n t h i s c o m p u t e d c e n t r e and t h e a c t u a l l o c a t i o n i s i n t e r p r e t e d as a m e a s u r e o f t h e l o c a t i o n a l i n e f f i c i e n c y o f t h e h o s p i t a l i n t e r m s o f t h e a r e a i t s e r v e s , S e v e r a l a s p e c t s o f t h i s c o n c e p t u a l i z a t i o n d e s e r v e f u r t h e r c o n s i d e r a t i o n . The Kuhn a l g o r i t h m r e q u i r e s E u c l i d e a n d i s t a n c e m e a s u r e s as i t s i n p u t s , a n d w h i l e t h e s e may be a p p r o p r i a t e i n t h e u r b a n c o n t e x t i t i s u n l i k e l y t h a t t h e y w i l l ' b e so i n a r u r a l s e t t i n g . T h e r e i s no g u a r a n t e e e i t h e r t h a t t h e o p t i m a l s o l u t i o n as c o m p u t e d by t h i s m e t h o d w i l l be c o n s t r a i n e d t o a s e t t l e m e n t o r e v e n a d j a c e n t t o a r o a d . F i n a l l y t h i s a p p r o a c h c a n c o n s i d e r o n l y one h o s p i t a l a t a t i m e a l t h o u g h h e u r i s t i c s h a v e b e e n d e v e l o p e d by C o o p e r f o r a p p l y i n g t h e a l g o r i t h m t o m u l t i p l e p l a n t l o c a t i o n p r o b -1 ems ( C o o p e r , 1 9 6 7 ) . In a l a t e r m o d e l S c h n e i d e r e x t e n d s h i s a n a l y s i s t o l o c a t i o n a l i m b a l a n c e s i n h o s p i t a l s y s t e m s . F o r e a c h m e d i c a l c a t e g o r y t h e c e n t r e o f m i n i m u m t o t a l t r a v e l b a s e d on p a t i e n t r e s i d e n c e s i s c o m p u t e d as b e f o r e . N e x t t h i s c e n t r e i s f o u n d w i t h r e s p e c t t o h o s p i t a l l o c a t i o n s , t h e w e i g h t s i n t h i s i n s t a n c e b e i n g t h e h o s p i t a l a d m i s s i o n f i g u r e s . The d i f f e r -e n c e s b e t w e e n t h e two p o i n t s i s t a k e n as an i n i t i a l m e a s u r e o f i m b a l a n c e i n t h e s y s t e m t o w h i c h i s a d d e d t h e d i s p a r i t y b e t w e e n t h e r e l a t i v e d i s p e r s i o n s ( s t a n d a r d d i s t a n c e s ) a b o u t t h e two c e n t r e s , u n d e r t h e a s s u m p t i o n t h a t t h e g r e a t e r t h e d e g r e e o f p a t i e n t d i s p e r s a l r e l a t i v e t o h o s p i t a l d i s p e r s a l , t h e g r e a t e r t h e s p a t i a l i n e f f i c i e n c y o f l o c a t i o n . A l t h o u g h t h i s f o r m u l a t i o n has m e r i t o v e r t h e p r e v i o u s one i t s u f f e r s f r o m t h e d i s a d v a n t a g e , as n o t e d by t h e a u t h o r , o f e x a g g e r a t -i n g t h e s t a n d a r d d i s t a n c e m e a s u r e s i n c a s e s w h e r e p a t i e n t demand i s c l u s t e r e d i n t o two o r more s p a t i a l l y d i s t i n c t a r e a s ( S c h n e i d e r , 1 9 6 8 ) . M o r e o v e r , w h i l e t h i s a p p r o a c h w i l l g i v e i n s i g h t s i n t o i m b a l a n c e s i n t h e p r e s e n t s y s t e m i t has l e s s t o s a y on w h e r e t h e b e s t s e t o f l o c a t i o n s m i g h t be i f more t h a n one i s d e s i r a b l e . I t i s c l e a r l y p o s s i b l e t o p o s t u l a t e new l o c a t i o n s and r e - c y c l e t h e a l g o r i t h m i n an a t t e m p t t o r e d u c e t h e i m b a l a n c e m e a s u r e , b u t i f t h e number o f p o s s i b l e s i t e s i s l a r g e a c o m b i n a t o r i a l p r o b l e m may be e n c o u n t e r e d w h i c h has no f e a s i b l e c o m p u t a t i o n a l s o l u t i o n . G o u l d and L e i n b a c h s t u d i e d t h e p r o b l e m o f r u r a l h o s p i t a l l o c a t i o n s i n G u a t e m a l a . F rom f i v e p o s s i b l e l o c a -t i o n s t h e y s e t o u t t o f i n d t h o s e t h r e e w h i c h w o u l d m i n i m i z e t o t a l p a t i e n t d i s t a n c e t r a v e l l e d . T h e y b a s e d t h e i r a p p r o a c h upon t h e c l a s s i c t r a n s p o r t a t i o n m o d e l . S e t t i n g i n i t i a l h o s p i t a l c a p a c i t y t o be e q u a l a t e a c h s i t e , t h e y e v a l u a t e d e v e r y p o s s i b l e c o m b i n a t i o n o f t h r e e o u t o f t h e f i v e p o t e n t i a l l o c a t i o n s , a s s i g n i n g t h e f l o w s f r o m e a c h p a t i e n t a r e a t o t h e c u r r e n t s e t o f h o s p i t a l s i n s o l u t i o n by means o f a l i n e a r p r o g r a m m i n g a l g o r i t h m . By i n s p e c t i n g t h e b e s t s o l u t i o n so o b t a i n e d , i n e f f i c i e n t c r o s s f l o w s w e r e e l i m i n a t e d by a d j u s t -i n g t h e h o s p i t a l s i z e i n p u t s and r e - c y c l i n g t h e a l g o r i t h m u n t i l a s a t i s f a c t o r y f l o w p a t t e r n e m e r g e d ( G o u l d and L e i n b a c h , 1 9 6 6 ) . In f a c t , i f an a l l c o m b i n a t i o n s a p p r o a c h i s a d o p t e d t h e n t h e l i n e a r p r o g r a m m i n g a s p e c t o f t h i s s o l u -t i o n p r o c e d u r e i s q u i t e u n n e c e s s a r y . A l l t h a t i s r e q u i r e d i s t o a s s i g n t h e demand a t e a c h p a t i e n t a r e a t o t h e c l o s e s t one o f t h e t h r e e h o s p i t a l s c u r r e n t l y i n t h e s o l u t i o n . T h i s p r o c e d u r e w i l l g i v e t h e o p t i m a l s o l u t i o n , and p r o v i d e i n a d d i t i o n t h e a p p r o p r i a t e h o s p i t a l s i z e w h i c h w i l l n o t n e c e s s a r i l y be f o u n d by t h e G o u l d / L e i n b a c h m e t h o d . H o w e v e r , s i n c e t h e number o f c o m b i n a t i o n s r i s e s v e r y r a p i d l y b o t h as a f u n c t i o n o f t h e number o f l o c a t i o n s r e q u i r e d and o f t h e number o f p o t e n t i a l l o c a t i o n s , t h i s a p p r o a c h w i l l r a p i d l y become i n f e a s i b l e f o r e v e n m o d e r a t e l y s i z e d p r o b l e m s . M o r r i l l and K e l l e y p r e s e n t a d i f f e r e n t m o d e l whose b a s i c o b j e c t i v e i s t o m e a s u r e t h e e f f i c i e n c y o f t h e l o c a t i o n o f c a p a c i t y o v e r an o p e r a t i n g s y s t e m o f h o s p i t a l s i t e s . The a u t h o r s make u s e o f a m o d i f i e d g r a v i t y i n t e r a c t a n c e a l g o r -i t h m i n w h i c h p a t i e n t f l o w s f r o m a r e a s t o h o s p i t a l s a r e c o n c e i v e d t o be n o t o n l y an i n v e r s e f u n c t i o n o f d i s t a n c e , b u t a r e a l s o r e l a t e d d i r e c t l y t o h o s p i t a l s i z e and t o t h e s i z e s and d i s t a n c e s away o f i n t e r v e n i n g o p p o r t u n i t i e s ( M o r r i l l and K e l l e y , 1 9 7 0 ) . T h i s c o n c e p t u a l i z a t i o n has b e h a v i o u r a l m e r i t i n t h e s e n s e t h a t s i z e c a n be c o n s i d e r e d t o be a c r u d e r e f l e c t i o n o f t h e h o s p i t a l r e f e r r a l f u n c t i o n and as s u c h p a t i e n t s w i l l n o t n e c e s s a r i l y be c o n s t r a i n e d t o a d m i t t a n c e a t t h e i r n e a r e s t f a c i l i t y . As w i t h t h e H u f f ( 1 9 6 3 ) and L a c k s h m a n a n and H a n s e n ( 1 9 6 6 ) m o d e l i n i t i a l u s e i s made o f t h e i n t e r a c t i o n f o r m u l a t i o n t o p r e d i c t f l o w s b e t w e e n r e s i -d e n c e s a n d f a c i l i t i e s . The d i v e r g e n c e o f t h e p r e d i c t e d f r o m a c t u a l h o s p i t a l c a p a c i t y i s c o n s i d e r e d by t h e a u t h o r s t o be a m e a s u r e o f t h e i m b a l a n c e o f t h e l o c a t i o n o f c a p a c i t y . The p a r t i c u l a r c o n t r i b u t i o n o f M o r r i l l a n d K e l l e y i s i n t h e a l g o r i t h m by w h i c h h o s p i t a l c a p a c i t y i s r e l o c a t e d so t h a t a g g r e g a t e p a t i e n t t r a v e l i s r e d u c e d b u t i n a m a n n e r w h i c h i s s t i l l c o n s i s t e n t w i t h t h e b a s i c s t r u c t u r e o f t h e i n t e r -a c t a n c e m o d e l . In t h e i r w o r d s A n a l l o c a t i o n c a n b e c o n s i d e r e d o p t i m a l i f t h e m o d e l i s i n t e r p r e t e d . a s m a x i m i z i n g t h e j o i n t s a t i s f a c t i o n o f a h e t e r o g e n e o u s p o p u l a t i o n a t m i n i m u m d i s t a n c e 3 b u t w h e r e s a t i s f a c t i o n r e -q u i r e s t h e u s e by s o m e p e o p l e , p e r h a p s p a r t o f t h e t i m e , o f a v a r i e t y o f o p p o r t u n i t i e s . ( M o r r i l l and K e l l e y , 1 9 7 0 ; p . 2 8 6 ) . The r e - a l l o c a t i n g a l g o r i t h m a d j u s t s t h e s i z e s o f t h e h o s p i -t a l by r e d u c i n g c a p a c i t y a t u n d e r p r e d i c t e d l o c a t i o n s and i n c r e a s i n g i t a t o v e r p r e d i c t e d o n e s . T h i s s t e p w h i c h a m o u n t s t o u s i n g t h e p r e d i c t e d c a p a c i t i e s as i n p u t s , i s r e p e a t e d u n t i l some a c c e p t a b l e l e v e l o f c o n v e r g e n c e i s a t t a i n e d b e t w e e n two s u c c e s s i v e i t e r a t i o n s . I t s h o u l d be n o t e d h o w e v e r , t h a t a f t e r t h e i n i t i a l a l l o c a t i o n t h e c a l c u l a t e d d i v e r g e n c e may be a s much due t o i m b a l a n c e s i n t h e g r a v i t y m o d e l as t o a c t u a l i m b a l a n c e s i n t h e l o c a t i o n o f c a p a c i t y . F u r t h e r m o r e t h e r e d u c t i o n i n a g g r e g a t e p a t i e n t t r a v e l w h i c h m i g h t be a c h i e v e d w i l l n o t n e c e s s a r i l y be one i n w h i c h t h e c a p a c i t y o f l o c a t i o n has b e e n o p t i m a l l y r e d i s t r i b u t e d . S e v e r a l e q u i l i b r i u m s t a t e s may e x i s t and i t i s p o s s i b l e t o d e v i s e n e t w o r k s t r u c t u r e s w h e r e t h i s o c c u r s t o w i t h i n q u i t e l o w t o l e r a n c e l e v e l s ' ( u n d e r 5 p e r c e n t ) , and w h e r e t h e a l g o r i t h m c o n v e r g e s f r o m a g i v e n s t a r t i n g p o i n t t o some l o c a l m i n i m u m . An a l t e r n a -t i v e m e t h o d s u g g e s t e d h e r e i s t o b e g i n t h e c y c l i n g w i t h an i n i t i a l s e t o f h o s p i t a l s i z e s b a s e d n o t on t h e c u r r e n t o p e r a t i n g l e v e l s b u t upon a s e t o f v a l u e s d e r i v e d f r o m a l l o c a t i n g e a c h a r e a o f p a t i e n t demand t o i t s n e a r e s t t r e a t ment c e n t r e . I t i s c o n j e c t u r e d t h a t t h i s p r o c e d u r e w i l l t e n d t o g i v e a b e t t e r r e d i s t r i b u t i o n o f c a p a c i t y s i n c e t h e d i r e c t i o n o f a d j u s t m e n t t o a c h i e v e c o n s i s t e n c y w i t h t h e g r a v i t y s t r u c t u r e i s a l w a y s away f r o m t h e m i n i m u m d i s t a n c e a l l o c a t i o n . T h i s s u g g e s t s t h a t t h e e q u i l i b r i u m o b t a i n e d upon t e r m i n a t i o n o f t h e i t e r a t i o n s w i l l t e n d t o be n e a r e r t h e d e s i r e d g o a l t h a n a n y o t h e r f e a s i b l e s o l u t i o n . E x p e r i -m e n t s w i t h a v e r y l i m i t e d number o f s m a l l h y p o t h e t i c a l e x a m p l e s h a v e b o r n e t h i s o u t i n some i n s t a n c e s , and i n o t h e r c a s e s w h e r e b o t h m e t h o d s c o n v e r g e d t o t h e same a p p r o x i m a t e s o l u t i o n t h e p r o p o s e d a l g o r i t h m r e q u i r e d f e w e r i t e r a -t i o n s t o c o n v e r g e , p a r t i c u l a r l y w h e r e t h e s t a r t i n g p o i n t was b a d l y o u t o f b a l a n c e . I t i s p o s s i b l e t o t e s t t h e v i a b i l i t y o f new f a c i l i -t i e s w i t h t h e M o r r i 1 1 - K e l 1 e y a l g o r i t h m by i n t r o d u c i n g p o t e n t i a l s i t e s i n t o t h e n e t w o r k a n d r e - a l l o c a t i n g f l o w s a c c o r d i n g t o t h e i n t e r a c t a n c e m e c h a n i s m i n much t h e same way as was p r o p o s e d by ( H u f f , 1 9 6 6 ) f o r r e t a i l o u t l e t s . T h i s m o d e l c a n n o t h o w e v e r , t e s t f o r t h e d i r e c t e f f i c i e n c y o f l o c a t i o n i n t h e s e n s e o f s o l v i n g f o r t h e o p t i m a l s y s t e m o f l o c a t i o n s w i t h r e s p e c t t o a l l p o s s i b l e s i t e s . I t i s t h i s p a r t i c u l a r p r o b l e m w i t h w h i c h t h i s s t u d y i s p r i n c i p a l l y c o n c e r n e d . A G e n e r a l A p p r o a c h t o R u r a l H o s p i t a l L o c a t i o n S y s t e m s As s u g g e s t e d p r e v i o u s l y i n t h i s c h a p t e r t h e i n t e r -r e l a t i o n s h i p s among t h e many c o m p o n e n t s o f t h e h e a l t h c a r e s y s t e m t h r o u g h w h i c h t h e p a t i e n t r e c e i v e s m e d i c a l c a r e a r e v e r y c o m p l e x and n o t f u l l y r e s o l v e d as y e t . The p a t h o f some p a t i e n t t o t r e a t m e n t a t a n y h o s p i t a l may be t h e c u l m i -n a t i o n o f a c o m p l i c a t e d c h a i n o f e v e n t s . The r o l e o f a c c e s s i b i l i t y i n t h i s p r o c e s s h a s r e c e i v e d c o n s i d e r a b l e d o c u m e n t a t i o n i h t h e p a s t ( M o r r i l l and E a r i c k s o n , 1 9 7 0 ; M a r r i n s o n , 1 9 6 6 ) . Even i n m e t r o p o l i t a n r e g i o n s , w h e r e t h e p a t t e r n o f c r o s s f l o w s c a n be e x p e c t e d t o be c o m p l e x , d i s -t a n c e m i n i m i z i n g t e n d e n c i e s e x e r t a s i g n i f i c a n t i n f l u e n c e ( L u b i n e t . a l . , 1 9 6 6 ) . T h i s p r i n c i p i e i s h o w e v e r , f r e -q u e n t l y v i o l a t e d by o t h e r o f f s e t t i n g f a c t o r s . F o r t h e p h y s i c i a n w i t h h o s p i t a l p r i v i l e g e s t h e c h o i c e o f p r e f e r r e d o f f i c e l o c a t i o n w i l l t e n d t o be i n f l u e n c e d by a c c e s s i b i l i t y t o t h e h o s p i t a l i n w h i c h he p r a c t i c e s , and w h e r e t h i s c o i n -c i d e s w i t h a b u s i n e s s d i s t r i c t a c o n s i d e r a b l e p r o p o r t i o n o f t h e c l i e n t e l e may be e x p e c t e d t o be i n a t t e n d a n c e d u r i n g t h e i r w o r k i n g h o u r s . S i n c e t h e c h o i c e o f h o s p i t a l f o r t r e a t m e n t i s n o r m a l l y d i c t a t e d by t h e p h y s i c i a n ' s p r e f e r e n c e t h i s c a n n o t a l w a y s be e x p e c t e d t o c o i n c i d e w i t h t h e p a t i e n t ' s g e n e r a l a r e a o f r e s i d e n c e . C a s e m i x d i f f e r e n c e s a r e a l s o o f i m p o r t a n c e p a r t i c u l a r l y w i t h r e s p e c t t o t h e l e s s f r e -q u e n t l y u t i l i z e d s p e c t r u m o f s e r v i c e s . R e f e r r a l s t o s p e c -i a l i s t s and t o h o s p i t a l s f o r s p e c i a l i s e d t r e a t m e n t s w i l l c l e a r l y t e n d t o o f f s e t t h e d i s t a n c e m i n i m i z i n g p r i n c i p l e . O t h e r h o s p i t a l s w h i c h h a v e p a r t i c u l a r r e l i g i o u s o r e t h n i c a f f i l i a t i o n s may d r a w p a t i e n t s f r o m a n y p a r t o f t h e c i t y . F i n a l l y t h e r e a r e a w h o l e r a n g e o f l e s s w e l l d e f i n e d v a r i -a b l e s r e l a t e d t o t h e q u a l i t y o f a v a i l a b l e s e r v i c e s and t h e p r e s t i g e o f i n s t i t u t i o n s and p r a c t i t i o n e r s , w h i c h c a n and do c o m p l i c a t e t h e w h o l e p a t t e r n o f p a t i e n t f l o w s . S u c h i n t e r v e n i n g o p p o r t u n i t i e s a r e , a s i s t o be e x p e c t e d , much l e s s i n e v i d e n c e i n t h e r u r a l c o n t e x t , w h e r e d i s t a n c e s b e t w e e n c o m m u n i t i e s r e s o l v e t h e p r o b l e m i n t o a s t u d y o f i n t e r r a t h e r t h a n i n t r a u r b a n l o c a t i o n s . A p a r t f r o m s p e c i a l i s t r e f e r r a l s ( l e s s t h a n 10 p e r c e n t o f a l l p a t i e n t s ) t h e a c c e s s i b i l i t y v a r i a b l e a p p e a r s , w i t h some e x c e p t i o n s , t o be much t h e m o s t d o m i n a n t f o r c e . S i n c e t h i s i s g e n e r a l l y so t h e p r o b l e m o f o p t i m i z i n g h o s p i t a l l o c a t i o n s i n r u r a l a r e a s c a n be c o n c e p t u a l l y r e d u c e d t o one o f f i n d i n g t h e m e d i a n s o f a w e i g h t e d g r a p h s u b j e c t t o c e r t a i n s i d e c o n s t r a i n t s . S u c h a m o d e l i s d e v i s e d i n t h e f o l l o w i n g c h a p t e r and t e s t e d i n t h e f o u r t h . The t e s t i n g p r o c e d u r e f o r a s s e s s i n g t h e l o c a t i o n a l e f f i c i e n c y o f a s e t o f s i t e s o v e r a g i v e n n e t w o r k i s t o m e a s u r e t h e d i f f e r e n c e s i n t o t a l d i s t a n c e t r a v e l l e d b e t w e e n two s o l u t i o n s , one c o n s t r a i n e d t o t h e e x i s t i n g s y s t e m o f l o c a t i o n s a n d t h e o t h e r u n c o n -s t r a i n e d . I f a more r e a l i s t i c t r i p d i s t r i b u t i o n p a t t e r n i s d e s i r e d , one i n w h i c h p a t i e n t a r e a s a r e n o t a l l o c a t e d i n b l o c k t o a s i n g l e c e n t r e , t h e n t h e s e s o l u t i o n s s h o u l d be m o d i f i e d by an i n t e r a c t a n c e m o d e l . T h e s e v a r i o u s a p p l i c a t i o n s a r e u n d e r t a k e n i n C h a p t e r I V , t o g e t h e r w i t h a p r a c t i c a l c o m p a r -i s o n o f t h e two m e t h o d s p r e v i o u s l y d i s c u s s e d f o r r e d i s t r i b u t -i n g c a p a c i t y o v e r an o p e r a t i n g l o c a t i o n s y s t e m . CHAPTER I I I AN ALGORITHM FOR EST IMATING THE MEDIANS OF A  WEIGHTED GRAPH SUBJECT TO S IDE CONSTRAINTS P r o b l e m S t a t e m e n t I f h o s p i t a l l o c a t i o n c a n be r e g a r d e d as a s p e c i a l c a s e o f t h e g e n e r a l i z e d p l a n t l o c a t i o n p r o b l e m t h e n t h e s o l u t i o n r e q u i r e s t h e d e t e r m i n a t i o n o f t h e n u m b e r , s i z e s and l o c a t i o n s o f p l a n t s w h i c h w i l l s e r v i c e a t a m i n i m u m o v e r a l l c o s t , a g i v e n s e t o f demands a t d i s c r e t e p o i n t s o v e r s p a c e . The u n i t t r a n s p o r t c o s t s o r d i s t a n c e s b e t w e e n e a c h demand p o i n t and p o t e n t i a l l o c a t i o n s i t e m u s t be known i n a d v a n c e , and i f t h e number o f s p e c i f i e d p l a n t s i s n o t known a t t h e o u t s e t i t w i l l be a s s u m e d t h a t a p r o d u c t i o n c o s t f u n c t i o n i s a s s o c i a t e d w i t h e a c h p o t e n t i a l p l a n t l o c a -t i o n . B e c a u s e o f t h e e c o n o m i e s o f s c a l e i n h e r e n t i n p l a n t p r o d u c t i o n c o s t s t h i s p r o b l e m , b o t h t h e o r e t i c a l l y and c o m -p u t a t i o n a l l y , i s one o f c o n s i d e r a b l e c o m p l e x i t y . A t t h e p r e s e n t t i m e no u n d i s p u t e d e x a c t a n a l y t i c a l s o l u t i o n h a s b e e n d e v e l o p e d f o r t h e c a s e w h e r e t h e p l a n t c o s t f u n c t i o n s a r e c o n t i n u o u s a n d c o n c a v e o v e r t h e w h o l e o f t h e i r f e a s i b l e r a n g e . I t i s known h o w e v e r , t h a t u n d e r s c a l e e c o n o m i e s u n i t p r o d u c t i o n c o s t s w i l l r i s e as t h e number o f p l a n t s i n c r e a s e s , w h e r e a s c o n v e r s e l y , t h e t r a n s p o r t a t i o n c o s t s w i l l f a l l ( F i g u r e 1 ) . In t h e t w i n c o s t p l a n t l o c a t i o n p r o b l e m t h e g o a l t h e r e f o r e , i s t o a c h i e v e an e q u i l i b r i u m among t h e s e two o p p o s i n g s e t s o f f u n c t i o n s s u c h t h a t t h e t o t a l c o s t l e v e l i s m i n i m i z e d . The r e a l n a t u r e o f many p l a n t l o c a t i o n p r o b l e m s i s s u c h t h a t i t i s f r e q u e n t l y d e s i r a b l e t o i m p o s e , w h e r e p o s -s i b l e , a f u r t h e r s e t o f c o n d i t i o n s and c o n s t r a i n t s upon t h e s y s t e m . S u c h c o n s t r a i n t s c a n t a k e a w i d e v a r i e t y o f f o r m s and may i n c l u d e r e s t r i c t i o n s p l a c e d upon t h e s i z e o f p l a n t o p e r a t i o n , upon t h e maximum f l o w a l l o w a b l e o v e r a n y one l i n k , o r upon some p r e d e t e r m i n e d maximum d i s t a n c e any demand p o i n t may be f r o m some s u p p l y i n g s o u r c e . The number o f p o t e n t i a l l o c a t i o n s may be l e s s t h a n t h e number o f n e t w o r k n o d e s , o r a g a i n t h e r e may be a p r i o r i r e a s o n s f o r f o r c i n g c e r t a i n l o c a t i o n s i n t o t h e s o l u t i o n a s , f o r e x a m p l e , when e x t e n s i o n s t o an e x i s t i n g d i s t r i b u t i o n s y s t e m a r e r e q u i r e d . M u l t i p l e p r o d u c t a l l o c a t i o n p r o b l e m s , s t o c h a s t i c demand f u n c t i o n s and w a r e h o u s e l o c a t i o n s , i n t e r m e d i a t e b e t w e e n p l a n t s o u r c e s and c u s t o m e r s , r e p r e s e n t f u r t h e r p o t e n t i a l v a r i a -t i o n s upon t h e g e n e r a l f o r m u l a t i o n . In t h i s c h a p t e r a n u m e r i c a l m e t h o d i s d e v e l o p e d f o r o b t a i n i n g o p t i m a l o r n e a r o p t i m a l s o l u t i o n s t o t h e p l a n t 1 o c a t i o n . p r o b l e m , s u b j e c t w h e r e a p p l i c a b l e , t o a s p e c i f i c s u b s e t o f a l l f e a s i b l e c o n s t r a i n t s . Some a d v a n t a g e i s t a k e n i n t h i s m o d e l o f an i n g e n i o u s p a r t i t i o n i n g h e u r i s t i c g i v e n \jransportation \ Cost "'Plant Cost c o Unit Cost Funct ! I I 1 Minimum Cost Solution 1 1 1 1 1 Number of Plants F i g u r e 1 R e l a t i o n s h i p B e t w e e n P l a n t and T r a n s p o r t a t i o n C o s t F u n c t i o n • Variable ^ - s » " C o s t ^ r ^ it Cost — y? >Vi__ 1 —Piecewise Linear Cost Approximation o OL — / > / • Ss S' / / /' > Fixed Cost j 1 1 1 1 1 1 1 Plant Size F i g u r e 2 P i e c e W i s e L i n e a r A p p r o x i m a t i o n o f a C o n c a v e P l a n t C o s t F u n c t i o n by ( S i n g e r , 1 9 6 8 ) w h o s e c o n t r i b u t i o n w i l l s u b s e q u e n t l y be r e v i e w e d . The p a r t i c u l a r p r o b l e m s p e c i f i c a t i o n t o be c o n -s i d e r e d h e r e c a n be e x p r e s s e d as f o l l o w s : G i v e n ; (1) The l o c a t i o n o f each d e s t i n a t i o n . (2) The demand a t each d e s t i n a t i o n f o r a s i n g l e p r o d u c t commodity c o n s i d e r e d as a f i x e d n o n - s t o c h a s t i c v a r i a b l e . (3) A v e c t o r o f p o t e n t i a l p l a n t s i t e l o c a t i o n s . (4) A subset of l o c a t i o n s t o be f o r c e d i n t o the s o l u t i o n — i f n e c e s s a r y . (5) An upper bound on the s i z e o f each p o t e n t i a l p l a n t s i t e — i f o p t i o n e d . (6) A c o n s t r a i n t on the maximum d i s t a n c e any d e s t i n a t i o n may be from some s u p p l y s o u r c e — i f o p t i o n e d . (7) A s e t o f u n i t t r a n s p o r t c o s t s ( c u r r e n c y , time or d i s t a n c e ) between each d e s t i n a t i o n and each p o t e n t i a l s u p p l y p o i n t w h i c h , s i n c e they a re averaged u n i t v a l u e s , may r e p r e s e n t n o n - l i n e a r i t i e s . ( 8 ) A v e c t o r o f p r o d u c t i o n f u n c t i o n s — i f o p t i o n e d — o f any t y p e , which may o r may not be i d e n t i c a l f o r each p r o s p e c t i v e s i t e . Then d e t e r m i n e ; (1) The number o f s u p p l y s o u r c e s r e q u i r e d (unknown o n l y i f p l a n t c o s t s a re o p t i o n e d ) . ( 2 ) The l o c a t i o n o f each s u p p l y s o u r c e . (3) The s i z e o f each s u p p l y s o u r c e . (4) The a l l o c a t i o n o f d e s t i n a t i o n s t o each s u p p l y p o i n t . (5) The amount s u p p l i e d from each p l a n t s o u r c e so chosen t o each d e s t i n a t i o n so a l l o c a t e d (may be f r a c t i o n a l o n l y i f p l a n t s i z e upper bounds are imposed). M a t h e m a t i c a l l y t h i s i s a- m i x e d i n t e g e r p r o g r a m -m i n g p r o b l e m o f t h e f o r m : ( 1 ) Min = s s u b j e c t t o m n m = T y X.. D.. X. + I f . (T.) X. 1=1 j i i 1 J 1 J 1 i = i 1 1 1 ( 2 ) X ^ 0 ( 1=1 , •••,,m ; j = l , . " , n ) _L J m ( 3 ) I X = B (j=l,.,..,n) 1=1 1 J J (4.) I X = T (1 = 1,.-.,m) j = l 3 1 m n m n ( 5 ) I T = I B = I I X (1=1,,..,m ; 1 = 1 j = l J 1 = 1 j = l 1 J j = l , • • • , n ) ( 6 ) T. ^  C. (1=1, . . . ,m) ( 7 ) D i j " v j (1=1,••• sm ; j = l , . . . , n ) (8.) X± = 1 (1=1,-..,k) (9) x± = (l=k+l,...,m) Where n = The number o f d e s t i n a t i o n s k = The number o f f o r c e d p l a n t s i n s o l u t i o n ( i f any) m = The t o t a l number o f p o t e n t i a l p l a n t s i t e s ( i n c l u d i n g f o r c e d nodes) X. . .= The commodity f l o w from p l a n t i t o des-^ t i n a t i o n j .T. = The volume o f p l a n t i D. . = The c o s t o f t r a n s p o r t i n g one u n i t o f X.. 1^ from p l a n t i t o d e s t i n a t i o n j . x^ C. = The p l a n t s i z e upper bound c o n s t r a i n t on 1 p l a n t i B . = The demand a t d e s t i n a t i o n j 3 V. = Maximum d i s t a n c e ( t r a n s p o r t c o s t ) c o n s t r a i n t ^ on d e s t i n a t i o n j f . =• P l a n t c o s t f u n c t i o n a t p l a n t i A. = A zero one v a r i a b l e ; a one i n d i c a t i n g a 1 p l a n t s i t e i s i n the s o l u t i o n , a ze r o i n d i c a t i n g i t i s n o t . F o r m u l a t i o n o f t h e P r o b l e m i n T e r m s o f a W e i g h t e d G r a p h F o r t h e p u r p o s e o f s u b s e q u e n t e x p o s i t i o n t h i s m a t h e -m a t i c a l s p e c i f i c a t i o n o f t h e p r o b l e m may be r e p h r a s e d i n t e r m s o f a w e i g h t e d g r a p h . C o n s i d e r a c o n n e c t e d g r a p h G w i t h n v e r t i c e s l a b e l l e d v . , i = l , - - - , n and k b r a n c h e s o r l i n k s . E a c h l i n k has a l e n g t h >0 a s s o c i a t e d w i t h i t s u c h t h a t t h i s l e n g t h i s t h e d i s t a n c e f r o m some v . t o some v . . • i J In a d d i t i o n e a c h node h a s a t t a c h e d t o i t a w e i g h t b^ w h e r e b . >0 and i s t h e demand a t v . . I f b . >1 f o r a t l e a s t o n e node on t h e n e t w o r k a w e i g h t e d g r a p h e x i s t s ; o t h e r w i s e t h e g r a p h i s s a i d t o be n o n - w e i g h t e d . The b a s i c p r o b l e m , w i t h -o u t c o n s t r a i n t s , i s t o f i n d a s e t o f p n o d e s on t h e g r a p h s u c h t h a t t h e w e i g h t e d d i s t a n c e s f r o m t h i s s e t t o a l l o t h e r n o d e s on t h e g r a p h i s a m i n i m u m o v e r a n y s u c h s e t o f p . D e s i g n a t e a n y s e t o f p n o d e s on G ( x 1 , x 2 , , , , , x ) as Xp and l e t D = | d ( v ^ v. . ) | an n by m d i s t a n c e m a t r i x i n w h i c h e a c h e l e m e n t d ( v . v . ) i s t h e v a l u e o f t h e s h o r t e s t p a t h f r o m v e r t e x i , i = 1 , • - • , n t o v e r t e x j , j = l , w h e r e m i s t h e number o f p o t e n t i a l p l a n t s i t e s m s n . F rom |D| g e n e r a t e a new m a t r i x |W| s u c h t h a t w(v.. v . ) = d (v . . v j ) D . j o v e r a l l i and j . T h i s w e i g h t e d d i s t a n c e m a t r i x |W| i s t h e n , e q u i v a l e n t t o t h e m u l t i p l i c a t i o n o f e a c h row i i n |D| by t h e c o r r e s p o n d i n g w e i g h t b . o f t h e v e r t e x v . . F o r e a c h n o d e ' v . l e t ( 1 0 ) w(v. X ) = min l p w(v. X , ) ,. W ( V . X „ ) , ' ' ' , W ( V . x ) l 1 3 l 2 ' ' I p t h a t i s , f i n d f o r e a c h node i t s m i n i m u m w e i g h t e d d i s t a n c e t o some member o f t h e s e t X . • The r e q u i r e d s e t o f p n o d e s Xp on G i s t h e p m e d i a n o f t h e n e t w o r k i f , f o r e v e r y o t h e r p o s s i b l e s e t o f X on G . P n n ( 1 1 ) I w(v . X ) > T w ( v . X*) i = l i P ± ^ 1 w i P y As S c o t t h a s r e c e n t l y p r o v i d e d an e x c e l l e n t c r i t i q u e o f p l a n t l o c a t i o n a l g o r i t h m s o n l y a s h o r t s u r v e y o f t h e c u r r e n t m e t h o d s w i l l be r e v i e w e d h e r e ( S c o t t , 1 9 7 0 ) . T h e r e a r e , b r i e f l y , two p o s s i b l e a p p r o a c h e s t o t h e p r o b l e m ; e x a c t s o l u t i o n t e c h n i q u e s a n d h e u r i s t i c n u m e r i c a l m e t h o d s . ( a ) E x a c t S o l u t i o n M e t h o d s I f p l a n t c o s t s a r e i g n o r e d t h e n t h e b a s i c l o c a t i o n p r o b l e m f o r a n y f i x e d v a l u e o f p c a n be n e a t l y s t r u c t u r e d and s o l v e d i n a l i n e a r p r o g r a m m i n g f o r m a t ( R e v e l l e and S w a i n , 1 9 7 0 ) . The i n c l u s i o n o f p r o d u c t i o n c o s t s a d d s c o n -s i d e r a b l y t o t h e c o m p l e x i t y o f t h e p r o b l e m , and as p r e v i o u s l y n o t e d , t h e v a l u e o f p i s n o t known a t t h e o u t s e t . An e x a c t s o l u t i o n m e t h o d f o r c a s e s w h e r e t h e p l a n t c o s t s a r e c o n t i n u -o u s l y c o n c a v e o v e r t h e i r w h o l e r a n g e i s g i v e n by R u t e n b e r g ( R u t e n b e r g , 1 9 6 8 ) , b u t i t r e m a i n s , t o be c o n c l u s i v e l y shown t h a t t h i s a l g o r i t h m w i l l a l w a y s c o n v e r g e t o t h e o p t i m a l s o l u t i o n . E f r o y m s o n and Ray ( 1 9 6 6 ) t a k e a d i f f e r e n t , and c o m p u t a t i o n a l l y f a s t e r a p p r o a c h by a p p r o x i m a t i n g t h e c o n c a v e c o s t f u n c t i o n w i t h p i e c e w i s e l i n e a r s e g m e n t s ( F i g u r e 2 ) , and s o l v i n g a g a i n i n an e x t e n d e d p r o g r a m m i n g f o r m . As a . g e n e r a l r u l e t h e i n i t i a l l i n e a r p r o g r a m m i n g s o l u t i o n w i l l n o t meet t h e i n t e g e r r e s t r i c t i o n w h i c h r e q u i r e s e a c h p o -t e n t i a l p l a n t s i t e t o h a v e a v a l u e o f z e r o o r one d e p e n d i n g upon w h e t h e r i t i s , o r i s n o t , a c t i v e i n t h e f i n a l s o l u t i o n . I f e a c h v a r i a b l e n o t s a t i s f y i n g t h i s c o n d i t o n i s now g i v e n one o f t h e s e v a l u e s , and i f e v e r y a l t e r n a t i v e i s c o n s i d e r e d , a c o m b i n a t o r i a l t r e e c a n be c o n s t r u c t e d w h i c h r e p r e s e n t s t h e e n t i r e s p a c e o f a l l f e a s i b l e s o l u t i o n s ! The g e n e r a t i o n o f s u c h a t r e e e s t a b l i s h e s a s e t o f t e r m i n a l s t a t e s , e a c h one o f w h i c h i s an i n t e g e r s o l u t i o n . In p r a c t i c e i t i s n o t p o s s i b l e t o s e a r c h a l l o f t h e s e , a n d t h e a p p r o a c h t a k e n by E f r o y m s o n and Ray i s t o d e v i s e a b r a n c h - b o u n d a l g o r i t h m w h i c h w i l l i g n o r e e v a l u a t i o n o f a l l s u c h a l t e r n a t i v e s as l i e a l o n g t h o s e b r a n c h e s o f t h e t r e e w h o s e d e s c e n d e n t s o l u t i o n s a r e k n o w n , a p r i o r i , t o be i n f e a s i b l e by v i r t u e o f some p r e -v i o u s l y e s t a b l i s h e d l o w e r b o u n d i n t e g e r s o l u t i o n . The g e n e r a l n a t u r e o f b r a n c h - b o u n d a l g o r i t h m s c a n be r e a d l y i l l u s t r a t e d w i t h r e f e r e n c e t o F i g u r e 3 , a l t h o u g h t h i s i s n o t p r e c i s e l y t h e m e t h o d u s e d by E f r o y m s o n and R a y . C o n s i d e r a p r o b l e m w i t h n d e s t i n a t i o n s and m p o s s i b l e p l a n t s i t e s w h e r e m s n . L e t m h a v e t h e v a l u e o f 3 and d e s i g n a t e t h e 3 p o s s i b l e s i t e s as X-j X 2 and X 3 . Any s u b s e t o f t h i s s e t may c o n t a i n t h e o p t i m a l s o l u t i o n . S o l v e t h e p r o b l e m f i r s t as an i n t e g e r u n c o n s t r a i n e d l i n e a r p r o g r a m w i t h a m i n i m u m c o s t o f , s a y , e q u a l t o 7 0 . A l l s u b s e q u e n t i n t e g e r r e s t r i c t e d s o l u t i o n s m u s t be a t l e a s t as g r e a t as t h i s . S e t X-j e q u a l t o 1 and s o l v e a g a i n t o o b t a i n a s o l u t i o n o f 7 5 . A t t h i s s t a g e a s s u m e t h a t n e i t h e r X 2 n o r Xg h a s i n t e g e r c o e f f i c i e n t s . B r a n c h n e x t on X 2 = 1 and o b t a i n a b o u n d o f 7 8 . S e t Xg = 1 and s o l v e a g a i n w i t h S = 8 5 . T h i s i s one t e r m i n a l s t a t e on t h e t r e e , an i n t e g e r f e a s i b l e s o l u t i o n . Unconstrained Solution S = 70 X, =0 S = 83 S = 85 F i g u r e 3 P r i n c i p l e s o f t h e A p p l i c a t i o n o f a B r a n c h - B o u n d A l g o r i t h m t o P l a n t L o c a t i o n Now b a c k t r a c k down t h e t r e e and s e t Xg = 0 , k e e p i n g = 1 and X 2 = 1 . A new l o w e r b o u n d w i t h t h e o b j e c t i v e f u n c t i o n S = 83 i s e s t a b l i s h e d . N e x t r e l a x t h e e x p l i c i t i n t e g e r c o n s t r a i n t on X g , c o n t i n u e down t h e t r e e and s e t X 2 = 0 . In s o l v i n g t h i s l i n e a r p r o g r a m w i t h X-j = 1 , X 2 = 0 and Xg u n c o n s t r a i n e d t h e o b t a i n e d s o l u t i o n i s 8 4 . I t i s n o t n e c e s -s a r y t h e r e f o r e t o c o n t i n u e t h e s e a r c h a l o n g t h i s b r a n c h o f i t h e t r e e s i n c e r e s t r a i n i n g Xg t o e i t h e r 1 o r 0' m u s t r e s u l t i n a s o l u t i o n > t o 8 4 , t h e b o u n d o b t a i n e d when Xg was u n -c o n s t r a i n e d . When t h e c o n s t r a i n t s a r e r e l a x e d on b o t h X 2 and X g , a n d X-j i s s e t = 0 , t h e u n c o n s t r a i n e d s o l u t i o n ( i n t e r m s o f X 2 and Xg) a g a i n e x c e e d s t h e l o w e r b o u n d on t h e b e s t p r e v i o u s l y e s t a b l i s h e d i n t e g e r s o l u t i o n f o r w h i c h S = 8 3 , and a n y f u r t h e r e v a l u a t i o n i n t h i s d i r e c t i o n i s t h e r e f o r e q u i t e u n n e c e s s a r y ( s e e a l s o S c o t t , 1 9 6 9 ; W h i t e , 1 9 6 9 ) . 1 The d i s a d v a n t a g e s o f t h e e x a c t s o l u t i o n a p p r o a c h a r e g e n e r a l l y a c k n o w l e d g e d t o be t w o - f o l d . A l t h o u g h t h e l i n e a r a p p r o x i m a t i o n o f t h e c o n c a v e c o s t c u r v e may be a c h i e v e d t o a n y d e s i r e d d e g r e e o f a c c u r a c y , c o m p u t e r s t o r a g e r e q u i r e -m e n t s t e n d t o become e x c e s s i v e as t h e number o f s e g m e n t s i n c r e a s e s . I t i s more d i f f i c u l t t o g e n e r a l i z e upon t h e c o m -p u t a t i o n a l e f f i c i e n c y o f b r a n c h b o u n d a l g o r i t h m s s i n c e i f a g o o d i n t e g e r s o l u t i o n i s q u i c k l y f o u n d t h e a m o u n t o f s e a r c h i n g a l o n g t h e b r a n c h e s o f t h e t r e e c a n be g r e a t l y c u r t a i l e d , and t h e a l g o r i t h m may r a p i d l y c o n v e r g e t o t h e g l o b a l m i n i m u m . As a r u l e h o w e v e r , t h e s e m o d e l s t e n d t o become c o m p u t a t i o n a l l y o v e r - e x t e n d e d f o r t h e l a r g e r s i z e d p r o b l e m s , and t h i s t e n d e n c y i s c o n s i d e r a b l y e n h a n c e d w i t h e a c h a d d i t i o n a l c o n s t r a i n t t h a t i s p l a c e d upon t h e s y s t e m . ( b ) . H e u r i s t i c S o l u t i o n s H e u r i s t i c a l g o r i t h m s g e n e r a t e , by c o n t r a s t , s o l u t i o n s c h a r a c t e r i z e d by a much g r e a t e r d e g r e e o f c o m p u t a t i o n a l e f f i c i e n c y a t t h e e x p e n s e o f known a c c u r a c y i n t h e f i n a l p r o d u c t . The two a p p r o a c h e s c a n be u s e d t o c o m p l e m e n t e a c h o t h e r i f t h e b r a n c h - b o u n d a l g o r i t h m c a n be s t r u c t u r e d i n s u c h a way a s t o t a k e a d v a n t a g e o f t h e h e u r i s t i c s o l u t i o n as an i n i t i a l s t a r t i n g b o u n d . In t h i s m a n n e r t h e number o f a l t e r n a t i v e s e v a l u a t e d by t h e e x a c t s o l u t i o n m e t h o d may be g r e a t l y r e d u c e d i f t h e h e u r i s t i c s o l u t i o n i s n e a r o p t i m a l . I t i s e v e n p o s s i b l e , s i n c e t h e c o n c a v e c o s t f u n c -t i o n s n e e d n o t be a p p r o x i m a t e d i n h e u r i s t i c m o d e l s , t h a t t h i s a p p r o a c h may a c t u a l l y f i n d a l o w e r c o s t s o l u t i o n . Kuhn and H a m b u r g e r ( 1 9 6 3 ) s u g g e s t an a l g o r i t h m i n w h i c h i t i s a s s u m e d t h a t t h e b e s t p s o u r c e s w i l l be c o n -t a i n e d among t h e b e s t p+1 l o c a t i o n s . T h i s m e t h o d l o c a t e s t h e p p l a n t a t t h a t s i t e w h i c h o f f e r s t h e l a r g e s t c o s t s a v i n g s o v e r t h e s e t o f s o u r c e s c o n t a i n e d i n t h e p - 1 s o l u -t i o n . The v a l u e o f p i s i n c r e m e n t e d s e q u e n t i a l l y u n t i l s u c h t i m e as t h e p+1 s o l u t i o n e x c e e d s i n c o s t t h e p s o l u t i o n , a t w h i c h p o i n t t h e a l g o r i t h m t e r m i n a t e s . A t e a c h s t a g e i n t h e c y c l i n g p r o c e d u r e some m o d i f i c a t i o n o f t h e c u r r e n t l o c a t i o n s y s t e m i s a l l o w e d f o r by a b u m p - s h i f t r o u t i n e w h i c h e l i m i n a t e s a l l s u c h p l a n t s i n s o l u t i o n as h a v e become u n e c o n o m i c a l as a r e s u l t o f t h e g e n e r a t i o n o f s u b s e q u e n t p l a n t s . F e l d m a n e t . a l . ( 1 9 6 6 ) c o n s i d e r t h e p r o b l e m f r o m t h e r e v e r s e o r d e r and b e g i n w i t h a l l i n p o t e n t i a l p l a n t s i t e s i n s o l u t i o n . A t e a c h s t e p one p l a n t i s e l i m i n a t e d u n t i l no f u r t h e r c o s t r e d u c t i o n i s a t t a i n e d . T h i s i s t h e p o i n t a t w h i c h s c a l e e c o n o m i e s a c h i e v e d by i n c r e a s i n g p l a n t s i z e as t h e number o f s o u r c e s d i m i n i s h e s i s o f f s e t by g r e a t e r t r a n s p o r t a t i o n c o s t s w h i c h a c c r u e f r o m t h e s m a l l e r number o f p l a n t s now i n s o l u t i o n . An i n i t i a l e s t i m a t e o f p l a n t s i z e , n e c e s s a r y f o r t h e b e g i n n i n g s o l u t i o n , i s d e r i v e d f r o m e a c h p l a n t ' s l o c a l c u s t o m e r s e t d e f i n e d as b e i n g t h o s e d e s t i n a t i o n s t o w h i c h e a c h p l a n t i s c l o s e s t t o i n t e r m s o f t r a n s p o r t a t i o n c o s t s a l o n e . The b a s i c d i s a d v a n t a g e o f t h e s e m e t h o d s i s t h a t on a n y c y c l e i n w h i c h t h e v a l u e o f p c h a n g e s t h e s e t o f c u r r e n t s o u r c e s may t e n d t o be t o o d e p e n d e n t upon t h e v e c t o r o f p l a n t s d e f i n e d i n t h e a n t e c e d e n t s t e p . Two o t h e r f u n d a -m e n t a l a p p r o a c h e s t o t h e p r o b l e m h a v e b e e n a d v o c a t e d i n w h i c h f o r a n y v a l u e p t h e n e t w o r k may be i n d e p e n d e n t l y b r o k e n i n t o p d e s c e n d e n t s u b g r a p h s . T h e r e a r e t h e v e r t e x s u b s t i t u t i o n m e t h o d and t h e p a r t i t i o n a p p r o a c h . In t h e f i r s t o f t h e s e a s e t o f p p o i n t s i s c h o s e n ( o f t e n a r b i t r a r i l y ) a s t h e i n i t i a l s o u r c e c o n f i g u r a t i o n . E a c h node i s now a s s i g n e d t o t h a t member o f t h e s e t t o w h i c h i t i s c l o s e s t and a l o w e r c o s t b o u n d S , t h e o b j e c t i v e f u n c t i o n i s c a l c u l a t e d f o r t h i s s e t o f p , X p . Some v e r t e x My n o t a member o f Xp i s now s e l e c t e d and i n t u r n e a c h v . e X i s r e p l a c e d by v . . I f a l o w e r b o u n d o f S i s f o u n d d u r i n g a n y one o f t h e s e p s u b s t i t u t i o n s t h e n t h i s s e t r e -p l a c e s X p . I f n o t , t h e n some o t h e r v^. i s c h o s e n and t h e p r o c e s s i s r e p e a t e d . The a l g o r i t h m t e r m i n a t e s when a c o m -p l e t e c y c l e h a s b e e n made d u r i n g w h i c h no r e d u c t i o n i n S i s f o u n d . V a r i a t i o n s on t h i s m e t h o d has b e e n p r o p o s e d by T e i t z and B a r t ( 1 9 6 8 ) , and by S c o t t ( 1 9 6 9 ) . N o t e t h a t t h i s m e t h o d i n v o l v e s a f a i r l y e x t e n s i v e s e a r c h p a t t e r n , w h i c h , w h i l e f a l l i n g f a r s h o r t o f a d i r e c t e n u m e r a t i o n o f a l l p c o m b i n a t i o n s on t h e g r a p h , i s l i k e l y t o become c o m p u t a -t i o n a l l y p r o h i b i t i v e f o r t h e l a r g e r s c a l e p r o b l e m w i t h s e v e r a l h u n d r e d n o d e s . F o r n e t w o r k s o f t h i s m a g n i t u d e t h e p a r t i t i o n i n g a p p r o a c h a p p e a r s t o o f f e r t h e b e s t a l t e r n a t i v e f o r e s t i m a t -i n g t h e r e q u i r e d p m e d i a n . S u c h m e t h o d s c a n be g e n e r a l i z e d i n t e r m s o f t h r e e b a s i c s t e p s . i ) A p p l y some c r i t e r i o n t o p a r t i t i o n t h e g r a p h i n t o some i n i t i a l s e t o f X p n o d e s . C a l l t h i s s e t t h e b e g i n n i n g s e t X - . i i ) N e x t e v e r y o t h e r node o r d e s t i n a t i o n i s a s -s i g n e d t o some member o f X - by ( 1 0 ) . E a c h s u b g r a p h , i = p - n o w h a s an a s s o c i a t e d s u b s e t o f n o d e s v r e G . . and f o r e a c h s u c h s u b g r a p h t h e r e e x i s t s some node v ^ . ( n o t n e c e s s a r i l y u n i q u e ) f o r w h i c h (12) e Q_ w(v r v k ± ) = min = S. w h e r e v k i i s d e s i g n a t e d as t h e m e d i a n o f s u b g r a p h G^ and i s t h e v a l u e o f v ^ . , o r t h e a g g r e g a t e w e i g h t e d d i s t a n c e f r o m v ^ . t o a l l o t h e r n o d e s i n G^ w h e r e t h i s v a l u e i s a t a m i n i m u m f o r one s u c h node ( v ^ - ) i n G.j . The v a l u e $13) S = I S. 1=1 o v e r a l l p o s s i b l e p a r t i t i o n i n g s o f G i n t o p s u b g r a p h s i s t h e v a l u e w h o s e o v e r a l l m i n i m u m i s s o u g h t ^ To t e s t , a t a n y s t a g e , w h e t h e r a t l e a s t a l o c a l m i n i m u m o b t a i n s t h e f o l l o w -i n g c o n d i t i o n , (14) w(v r v k . ) > w(v r v k ) j = l , - - - , p - l ; j T i i s e x a m i n e d f o r e a c h G^ and f o r e a c h node v r e G ^ . I f t h i s c o n d i t i o n h o l d s t h e n v r may be s w i t c h e d f r o m G. t o G . , n o t a b l y t o t h a t G . f o r w h i c h w(v v . . ) i s a m i n i -l j j r K J mum i f . i s n o t u n i q u e . The v a l u e o f S so o b t a i n e d w i l l be l e s s t h a n t h e p r e v i o u s v a l u e o f S i f ( 1 4 ) i s a s t r i c t i n e q u a l i t y , and w i l l be l e s s t h a n o r e q u a l t o i t i f ( 1 4 ) i s s a t i s f i e d as an e q u a l i t y . S t e p ( i i ) i s r e i t e r a t e d u n t i l s u c h t i m e a s (15) w(v. v, . ) 1 w(v V, . ) r k i r kj i s met f o r a l l o v e r a l l . A l t e r n a t i v e s o l u t i o n s may be o b t a i n e d i n i n s t a n c e s w h e r e c o n d i t i o n ( 1 4 ) e x i s t s as an e q u a l i t y , and w i l l o c c u r when t h e m e d i a n o f some s u b g r a p h ( s ) i s n o t u n i q u e and c o n d i t i o n ( 1 5 ) i s met f o r a l l t h e c o m b i -n a t i o n s o f t h e p m e d i a n s o f t h e s u b g r a p h s . i i i ) The p r o c e d u r e s o f s t e p i i ) e n s u r e a t l e a s t a l o c a l m i n i m u m w h i c h may o r may n o t be t h e o p t i m a l s o l u t i o n . M o s t p a r t i t i o n i n g m e t h o d s c o n t a i n some means o f t e s t i n g t h i s l o w e r b o u n d by p e r t u r b i n g t h e s o l u t i o n ; t h a t i s s w i t c h -i n g n o d e s a c c o r d i n g t o some r u l e and r e - c a l c u l a t i n g s t e p i i ) : The a c t u a l p e r f o r m a n c e o f a n y p a r t i t i o n i n g a l g o r -i t h m i s c l e a r l y d e p e n d e n t upon t h e l o g i c a l s t r e n g t h o f t h e h e u r i s t i c s a d o p t e d f o r s t e p s i ) and i i i ) . A l t h o u g h r a n d o m s t a r t s w i l l p r o v i d e f a s t s o l u t i o n s t h e y a r e g e n e r a l l y much l e s s d e s i r a b l e t h a n s y s t e m a t i c p a r t i t i o n i n g p r o c e d u r e s . A h e u r i s t i c g i v e n by C o o p e r c o n s i s t s o f b e g i n n i n g w i t h t h e s e t o f a l l p o t e n t i a l s i t e s and e l i m i n a t i n g one member a t e a c h s t e p u n t i l o n l y p n o d e s a r e l e f t ( C o o p e r , 1 9 6 7 ) . The e l i m i n a t i o n c r i t e r i o n ( f o r ' C o o p e r ' s g r a p h i s n o n -w e i g h t e d ) i s t o t a k e t h e c l o s e s t p a i r o f p o i n t s on a n y i t e r a t i o n and t o d r o p t h a t one w h o s e e s t i m a t e o f t h e s i n g l e p m e d i a n o f t h e n e t w o r k i s t h e h i g h e r . A f t e r c o m p l e t i o n o f s t e p i i ) t h e p e r t u r b a t i o n t e s t on t h e l o w e r b o u n d f i r s t f i n d s some G . s u c h t h a t f o r a n y two n o d e s v . a n d v . , 1 "I J d ( v . v . ) i s a maximum o v e r a l l G . . T h i s m e d i a n i s t h e n d r o p p e d t o be r e p l a c e d by b o t h v . and v . . S i n c e X now 1 J P c o n t a i n s p+1 members t h e two c l o s e s t m e d i a n s a r e s e l e c t e d and i n t u r n e a c h one i s d r o p p e d w h i l e s t e p i i ) i s r e p e a t e d . The S i n g e r A l g o r i t h m S i n g e r has d e v e l o p e d an a l t e r n a t i v e a l g o r i t h m o f v e r y c o n s i d e r a b l e m e r i t i n w h i c h he p r e s e n t s an i n g e n i o u s m e t h o d f o r o b t a i n i n g X - b a s e d upon t h e a t t a i n m e n t o f a s e t o f p n o d e s w h i c h a r e t h e maximum w e i g h t e d d i s t a n c e a p a r t . L e t |A| be an n by n ( o r m by m i f t h e number o f p o t e n t i a l s i t e s i s l e s s t h a n n) s y m m e t r i c m a t r i x i n w h i c h e a c h a . , e l e m e n t i s t h e w e i g h t e d d i s t a n c e a p a r t o f some v . and some v . , V 1 * V j . Then s e t = dCv, V j ) b . b j ./ ( b . + b . ) w h i c h i s e q u i v a l e n t t o a . . = w ( v . e) = w ( v . e) w h e r e e i s t h e f u l c r u m b e t w e e n e a c h o f t h e p a i r w i s e w e i g h t e d v e r t i c e s . F o r t h e n o n - w e i g h t e d g r a p h i n w h i c h a l l b^ = 1 , e a c h a^ . v a l u e i s s i m p l y e q u a l t o o n e - h a l f o f t h e s h o r t e s t d i s t a n c e b e t w e e n e a c h p a i r o f v e r t i c e s ( S i n g e r , 1 9 6 8 ) . The a i m b e h i n d t h i s and m o s t o t h e r p a r t i t i o n i n g c r i t e r i a i s t o o b t a i n t h a t s e t o f X - n o d e s w h i c h , c e t e r i s P p a r i b u s , a r e t h e l e a s t l i k e l y i n t h e f i n a l s o l u t i o n t o be i n t h e same s u b g r a p h s e t . S i n c e node w e i g h t s a r e t r a d e d a g a i n s t t h e d i s t a n c e s b e t w e e n n o d e s , t h e S i n g e r c r i t e r i o n p r o v i d e s a l o g i c a l l y s o u n d s p r i n g b o a r d f r o m w h i c h t o b e g i n t h e s e a r c h p r o c e d u r e . In m a t r i x |A| t h e maximum a . , e l e m e n t w i l l d e s i g n a t e t h e two n o d e s and v .^ w h i c h a r e t h e m a x i -mum w e i g h t e d d i s t a n c e a p a r t . F o r p > 2 X - i s t h e r e q u i r e d s e t o f p n o d e s i f , and o n l y i f , t h e m i n i m u m p a i r w i s e a . , v a l u e s among a l l p a i r s o f n o d e s v . , v . ' e X i s a maximum o v e r a l l s u c h m i n i m u m v a l u e s i n e v e r y o t h e r s e t Xp i n G . A c o m p u t a t i o n a l m e t h o d f o r o b t a i n i n g Xp when p>2 ( e v a l u a t i o n o f a l l c o m b i n a t i o n s o f X b e i n g g e n e r a l l y i n -f e a s i b l e ) i s g i v e n by S i n g e r and may be i l l u s t r a t e d f o r t h e c a s e o f p = 3. i ) L e t a . , be t h e maximum v a l u e i n row i i n t h e A m a t r i x . i i ) S c a n down t h e rows o f jA| and c h o o s e some y row r f o r w h i c h maxmin (a . y a . ) i s s a t i s f i e d . T h a t i s r i A r j p i c k t h e row i n w h i c h t h e m i n i m u m o f a r | . o r & r - i s a m a x i -mum o v e r a l l r ( r = n ) r o w s . The a s s o c i a t e d node v^ i s t h e t e n t a t i v e t h i r d n o d e . i i i ) D e s i g n a t e a g as t h e m i n i m u m o f ( a j j)< a r j X a r j ' ' i v ) R e p e a t s t e p s i i ) and i i i ) f o r a l l p a i r s o f n o d e s v^ and v .^ i n row i f o r w h i c h a ^ . > a s . The s e t o f p n o d e s f o r w h i c h t h e m i n i m u m v a l u e o f a i s a maximum i s t h e s c a n d i d a t e s e t f r o m t h i s r o w . v ) R e p e a t s t e p s i ) t o i v ) f o r a l l r ows and c h o o s e t h a t s e t o f 3 n o d e s ( v . v . v ) f o r w h i c h a , as d e f i n e d , i s i J i s a maximum o v e r a l l r o w s . G e n e r a l i z i n g t o p > 3 , f i n d i n e a c h row i t h e p - 2 maximum e l e m e n t s . T h e r e a r e p - 1 i n d i c e s a s s o c i a t e d w i t h t h e s e e l e m e n t s , t h a t i s p - 2 i n d i c e s w h i c h c o r r e s p o n d t o t h e c o l u m n s i n |A| c o n t a i n i n g t h e maximum e l e m e n t s , p l u s t h e i n d e x o f t h e row i t s e l f . I t i s p o s s i b l e t o s p e e d up S i n g e r ' s c o m p u t a t i o n a l p r o c e d u r e by o m i t t i n g s t e p i v ) u n t i l a l l r o w s o f t h e m a t r i x h a v e b e e n s c a n n e d a t l e a s t o n c e . T h i s i n c r e a s e s t h e p r o b a b i l i t y o f f i n d i n g an i n i t i a l l y g o o d v a l u e f o r a g . H o w e v e r i n t h e a u t h o r ' s e x p e r i e n c e e v e n t h i s i m p r o v e m e n t c a n be c o m p u t a t i o n a l l y e x a c t i n g i f n o t i n -f e a s i b l e , s h o u l d t h e a c t u a l r e q u i r e d maximum v a l u e o f a $ h a p p e n t o be r e l a t i v e l y s m a l l , o r s h o u l d a good u p p e r b o u n d f o r i t n o t be e s t a b l i s h e d e a r l y i n t h e s e a r c h . C o n s i d e r t h e e x a m p l e o f a 100 node n e t w o r k w i t h p=6 . F o r a n y row i a f t e r one c o m p l e t e s c a n t h e r e may be many n o d e s , i n t e r m s o f s t e p i v ) , w h i c h e x c e e d t h e c u r r e n t v a l u e o f a g . T h i s i m p l i e s t h e r e f o r e t h a t t h e r e may be many c o m b i n a t i o n s o f p - 1 n o d e s ( t h e f i r s t member o f w h i c h i s a l w a y s t h e row i n d e x ) t h a t r e q u i r e r e i t e r a t i o n o f s t e p s i ) t o i i i ) . I f t h e v a l u e o f p i s l a r g e i n b o t h a b s o l u t e t e r m s and p e r h a p s more s i g n i f i c a n t l y r e l a t i v e t o m, t h e n t h e number o f c o m -b i n a t i o n s t o be e v a l u a t e d c a n e a s i l y become c o m p u t a t i o n a l l y p r o h i b i t i v e . An a l t e r n a t i v e m e t h o d f o r o b t a i n i n g a good e s t i m a t e o f X" i n t e r m s o f t h e S i n g e r c r i t e r i o n and w h i c h a v o i d s t h e a b o v e d i f f i c u l t i e s may now be g i v e n . The p r i n c i p l e i s t o a d d n o d e s one a t a t i m e a p p l y i n g t h e maxmin c r i t e r i o n a t e a c h s t e p . S a t i s f a c t i o n o f t h i s c r i t e r i o n w i l l d e t e r m i n e t h e n e x t v e r t e x t o be a d d e d . N o t e t h a t i n t h e S i n g e r m e t h o d p - 1 e l e m e n t s a r e c h o s e n p r i o r t o t h e a p p l i c a t i o n o f t h e maximum c r i t e r i o n w h i c h i s u t i l i z e d o n l y t o f i n d t h e p*'1 node f o r t h e c a n d i d a t e s e t f r o m t h a t r o w . In a d d i n g e a c h new node by t h i s s u g g e s t e d m e t h o d i t i s n o t n e c e s s a r y i n u p d a t i n g a g t o r e - s c a n t h e c o l u m n s o f |A| b e l o n g i n g t o t h o s e n o d e s a l r e a d y a s s i g n e d t o t h e c a n d i d a t e s e t , i f t h e f o l l o w i n g p r o c e d u r e i s a d o p t e d . F o r e a c h row i ( l e t p = 3 f o r i l l u s t r a t i o n ) . i ) S c a n t h e i c o l u m n i n |A| and s t o r e a l l a . , v a l u e s , j = l , - - - , m , j =f i , i n some among s a y | BC | . 3 P i c k t h e maximum a . , v a l u e = a and s t o r e t h e a s s o c i a t e d J i s node v w h e r e r d e n o t e s t h e row i n d e x i n w h i c h a „ was r s f o u n d . The c a n d i d a t e s e t row h a s two m e m b e r s , v^ and V i i ) B r a n c h n e x t on v^ a l o n e ; t h a t i s s c a n o n l y c o l u m n r i n | A | . C o n s i d e r a g a i n a l l e l e m e n t s a j r > j = l , - - - , m , j f i and j =(  r . T h a t i s i g n o r e t h e row i n d i c e s o f a l l t h e c u r r e n t members o f t h e c a n d i d a t e s e t . Now i f some a . < b e . t h e n r e p l a c e t h e j e l e m e n t i n IBCI w i t h t h a t a . v a l u e . Thus t h e v a l u e s now s t o r e d i n | BC | o f t h o s e n o d e s n o t y e t i n t h e b e g i n n i n g s e t a r e a l w a y s t h e m i n i m u m v a l u e s i n |A| w i t h r e s p e c t t o t h o s e members w h i c h a r e c u r r e n t l y i n X - . F o r a n y node v . t h e n , P J v.j f V j and v .^ =)  v r i t s v a l u e i n |BC| i s a l w a y s m i n ( a j i 5 a j r ) . Now s e a r c h among BC e x c l u d i n g bc^ a n d b c ^ and f i n d some b e . = m a x , s t o r i n g v 0 t h e a s s o c i a t e d n o d e . L e t a „ = b e . max = t h e m i n i m u m o f ( a . , a „ . , a n ^ ) . T h i s i s s j 1 l r £1 i r t h e maximum o f t h e m i n i m u m p a i r w i s e v a l u e s o f a n y o t h e r I w i t h i and r i n | A | . i i i ) I f p > 3 r e p e a t s t e p i i ) u n t i l a l l p v e r t i c e s h a v e b e e n c h o s e n , t h e c a n d i d a t e s e t Xp f o r t h i s r o w . L e t a = a t h e maximum v a l u e o v e r p e l e m e n t s m s ( a g a l w a y s d e c r e a s e s w i t h e a c h a d d i t i o n a l n o d e ) . i v ) R e p e a t s t e p s i t o i i i ) f o r a l l i r o w s . I f t h e l a s t v a l u e f o r a „ f o r a n y row i s > a r e p l a c e s J m r a m w i t h a g and s t o r e t h i s s e t o f n o d e s i n p r e f e r e n c e . I f a s = a m s t o r e b o t h s e t s o f n o d e s i f t h e y d i f f e r , f o r t h e s e a r e a l t e r n a t e s e t s o f X - . P T h i s p a r t i t i o n i n g h e u r i s t i c may be i l l u s t r a t e d w i t h r e f e r e n c e t o t h e n e t w o r k d i s p l a y e d i n F i g u r e 4. In t h i s and a l l s u b s e q u e n t e x a m p l e t h e node r e f e r e n c e s a r e d e s i g n a t e d as v . , i = l , - - - , n , t h e b . v a l u e o f e a c h v e r t e x i s e n c i r c l e d , and t h e d i s t a n c e b e t w e e n e a c h p a i r o f n o d e s i s t h e f i g u r e l y i n g a d j a c e n t t o e a c h l i n k . The d i s t a n c e o f e a c h v e r t e x w i t h i t s e l f i s a s s u m e d t o be z e r o and t h e s h o r t d i s t a n c e and A m a t r i c e s f o r t h i s F i g u r e 4 Table 1 SHORT DISTANCE MATRIX FOR FIGURE 4 0.0 4.0 5.0 3.0 1.0 3.0 2.0 4.0 5.0 6.0 4.0 0.0 3.0 2.0 4.0 5.0 5.0 4.0 5.0 3.0 5.0 3.0 0.0 5.0 4.0 2.0 3.0 1.0 2.0 1.0 3.0 2.0 5.0 0.0 2.0 4.0 3.0 5.0 6.0 5.0 1.0 4.0 4.0 2.0 0.0 2.0 1.0 3.0 4.0 5.0 3.0 5.0 2.0 4.0 2.0 0.0 3.0 1.0 2.0 3.0 2.0 5.0 3.0 3.0 1.0 3.0 0.0 2.0 3.0 4.0 4.0 4.0 1.0 5.0 3.0 1.0 2.0 0.0 1.0 2.0 5.0 5.0 2.0 6.0 4.0 2.0 3.0 1.0 0.0 2.0 6.0 3.0 1.0 5.0 5.0 3.0 4.0 2.0 2.0 0.0 T a b l e 2 THE |AI MATRIX FOR FIGURE 4 0 . 0 3 . 0 2 . 1 4 . 5 1 . 2 5 . 1 3 . 7 3 . 0 7 . 5 7 . 2 3 . 0 0 . 0 1 . 0 1 . 5 2 . 7 4 . 0 4 . 2 2 . 0 3 . 8 2 . 0 2 . 1 1 . 0 0 . 0 2 . 1 1 . 6 0 . 9 1 . 4 0 . 3 0 . 9 0 . 4 4 . 5 1 . 5 2 . 1 0 . 0 2 . 4 6 . 9 5 . 6 3 . 8 9 . 0 6 . 0 1 . 2 2 . 7 1 . 6 2 . 4 0 . 0 2 . 7 1 . 4 2 . 0 4 . 8 5 . 0 5 . 1 4 . 0 0 . 9 6 . 9 2 . 7 0 . 0 6 . 7 0 . 9 3 . 4 4 . 0 3 . 8 4 . 2 1 . 4 5 . 6 1 . 4 6 . 7 0 . 0 1 . 7 5 . 6 5 . 7 3 . 0 2 . 0 0 . 3 3 . 8 2 . 0 0 . 8 1 . 7 0 . 0 0 . 8 1 . 3 7 . 5 3 . 8 0 . 9 9 . 0 4 . 8 3 . 4 5 . 6 0 . 8 0 . 0 2 . 4 7 . 2 2 . 0 0 . 4 6 . 0 5 . 0 4 . 0 5 . 7 1 . 3 2 . 4 0.0 ' F i g u r e 6 F i g u r e 7 PBOBLEB BASE FIGURE 8(A) 2 HEOIAI! SOLUTION ?OS FIGUSE 4 ( U I I LOCATION-ALLOCATION 2 BEDIAB SOLUTION BEDIAB PLAST SIZE FR EIGHT COST PLAST COST TOTAL COST BODES SERVICED TOLO.IE SUPPLIED TO . EACH BEDIAB 1 5 10.00 18.00 0.00 18.00 1 2 4 5 7 3.0 1.0 3.0 2.0 5.0 2 8 10.50 1 1.50 0.00 1 1. 50 3 6 8 9 10 0.5 4.0 1.0 3.0 2.0 lETaoaK 24.50 29.SO 0.00 29.50 " PLAKT LOCATION-ALLOCATION 2 BEDIAS SOLUTIOS H E Dl AK PLAKT SIZE FREIGHT COST PLAST COST TOTAL COST BODES SERVICED VOLOBE StJPPLISD TO EACH BEDIAB 1 S 13.00 14.00 0.00 14.00 1 ( 5 7 3.0 3.0 2.0 5.0 2 8 11.50 15.50 0.00 15.50 2 3 6 8 9 1.0 0.5 4.0 1.0 3.0 10 2.0 BETWORK 24.50 29.50 0.00 29.50 • ll4*t*ittttlM«t4*t>4*.*tt4tttttt(t*»ttilttt**»tt4«*t*tt.*ttttt*t..t*».t*ttdrt.*,*»*»»,ttt*ttt*t«tlttt**(t)*(?t*t« PBOBLE.! BARE FIGURE 8(B) 3 BEDIAS SOLUTION FOR FIGURE 4 • • * < * t * » * * t 4 « « * * * . * « » * * * * * * t * * * * * * 4 * * * * * * * * 4 * * * * » t « * » * * * * * * * * t * * * » * * * » * * » * * « » 4 4 ! * * * PLAKT LOCATION-ALLOCATION 3 BEDIA1C SOLOTIOS BEDIAB PLANT SIZE FREIGHT COST PLANT COST TOTAL COST BODES SERVICED VOLOBE SUPPLIED TO EACH REDIiN 4.00 10.00 10.50 24.50 2.00 8.00 1 1. 50 21.50 0.00 0.00 0.00 0.00 2.00 8.00 11.50 21.50 10 1.0 3.0 O.S 3.0 2.0 4.0 5.0 1.0 3.0 2.0 1 ET'WC RK t » * » 4 4 4 4 44 » 44.4 444 4 * * 4 4 4 4 4 4 4 4 4 4 4 4 4 * 4 4 4 4 * 4 4 > » * 4 * * * * 4 4 4 * 4 * * * 4 * 4 * * * 4 4 4 * 4 4 4 * 4 4 4 * * 4 * 4 4 * 4 4 4 4 4 * 4 * * * * * * 4 * * * « * * * « * * * * * * 4 4 PLABT LOCATIOB'ALLOCATIOB 3 B EDIAN SOL0TIOB BEDIAB PLABT SIZE FREIGHT COST PLABT COST' TOTAL COST BODES SERVICED VOLOBE SUPPLIED TO EACH HEDIAN 4.00 10.00 10.50 24.50 2.00 8.00 11.50 21. 50 0.00 0.00 0.00 0.00 2.00 8.00 11.50 21.50 1.0 3.0 0.5 3.0 2.0 4.0 5.0 1.0 BET80EK • 4 4 4 * * * * 4 * 4 4 * 4 * * * * * * 4 4 * * t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * t * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * PROBLEM BABE FIGURE 8(C) 3 BEDIAB SOLUTIOB FOR FIGURE 5 * 4 * * * * * * * * 4 4 * t * * * * * * * « * 4 * « * 4 4 * * * * * * * * * * 4 * * * * * * * * * * * * * * * * 4 4 4 4 * * 4 * * * * * * * * * 4 4 * * * * * * * * * * » * * * * * 4 « * » * * * * * * * * * * * * * * * * * * * * * * PLABT LOCATIOS-ALLOCAIION 3 BEDIAB SOLUTIOB BEDIAB PLABT SIZE FREIGHT COST PLAST COST TOTAL COST BODES SERVICED VOLOBE SUPPLIED TO EACH BEDIAB 1 3 22.00 10.00 0.00 10.00 1 2 3 1.0 1.0 20.0 2 4 25.00 137.00 0.00 137.00 4 5 6 7 8 20.0 1.0 1.0 1.0 1.0 9 1.0 3 10 2.00 1.00 0.00 1.00 10 11 1.0 1.0 SET tfOBK 49.00 143.00 0.00 148.00 • * 4 4 * * * * * * * * * * * * * * * * * * * * * t * * * * * « * * * « * * * * * * * * * * * * * * * * * * * * * * * « * * * * 4 * * * * * * * * 4 * * * * * * * * * * * * * * * * * * * « « * * * * * * * * * * * * 4 4 * * * * 4 « 4 PLABT LOCATIOB-ALLOCATIOB 3 BEDIAB SOLUTIOB BEDIAB PLAST SIZE FREIGHT COST PLAST COST TOTAL COST BODES SERVICED VOLOBE SOPPLIED TO EACH BEDIAB 1 3 22.00 10.00 0.00 10.00 1 2 3 1.0 1.0 20.0 2 4 25.00 137.00 0.00 137.00 4 5 6 7 8 20.0 1.0 1.0 1.0 1.0 9 1.0 3 11 2.00 1.00 0.00 1.00 10 11 1.0 1.0 •ETBOBK 49.00 148.00 0.00 148.00 • • • • • • * * * * * * * « * * * * * * * * « * * « * * * * * * « * * * * « * « « * * « * « * « * « « * * « » * * 4 * * « 4 4 « * 4 * * « * * * « * * * * * * 4 4 * * * « * 4 4 4 * 4 * * « * * « * * * * a « « * * * « * * * « * * * * PLAST LOCATION-ALLOCATION 3 BEDIAB SOLUTIOS BEDIAB PLAST SIZE FREIGHT COST PLAST COST TOTAL COST BODES SERVICEO VOLOBE SOPPLIED TO EACH BEDIAB 1 3 22.00 10.00 0.00 10.00 1 2 3 1.0 1.0 20.0 2 4 23.00 80.00 0.00 80.00 4 5 10 11 20.0 1.0 1.0 1.0 3 6 4.00 16.00 0.00 16.00 6 7 8 9 1.0 1.0 1.0 1.0 BETVOSK 49.00 106.00 0.00 106.00 • « 4 * 4 * 4 4 4 * * 4 4 4 4 * 4 4 * * 4 * * * * « * * t * 4 4 4 4 * 4 « 4 4 * * 4 « * . « 4 * * * * * 4 4 4 * « 4 4 4 4 4 * * * * * * * * 4 4 4 * 4 4 * * * * * « * » * * 4 * * * « 4 4 * 4 * * * * * t * « t 4 * * * * * « « * * * * F i g u r e 8 g r a p h a r e g i v e n i n T a b l e s 1 and 2 r e s p e c t i v e l y . F o r t h e two m e d i a n s o l u t i o n o f t h i s n e t w o r k t h e members o f t h e s e t Xp- f o r w h i c h a ^ . i s a maximum a r e v^ and Vg w i t h a v a l u e o f 9 . F o r t h e t h r e e m e d i a n s o f t h i s g r a p h t h e i n i t i a l p a r t i t i o n i n g i s b a s e d upon e a c h o f t h e t h r e e s e t s o f X - ( v 4 v ? v g ) , ( v 4 v g v ? ) ; o r ( v 4 v ? v 1 Q ) . The m i n i m u m p a i r w i s e d i s t a n c e b e t w e e n a n y two n o d e s i n t h e s e t h r e e s e t s i s a maximum o v e r a l l s u c h s e t s i n t h e n e t w o r k w i t h a v a l u e o f 5 - 6 2 5 . The l o w e r b o u n d p e r t u r b a t i o n t e s t s u g g e s t e d by S i n g e r i s t o a r r a y f i r s t t h e s u b g r a p h s i n d e c r e a s i n g o r d e r o f t h e v a l u e s S-. Then f o r e a c h . s u b g r a p h , t h a t node v.. e G^ w h o s e w e i g h t e d d i s t a n c e t o t h e m e d i a n o f some o t h e r s u b g r a p h G . i s l e a s t , i . e . f o r w h i c h w ( v . v . . ) J 1 K j i s a m i n i m u m , i s s w i t c h e d t o s u b g r a p h G . and t h e r e -3 s u l t a n t s u b g r a p h s a r e m o d i f i e d u n t i l c o n d i t i o n ( 1 5 ) i s s a t i s f i e d . T h i s p r o c e d u r e i s r e p e a t e d p t i m e s m o v i n g one node on e a c h o c c a s i o n . I f a l o w e r b o u n d i s f o u n d on a n y i t e r a t i o n t h e n t h e w h o l e t e s t i n g p r o c e s s i s a g a i n r e - c y c l e d . The s o l u t i o n s t o t h e two and t h r e e m e d i a n s o f F i g u r e 4 w e r e a c t u a l l y o b t a i n e d w i t h o u t r e c o u r s e t o t h e p e r t u r b a t i o n t e s t a n d a r e shown i n F i g u r e s 8a and 8 b . Some Comments on T h e s e A l g o r i t h m s One c o n s i d e r a b l e a d v a n t a g e o f S i n g e r ' s a l g o r i t h m i s t h a t f o r e a c h v a l u e o f p r e q u i r e d , t h e g r a p h i s r e p a r t i t i o n e d . T h i s means t h a t t h e X - + 1 o r X - - 1 P P b e g i n n i n g s e t i s n o t n e c e s s a r i l y d e p e n d e n t upon t h e s e t o f X - p r e s e n t i n t h e pzfn o r a n t e c e d e n t s t e p . I t i s n o t t o o d i f f i c u l t i n p r a c t i c e h o w e v e r , t o d e s i g n n e t w o r k s i n w h i c h n e i t h e r t h e S i n g e r n o r C o o p e r c r i t e r i o n p r o v i d e s a g o o d i n i t i a l s t a r t i n g b a s i s . C o n s i d e r t h e n e t w o r k shown i n F i g u r e 5 f o r w h i c h t h e t h r e e m e d i a n s o l u t i o n i s r e q u i r e d . B o t h o f t h e s e m e t h o d s ( i f w e i g h t e d r a t h e r t h a n s i m p l e d i s t a n c e s a r e c o n s i d e r e d f o r C o o p e r ' s c r i t e r i o n ) g i v e a s e t o f Xp c o n t a i n i n g n o d e s v ^ , and v - ^ , w h i c h c o n v e r g e t o a l o c a l m i n i m u m o f S = 1 4 8 - 0 . As i t h a p p e n s C o o p e r ' s p e r t u r b a t i o n t e s t f i n d s t h e g l o b a l o p t i m u m ( s e e F i g u r e 8 c ) w h e r e a s S i n g e r ' s t e s t d o e s n o t . In t h i s r e s p e c t C o o p e r ' s m e t h o d seems t o be c o n s i s -t e n t l y t h e b e t t e r o f t h e two a p p r o a c h e s . S i n g e r ' s p e r t u r b a t i o n r o u t i n e s u f f e r s f r o m t h e f u r t h e r d i s -a d v a n t a g e t h a t a t l e a s t p r e - a s s i g n m e n t s and m o d i f i -c a t i o n s m u s t be made on e a c h t e s t i n g c y c l e , so t h a t f o r b i g n e t w o r k s w h e r e p i s a l s o l a r g e , t h i s t a s k c o u l d c o n c e i v a b l y become o v e r - e x t e n d e d . The b a s i c p r o b l e m w i t h b o t h t h e p a r t i t i o n i n g and p e r t u r b a t i o n h e u r i s t i c s o f t h e s e and o t h e r a l g o r i t h m s i s t h a t n o d e s a r e e v a l u a t e d i n d i v i d u a l l y a n d w i t h o u t s u f f i c i e n t c o n s i d e r a t i o n o f t h e w e i g h t s and d i s t a n c e s away o f t h e i r i m m e d i a t e n e i g h b o u r s . As a c o n s e q u e n c e b i a s e s c a n o c c u r t o w a r d s t h e i n c l u s i o n o f s o l i t a r y n o d e s w h i c h l i e f a r r e m o v e d a t t h e p e r i p h e r y o f t h e n e t w o r k . T h i s c i r c u m s t a n c e i s s o m e t i m e s p r e v a l e n t w i t h n e t w o r k s o f a t r e e o r d e n d r i t i c d e s i g n . Some c l u s t e r i n g p r o c e d u r e w h i c h c o n s i d e r s t h e r e l a t i o n s h i p o f e a c h node t o i t s s u r -r o u n d i n g n e i g h b o u r s i s c l e a r l y d e s i r a b l e i f t h i s e f f e c t i s t o be d i m i n i s h e d . In d e r i v i n g - X - i t i s a l w a y s p o s s i b l e t o a r t i f i c i a l l y i n c r e a s e t h e a t t r a c t i o n o f some p a r t i c u l a r s i t e by a d d i n g t o i t s w e i g h t t h e demands o f a l l s u c h s u r -r o u n d i n g n o d e s , n o t t h e m s e l v e s c o n t e n d e r s f o r a p l a n t l o c a -t i o n , w h i c h a r e c l o s e r t o i t t h a n t o some o t h e r p o t e n t i a l s i t e . T h i s p r o c e d u r e c a n be a d o p t e d w h e r e v e r t h e number o f n e t w o r k n o d e s i s g r e a t e r t h a n t h e number o f f e a s i b l e s i t e s , b u t w h e r e t h i s i s n o t so some o t h e r p r o c e d u r e i s r e q u i r e d . Some A l t e r n a t i v e P a r t i t i o n i n g and P e r t u r b a t i o n M e t h o d s The p r o b l e m o f node a g g l o m e r a t i o n c a n be a p p r o a c h e d by a d o p t i n g t h e f o l l o w i n g p r o c e d u r e w h i c h w i l l be known h e r e a f t e r as t h e w e i g h t s h i f t e l i m i n a t i o n m e t h o d . The d e t a i l s f o r d e r i v i n g X - w i t h t h i s h e u r i s t i c may be g i v e n as i ) F o r e a c h node on t h e n e t w o r k f i n d some o t h e r n o d e v - f o r w h i c h d ( v . v . ) i s a m i n i m u m . S t o r e a l l v . i n some a r r a y s a y | A B | , a l l v . i n |AC| and a l l d ( v . v . ) i n a r r a y | A D | . L e t some v e c t o r |B| c o n t a i n t h e l i s t o f node w e i g h t s f o r e a c h p o t e n t i a l s i t e tK , i = 1 , ••• , m . i i ) S c a n a r r a y |A D j and f i n d some e l e m e n t , t h e r^ *1 e l e m e n t , f o r w h i c h d ( v . v . ) b . = w ( v . v . ) i s a m i n i -mum. D e s i g n a t e t h e a s s o c i a t e d node i n |AB|as v^ and t h a t i n AC as v , . s i i i ) E l i m i n a t e node v^ f r o m | A B | . R e w r i t e a r r a y s |AB|, |AC| a n d |AD| o m i t t i n g t h e r t h e l e m e n t i n e a c h and r e d u c e n t o n - 1 w h i c h i s t h e number o f n o d e s now e x t a n t i n X - . i v ) L e t b be t h e w e i g h t o f node v and b „ t h e r 3 r s w e i g h t o f node v . Then c h a n g e b so t h a t b = b + b . • s - s s s r In t h i s s t e p a d v a n t a g e i s t h u s t a k e n o f t h e w e i g h t o f t h e node v w h i c h i s d r o p p e d on e a c h i t e r a t i o n . S p e c i f i c a l l y i t s w e i g h t i s a d d e d t o t h e w e i g h t o f t h a t node v g t o w h i c h i t was c l o s e s t b e f o r e e l i m i n a t i o n . E a c h t i m e d ( v . v . ) b i i s c a l c u l a t e d a s i n s t e p i i ) t h e v a l u e o f b^ i n t h e c o m p u -t a t i o n may t h e r e f o r e be g r e a t e r t h a n t h e a c t u a l w e i g h t o f node v.j on t h e n e t w o r k . As n o d e s a r e p r o g r e s s i v e l y e l i m i -n a t e d t h e w e i g h t t r a n s f e r s a r e . c u m u l a t i v e , t h e w e i g h t o f e a c h d r o p p e d node c o n s i s t i n g o f t h e sum o f a l l w e i g h t s p r e -v i o u s l y a s s i g n e d t o i t i n t u r n by n o d e s w h i c h w e r e d r o p p e d on an e a r l i e r i t e r a t i o n . The n e t r e s u l t o f t h i s c u m u l a t i v e w e i g h t s h i f t i n g i s t o p r o v i d e i n e f f e c t a m e a s u r e o f node a g g l o m e r a t i o n . v ) In t h e f i n a l s t e p a r r a y |AC| i s s e a r c h e d and f o r a l l n o d e s v . i n 1AC1 f o r w h i c h v . = v c a l c u l a t e a new J J r v a l u e d ( v . v . ) v . f v . R e p l a c e t h e a p p r o p r i a t e e l e m e n t s ' 3 3 ^ i n |AC| and |AD| w i t h t h e new v a l u e s o f v . a n d d ( v . v . ) 3 J r e s p e c t i v e l y . In o t h e r w o r d s a new n e a r e s t n e i g h b o u r v . i s f o u n d f o r a l l n o d e s v . whose c l o s e s t node was v b e f o r e i r i t s e l i m i n a t i o n . S t e p s i i ) t o v ) a r e r e p e a t e d u n t i l X - c o n t a i n s p n o d e s . T h i s i s t h e c a n d i d a t e s e t o f b e g i n n i n g n o d e s b a s e d upon t h i s h e u r i s t i c . I f t h e t h r e e m e d i a n p r o b l e m o f F i g u r e 5 i s r e c o n s i d e r e d t h e s e t o f X - d e r i v e d by t h i s p r o c e d u r e i s v 3 , v^ and v g w h i c h i s c l e a r l y a b e t t e r i n i t i a l f e a s i b l e s o l u t i o n t h a n t h o s e g i v e n by t h e o t h e r m e t h o d s . F o r p e r t u r b i n g t h e l o w e r b o u n d s o l u t i o n t h e f o l l o w -i n g h e u r i s t i c i s s u g g e s t e d . D rop i n s e q u e n c e f r o m one t o t h r e e o f t h e c u r r e n t m e d i a n s and r e p l a c e them i n t u r n w i t h some o t h e r p o t e n t i a l p l a n t s i t e n o t p r e s e n t l y i n t h e l o w e r b o u n d s o l u t i o n . From among t h e s e n o d e s v.. , i = l , - - - , m - p t h e r e e x i s t s one s u c h member v f o r w h i c h p (16) max 7 y. . w(v.. v ) - w(v. v, . ) . L . H e G. j v j kx w h e r e ' v . i s a member o f G . s u c h t h a t w ( v - v „ ) < w ( v - V i , - ) -j i j r j K l T h i s e n s u r e s t h e c h o i c e o f t h a t node f o r w h i c h t h e sum o f t h e d i f f e r e n c e s o f t h e w e i g h t e d d i s t a n c e s o f a l l n o d e s w h i c h a r e n e a r e r t o i t t h a n t o t h e i r a s s i g n e d m e d i a n , i s a m a x i m u m . T h i s node i s t h e n s u b s t i t u t e d i n t u r n f o r t h e f o l -l o w i n g t h r e e m e d i a n s , a s s u m i n g a d i f f e r e n t m e d i a n i n e a c h c a s e . F i r s t d r o p t h e m e d i a n o f t h a t G^ f o r w h i c h i s a m i n i m u m b e f o r e ( T 6 ) i s e x e c u t e d . The s e c o n d c a n d i d a t e m e d i a n f o r e l i m i n a t i o n b e l o n g s t o t h a t s u b g r a p h whose S. v a l u e i s a t a m i n i m u m a f t e r c o m p l e t i o n o f ( 1 6 ) . F i n a l l y t h e t h i r d c a n d i d a t e m e d i a n i s t h e one f o r w h i c h wCv^.. v^j) i s l e a s t . T h i s i s t h e m e d i a n whose w e i g h t e d d i s t a n c e t o some o t h e r m e d i a n i s a m i n i m u m o v e r t h e n e t w o r k . An A l g o r i t h m f o r E s t i m a t i n g t h e M e d i a n s o f a W e i g h t e d G r a p h T h i s a l g o r i t h m u t i l i z e s f r o m one t o f o u r m e t h o d s ( a t t h e u s e r ' s d i s c r e t i o n ) f o r o b t a i n i n g t h e i n i t i a l p a r -t i t i o n i n g o f t h e g r a p h . The b e s t d e s c e n d a n t s o l u t i o n f r o m a n y o n e o f t h e s e m e t h o d s i s r e t a i n e d a s t h e e s t i m a t e d m i n i -mum b o u n d . The f i r s t p a r t i t i o n i n g h e u r i s t i c makes u s e o f t h e S i n g e r c r i t e r i o n as p r e v i o u s l y d e s c r i b e d . I t i s w o r t h n o t i n g t h a t s i n c e a s e t o f X - i s d e r i v e d f o r e a c h r o w , f o r l a r g e s c a l e p r o b l e m s w i t h many h u n d r e d s o f n o d e s c o m -p u t a t i o n c a n become e x a c t i n g . F o r s u c h c a s e s an a l t e r n a t i v e p r o c e d u r e i s t o o b t a i n j u s t one c a n d i d a t e s e t w h i c h i s r e s o l v e d f r o m b r a n c h i n g on t h a t p a i r o f n o d e s f o r w h i c h a ^ . i s a maximum o v e r a l l s u c h e l e m e n t s i n | A | . ]A s e c o n d p a r t i -t i o n i n g h e u r i s t i c c o n s i s t s o f a d d i n g n o d e s one a t a t i m e a c c o r d i n g t o t h e m a x m i n • p r i n c i p i e s u c h t h a t v t h e n e x t node a d d e d , i s t h a t one f o r w h i c h t h e m i n i m u m o f w ( v r X p ) i s a maximum o v e r a n y s u c h v.. , w h e r e Xp c o n t a i n s n o d e s p r e v i o u s l y a s s i g n e d t o t h e c a n d i d a t e s e t and V . f X p . T h i s i s r e a l l y a s o r t o f s e c o n d - h a n d v e r s i o n o f t h e S i n g e r c r i t e r i o n , i t s m a i n b e n e f i t b e i n g t o p r o v i d e a q u i c k a l t e r -n a t i v e s e t o f Xp f o r t h e l a r g e r s c a l e p r o b l e m . The t h i r d p a r t i t i o n i n g h e u r i s t i c i s t h e w e i g h t s h i f t e l i m i n a t i o n m e t h o d , and t h e f o u r t h i s s i m i l a r t o t h e t h i r d b u t w i t h s t e p i v ) o m i t t e d , and i s t h e r e f o r e t a n t a m o u n t t o a w e i g h t e d v e r s i o n o f C o o p e r ' s e l i m i n a t i o n p r o c e d u r e . Any one o r a l l o f t h e s e p a r t i t i o n i n g h e u r i s t i c s may be u s e d d e p e n d i n g on t h e s i z e o f t h e n e t w o r k and t h e a m o u n t o f c o m p u t i n g t i m e a v a i l a b l e . The w e i g h t s h i f t e l i m -i n a t i o n m e t h o d g i v e s , i n t h e a u t h o r ' s e x p e r i e n c e , t h e b e s t i n i t i a l s t a r t i n g s o l u t i o n . The s h o r t v e r s i o n o f t h e S i n g e r c r i t e r i o n c a n be r e c o m m e n d e d f o r t h e l a r g e n e t w o r k p r o b l e m , s i n c e t h e d e r i v a t i o n o f Xp b a s e d on t h i s h e u r i s t i c i s c o m -p u t a t i o n a l l y f a s t e r t h a n f o r a n y o f t h e o t h e r a l t e r n a t i v e m e t h o d s . W i t h t h e p r e s e n t a t i o n o f p a r t i t i o n i n g and p e r t u r b a -t i o n m e t h o d s now c o m p l e t e a f o r m a l d e s c r i p t i o n o f t h e b a s i c a l g o r i t h m may now be g i v e n . S t e p 1 ) . I n p u t t h e d a t a c o n s i s t i n g o f a s h o r t p a t h c o s t m a t r i x and a v e c t o r o f node d e m a n d s , t h e v a l u e s . A p p e n d i x 3 g i v e s a F o r t r a n IV c o m p u t e r p r o g r a m ( t h e D a n t z i g a l g o r i t h m ) f o r c a l c u l a t i n g a l l t h e s h o r t e s t p a t h s t h r o u g h a n e t w o r k . The m i n i m u m l o w e r b o u n d v a l u e S i s i n i t i a l i z e d w i t h an a r b i t r a r i l y h i g h n u m b e r , and some i n d e x , s a y I , i s s e t e q u a l t o z e r o . S t e p 2 ) . P a r t i t i o n t h e g r a p h a c c o r d i n g t o a n y o r a l l o f t h e h e u r i s t i c s j u s t g i v e n . I f , f o r m e t h o d s one and t w o , more t h a n one i n i t i a l s e t o f i s f o u n d , o n l y t h e f i r s t two a r e r e t a i n e d . The u s e r may w i s h t o s u p p l y , i n a d d i t i o n , h i s own s e t o f b e g i n n i n g w e i g h t s a n d / o r an i n i t i a l g u e s s a t t h e o p t i m a l s o l u t i o n . L e t J = t h e t o t a l number o f X-s e t s . S t e p 3) . I n c r e m e n t I by o n e . C o n s i d e r t h e I**1 s e t of -Xp n o d e s and l e t t h e s e be d e s i g n a t e d as t h e i n i t i a l p m e d i a n s o f t h e n e t w o r k . S t e p 4 ) . A s s i g n n o d e s t o t h e c u r r e n t s e t o f m e d i a n s by e q u a t i o n ( 1 0 ) , f i n d a new s e t o f m e d i a n s as i n ( 1 2 ) and an o b j e c t i v e f u n c t i o n <p a s i n ( 1 3 ) . R e p e a t t h i s s t e p f o r a l l c o m b i n a t i o n o f m e d i a n s i n t h e c u r r e n t l i s t , a n d f o r a l i m i t e d number o f c o m b i n a t i o n s o f s h a r e d n o d e s i f a n y , ( i . e . n o d e s e q u i d i s t a n t t o m e d i a n s ) . S t o r e t h e c u r r e n t b e s t s o l u t i o n . S t e p 5) . C o m p a r e cj> and S . I f cf> < S , l e t S = <j> and go t o s t e p 4 . I f cf) = S p r i n t t h e r e s u l t s f o r a l l c o m -b i n a t i o n s i f a n y , and go t o s t e p 6 . S t e p 6 ) . D r o p one o f t h e s p e c i f i e d c u r r e n t b e s t m e d i a n s and r e p l a c e i t w i t h a s u b s t i t u t e as p r e v i o u s l y d e -f i n e d . Go t o s t e p 4. R e p e a t t h i s s t e p d r o p p i n g t h e o t h e r two d e s i g n a t e d m e d i a n s i f d i f f e r e n t f r o m t h e f i r s t and f r o m e a c h o t h e r . Go t o s t e p 7 a f t e r e v a l u a t i o n o f a l l s u b s t i t u t e s . S t e p 7 ) . I f I = J t h e a l g o r i t h m h a s c o m p l e t e d i t s s e a r c h f o r c o n d i t i o n ( 1 1 ) and t e r m i n a t e s . O t h e r w i s e p r o -c e e d t o s t e p 3 . R e s t r i c t i o n s on P o t e n t i a l P l a n t S i t e s In some s p e c i f i c a t i o n s o f t h e p l a n t l o c a t i o n p r o b -l e m n o t e v e r y node n e e d be c o n s i d e r e d as a p o t e n t i a l p l a n t s i t e . Many l o c a t i o n s b e c a u s e o f t h e i r r e m o t e n e s s o r l a c k o f a m e n i t i e s c a n be s a f e l y e x c l u d e d f r o m c o n s i d e r a t i o n . The r e q u i r e d s e t o f p n o d e s i s t h e r e f o r e t o be c h o s e n f r o m among some s u b s e t m o f a l l n n o d e s . C o m p u t a t i o n a l l y t h i s r e s t r i c t i o n i s d e s i r a b l e s i n c e i t has t h e a d v a n t a g e o f b o t h a l l o w i n g l a r g e r n e t w o r k s t o be h a n d l e d and o f c o n s i d e r a b l y s p e e d i n g e x e c u t i o n t i m e s p a r t i c u l a r l y w i t h r e s p e c t t o s t e p 2 i n t h e m a i n a l g o r i t h m . In o t h e r v a r i a t i o n s o f t h e p r o b l e m some e x p a n d i n g o p e r a t i o n may be f a c e d w i t h t h e t a s k o f a d d i n g new f a c i l -i t i e s t o an e x i s t i n g d i s t r i b u t i o n s y s t e m w h e r e k p l a n t s a r e a l r e a d y i n o p e r a t i o n . C o n c e p t u a l l y p - k new p l a n t s . a r e n e e d e d f r o m among t h e m p o t e n t i a l s i t e s w h e r e m < n . 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I f k > 0 t h e n k n o d e s a r e f o r c e d i n t o t h e s o l u t i o n , and t h e s e a r c h f o r t h e r e m a i n i n g p - k n o d e s i s c o n d u c t e d among t h a t s u b s e t m - k , w h e r e t h e k d e s i g n a t e d n o d e s a r e some o f t h o s e o r i g i n a l l y s p e c i f i e d i n s u b s e t m. T h i s r e s t r i c t i o n w i l l e n s u r e t h a t e a c h node l a b e l l e d k ( w h e r e k > 0 ) w i l l be a s s i g n e d t o a s e p a r a t e s u b g r a p h , and t h a t t h e r e m a i n i n g p - k s u b g r a p h s w i l l c o n t a i n a t l e a s t one member o f s u b s e t m - k . The n n o d e s o r m a r k e t s a r e t h e n a s s i g n e d by ( 1 0 ) , a n d t h e c h o i c e o f m e d i a n f o r e a c h G. i s c o n s t r a i n e d t o n o d e s i n t h o s e s u b g r a p h s w h e r e s u c h a k l a b e l l e d node i s p r e s e n t , and f r o m among s u c h m-k members as a r e p r e s e n t i n e a c h s u b -g r a p h w h e r e no node e x i s t s . T h e s e c o n c e p t s may be i l l u s t r a t e d w i t h r e f e r e n c e t o F i g u r e 4 . L e t m, t h e s u b s e t o f p o t e n t i a l p l a n t s i t e s , c o n -t a i n members v^ v^ Vg a n d w i t h t h e a d d e d r e s t r i c -t i o n t h a t = v ^ , t h e d e s i g n a t e d f o r c e d n o d e . T h e n , f o r t h e two m e d i a n o f t h i s g r a p h c o n t a i n s a n d Vg by t h e f i r s t h e u r i s t i c and and Vy by t h e t h i r d , and t h e f i n a l s o l u t i o n i s shown i n F i g u r e 9 . P l a n t S i z e U p p e r B o u n d s Z a n g w i l l ( 1 9 6 8 ) has shown t h a t u n d e r c o n d i t i o n s o f u n l i m i t e d p l a n t s i z e e a c h d e s t i n a t i o n w i l l , i n t h e o p t i m a l s o l u t i o n , be s u p p l i e d by one and o n l y one s o u r c e . In p r a c -t i c e h o w e v e r , i t may h a p p e n t h a t s c a l e d i s e c o n o m i e s i n p l a n t p r o d u c t i o n c o s t s o r r e s t r i c t i o n s on t h e a v a i l a b i l i t y o f l a b o u r o r o t h e r i n p u t s f u n d a m e n t a l t o t h e p r o d u c t i o n p r o c e s s n e c e s s i t a t e t h e i m p o s i t i o n o f u p p e r b o u n d s on t h e p o t e n t i a l c a p a c i t y o f some p r o s p e c t i v e l o c a t i o n s i t e . S u c h p l a n t s i t e c o n s t r a i n t s c a n i m p l y t h a t i n t h e g l o b a l m i n i m u m one o r more m a r k e t s may be s u p p l i e d f r o m more t h a n one s o u r c e . The c a p a c i t y r e s t r i c t e d p l a n t l o c a t i o n p r o b l e m h a s r e c e i v e d l i t t l e p r e v i o u s a t t e n t i o n . E f r o y m s o n and Ray i g n o r e t h i s c o n s t r a i n e d s i t u a t i o n w h i l e R u t e n b e r g ' s a p p r o a c h i s v a l i d o n l y f o r a p p r o x i m a t i n g p l a n t s i z e u p p e r b o u n d s . Among t h e n u m e r i c a l s o l u t i o n s o n l y C o o p e r ( 1 9 6 1 ) h a s c o n -s i d e r e d t h i s r e s t r i c t i o n and h i s h e u r s i t i c s w i l l be e x a m i n e d h e r e i n some d e t a i l . C o o p e r ' s a l g o r i t h m c o v e r s some f o u r b a s i c s t e p s as f o l l o w s : i ) The problem i s s o l v e d f i r s t f o r the u n r e -s t r i c t e d c a s e . A s s o c i a t e d w i t h each o f the r e s u l t i n g medians i s a subset o f demand'which may exceed the median p l a n t s i z e upper bound ( c a p a c i t y d e f i c i t ) o r may be l e s s ( c a p a c i t y s u r p l u s ) . i i ) N e x t , i n Cooper's words " l e t I - be any s e t w i t h a c a p a c i t y d e f i c i t and l e t 1+ be any s e t w i t h a capa-c i t y s u r p l u s . Choose a d e s t i n a t i o n p o i n t i n the s e t I -such t h a t the d i f f e r e n c e i n the d i s t a n c e s from v t o the r s o u r c e o f I - and from v t o the source o f 1+ i s a minimum." r i i i ) As much as p o s s i b l e of the demand of v'r i s now r e - a l l o c a t e d from I - t o 1+ and a new s e t o f medians i s computed f o r each subgraph. i v ) Each subgraph i s next examined f o r any member of the s e t which may be c l o s e r t o some o t h e r median w i t h a c a p a c i t y s u r p l u s , and i f t h i s c o n d i t i o n h o l d s , f u r t h e r r e - a l l o c a t i o n o c c u r s . Steps i ) and i v ) are r e p e a t e d u n t i l a l l d e f i c i t s have been e l i m i n a t e d . N o t w i t h s t a n d i n g t h e g o o d r e s u l t s s e e m i n g l y o b t a i n e d by C o o p e r w i t h t h i s m e t h o d two d i s a d v a n t a g e s o f t h e f o r m u -l a t i o n a r e r e a d i l y a p p a r e n t . The f i r s t o f t h e s e i s s i m p l y t h a t t h e r e seems t o be no g u a r a n t e e t h a t t h e m e d i a n s o f t h e u n c o n s t r a i n e d s o l u t i o n c a n t o g e t h e r , s a t i s f y t h e t o t a l n e t w o r k d e m a n d . N e i t h e r i s t h e r e any c e r t a i n t y t h a t e a c h s u b g r a p h c o n t a i n s a t l e a s t one p o t e n t i a l m e d i a n w h i c h w i l l e n s u r e t h i s o c c u r r e n c e . I f t h e u p p e r b o u n d c o n s t r a i n t s a r e s u f f i c i e n t l y r e s t r i c t i v e i t i s n o t i m p o s s i b l e f o r t h e o n l y p n o d e s c a p a b l e o f s a t i s f y i n g t o t a l node r e q u i r e m e n t s , t o be c l u s t e r e d t o g e t h e r w i t h i n t h e same s u b g r a p h upon t e r m i n a t i o n o f t h e u n c o n s t r a i n e d s o l u t i o n . The s e c o n d d i s a d v a n t a g e i s a d i r e c t c o n s e q u e n c e o f t h e f a c t t h a t s t e p i v ) i n C o o p e r ' s a l g o r i t h m o n l y r e -a l l o c a t e s node demands t o m e d i a n s w i t h e x c e s s c a p a c i t y . U n d e r t h e s e c i r c u m s t a n c e s t h e a r b i t r a r y o r d e r i n w h i c h t h e I- and 1+ s o u r c e s a r e a p p a r e n t l y c h o s e n c a n be o f c o n s i d e r a b l e s i g n i f i c a n c e . C o n s i d e r t h e n e t w o r k g i v e n i n F i g u r e 6 f o r w h i c h a 4 m e d i a n s o l u t i o n i s r e q u i r e d . L e t t h e p l a n t s i z e u p p e r b o u n d f o r e a c h o f t h e 10 p l a n t s be i n s e q u e n c e 8 , 1 3 , 1 0 , 1 2 , 1 1 , 1 1 , 7 , 1 5 , 8 and 9 u n i t s . T a b l e 3 P O S S I B L E SOLUTIONS TO THE 4 MEDIAN PROBLEM OF F IGURE 6 BY C O O P E R ' S H E U R I S T I C S a) Unconstrained Solut ion M e d i an A s s o c i a t e d Node Node V o l u m e s P l a n t S i z e U p p e r B o u n d I- 1 + C o s t v 2 V ! , V 2 , V 3 1 ,10,4 15 1 3 -2 21 10 10 1 2 + 2 0 v 6 v 5 , v 6 , v 7 2,10,1 13 11 -2 7 v 8 ! V 8 , V 9 , V i o 10,1 ,1 1 2 14 + 2 2 TOTAL 50 50 -4 + 4 30 Plant Size Constrained Solut ion with "V\ and V 6 .the f i r s t 1+ and 1 - 1 sources r e s p e c t i v e l y . as v 2 V i , V 2 , V 3 1 ,10,2 13 , 13 < 11 V - , V 5 10,2 12 12 10 v 6 c v 6 , v 7 10,1 11 11 1 V 8 v 3 , v 8 , v 9 , v 1 0 2,10,1,1 14 14 40 TOTAL 50 50 62 In t h e u n c o n s t r a i n e d s o l u t i o n ( T a b l e 3) m e d i a n s v 2 and v g h a v e c a p a c i t y d e f i c i t s and m e d i a n s v^ and Vg c a p a c i t y s u r p l u s e s . I f t h e I- s o u r c e c h o s e n i s Vg and t h e 1+ s o u r c e i s v^ t h e n , by t h e d e f i n i t i o n o f s t e p i i ) , node Vg i s s w i t c h e d f r o m Vg t o v^ so t h a t t h e c a p a c i t y c o n s t r a i n t s a r e e x p l i c i t l y s a t i s f i e d a t b o t h o f t h e s e p l a n t s . M e d i a n v^ s t i l l e x c e e d s i t s u p p e r b o u n d so t h a t on t h e n e x t i t e r a -t i o n two u n i t s f r o m node v^ a r e s w i t c h e d t o m e d i a n V g . S i n c e e a c h u p p e r b o u n d i s e x a c t l y s a t i s f i e d no f u r t h e r r e - a l l o c a t i o n o c c u r s and t h e a l g o r i t h m t e r m i n a t e s w i t h a s o l u t i o n o f S = 6 2 - 0 . T h i s i s n o t h o w e v e r t h e g l o b a l m i n -i m u m , f o r i f m e d i a n v^ i s c h o s e n as t h e f i r s t I- s o u r c e i n s t e a d o f Vg t h e n t h e o p t i m a l b o u n d r e d u c e s t o S = 5 0 - 0 ( F i g u r e 1 0 a ) . Two new s e t s o f h e u r i s t i c s may now be p r e s e n t e d w h i c h seem t o o f f e r c o n s i s t e n t l y b e t t e r r e s u l t s . In t h e f i r s t o f t h e s e , t h e c o n s t r a i n e d s o l u t i o n a p p r o a c h , t h e c a p a c i t y r e s t r i c t i o n s a r e s a t i s f i e d a t e a c h s t e p so t h a t a f e a s i b l e s o l u t i o n i s e n s u r e d a t e v e r y s t a g e i n t h e c o m p u -t a t i o n . L e t t h e u p p e r b o u n d i m p o s e d on e a c h p o t e n t i a l p l a n t s i t e be l a b e l l e d c ^ , i = l , - - - , m . T h e n i n any f e a s i b l e s o l u t i o n , t h e r e s t r i c t i o n t . < c. i = l , - - - , p m u s t a p p l y w h e r e t . i s t h e c u r r e n t p l a n t s i z e f o r m e d i a n v ^ . . Now p r o c e e d i n t h e f o l l o w i n g s t e p s . i ) F i r s t e n s u r e t h a t f o r a g i v e n v a l u e o f p t h e r e a r e p p l a n t s w h o s e c o m b i n e d u p p e r b o u n d s a r e a t l e a s t e q u a l t o n 9 = I b j = l J t h e t o t a l demand on t h e n e t w o r k . S p e c i f i c a l l y f i n d t h e p p o t e n t i a l p l a n t s i t e s w h o s e c^ v a l u e s a r e a m a x i m u m . T e s t w h e t h e r t h e s u m m a t i o n o f t h e s e c . v a l u e s > 6. I f s o , t h e n a f e a s i b l e s o l u t i o n e x i s t s . i i ) Among t h e m p l a n t s i t e s f i n d a l o w e r c . v a l u e 4> s u c h t h a t any p c o m b i n a t i o n o f c . v a l u e s e x c e e d i n g <j> w i l l , when s u m m e d , be > 0. Of c o u r s e t h e r e may ( w i l l p r o b a b l y ) e x i s t c o m b i n a t i o n s o f p n o d e s some o f w h o s e i n d i v i d u a l c . v a l u e s a r e l e s s t h a n <f> w h i c h a r e a l s o f e a s -i b l e w i t h r e s p e c t t o G. The c o m p u t a t i o n o f <f> w i l l a c t as an i n s u r a n c e p o l i c y t o k e e p t h e s o l u t i o n f e a s i b l e , i f n e c e s s a r y , a t some s t a g e s i n t h e e n s u i n g c o m p u t a t i o n . i i i ) B r e a k o p e n t h e n e t w o r k a c c o r d i n g t o a n y o f t h e p a r t i t i o n i n g h e u r i s t i c s p r e v i o u s l y g i v e n . R e t a i n t h i s s e t o f Xp o n l y i f t h e sum o f t h e c o r r e s p o n d i n g c. v a l u e s e x c e e d s 9. O t h e r w i s e r e p e a t t h e s t e p r e s t r i c t i n g t h e c h o i c e o f Xp f r o m among o n l y t h o s e n o d e s w h o s e c^ v a l u e s a r e > <j>. T h i s p r o c e d u r e e n s u r e s t h e c h o i c e o f an i n i t i a l s e t o f m e d i a n s c a p a b l e o f s a t i s f y i n g t h e t o t a l n e t w o r k r e q u i r e m e n t s . i v ) D e f i n e some v e c t o r s a y z , w h i c h c o n t a i n s t h e c u r r e n t a m o u n t o f u n f i l l e d s t o r a g e a v a i l a b l e a t e a c h m e d i a n i n s o l u t i o n . I n i t i a l l y t h e z v a l u e s w i l l be s e t e q u a l t o c., i = 1,•• • ,p . v ) Now a l l o c a t e t h e demand o f i n d i v i d u a l n o d e s t o some one o r more o f t h e c u r r e n t m e d i a n s e t s . The f o l -l o w i n g p r o c e d u r e w i l l p r o v i d e a good b a s i c s t a r t . D e f i n e f o r e a c h s u c h node v^ , i = n some v a l u e x s u c h t h a t x^ = d(v . j v k l ) - d ( v . j v k 2 ) w h e r e v ^ and v k 2 a r e t h e c l o s e s t two m e d i a n s f o r t h a t v^ r e s p e c t i v e l y . S t o r e a l l s u c h x^ v a l u e s and r a n k them i n d e s c e n d i n g o r d e r o f m a g -n i t u d e . The a s s o c i a t e d l a b e l l e d n o d e s ( c o r r e s p o n d i n g t o t h e x.j v a l u e s ) w i l l now be a s s i g n e d i n t h i s o r d e r , e a c h o n e , o r as much as p o s s i b l e o f t h e demand a t e a c h o n e , t o i t s p a r t i c u l a r v ^ m e d i a n . A f t e r e a c h a s s i g n m e n t t h e v a l u e o f t h e a p p r o p r i a t e z e l e m e n t i s r e d u c e d a c c o r d i n g t o t h e a m o u n t a l l o c a t e d . I f z = 0 f o r any m e d i a n t h e u p p e r b o u n d c o n s t r a i n t has b e e n e x p l i c i t l y s a t i s f i e d and no more d e s t i n a t i o n r e q u i r e m e n t s c a n be s u p p l i e d f r o m t h a t s o u r c e . The a r r a y x a c t s i n e f f e c t a s a p e n a l t y v e c t o r s u c h t h a t t h e node f i r s t a s s i g n e d i s a l w a y s t h a t one w h o s e n o n - a l l o c a t i o n t o i t s d e s i g n a t e d n e a r e s t m e d i a n m i g h t p r o v e , s h o u l d z become z e r o , t o be m o s t e x p e n s i v e i n t e r m s o f t r a n s p o r t a t i o n c o s t s . W h e n e v e r some z v a l u e i s r e d u c e d t o z e r o t h e n t h i s s t e p i s r e p e a t e d f o r a l l n o d e s wh ose demand i s n o t as y e t t o t a l l y a s s i g n e d , o m i t t i n g f r o m c o n s i d e r a t i o n t h e a s s o c i a t e d m e d i a n and a l l o t h e r s wh ose a v a i l a b l e c a p a c i t y h a s b e e n t o t a l l y f i l l e d by p r e v i o u s a l l o c a t i o n s . v i ) T h i s s t e p r e c o n s i d e r s t h e a l l o c a t i o n s d e f i n e d i n t h e p r e v i o u s s t a g e and makes r e a d j u s t m e n t s w h e r e n e c e s -s a r y . C o n s i d e r e a c h s u b g r a p h i n t u r n and s e a r c h f o r some node v r e G . . , w h i c h i s n e a r e r t o t h e m e d i a n o f some o t h e r s u b g r a p h G - . S y m b o l i c a l l y i f t h e r e e x i s t s some v 3 ^* s u c h t h a t d ( v r v k i ) > d ( v r v k j ) t h e n l e t = d ( v r v k . ) - d ( v r v k j ) w h e r e y 1 i s t h e p o t e n t i a l s a v i n g a s s o c i a t e d w i t h t h e t r a n s -f e r o f one u n i t o f v^ f r o m G| t o G . . Now e x a m i n e t h e r v j s u b s e t o f n o d e s e G . f o r some node v 0 w h i c h i s c l o s e r t o v k i t h a n i s v r . I f d ( v £ v k i ) < d ( v r v k - ) t h e n c a l c u l a t e y 2 = d ( % v k i ) - d ( v A v k j ) w h e r e y 2 i s t h e p o t e n t i a l l o s s ( i n t e r m s o f v^ ) o f t r a n s -f e r r i n g b a c k one u n i t o f V A f r o m G . t o G . . Now c o m p u t e , y = y-j - y 2 and i f y > 0 t h e n t h e s a v i n g s t o be made f r o m r e - a l l o c a t i n g v r ( o r t h a t p o r t i o n o f t h e demand o f v r i n G.j) f r o m G.j t o G .^ more t h a n o f f s e t t h e l o s s e s o f r e t u r n i n g a l i k e a m o u n t o f t h e demand o f v 0 f r o m G . t o G . . The a c t u a l a m o u n t r e - a l l o c a t e d w i l l c l e a r l y d e p e n d upon t h e r e l a t i v e s i z e s o f t h e demands a t v and v „ . I f b = t h e r l r demand o f v „ i n G . and b „ = t h e demand o f v „ i n G . t h e n r i I & 3 when b ^ b 0 an amount e q u a l t o b i s s w i t c h e d f r o m F X / i 6 . t o G . . Where b > b h o w e v e r , t h e a m o u n t t r a n s f e r r e d i s e q u a l o n l y t o b ^ . In e a c h c a s e o f c o u r s e an e q u a l a m o u n t i s r e t u r n e d f r o m G^ t o G^ (by r o u t e v^ t o v ^ . ) t o m a i n t a i n t h e e q u i l i b r i u m o f s u p p l y and d e m a n d . T h i s s t e p i s r e p e a t e d u n t i l no f u r t h e r r e a d j u s t m e n t s a r e p o s s i b l e , and t h e c h o i c e o f and v^ on a n y c y c l e i s d e t e r m i n e d by t h e maximum v a l u e o f y . The c o n c e p t s d i s c u s s e d so f a r c a n be b r i e f l y i l l u -s t r a t e d w i t h r e f e r e n c e a g a i n t o F i g u r e 6 . A t t e r m i n a t i o n o f s t e p i i i ) c o n s i s t s o f v 2 v^ v g a n d v g whose p l a n t s i z e s t o g e t h e r s a t i s f y 6, t h e t o t a l n e t w o r k d e m a n d . By t h e d e t a i l s o f s t e p v ) t h e node o r d e r f o r a s s i g n m e n t i s v - j , v 2 , v g , v g , v - | 0 , v 4 , v 6 , v . , , v 5 , v 3 s i n c e t h e c o r r e -s p o n d i n g v a l u e s o f x a r e , i n d e s c e n d i n g o r d e r , I T , 1 1 , 9 , 9 , 9 , 4 , 4 , 4 , 2 , and 1 . Node v-j i s t h u s a s s i g n e d f i r s t s i n c e t h e d i f f e r e n c e i n t h e d i s t a n c e s t o i t s n e a r e s t two m e d i a n s ( v 2 and v^O i s 1 1 , w h i c h i s a m a x i m u m , ( t h o u g h n o t u n i q u e ) , f o r a n y node on t h e n e t w o r k . The i n i t i a l b a s i c s t a r t g i v e n by s t e p v ) i s shown i n T a b l e 4 ( b ) w i t h S r= 6 2 - 0 . In e x a m i n i n g t h i s t a b l e a u n o t e t h a t 2 u n i t s o f v 3 a r e a s s i g n e d t o G ^ . The m e d i a n s o f a l l o t h e r t h r e e s u b g r a p h s a r e a c t u a l l y c l o s e r t o v^ t h a n i s Vg, t h e m e d i a n o f G ^ . C o n s i d e r G 2 w h i c h c o n t a i n s n o d e s v^ ( t h e m e d i a n ) and Vg. To t r a n s f e r one u n i t o f v^ f r o m Vg t o v^ w i l l r e d u c e t h e c o s t ( d i s t a n c e ) f r o m 19 t o 6 so t h a t y-| = 1 3 . T r a n s f e r r i n g one u n i t o f v^ b a c k t o G^ f r o m G 2 i n c r e a s e s t h e c o s t o f s u p p l y i n g t h a t u n i t f r o m 5 ( d ( v 5 v^)) t o 1 2 ( d ( v 5 v , g ) ) s o t h a t t h e l o s s y 2 = 1 2 - 5 = 7 . B u t t h e o v e r a l l g a i n a s s o c i a t e d w i t h t h i s r e - a l l o c a t i o n i s p o s i t i v e , y = 13 - 7 = 6 f o r e a c h u n i t , and s i n c e two u n i t s c a n be t r a n s f e r r e d t h e t o t a l t r a n s p o r t c o s t ( d i s t a n c e ) c a n be r e d u c e d by 1 2 . S i n c e no f u r t h e r r e a d j u s t m e n t s c a n be made t h i s i s t h e o p t i m a l a l l o c a t i o n ( T a b l e 4 o , F i g u r e 1 0 a ) . N o t e t h a t s t e p s v ) and v i ) a r e i n f a c t a c t u a l l y a r e s t a t e m e n t o f t h e c l a s s i c a l t r a n s p o r t a t i o n p r o b l e m b e c a u s e t h e t o t a l n e t w o r k demand 0 c a n a l w a y s be made e x a c t l y e q u i v a l e n t t o t h e c o m b i n e d c a p a c i t y o f t h e m e d i a n s i n s o l u -t i o n by t h e i n t r o d u c t i o n o f s l a c k v a r i a b l e s . The p r o c e d u r e o f v ) w i l l a l w a y s s e r v e t o p r o v i d e a good i n i t i a l b a s i s , and a l t h o u g h t h e m e t h o d o f s t e p v i ) f o r r e - a l l o c a t i n g f l o w s w i l l n o r m a l l y be l e s s e f f i c i e n t t h a n e x i s t i n g a l g o r i t h m s w h i c h e m p l o y t h e m e t h o d o f s h a d o w c o s t s , t h e p r o b l e m o f d e g e n e r a c y d o e s n o t a p p e a r t o o c c u r so t h a t c o d i n g i s c o n -s i d e r a b l y s i m p l i f i e d . v i i ) L e t some v a l u e t^ be t h e t o t a l s u b g r a p h demand so t h a t 0 t h e o v e r a l l n e t w o r k r e q u i r e m e n t . The m e d i a n s o f e a c h s u b -g r a p h a r e now r e - c a l c u l a t e d a c c o r d i n g t o ( 1 2 ) s u b j e c t t o f u r t h e r r e s t r i c t i o n t h a t o n l y t h o s e p o t e n t i a l s i t e s y e G± wh ose u p p e r b o u n d s c.. a r e > t . w i l l be c o n s i d e r e d as c a n -d i d a t e s . T h i s c o n s t r a i n t e n s u r e s c h o i c e f r o m among o n l y T a b l e 4 SOLUTION TABLEAU FOR F IGURE 6 (a) D i s t a n c e ( T r a n s p o r t a t i o n Cost) M a t r i x G V i V2 v 3 V 5 V 6 v 7 v 8 v 9 Vio C a p a c i t y 1 v 2 T 0 5 1 1 16 1 5 1 6 24 25 25 13 2 1 2 1 1 6 0 5 4 5 13 14 14 12 3 v 6 16 15 10 4 3 0 1 9 10 10 11 4 v 8 25 24 19 13 12 9 10 0 1 1 1 5 Demand 1 10 4 10 2 10 1 10 1 1 50 (b) I n i t i a l A l l o c a t i o n (S'tep v ) G V i v 2 v 3 V. v 5 Ve v 7 v 8 v 9 Vio C a p a c i t y C o s t 1 v 2 1 10 2 1 3 11 2 v . 10 2 1 2 10 3 v 6 10 1 11 1 4 v 8 2 10 1 1 14 40 Demand 1 10 4 10 2 10 1 10 1 1 50 62 (c) O p t i m a l A l l o c a t i o n (Step v i ) G v 2 v 3 V - V 5 V 6 v 7 v 8 v 9 Vio C a p a c i t y C o s t 1 V 2 1 2 1 3 11 2 V,* 2 10 1 2 12 3 V 6 10 1 11 1 4 V 8 2 10 : i 1 14 26 De ;mand 1 10 4 10 2 10 1 10 i 1 50 50 P808LSH BARE FIGURE 10(A) 1 BEDIAS SOLUTION (PLAST SIZE COKSTRAIHTS) FOB PIGU9E 6 • • 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 * 4 . 4 4 4 4 4 4 4 « 4 4 4 4 4 4 4 4 4 * * 4 4 4 4 4 4 4 4 4 « 4 4 4 4 4 * 4 4 4 4 4 * 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 * 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 PLABT LOCATION-ALLOCATION 0 BEDIAB SOLUTIOB BEDIAB PLAST SIZE FREIGHT COST PLAST COST TOTAL COST BODES SERVICED VOLOBE SOPPLIED TO EACH BEDIAB 1 2 13.00 11.00 0.00 11.00 1 2 3 1.0 10.0 2.0 2 0 12.00 12.00 0.00 12.00 3 Q 2.0 10.0 3 6 11.00 1.00 0.00 1.00 6 7 10.0 1.0 0 8 11.00 26.00 0.00 26.00. 5 8 9 10 2.0 10.0 1.0 1.0 ARTWORK' 50.00 50.00- 0.00 50.00 • • 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 * 4 4 4 4 * 4 4 4 4 4 4 4 4 4 4 4 4 4 * 4 4 4 M * t . t t t i t » f t t * M t * M » « t l l * t « » t « l ( t l » » l t t * t l t t * 4 : t t f * « t t « * t * t . » t PB08LER BABE FIGURE 10(B) 2 ASD 3 BEDIAS'SOLUTIONS (PLABT SIZE COBSTRAIBTS) FOR FIGURE 0 CAPACITI CONSTRAINT CRITERION PREVENTS A 2 LOCATIOS SOLOTIOB • • • • 4 4 * * * * * * « 4 4 4 4 4 4 4 4 4 4 4 4 * * * * * * 4 4 * * * * * 4 « * 4 * * 4 * * * * * * 4 * 4 * 4 4 4 4 4 4 4 4 4 * * » « 4 * * * * * 4 4 4 4 4 4 4 4 * 4 4 4 4 4 4 4 * * * * * * * 4 4 * 4 * * * « * * * * * * * « * « * PLAST LOCATIOS-ALLOCATION 3 B ED IAN SOLOTIOB BEDIAB PLANT SIZE FREIGHT COST PLANT COST TOTAL COST BODES SERVICED VOLDBE SUPPLIED TO EACH BED IAN 1- 5 9 . 0 0 10 . 0 0 0 . 0 0 10 . 0 0 1 0 5 7 3 . 0 3 . 0 2 . 0 1 . 0 2 8 8 . 0 0 10.00 0 . 0 0 10 . 0 0 6 7 8 0 . 0 3 . 0 1 . 0 3 10 7.50 13.50 0 . 0 0 13.50 2 3 7 9 10 1 . 0 0 . 5 1 . 0 3 . 0 2 . 0 B ETHORK 20.50 3 3 . 50 0 . 0 0 33.50 « 4 4 4 4 4 4 4 4 4 4 4 4 4 * * t * « * * « * * * « * * * * * * « * * * * « * * * * * * « * * « * * * * * * * * * * 4 t * * * * 4 4 4 4 t 4 4 4 4 4 4 4 4 4 4 4 4 * * * * * * * * * * « * * * » * * * * * 4 * * * * * * * « * * * * « « PLABT LOCATIOS-ALLOCATIOB 3 BEDIAB SOLUTION BEOIAS PLANT SIZE FBEIGHT COST PLANT COST TOTAL COST BODES SERVICED VOLURE SOPPLIED TO EACH BEDIAN " 1 1 7.50 1 1 . 0 0 0 . 0 0 1 1 . 0 0 1 2 7 3 . 0 1 . 0 3 . 5 2 5 9.00 12. 50 0 . 0 0 12.50 0 5 6 7 3 . 0 2 . 0 2 . 5 1 . 5 3 8 8.00 9.00 0.00 9.00 3 6 8 9 10 0 . 5 1 .5 1.0 3 . 0 2 . 0 •ETsoar 2 0 . S 0 3 2 . 5 0 0 . 0 0 3 2 . 5 0 • * 4 « « « « * * * * « « « * * 4 4 4 4 4 4 « « 4 * 4 « * * * * * < * « * * * * * * « * « * * * * « * * * * * « * * * * 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 « 4 * 4 4 « * * 4 4 4 4 * * * * * * 4 4 4 « 4 4 4 * 4 « « * 4 » * 4 * « 4 * * « PROBLEM BABE FIGURE 10(C) 2 BEDIAB SOLOTIOB (BAIIBOB OISTASCE CONST RAIHT) FOR FIGUBE 0 • 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 * 4 4 * 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 PLABT LOCATIOS-ALLOCATIOB 2 BEDIAB SOLOTIOB BEDIAB PLAST SIZE FREIGHT COST PLANT COST TOTAL COST BODES SERVICED VOLOBE SOPPLIED TO EACH BEDIAB 1 1 10.00 25.00 0.00 25.00 1 2 0 5 7 3.0 1.0 3.0 2.0 5.0 2 6 10.SO 10.00 0.00 10.00 3 6 • 8 9 10 0.5 0.0 1.0 3.0 2.0 ISTilOBK 20.50 39.00 0.00 39.00 • * 4 * * * * 4 4 4 * 4 4 4 4 * * * * 4 * * * « * * * * * * * * * * * * * * * 4 4 4 4 4 4 4 4 4 * * * « * e * * * * « » « * * 4 * * * 4 4 « 4 4 4 « « 4 * * * * * * a * * t * 4 4 4 4 * 4 * 4 4 4 4 4 4 * * * * * * * * * 4 4 * * * * 4 BO SOLOTIOB FOUND OB TBIS ITERATION SATISFYING BAIIBUB OISTABCE COBSTBAIST FOB A 2 LOCATIOH SOLUTIOB F i g u r e 10 t h o s e n o d e s c a p a b l e o f s a t i s f y i n g t h e s u b g r a p h ' s d e m a n d . S i n c e f r o m t h e p r e v i o u s s t e p s t h e r e i s a t l e a s t one s u c h member i n e a c h G.. t h e s o l u t i o n w i l l a l w a y s r e m a i n f e a s i b l e . v i i i . ) S t e p s i v ) t o v i i ) a r e r e i t e r a t e d u n t i l one f u l l c y c l e f a i l s t o r e d u c e t h e o b j e c t i v e f u n c t i o n S . In t h e s e c o n d s e t o f h e u r i s t i c s , t h e u n c o n s t r a i n e d s o l u t i o n a p p r o a c h , t h e o b j e c t i v e i s t o s o l v e f i r s t t h e m i n i m u m t r a n s p o r t c o s t s o l u t i o n w i t h o u t p l a n t s i z e c o n -s t r a i n t s , b u t r e s t r i c t i n g c h o i c e o f m e d i a n s t o some s e t o f p , known a p r i o r i , t o be c a p a b l e t o g e t h e r o f s a t i s f y i n g 9 . T h i s s o l u t i o n i s t h e n s u b s e q u e n t l y m o d i f i e d t o s a t i s f y t h e p l a n t u p p e r b o u n d r e s t r i c t i o n s . T h e s e g o a l s a r e s i m p l y a c h i e v e d as f o l l o w s : 1) In p a r t i t i o n i n g t h e n e t w o r k c h o i c e o f n o d e s i s e x c l u s i v e l y r e s t r i c t e d t o t h o s e p o t e n t i a l s i t e s w h o s e c . v a l ues e x c e e d <f>. • 2) E a c h node on t h e n e t w o r k y . , i = l , - - - , n i s a s s i g n e d by ( 1 0 ) t o i t s c l o s e s t member o f U p p e r p l a n t b o u n d v i o l a t i o n s a r e i g n o r e d f o r t h e m o m e n t . 3) The s e a r c h f o r t h e m e d i a n o f e a c h s u b g r a p h i s a g a i n r e s t r i c t e d t h i s t i m e t o t h o s e members v . e G . f o r w h i c h c > <j>. F rom t h e p r e v i o u s r e s t r i c t i o n on s t e p 1) t h e r e m u s t be a t l e a s t one s u c h node i n e a c h s u b s e t . 4 ) S t e p s 2 and 3 a r e r e p e a t e d and t h e l o w e r b o u n d t e s t a p p l i e d u n t i l a s t a b l e s o l u t i o n i s a t t a i n e d . ( N . B . In e a c h o f t h e two s e t s o f h e u r i s t i c s n o d e s a r e o n l y c o n s i d e r e d f o r m e d i a n r e p l a c e m e n t d u r i n g t h e l o w e r b o u n d t e s t , i f and o n l y i f , i t i s known t h a t t h e y c a n g u a r a n t e e t h e c o n t i n u e d f e a s i b i l i t y o f t h e s o l u t i o n ) . 5) U s i n g t h i s l o c a l m i n i m u m ( w h i c h may o r may n o t be o p t i m a l ) as a new s t a r t i n g b a s i s s t e p s i v ) t o v i i i ) o f t h e p r e v i o u s h e u r i s t i c s e t a r e a c t i v a t e d u n t i l no f u r t h e r i m p r o v e m e n t i s f o u n d . T h i s s o l u t i o n c h o s e n i s , o f c o u r s e , t h e b e s t d e s c e n d e n t one f r o m a n y o f t h e two g i v e n m e t h o d s i f b o t h a r e o p t i o n e d . I t i s p o s s i b l e t o s o l v e some h i g h l y c o n s t r a i n e d p r o b l e m s w i t h t h e s e s e t s o f h e u r i s t i c s . C o n s i d e r a g a i n t h e g r a p h o f F i g u r e 4 i n w h i c h , f o r t h i s e x a m p l e , t h e m p o t e n -t i a l p l a n t s i t e s a r e r e s t r i c t e d t o v^ , v ^ , V g , V g , Vg and V-|Q a n d , as a f u r t h e r p r o v i s o , node V;g i s t o be f o r c e d i n t o t h e s o l u t i o n . L e t t h e u p p e r b o u n d s on e a c h o f t h e m s i t e s be s e t e q u a l t o 7 , 7 , 9 , 5 , 8 and 7*5 r e s p e c t i v e l y . F o r t h e 3 m e d i a n o f t h i s n e t w o r k t h e b e g i n n i n g s e t c o n s i s t s o f e i t h e r v-j , V g , Vg o r v ^ , v 5 > ' v 5 - E a c h o f t h e two h e u r i s t i c s e t s c o n v e r g e s t o a l o c a l m i n i m a o f S = 3 3 * 5 w i t h n o d e s V g , Vg and v 1 Q as m e d i a n s ( F i g u r e 1 0 b ) . By t h e l o w e r b o u n d t e s t V-JQ i s d r o p p e d f r o m s o l u t i o n and node v-j i s i n t r o d u c e d as t h e new c a n d i d a t e m e d i a n . The s o l u t i o n a t t a i n e d on t e r -m i n a t i o n i s v-| , Vg and v g w i t h S = 3 2 - 5 ( F i g u r e 1 0 b ) w h i c h a p p e a r s t o be t h e o p t i m a l l o w e r b o u n d . I t seems f a i r t o s a y t h a t t h i s r e s u l t i s n o t o b v i o u s f r o m i n s p e c t i o n . F o r t h i s o p t i o n t h e h e u r i s t i c s p r e s e n t e d may be u n s u c -c e s s f u l s i n c e s o l u t i o n s c a n s o m e t i m e s e x i s t w h i c h a r e n o t f o u n d d u r i n g c o m p u t a t i o n . The s u g g e s t e d m e t h o d i s s i m p l y t o c o n s t r a i n t h e s e a r c h f o r t h e m e d i a n o f e a c h s u b g r a p h G . , i = l , - - - , p t o t h o s e n o d e s v e G . f o r w h i c h d ( v . v )<V. f o r a l l v . e G . , w h e r e V . i s t h e maximum a l l o w a b l e d i s t a n c e o f v . t o some m e d i a n on the n e t w o r k . For a n y p o t e n t i a l m e d i a n v r n o t s a t i s f y i n g t h i s c o n d i t i o n t h e n t h e r e e x i s t s a t l e a s t one other node v . whose d i s t a n c e t o v r i s g r e a t e r than the s t i p u l a t e d m a x -i m u m . Th is a m o u n t t o s a t i s f y i n g e q u a t i o n (12) subject i f possible t o (17) d(v. v, . ) < V. f o r a l l v. e G. As an e x a m p l e r e c o n s i d e r F i g u r e 4 and p l a c e t h e f o l -l o w i n g maximum d i s t a n c e c o n s t r a i n t s on e a c h node 0 - 5 , 4 , 3 , 7 , 2 , 0 - 5 , 3 , 4 , 8 and 5 . S u b j e c t t o t h e s e c o n s t r a i n t s t h e 2 m e d i a n s o l u t i o n c o n s i s t s o f v-| a n d Vg, w h i c h a r e i n f a c t , t h e o n l y two v e r t i c e s c a p a b l e o f s a t i s f y i n g c o n d i t i o n ( 1 7 ) . ( F i g u r e 1 0 c ) . In t h e e v e n t , on a n y i t e r a t i o n , t h a t p m e d i a n s s a t i s f y i n g t h e c o n s t r a i n t c a n n o t be f o u n d , t h e l e a s t c o s t t r a n s p o r t s o l u t i o n i s c h o s e n i n s t e a d f o r t h e n e x t c y c l e , a n d a m e s s a g e i s p r i n t e d t o t h i s e f f e c t . In e x t r e m e c a s e s t h e a l g o r -i t h m may t h e r e f o r e t e r m i n a t e w i t h o u t p r i n t i n g a f e a s i b l e s o l u -t i o n e v e n t h o u g h one e x i s t s . One p a r t i a l way o u t o f t h i s i m -p a s s e m i g h t be t o c h o o s e , f o r a l l s u b g r a p h s w h e r e the constraint cannot be c u r r e n t l y s a t i s f i e d , that node v . e G . whose value i s a m i n i m u m i n t h e s e t , and c o n t i n u e f r o m t h e r e . H o w e v e r e v e n t h i s s c h e m e w i l l n o t g u a r a n t e e s u c c e s s and i s n o t a s y e t i n c o r p o r a t e d i n t o t h e c o m p u t e r c o d e f o r t h i s a l g o r i t h m ( s e e A p p e n d i x 5). I t i s p o s s i b l e , e v e n d e s i r a b l e , t o f o r c e i n t o t h e s o l u t i o n t h o s e n o d e s w h o s e d i s t a n c e c o n s t r a i n t s a r e v e r y r e s t r i c t i v e , so t h a t t h e l i k l i h o o d o f n o t o b t a i n i n g a f e a s i b l e s o l u t i o n i s s h a r p l y r e d u c e d . F o r many n e t w o r k s t h i s p r o c e d u r e w i l l p r o v i d e t h e b e s t a l t e r n a t i v e , s i n c e t h e r e seems t o be no o b v i o u s way o f e n s u r i n g e v e n an i n i t i a l s e t o f Xp w h i c h i s f e a s i b l e . In o t h e r c a s e s i t may w e l l be t h a t t h e p r o b l e m i s s o l u b l e o n l y i f s t r u c t u r e d as a l i n e a r p r o g r a m . P l a n t C o s t s F o r t h e p l a n t 1 o c a t i o n - a l 1 o c a t i o n p r o b l e m w h i c h r e -q u i r e s t h e s i m u l t a n e o u s m i n i m i z a t i o n o f b o t h p l a n t and t r a n s p o r t c o s t s , t h e v a l u e o f p f o r w h i c h t h i s i s a m i n i m u m w i l l n o t u s u a l l y be known a t t h e o u t s e t . In t h e m e t h o d g i v e n h e r e t h e g e n e r a l s t r a t e g y w i l l be t o c a l c u l a t e t h e t o t a l c o s t f u n c t i o n f o r s e v e r a l f i x e d a r b i t r a r y v a l u e s . o f p , c h o o s e t h e two s o l u t i o n s w i t h i n w h i c h t h e o p t i m u m m u s t l i e , and b e g i n n i n g w i t h t h e l o w e r v a l u e o f p , i n c r e m e n t i t s e q u e n t i a l l y by one u n t i l t h e p+1 s o l u t i o n e x c e e d s t h e p s o l u t i o n . As w i t h t h e p l a n t s i z e c o n s t r a i n t o p t i o n s two s e t s o f h e r u i s t i c s w i l l be g i v e n one o f w h i c h w i l l y i e l d a b e t t e r e s t i m a t e o f t h e t r u e l o w e r b o u n d . In t h e f i r s t o f t h e s e two m e t h o d s t h e p r o c e d u r e i s a s f o l l o w s : i ) S o l v e f i r s t t h e m i n i m u m t r a n s p o r t a t i o n c o s t s o -l u t i o n . D e t e r m i n a t i o n o f t h i s p m e d i a n w i l l a u t o m a t i c a l l y d e f i n e t h e l e a s t c o s t a l l o c a t i o n o f m a r k e t s t o s o u r c e s , so t h a t t h e p l a n t s i z e t . o f e a c h s o u r c e i s a l s o g i v e n . F rom t h i s t h e a s s o c i a t e d p l a n t c o s t s c a n be r e a d i l y e s t a b l i s h e d . T h i s i s t h e a p p r o a c h t a k e n by S i n g e r b u t h i s a l g o r i t h m t e r m i n a t e s a t t h i s p o i n t p r e s u m a b l y u n d e r t h e a s s u m p t i o n t h a t t h e m i n i m u m t r a n s p o r t a t i o n c o s t s o l u t i o n w i l l a l s o d e f i n e , f o r a n y p , t h e t o t a l l e a s t c o s t b o u n d . W h i l e t h i s may g e n e r a l l y be t r u e i t i s n o t d i f f i c u l t t o v i s u a l i z e s i t u a t i o n s i n w h i c h s c a l e e c o n o m i e s a t some p l a n t s may be more t h a n s u f f i c i e n t t o o f f s e t a n o n - o p t i m a l t r a n s p o r t a t i o n c o s t s o l u t i o n . i i ) T h i s s t e p d i f f e r s f r o m t h e p r e v i o u s one o n l y i n t e r m s o f t h e m e d i a n c a l c u l a t i o n . S p e c i f i c a l l y , c h o i c e o f m e d i a n on e a c h s u b s e q u e n t i t e r a t i o n i s now d e t e r m i n e d by an a p p r a i s a l o f b o t h p l a n t and t r a n s p o r t a t i o n c o s t s t o g e t h e r , r a t h e r t h a n on t h e l a t t e r b a s i s a l o n e . Thus f o r e a c h G . , S . now e q u a l s (18) I. £=G_ w(v. v k . ) + f . ( t . ) where f i i s some f u n c t i o n o f t . , t h e c u r r e n t s o u r c e s i z e , s u c h t h a t f . ( t . ) i s t h e p l a n t c o s t f o r t h a t s o u r c e a t i t s p r e s e n t l e v e l o f o p e r a t i o n . In p r a c t i c e t h i s f u n c t i o n m u s t be s u p -p l i e d f o r e a c h p o t e n t i a l s i t e by t h e u s e r as a s u b r o u t i n e . i i i ) A f t e r r e p e a t i n g s t e p i i ) t o a c h i e v e c o n v e r -g e n c e t h e r e s t i l l r e m a i n s t h e p o s s i b i l i t y t h a t f u r t h e r a d v a n t a g e o f t h e p l a n t e c o n o m i e s o f s c a l e c a n be g a i n e d by j u d i c i o u s m a n i p u l a t i o n s o f t h e c u r r e n t v e c t o r o f a l l o -c a t i o n s . A t e s t i s t h e r e f o r e made i n w h i c h m a r k e t s ( o r f r a c t i o n s o f m a r k e t s i f s i z e c o n s t r a i n t s d i c t a t e ) a r e f l i p p e d o n e by one f r o m e a c h s o u r c e c u r r e n t l y a s s i g n e d t o i t t o t h o s e w h i c h a r e n o t . L e t y-j be t h e d e c r e a s e i n t o t a l c o s t s ( o v e r some G^) a c h i e v e d by r e m o v i n g some m a r k e t v f f r o m m e d i a n v ^ .. S i m i l a r l y l e t be t h e i n c r e a s e i n t o t a l c o s t s ( o v e r some G . ) e f f e c t e d by t h e a d d i t i o n o f v t o 3 ^* some v k j . I f f o r a n y m e d i a n v ^ j ? y^ < y-j t h e n some r e d u c t i o n i n t o t a l c o s t ( o v e r t h e n e t w o r k ) i s f e a s i b l e . C h o i c e o f and y . i s d e p e n d e n t upon t h e maximum d i f f e r e n c e y - j - y ^ ( i f a n y ) f o u n d . T h i s s t e p i s r e p e a t e d u n t i l no f u r t h e r i m p r o v e m e n t i s f o u n d , s u b j e c t t o t h e m a i n t e n a n c e o f a t l e a s t one u n i t o f demand a t e a c h s o u r c e . The s e c o n d s e t o f p l a n t c o s t h e u r i s t i c s d i f f e r s f r o m t h e f i r s t i n t h a t n o d e s a r e a s s i g n e d / t o m e d i a n s on t h e b a s i s o f a t l e a s t a p a r t i a l e v a l u a t i o n o f p l a n t ( i n a d d i t i o n t o t r a n s p o r t a t i o n ) c o s t s . C o n s i s t e n t w i t h t h e a p p r o a c h t a k e n t h r o u g h o u t t h i s c h a p t e r t h e s u g g e s t e d p r o c e d u r e w i l l be o u t l i n e d i n s t e p s . 1) F o r e a c h node v . , j = 1 , • • • 9 n f i n d t h e c u r r e n t m e d i a n v^. , i = l , - - - , p f o r w h i c h d (v . . v k i ) i s a m i n i m u m . 2) E a c h s u b g r a p h , i = 1 , • • • ,p now has a s u b s e t o f n o d e s v . e G . w h i c h s a t i s f y S t e p 1 ) . Rank e a c h member o f e a c h s u b s e t i n a s c e n d i n g o r d e r i n t e r m s o f d ( v . v, . ) . J K1 3) A s s i g n f i r s t t h a t member o f G^ w h i c h i s c l o s e s t t o v^-j . S e c o n d l y a s s i g n t o v ^ t h a t member o f w h i c h i s c l o s e s t t o and so on u n t i l p s u c h a s s i g n m e n t s h a v e b e e n m a d e . T h i s o p e r a t i o n e n s u r e s t h e a l l o c a t i o n o f a t l e a s t one d e s t i n a t i o n t o e a c h s o u r c e and y i e l d s a b e g i n n i n g p l a n t s i z e f o r e a c h m e d i a n . 4 ) T e n t a t i v e l y a s s i g n t o t h e member i n t h e s e t o f G-j w h i c h i s s e c o n d c l o s e s t t o . B u t b e f o r e t h i s a l l o c a t i o n i s c o n f i r m e d t h e t e s t g i v e n i n s t e p i i i ) o f t h e p r e v i o u s h e u r i s t i c s e t i s m a d e . R e p e a t t h i s p r o c e d u r e f o r a l l r e m a i n i n g node a s s i g n m e n t s , t h e c h o i c e o f o r d e r b e i n g d e t e r m i n e d by t h e r a n k i n g o f t h e n o d e s i n e a c h s u b s e t . E a c h s e t o f p a s s i g n m e n t s w i l l t h e n , t a k e one member f r o m e a c h s u b s e t , b u t t h e t e s t i s a c t i v a t e d o n l y f o r t h e s e c o n d and s u b s e q u e n t s e r i e s o f a l l o c a t i o n s . On any one s e r i e s o f s u c h a s s i g n m e n t s a f t e r t h e i n i t i a l a l l o c a t i o n s h a v e b e e n made i t i s f e a s i b l e t h a t some s u b s e t s may become e x h a u s t e d o f m e m b e r s . On t e r m i n a t i o n a l l s e t s w i l l o f c o u r s e be e m p t y . 5) The m e d i a n s o f e a c h G . a r e now c o m p u t e d a c c o r d -i n g t o t h e t o t a l c o s t c r i t e r i o n o f s t e p i i ) , and s t e p s 1 t o 5 a r e a g a i n r e p e a t e d u n t i l t h e l o w e s t b o u n d i s f o u n d . A p p l y i n g t h e s e h e u r i s t i c s a g a i n t o F i g u r e 4 w i t h p l a n t c o s t s s e t e q u a l on a n y i t e r a t i o n t o 1 0 / t T ( t . = c u r r e n t s i z e o f p l a n t i ) a t n o d e s , v ^ , V g , v g a n d v - j 0 and t o 7 / t T f o r t h e o t h e r f i v e p o t e n t i a l s o u r c e s , t h e l o w e s t c a l c u l a t e d b o u n d o f S = 8 6 - 8 o c c u r s when p = 3 , w i t h m e d i a n s v-| , v ^ , Vg o r v-| , v ^ , Vg i n s o l u t i o n ( F i g u r e 1 1 a ) . W i t h t h e same p l a n t c o s t f u n c t i o n s b u t s i z e u p p e r b o u n d s o f 4 , 6 , 7 , 1 0 , 1 2 , 1 1 , 8 , 1 3 , 9 and 10 r e s p e c t i v e l y i m p o s e d upon e a c h n o d e , t h e t h r e e m e d i a n s o l u t i o n o f t h e g r a p h c o n s i s t s o f v ^ , Vg and Vg w i t h S = 9 2 - 3 ( F i g u r e l i b ) . D y n a m i c L o c a t i o n - A l 1 o c a t i o n S y s t e m s The h e u r i s t i c s f o r s y s t e m s o f s t a t i c l o c a t i o n - a l l o -c a t i o n p r o b l e m s g i v e n p r e v i o u s l y i n t h i s c h a p t e r a r e r e a d i l y a d a p t a b l e t o t h e d y n a m i c s p h e r e o f o p e r a t i o n s . T h i s e x t e n -s i o n demands t h e s e q u e n t i a l p a r t i t i o n i n g o f a n e t w o r k o v e r d i s c r e t e i n t e r v a l s o f t i m e . R e d e f i n i n g t . as some t i m e J p e r i o d j = 1 , 2 , - ' - , r , t h e n g i v e n some l o n g t e r m p l a n n i n g h o r i z o n - t , and a b u d g e t c o n s t r a i n t w h i c h a l l o w s f o r t h e c o n s t r u c t i o n o f one o r more f a c i l i t i e s i n any t i m e i n t e r v a l t . , t h e p r o b l e m i s t o d e c i d e i n w h i c h l o c a t i o n s and i n w h a t 3 o r d e r t h e p l a n t s s h o u l d be b u i l t s u c h t h a t t h e o v e r a l l c o s t s ( a g g r e g a t e d i s t a n c e s ) a r e a t a m i n i m u m o v e r a l l p e r i o d s u n d e r q u e s t i o n . T h i s i s a c o m b i n a t o r i a l p r o g r a m m i n g p r o b l e m w h i c h c a n a l w a y s be e x p l i c i t l y s o l v e d by d i r e c t e n u m e r a t i o n o f PB03LEH BASE FIGURE 11(A) BIHIBOS COHBISED PLAST-TBASSPORIATI0« SOLUTION FOB PISU B E 0 PLANT LOCATIOS-ALLOCATIOH 1 BEDIAS SOLUTION BODES SEBVICED 1 2 3 0 VOLOBE SOPPLIED TO EACH BEDIAS 3.0 e.o 1.0 5.0 0.5 1.0 3 . 0 3-0 2.0 2.0 BEDIAS PLAST SIZE FREIGHT COST PLAST COST TOTAL COST 1 6 29.50 59.00 30.65 93.65 IETKOBK 20.50 59.00 30.65 93.65 • * « • « • « * • * « « • • * • * • * • • » • * • * • » • # » • • » • » * « ( • • * » • « • • • » » * * « 4 « « t t * . A » « t » * * t * « » t * 4 « « * f * 4 ? « t * » * * * * * « * * * « * * * « t « t * * * * * * « * * * * * * * PLAST LOCATIOH-ALLOCATIOH 2 BEDIAS SOLOTIOS HEDIAS PLAST SIZE FBEIGHI COST PLAST COST TOTAL COST BODES SERVICED 2 0 VOLOBE SOPPLIED TO EACH BEDIAH 1 1 2 6 BETSOBK •«*••***«« 10.00 10.50 20.50 25.00 10.00 39.00 26. 19 22.68 OS.87 51. 19 36.68 87.87 5 7 9 10 3.0 0.5 1.0 0.0 3.0 1.0 2.0 3.0 5.0 2.0 PLAST LOCATIOH-ALLOCATIOS 2 BEDIAH SOLOTIOS BEDIAS PLAST SIZE FREIGHT COST PLAST COST TOTAL COST BODES SERVICED 2 0 5 VOLOBE SUPPLIED TO EACH BEDIAH 1 1 2 9 BETSOBK 10.00 10.50 20. 50 •*••«•*** 25.00 10.00 39.00 26. 19 22.66 08. 87 51. 19 36.68 87.87 7 10 3.0 0.5 1.0 0.0 3. 0 1.0 2.0 3.0 5.0 2.0 PLAST LOCATIOH-ALLOCATIOS 3 BEDIAS SOLOTIOS BEDIAS PLAST SIZE FREIGHT COST PLAST COST TOTAL COST HODES SERVICED 1 10.00 12.00 22.10 30.10 1 5 7 1 5 3.0 2' 0 0.00 2.00 10.00 16.00 2 0 1.0 3 6 10. 50 10.00 22.68 36.68 3 6 8 9 10 0.5 •ETWORK 20.50 28.00 58.82 86.82 * • * • « • • « « • * • 4 * • • • • * * * VOLOBE SUPPLIED TO EACH BEDIAH 5.0 2.0 3.0 0.0 1.0 3.0 2.0 PLAST LOCATIOH-ALLOCATIOS 3 BEDIAH SOLOTIOS BEDIAH PLAST SIZE FBEIGHT COST PLAST COST TOTAL COST BODES SERVICED 5 7 0 6 8 9 10 1 1 2 0 3 9 BET WORK 10.00 0.00 10. 50 20.50 12.00 2.00 10.00, 28.00 22. 10 10.00 22.68 58.82 30. 10 16.00 36.68 86.82 VOLOBE SOPPLIED TO EACH BEDIAS 5.0 1.0 ' 3.0 2.0 3.0 1.0 0.5 2.0 3.0 0.0 • * l * t * » M * * t t M * 4 * * t t i * * 1 * 4 l l l * * t t l t * 4 l l * • • • • • • * * * » * » « » » * 4 i * « * * « » * * * * « « « « « * * » * * 4 « * * r l i « * * * * * * * * * * » * « * * * * 4 > « * » * * * * » t * * * PEOBLEB BABE FIGURE 11(B) 3 BEDIAS SOLUTIOS (PLAST SIZE COHSTBAISTS) FOR FIGDBE 0 • t « * * 4 « 4 « t * t * « « * « * * 4 * * « » * » « * * * * « • • • « • • * * • • < * » * * • • • « « • • « « • • • « * * • » • * * * * • « • * « ( > * • * « * * * * • « * « . « » « * • . * * • * « • * * * * • * * * * * « PLAHT LOCATIOH-ALLOCATIOH 3 BEDIAH SOLUTIOH • • BEDIAH PLAST SIZE FBEIGHT COSI PLAHT COST TOTAL COST IODES SEBVICED VOLOBE SUPPLIED TO EACH BEDIAH 1 0 6.00 6.00 17.15 23.15 2 0 5 1.0 3.0 2.0 2 6 10.50 10.00 22.68 36.68 3 6 8 9 10 0.5 0.0 1.0 3.0 2.0 3 7 8.00 6.00 28.28 30.28 1 7 3.0 5.0 IETSOBK 20.50 26.00 68.11' 90.11 PLAST LOCATIOH-ALLOCATIOS 3 BEDIAH SOLUTIOH BEDIAS PLAHT SIZE FSEIGHT COSI PLAST COST TOTAL COST BODES SERVICED VOLOBE SUPPLIED TO BACH BEDIAH 1 0 0.00 2.00 10.00 16.00 2 tt 1.0 3.0 2 5 10.00 8.00 31.62 39.62 1 5 7 3.0 2.0 5.0 3 6 10. 50 10.00 22.68 36.68 3 6 8 9 10 0.5 0.0 1.0 3.0 2.0 BETBOBK 20.50 20.00 68.31 92.31 F i g u r e 11 a l l p o s s i b i l i t i e s . In p r a c t i c e t h i s i s n o r m a l l y q u i t e i n -f e a s i b l e , and c o n s i s t e n t w i t h t h e h e u r i s t i c a p p r o a c h t a k e n h e r e , f a r f e w e r e v a l u a t i o n s w i l l be n e e d e d . W i t h r e s p e c t t o t h e p a r t i c u l a r f o r m u l a t i o n u n d e r c o n s i d e r a t i o n some a d d i t i o n a l a s s u m p t i o n s w i l l be m a d e . F i r s t t h e n e t w o r k o f a l l n d e s t i n a t i o n s w i l l be a s s u m e d t o be u n c h a n g e d o v e r a l l t i m e p e r i o d s i n q u e s t i o n . N e x t p l a n t s ( m e d i a n s ) w i l l be a d d e d one a t a t i m e d u r i n g e a c h t e m p o r a l i n t e r v a l , and t h i r d l y o n c e c h o s e n , e a c h s u c h s o u r c e w i l l be a s s u m e d t o r e m a i n f i x e d a t i t s o r i g i n a l l o c a t i o n o v e r a l l s u c c e e d i n g p e r i o d s . A c t u a l l y t h e f i r s t two a s s u m p t i o n s c a n be r e l a x e d w i t h o n l y m i n o r m o d i f i c a t i o n s t o t h e e x i s t i n g p r o g r a m , b u t t h e t h i r d r e m a i n s m o r e r e s t r i c t i v e r e f l e c t i n g as i t d o e s t h e q u e s t i o n o f r e l o c a t i o n c o s t s and t h e t r o u b l e s o m e p r o b -l e m s o f l o c a t i o n a l o b s o l e s c e n c e . Of t h e two p o s s i b l e a p p r o a c h e s t o t h i s p r o b l e m t h e f i r s t makes u s e o f a f o r w a r d r e c u r s i v e p r o c e d u r e i n w h i c h , b e g i n n i n g w i t h t i m e p e r i o d t-j , a s i n g l e m e d i a n i s l o c a t e d w h i c h i s o p t i m a l w i t h r e s p e c t t o a l l w e i g h t e d d i s t a n c e s o v e r t h e n e t w o r k . A s e c o n d f a c i l i t y i s a d d e d n e x t i n w h i c h m i n i m i z e s t h e sum o f t h e a g g r e g a t e f l o w s s u b j e c t t o t h e m a i n t e n a n c e o f p l a n t 1 a t i t s p r e v i o u s l y l o c a t e d s i t e . F o r a n y t i m e p e r i o d t+1 t h e n , t h e n e x t s o u r c e t o be a d d e d i s t h a t one w h i c h m i n i m i z e s t h e f u n c t i o n n t n ( 1 9 ) s " J x & x±i D i j + ± l x i t + i D i t + i w h e r e t l a b e l s t h e t p l a n t s e s t a b l i s h e d i n t h e p r e c e e d i n g t i m e p e r i o d s 1 t h r o u g h t . and X . . and D . . a r e a s p r e v i o u s l y d e f i n e d . C o n s i d e r t h i s a p p r o a c h i n t e r m s o f t h e t w e n t y node n e t w o r k o f F i g u r e 7 w h e r e t h e l o n g t e r m p l a n n i n g g o a l t r = 3 . The s i n g l e m e d i a n o f t h i s w e i g h t e d g r a p h i s p e r i o d t-j i s e i t h e r v g o r v g w i t h S = 1 8 4 - 0 . B r a n c h i n g f i r s t on m e d i a n Vg, node Vg i s a d d e d i n t^, and e i t h e r v-|-| o r v q 3 i n t g w i t h S = 6 8 - 0 . The c u m u l a t e d c o s t s o v e r a l l t h r e e p e r i o d s t o t a l t o 3 6 0 - 0 . A l t e r n a t i v e b r a n c h i n g on v g when t = l a d d s v-|7 i n t h e s e c o n d p e r i o d and Vg i n t h e t h i r d w i t h S = 8 1 - 0 and c u m u l a t e d c o s t s e q u a l t o 3 8 0 - 0 ( s e e F i g u r e s 12 a n d 1 3 ) . A l t h o u g h t h i s a p p r o a c h i s a l w a y s o p t i m a l w i t h r e s p e c t t o p e r i o d t-| , on s u b s e q u e n t i t e r a t i o n s o v e r p e r i o d s t^*'''*?* t h e p r o b a b i l i t y i s v e r y h i g h t h a t i t w i l l become i n c r e a s i n g l y s u b - o p t i m a l ( T e i t z , 1 9 6 8 ) . To g u a r a n t e e t h e t r u e g l o b a l m i n i m u m i t i s n e c e s s a r y t o r e s o r t t o d i s c r e t e d y n a m i c p r o g r a m m i n g m e t h o d s w h e r e t h e r e c u r s i o n p r o c e d u r e w i l l be b a c k w a r d s r a t h e r t h a n f o r w a r d s i n t i m e . The a d o p t i o n o f a b a c k w a r d s r e c u r s i o n d e p e n d s upon t h e p r i n c i p l e t h a t a d y n a m i c a l l y o p t i m a l s y s t e m s h o u l d a l w a y s , a t t h e f u l f i l l m e n t o f t h e l o n g t e r m p l a n n i n g h o r i z o n , h a v e a PBOBLEn S A* 2 .. FIGURE 12 FOR•A BD RECURSIVE DJSABIC SOLUTION FOR PICDBE 7 «ltt*tt*lttltl»tM4tl*tllttHllttlH4tMMItMttlttl(tlt«ltltMtM)ttf**MtlflltMM*»tttl»tMttttMfttttftl*ttf«ft* PLANT LOCATION-ALLOCATION 1 BEDIAN SOLUTION BED IAN PLANT SIZE FREIGHT COST PLANT COST TOTAL COST • ODES SERVICED VOLUBE SOPPLIED TO EACH BED IAS 1 9 50.00 181.00 0.00 184.00 1 2 3 4 '5 3.0 1.0 4.0 2.0 3.0 6 7 8 9 10 4.0 4.0 3.0 1.0 2.0 • 11 12 13 14 15 2.0 1.0 2.0 3.0 3.0 16 17 18 19 20 1.0 4.0 2.0 3.0 2.0 BET WORK 50.00 181.00 0.00 184.00 • • * « « 4 * a * * * 4 * * t * « * * « « t * *•««*«••.«••*•*»«•* *•««•«•*«»*•••»**•**•***««•**»•***• ••«»«* PLANT LOCATION-ALLOCATION 2 BEDIAN SOLUTIOB BEDIAN PLANT SIZE FREIGHT COST PLANT COST TOTAL COST BODES SERVICED VOLOBE SUPPLIED TO EACH BEDIAN 1 9 28.00 85.00 0.00 85.00 1 2 3 4 5 3.0 1.0 4.0 2.0 3.0 6 9 10 11 12 . 4.0 1.0 2.0 2.0 1.0 13 11 2.0 3.0 2 17 22.00 JO.00 0.00 30.00 7 8 15 16 17 4.0 3.0 3.0 1.0 4.0 18 19 20 2.0 3.0 2.0 . BETHORK 50.00 115.00 0.00 115.00 * * « * * * * « « # * « « 4 * * t « t * 4 « * « « 4 « * « « PLAST LOCATION-ALLOCATION 3 BEDIAN SOLUTION BEDIAN PLANT SIZE FREIGHT COST PLANT COST TOTAL COST 17 19 BETSORK 17.00 a 5.00 18.00 0.00 0.00 85.00 18.00 2.00 105.00 0.00 0.00 2.00 105.00 BODES SERVICED 1 2 3 4 5 6 9 10 11 12 13 14 7 8 15 16 17 18 19 20 VOLOBE SUPPLIED TO EACH BEDIAS 3.0 4.0 2.0 4.0 2.0 3.0 50.00 *•••*•«•«•*«*••***•*••**•*••«•••*•**•*•*•**•••••*•#••**•••»*•»•*•**.**.«*»•«•**•••» PLABT LOCATIOS-ALLOCATIOB 3 BEDIAS SOLOTIOS 1.0 1.0 3.0 3.0 2.0 0.0 2.0 2.0 2.0 3.0 1.0 BEDIAS PLABT SIZE FREIGHT COST PLABT COST TOTAL COST 1 9 28.00 85.00 -' 0.00 85.00 BODES SERVICED VOLOBE SUPPLIED TO EACH BEDIAB 3 4 10 11 2 17 3 19 • ETSORK 14.00 8.00 50.00 15.00 5.00 105.00 0.00 0.00 0.00 15.00 5.00 105.00 1 2 6 9 13 14 7 8 16 17 18 IS 19 20 3.0 4.0 2.0 4.0 3.0 1.0 1.0 3.0 3.0 3.0 4.0 2.0 1.0 2.0 2.0 2.0 3.0 1.0 4.0 2.0 *••«*»••***•*****«*•****** *>•«••••.«»*»•••»*«**•*•«**•*••••••*«»•«•**•••*•«•»«»#•«••*••*••*«»•*•« PLABT LOCATIOS-ALLOCATIOB 3 BEDIAB SOLUTIOB BEDIAB PLABT SIZE PREIGHT COST PLANT COST TOTAL COST 1 6 2 9 3 17 B ETBORK 17.00 11.00 22.00 50.00 20.00 31.00 30.00 81.00 0.00 . 0.00 0.00 0.00 20.00 31.00 30.00 81.00 •ODES SERVICED 2 3 4 1 6 9 14 7 8 18 19 10 11 12 13 15 16 17 20 VOLOBE SOPPLIED TO EACH BEDIAS 1.0 4.0 2.0 3.0 2.0 2.0 3.0 4.0 1.0 3.0 4.0 2.0 3.0 3.0 3.0 2.0 1.0 1.0 2.0 4.0 F i g u r e 12 PROBLEB SAKE FIGURE 13 FORWAItD R.ECUaSIV: DYSAJIIC SOLUTION (BODE 9 FORCED) FOB FIGURE 7 • . • 4 * 4 * * * 4 * * i a * 4 4 4 4 4 4 4 4 4 4 4 « 4 4 4 4 4 4 4 4 4 4 4 t « * * 4 4 4 4 * 4 4 « 4 4 * * * * * * * 4 * « 4 * * * a * * * * * 4 « « « * * * * * * * * * * « * 4 * * * * * 4 * * * * * * . 4 * * * * * t * t 4 * * * * * PIAST LOCATIOH-ALLOCATIOH 1 HEDIAS SOLUTION BEDIAS PLAHT SIZE FB EIGHT COST PLANT COST TOTAL COST BODES SERVICED VOLOBE SOPPLIED TO EACH BED IAN 1 8 50.00 180.00 0.00 180.00 1 2 3 . 0 5 3.0 1.0 0.0 2.0 3.0 « 7 8 9 10 0.0 0.0 3.0 1.0 2.0 11 12 13 10 15 2.0 1.0 2.0 3.0 3.0 16 17 18 19 20 1.0 0.0 2.0 3.0 2.0 IETVOBK 50.00 180.00 0.00 180.00 • • t * * * * * * * * * * 4 * * * » * * * * * * * * * * 4 4 4 « 4 * * * * * * « * 4 4 4 * * * « * « * * * * * * * * * * * * * * * * » * « « * * * * * » 4 * * * * * * * 4 * » * * « * * * » * * » * * * * * » * * * * * * * * * * * * * PLANT LOCATION-ALLOCATION 1 BEDIAS SOLUTIOH • BEDIAH PLAHT SIZE FREIGHT COST PLAHT COST TOTAL COST BODES SEBVICED VOLUBS SUFPLIED TO EACH BEDIAH 1 9 50.00 180.00 0.00 180.00 1 2 3 0 5 3.0 1.0 0.0 2.0 3.0 6 7 8 9 10 0.0 0.0 3.0 1.0 2.0 11 12 13 10 15 2.0 1.0 2.0 3.0 3.0 16 17 16 19 20 1.0 0.0 2.0 3.0 2.0 IETVOBK 50.00 180.00 0.00 160.00 PLANT LOCATION-ALLOCATION 2 BEDIAH SOLUTI03 BEDIAH PLANT SIZE FREIGHT COST PLAHT COST TOTAL COST BODES SERVICED VOLU r.E SUPPLIED 1 C EACH "EDI?.! 1 6 28.00 73.00 0.00 73.00 1 2 3 0 5 3.0 1.0 o.o 2.0 3.0 6 9 10 11 12 s.o 1.0 2.0 2.0 1.0 13 10 2.0 3.0 2 8 22.00 35.00 0.00 35.00 7 8 15 16 17 0.0 3.0 3.0 1.0 0.0 18 19 20 2.0 3.0 2.0 BETSOBK 50.00 108.00 0.00 108.00 I * * * * * * * * * * « 4 4 4 4 4 4 4 4 4 « 4 4 4 4 4 4 4 4 4 4 4 4 * 4 4 4 4 4 4 4 4 4 4 4 * 4 * * * * 4 * * * * * * * * * » * * * * « » * * » 4 » * 4 * * * * « * * • ***•*. 4 * 4 * 4 4 * * * * * 4 4 4 4 4 4 4 * * • « » • • 1 PLAHT LOCATIOH-ALLOCATIOS 2 BEDIAH SOLUTION BEDIAH PLAHT SIZE FB EIGHT COST PLAHT COST TOTAL COST BODES SEBVICED VOLOBE SUPPLIED TO EACH BEOIAS 1 6 25.00 55.00 0.00 55.00 1 2 3 0 5 3.0 1.0 0.0 2.0 6 9 10 11 12 0.0 1.0 2.0 2.0 13 2.0 2 8 25.00 53.00 0.00 53.00 7 8 10 15 16 0.0 3.0 3.0 3.0 17 18 19 20 0.0 2.0 3.0 2.0 • ETVORK 50.00 108.00 0.00 108.00 4 > « * « * » * * * * 4 4 4 4 4 4 4 * * « * * * * * * * 4 4 4 4 4 4 4 * * * 4 4 * 4 « * * * * * * * l >••***•*•*• ***•«•« **»• * * * * * * * * * * * * * * » * * 4 * «*»*•• *•«»**•*«**< PLAST LOCATIOS-ALLOCATIOH 3 B ED IAB SOLUTIOH BEDIAH PLAHT SIZE FREIGHT COST PLAST COST TOTAL COST BODES SEBVICED VOLUBE SOPPLIED TO EACH BED II 1 6 18.00 22.00 0.00 22.00 1 2 3 0 5 3.0 1.0 0.0 2.0 3.0 6 9 0.0 1.0 2 8 22.00 35.00 0.00 35.00 7 8 15 16 17 o.o 3.0 3.0 1.0 o.o 18 19 20 2.0 3.0 2.0 3 11 10.00 11.00 0.00 11.00 10 11 12 13 10 2.0 2.0 1.0 2.0 3.0 IETVOBK 50.00 68.00 0.00 68.00 • * « 4 « « * * 4 * * « 4 * « * * * * 4 4 * « t * * * 4 4 * * « * * * * * * * * * 4 * * * * * * * * * * * * » * * * * * * * 4 * * * * * » * * * * * * * * * * * * * * « * * * * * * * * « * * * c * * 4 » « * « * « < * « * * * * * « * PLAHT LOCATION-ALLOCATIOS 3 BEDIAH SOLOTIOS BEDIAH PLAHT SIZE FREIGHT COST PLAHT COST TOtAL COST BODES SERVICEC VOLOBE SUPPLIED TO EACH P.EDIAH 1 6 17.00 20.00 . 0.00 20.00 1 6 2 3 0 5 3.0 0.0 1.0 0.0 2.0 3.0 2 8 22.00 35.00 0.00" 35.00 7 18 8 19 15 20 16 17 0.0 2.0 3.0 3.0 3.0 2.0 .1.0 0.0 3 11 11.00 13.00 0.00 13.00 9 10 10 11 12 13 1.0 3.0 2.0 2.0 1.0 2.0 I ETtfORK SO.00 68.00 0.00 68.00 t * 4 4 * * 4 4 * * 4 4 4 4 4 4 * 4 4 * * * 4 t 4 * * 4 « i 4 4 l t » 4 t t l l l * 4 4 4 ( l l * 4 t 4 l * ) * 4 t t l * » * 4 4 4 4 t 4 4 * 4 4 * * t * * 4 4 4 4 4 t * * 4 4 4 4 * * * 4 4 f 4 * t 4 * 4 4 * 4 * * 4 4 * 4 * 4 * 4 » PLAHT LOCATIOH-ALLOCATIOH 3 BEDIAH SOLUTIOH BEDIAS PLAHT SIZE FREIGHT COST PLAHT COST TOTAL COST BODES SERVICED VOLUBE SOPPLIED TO EACH BEDIAH 1 6 18.00 22.00 0.00 22.00 1 2 3 0 5 3.0 1.0 0.0 2.0 3.0 6 9 B.O 1.0 2 8 22.00 35.00 0.00 35.00 7 8 IS 16 17 0.0 3.0 3.0 1.0 0.0 18 19 20 2.0 3.0 2.0 3 13 10. C> 11.00 0.00 1 1.00 10 11 12 13 10 2.0 2.0 1.0 2.0 3.0 •ETVORK 50.00 68.00 0.00 68.00 * « * * « * * * * * * « 4 * * * * * * « * * * * * * * * « « * * * * * * * * * * « * * 4 * * * * * * * * * * * « * « * * * 4 * * * * * * * « « * < * 4 4 * * * * * * * * * * « 4 * * * * * * * * * * * * * * * « * * * * * * * * * * * * F i g u r e 13 l o c a t i o n a l s t r u c t u r e w h i c h i s e q u i v a l e n t t o t h a t o f a s t a t i c s y s t e m w h e r e p = t t h e l o n g r a n g e g o a l . H e n c e a t t h e o u t s e t i t i s n e c e s s a r y t o m i n i m i z e t h e f o l l o w i n g f u n c t i o n . n r w h e r e M i s the s e t of a l l p o t e n t i a l p l a n t s i t e s . T h i s i s t h e s o l u t i o n f o r t h e s t a t i c c a s e when p = r , and i s t a n t a -m o u n t t o ( 1 ) w i t h o u t a l l t h e s i d e c o n s t r a i n t s . W h e t h e r t h i s s o l u t i o n i s a c h i e v e d by t h e h e u r i s t i c m e t h o d s g i v e n p r e v i o u s l y i n t h i s c h a p t e r o r by some e x a c t b r a n c h - b o u n d a l g o r i t h m , i t now r e m a i n s o n l y t o o b t a i n an o r d e r l y s e -q u e n c i n g o f t h e p l a n t s g i v e n by ( 2 0 ) s u c h t h a t t h e l o n g t e r m c o s t a c c u m u l a t i o n i s m i n i m i z e d . I t i s t h i s s e q u e n c i n g s o l u t i o n w h i c h i s n o r m a l l y s o u g h t by d y n a m i c p r o g r a m m i n g , and t h e a p p r o a c h w i l l u s u a l l y be f e a s i b l e up t o a l i m i t o f a b o u t n i n e o r t e n t i m e p e r i o d s d e p e n d i n g upon t h e s i z e o f t h e n e t w o r k . B e y o n d t h i s , as w i l l become a p p a r e n t , d y n a m i c p r o g r a m m i n g t e n d s t o w a r d s i n f e a s i b i 1 i t y s i n c e c o m p u t a t i o n i n c r e a s e s v e r y r a p i d l y as a n o n - l i n e a r f u n c t i o n o f r . T h u s a t a n y s t a g e t b a c k w a r d s f r o m t h e s o l u t i o n o f ( 2 0 ) t h e r e a r e ( t ) o r r ! / t ! ( r - t ) ! c o m b i n a t i o n s o f p o s s i b l e l o c a t i o n s . O v e r a l l t i m e p e r i o d s t . , j = l , - - - , r , t h s r e a r e t h e n J ( 2 1 ) I t = l C l e a r l y t h e a m o u n t o f c a l c u l a t i o n i n v o l v e d w i l l be a f u n c -t i o n o f b o t h t h e s i z e o f t h e n e t w o r k ( s i n c e e a c h node m u s t be a s s i g n e d t o one o f t h e t l o c a t i o n s ) , and o f t h e s i z e o f r w h i c h i s t h e maximum number o f p e r i o d s i n v o l v e d . I f r = 10 f o r e x a m p l e , then by ( 2 1 ) t h e number o f l o c a t i o n a l c o m b i n a t i o n s w h i c h m u s t - be e v a l u a t e d i s i n e x c e s s o f 1 0 0 0 . M o r e o v e r o n c e t h e s e s o l u t i o n s h a v e b e e n o b t a i n e d t h e r e s t i l l r e m a i n some r ! p o s s i b l e ways t o o r d e r t h e m , one o f w h i c h w i l l be t h e o p t i m a l s e q u e n c e r e q u i r e d . F o r r = 10 t h e number o f s u c h p o s s i b l e s e q u e n c e s o r p e r m u t a t i o n s i s o v e r one m i l l i o n . A l t h o u g h d y n a m i c p r o g r a m m i n g w i l l a c t = u a l l y e v a l u t e f o r f e w e r s e q u e n c e s ; by v i r t u e o f t h e c o m p u -t a t i o n a l e f f o r t i m p l i e d i n ( 2 1 ) , r n e e d n e v e r be v e r y l a r g e b e f o r e t h e p r o b l e m i n p r a c t i c e b e c o m e s h o p e l e s s l y o v e r -e x t e n d e d . The p r e s e n t a t i o n o f a d y n a m i c p r o g r a m f o r t h e o p t i m a l s e q u e n c i n g o f p l a n t l o c a t i o n s o v e r t i m e w i l l be d e f e r r e d t o t h e n e x t s e c t i o n . The r e m a i n d e r o f t h i s one w i l l be d e v o t e d t o e x p l o r i n g h e u r i s t i c s f o r t h e v e r y l o n g t e r m p l a n n i n g h o r i z o n p r o b l e m when r i s i n e x c e s s o f s a y n i n e o r t e n . The g e n e r a l p r o c e d u r e i s a c t u a l l y s i m p l e and w i l l ' be t o r e c u r s e b a c k w a r d s t a k i n g a d v a n t a g e o f t h e s o l u t i o n o f e a c h p r e c e e d i n g s t e p . B e g i n n i n g w i t h t h e s o l u t i o n t o such solut ions to be evaluated ( 2 0 ) d e f i n e R as t h e i n i t i a l s u b s e t o f t = r m e d i a n s so o b t a i n e d . T h e n , w o r k i n g b a c k w a r d s , and u s i n g t h e h e u r i s t i p r e v i o u s l y g i v e n f o r t h e e s t i m a t i o n o f t h e p m e d i a n o f a w e i g h t e d g r a p h s o l v e f o r a n y s u b s e q u e n t t i m e p e r i o d t . , t <= r - 1 , r - 2 , - - - , l , t h e f o l l o w i n g e x p r e s s i o n n t ( 2 2 ) = y y . x..D. z i= i j = i J £ R 1 J xs T h a t i s t h e s o l u t i o n i s c o n s t r a i n e d t o a s e a r c h among t h e members ( m e d i a n s ) o f t h e s e t R w h i c h , a t a n y s t a g e , c o n t a i n s t+1 m e d i a n s . A l t h o u g h i t i s a l w a y s p o s s i b l e t o f i n d c o u n t e r -e x a m p l e s , t h i s m e t h o d w i l l u s u a l l y g i v e s a t i s f a c t o r y r e -s u l t s . R e f e r r i n g b a c k ' t o t h e n e t w o r k o f F i g u r e 7 , t h e r e s u l t s o f t h e b a c k w a r d r e c u r s i o n c a n be s e e n i n F i g u r e 1 4 - A t c o m p l e t i o n o f s t a g e 1 , t=3 and t h e m i n i m u m c o s t f o r t h r e e m e d i a n s o l u t i o n o f Vg, v-j -| and v - ^ i s Sg = 6 3 - 0 D u r i n g p e r i o d s t 2 and t g m e d i a n s v^-j and v-j 7 - a r e ' d r o p p e d l e a v i n g Vg w i t h = 6 0 - 0 . The c u m u l a t e d c o s t s f o r a l l t h r e e p e r i o d s a r e 3 ( 2 3 ) S = I S. = 3 5 0 - 0 j=l w h i c h i s l e s s t h a n e i t h e r f o r w a r d r e c u r s i v e m e t h o d . o f t h e two r e s u l t s g i v e n by t h e PR08LEB BABE FIGURE 14 BACKWARD RECURSIVE DYKA3IC SOLUTIOB FOR FIGURE 7 4 4 * * 4 * * 4 4 * * t * * « * * 4 * • « * * * * * * * * 4 4 4 * * * * * * * 4 4 4 4 4 4 4 4 * 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 J PLAST LOCATIOS-ALLOCATIOB 3 BEDIAS SOLUTIOB BEDIAS PLABT SIZE FREIGHT COST PLAST COST TOTAL COST BODES SERVICED 4 4 * 4 4 4 4 4 4 4 4 4 4 4 * * * 4 « * * * * * * * » * * 4 VOLOBE SUPPLIED TO EACH BEDIAS 1 « 18.00 22.00 0.00 22.00 1 2 3 . 4 5 3.0 1.0 4.0 2.0 3.0 6 9 4.0 1.0 2 11 10.00 11.00 0.00 11.00 10 11 12 13 14 2.0 2.0 1.0 2.0 3.0 3 17 22.00 30.00 0.00 30.00 7 8 15 IS 17 4.0 3.0 3.0 1.0 4.0 18 19 20 2.0 3.0 2.0 lETt-OBIt 50.00 6 3.00 0.00 63.00 • • 4 4 4 4 4 4 4 4 * * 4 4 4 * 4 * * « * * t « t l l l > 1 M t > t U 4 t t ) t M M t t M t * * * « * * * * 4 4 4 4 * * * « 4 4 * 4 4 * * 4 4 4 4 « 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 '*?*** * * 4 * * * * * * * * * * PLABT LOCATIOS-ALLOCATIOB 3 BEDIAS SOLUTIOB BEDIAB PLANT SIZE FREIGHT COST PLAST COST .TOTAL COST BODES SERVICED VOLOBE SUPPLIED TO EACH BEDIAB 1 .« 17.00 20.00 0.00 20.00 1 2 3 . 4 5 3.0 1.0 4.0 2.0 3.0 6 4.0 2 11 11.00 13.00 0.00 13.00 9 10 11 12 13 1.0 2.0 2.0 1.0 2.0 11 3.0 3 17 22.00 30.00 0.00 . 30.00 7 8 15 16 17 4.0 3.0 3.0 1.0 4.0 18 19 20 2.0 3.0 2.0 •ETUOSK 50.00 63.00 0.00 63.00 • 4 4 4 4 4 4 4 4 4 4 4 * . * * * * * * » * 4 « 4 * * * * * * * * * « ( >«*•*•*•«•••* • * « » * * « * * 4 4 * * * » »* * • • * 4 4 » • 4 4 4 4 4 4 4 4 4 4 * 4 * 4 4 4 4 4 4 4 4 * 4 4 * 4 * * * 4 4 4 ( 4 4 * * 4 4 4 ( 4 4 PLABT LOCATIOS-ALLOCATION 3 BEDIAS SOLUTION BEDIAS PLABT SIZE FREIGHT COST PLAST COST TOTAL COST BODES SERVICED VOLOBE SOPPLI ED TO EACH BEDIAS 1 6 18.00 22.00 0.00 22.00 1 2 3 4 5 3.0 1.0 4.0 2.0 3.0 6 9 4.0 1.0 2 13 10.00 1 1.00 0.00 11.00 10 1 1 12 13 14 2.0 2.0 1.0 2.0 3.0 3 17 22.00 30.00 0.00 30.00 7 8 15 16 17 4.0 3.0 3.0 1.0 4.0 18 19 20 2.0 3.0 2.0 BETBOSK 50.00 63.00 0.00 63.00 • * 4 4 4 4 * 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 * 4 4 *••*••**«•* 4 * 4 « * * * * * * * 4 4 4 4 * * 4 4 4 4 4 4 4 4 4 4 4 4 4 4 * 4 4 4 * * 4 » 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 * 4 * « 4 * « 4 * 4 4 4 * 4 * PLABT LOCATIOS-ALLOCATIOB 2 BEDIAB SOLUTION BEDIAS PLAST SIZE FREIGHT COST PLAST COST TOTAL COST BOOES SERVICED VOLOBE SUPPLIED TO EACH BEDIAB 1 6 25.00 55.00 0.00 55.00 1 2 3 4 5 3.0 1.0 4.0 2.0 3.0 6 9 10 11 12 4.0 1.0 2.0 2.0 1.0 13 2.0 2 17 25.00 45.00 0.00 45.00 7 8 14 15 16 4.0 3.0 3.0 3.0 1.0 17 18 19 20 4.0 2.0 3.0 2.0 •ETWOBK 50.00 100.00 0.00 100.00 ((**(***(**( • 4 * 4 4 4 4 4 * 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ( 4 4 * 4 4 4 4 * 4 4 1 1 ( 4 ( 4 4 4 4 4 * 4 * * * * 4 4 4 4 4 4 4 4 4 4 4 * 4 * 4 * * * * * 4 4 4 4 4 * 4 4 ' ( 4 4 4 4 4 4 4 * * » * * * 4 t * • *••*' PLAST LOCATIOB-ALLOCATIOS BEDIAS PLANT SIZE FREIGHT COST PLAST COST TOTAL COST 1 6 50.00 137.00 0.00 187.00 1 BEDIAB SOLUTIOB BODES SERVICED VOLUBE SOPPLIED TO EACH BEDIAS 1 2 3 4 5 3.0 1.0 4.0 2.0 3. .0 6 7 8 9 10 4.0 4.0 3.0. 1.0 2. .0 11 12 13 14 15 2.0 1..0 2.0 3.0 3. .0 16 17 18 19 20 1.0 4.0 2.0 3.0 2. .0 BETH08K 50.00 187.00 0.00 187.00 F i g u r e 14 In some i n s t a n c e s i t may be p o s s i b l e t o c o m b i n e t h e r e s u l t s o f b o t h t h e f o r w a r d and b a c k w a r d r e c u r s i v e s o l u -t i o n s . L e t t . ( i f i t e x i s t s f o r a p a r t i c u l a r c a s e ) be some t i m e p e r i o d , 0 > t . - r , i n w h i c h t h e s o l u t i o n s o f b o t h m e t h o d s a r e i d e n t i c a l . Then t h e s e q u e n c i n g o f t h e f o r w a r d r e c u r s i v e m e t h o d f r o m p e r i o d t-, t o t . m u s t a l w a y s be a t l e a s t a s g o o d as t h a t g i v e n by t h e b a c k w a r d r e c u r s i v e m e t h o d o v e r t h e same t i m e i n t e r v a l s . A l t e r n a t i v e l y , f o r a n y t i m e p e r i o d t b a c k down t o t-j , t h e o p t i m a l s e q u e n c i n g c a n a l w a y s be e x p l i c i t l y s o l v e d by t h e d y n a m i c p r o g r a m m i n g m e t h o d s o f t h e f o l l o w i n g s e c t i o n p r o v i d i n g r i s n o t t o o l a r g e . A D y n a m i c P r o g r a m f o r t h e O p t i m a l S e q u e n c i n g o f P l a n t  L o c a t i o n s O v e r T i m e The a p p l i c a t i o n o f d y n a m i c p r o g r a m m i n g t o p l a n t l o c a t i o n t h r o u g h t i m e h a s r e c e i v e d s u r p r i s i n g l y l i t t l e a t t e n -t i o n i n t h e p a s t . B e l l m a n ( 1 9 6 5 ) has d e m o n s t r a t e d how t h i s a p p r o a c h may be a p p l i e d t o t h e c a s e o f s t a t i c s y s t e m s b u t , t o t h e a u t h o r ' s k n o w l e d g e , o n l y S c o t t ( 1 9 6 9 ) h a s e x p l i c i t l y c o n s i d e r e d t h e p r o b l e m i n i t s t e m p o r a l c o n t e x t . S c o t t has p r o v i d e d a c a r e f u l a n a l y s i s o f t h e p o t e n t i a l a d v a n t a g e s o f d y n a m i c p r o g r a m m i n g o v e r f o r w a r d r e c u r s i v e p r o c e d u r e s , b u t s u g g e s t s t h a t i t i s n e c e s s a r y t o e v a l u a t e a l l r ! p o s s i b l e s e q u e n c e s t o f i n d t h e g l o b a l m i n i m u m , a f t e r s o l v i n g f o r t h e s o l u t i o n s i m p l i c i t i n e q u a t i o n ( 2 0 ) and ( 2 1 ) . A c t u a l l y , as will be d e m o n s t r a t e d , f a r f e w e r e v a l u a t i o n s a r e n e e d e d . I n d e e d i f r = 1 0 , o n l y a f e w t h o u s a n d c a l c u l a t i o n s a r e r e q u i r e d t o s o l v e t h e m i n i m u m c o s t s e q u e n c e c o m p a r e d t o s e v e r a l m i l l i o n i f a l l p o s s i b l e o r d e r i n g s a r e e x a m i n e d . As i t h a p p e n s t h e c o m p u t a t i o n a l d i f f i c u l t i e s a s s o c i a t e d w i t h a p p l y i n g d y n a m i c p r o g r a m m i n g t o p l a n t l o c a t i o n o v e r l o n g p e r i o d s o f t i m e l i e l e s s i n t h e s e q u e n c i n g p r o b l e m t h a n i n t h e s e e m i n g l y u n -a v o i d a b l e n e e d ^ t o e v a l u a t e a m i n i m u m c o s t a l l o c a t i o n f o r a l l n e t w o r k n o d e s w i t h r e s p e c t t o a l l ( t ) s e t s o f p o s s i b l e l o - • c a t i o n s a t e a c h p e r i o d t i n t h e a n a l y s i s . The f u n d a m e n t a l r e c u r s i v e r e l a t i o n s h i p o f d i s c r e t e d y n a m i c p r o g r a m m i n g i s due o r i g i n a l l y t o B e l l m a n ( 1 9 5 7 ) . Im p r o v i s i n g on a g e n e r a l f o r m a t g i v e n by W a g n e r , (1 9 6 9 ) t h i s r e l a t i o n s h i p may be e x p l i c i t l y e x p r e s s e d i n t e r m s o f t h e p l a n t l o c a t i o n p r o b l e m as f o l l o w s : (24) f k ( s ) = min f ^ ( s ) i s a m i n i m u m c o s t ( a g g r e g a t e d i s t a n c e ) p o l i c y d e c i s i o n t a k e n i n s t a t e s w i t h k more p l a n t s t o l o c a t e . S g i s t h e c o s t o f s e r v i n g t h e n e t w o r k i n s t a t e s w i t h k l o c a t i o n s i n s o l u t i o n and f| c _-|( j ) i s t h e o p t i m a l a g g r e g a t e l o c a t i o n c o s t s b a c k o u t o f s t a t e j w i t h k - 1 p l a n t s t o l o c a t e i . for a l l . k=l ,2 , w h e r e ( a ) ( b ) ( c ) o p t i m a l f r o m s t a t e j b a c k t h r o u g h a l l a n t e c e d e n t p l a n t s e q u e n c i n g d e c i s i o n s i n a l l p r e v i o u s s t a g e s t o t h e commencement o f s t a g e 1 w h e r e r l o c a t i o n s a r e i n s o l u t i o n . S i n c e t h e a p p l i c a t i o n o f t h i s r e c u r s i o n i s b a c k w a r d s i n t i m e a n y s t a t e j w i l l c o n t a i n one more l o c a t i o n t h a n a n y e n t e r i n g s t a t e s , s o t h a t a t t h e b e g i n n i n g o f s t a g e k=l ( t i m e p e r i o d t = r ) a l l p r e - d e t e r m i n e d r l o c a t i o n s a r e p r e s e n t , and a t t h e t e r m i n a t i o n o f s t a g e k=r ( t i m e p e r i o d t = l ) e a c h s t a t e j w i l l c o n t a i n o n l y one l o c a t i o n . A t r r-k new s t a t e s e v e r y s t a g e K i n t h e d y n a m i c p r o c e s s e n t e r i n t o t h e s o l u t i o n , e a c h s u c h s t a t e s b e i n g a u n i q u e c o m b i n a t i o n o f t h e o r i g i n a l r l o c a t i o n s w i t h r - k t e r m s . F o r e a c h one t h e d e c i s i o n f ^ ( s ) t a k e n w i t h r e s p e c t t o some p a r t i c u l a r s t a t e j w i t h r - k + 1 p l a n t s ( o f w h i c h t h e l o c a -t i o n s b e l o n g i n g t o s t a t e s m u s t be a s u b s e t ) i s s u c h t h a t t h e a c t o f t r a n s f e r r i n g t o o r r o u t i n g t h r o u g h s t a t e j c o n s t i t u t e s a l e a s t c o s t o p t i m a l p l a n t s e q u e n c e f o r s t a t e j back , t o s t a g e o n e . S i n c e t h e f u n c t i o n f^ - . ( j ) e m b o d i e s t h e m i n i m u m c o s t p l a n t s e q u e n c e f o r some s t a t e j b a c k t t h r o u g h a l l p r e c e e d i n g s t a g e s t o k = l , t h e d e c i s i o n s o u g h t f o r s t a t e s a t s t a g e k i s s i m p l y t h e m i n i m u m o f S + f v n ( j ) o v e r a l l j f o r w h i c h s t a t e s i s a s u b s e t . T h i s e x p l a n a t i o n i s a p t l y s u m m a r i z e d by t h e p r i n c i p l e o f o p t i m a l i t y w h i c h s t a t e s t h a t : A n o p t i m a l p o l i c y m u s t h a v e t h e p r o p e r t y t h a t r e g a r d l e s s o f t h e r o u t e t a k e n to e n t e r a p a r -t i c u l a r s t a t e t t h e r e m a i n i n g d e c i s i o n m u s t c o n s t i t u t e a n o p t i m a l p o l i c y f o r l e a v i n g t h a t s t a t e , ( W a g n e r , 1 9 6 9 , p . 2 5 7 ) The c h o i c e o f some s t a t e j f o r a n y s t a t e s c o n s t i t u t e s t h e d e c i s i o n by w h i c h s t a t e s i s l e f t . T h e s e c o n c e p t s c a n be c l a r i f i e d w i t h r e f e r e n c e t o t h e h y p o t h e t i c a l e x a m p l e o f f o u r s t a g e s shown i n T a b l e 5 . A s s u m e f o r some n e t w o r k a f o u r l o c a t i o n s o l u t i o n g i v e n by ( 2 0 ) w i t h n o d e s 1 , 2 , 3 and 4 as t h e m e d i a n s o f t h e g r a p h and S ^ , t h e a g g r e g a t e c o s t o f s u p p l y i n g e a c h d e s t i n a t i o n f r o m i t s c l o s e s t m e d i a n , as 1 0 . The o b j e c t i v e i s t o o r d e r t h e s e q u e n c i n g o f t h e p l a n t s o v e r t i m e s u c h t h a t t h e c u m u -l a t i v e l o n g t e r m c o s t f u n c t i o n o v e r a l l p e r i o d s i s m i n i -m i z e d . In T a b l e 5 t h e r e a r e f o u r m a t r i c e s e a c h c o r r e s p o n d -i n g t o one s t a g e i n t h e a n a l y s i s w i t h t h e c o l u m n s r e f e r -e n c i n g some s t a t e j and e a c h row some e n t e r i n g s t a t e s . F o r a n y s t a g e k t h e number o f l o c a t i o n s i n s t a t e s w i l l i n d i c a t e t h e t i m e p e r i o d r - k , w h i l e t h o s e f o r s t a t e j h a v e r - k + 1 t e r m s , e a c h j b e i n g t h e s i g n p o s t t o t h e s= j e n t e r i n g s t a t e o f p e r i o d t+1 . The number o f e n t e r i n g s t a t e s a t any s t e p w i l l be a l l t h e r - k c o m b i n a t i o n s o f r , i . e r r-k w h i l e f o r any s t a g e k+1 t h e j s t a t e s w i l l be t h e e n t e r i n g s t a t e s o f t h e k s t a g e . A t t h e commencement o f s t a g e 1 , s t a t e j i s t h e m e d i a n s e r i e s 1 , 2 , 3 and 4 , and t h e e n t e r i n g s t a t e s a r e 1 23 , 1 2 4 , 1 3 4 , a n d 2 3 4 , t h e c o m b i n a t o r i a l . " s e r i e s - down t h e rows o f T a b l e 5 ( a ) . L e t t h e m i n i m u m c o s t S g o f s u p p l y i n g t h e n e t w o r k f o r e a c h one o f t h e s e c o m b i n a t i o n s o r l o c a t i o n s b e S 1 2 3 = S 1 2 4 = 1 8 ' S 1 3 4 = ^ 5 a n d S 2 3 4 = ^ r e s p e c -t i v e l y . The c o s t e n t r y i n t h e f i r s t c e l l ( 1 6 - K 1 0 ) r e p r e -s e n t s t h e c u m u l a t e d c o s t s o f s e r v i n g t h e g r a p h w i t h m e d i a n s 123 i n t i m e p e r i o d 3 and 1 2 3 4 i n t h e f o u r t h o r f i n a l p e r i o d . The i n s e t box on t h e l e f t c o n t a i n s t h e i n t e g e r 4 , w h i c h ": l a b e l s t h e m e d i a n w h i c h w i l l be a d d e d t o t h e e n t e r i n g s t a t e 123 on i t s e x i t f r o m t h i s s t a g e s h o u l d i t p r o v e s u b s e q u e n t l y t o be p a r t o f t h e o p t i m a l s e q u e n c i n g r o u t e . The i n s e t box on t h e r i g h t o f e a c h c e l l c o n t a i n s a c o u n t e r w h i c h r e f e r -e n c e s t h e row i n t h e p r e c e e d i n g s t a g e m a t r i x w h e r e i n c a n be f o u n d t h a t member o f t h e l e f t i n s e t box w h i c h i s t o be a d d e d i n t h e n e x t t i m e p e r i o d t + 1 . The o p t i m a l d e c i s i o n p o l i c y f o r a n y e n t e r i n g s t a t e w i t h r e s p e c t t o i t s f e a s i b l e j e x i t s i s s t o r e d i n t h e d e c i s i o n c e l l , t h e p e n u l t i m a t e r i g h t m o s t c o l u m n o f e a c h m a t r i x . S i n c e f o r s t a g e 1 t h e r e i s o n l y one e x i t s t a t e j no c h o i c e i s i n v o l v e d i n t h e c o n -t e n t s o f t h i s d e c i s i o n c e l l . F o r s t a g e k=2 t h e e n t e r i n g s t a t e s ( t h e r e a r e o f them) r e p r e s e n t a l l t h e p o s s i b l e l o c a t i o n a l c o m b i n a t i o n s o f t i m e p e r i o d 2 , and t h e l e a v i n g s t a t e s i n d i c a t e t h e r o u t e s t o a l l t h e c o m b i n a t i o n s o f e n t e r i n g s t a t e s i n p e r i o d t=3 ( k = l ) . C o n s i d e r t h e v a l u e o f S ^ = 2 5 , t h e m i n i m u m c o s t o f s u p p l y i n g a l ' l n e t w o r k d e s t i n a t i o n s f r o m m e d i a n s l a n d 2 . F o r t h i s e n t e r i n g s t a t e e i t h e r m e d i a n 3 o r 4 w i l l be a d d e d a t t h e r-k k T a b l e 5 OPTIMAL SEQUENCING OF PLANT LOCATIONS FOR A HYPOTHETICAL EXAMPLE (a) k=l t=3 to 4 E x i t S t a t e j 1 2 3 4 D e c i s i o n C o s t 1 23 E n t e r i n g 124 S t a t e .s 1 34 234 4 16 + 10 1 4 26 1 16 3 18 + 10 1 3 28 1 18 2 1 5 + 10 1 2 25 1 1 5 1 19 + 10 1 1 29 1 19 (b) k=2 t=2 to 3 E x i t S t a t e j E n t . S t . s 123 1 24 134 234 Dec C o s t 1 2 3 1 4 2 3 1 25 + 26 25+28 X X 51 25 13 2 27+26 1 X 4 27 + 25 3 X 4 52 3 27 14 2 2 3 3 3 3 X 28+28 28+25 X 53 28 23 I J 24 + 26 1 X X 4 24+29 4 1 50 1 24 1 2 3 4 1 2 24 X 26 + 28 X 26 + 29 54 26 34 1 3 2 4 1 3 X X 27 + 25 27 + 29 52 27 1 2 1 3 14 23 24 34 Dec C o s t 2 1 3 2 4 3 1 2 1 34+51 34+52 34+53 X X X 85 34 1 1 3 4 4 5 1 3 4 33 + 51 X X 33+50 33 + 54 X 1 8 3 33 X 1 35+52 2 X 2 35+50 4 X 4 35 + 52 6 «S 2 85 4 35 1 3 2 5 3 6 % tk 3 6 X X . 32+53 X 32+54 32+52 P 84 32 (d) k=4 t = 0 to- 1 1 2 3 4 D e c i s i o n C o s t E n t e r i n g 1 1 2 2 3 3 4 2 2 S t a t e s 0 . 0 + 85 0 + 83 0 + 85 0 + 84 i 83 0 T a b l e 5 c o n t 1 d . t e r m i n a t i o n o f t h e s t a g e . I f t h e c h o i c e i s v i a s t a t e j , , c o n t a i n i n g p l a n t s 1 , 2 and 3 , t h e n f r o m t h e p r e v i o u s s t a g e k - 1 = 1 , i t i s known t h a t t h e c o m b i n e d c o s t s o f s o l u t i o n s 123 i n t 3 and 1 2 3 4 i n t ^ a r e 26 as i n d i c a t e d by t h e d e c i s i o n c e l l i n row 1 o f s t a g e 1 . S i m i l a r l y i f t h e r o u t i n g i s t h r o u g h s t a t e j 2 = 1 2 4 , t h e c o s t w i t h r e s p e c t t o t h i s s t a t e i s 2 8 . T h e r e f o r e , i f l o c a t i o n s 1 and 2 p r o v e l a t e r t o be i n t h e o p t i m a l s e q u e n c e i n t i m e p e r i o d 2 , t h e l e a s t c o s t d e c i s i o n , f 2 s , w i l l be t o a d d m e d i a n 3 r a t h e r t h a n 4 a t t h e t e r m i n a t i o n o f t h i s s t a g e , s i n c e 25 + 26 < 25 + 2 8 , g o v e r n s t h e m i n i m u m o r d e r f o r t h i s s t a t e s b a c k t o t = 4 ( k = l ) . The d e c i s i o n c e l l c o n t a i n s t h e v a l u e o f 5 1 , t h e c u m u l a t i v e c o s t s o f t h e t h r e e p e r i o d s o l u t i o n s o f 1 , 2 ; 1 , 2 and 3 ; and 1 , 2 , 3 and 4 . The l e f t i n s e t b o x c o n -t a i n s a 3 , t h e r e q u i s i t e b r a n c h i n g d e c i s i o n , and t h e r i g h t i n s e t box t h e v a l u e o f 1 w h i c h i s t h e c o l u m n i n d e x o f s t a t e j = 1 2 3 . N o t e t h a t s i n c e t h e j s t a t e s 134 and 234 do n o t i n c l u d e b o t h m e d i a n s 1 and 2 as s u b s e t s , t h e y a r e n o t f e a s i b l e e x i t p a t h s f o r s t a t e s = l , 2 . A t s t a g e k = 4 no s t a t e s a c t u a l l y e.n.ter i n t o t h e s o l u t i o n s i n c e t h e t i m e p e r i o d r e f e r e n c e i s r - k = 0 . B u t f r o m t h i s s t a t e t h e r e a r e f o u r p o s s i b l e e x i t s , m e d i a n s 1 , 2 , 3 o r 4 . w h i c h may be e n t e r e d a t commencement o f t h e n e x t t i m e p e r i o d t = l ( k = 3 ) . I t i s w o r t h e m p h a s i z i n g t h a t a t a n y s t e p i n t h e d y n a m i c p r o c e s s t h e o n l y i n f o r m a t i o n r e q u i r e d f r o m t h e p r e v i o u s s t a g e i s t h e v a l u e s c o n t a i n e d i n t h e d e c i s o n c e l l s o f t h a t m a t r i x . T h i s i l l u s t r a t e s t h e k e y p r i n c i p l e o f d y n a m i c p r o g r a m m i n g t h a t f o r any s t a t e s t h e l e a s t c o s t p a t h w i t h k s t a g e s r e m a i n i n g i s d e p e n d e n t o n l y upon t h e r e s u l t o f t h e i m m e d i a t e d e c i s i o n , and some a p p r o p r i a t e o p t i m a l p o l i c y e s t a b l i s h e d w i t h k - 1 s t a g e s l e f t . Upon c o m p l e t i o n o f a l l s t a g e s t h e v a l u e o f t h e m i n i m u m c o s t s e q u e n c i n g , g i v e n i n t h e s i n g l e d e c i s i o n e n t r y o f T a b l e 5 ( d ) , i s 8 3 . To t r a c e b a c k t h e a c t u a l s e q u e n c e w h i c h r e f l e c t s t h i s o p t i m u m t h r o u g h a l l f o u r t i m e p e r i o d s i s now an e a s y m a t t e r . The l e f t i n s e t box o f t h e d e c i s i o n c e l l o f 5 ( d ) i n d i c a t e s t h a t m e d i a n 2 s h o u l d be t h e f i r s t t o be l o c a t e d i n p e r i o d t^ . The r i g h t i n s e t box c o n t a i n s t h e i n t e g e r 2 and g i v e s t h e a p p r o p r i a t e d e c i s i o n ce l l t o r e f e r e n c e i n s t a g e 3 . E n t r a n c e t o t h i s c e l l i n row 2 o f 5 ( c ) l e a d s t o t h e s e l e c t i o n o f m e d i a n number 3 as t h e n e x t p l a n t t o be s i t e d . C o n t i n u i n g b a c k i n t h i s m a n n e r r e s u l t s i n t h e a d d i t i o n o f m e d i a n 1 a t t h e c o m m e n c e -ment o f t = 3 , and f i n a l l y s o u r c e number 4 c o m p l e t e s t h e o r d e r i n g i n t h e l a s t t i m e p e r i o d . The o p t i m a l s e q u e n c e t h e r e f o r e l o c a t e s p l a n t s 2 , 3 , 1 and 4 i n t h a t o r d e r . The c o s t o f t h e a l l o c a t i o n s i n e a c h t i m e p e r i o d a r e s t o r e d i n t h o s e c e l l s o f t h e c o s t c o l u m n w h i c h a r e a p p r o p r i a t e f o r t h e e n t e r i n g s t a t e s , and f o r t h i s e x a m p l e a r e 33 + 24 + 16 + 10 = 8 3 . A c o m p u t a t i o n a l s h o r t c u t c a n now be i n t r o d u c e d f o r f i n d i n g a l l t h e f e a s i b l e p a t h s o f t h e m a t r i x f o r s o m e s t a g e k. Remember t h a t a f e a s i b l e p a t h e x i s t s o n l y i f s t a t e s i s a s u b s e t o f s t a t e j . The number o f s u c h p a t h s f o r a n y k i s g i v e n by r - k + 1 r r - k r - k + 1 w h i c h c o m p a r e s t o f *\ r f \ r r - k + 1 X r - k t o t a l e l e m e n t s i n any m a t r i x . The number o f a c t i v e c e l l s down e a c h c o l u m n i s 7 - k + r r - k and a c r o s s e a c h row i s k . In t h e e x a m p l e o f T a b l e 5 f e a s i b l e p a t h s a c c o u n t f o r h a l f o r more o f a l l c e l l s f o r a n y s t a g e i n t h e c y c l e . F o r t h e l a r g e r p r o b l e m h o w e v e r , t h i s r a t i o may be v e r y much l e s s . I f r = 10 t h e n t h e t o t a l number o f f e a s i b l e r o u t e s ( a b o u t 4 0 0 0 ) o v e r a l l r t i m e p e r i o d s i s l e s s t h a n one f o u r t i e t h o f a l l p o s s i b l e p a t h s . B e c a u s e k w i l l be a t l e a s t e q u a l t o ( a n d may be v e r y much l e s s ) t h a n j a t any s t a g e , i t i s c o m p u t a t i o n a l l y i n e f f i c i e n t t o e v a l u a t e by d i r e c t c o m p a r i s o n e a c h member o f some e n t e r i n g s t a t e ' s a g a i n s t e v e r y member o f e a c h e x i t s t a t e j t o t e s t f o r f e a s -i b i l i t y . E v e n i f t h e e x e c u t i o n f o r a n y row c a n be t e r m i n a t e d upon c o u n t i n g k a c t i v e c e l l s , more t h a n 50 p e r ' c e n t o f a l l i n f e a s i b l e p a t h s w i l l s t i l l n e e d be e x a m i n e d d u r i n g t h e a n a l y s i s . S i n c e b e t w e e n h a l f a d o z e n t o more t h a n one h u n d r e d o p e r a t i o n s ( d e p e n d i n g upon t h e s i z e s o f r and k) may be n e c e s s a r y f o r t h e c o m p l e t e e v a l u a t i o n o f e a c h c e l l , s i g n i f i c a n t c o m p u t a t i o n a l s a v i n g s w i l l be r e a l i s e d i f a l l f e a s i b l e p a t h s c a n be d i r e c t l y o b t a i n e d . T h i s t a s k , i n w h i c h t h e d i r e c t i o n o f t h e a n a l y s i s w i l l be down c o l u m n s i n s t e a d o f a c r o s s r o w s , c a n be a c c o m p l i s h e d as f o l l o w s . C o n s i d e r s t a t e j = 1 2 3 i n s t a g e k=2 o f T a b l e 5 ( b ) . T h e r e a r e t h r e e d e s c e n d e n t s u b s e t s o f r - k t e r m s e a c h , 1 2 , 13 and 23 r e s p e c t i v e l y . T h e s e c o m b i n a t i o n s o c c u p y rows 1 , 2 , and 4 o f t h e s i x c e l l s i n c o l u m n 1 , e a c h one o f w h i c h r e f e r e n c e s a u n i q u e d e s c e n d a n t o f a l l t h e ( ^) p o s s i b l e c o m b i n a t i o n s o f e n t e r i n g s t a t e s f o r t h a t s t a g e . F u r t h e r -more e v e r y one o f t h e d e s c e n d a n t s c a n be d e t e r m i n e d i n s t r i c t L e x i c o g r a p h i c a l o r d e r . I f e a c h one o f t h e t h r e e s u b s e t s o f j = 1 2 3 c a n be g i v e n a v a l u e w h i c h c o r r e s p o n d s e x a c t l y t o t h e r a n k o f t h a t c o m b i n a t i o n i n t e r m s o f a l l Y* ( r _ l < ) d e s c e n d a n t s , t h e n t h e c e l l w h i c h e a c h s h o u l d o c c u p y w i l l be u n i q u e l y d e t e r m i n e d . The r a n k o f a c o m b i n a t i o n c a n be o b t a i n e d by means o f t h e f o l l o w i n g m e t h o d . L e t ( m ) = ( p . - ^ ) be t h e number o f c o m b i n a t i o n s o f l o c a t i o n s o r e n t e r i n g s t a t e s a t e a c h s t a g e i n t h e a n a l y s i s . L e t ( b , , b 9 , - - - , b ) d e n o t e one o f t h e s e c o m b i n a t i o n s o f t h e r i n t e g e r s , a n d f u r t h e r d e f i n e (a- , a 2 ' " " > a ^ ) a s a c o m ~ b i n a t i o n o f r i n t e g e r s 0 , l , - - - , r - l s u c h t h a t (25) 0 < a n < a 0 • • • < a, < r w h i c h i m p l i e s t h a t e a c h e l e m e n t o f t h e s e r i e s (a-j , a 2 , ' - ' , a m ) i s one l e s s t h a n i t s c o r r e s p o n d i n g c o u n t e r -p a r t i n (b-j , b 2 , * " ' b m ) . I f R y i s d e f i n e d as t h e r e q u i r e d r a n k o f t h e s e r i e s (b-. , b 2 , • • • , b m ) t h e n m (26) R = ( r ) - I v m . L . J=l T h i s d e r i v a t i o n a d a p t e d f r o m ( L e h m e r , 1 9 6 4 ) c o n c l u d e s t h i s s e c t i o n on d y n a m i c p l a n t l o c a t i o n - o v e r t i m e . am-j+l j CHAPTER IV AN A P P L I C A T I O N OF THE MEDIAN ALGORITHM TO RURAL HOSPITAL LOCATIONS IN A S U B - R E G I O N OF B R I T I S H COLUMBIA In t h e p r e c e e d i n g c h a p t e r a d e t a i l e d d e s c r i p t i o n o f an a l g o r i t h m f o r e s t i m a t i n g t h e m e d i a n s o f a w e i g h t e d g r a p h s u b j e c t t o s i d e c o n s t r a i n t s was p r e s e n t e d . In t h i s c h a p t e r on a p p l i c a t i o n o f t h e b a s i c m o d e l ( w i t h o u t c o n s t r a i n t s ) t o h o s p i t a l l o c a t i o n s i n a r u r a l s u b - r e g i o n o f B r i t i s h C o l u m b i a i s g i v e n . A l t h o u g h t h e a l g o r i t h m i s c o m p u t a t i o n a l l y f e a s i b l e f o r l a r g e s c a l e n e t w o r k p r o b l e m s , i f j u d i c i o u s c h o i c e o f t h e p a r t i t i o n i n g h e u r i s t i c s i s m a d e , o n l y one p a r t o f t h e p r o v i n c e w i l l be s u b j e c t t o a d e t a i l e d a n a l y s i s h e r e . The o u t l i n e f o r t h i s c h a p t e r i s as f o l l o w s : 1) A s u i t a b l e s u b - r e g i o n f o r a n a l y s i s i s f i r s t chosen. 2 ) S p e c i f i e d m e d i c a l d a t a p e r t a i n i n g t o t h i s r e g i o n i s summarized i n terms o f a network of nodes and l i n k s where each node, the c e n t r e f o r some d i s c r e t e s u r r o u n d i n g a r e a i n s p ace, r e p r e s e n t s i n t u r n the demand over a one-year p e r i o d f o r a d u l t m e d i c a l , a d u l t s u r g i c a l and o b s t e t r i c a l c a r e , and each l i n k r e p r e s e n t s the d i s t a n c e ( n o r m a l l y road' d i s t a n c e s ) between some p a i r o f nodes. A subset o f o f t h e s e nodes i s f u r t h e r d e s i g n a t e d as p o t e n t i a l s i t e s . 3) The a l g o r i t h m i s t e s t e d upon the network i n accordance w i t h the s t a t e d aims and o b j e c t i v e s d e t a i l e d i n Chapter I . 4) A s h o r t d i s c u s s i o n of the r e s u l t s f o l l o w s t o n g e t h e r w i t h an a p p r a i s a l o f the v a l i d i t y o f the g e n e r a l model f o r m u l a t i o n f o r t h i s type of l o c a t i o n problem. C h o i c e o f R e g i o n a l S t u d y A r e a The o b j e c t i v e o f t h i s s e c t i o n was t o a t t a i n a p a r -t i t i o n i n g o f t h e p r o v i n c e o f B r i t i s h C o l u m b i a i n t o s u b -r e g i o n s s u c h t h a t e a c h one f o r m e d a r e l a t i v e l y s e l f - c o n -t a i n e d u n i t w i t h r e s p e c t t o t h e m a g n i t u d e and d i r e c t i o n o f t h e l i n k a g e s and f l o w s w i t h i n i t s b o u n d s . To a v o i d p o s -s i b l e b i a s w h i c h m i g h t be p r e s e n t i f t h e a c t u a l p a t t e r n o f h o s p i t a l f l o w s c u t s a c r o s s a n y n a t u r a l n o d a l r e g i o n , t r a f f i c f l o w d a t a , c o l l e c t e d by t h e p r o v i n c i a l g o v e r n m e n t d u r i n g t h e summer o f 1 9 6 6 , was u t i l i z e d a s t h e c r i t e r i o n upon w h i c h t h e p a r t i t i o n i n g was b a s e d . N o t w i t h s t a n d i n g some b i a s i n h e r e n t i n t h e r e c o g n i t i o n t h a t t h i s d a t a was and i s c o l l e c t e d p r i m a r i l y f o r t h e p u r p o s e s o f t o u r i s m , i t was f e l t t h a t t h i s i n f o r m a t i o n was t h e b e s t r e a d i l y a v a i l a b l e s o u r c e f o r f o r m i n g t h e c r u d e l y d e f i n e d n o d a l r e g i o n s n e c e s -s a r y f o r t h i s s t a g e o f t h e a n a l y s i s . T r a f f i c f l o w d a t a was g a t h e r e d f o r e i g h t y c o n t r o l p o i n t s d i s t r i b u t e d r a n d o m l y t h r o u g h o u t t h e p r o v i n c e . G i v e n t h e t r a f f i c f l o w a c r o s s e a c h o f t h e . d e s i g n a t e d c o n t r o l p o i n t s , t h e d i s t a n c e o f t h e l i n k b e t w e e n a n y two p o i n t s , and an a s s e s s m e n t o f t h e a v e r a g e t r a f f i c s p e e d o v e r a n y l i n k ( t a k e n s o m e w h a t a r b i t r a r i l y t o be 35 m . p . h . , w i t h 25 m . p . h . on g r a v e l r o a d s and 4 0 - 4 5 m . p . h . on f r e e w a y s ) , an e s t i m a t e o f r o a d t r a f f i c d e n s i t y , o r more s p e c i f i c a l l y t h e a v e r a g e d i s t a n c e b e t w e e n v e h i c l e s o v e r a n y l i n k , may r e a d i l y be o b t a i n e d . F o r e a c h s u c h c o n t r o l p o i n t t h e w e i g h t e d a v e r a g e o f t h i s m e a s u r e o v e r a l l l i n k s c o n n e c t e d t o i t was c a l c u l a t e d . G i v e n one s u c h v a l u e f o r e a c h c o n t r o l s t a t i o n a g r o u p i n g o f t h e p o i n t s was s o u g h t , s u b j e c t t o a c o n t i g u i t y c o n s t r a i n t , f o r w h i c h t h e a v e r a g e i n t r a - g r o u p d i f f e r e n c e s among t h e p o i n t s m i g h t be m i n i m i z e d w h i l e m a x i m i z i n g t h e a v e r a g e i n t e r - g r o u p d i s p e r s i o n . A l t h o u g h t h i s o b j e c t i v e f u n c t i o n i s n o t e a s i l y r e a l i s e d t h e w e l l -known h i e r a r c h i c a l s t e p g r o u p i n g a l g o r i t h m o f ( W a r d , 1 9 6 3 ) w i l l t e n d t o p r o v i d e a good e s t i m a t e o f t h e o p t i m a l g r o u p i n g , p a r t i c u l a r l y w h e r e t h e n a t u r a l c l u s t e r i n g i s s t r o n g . A p p l i c a t i o n o f t h i s a l g o r i t h m ( f o r w h i c h an e f f i c i e n t c o m p u t e r c o d e i n c o r p o r a t i n g a d e n d o g r a m i s g i v e n i n A p p e n d i x 4) t o t h e v e h i c l e s p a c i n g m e a s u r e r e v e a l e d a w e l l d e f i n e d r e g i o n a l i z a t i o n w i t h s i x g r o u p s r e m a i n i n g . T h e s e r e g i o n s w e r e i d e n t i f i a b l e as a) V i c t o r i a and i t s e n v i r o n s , b) n o r t h -c e n t r a l and n o r t h V a n c o u v e r I s l a n d , c ) V a n c o u v e r and t h e l o w e r m a i n l a n d as f a r e a s t as H o p e , d) t h e S e c h e l t p e n i n -s u l a , e ) t h e m a i n l a n d n o r t h o f L y t t o n t o P r i n c e G e o r g e a n d w e s t t o P r i n c e R u p e r t and f ) s o u t h - e a s t e r n B r i t i s h C o l u m b i a f r o m Hope t o t h e A l b e r t a b o r d e r , b o u n d e d on t h e n o r t h by t h e T r a n s - C a n a d a h i g h w a y and on t h e s o u t h by t h e b o u n d a r y w i t h t h e U n i t e d S t a t e s . The l a s t o f t h e s e r e g i o n s , w h e r e i n c o n t r a s t s e x i s t b e t w e e n c l o s e l y c l u s t e r e d s e t s o f commun-i t i e s w h e r e o p p o r t u n i t i e s f o r i n - p a t i e n t h o s p i t a l c a r e a r e v a r i e d , and o t h e r a r e a s o f more d i s p e r s e d s e t t l e m e n t w h e r e d i s t a n c e s b e t w e e n t o w n s c o r r e s p o n d i n g l y r e d u c e s u c h c h o i c e s , s e e m e d t o o f f e r t h e b e s t o p p o r t u n i t y f o r a r i g o r o u s t e s t i n g o f t h e m o d e l , and was a c c o r d i n g l y c h o s e n as t h e s u b - a r e a f o r a n a l y s i s . G e n e r a t i o n o f a M e d i c a l Demand N e t w o r k f o r S o u t h - E a s t e r n  B r i t i s h C o l u m b i a As shown i n F i g u r e 15 t h e d e f i n e d s t u d y a r e a o f • s o u t h -e a s t B r i t i s h C o l u m b i a c o v e r s some t w e n t y - n i n e s c h o o l d i s -t r i c t s w h i c h a r e t h e s m a l l e s t g e o g r a p h i c a l b a s e f o r w h i c h m e d i c a l d a t a c a n be a s s e m b l e d f r o m c o l l e c t e d i n f o r m a t i o n . F rom 1966 d a t a made a v a i l a b l e by t h e B r i t i s h C o l u m b i a H o s p i t a l I n s u r a n c e Scheme i n V i c t o r i a , h o s p i t a l p a t i e n t s e p a r a t i o n s ( d i a g n o s i s on d i s c h a r g e ) w e r e o b t a i n e d d i s -a g g r e g a t e d by m e d i c a l , s u r g i c a l and m a t e r n i t y c a t e g o r i e s . F o r e a c h s c h o o l d i s t r i c t t h e f o l l o w i n g i n f o r m a t i o n was c o m p i l e d : 1) The number o f s e p a r a t i o n s f o r each o f the t h r e e m e d i c a l c a t e g o r i e s over the one year p e r i o d ( T a b l e 6 ) . -E2L-T a b l e 6 NUMBER OF SCHOOL D I S T R I C T SEPARATIONS BY MEDICAL CATEGORIES S . D . P o p u l a t i o n M e d i c a l S u r g i c a l M a t e r n i t y 1 6 , 6 2 4 521 4 7 3 1 26 2 1 1 , 0 5 1 6 4 5 625 267 3 8 , 6 6 5 389 8 2 5 1 50 4 4 , 4 6 1 352 314 93 7 1 6 , 0 1 1 8 7 5 91 5 305 8 3 , 3 3 6 256 255 71 9 11 , 3 1 9 533 790 270 10 3 , 2 6 9 1 64 215 63 11 2 4 , 1 3 8 9 8 4 1 9 2 2 4 6 3 12 5 , 3 0 0 291 271 91 13 2 , 7 3 9 97 1 63 45 14 8 , 1 1 6 524 541 1 39 15 1 7 , 7 8 3 966 1 1 4 0 324 16 2 , 4 3 8 194 203 37 1 7 2 , 9 9 8 252 215 46 18 5 , 6 9 6 314 265 1 72 19 8 , 2 3 7 382 379 197 20 11 , 9 1 1 1 0 3 8 801 214 21 3 , 5 0 2 272 186 57 22 21 , 3 3 6 889 1 1 1 2 3 5 5 23 3 3 , 7 3 0 1 3 5 0 1 770 556 24 3 9 , 2 4 9 1189 1661 8 3 2 25 1 , 8 9 5 61 90 62 30 6 , 5 1 9 497 4 6 2 184 31 7 , 4 6 5 419 51 3 196 S . D . P o p u l a t i o n M e d i c a l S u r g i c a l M a t e r n i t y 32 5 , 8 7 8 426 3 2 2 139 77 4 , 6 9 1 4 2 2 321 59 78 3 , 0 5 5 264 230 47 86 11 , 1 4 1 661 690 197 2) The number of s e p a r a t i o n s d i s c h a r g e d from each of t h e t h i r t y - t h r e e h o s p i t a l s o p e r a t i n g i n the s u b - r e g i o n , t h a t i s the p a t i e n t f l o w from each s c h o o l d i s t r i c t t o each h o s p i t a l . 3) The t o t a l p o p u l a t i o n o f each s c h o o l d i s t r i c t . S i n c e t h e s c a l e o f m o s t o f t h e s c h o o l d i s t r i c t s p r o v e d t o be t o o l a r g e t o be o f much p r a c t i c a l v a l u e f o r a n a l y s i s , i t was n e c e s s a r y t o d i s a g g r e g a t e t h i s i n f o r m a -t i o n i n t o a much f i n e r n e t , one w h e r e e a c h u n i t c o u l d be s u m m a r i z e d by a s i n g l e p o i n t o r n o d e . T h i s was s u c c e s s i v e l y a c h i e v e d i n t h e f o l l o w i n g s t a g e s , e a c h one n e c e s s a r i l y e n t a i l i n g a c e r t a i n m e a s u r e o f e s t i m a t i o n a t e a c h s t e p . a ) U s i n g e n u m e r a t i o n d i s t r i c t p o p u l a t i o n maps c o m p i l e d f r o m t h e N a t i o n a l C e n s u s o f 1 9 6 6 , e a c h s c h o o l d i s t r i c t was d i s a g g r e g a t e d i n t o t h e s e s m a l l e r g e o g r a p h i c a l u n i t s f o r w h i c h t h e e x a c t p o p u l a t i o n f i g u r e s w e r e k n o w n . In a f e w i n s t a n c e s t h e b o u n d a r i e s o f t h e e n u m e r a t i o n and s c h o o l d i s t r i c t s do n o t c o i n c i d e and i t was n e c e s s a r y t o make an e s t i m a t e o f t h e p r o p o r t i o n o f e n u m e r a t i o n d i s t r i c t p o p u l a t i o n w h i c h l a y upon e a c h s i d e o f t h e s c h o o l d i s t r i c t b o u n d a r y . T h e s e e s t i m a t e s w e r e a d j u s t e d t o e n s u r e t h a t t h e sum o f a l l e n u m e r a t i o n d i s t r i c t p o p u l a t i o n s w i t h i n t h e b o u n d s o f a g i v e n s c h o o l d i s t r i c t was s e t e q u a l t o t h e known p o p u l a t i o n t o t a l o f t h a t s c h o o l d i s t r i c t . b) The c e n t r e s o f p o p u l a t i o n g r a v i t y f o r e a c h o f t h e e n u m e r a t i o n d i s t r i c t s w e r e a p p r o x i m a t e d w i t h t h e a i d o f a p o p u l a t i o n d o t d i s t r i b u t i o n map o f B r i t i s h C o l u m b i a p r e p a r e d f o l l o w i n g t h e 1961 c e n s u s by ( F a r l e y , e t . a l . , 1 9 6 1 ) . In some c a s e s d e p e n d i n g upon t h e l o c a l d i s t r i b u t i o n two o r more p o i n t s w e r e a s s i g n e d t o t h e same e n u m e r a t i o n d i s t r i c t , w h i l e i n o t h e r s one p o i n t s e r v e d t o s u m m a r i z e t h e p o p u l a -t i o n o f more t h a n one d i s t r i c t o r p a r t s o f s e v e r a l , i f p o p u l a t i o n c o n c e n t r a t i o n s l a y on e i t h e r s i d e o f an e n u m e r -a t i o n d i s t r i c t b o u n d a r y . A s e t o f g e o g r a p h i c a l c o - o r d i n a t e s w e r e t h e n r e c o r d e d f o r e a c h o f t h e 168 p o i n t s c o m p i l e d . c ) T h e s e c o - o r d i n a t e s t o g e t h e r w i t h t h e i r a t t a c h e d p o p u l a t i o n w e i g h t s e n a b l e d t h e c e n t r o i d s o f t h e s c h o o l d i s t r i c t s t h e m s e l v e s t o be a s c e r t a i n e d i n t e r m s o f a l l t h e p o i n t s l y i n g w i t h i n t h e i r b o u n d a r i e s . T h i s a n a l y s i s g a v e t h i r t y - f i v e s u c h v a l u e s o f w h i c h s i x r e p r e s e n t e d s c h o o l d i s t r i c t s s u r r o u n d i n g t h e d e f i n e d s u b - r e g i o n . d) I n c i d e n c e r a t e s f o r e a c h o f t h e s c h o o l d i s t r i c t s w e r e n e x t c o m p i l e d d i s a g g r e g a t e d by m e d i c a l c a t e g o r y . T h e s e r a t e s w e r e b a s e d upon t h e number o f p a t i e n t s e p a r a t i o n s p e r 1 0 0 0 p o p u l a t i o n e x c l u d i n g a l l c a s e s r e f e r r e d t o h o s p i t a l s b e y o n d t h e r e g i o n e x c e p t t h o s e t o C h i l l i w a c k , L i l l o o e t and 100 m i l e h o u s e . e.) F i n a l l y an e s t i m a t e o f t h e demand o r a c t u a l number o f p a t i e n t s e p a r a t i o n s g e n e r a t e d by e a c h e n u m e r a t i o n d i s t r i c t was r e q u i r e d In u t i l i z i n g t h e g i v e n c o - o r d i n a t e s o f th-.e s c h o o l d i s t r i c t p o p u l a t i on c e n t r o i d s " , t r e n d s u r f a c e a n a l y -s e s w e r e a p p l i e d t o a l l t h r e e s e t s o f m e d i c a l . i n c i d e n c e d a t a . S i n c e o n l y t h i r t y - f i v e p o i n t s w e r e i n v o l v e d a s i x t h o r d e r t r e n d s u r f a c e a n a l y s i s a c c o u n t e d i n e v e r y c a s e , a s m i g h t be e x p e c t e d w i t h so f e w v a l u e s , f o r more t h a n 90 p e r c e n t o f t h e v a r i a t i o n i n e a c h c a s e . A t t e m p t s t o u s e t h e s i x t h o r d e r t r e n d e q u a t i o n as a means o f e s t i m a t i n g t h e i n c i d e n c e r a t e a t e a c h e n u m e r a t i o n d i s t r i c t c e n t r o i d p r o v e d t o be i l l u s o r y h o w e v e r , o w i n g t o t h e e x t r e m e i n s t a b i l i t y o f t h e s u r f a c e s a t t h e i r e d g e s . The t r e n d s p r o v e d t o be i l l -d e f i n e d up t o t h e f i f t h o r d e r , l i t t l e more t h a n 50 p e r c e n t o f t h e v a r i a t i o n b e i n g e x p l a i n e d a t t h i s l e v e l . As a c o n s e q u e n c e t h e g e n e r a t e d s u r f a c e s d i d n o t p r o v i d e a s a t i s f a c t o r y a p p r o x i m a t i o n t o t h e t r u e o n e s so t h a t t h e c e n t r o i d e s t i m a t i o n s , p a r t i c u l a r l y a t t h e e d g e s o f t h e r e g i o n , w e r e s u b j e c t t o g r o s s d i s t o r t i o n s . The r e q u i r e d c e n t r o i d i n c i d e n c e r a t e s w e r e f i n a l l y i n t e r p o l a t e d f r o m c o n t o u r maps g e n e r a t e d f o r e a c h o f t h e t h r e e s e t s o f s c h o o l d i s t r i c t d a t a . G i v e n t h e e n u m e r a t i o n c e n t r o i d p o p u l a t i o n s i t was t h e n a s h o r t s t e p t o c a l c u l a t e t h e e s t i m a t e d demand over each of the 168 e n u m e r a t i o n n o d e s ( A p p e n d i x 1 ) . F i g u r e 16 g i v e s t h e p o s i t i o n o f a l l t h e c e n t r o i d s ( n o d e s ) w i t h r e s p e c t t o t h e i r a p p r o p r i a t e s c h o o l s d i s t r i c t s , t o g e t h e r w i t h t h e l o c a t i o n s o f t h e t h r e e h o s p i t a l s on t h e p e r i p h e r y o f t h e r e g i o n . ro to A M e a s u r e o f t h e L o c a t i o n a l E f f i c i e n c y o f t h e H o s p i t a l  S y s t e m o f S o u t h - E a s t B r i t i s h C o l u m b i a By t h e m e t h o d s s p e c i f i e d i n t h e p r e c e e d i n g s e c t i o n t h r e e m e d i c a l demand n e t w o r k s w e r e c o n d e n s e d f r o m p o p u l a -t i o n c e n s u s d a t a and h o s p i t a l s e p a r a t i o n t a p e s . F i g u r e 16 p r o v i d e s t h e c a r t o g r a p h i c r e p r e s e n t a t i o n o f t h e s e n e t -w o r k s , e a c h one c o n s i s t i n g o f t h e same 171 n o d e s , o f w h i c h 52 w e r e a s s i g n e d t h e s t a t u s o f p o t e n t i a l h o s p i t a l s i t e s i n c l u d i n g t h e 33 f a c i l i t i e s w h i c h a r e c u r r e n t l y i n o p e r a -t i o n i n t h i s r e g i o n . The c h o i c e o f e a c h p o t e n t i a l l o c a t i o n was s i m p l y made on t h e b a s i s o f i n c l u d i n g n e a r l y e v e r y node i n t h e n e t w o r k who se p o p u l a t i o n e x c e e d e d 1 0 0 0 p e o p l e . A s h o r t p a t h m a t r i x ( s e e A p p e n d i x 2) was n e x t p r e -p a r e d b e t w e e n e v e r y node and e a c h p o t e n t i a l s i t e , t h e d i s t a n c e b e t w e e n e a c h node and i t s e l f b e i n g a s s i g n e d , w h e r e a p p l i c a b l e , a v a l u e o f o n e . In t h e f i r s t t h r e e r u n s ( o n e f o r e a c h m e d i c a l c a t e g o r y ) t h e s o l u t i o n was c o n s t r a i n e d t o l o c a t i o n s c u r r e n t l y o p e r a t i n g i n and a t t h e p e r i p h e r y o f t h e a r e a . T h e s e c o n s t r a i n t s w e r e r e l a x e d i n t h e s e c o n d s e r i e s o f a n a l y s e s , t h e o b j e c t i v e b e i n g t o t e s t t h e a b i l i t y o f t h e m e d i a n a l g o r i t h m t o f i n d s e t s o f l o c a t i o n s f o r w h i c h t h e t o t a l p a t i e n t m i l e a g e on t h e n e t w o r k was l e s s t h a n t h a t g i v e n by t h e c o n s t r a i n e d s o l u t i o n s . The r e s u l t s o f t h e s e a n a l y s e s ( p r e s e n t e d i n T a b l e 7) show t h a t t h e m e d i a n a l g o r i t h m s o l u t i o n s w e r e c o n s i s t e n t l y b e t t e r t h a n t h e c o n s t r a i n e d o n e s t o t h e e x t e n t o f 15 t o COMPARISON OF DISTANCE M I N I M I Z I N G R E S U L T S : CONSTRAINED AND UNCONSTRAINED SOLUTION C a t e g o r y T o t a l P a t i e n t M i l e s Cons trained Uncons trained M e d i c a l 8 9 , 0 2 1 7 6 , 4 9 4 S u r g i c a l 9 6 , 8 6 0 8 3 , 0 7 7 M a t e r n i t y 3 2 , 2 8 8 2 6 , 1 4 9 20 p e r c e n t i n e a c h c a s e . T a b l e 8 p r o v i d e s d e t a i l s on t h e d i f f e r e n t h o s p i t a l s e t s o b t a i n e d i n e a c h s o l u t i o n . S i n c e t h e a l g o r i t h m i s h e u r i s t i c i n d e s i g n t h e r e c a n be no g u a r a n t e e t h a t t h e s e p a t t e r n s o f l o c a t i o n s a r e o p t i m a l , b u t a c a r e f u l s c r u t i n y o f d a t a and r e s u l t s s u g g e s t no o b v i o u s i m p r o v e m e n t s and i t i s c o n j e c t u r e d t h a t t h e o b -t a i n e d s o l u t i o n s a r e a t l e a s t c l o s e t o t h e g l o b a l m i n i m a . N o t e t h a t t h e t h r e e p e r i p h e r a l l o c a t i o n s o f C h i l l i w a c k , L i l l o o e t and 100 M i l e H o u s e a r e d r o p p e d i n t h e u n c o n -s t r a i n e d s o l u t i o n s i n e a c h c a s e . A c t u a l l y , s i n c e t h e s e f a c i l i t i e s a r e i n v i a b l e l o c a t i o n s w i t h r e s p e c t t o t h e s u r r o u n d i n g c o m m u n i t i e s t h e y s e r v e o u t s i d e t h e b o u n d s o f t h e s p e c i f i e d r e g i o n , t h e a c t u a l m e a s u r e o f l o c a t i o n a l i n e f f i c i e n c y i m p l i c i t i n t h e r e s u l t s w i l l be s o m e w h a t e x a g g e r a t e d . P r e d i c t i o n o f A c t u a l F l o w P a t t e r n s W i t h i n t h e S u b - R e g i o n A l t h o u g h t h e r e s u l t s g i v e n by t h e m e d i a n a l g o r i t h m r e f l e c t f a v o u r a b l y upon t h e i n t r i n s i c s t r u c t u r a l s t r e n g t h o f t h e m o d e l f o r m u l a t i o n , i t s a p p l i c a b i l i t y t o t h e p r o b l e m a t h a n d s t i l l r e q u i r e s some f u r t h e r j u s t i f i c a t i o n . Im -p l i c i t i n t h e a n a l y s i s so f a r has b e e n t h e c e n t r a l p o s i t i o n o f d i s t a n c e , f o r i t i s upon t h i s v a r i a b l e t h a t t h e l o c a -t i o n a l s t r u c t u r e o f t h e r e g i o n a l h o s p i t a l s y s t e m has b e e n e s s e n t i a l l y a s s e s s e d . I Y e t t h e s i g n i f i c a n c e o f t h i s f a c t o r as a d e t e r m i n a n t o f t h e a c t u a l p a t t e r n o f t r i p d i s t r i b u -t i o n s s t i l l r e m a i n s t o be p o s i t i v e l y d e m o n s t r a t e d f o r t h i s COMPARISON OF CONSTRAINED AND UNCONSTRAINED LOCATION SYSTEMS H o s p i t a l C o n s t . S o l u t i o n U n c o n s t r a i n e d S o l u t i o n Medical Surgical Maternity G o l d e n X X X X I n v e r m e r e X . X X X K i m b e r l y X X X X C r a n b r o o k . X X X X F e r n i e X X X X M i c h e l X X C r e s t o n X X X X K o o t e n a y Bay X Kas 1 o X X X X N e l s o n X X X X S a l m o X X T r a i 1 X X X X R o s s 1 and X X C a s t ! e g a r X X X X New D e n v e r X X X X N a k u s p X X X X G r a n d F o r k s X X X X G r e e n w o o d X O s o y o o s X X O l i v e r X X K e r e m e o s X X X P e n t i c t o n X X X X S u m m e r l a n d X X X X K e l o w n a X X X X T a b l e 8 c o n t ' d . H o s p i t a l C o n s t . S o l u t i o n U n c o n s t r a i n e d S o l u t i o n Medical Surgical Maternity Vermon X X X X A r m s t r o n g X E n d e r b y X X X X R e v e l s t o k e X X X X S a l m o n Arm X X X X C h a s e X X X Kami o o p s X X X X B a r r i e r e X X M e r r i t t X X X : X P r i n e e t o n X X X X Hope X X X X L y t t o n X X X X A s h c r o f t . X X X X C l i n t o n X X X L i 11 o o e t X C h i 1 1 i w a c k X 100 M i l e H s . X e x a m p l e . To t e s t t h i s q u e s t i o n t h e number o f s e p a r a t i o n s f r o m e a c h node a t e a c h o f t h e 33 o p e r a t i n g h o s p i t a l s w e r e a g g r e g a t e d by t h e i r s c h o o l d i s t r i c t d e s i g n a t i o n s , and t h e n c o m p a r e d t o t h e a c t u a l f l o w p a t t e r n s c o n d e n s e d f r o m t h e h o s p i t a l t a p e s . T h e s e r e s u l t s a r e shown i n T a b l e 9 i n w h i c h t h e e n t r i e s down e a c h c o l u m n a r e c o m p u t e d p e r c e n t a g e s o b t a i n e d f r o m s o l u t i o n o f t h e e x p r e s s i o n (a - b ) / a x 1 0 0 , w h e r e a i s t h e number o f s e p a r a t i o n s f r o m any s c h o o l d i s t r i c t and b i s t h e sum o f t h e d e v i a t i o n s b e t w e e n a c t u a l and p r e d i c t e d s e p a r a t i o n s o v e r a l l h o s p i t a l s . The o v e r a l l r e s u l t s w e r e e n c o u r a g i n g t o t h e e x t e n t t h a t t h e r a n g e o f p r e d i c t i o n v a r i e d f r o m a l o w o f 7 2 . 9 p e r c e n t f o r s u r g i c a l c a r e t o a h i g h o f 8 6 . 7 p e r c e n t f o r m e d i c a l . T h e s e r e s u l t s t e n d t o c o n f i r m t h a t a f u n d a m e n t a l r o l e i s p l a y e d by a c c e s s i b i l i t y i n r u r a l ' m e d i c a l t r i p p a t t e r n g e n e r a t i o n , and so g i v e s s t r e n g t h t o t h e v e r i f i c a t i o n o f t h e m e d i a n a l g o r i t h m as a t o o l f o r h o s p i t a l l o c a t i o n p l a n n i n g i n t h e r u r a l i n t e r -u r b a n c o n t e x t . N o t a l l f l o w s o v e r t h e r e g i o n a r e d i s t a n c e d e t e r -m i n e d h o w e v e r , as i s shown by i n d i v i d u a l a n o m a l i e s i n t h e v a l u e s o f p a r t i c u l a r s c h o o l d i s t r i c t s . C o n s i d e r t h e e x a m p l e o f s c h o o l d i s t r i c t 8 w h e r e t h e c o m p u t e d v a l u e s a r e 31 and - 3 3 . 3 f o r m e d i c a l and s u r g i c a l c a r e r e s p e c t i v e l y . The n e g a t i v e v a l u e s f o r s u r g i c a l a c t u a l l y i n d i c a t e t h a t t h e sum o f t h e f l o w d e v i a t i o n s i s g r e a t e r t h a n t h e a c t u a l number o f FLOW PATTERN P R E D I C T I O N : DISTANCE M I N I M I Z I N G AND INTERACTANCE MODELS S . D . D i s t a t i c * 2 M i n i m i z i ng M o d e l G r a v i t y I n t e r a c t a n c e M o d e l Medical Surg ica l Maternity Medical Surg ica l Maternity 1 9 0 - 4 7 8 - 4 91 - 0 91 - 1 7 6 - 0 91 - 5 2 92 r 6 7 8 - 2 9 2 - 4 9 0 - 0 7 8 - 6 9 2 - 5 3 8 8 - 7 9 1 - 2 9 6 - 4 9 4 - 2 9 0 - 0 9 6 - 7 4 86 -,3 8 2 - 8 83 r 1 8 7 - 7 9 2 - 1 8 3 - 6 7 8 3 - 8 91 - 5 9 3 - 9 9 5 - 7 91 - 3 9 6 - 0 8 3 - 1 - 3 3 r 3 3 2 : 9 73>0 6 1 - 7 5 3 - 0 9 8 3 - 5 4 5 - 3 8 6 - 5 8 0 - 5 4 8 - 4 8 7 - 3 10 41 - 5 - 2 7 - 4 6 5 - 0 5 0 - 6 2 0 - 2 6 7 : 1 11 9 3 - 5 9 5 - 4 8 2 - 2 93 7 9 5 - 4 8 2 - 3 1 2 8 4 - 9 4-/I 8 3 - 5 8 6 - 6 21 - 2 8 3 - 9 13 6 2 - 9 4 2 - 3 7 0 - 9 7 5 - 6 2 6 - 1 7 1 - 7 14 7 9 - 3 61 - 9 7 7 - 3 8 0 * 0 7 1 - 3 7 7 - 4 1 5 9 6 - 9 90 f.4 9 6 - 5 97 r 1 9 0 - 7 9 6 - 4 16 8 0 - 4 6 2 - 6 6 9 - 8 71 tO 67 • 3 7 2 - 8 ! 17 8 9 - 7 6 8 - 4 8 5 - 4 9 2 ^ 3 81 - 7 8 5 - 9 18 8 9 - 2 8 1 - 9 9 4 * 2 8 9 - 6 8 4 * 7 9 4 - 2 19 84 - 8 47 •7 9 2 - 0 8 5 ? 9 51 - 9 9 2 - 0 20 8 4 - 4 51 - 6 6 6 - 7 8 2 : 7 6 0 - 7 7 0 - 0 21 9 4 - 1 2 8 : 0 2 8 - 8 9 3 - 9 2 8 - 2 2 9 - 3 22 9 2 : 8 8 7 : 4 9 3 - 9 9 4 - 4 8 9 - 1 9 3 - 8 23 9 3 - 3 9 2 - 8 9 8 - 5 9 6 - 3 9 3 : 0 9 8 : 3 24 9 1 - 1 91 - 0 9 5 - 4 9 5 - 8 91 - 6 9 5 - 4 25 8 6 - 9 1 0 0 * 0 9 9 * 3 8 7 - 3 9 8 : 7 9 9 - 5 30 8 9 * 9 6 2 - 8 75 r 1 8 6 * 6 8 0 - 7 7 5 - 1 31 91 - 4 6 5 - 3 8 6 - 3 9 4 - 3 8 3 - 1 87 - ,1 T a b l e 9 c o n t ' d . S . D . D i s t a n c e M i n i m i z i n g M o d e l G r a v i t y I n t e r a c t a n c e M o d e l M e d i c a l S u r g i c a l M a t e r n i t y M e d i c a l S u r g i c a l M a t e r n i t y 32 91 - 1 8 4 - 5 8 9 - 2 9 3 - 5 8 3 - 5 9 2 - 3 77 8 8 - 2 2 0 - 2 41 - 4 8 9 - 2 3 0 - 5 4 3 - 1 78 8 0 - 3 3 3 - 0 7 9 - 6 8 7 - 1 5 8 - 7 7 9 - 2 86- 7 6 - 4 42 • 3 6 7 - 1 7 6 - 8 61 - 4 7 0 - 5 TOTAL 8 6 - 7 7 2 - 9 8 5 - 2 8 9 - 6 7 8 - 7 8 7 - 7 t r i p s . T h i s s c h o o l d i s t r i c t c o n t a i n s t h e f o l l o w i n g l a b e l l e d n o d e s 4 7 , 5 8 , 5 9 , 6 0 , 61 and 6 2 , o f w h i c h o n l y t h e l a s t two a r e a s s i g n e d t o New D e n v e r h o s p i t a l . Tlfe o t h e r f o u r a r e d i r e c t e d s o u t h i n e a c h c a s e t o C a s t l e g a r w h i c h i s a p p r o x i -m a t e l y o n l y one m i l e n e a r e r t h a n t h e l a r g e r . f a c i l i t y a t N e l s o n . In f a c t more t h a n 50 p e r c e n t o f t h e s e s e p a r a t i o n s w e r e t r e a t e d a t t h e N e l s o n f a c i l i t y . T h i s i l l u s t r a t e s a g a i n t h e c a r d i n a l d i s a d v a n t a g e o f s i m p l e d i s t a n c e m i n i -m i z i n g m o d e l s i n s i t u a t i o n s w h e r e , f o r a n y a r e a o f p a t i e n t d e m a n d , more t h a n one t r e a t m e n t c e n t r e p r o v i d e s a f e a s i b l e a l t e r n a t i v e f o r c a r e . T h i s t y p e o f s i t u a t i o n o c c u r s c o m -p a r a t i v e l y i n f r e q u e n t l y i n t h i s p a r t i c u l a r a p p l i c a t i o n s i n c e t h e r u r a l c o n t e x t o f t h e s t u d y , w h e r e d i s t a n c e s b e t w e e n c o m m u n i t i e s a r e r e l a t i v e l y g r e a t , t e n d s t o r e d u c e t h e v i a b l e a l t e r n a t i v e s f o r p a t i e n t c a r e e x c e p t i n s p e c i a l c a s e s w h e r e t r e a t m e n t demands r e f e r r a l t o t h o s e l a r g e r h o s p i t a l s w h e r e t h e r e q u i s i t e f a c i l i t i e s a r e a v a i l a b l e . N o t w i t h s t a n d i n g t h i s c a v e a t i t i s f a i r t o s a y t h a t , f o r t h e p u r p o s e s o f o p t i m i z i n g t r i p d i s t r i b u t i o n p r e d i c t i o n , t h e i n t e r a c t a n c e m o d e l o f f e r s a p o t e n t i a l l y more f r u i t f u l a p p r o a c h . The s t r u c t u r e o f t h i s m o d e l i s b e h a v i o u r a l l y more r e a l i s t i c f o r i t a l l o w s f o r t h e p o s s i b i l i t y o f a s s i g n -i n g p a t i e n t demand f r o m a n y one a r e a t o more t h a n one h o s p i t a l . A g r a v i t y m o d e l o f t h e g e n r e p r o p o s e d b y - f H u f f , ( 1 9 6 3 ) was a c c o r d i n g l y s t r u c t u r e d t o t e s t f o r i m p r o v e d t r i p d i s t r i b u t i o n p r e d i c t i o n , t h o u g h i t s h o u l d be e m p h a s i z e d a g a i n a t t h i s p o i n t t h a t t h e i n t e r a c t a n c e m o d e l i s n o t a t a l l s u i t e d f o r o p t i m i z i n g l o c a t i o n s o v e r a n e t w o r k , w h i c h i s t h e c e n t r a l c o n c e r n o f t h i s s t u d y . The H u f f m o d e l i s o f t h e g e n e r a l f o r m P . i s t h e s i z e o f h o s p i t a l , m e a s u r e d h e r e J i n t e r m s o f t h e number o f s e p a r a t i o n s t r e a t e d . d . . i s t h e d i s t a n c e f r o m node i t o h o s p i t a l j C . i s t h e e s t i m a t e d demand f o r t r e a t m e n t by m e d i c a l c a t e g o r y a t node i and g and m a r e b e s t f i t t i n g e x p o n e n t i a l p a r a m e t e r s d e t e r m i n e d e m p i r i c a l l y i n t h i s a p p l i c a t i o n . The r a t i o n a l e f o r t h i s m o d e l e x p r e s s e d as i t i s h e r e i n i t s s i m p l e s t f o r m i s t h a t , c e t e r i s p a r i b u s , t h e t r i p d i s t r i -b u t i o n b e t w e e n a n y node and h o s p i t a l c a n be e x p e c t e d t o v a r y as a d i r e c t f u n c t i o n o f h o s p i t a l s i z e a n d a s an i n v e r s e f u n c t i o n o f i t s a c c e s s i b i l i t y m e a s u r e d i n s i m p l e d i s t a n c e . A p p l i c a t i o n o f t h i s m o d e l t o t h e d a t a g a v e t h e f o l l o w -i n g r e s u l t s . The p a r a m e t e r g was c a l c u l a t e d t o be 1 - 5 , 2 - 3 , and 1 - 4 f o r m e d i c a l , s u r g i c a l and m a t e r n i t y c a r e r e s p e c -t i v e l y , w h i l e 5 - 5 , 4 - 4 , and 5 - 7 p r o v e d t o be t h e b e s t f i t -t i n g v a l u e s f o r m. A g g r a g a t i n g t h e n o d e s by s c h o o l d i s t r i c t g a v e p r e d i c t i o n s o f 8 9 - 6 , 7 8 - 7 and 8 7 - 7 p e r c e n t f o r e a c h o f (27) T . . w h e r e i s t h e f l o w f r o m node i t o h o s p i t a l j t h e t h r e e m e d i c a l c a t e g o r i e s w h i c h , as T a b l e 9 s h o w s , p r o v e d t o be a s l i g h t i m p r o v e m e n t on t h e o v e r a l l p e r f o r m a n c e o f t h e d i s t a n c e m i n i m i z i n g m o d e l . I n s p e c t i o n o f t h e r e s u l t s f o r i n d i v i d u a l s c h o o l d i s t r i c t s shows s t r i k i n g i m p r o v e m e n t s f o r s c h o o l d i s t r i c t 8, a l t h o u g h f o r t h e s u r g i c a l d a t a t h e g e n e r a l i n c r e a s e s w e r e s o m e t i m e s l e s s s p e c t a c u l a r , r e m a i n i n g , on o c c a s i o n , b e l o w t h e t h i r t y p e r c e n t p r e d i c t e d l e v e l . I t i s d i f f i c u l t t o p r o v i d e s a t i s f a c t o r y e x p l a n a t i o n s f o r t h e s e p a r t i c u l a r r e s u l t s w i t h o u t r e c o u r s e t o d e t a i l e d k n o w l e d g e o f t h e m e d i c a l p a t t e r n s and l i n k a g e s o v e r t h e s e a r e a s . One o f t h e w e a k n e s s e s o f t h i s i n t e r a c t a n c e m o d e l a p p l i c a t i o n and i n d e e d o f m o s t s u c h a l g o r i t h m s w h e r e t h e p a r a m e t e r s a r e e m p i r i c a l l y d e t e r m i n e d as i s n o r m a l l y t h e c a s e , i s t h a t t h e r e i s u s u a l l y l i t t l e a p r i o r i t h e o r e t i c a l j u s t i f i c a t i o n w h i c h c a n be a s c r i b e d t o t h e f u n c t i o n s g and m. The m e a s u r e f o r h o s p i t a l s i z e i s a t b e s t a c r u d e i n d i c a t o r and i s c e r t a i n l y much l e s s s a t i s f a c t o r y t h a n one w h i c h m i g h t be i n c l u s i v e o f t h e r a n g e and s c o p e o f f a c i l i t i e s p r e s e n t a t e a c h i n s t i t u t i o n . N o t e t o o t h a t t h e d i s t a n c e m e a s u r e g i v e s no r e c o g n i t i o n a t a l l t o t h e p a r t i c u l a r p r o f e s s i o n a l l i n k a g e s and p a t t e r n s o f r e f e r r a l w h i c h m i g h t p r e v a i l o v e r c e r t a i n p a r t s o f t h e r e g i o n . M o r e o v e r i t i s s o m e w h a t l i m i t i n g t o s u p p o s e t h a t t h e e x p o n e n t i a l e f f e c t o f d i s t a n c e d e c a y i n c o r p o r a t e d i n t o t h i s m o d e l i s n e c e s s a r i l y c o n s t a n t o v e r i t s w h o l e r a n g e . Many d e c a y f u n c t i o n s d i s p l a y a r a p i d d e c l i n e o v e r t h e f i r s t f e w m i l e s and amore g r a d u a l t a i l i n g o f f t h e r e a f t e r . I n d e e d t h e i n t e r e s t i n g a n a l y s i s o f ( M o r r i l l and K e l l e y , 1 9 7 0 ) seems t o b e a r t h i s o u t f o r h o s p i t a l u s a g e , a n d t h e f u n c t i o n s t h e y t e s t e d w e r e c e r t a i n l y o f a more v a r i e d and c o m p l e x f o r m t h a n t h e s i m p l e e x p o n e n t i a l e f f e c t u s e d h e r e . A l t h o u g h a l l t h e s e c o n s i d e r a t i o n s a r e i n d i c a t i v e o f r e a l w e a k n e s s e s i n t h i s p a r t i c u l a r f o r m u l a t i o n , t h e o b t a i n e d r e s u l t s n e v e r t h e l e s s s u g g e s t t h a t i n t e r a c t a n c e s t r u c t u r e s a r e a p p r o p r i a t e t o t h i s t y p e o f p r o b l e m i n t h i s s e t t i n g . Where p o s s i b l e h o w e v e r , t h e d e t e r m i n a t i o n o f a p r i o r i p a r a -m e t e r s b a s e d upon i n - d e p t h b e h a v i o u r a l s t u d i e s o f t h e m e d i c a l p a t t e r n s o f a r e g i o n ' i s c e r t a i n l y t o be p r e f e r r e d t o a p o s t e r i o r i p a r a m e t e r s on b o t h t h e o r e t i c a l and p r a c t i c a l g r o u n d s . S u c h an a p p r o a c h w o u l d s u r e l y h e l p t o c l a r i f y p a r t i c u l a r i d i o s y n c r a c i e s s u c h as t h o s e r e f l e c t e d i n t h e l o w e r p r e d i c t e d v a l u e s o f T a b l e 9 . H o w e v e r t h e a c q u i s i t i o n o f s u c h d e t a i l e d k n o w l e d g e i s b o t h e x p e n s i v e and t i m e c o n -s u m i n g and f o r t h i s a p p l i c a t i o n h a s l a i n b e y o n d t h e s c o p e o f t h e a n a l y s i s . A M e t h o d f o r E s t i m a t i n g O p t i m a l P l a n t S i z e O v e r t h e C u r r e n t  S y s t e m o f H o s p i t a l s i n t h e R e g i o n As a c o r o l l a r y t o t h e m a i n a i m s o f t h i s c h a p t e r , a m e t h o d f o r t e s t i n g and r e l o c a t i n g t h e i m b a l a n c e o f c a p a c i t y o v e r e x i s t i n g h o s p i t a l s on a n e t w o r k i s s u g g e s t e d by c o m b i n e d u s a g e o f f t h e m e d i a n and i n t e r a c t a n c e m o d e l s . In t h i s p r o b -l e m as f o r m u l a t e d by M o r r i l l and K e l l e y ( 1 9 7 0 ) , and as d e s c r i b e d i n C h a p t e r I I , t h e a i m i s t o r e d i s t r i b u t e c a p a c i t y ( h o s p i t a l s i z e ) o v e r a g i v e n s e t o f c e n t r e s so t h a t p a t i e n t t r a v e l i s m i n i m i z e d s u b j e c t t o t h e b e h a v i o u r a l l y more r e a l i s t i c c o n s t r a i n t s o f an i n t e r a c t a n c e m o d e l s t r u c t u r e . The M o r r i 1 1 - K e l 1 e y m e t h o d b e g i n s w i t h some g i v e n m e a s u r e o f h o s p i t a l s i z e , a l l o c a t e s p a t i e n t s w i t h t h e i n t e r a c t a n c e m o d e l among t h e s e t o f h o s p i t a l s , c a l c u l a t e s a new s e t o f s i z e s b a s e d upon t h e s e a l l o c a t i o n s , a n d c o n -t i n u e s t h e i t e r a t i o n u n t i l an a c c e p t a b l e l e v e l o f d i v e r g e n c e i s o b t a i n e d b e t w e e n two s u c c e s s i v e c y c l e s . The m e t h o d p r o p o s e d h e r e d i f f e r s f r o m t h i s p r o c e d u r e i n t h a t t h e s e t o f i n i t i a l h o s p i t a l s i z e s i s n o t t a k e n as g i v e n , b u t i s d e r i v e d f r o m some m e a s u r e , e i t h e r d i r e c t l y o r i n d i r e c t l y , b a s e d u p o n t h e p r i o r a s s i g n m e n t o f e a c h p a t i e n t demand a r e a t o i t s n e a r e s t f a c i l i t y . I t was s u g g e s t e d i n C h a p t e r I I t h a t t h i s m e t h o d s h o u l d t e n d t o g i v e a b e t t e r r e d i s t r i b u -t i o n , and e x p e r i m e n t s w i t h some s m a l l s a m p l e n e t w o r k s h a v e shown t h i s t o be so on o c c a s i o n . The r e s u l t s o f an a n a l y s i s u s i n g b o t h m e t h o d s on t h e g e n e r a t e d m e d i c a l and s u r g i c a l d a t a p r o v e d h o w e v e r , t o be i n c o n c l u s i v e i n t h i s r e s p e c t , s i n c e b o t h a l g o r i t h m s c o n v e r g e d t o a p p r o x i m a t e l y t h e same s o l u t i o n s . A c t u a l l y t h e m e a s u r e d i m p r o v e m e n t was l e s s t h a n 10 p e r c e n t i n b o t h c a s e s , a l t h o u g h t h e p r o p o s e d m e t h o d r e q u i r e d s l i g h t l y f e w e r (3 t o 4 ) i t e r a t i o n s t o c o n v e r g e . P e r h a p s t h e s e r e s u l t s a r e n o t t o o s u r p r i s i n g on a n e t w o r k w h e r e n e a r l y e v e r y i i n c o r p o r a t e d c o m m u n i t y has i t s own h o s p i t a l and w h e r e , as a c o n s e q u e n c e , t h e d i s t r i b u t i o n o f p l a n t c a p a c i t y i s n e a r o p t i m u m a t t h e o u t s e t . C h a p t e r V SUMMARY AND CONCLUSION R a p i d c h a n g e s i n m e d i c a l t e c h n o l o g y and i n t h e m a t e r i a l e x p e c t a t i o n s o f s o c i e t i e s o v e r t h e l a s t d e c a d e h a v e c r e a t e d demands f o r f u n d a m e n t a l c h a n g e s i n t h e o r g a n i -z a t i o n o f h e a l t h c a r e . A r e v i e w o f t h e p e r t i n e n t l i t e r a -t u r e o v e r t h i s p e r i o d r e v e a l s a q u i c k e n i n g i n t e r e s t i n t h e c o n c e p t o f r e g i o n a l p l a n n i n g as a s u i t a b l e f r a m e w o r k w i t h i n w h i c h t o i n s t i t u t e p o l i c i e s , made u r g e n t by t h e g r o s s i n f l a t i o n a r y n a t u r e o f m e d i c a l c o s t s , and by t h e g r o w i n g a w a r e n e s s o f t h e m a r k e d d i s c r e p a n c i e s w h i c h e x i s t i n t h e r e c e i p t o f m e d i c a l c a r e among v a r i o u s s e g m e n t s o f t h e p o p u l a t i o n . The t r a n s l a t i o n o f t h e s e d e s i r e s i n t o o p e r a -t i o n a l p l a n n i n g t o o l s has b e e n t h e f o c u s o f much o n - g o i n g e f f o r t i n r e c e n t y e a r s . Among t h e s p a t i a l c o m p o n e n t s o f t h i s f i e l d t h e p r o b l e m o f d e s i g n i n g e f f i c i e n t h o s p i t a l l o c a t i o n s y s t e m s f o r r u r a l and d e v e l o p i n g a r e a s , w h e r e b u d g e t a r y l i m i t a t i o n s p l a c e s e v e r e c o n s t r a i n t s upon t h e number o f a v a i l a b l e f a c i l i t i e s , has b e e n deemed a m a t t e r o f some s i g n i f i c a n c e ( L o r d L I e w e l y n - D a v i e s , 1 9 6 6 ) . The r e s u l t s o f t h i s s t u d y s t r o n g l y s u g g e s t t h a t i n t h e i n t e r - u r b a n c o n t e x t a t l e a s t , t h e r o l e o f a c c e s s i b i l i t y i s o f f u n d a m e n t a l i m p o r t a n c e i n c o n d i t i o n i n g t h e m a g n i t u d e and d i r e c t i o n o f many p a t i e n t t o h o s p i t a l t r i p s . T h i s f a c t g i v e s some p r i o r j u s t i f i c a t i o n f o r c o n s i d e r i n g h o s p i t a l l o c a t i o n as a p a r t i c u l a r c a s e o f t h e more g e n e r a l i z e d p r o b -l e m o f f i n d i n g t h e m e d i a n s o f a w e i g h t e d g r a p h s u b j e c t t o c e r t a i n s i d e c o n s t r a i n t s . An a l g o r i t h m f o r e s t i m a t i n g t h e s e m e d i a n s was p r e s e n t e d i n some d e t a i l i n C h a p t e r I I I . The h e u r i s t i c s d e v i s e d t e n t a t i v e l y a p p e a r t o be a t l e a s t as s u c c e s s f u l as t h o s e o f o t h e r n u m e r i c a l a l g o r i t h m s , and a r e c o m p u t a t i o n a l l y more f e a s i b l e t h a n many f o r t h e l a r g e s c a l e p r o b l e m . I t i s c o n j e c t u r e d , t h o u g h by i n s p e c t i o n r a t h e r t h a n d e f i n i t i v e p r o o f , t h a t t h e y w i l l g e n e r a l l y l e a d t o c o n v e r g e n c e w i t h o p t i m a l o r n e a r o p t i m a l s o l u t i o n s . I m p r o v e m e n t s i n t h e m e d i a n a l g o r i t h m c a n a r i s e o n l y f r o m t h e d e v e l o p m e n t o f more e f f i c i e n t p a r t i t i o n i n g a n d p e r t u r b a t i o n t e c h n i q u e s . The p a r t i t i o n i n g p r o c e d u r e s o f t h i s m o d e l seem t o be p a r t i c u l a r l y s u c c e s s f u l f o r b r e a k i n g o p e n t h e g r a p h i n ways w h i c h w i l l t e n d t o l e a d t o s u b s e q u e n t r a p i d c o n v e r g e n c e . The p e r t u r b a t i o n t e c h n i q u e s c a n c e r -t a i n l y be i m p r o v e d , b u t o n l y a t t h e e x p e n s e o f c o n s i d e r a b l y e x p a n d i n g t h e s e a r c h p a t t e r n t o t a k e a c c o u n t o f p a r t i c u l a r , and p e r h a p s u n u s u a l , n e t w o r k d e s i g n s . The w i s d o m o f s u c h an e x p a n s i o n m u s t h o w e v e r , a l w a y s be j u d g e d a g a i n s t t h e p r i m e a d v a n t a g e o f h e u r i s t i c o v e r e x a c t s o l u t i o n m e t h o d s , t h a t o f more r a p i d c o m p u t a t i o n a l c o n v e r g e n c e o v e r e x t e n s i v e n e t w o r k s t r u c t u r e s . The p a r t i t i o n i n g p r o c e d u r e s a l o n e s h o u l d c o n v e r g e t o r e s u l t s s u f f i c i e n t l y g o o d f o r t h e m a j o r -i t y o f p r a c t i c a l a p p l i c a t i o n s . The a p p l i c a t i o n o f t h e m e d i a n a l g o r i t h m t o t h e c u r -r e n t h o s p i t a l s y s t e m o f S o u t h - E a s t B r i t i s h C o l u m b i a s u g g e s t s t h a t t h e c u r r e n t p a t t e r n o f l o c a t i o n s i s n o t an o p t i m a l o n e . The i n e f f i c i e n c y o f t h e p r e s e n t l o c a t i o n a l s t r u c t u r e w i t h r e s p e c t t o t h e m i n i m i z a t i o n o f p a t i e n t t r a v e l o v e r t h e a r e a , a m o u n t s t o a t l e a s t 10 p e r c e n t f o r e a c h o f t h e t h r e e t y p e s o f c a r e c o n s i d e r e d . The d i s a d v a n t a g e o f t h e a l g o r -i t h m i c m e c h a n i s m i n r e s t r i c t i n g p a t i e n t a s s i g n m e n t s t o o n l y one a r e a , c a n be t e m p e r e d by m o d i f y i n g t h e t r i p d i s t r i b u -t i o n w i t h some s u i t a b l y d e s i g n e d i n t e r a c t a n c e m o d e l w h i c h , as d e m o n s t r a t e d h e r e and p r e v i o u s l y by M o r r i l l and K e l l e y ( 1 9 7 0 ) , i s b e h a v i o u r a l l y more r e a l i s t i c i n t h i s s e n s e . I f r e s o u r c e s p e r m i t , t h e n t h e a p r i o r i e s t i m a t i o n o f t h e i n t e r a c t a n c e p a r a m e t e r s i s t h e o r e t i c a l l y t o be p r e f e r r e d t o e m p i r i c a l l y d e t e r m i n e d f u n c t i o n s . I n t e r a c t a n c e m o d e l s c a n n o t h o w e v e r , be u s e d t o s o l v e t h e q u e s t f o r d i r e c t m u l t i - l o c a t i o n o p t i m i z a t i o n on a c o n n e c t e d w e i g h t e d g r a p h . D i s t a n c e m i n i m i z i n g a l g o r i t h m o f t h e g e n r e p r o p o s e d h e r e seem t o o f f e r a t p r e s e n t t h e b e s t p o t e n t i a l t o o l f o r s u c h p r o b l e m s , i f a c c e s s i b i l i t y c a n be shown t o be a k e y v a r i a b l e and i ts~mwiTnTZci - t ion t h e p r i m e g o a l . The o p e r a t i o n a l a l g o r i t h m d e v e l o p e d i n t h i s s t u d y s h o u l d p r o v i d e one d e c i s i o n - m a k i n g b a s i s f r o m w h i c h t o t e s t t h e f e a s i b i l i t y o f p r o p o s e d f u t u r e h o s p i t a l l o c a -t i o n s w i t h i n a r u r a l c o n t e x t , and u n d e r a w i d e v a r i e t y o f p o t e n t i a l l y m e a s u r a b l e c o n s t r a i n t s . L I T E R A T U R E CITED A n o n . " R e g i o n a l H o s p i t a l D i s t r i c t s . " B r i t i s h C o l u m b i a  M e d i c a l J o u r n a l , 9 ( 1 9 6 7 ) , 4 7 3 - 4 . A n d e r s e n , R. and H u l l , J . T . " H o s p i t a l U t i l i z a t i o n and C o s t T r e n d s i n C a n a d a and t h e U n i t e d S t a t e s , " s u p p l e -ment t o M e d i c a l C a r e , 7 ( 1 9 6 9 ) , 4 - 2 3 . A r n o l d , M . F . and H i n k , D . L . " A g e n c y P r o b l e m s i n P l a n n i n g f o r C o m m u n i t y H e a l t h N e e d s , " M e d i c a l C a r e , 6 ( 1 9 6 8 ) , 4 5 4 - 4 6 6 . B a t t i s t e l l a , R . M . D i r e c t i o n s i n R e g i o n a l P l a n n i n g , M i m e o -g r a p h e d , (1 9 6 6 ) . B e l l m a n , R. D y n a m i c P r o g r a m m i n g , P r i n c e t o n , New J e r s e y : P r i n c e t o n U n i v e r s i t y P r e s s , ( 1 9 5 7 ) . B e l l m a n , R. "An A p p l i c a t i o n o f D y n a m i c P r o g r a m m i n g t o L o c a t i o n - A l l o c a t i o n P r o b l e m s , " SIAM R e v i e w , 7 ( 1 9 6 5 ) , 1 2 6 - 8 . B e r r y , R . E . " R e t u r n s t o S c a l e i n t h e P r o d u c t i o n o f H o s p i t a l S e r v i c e s , " H e a l t h S e r v i c e s R e s e a r c h , 2 ( 1 9 6 7 ) , 1 2 3 - 1 3 9 . B o d e n h e i m e r , T . S . " M o b i l e U n i t s : A S o l u t i o n t o t h e R u r a l H e a l t h P r o b l e m , " M e d i c a l C a r e , 7 ( 1 9 6 9 ) , 1 4 4 - 1 5 4 . B r i d g m a n , R . F . " I n t e g r a t i o n o f t h e O r g a n i z a t i o n o f M e d i c a l C a r e i n t o H e a l t h and Town P l a n n i n g , " W o r l d  H o s p i t a l , 3 ( 1 9 6 7 ) 5 0 - 5 3 . C a n a d i a n M e d i c a l A s s o c i a t i o n . G r o u p P r a c t i c e i n C a n a d a , T o r o n t o : R y e r s o n P r e s s , (1 9 6 7 ) . C a r d w e l l , R. and K u c k a , K. "The C a s e f o r C o - o r d i n a t e d H o s p i t a l P l a n n i n g , " H o s p i t a l , 35 (1 961 ) , 4 3 - 3 5 . C o l e m a n , J . V . " P s y c h i a t r i c S t u d i e s o f P a t i e n t N e e d s i n E m e r g e n c y S e r v i c e s o f G e n e r a l H o s p i t a l s , " M e d i c a l  C a r e , 5 (1 9 6 7 ) , 2 5 5 - 2 5 9 . C o o p e r , L . " S o l u t i o n s o f G e n e r a l i z e d L o c a t i o n a l E q u i l i b r i u m M o d e l s , " J o u r n a l o f R e g i o n a l S c i e n c e , 7 ( 1 9 6 7 ) , 1 - 1 9 . C o u g h l i n , R . E . H o s p i t a l C o m p l e x A n a l y s i s : An A p p r o a c h t o  A n a l y s i s f o r P l a n n i n g a M e t r o p o l i t a n S y s t e m o f  S e r v i c e F a c i l i t i e s , U n p u b l i s h e d P h ' d D i s s e r t a t i o n , P h i l a d e l p h i a ; U n i v e r s i t y o f P e n n s y l v a n i a , ( 1 9 6 4 ) . D o n a b e d i a n , A . and A x e l r o d , S . J . " O r g a n i z i n g M e d i c a l C a r e P r o g r a m s t o M e e t H e a l t h N e e d s , " A n n a l s o f t h e  A m e r i c a n A c a d e m y o f P o l i t i c a l and S o c i a l S c i e n c e , 337 ( 1 9 6 1 ) , 4 5 - 5 6 . D o n a b e d i a n , A . " P r o m o t i n g Q u a l i t y T h r o u g h E v a l u a t i n g t h e P r o c e s s o f P a t i e n t C a r e , " M e d i c a l C a r e , 6 ( 1 9 6 8 ) , 1 8 1 - 2 0 2 . D o r s e y , J . L . " P h y s i c a l D i s t r i b u t i o n i n B o s t o n and B r o o k l i n e 1 9 4 0 and 1 9 6 1 , " M e d i c a l C a r e , 7 ( 1 9 6 9 ) , 4 2 9 - 4 4 0 . D u n c a n , J . " F l y i n g D o c t o r S e r v i c e o f A f r i c e , " W o r l d M e d i c a l  J o u r n a l , 1 3 (1 9 6 6 ) , 1 0 2 - 1 2 2 . E f r o y m s o n , M . A . and R a y , T . L . "A B r a n c h - B o u n d A l g o r i t h m f o r P l a n t L o c a t i o n , " O p e r a t i o n s R e s e a r c h , 14 (1 9 6 6 ) , 361 - 3 6 8 . F a r l e y , A . e t . a l . B r i t i s h C o l u m b i a : P o p u l a t i o n D i s t r i b u -t i o n 1 9 6 1 . D e p a r t m e n t o f G e o g r a p h y , U n i v e r s i t y o f B r i t i s h C o l u m b i a . F e l d m a n , E. e t . a l . " W a r e h o u s e L o c a t i o n U n d e r C o n t i n u o u s E c o n o m i e s o f S c a l e , " M a n a g e m e n t S c i e n c e , 12 (1 9 6 6 ) , 6 7 0 - 6 8 4 . F e l d s t e i n , M . S . E c o n o m i c A n a l y s i s f o r H e a l t h S e r v i c e E f f i c i e n c y . A m s t e r d a m : N o r t h H o l l a n d P u b l i s h i n g C o m p a n y , ( 1 9 6 7 ) . F e l d s t e i n , P . J . " R e s e a r c h on t h e Demand f o r H e a l t h S e r v i c e s , " M i l b a n k M e m o r i a l Fund Q u a r t e r l y , 44 ( 1 9 6 6 ) , 1 2 8 - 1 6 5 . F r i e d s o n , E . P a t i e n t s V i e w s o f M e d i c a l P r a c t i c e , New Y o r k : R u s s e l S a g e F o u n d a t i o n , ( 1 9 6 1 ) . G o u l d , P . R . and L e i n b a c h , T . R . "An A p p r o a c h t o t h e G e o -g r a p h i c A s s i g n m e n t o f H o s p i t a l S e r v i c e s , " T i j d s c h n i f t V o o r E c o n o m i q u e en S o c i a l e G e o g r a f i e , 57 ( 1 9 6 6 ) , 2 0 3 - 2 0 6 . H a k i m i , S . L . " O p t i m u m L o c a t i o n o f S w i t c h i n g C e n t e r s and t h e A b s o l u t e C e n t e r s and M e d i a n s ' o f a G r a p h , " O p e r a t i o n s R e s e a r c h , 12 ( 1 9 6 4 ) , 4 5 0 - 4 5 9 . H i l l e b o e , H. " H e a l t h P l a n n i n g on a C o m m u n i t y B a s i s , " M e d i c a l C a r e , 6 ( 1 9 6 8 ) , 2 0 3 - 2 1 4 . H o g e , V . M . " P l a n n i n g , D e s i g n and C o n s t r u c t i o n o f H o s p i t a l s -A n a l y s i s o f C o m m u n i t y N e e d s , " W o r l d H o s p i t a l s , 3 ( 1 9 6 7 ) , 1 1 - 2 0 . H u f f , D . L . " D e f i n i n g and E s t i m a t i n g a T r a d i n g A r e a , " J o u r n a l o f M a r k e t i n g , 28 ( 1 9 6 4 ) , 3 4 - 3 8 . H u f f , D . L . "A P r o g r a m m e d S o l u t i o n f o r A p p r o x i m a t i n g an O p t i m a l R e t a i l L o c a t i o n , " L a n d E c o n o m i c s , 42 (1 9 6 6 ) , 2 9 3 - 3 0 3 . J a c o b s , A . R . and M c L a u g h l i n , C P . " A n a l y s i n g t h e R o l e o f t h e H e l i c o p t e r i n E m e r g e n c y M e d i c a l C a r e f o r a C o m m u n i t y , " M e d i c a l C a r e , 5 ( 1 9 6 7 ) , 3 4 3 - 3 5 0 . J a m e s , 6 . " H e a l t h S e r v i c e s and C o m m u n i t y N e e d s , " M e d i c a l  C a r e , 5 ( 1 9 6 7 ) , 6 2 - 6 9 . K i l s c l o n k , R. " T r a v e l l i n g C l i n i c s B r i n g C a r e t o C a r d s t o n , " C a n a d i a n H o s p i t a l , 43 (1 9 6 6 ) , 3 9 - 4 1 . K i s s i c k , W . L . " P l a n n i n g P r o g r a m m i n g and B u d g e t i n g i n H e a l t h , " M e d i c a l C a r e , 5 ( 1 9 6 7 ) , 2 0 1 - 2 2 0 . K l a r m a n , H . E . "Some T e c h n i c a l P r o b l e m s i n A r e a w i d e P l a n -n i n g f o r H o s p i t a l C a r e , " J o u r n a l o f C h r o n i c  D i s e a s e s , 1 7 ( 1 9 6 4 ) , 7 3 5 - 7 4 7 . K l a r m a n , H . E . " A p p r o a c h e s t o M o d e r a t i n g t h e I n c r e a s e s i n M e d i c a l C a r e C o s t s , " M e d i c a l C a r e , 7 ( 1 9 6 9 ) , 4 7 - 5 2 . K u e h n , A . A . and H a m b u r g e r , M. "A H e u r i s t i c P r o g r a m f o r L o c a t i n g W a r e h o u s e s , " M a n a g e m e n t S c i e n c e , 9 ( 1 9 6 3 ) , 6 4 3 - 6 6 . K u h n , H.W. and K u e n n e , R . E . "An E f f i c i e n t A l g o r i t h m f o r t h e N u m e r i c a l S o l u t i o n o f t h e G e n e r a l i z e d Weber P r o b l e m i n S p a t i a l E c o n o m i c s , " J o u r n a l o f R e g i o n a l  S c i e n c e , 4 (1 9 6 2 ) , 21 - 3 4 . L a k s h m a n a n , T . R . and H a n s e n , W. "A R e t a i l M a r k e t P o t e n t i a l M o d e l , " J o u r n a l o f t h e A m e r i c a n I n s t i t u t e o f P l a n n e r s , 31 (1 9 6 6 ) , 1 3 4 - 1 4 3 . L e h m e r , D . H . " The M a c h i n e T o o l s o f C o m b i n a t o r i c s , " C h a p t e r 1 i n : B e c k e n b a c h , C F . ( e d ) . A p p l i e d C o m b i n a -t o r i c a l A n a l y s i s . New Y o r k : W i 1 e y , (1 9 6 4 ) . L e m b c k e , P . A . " R e g i o n a l O r g a n i z a t i o n o f H o s p i t a l s , " A n n a ! s  o f The A m e r i c a n A c a d e m y o f P o l i t i c a l and S o c i a l  S c i e n c e s , 2 7 3 ( 1 9 5 1 ) , 5 3 - 6 1 . L e t o u r n e a u , C " R e g i o n a l H o s p i t a l P l a n n i n g , " H o s p i t a l  M a n a g e m e n t , 99 ( 1 9 6 5 ) , 41 - 4 4 . L o r d H e w e l y n - D a v i e s . " F a c i l i t i e s and E q u i p m e n t f o r H e a l t h S e r v i c e s : N e e d e d R e s e a r c h , " M i l l bank M e m o r i a l  F u n d Q u a r t e r l y , 44 ( 1 9 6 6 ) , 2 4 9 - 2 7 2 . L u b i n , J . W . e t . a l . "How D i s t a n c e A f f e c t s P h y s i c i a n A c t i v i t y , " M o d e r n H o s p i t a l , 107 (1 9 6 6 ) 8 0 - 8 2 . M a r r i n s o n , R. " H o s p i t a l S e r v i c e A r e a s , " H o s p i t a l s , 38 ( 1 9 6 4 ) , 5 2 - 5 4 . M c G i b o n y , J . R . and B l o c k , L . " B e t t e r P a t i e n t C a r e T h r o u g h C o - o r d i n a t i o n , " P u b l i c H e a l t h R e p o r t s , 63 ( 1 9 4 9 ) , 1 4 9 9 - 1 5 2 7 . M c K e o w n , T . "The C o n c e p t o f t h e D i s t r i c t G e n e r a l H o s p i t a l , " The L a n c e t , ( 1 9 6 6 ) , 8 6 9 - 8 7 1 . M c N e r n e y , W. R e g i o n a l i z a t i o n and R u r a l H e a l t h C a r e , Ann A r b o r : U n i v e r s i t y o f M i c h i g a n , (1 9 6 2 ) . M c L a c h l a n , G . " H e a l t h S e r v i c e s : P u b l i c P o l i c y I s s u e s , " M e d i c a l C a r e , 5 ( 1 9 6 7 ) , 5 5 - 6 1 . M e t r o p o l i t a n S t . P a u l H o s p i t a l P l a n n i n g C o u n c i l . P h y s i c i an  A d m i t t i n g P a t t e r n s i n M e t r o p o l i t a n S t . P a u l , S t . P a u l : The C o u n c i 1 , 1 9 6 5 . M i n i s t r y o f H e a l t h o f G r e a t B r i t a i n . I n t e r i m R e p o r t on  t h e F u t u r e P r o v i s i o n o f M e d i c a l and A l l i e d  S e r v i c e s , L o n d o n l H . M . S . O . P a p e r 6 9 3 , 1 9 2 0 . M o r r i l l , R. and E a r i c k s o n , R. " H o s p i t a l V a r i a t i o n and P a t i e n t T r a v e l D i s t a n c e , " I n g u i r y , 5 (1 9 6 9 ) , 2 6 - 3 4 . M o r r i l l , R. e t . a l . " F a c t o r s I n f l u e n c i n g D i s t a n c e s T r a v e l l e d t o H o s p i t a l s , " E c o n o m i c G e o g r a p h y , 46 ( 1 9 7 0 ) , 1 6 1 - 1 7 1 . M o r r i l l , R. a n d K e l l e y , M. "The S i m u l a t i o n o f H o s p i t a l Use and t h e E s t i m a t i o n o f L o c a t i o n a l E f f i c i e n c y , " G e o g r a p h i c a l A n a l y s i s , 2 ( 1 9 7 0 ) , 2 8 3 - 3 0 0 . M o t t , F . D . and R o e m e r , M . J . R u r a l H e a l t h and M e d i c a l C a r e , New Y o r k : M c G r a w - H i 1 1 , ( 1 9 4 8 ) , 5 0 - 7 3 . M y e r s , R . S . " A r e a w i d e P l a n n i n g f o r H o s p i t a l s L e a d s t o Good M e d i c a l S e r v i c e s , " The M o d e r n H o s p i t a l , 97 (1 961 ) , 1 1 3 - 1 1 6 . O d o r o f f , M . E . and A b b e , L . M . " U s e o f G e n e r a l H o s p i t a l s ; D e m o g r a p h i c and E c o l o g i c F a c t o r s , " P u b l i c H e a l t h  R e p o r t s , 72 ( 1 9 5 7 ) , 3 9 7 - 4 0 3 . P e t e r s o n , G . W . " U n f o r t u n a t e T r e n d s E m e r g i n g as G o v e r n m e n t I n s u r a n c e P l a n s G a i n E x p e r i e n c e , " H o s p i t a l  A d m i n i s t r a t i o n i n C a n a d a , 9 ( 1 9 6 7 ) , 3 2 - 3 3 . R e v e l l e , C . S . and S w a i n , R .W. " C e n t r a l F a c i l i t i e s L o c a t i o n , " G e o g r a p h i c a l A n a l y s i s , 2 ( 1 9 7 0 ) , 3 0 - 4 2 . R o e m e r , M . I . " B e d S u p p l y and H o s p i t a l U t i l i z a t i o n : A N a t u r a l E x p e r i m e n t , " H o s p i t a l s , 35 (1 961 ) , 3 6 - 4 2 . R o e m e r , M . I . a n d A n z e l , D . M . " H e a l t h N e e d s and S e r v i c e s o f t h e R u r a l P o o r , " M e d i c a l C a r e R e v i e w , 25 ( 1 9 6 8 ) , 3 7 7 - 3 9 0 . R o s e n s t o c k , I . M . "Why P e o p l e U s e . H e a l t h S e r v i c e s , " Mi 1 b a n k  M e m o r i a l F u n d Q u a r t e r l y , 4 4 ( 1 9 6 6 ) , 9 4 - 1 2 7 . R o s e n t h a l , G . D . The Demand f o r G e n e r a l H o s p i t a l F a c i l i t i e s , C h i c a g o : The A m e r i c a n H o s p i t a l A s s o c i a t i o n , M o n o g r a p h N o . 1 4 , ( 1 9 6 4 ) . R u t e n b e r g , D . P . P l a n t L o c a t i o n U n d e r E c o n o m i e s o f S c a l e i n  P r o d u c t i o n and T r a n s p o r t a t i o n , P i t t s b u r g : G r a d u a t e S c h o o l o f I n d u s t r i a l A d m i n i s t r a t i o n , C a r n e g i e - M e l l o n U n i v e r s i t y , ( 1 9 6 8 ) . S a n a z a r o , P . J . "The E v a l u a t i o n o f M e d i c a l C a r e U n d e r P u b l i c Law 8 9 - 2 3 9 , " M e d i c a l C a r e , 5 ( 1 9 6 7 ) , 1 6 2 - 1 6 8 . S c h n e i d e r , J . B . " M e a s u r i n g t h e L o c a t i o n a l E f f i c i e n c y o f t h e U r b a n H o s p i t a l , " H e a l t h S e r v i c e s R e s e a r c h , 2 ( 1 9 6 7 ) , 1 5 4 - 1 6 9 . S c h n e i d e r J . B . " M e a s u r i n g , E v a l u a t i n g and R e d e s i g n i n g H o s p i t a l - P h y s i c i a n P a t i e n t S p a t i a l R e l a t i o n s h i p s i n M e t r o p o l i t a n A r e a s , " I n q u i r y , 5 ( 1 9 6 8 ) , 2 4 -4 3 . S c o t t , A . J . "On t h e O p t i m a l P a r t i t i o n i n g o f S p a t i a l l y D i s t r i b u t e d P o i n t S e t s , " i n : S c o t t , A . J . ( e d . ) : S t u d i e s i n R e g i o n a l S c i e n c e , L o n d o n , P i o n , ( 1 9 6 9 ) , 5 7 - 7 2 . S c o t t , A . J . " L o c a t i o n - A l l o c a t i o n S y s t e m s : A R e v i e w , " G e o g r a p h i c a l A n a l y s i s , 2 ( 1 9 7 0 ) , 9 5 - 1 1 9 . S c o t t , A . J . D y n a m i c L o c a t i o n - A l l o c a t i o n S y s t e m s : Some B a s i c P l a n n i n g S t r a t e g i e s , D e p a r t m e n t o f G e o g r a p h y D i s c u s s i o n P a p e r N o . 3 , T o r o n t o , U n i v e r s i t y o f T o r o n t o , ( 1 9 6 9 ) . S i g m o n d , R . M . " H e a l t h P l a n n i n g , " M e d i c a l C a r e , 5 ( 1 9 6 7 ) , 1 1 7 - 1 2 8 . S i n g e r , S . M u l t i - C e n t e r s and M u l t i - M e d i a n s o f a G r a p h w i t h an A p p l i c a t i o n t o O p t i m a l W a r e h o u s e L o c a -t i o n , M i m e o g r a p h e d P a p e r P r e s e n t e d a t t h e A n n u a l M e e t i n g o f t h e M a n a g e m e n t S c i e n c e and O p e r a t i o n s R e s e a r c h A s s o c i a t i o n , San F r a n c i s c o , J u n e ( 1 9 6 8 ) . S o m e r s , H . M . and S o m e r s , A . R . D o c t o r s , P a t i e n t s , a n d H e a l t h  I n s u r a n c e , New Y o r k : D o u b l e d a y , ( 1 9 6 2 ) . S t e w a r t , W . H . R e p o r t on R e g i o n a l M e d i c a l P r o g r a m s t o t h e  P r e s i d e n t and t h e C o n g r e s s , W a s h i n g t o n : U . S . D e p a r t m e n t o f H e a l t h , E d u c a t i o n , a n d W e l f a r e , ( 1 9 6 7 ) . S u c h m a n , E . A . S o c i a l P a t t e r n s o f I l l n e s s and M e d i c a l C a r e , " J o u r n a l o f H e a l t h and Human B e h a v i o u r , 6 ( 1 9 6 5 ) , 2 - 1 6 . T a s k F o r c e R e p o r t . The C o s t o f H e a l t h S e r v i c e s i n C a n a d a , O t t a w a : The Queens P r i n t e r , 1 9 7 0 . T e i t z , M . B . " T o w a r d s a T h e o r y o f t h e U r b a n P u b l i c F a c i l i t y L o c a t i o n , " P a p e r s o f t h e R e g i o n a l S c i e n c e A s s o -c i a t i o n , 21 ( 1 9 6 8 ) , 3 5 - 5 1 . T e i t z , M . B . and B a r t , P . " H e u r i s t i c M e t h o d s f o r E s t i m a t i n g t h e G e n e r a l i z e d V e r t e x M e d i a n o f a W e i g h t e d G r a p h , " O p e r a t i o n s R e s e a r c h , 16 ( 1 9 6 8 ) , 9 5 5 - 9 6 8 . W a g n e r , H . M . P r i n c i p l e s o f O p e r a t i o n s R e s e a r c h . E n g l e w o o d C l i f f s : P r e n t i c e - H a l l I n c . , 1 9 6 9 . W a h n , E . V . " R e g i o n a l H o s p i t a l P l a n n i n g : V o l u n t a r y o r C o m p u l s o r y ? " H o s p i t a l A d m i n i s t r a t i o n i n  C a n a d a , 1 1 ( 1 9 6 9 ) , 5 8 - 6 0 . W a r d , J . H . " H i e r a r c h i c a l G r o u p i n g t o O p t i m i z e an O b j e c t i v e F u n c t i o n , " J . A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n  J o u r n a l , 58 ( 1 9 6 3 ) , 2 3 6 - 2 4 4 . W e i n e r m a n , E . R . ( e d . ) . R e a d i n g s i n M e d i c a l C a r e , C h a p e l H i l l : U n i v e r s i t y o f N o r t h C a r o l i n a P r e s s , ( 1 9 5 1 ) . W e i n e r m a n , " G r o u p P r a c t i c e R e v i s t e d , " M e d i c a l C a r e , 7 ( 1 9 6 9 ) , 1 7 3 - 1 7 4 . W h i t e , C H . " P r o d u c t i o n - A l l o c a t i o n w i t h S e t - U p P e n a l t i e s and C o n c a v e M a t e r i a l C o s t s , " O p e r a t i o n s R e s e a r c h , 18 ( 1 9 7 0 ) , 9 5 9 - 9 7 2 . W h i t e , K . L . " P a t t e r n s o f M e d i c a l P r a c t i c e , " C h a p t e r 43 i n : C l a r k , W. and M a c M a h o n , B . ( e d . ) . P r e v e n t i v e  M e d i c i n e , B o s t o n : L i t t l e , B rown and C o . , ( 1 9 6 7 ) . W o r l d H e a l t h O r g a n i z a t i o n . M e t h o d o l o g y o f P l a n n i n g an  I n t e g r a t e d H e a l t h P r o g r a m m e f o r R u r a l A r e a s , T e c h n i c a l R e p o r t S e r i e s N o . 8 3 , G e n e v a : W . . H . 0 . ( 1 9 5 4 ) . Y e r b y , A . S . " H e a l t h D e p a r t m e n t s , H o s p i t a l s and H e a l t h S e r v i c e s , " M e d i c a l C a r e , 5 ( 1 9 6 7 ) , 7 0 - 7 4 . Z a n g w i l l , W . I . " M i n i m u m C o n c a v e C o s t F l o w s i n C e r t a i n N e t w o r k s , " M a n a g e m e n t S c i e n c e , 14 ( 1 9 6 8 ) , 4 2 9 - 4 5 0 . IODE POPULATIONS AND DEB AN DS BI SF.OICAL CATEGORIES • ODE POPULATION MEDICAL SURGICAL HATERNITT NODE POPOLATIOS 3EDICAL SUR3ICAL SATERNITY 1. 177 9 8 5 85- 4200 333 291 53 2 • 3467 192 161 107 86 291 - 22 17 4 3 870 46 40 27 87 750 95 50 12 4 476 27 22 10 88 1395 62 77 22 5 638 36 31 18 89 26777 1047 1393 444 6 63 4 3 1 90 1693 70 90 27 7 370 28 22 9 91 607 22 3D 9 8v 300- 24 19 6 92 1067 41 59 18 9 734- . 60 51 17 93 903 33 47 15 IO- 1529 127 1 10 33 94 538 29 27 9 I I 492 39 30 10 95 514 20 26 8 12 129 9 9 2 96 2660 109 132 4 5 13 547 41 41 10 97 1479 65 77 24 19 " 260 18 20 5 98 550 27 31 9 15 100 6 8 1 99 059 23 26 8 16 167 9 13 3 100 400 26 22 6 17 307 15 29 5 101 10795 594 770 248 18 7691 305 737 130 102 479 20 24 7 19 200 8 18 3 103 000 30 19 6 20 300 12 28 5 104 2626 203 142 44 21 270 15 17 6 105 076 33 25 7 22 417 25 22 9 106 2314 203 171 37 23 200 10 13 4 107 306 23 23 4 24 9205 541 522 227 108 035 38 36 6 25 798 45 03 20 109 300 23 20 5 26 161 9 8 3 110 816 67 S3 16 27 287 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3 0 1 8 9 1 6 5 1 5 3 1 3 6 2 0 0 176 1 6 0 1 0 7 20 7 1 5 5 1 6 1 1 0 0 2 1 7 1 6 9 1 7 5 1 5 8 2 2 2 1 7 0 1 7 6 1 5 9 2 0 0 1 3 3 1 9 0 1 7 7 2 3 1 1 7 9 1 8 5 1 6 8 2 2 6 178 1 6 0 1 6 7 2 3 5 1 8 7 1 9 3 1 7 6 2 0 S 1 9 7 2 0 3 1 8 6 2 5 6 2 0 8 2 1 0 1 9 7 1 7 5 1 2 3 1 2 9 1 1 2 192 1 0 0 1 0 6 1 2 9 1 9 0 138 : oo 1 2 7 1 8 6 130 1 0 0 1 2 3 1 7 0 1 1 8 1 2 0 1 0 7 1 7 5 1 2 3 1 2 9 1 1 2 1 6 9 1 1 7 1 2 3 1 0 6 1 5 0 1 0 2 1 0 8 9 1 1 6 0 108 1 1 0 9 7 1 0 0 9 2 9 8 8 1 1 3 0 82 8 8 71 1 2 0 72 7 8 61 1 3 0 8 5 8 0 6 3 121 7 9 6 7 SO 1 0 0 9 2 9 8 81 1 3 0 132 1 3 8 1 2 1 2 3 2 1 8 0 1 8 6 1 6 9 2 0 7 1 5 S 1 6 1 1 0 0 2 2 2 1 7 0 1 7 6 1 5 9 2 3 5 1 8 3 1 8 9 1 7 2 2 0 8 196 2 0 2 1 8 5 2 3 3 181 1 8 7 1 7 0 2 5 8 2 0 6 2 1 2 1 9 5 2 2 3 171 1 7 7 1 6 0 2 1 0 1 5 8 1 6 0 1 0 7 1 9 6 1 0 0 1 5 0 1 3 3 2 2 5 1 7 3 1 7 9 1 6 2 2 2 3 171 1 7 7 1 6 0 2 2 8 176 1 8 2 1 6 5 2 5 3 2 0 1 2 0 7 1 9 0 2 8 3 2 3 1 2 3 7 2 2 0 2 6 1 2 0 9 2 1 5 1 9 8 2 7 3 2 2 1 2 2 7 2 1 0 • O D E 7 9 8 2 8 3 e s 8 7 8 9 101 1 0 0 106 1 t o 1 1 3 1 1 9 1 2 3 128 1 3 5 i o o 1 0 7 1 5 3 1 5 6 1 5 7 1 6 3 1 6 0 1 6 5 1 6 9 1 7 0 171 8 5 3 9 11 2 2 i 13 22 31 0 5 5 3 78 122 6 7 7 7 1 0 0 139 102 56 139 1 8 0 1 6 5 1 5 5 150 1 7 5 1 9 7 1 7 7 2 0 9 8 6 5 1 2 3 30 12 2 5 30 0 3 5 7 6 5 90 1 3 0 7 9 6 9 112 151 90 0 0 127 163 1 5 3 1 0 3 1 0 8 1 7 3 195 1 6 5 2 0 7 8 7 5 2 20 35 13 1 15 00 5 8 66 9 1 1 3 5 ' 8 0 9 0 1 13 152 1 1 5 6 9 152 193 1 7 8 1 6 3 1 6 3 1 8 3 2 1 0 1 9 0 2 2 2 8 8 5 9 31 0 2 2 0 7 8 37 51 5 9 80 128 7 3 8 3 106 1 0 5 122 7 6 159 2 0 0 1 8 5 1 6 1 156 131 20 3 1 9 7 2 1 5 8 9 6 7 39 50 2 8 15 1 29 0 3 51 76 120 6 5 7 5 98 1 3 7 1 3 0 80 1 6 7 2 0 3 1 9 3 1 5 3 108 1 7 3 1 9 5 2 0 5 2 0 7 9 0 1 0 0 6 57 3 5 22 7 36 5 0 S 3 8 3 1 2 7 72 8 2 1 0 5 1 0 0 1 3 7 9 1 1 7 0 2 1 5 2 0 0 1 6 0 1 5 5 1 8 0 2 0 2 2 1 2 2 1 0 9 1 6 0 36 0 7 2 5 12 3 32 0 6 50 7 9 1 2 3 6 8 7 8 101 1 0 0 1 2 7 6 1 1 6 0 2 0 5 1 9 0 156 151 1 7 6 1 9 8 2 02 2 1 0 9 2 7 7 0 9 60 38 2 5 10 19 3 3 01 66 110 5 5 6 5 88 1 2 7 1 2 9 90 1 7 7 2 1 7 191 1 0 3 138 1 6 3 185 2 1 5 1 9 7 9 3 8 3 5 5 ' 6 6 00 3 1 22 0 3 0 5 S3 7 8 122 6 7 7 7 1 0 0 1 3 9 10 1 1 0 0 133 2 2 0 2 0 3 1 5 5 1 5 0 1 7 5 1 S 7 2 2 1 2 0 9 9 0 1 3 9 111 122 1 0 0 8 7 7 8 - 13 2 7 3 5 60 100 0 9 5 9 8 2 121 123 156 2 3 9 2 1 1 1 8 5 1 3 7 132 1 5 7 1 7 9 2 7 7 191 9 5 - 1 3 2 1 0 0 1 1 5 9 3 60 7 1 6 20 28 53 9 7 0 2 5 2 7 5 110 1 16 1 0 9 2 3 2 2 0 0 1 7 8 1 3 0 1 2 5 1 5 0 172 2 7 0 180 9 6 1 3 1 1 0 3 1 1 0 9 2 7 9 7 0 5 19 27 5 2 9 6 01 S I 70 1 1 3 I I S 1 0 8 2 3 1 2 0 3 1 7 7 1 2 9 1 2 0 109 171 2 6 9 1 8 3 9 7 1 0 2 110 1 2 5 1 0 3 9 0 8 1 16 30 38 6 3 1 0 7 52 6 2 8 5 120 126 159 2 0 2 2 1 0 1S8 1 0 0 1 3 5 1 6 0 182 2 8 0 190 9 3 1 5 6 128 139 1 1 7 100 9 5 30 00 S2 77 121 6 6 76 9 9 1 3 8 1 0 0 1 7 3 2 5 6 2 2 8 2 0 2 1 5 0 1 0 9 1 7 0 196 2 9 0 2 0 3 9 9 1 6 6 138 109 127 1 10 1 0 5 00 5 0 6 2 87 131 76 8 6 109 1 0 8 1 5 0 1 8 3 2 6 6 2 3 3 2 1 2 1 6 0 1 5 9 1 8 0 2 0 6 3 0 0 2 1 8 1 0 0 1 6 0 1 3 2 1 0 3 121 108 9 9 3 0 0 8 5 6 81 125 7 0 80 1 0 3 102 100 177 2 6 0 2 3 2 2 0 6 158 1 5 3 178 2 0 0 2 9 8 2 1 2 1 0 1 1 2 6 98 1 0 9 8 7 70 6 5 1 10 22 0 7 9 1 36 0 6 6 9 1 0 8 1 1 0 1 0 3 2 2 6 1 3 3 172 120 119 100 1 6 6 2 6 0 178 1 0 2 1 0 3 7 5 8 6 60 51 0 2 23 2 5 3 3 5 8 10 2 0 7 5 7 8 0 1 1 9 1 2 1 1 2 0 2 0 3 2 0 9 1 8 3 1 3 5 1 3 0 1 5 5 1 7 7 2 0 1 1 8 9 1 0 3 1 2 0 9 2 1 0 3 81 6 8 5 9 6 8 16 01 B S 30 OO 6 3 102 100 1 3 7 2 20 192 166 118 1 1 3 138 1 6 0 2 5 3 172 i o o 1 2 8 1 0 0 111 8 9 76 6 7 10 1 8 3 3 7 7 20 0 8 71 110 112 1 0 5 2 2 8 2 0 0 1 7 0 1 2 6 121 106 1 6 8 2 6 6 1 8 0 1 0 5 1 3 3 1 0 5 116 9 0 81 7 2 19 5 3 28 72 19 50 7 3 112 1 1 0 ISO 2 3 3 2 0 2 1 7 6 1 2 8 1 2 3 1 0 8 170 2 7 1 182 1 0 6 1 3 6 108 119 9 7 80 7 5 22 8 1 2 5 6 9 16 0 7 70 109 111 1 0 7 2 3 0 199 1 7 3 1 2 5 1 2 0 1 0 5 1 6 7 2 6 8 1 7 9 1 0 7 1 5 1 123 1 3 0 1 1 2 9 9 9 0 37 2 3 15 0 0 8 0 31 62 8 5 120 126 162 2 0 5 2 1 0 1 9 8 100 1 3 5 160 132 2 8 3 190 1 0 8 1 0 3 1 1 5 126 1 0 0 91 8 2 29 IS 7 18 62 9 0 1 76 1 1 5 1 1 8 ISO 2 37 2 0 5 1 7 9 1 3 1 1 2 6 151 1 7 3 2 7 5 1 8 5 1 0 9 1 5 1 1 2 3 130 112 9 9 9 0 3 7 2 3 15 10 50 17 0 9 80 1 2 3 126 162 2 0 5 2 1 3 1 8 7 1 3 9 1 3 0 159 181 2 8 3 193 1 1 0 1 6 1 1 3 3 1 0 0 122 1 0 9 100 07 3 3 2 5 1 00 17 0 9 80 1 2 3 1 3 5 171 2 50 2 1 3 1 3 7 1 3 9 1 3 0 1 5 9 161 2 9 2 1 9 3 1 1 1 1 7 7 1 0 9 1 6 0 138 1 2 5 116 63 0 9 01 16 2 8 33 6 5 100 139 151 1 8 7 2 70 2 2 9 2 0 3 1 5 5 150 1 7 5 1 9 7 3 0 3 2 0 9 1 1 2 1 9 0 162 1 7 3 151 138 129 76 62 50 2 9 15 06 7 8 1 1 3 152 160 2 0 0 2 8 3 2 0 2 2 1 6 168 1 6 3 1 8 8 2 1 0 321 2 2 ? 1 1 3 2 0 5 177 138 166 153 100 91 7 7 6 9 0 0 1 6 1 9 3 128 1 6 7 179 2 1 5 2 9 8 2 5 7 2 3 1 1 8 3 178 2 0 3 2 2 5 3 3 6 2 3 7 1 1 0 2 3 0 2 0 2 2 1 3 191 178 169 1 1 6 1 0 2 90 6 9 25 8 6 118 153 192 2 0 0 2 0 0 3 2 3 2 8 2 2 5 6 2 0 8 2 0 3 2 2 8 2 5 0 361 2 6 2 1 1 5 2 2 0 196 2 0 7 1 8 5 1 7 2 163 113 9 6 38 6 3 19 80 1 1 2 1 0 7 186 . 1 9 8 2 3 0 3 1 7 2 7 6 2 5 0 2 0 2 1 9 7 2 2 2 2 0 0 3 5 5 2 5 6 1 1 6 2 0 0 2 1 6 2 2 7 2 0 5 192 183 130 1 16 1 0 3 8 3 39 100 132 167 2 0 6 2 1 8 2 5 0 3 3 7 2 9 6 2 7 0 2 2 2 2 1 7 2 0 2 2 6 0 3 7 5 2 76 1 1 7 2 3 3 2 0 5 2 1 6 190 18 1 172 119 1 0 5 97 72 28 8 9 121 1S6 1 9 5 2 0 7 2 0 3 3 2 6 2 3 5 2 5 9 2 1 1 2 0 6 2 3 1 2 5 3 3 6 0 2 6 5 1 1 8 2 0 3 2 1 5 2 2 6 2 0 0 191 182 129 1 1 5 107 82 38 9 9 1 3 1 166 2 0 5 2 1 7 2 5 3 3 3 6 2 9 5 2 6 9 2 2 1 2 1 6 2 0 1 2 6 3 3 7 0 2 7 5 1 1 9 1 5 0 122 1 3 3 1 1 1 9 8 8 9 36 20 16 17 6 1 1 32 67 106 1 13 1 5 0 2 3 7 196 1 7 0 1 2 2 1 17 1 0 2 160 2 7 5 176 1 2 0 1 0 2 1 10 1 2 5 1 0 3 9 0 81 28 30 2 9 S 3 9 7 3 6 0 2 01 60 82 1 18 2 0 1 170 1 0 0 96 91 1 1 6 1 3 8 2 3 9 1 5 0 1 2 1 1 2 8 1 0 0 111 8 9 76 67 10 16 15 3 9 .8 3 22 32 5 5 90 96 1 3 2 2 1 5 180 158 1 10 1 0 5 1 3 0 152 2 5 3 160 1 2 2 166 138 109 1 2 7 1 10 105 52 50 02 0 3 87 2 6 6 01 8 0 9 2 138 2 1 1 170 100 96 9 1 1 1 6 138 2 0 9 1 5 0 1 2 3 1 6 0 132 103 121 1 0 3 9 9 06 0 8 07 0 9 9 3 32 1. 3 5 70 85 1 3 2 20 5 160 1 3 8 9 0 8 5 1 10 132 2 0 3 100 1 2 0 1 0 9 121 1 3 2 1 1 0 9 7 88 35 37 36 6 0 100 0 3 3 5 30 7 3 7 5 1 1 1 190 1 6 3 1 3 7 8 9 80 1 0 9 131 2 3 2 1 0 3 1 2 5 1 5 5 1 2 7 138 116 1 0 3 9 0 01 0 3 02 66 110 09 2 9 2 8 6 7 79 1 1 7 1 9 3 1 5 7 131 8 3 7 3 1 0 3 1 2 5 2 3 6 1 3 7 1 2 6 1 5 6 136 1 0 9 1 2 7 1 10 1 0 5 52 50 53 6 7 111 50 18 17 5 6 6 8 1 1 0 1 8 7 1 0 6 1 2 0 7 2 6 7 9 2 110 2 2 5 126 1 2 7 . 1 2 5 . . 3 5 0 161 1 ? 9 1 0 6 1 3 7 80 86 8 5 9 9 103 82 5 0 15 50 07 8 3 166 136 1 1 0 7 0 6 5 9 0 1 1 2 2 0 0 120 1 2 8 1 3 9 1 5 5 166 1 0 0 131 122 69 71 70 30 1 2 3 6 7 3 5 1 39 5 1 9 7 1 7 0 129 1 0 3 5 5 5 0 7 5 9 7 2 0 8 1 0 9 1 2 9 1 0 0 1 6 5 176 1 5 0 158 109 9 6 98 97 1 1 1 1 5 5 90 6 2 27 66 52 9 3 103 102 7 6 28 2 3 0 8 7 0 181 82 1 3 0 1 S 8 1 8 3 1 3 0 172 1 6 5 156 1 0 3 1 0 5 1 0 0 1 1 3 1 6 2 101 6 9 30 7 3 7 0 116 153 112 8 6 3 3 3 3 5 8 63 191 92 1 3 1 1 0 9 173 180 1 6 2 109 1 0 0 87 8 9 88 102 106 8 5 5 3 18 5 7 61 1 0 7 152 111 8 5 37 3 2 5 7 7 9 190 9 1 1 3 2 1 0 8 1 6 0 1 7 5 1 5 3 1 0 0 1 31 7 8 80 79 9 3 1 3 7 76 0 0 9 3 0 60 106 179 138 1 1 2 60 5 9 80 106 2 1 7 1 1 8 1 3 3 1 5 7 1 7 3 1 8 0 1 6 2 1 0 9 i o o 87 8 9 8 8 1 0 2 106 8 S 5 3 18 21 6 9 1 1 5 188 1 0 7 1 2 1 7 3 6 8 9 3 1 1 5 2 2 6 1 2 7 1 3 0 1 6 7 1 8 3 1 9 0 172 1 5 9 1 5 0 97 .99 9 8 112 1 5 6 9 5 6 3 28 11 7 9 1 2 5 198 1 5 7 1 3 1 8 3 7 8 1 0 3 1 2 5 2 3 6 1 3 7 1 3 5 1 7 8 190 2 0 5 1 8 3 1 7 0 161 108 1 10 1 0 9 123 167 106 70 3 9 1 90 136 2 0 9 168 1 0 2 9 0 8 9 1 1 0 136 2 0 7 1 0 8 1 3 6 9 3 1 1 8 t 2 9 1 0 7 1 2 0 129 9 S 9 7 96 1 2 0 1 6 0 10 3 82 0 7 8 6 1 5 5 1 1 3 0 100 7 3 6 8 7 3 9 8 1 2 0 1 7 2 132 1 3 7 1 1 0 1 3 5 106 1 2 0 1 3 7 106 9 9 1 0 1 t o o 1 10 158 9 7 6 5 3 0 6 9 3 2 6 3 151 121 9 5 8 5 8 0 - 1 0 5 1 2 7 1 8 9 139 1 3 8 1 0 8 1 3 3 100 122 1 3 5 1 0 0 100 1 0 6 105 119 1 6 3 102 7 0 3 5 70 2 0 6 6 108 1 0 7 81 3 3 3 3 6 3 8 5 1 3 6 97 1 3 9 100 129 1 0 0 1 1 3 13 1 100 126 1 2 8 127 151 1 9 5 130 1 0 6 71 1 1 0 16 6 2 1 1 0 7 3 » 7 0 9 5 0 7 9 1 0 1 1 5 2 1 1 3 1 0 0 8 8 1 1 3 1 2 0 102 1 15 120 1 10 1 12 1 1 1 1 3 5 179 118 9 0 S S 9 0 1 0 6 1 2 9 8 9 6 3 S 3 5 3 8 3 1 0 5 1 6 7 1 1 7 1 0 1 9 3 1 1 8 1 2 9 1 0 7 1 2 0 123 1 2 5 1 2 7 126 150 190 133 1 0 7 72 111 17 5 1 130 106 8 0 7 0 7 5 1 0 0 122 1 7 2 130 1 0 2 8 7 1 1 2 1 2 3 101 1 1,0 123 131 1 3 3 132 156 2 0 0 t 3 9 1 1 5 80 1 1 9 2 5 0 5 128 110 8 8 7 6 8 3 1 0 8 1 3 0 1 6 6 102 1 0 3 7 2 97 1 0 8 85 9 9 108 116 1 18 t 17 10 1 1 8 5 120 1 0 3 6 8 1 0 7 16 30 1 1 3 1 0 5 7 9 6 9 7 0 9 9 121 151 1 3 3 1 0 0 7 8 1 0 3 1 1 0 9 2 1 0 5 1 10 1 10 1 12 111 1 3 5 179 118 9 7 6 2 101 10 36 1 1 9 9 9 7 3 6 3 6 8 9 3 1 1 5 1 5 7 1 2 7 1 0 5 62 87 9 8 76 8 9 98 107 121 127 1 5 1 1 9 5 130 1 1 3 78 1 1 7 26 20 1 0 3 1 1 5 8 9 7 9 8 0 1 0 9 131 1 0 1 1 0 3 106 5 2 77 8 8 6 6 7 9 8 3 97 1 1 1 1 1 9 100 188 1 3 1 1 2 3 8 3 1 2 7 36 10 9 3 1 2 5 9 9 8 9 90 1 19 101 131 1 5 3 1 0 7 0 2 6 7 7 8 56 6 9 7 3 87 101 109 130 178 1 2 3 1 3 3 9 3 1 3 7 06 1 8 3 1 2 0 1 0 9 9 9 1 0 0 1 2 9 151 121 1 6 3 1 0 8 S 5 50 6 5 0 3 56 6 5 70 8 3 96 121 1 6 5 1 1 0 1 2 0 1 11 1 5 0 5 9 13 9 6 1 3 7 122 1 12 1 17 102 160 1 3 0 176 1 0 9 6 8 0 1 5 2 3 0 0 3 5 2 61 7S 8 3 108 152 97 107 120 1 6 3 72 2 6 1 0 9 1 5 0 135 1 2 5 130 1 S 5 1 7 7 1 0 7 189 1 5 0 6 2 8 7 9 8 76 8 9 98 107 1 2 1 1 2 9 150 1 9 8 103 153 118 1 5 7 66 2 0 63 100 129 1 1 9 1 2 0 109 171 101 1 8 3 1 5 1 1 0 2 1 2 7 1 3 8 1 1 6 129 138 107 161 1 6 9 190 2 3 8 133 193 158 1 9 7 106 6 0 2 3 60 9 0 1 3 8 1 0 3 1 6 3 151 61 2 0 2 1 5 2 I S O 1 7 5 186 1 6 0 1 7 7 136 195 2 0 9 2 1 7 2 0 2 2 8 6 2 3 1 2 3 0 1 9 5 2 3 0 150 1 0 8 2 5 66 92 i o o 1 0 5 1 7 0 1 5 3 6 3 2 0 0 1 5 3 1 2 5 150 161 1 3 9 1 5 2 161 170 1 8 0 192 2 1 7 2 6 1 2 0 6 2 0 5 1 7 0 2 0 9 129 8 3 1 0 1 6 7 1 1 5 1 2 0 1 0 5 1 2 8 38 1 7 9 1 5 0 1 0 0 1 6 5 176 1 5 0 1 6 7 176 1 8 5 1 9 9 2 0 7 2 3 2 2 7 6 2 2 1 1 9 0 1 5 5 1 9 0 1 1 5 9 8 15 26 52 1 0 0 1 0 5 1 3 0 1 1 3 5 3 160 1 5 5 1 5 3 1 7 8 1 8 9 167 1 8 0 189 198 2 12 2 1 2 2 2 6 2 7 0 2 0 9 1 7 7 102 181 102 1 1 1 28 13 3 9 8 7 9 2 1 1 7 1 0 0 6 6 1 5 1 1 5 6 166 191 2 0 2 1 8 0 193 2 0 2 198 2 0 0 199 2 1 3 2 5 7 196 160 129 168 89 1 2 0 01 1 2 6 70 7 9 1 0 0 87 7 9 1 3 8 1 5 7 1 5 1 1 7 6 1 8 7 1 6 5 178 187 172 1 7 0 173 187 2 3 1 170 138 103 1 0 2 63 1 0 9 67 2 6 1 0 8 5 3 7 8 6 1 1 0 5 112 1 5 8 1 7 6 2 0 1 2 1 2 1 9 0 2 0 3 2 1 2 197 1 9 9 1 9 8 2 1 2 2 5 6 1 9 5 163 128 1 6 7 86 •130 92 51 2 5 7 3 7 8 81 36 1 3 0 1 1 5 1 5 9 l o t 1 6 6 1 7 7 1 5 5 1 6 8 177 162 1 60 163 177 2 2 1 1 6 0 128 9 3 132 5 3 9 9 77 36 10 38 0 3 6 8 7 1 1 1 5 102 1 6 0 1 2 8 . 1 5 3 160 102 1 5 5 160 109 1 5 1 1 5 0 160 2 0 6 107 1 1 5 80 119 00 86 90 0 9 2 3 2 5 30 5 5 7 7 1 2 8 8 9 1 6 1 1 1 0 1 3 9 1 5 0 1 2 3 1 0 1 ISO 1 3 6 1 3 3 1 3 7 161 2 0 S 100 116 81 1 2 0 26 7 2 100 6 3 37 39 0 0 6 9 9 1 1 0 2 1 0 3 1 6 2 1 0 3 1 6 8 1 7 9 1 5 7 170 179 130 1 3 6 1 3 5 109 1 9 3 1 3 2 1 0 0 6 5 i o o •55 101 1 0 5 60 38 10 1 5 0 0 6 2 1 0 3 70 1 6 3 1 0 1 1 6 6 1 7 7 1 5 5 168 177 1 2 0 1 2 6 125 139 1 8 3 122 9 0 5 5 9 0 5 3 9 9 1 1 5 70 0 8 1 5 30 5 2 1 5 3 60 1 6 0 1 0 6 171 132 1 6 0 1 7 3 172 119 1 2 1 1 2 0 1 1 0 1 7 3 117 8 5 5 0 8 9 5 8 1 0 0 1 2 0 7 9 S 3 5 1 2 S 0 7 1 5 8 S 9 1 6 5 1 7 1 196 2 0 7 185 193 197 1 0 0 1 0 6 1 0 5 159 2 0 3 102 1 1 0 7 S 1 1 0 •83 1 2 9 1 0 5 1 0 0 7 8 30 2 5 1 0 5 1 8 3 3 0 1 6 6 2 0 1 2 2 6 2 3 7 2 1 5 2 2 8 2 2 7 170 1 76 175 189 2 3 3 172 1 0 0 1 0 5 1 0 0 1 1 3 1 S 9 1 7 5 1 3 0 1 0 8 6 0 5 5 30 7 S 2 1 3 0 8 1 6 7 1 7 9 2 0 0 2 1 5 193 2 0 6 2 0 5 152 1 5 0 1 5 1 1 6 7 2 1 1 150 1 1 8 8 1 122 91 1 3 7 1 5 3 1 1 2 3 6 3 8 3 3 8 S 3 1 9 1 2 6 1 6 8 1 9 1 2 1 6 2 2 7 2 0 5 2 18 2 1 7 160 1 6 6 1 6 5 179 2 2 3 162 1 3 0 9 5 1 3 0 1 0 3 1 0 9 1 6 5 1 2 0 9 8 5 0 OS 2 0 6 5 2 0 3 10 APPENDIX I I I A C o m p u t e r P r o g r a m f o r C a l c u l a t i n g a l l o r Some o f t h e S h o r t e s t D i s t a n c e s T h r o u g h a N e t w o r k ( t h e D a n t z i g A l g o r i t h m ) C *** THIS PROGRAMME COMPUTES SHORTEST PATHS THROUGH A NETWORK. FOUR C *** OPTIONS ARE POSSIBLE C *** A). ALL ORIGINS TO ALL DESTINATIONS C *** B). ALL ORIGINS TO SOME DESTINATIONS C *»* C). SOKE ORIGINS TO ALL DESTINATIONS . C *.** 0). SOME ORIGINS TO SOME DESTINATIONS C *** N.B. SHORTEST PATHS TO ALL DESTINATIONS ARE.IN FACT COMPUTED BUT C *** PRINTING IS SUPPRESSED FOR THOSE DESTINATIONS NOT SPECIFIED C *** PUNCHED OUTPUT IS OPTIONED {IN FORMAT 8F10.2) C *** THE CANTZIG ALGORITHM IS USED C ******************************** ****************** **************** C *** N.B. NETWORK NODES SHOULD BE LABELLED IN SEQUENCE 1, ...,N C *****OECK SET UP AS FOLLOWS***** C *** l) , PROBLEM NAME CARD M A X I M U M . CF 7 2 COLUMNS C *** 2)t PROBLEM PARAMETER CARD C *** A), COLS 1-3 NUMBER OF NODES IN NETWORK {N ) C *** B). COLS 4-6 NUMBER OF ORIGINS FROM WHICH SHORTEST PATHS ARE TO C *** 8E COMPUTED INN) C *** C). COLS 7-9 NUMBER OF DESTINATIONS TO WHICH SHORTEST PATHS ARE C *** . TO BE COMPUTED (MM) C *** ' D), COL 10 PUNCH 1 FOR (A) SHORTEST PA TH NETWORK SUMMARY TABLE C *** AND (B) FOR SHORTEST PATH ROUTINGS. PUNCH 0 FOR C *** OPTION (A) ONLY -C *** E), COL LI PUNCH 1 IF PUNCHED OUTPUT IS OPTIONED C *** F). COL 15 SINCE AS MANY JOBS AS ARE REQUIRED MAY BE RUN C ***... . CONSECUTIVELY PUNCH IN 1 IF THIS IS THE LAST.(OR _ C *** ONLY) JOB IN THE RUN C *** 3). READ IN CARD(S) CONTAINING THE Ll'ST CF NODES FROM WHICH SHORT C *** DISTANCES ARE REQUIRED. 4 COL S PER NODE. (F THE TOTAL NUMBER C *** " OF- SUCH ORIGINS IS EQUAL TO THE TOTAL NUMBER OF NODES IN THE C *** NETWORK (I.E.IF N=NN) THEN IGNORE THIS INSTRUCTION C *** 4). READ IN CARO(S) CONTAINING T H F U $J OF DESTINATIONS TO WHICH C-*** SHORTEST DISTANCES ARE REQUIRED. 4 COLS PER NODE. IF THE TOTAL C *** NUMBER OF SUCH DESTINATIONS' I S EQUAL TO THE TOTAL NUMBER OF C *»* NODES IN THE NETWORK (I.E.IF N=MM) THEN IGNORE THIS  C *** INSTRUCTION C *** 5 ) . READ IN CARD(S) CONTAINING THE DISTANCE EACH NODE (FROM WHICH O*** THE SHORTEST PATHS TO OTHER NODES ARE REQUIRED) IS FROM t *** ITSELF.4 COLS PER NODE THE LAST COLUMN OF WHICH IS A DECIMAL= C *** FRACTION GREATER THAN OR EQUAL TO ZERO. E.G. IF THE DISTANCE C *** I S ZERO THEN P UNCH 0 00 0 IF TH E_D ISTANCE IS 1 THEN. PUNCH 0010 C *** OR IF THE DISTANCE I S'.~l. 5 THEN PUNCH 0015 C *** 6 ) . VARIABLE FORMAT CARD FOR READING IN DISTANCES FROM EACH NODE C . .TO ALL ADJOINING NODES. E.G, (5F5.0) MEANS A MAXIMUM OF. .5. SUCH C ***. VALUES (OR LESS IF THERE ARE FEWER CONNECTED NODES) PER CARD C *** OF 5 DIGITS EACH C »** 7). READ IN A VECTOR CONTAINING THE NUM3ER OF NODES EACH  C *** INDIVIDUAL NODE IN THE NETWORK IS CONNECTED TO. N.B.A NODE IS C *** NOT CONSIDERED TO BE CONNECTED To IT s E L F . 4 COLUMNS FOR EACH C *** VALUE. READ IN AIL VALUES (20. PER CARD) IN SEQUENCE I =1,N. C * * * 8 ) . READ TN DATA CARDS AS FOLLOWS; c * * * FOR EACH NODE I IN THE NETWORK IN REGULAR SEQUENCE BEGINNING C * * * ..WITH NODE NUMBER 1 C * * * A), READ IN CARD,S) CONTAINING A LIST OF ALL THE NODES TO C * * * WHICH NODE, I I S CONNECTED. 4 COLUMNS PER NODE. N.B. A NODE IS C NOT CONNECTED TO ITSELF ; C * * * B) . READ IN CARDIS ) UNDER THE SPECIFIED VARIABLE FORMAT CARD C * * * WHICH CONTAIN THE DISTANCES FROM NODE I TO ALL NODES SPECIFIED C * * * IN fl( A ] _ „•, _ ' •. . C * * * REPEAT INSTRUCTIONS 8(A) AND 8(B) IN SEQUENCE FOR ALL NOOES C * * * I = 1 , . . . »N £ ****************************************************************** _ _ _ _ _ . . . ; ; - -C * * * PROGRAMMER - R.A.WHITAKER ~ DEPARTMENT OF GEOGpAPHY,UNIVERSITY OF C * * * BRITISH COLUMBIA, JANUARY 1970 __ _ *** C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * DIMENSION S T O R E(1000)tLABEL( 1000),NCRlG( 1000)tMDEST< 1000) DIMENSION KS(IOOO),NS<1000),JS(lOOo),LS(1000) , I D<1000),MS(1000) D I MENS ION MD( 1000 ) , MIN ( 4000) , I S( 10 00) , U ( 1000) ,DS( 1000) ,PR03(18) DIMENSION FHT(20) ,NVEC(1000), 0IST< 1000) _ EQUIVALENCE (STORE(1),MIN(1)) REAL LA8EL C * * * * * * * * READ IN PROBLEM TITLE ANQ CONTROL CARD -2 FORMAT (313,211,12,12) 3 FORMAT (20 14) 11 FORMAT (18A4) . 13 FORMAT { 1H1/.16X, 17HPRGBLEM NAME ,18A4//) 14 FORMAT (7X,22HNO.OF'NODES IN NETWORK, I 5/6X,23HN3.0F ORIGINS SPECIF 1IED,I5/2BHNO.QF DESTINATIONS S PEC I F I ED, 15 / ) 15 FORMAT (20A4) * 17 FORMAT (20F4.1) 9 0 FORMAT ( 10X ,31HDISCOMMECTION IN NETWORK SYSTEM/lOX,2 2HCURRENT ORIG 1 IN IS "NODE, I5/10X, 29HN0DES LINKED BEFORE ERROR A RE , 2 01 4 / { 4 6 X , 20 I 4 ) 2) 135 FORMAT ( 1X,6H0RIGIN ,3X,11HDEST INATI ON,3X,8HDI STANCE,7X,14HR0UTED T 1HR0UGH/) 140 FORMAT ( 3 X , 1 4 , 6 X , 1 4 ,7X , F8.2 ,4X ,22 14/(40X,22 1 4 ) ) 147 FORMAT ( IH ) - . , . _ 155 FORMAT ( /3 X , 4HN0DE, 9 X , 27 HS HORT EST PATH NETWORK TABLE/) 160 FORMAT {7X.12I10) 170 FORMAT ( 17, 1 2F 10 . 2/ ( 7X , 1 2F 10... 2 ) )  175 FORMAT (15E6.0) 1 READ (5,11) (PROBi l ) , ! =1,18) READ (5,2) N ,MN ,MM ,KSW,KJV,NT,LFIN _ _ _ _ . WRITE ( 6 , 1 3 ) (PRC !B(I) ,I =1,18) WRITE ( 6 , 1 4 ) NtNN,MM IF ( N N . E Q . N ) GO TO 12 C * * * * * * * * READ IN VECTORS OF ORIGINS A N O DESTINATIONS IF OPTIONED RE AO (5,3) (NORlG(I)fl =1,NN) • GO TO 5 1 2 DO 4 I ~1,N 4 NORIG(I) =1 5 IF IMM.EQ.N) GO TO 8 _. DO 6 .! =1,N 6 MDEST(T) =0 R E A D ( 5 , 3 ) ( Mp( I ), I = 1 , H M ) ; . DO 7 I =1,MH J =MD(I) 7 M D E S T ( J ) = J • _• .,..„ . ;• GO TO 1 0 8 DO 9 I =1»N Mp( I , =1 ; . 9 M D E S T U ) =1 10 JJ = 0 R E A D ( 5 , 1 7 ) < O I S T(I),I •=lfNN)_ _ _ R E A D ( 5 , 1 5 ) ( F H T(I),I = 1 , 2 0 ) R E A D ( 5 , 3 ) ( N V E C ( I ) , I = 1 , N ) C * * * * * * * * R E A D I N A D J O I N I N G N O D E S A N D L I N K D I S T A N C E S  DO 4 0 I = 1 , N L = J J NT = N V E C ( I ) _ . .. . R E A D ( 5 , 3 ) ( K S ( J ) » J = 1 , N T ) R E A D ( 5 , FM T) ( S T O R E ( J ) , J = 1 , N T ) DO 2 0 J = 1 , N T . I F ( K S ( J ) . E Q . O ) GO TO 2 5 J J = J J + 1 I S ( J J ) = I __ _ ._ . I J ( J J ) = K S ( J ) 2 0 DS ( J J ) = S T O R E ( J ) C * * * * * * * * A R R A N G E E A C H MODES L I N K E D N E I G H B O U R S B Y A S C E N D I N G D I S T A N C E C * * * * * * * * O R D E R 2 5 L =L+1 I F ( L . F Q . J J ) G O TO 4 0 L J = J J - 1 2 7 I N D =1 DO 3 0 J ^ L , L J  I F ( D S ( J + 1 ) . G E . D S ( J ) ) GO TO 3 0 T E M P = DS ( J"+l ) NR =1 J ( J+l) _._ _._ „ _ D S ( J + 1 ) = O S ( J ) I J ( J + 1 ) - I J ( J ) D S ( J ) = TE H P J J ( J ) - N R IND = J 3 0 C O N T I N U E .... _ . . __. _ _ _ _ . . I F ( I N D - 1 ) 2 7 , 4 0 , 2 7 4 0 MS(I) - = J J . L V =0 ; DO 2 0 0 1 1 = 1 , N N C * * * * * * * * I N I T I A L I Z E A R R A Y S K - 0 D O 4 5 I = 1 , N . L S ( I ) = M S ( I ) - K ... . K . = M sen-...... ..: ID(I) = 0 4 5 L A B E L ( I) =10.0**1.0 R = 1 0 . 0 * * 1 0  K K = N O R I G ( T I ) K R = K K I D ( K K ) =0 _ • L A B E L ( K K ) = 0 . 0 D O 5 0 I =1 , J J 5 0 M I N ( I) = 0  N K = N - 1 . C * * * * * * * * C O M P U T E S H O R T E S T P A T H F O R S O M E N O D E . . . D O 1 0 0 . I N . = i ,N.K D O 6 5 I = 1 , J J I F ( I J ( I ) . N E . K K ) G O T O 6 5 M I N ( I ) = - 1  L J — I S ( I I L S ( L J ) = L S ( L J ) - l 6 5 C O N T I N U E _ T M I N = 1 0 . 0 * * 1 0 K V = 0 t .L = 0 :  D O 8 0 I K = I ,N L = L L + 1 L L = M S ( I K ) _ _ • I F ( L A B E L ( I K ) . E O . R ) G O T O 8 0 I F ( L S ( I K ) . E Q . O ) G O T O 8 0 D O 7 0 I - L ? L L  M J = I S ( I ) N J = I J ( I ) I F ( M I N ( I ) . N E . 0 ) G O T O 7 0 R M I N = " L A U E L ( M J ) + D S ( I ) I F ( R M I N . G E . T M I N ) G O T O 7 0 J R = 1  T M I N = R . M I N K K = N J K V = . M J . G O T O 3 0 7 0 C O N T I N U E 8 0 C O N T I N U E  I F ( K V . N E . O ) G O T O 9 5 C * * * * * * * * C H E C K E C R D I S C O N N E C T I O N K X = 0 . _ „ „......;. . : . _ . D O 8 5 I V = l , N I F ( I D ( I V ) . E Q . O ) G O T O 8 5 K X = K X + 1  M I N ( K X > = I 0 ( I V ) 8 5 C O N T I N U E I F ( K X . N E . 0 ) G O T O 8 7 ; KX =1 MIN(KX) = K R 8 7 WRITE (6,90) KR , ( MI N ( I V) , IV =1 ,KX .L GO TO 20 5 95 LABEL(KK) = T MIN NS(IN) = K V  JS(IN) = K K I D ( K K ) = IN MIN(JR) =1 100 CONTINUE KV =0 IE (KSW. EQ.OI GO TO 150  C * * * * * * * * TRACE SHORTEST PATH ROUTING DO 145 JX = l , N IF (MDEST(JX).EQ. 0) GO T 0 J 4 5 _ _ K =ID(JX) IF IK.EQ.O) GO TO 145 KS( 1 ) =J S( K)  KS(2) =NS(K) KK =2 LL =K-1 IF (LL.EQ.O) GO TO 125 DO 120 I = 1,LL K = K - 1 ;  IF (KS(KK) .NE.JS(K) ) GO TO 120 KK =KK+1 KS(KK) =NS(K) __ _ _ _ _ -120 CONTINUE C * * * * * * * * REARRANGE ROUTINGS IN INCREASING DISTANCE FROM ORIGIN 125 J =KK+l ; DO 130 I =1,KK * J -J-1 130 K[N(I). =KS(J) ._ . _ IF (KV.EQ.l) GO TO 133 K V =1 C * * * * * * * * PRINT OUT HEADINGS ' - WRITE (6,147) WR I TE (6,135) 133 WRITE (6, 140) KR , JX, LABEL C J.X) , (MIN(J) , J =1,KK) 145 CONTINUE 150 IF (KSW.NE.O) GO TO 153 IF (LV.NE.O? GO TO 162 •  LV =1 153 WR ITE ( 6, 155) 'WRITE (6,160) ( MD ( I ), I. = I» MM ) _/_ WRITE (6,147) 162 J =0 C * * * * * * * * PRINT SHORTEST PATHS FROM ORIGINS TO SPECIFIED DESTINATIONS DO 165 I =1,N IF (MDESTt I ) .EQ .6) GO TO 165 J =J + l STORE(J) = LABEL( I) IF (MDEST(I).eQ.NORIG( II })• S T O R E ( J ) =DISTCl-U 165 CONTINUE . . _ _„ ; ; WRITE (6,170) K R» ( STOR E ( I )* I = l f J ) IF ( KJV. EQ. 0) GO TO 200 WRITE (7,175) (STQRE(I)tl = 1»J) 200 CONTINUE. IF (LFIN)205,1,205 . 205... STOP. _• ; , _ END APPENDIX IV A C o m p u t e r P r o g r a m f o r W a r d ' s H i e r a r c h i c a l G r o u p i n g A l g o r i t h m w i t h D e n d o g r a m C *** c *** c *** c * * * ******************************* PROGRAM HIERARCHICAL GROUPING THIS PROGRAM PERFORMS A HIERARCHICAL GROUPING ANALYSIS ACCORDING... TO A CRITERION {MINIMIZATION. OF INCREMENT TO WITHIN GROUPS VARIATION AT EACH PAGES 236-244. IT STEP) GIVEN 8Y J.H.WARD.. SEE AMER IS ADAPTED FROM A BASIC CODING BY STAT ASSOC J . D. J . VELD MAN I fs C.**:* "FORTRAN PROGRAMMING FOR THE BEHAVIOURAL SCIENCES" CHANGES ARE AS C *** FOLLOWS C *** 1), SINGLE DIMENSIONED ARRAYS ARE USED TO DOUBLE THE SIZE OF.. _ C *** PROBLEM WHICH CAN BE HANDLED ON ANY MACHINE. ACCESSING THE I»J C *** VALUES IN MATRIX A MAY BE FRACTIONALLY INCREASED AS A" CONSEQUENCE. C *** 2), COMPUTING TIME FOR LARGE PROBLEMS IS APPROXIMATELY HALVED. C *** THIS IS ACHEIVED BY A), SHRINKING AT EACH STEP THE DOUBLE LOOP C *** SEARCH IN WHICH THE NEXT GROUP IS JOINED AND B), BY ELIMINATING C *** THE DCU3LE L OOP IN WHICH THE CURRENT MEMBERS. OF EACH GROUP ARE C *** FOUND, STORED, AND PRINTED. C *** 3), A CONTIGUITY CONSTRAINT IS INCLUDED.TWO GROUPS ARE CONSIDERED C *** CONTfGUOUS IF ANY MEMBER OF ONE GROUP IS CONTIGUOUS TO ANY MEMBER C *** OF ANOTHER GROUP C *** 4), A" DENDROGRAM IS PRINTED ON THE LINE PRINTER IF OPTIONED C ****************************** ********************** ************** C *****DECK SET UP AS FOLLOWS***** C *** 1), PROBLEM OR TITLE CARD. MAX! MUM OF 80 C *** 2), PROBLEM PARAMETER CARD COLUMNS ALLOWED C *** c * * * c * * * c * * * c * * * c * * * _ * * * c * * * c * * * c * * * c * * * COLS 1-3 NUMBER OF SUBJECTS COLS 4-6 NUMBER OF VARIA8LES COLS 7-9 LEVEL OF GROUPING TO BEGIN GROUP MEMBERSHIP PRINTING...... IF LEFT BLANK A CGMPLETE STEPWISE GROUPING IS PRINTED FOR ALL N-l STEPS - N.B. SINGLE MEMBER GROUPS ARE NOT PRINTED COL 10 PUNCH 1 IF STANDARDIZATION OF VARIABLES IS OPTIONED  COL 11 PUNCH 1 IF TREE GRAPH IS OPTIONED. MAX SIZE IS ARBITRARILY SET EQUAL TO 200 COL 12 PUNCH 1 IF CONTIGUITY CONSTRAINT IS IMPOSED COLS 13-15 MAXIMUM NUMBER OF CONTIGUITIES POSSESSED BY ANY SUBJECT.MAY BE LEFT BLANK IF 20 OR UNDER COL 16 PUNCH I IF IDENTIFICATION NUMBERS ARE TO BE READ IN. C C c C C c f -r *** *** * * * * * * * * THIS WILL BE NECESSARY IF RUN CONSECUTIVELY FROM 1,...,N COL 20 SINCE AS MANY DIFFERENT JOBS AS ARE CONSECUTIVELY PUNCH 1 IN COLUMN 20 FOR THE THE RUN 3), IDENTIFICATION NUMBERS OF ALL N SUBJECTS THE LABELS QF THE N SUBJECTS DO NOT REQUIRED LAST (GR MAY ..BE . RUN ONLY) JOB IN IF OPTIONED - 4 COL S • * * :** c. * * * C *** C *** c * * * PER SUBJECT 4) , VARIABLE FORMAT CARD FOR DATA E.G. (2F6.2 »3X,4F10.5 ) 5) , DATA CARDS—THESE MUST BE READ.IN IN VARIABLE ORDER I.E. READ.. IN AS AN N BY M MATRIX WHERE THE N ROW S ARE VARIABLES AND THE M COLUMNS ARE SUBJECTS • SUBJECTS C *** C *** C * * * I I I I I I I I VARIABLES I I 1 I I I I I I I I I I I I I C . * * * 6 ) . C O N T I G U I T Y M A T R I X . READ IN ONE ROW AT A T I M E C * * * PUNCH IN THE NUMBERS ( L A B E L S ) OF THOSE S U B J E C T S TO WHICH EACH.._ C *-*•* S U B J E C T I S CONTIGUOUS. 4 COLS PER NUMBER • E.G I F S U B J E C T NO 1 C * * * IS CONTIGUOUS TO S U B J E C T S 4 AND 7 THEN PUNCH I N 4 AND 7. I F THE C * * * MAXIMUM NUMBER OF S U B J E C T S ANY S U B J E C T I S CONTIGUOUS TO E X C E E D S 2< C * * * THEN TWO OR MORE CARDS W I L L BE R E Q U I R E D FOR EACH S U B J E C T C * * * * * * * * * * * * * * * * * * * * D I M E N S I O N I N G I N F O R M A T I O N * * * * * * * * * * * * * * * * * * * -C * * * I F N EQUALS MAXIMUM S I Z E OF PROBLEM WHICH CAN BE HANDLED ON ANY. C * * * PROCESSOR ( 1 9 9 FOR 32K STORAGE) THEN D I M E N S I O N S I Z E OF C * * * ARRAY A MUST BE SET EQUAL TO AT L E A S T ( ( N * N ) + N ) / 2 C * * * D I M E N S I O N S I Z E QF ARRAY KHOLD MUST BE AT L E A S T N*2  C * * * D I M E N S I O N S I Z E OF A L L OTHER A R R A Y S MUST BE EQUAL Trj N, E X C E P T FOR C * * * ARRAYS FMT AND T I T L WHICH R E M A I N AS D E F I N E D C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ^ c *** C * * * PROGRAMMER - R.A.WHITAKER - DEPARTMENT OF GEOGRAPHY,UNIVERSI T Y OF C »** B R I T I S H C O L U M B I A , A P R I L 1 9 6 9 •  C * * * c ****************************** ******************** ***************$ D I M E N S I O N DC 200) , KG ( 2 0 0 ) , W ( 2 00 ) , LC ( 2 00 ) ,CONT<200) _ DIMENS ION LAM ( 2 0 0 ) , NC! 2 0 0 ) , MC( 2 0 0 ) ,A( 2 0 0 0 0 ) DI MENS ION KHOLD ( 4 0 0 ) ,KST O R E ( 2 0 0 ) , F MT ( 20 ) ,T IT L ( 20 ), T 0 ( 2 0 0 ) E Q U I V A L E N C E (CONTf 1 ) ,D( 1 ) ) ' L FORMAT ( 3 1 3 , 3 1 1 , 1 3 , 1 1 , 1 4 ) 6 FORMAT ( 2 0 A 4 ) ._ 9. FORMAT. ( I X , .19HP.R0BLEM NAME.. *****... , 2 .0 A 4/.I 4 H NO .OF. S U B J E C T S , I 5 / 1 5 HN0. 10 F V A R l A B L E S , 1 4 ) 42 FORMAT ( 1 X , 1 7 H W [ T H C O N T I G U I T I E S / / / / ) 4 3 FORMAT ( I X , 2 0 H w I T H O U T C O N T I G U I T I E S / / / / ) 5 2 FORMAT ( 2 0 1 4 ) * 8 0 FORMAT ( / 5 H S T E P , I 4 , 2 X , I 4 , 1 X , 2 3 H G R O U P S A F T E R C O M B I N I N G GROUP, 1 4 , 1 0 I H AND .GROUP, I 4 , 2 O X , 6HERR0R= , F 1 3 . 4 , 5 X , I OHC UMULA TE D = , F 1 4 . 4 / ) 1 0 7 f OP MAT ( / / 1 X , 4 6 H C 0 N T I G U I T Y C R I T E R I O N P R E V E N T S FURTHER GROUPING) 1 1 0 FORMAT ( 8 X , 5HGR0UP t .1 5, I X , 1 H ( ,1 3 , 1 X , 6 H £ TE MS) , 1 2 1 4 / ( 3 0 X - , 1 2 1 4 ) ) 6 0 0 fQRMAT ( 1 H 1 ) \s -1- "i- 1^  * * * * * R E A 0 T I T L E CARD AND PROBLEM PARAMETER CARD 7 READ ( 5 , 6 ) ( T I T L ( I ) . I =1,20) R E AD ( 5 , 1 . ) NS ;,NV» K R - , K S , K T , KCONT, ML I J , K X N P =( ( NS*NS) + NS) /2 N H = N S * 2 WRITE ( 6 , 6 0 0 ) W R I TE ( 6 , 9 ) ( T I T L ( I ) , I = 1 , 2 0 ) , N S , N V IF (KCONT.EQ.O) GO TO 41 U P i T E ( 6 , 4 2 ) „ : _ . ; ;.. GO TO 45 4 1 WRITE ( 6 , 4 3 ) * * * * * R E A D I N I D E N T I F I C A T I O N NUMBERS I F O P T I O N E D 45 IF ( I J ) 4 4 , 4 7 , 4 -4. 4 7 D C 4 8 I --1 , NS 48 ID( I ) - I G O T O 4 6 R E A D ( 5 , 5 2 ) { 1 D { I ) , I = 1 , N S ) R E A D ( 5 , 6 ) ( F M T ( I ) , I = 1 , 2 0 ) I F { M L . E Q . 0 ) M L = 2 0 I F ( K P . E Q . O ) K P = N S - 1 I X = 0  L L = ( ( N S * N S ) + N S ) /2 D O 3 I = 1 , L L 3 A ( I ) = 0 . 0 _ _ L = 0 D O 2 4 J J = 1 , N V C * » » » * * » » R E A D I N S U B J E C T S I N V A R I A B L E O R D E R C * * * * * * * * I F O P T I O N E D ' R E A D ( 5 , F M T ) ( D ( J ) , J = 1 , N S ) . _ . I F { K S . E Q . O ) G O . T O 1 2 . . . B S U M = 0 . 0 S U M = 0 . 0 D O 4 I = 1 , N S  S U M = S U M * D ( I ) 4 B S U M = B S U M - t - ( 0 ( I ) * 0 ( I ) ) F M = N S -X B A R = S U M / F M S B A R = X 8 A R * X B A R S T D = S Q R T ( ( B S U M / F M ) - S B A R )  X = 1 . 0 / S T D D O 8 I = 1 , N S 8 DC I ) = ( D ( I ) - X B A R ) * X 1 2 N G = N S - 1 D O 1 7 1 = 1 , N G K. = 1 + 1  D O 1 7 J = K , N S X = D ( J ) - D ( I ) L = I + ( J * J ~ J ) / 2 „ A ( L ) = A ( L ) + ( X . * X ) C O N T I N U E C O N T I N U E  D O 3 2 I = 1 , N G K = 1 + 1 D O 3 2 J = K , N S _ _ L = I + ( J * J - J ) / 2 A ( L ) = A { L ) * 0 . 5 _ _ _ _ _ _ I F ( A ( L ) . EQ . 0 . 0 ) A ( L ) = 0 . 0 0 0 0 0 1  32 C O N T I N U E C * * * * * * * * R E A D I N C O N T I G U I T Y M A T R I X S T O R E A S M I N U S V A L U E S I N A I F ( K C O N T . E Q . O ) G O T O 5 5 D O 5 3 I = 1 , N S R E A D ( 5 , 5 2 ) ( K G ( J ) , J = l F M L ) D O 5 1 J J = 1 , M L :  I F ( K G ( J J ) . L E . I ) G O T O 5 1 J = K G ( J J ) ' • L L = ! + ( J * J - J ) / 2 . . . 4 4 4 6 - S T A N D A R D I Z E V A R I A B L E S 1 7 2 4 A ( L L ) = A ( L L ) * ( - 1 . 0 ) 5 1 C O N T I N U E 5 3 C O NT INU E„ _ ..... . ' _ _ ...... 5 5 -NG =NS KN =NS C * * * I N I T I A L I Z E A R R A Y S  D O 6 0 I = I t N S K G ( I ) = 1 .. KST O R E ( I ) = 1 L__ _. _ _ :_ NX ( I ) =1 LAM( I) =I 6 0 W( I ) = 1 . 0  C * * * * * * * * O B T A I N N E X T G R O U P T O B E J O I N E D KZ = 0 : . .... C U H = 0 . 0 _ _ _ _ . _ . . , 6 5 NG = NG - 1 I F ( N G . E Q . O ) G O T O 1 2 0 X = 1 0 . 0 * * 1 0 : • KV.' = KN-1 D O 7 5 II = 1 ,KW . . I =MC ( I I ) . . _ _ . _ _ M J = 1 1 + 1 ' 0 0 7 0 J J =MJ,KN j = M c t J j ) : L L = I + ( J * J - J ) / 2 I F ( K C O N T . E Q . O ) G O T O 6 7 I F ( A { L L ).. G E . 0 . 0 ) G O . T O . . 70.._ _ ._. 6 7 L J = I*( 1 + 1 ) / 2 L S = J * ( J+l ) / 2 D X --ABS( At L L ) » - A ( L J ) - A ( L S )  I F ( D X . G E . X ) G O T O 7 0 X = D X M = J 7 0 C O N T I N U E 7 5 C O N T I N U E • C * * * * * * * * C H E C K W H E T H E R C O N T I G U I T Y C R I T E R I G N ( I F O P T I O N E D ) P R E V E N T S C * * * * * * * * - F U R T H E R G R O U P I N G . . „ I F . ( K , C O N T . E Q . 0 . O R . X . N E . 1 C ..0** 1.0.) . . G O _ T A _ 7 6 . W R I T E ( 6 , 1 0 7 ) G O T O 1 2 0 7 6 K Z =Kl + l ; :  C U M ' = C U M + X I F ( K T . N E . l ) G O T O 77 K J - = N S + K Z , _ .; K H C L D ( K Z ) = L K H O L D I K J ) = M C O N T ( K Z ) = X  7 7 W R I T E ( 6 , 8 0 ) K Z » N G , L •» M t X » C U M C * * * * * * * * U P D A T E G R O U P M E M B E R S H I P A N D S E T A R R A Y F O R D E N D O G R A M L . J K = W ( L ) •" ; .. _ _ K = LAM ( M ) DO 85 I =ltNS IF ( K G ( I ) . N E . M ) GO TO 81 .„ _ KG ( I I =.L L'AK(I) = L A M ( L ) K S T C R E ( I ) =KSTQRE ( I ) +1. JK  GO TO 85 31 I F ( L A M ( I ) .GT.K) L A M ( I ) = L A M ( I ) - 1 85. CONTINUE .... . . . .. \ I F (KCONT.EQ.O) GO TO 89 C ******** ENSURE NEW GROUP HAS C O N T I G U I T I E S OF E L I M I N A T E D GROUP DO 8 7 I = 1,NS  I F ( M . GT . I ) GO TO 8 2 LL =M+(1*1-1 ) 7 2 GO TO .83 82 L L = I + ( M * M - M ) / 2 83 I F ( A ( L U . G E . O . O . C R . L . E Q . I ) GO TO 87 . IF (L .GT . I ) GO TO 84  LL =.L+( 1*1-1 )/2 GO To 86 84 L L = I + ( L * L - L ) / 2 86 I F ( A ( L L ) . L T . 0 . 0 ) GO TO 87 A(LL-) = A ( L L ) * ( - 1 . 0 ) 8 7 CONTINUE ; ; C ******** UPDATE ERROR P O T E N T I A L S - NEW GROUP TO R E M A I N I N G GROUPS 89 WS =W(L)+W(M) L L =L+(M*M-M)/2 „. _ _ X = A B S l A ( L L ) ) * W S . L J = L * { L + l ) / 2 LS = M » ( M - H ) / 2 .  Y = A B S ( A ( L J ) ) * W ( L ) + A B S ( A ( L S ) ) * W ( M ) A ( L J ) = A B S ( A ( L L ) ) DO 95 I - 1 ,NS . IF ( I .EQ. L .OR. K G ( l ) .NE. I ) GO TO 9 5 LL = I * ( I + 1 ) / 2 ' XY =ABS( A( L L ) ) »W( I )  I F ( I .GT .L ) GO TO 88 L L = I + ( L * L - L ) / 2 L S = I + ( M * M - M ) / 2 _ _ __ _ '_ _' _ _ ___ GO TO 91 88 I F ( I.LT.M) GO TO 9 0 LL =L + ( 1*1- I )/2 LS =M+( 1*1-1 )/.2 GO TO 9 1 90 . LL = L + ( I * I - n / 2 _ : LS =1+(M*M-M)/2 91 A TEMP = ( AB S ( A ( L L ) ) # I W' I ) +1-H-L ) ) +A8 S ( A ( L S ' ) * ( W (I ) + W ( M)') + X - Y - X Y ) / (W ( I 1 )+HS )  IF ( A ( L L ) . - L T . O . O l GO TO 93 A(L L) =A TEMP GO TO 9 5 9 3 . A ( L L ) = A T E M P * ( - 1 . 0 ) 9 5 C O N T I N U E W ( L ) = W S „ _ . : C * * * R E W R I T E A R R A Y O F R E M A I N I N G G R O U P S SANs T H E O N E J U S T E L I M I N A T E D J J = 0 D O 92 I •= h, K N :  I F ( M C ( I ) . ' E Q . M ) G O T O 9 2 J J = J J + 1 M C ( J J ) = M C ( T ) ; . ___ j = M c m N C ( J J ) = W ( J ) 9 ? C O N T I N U E ; :  K N = K N - 1 I F ( N G . G T . K P ) G O T O 6 5 C * * * * * * * * . O R D E R . M E M B E R S O F . E A C H . G R O U P R E A D Y F OK P R I N T I N G J J = 0 0 0 9 6 I = I , K N L L = N C ( i ) ; [ .  N c m = J J + I 9 6 J J = J J + L L . 0 0 1 0 0 I = 1 , N S •.. . J = L A M ( I ) K = N C ( J ) • L C ( K ) = 1 . _ _ 1 0 0 N C ( J ) = K + 1 C * * * * * * * * P R I N T G R O U P M E M B E R S H I P .. J J =0 . __ D O 1 1 5 I = 1 i K N J = J J + 1 . J J = N C C I ) - i : . L = ( . J J - J ) + 1 I F ( L . L E . l ) G O T O 1 1 5 L I . = L C ( J ) I F ( I J . E Q . O ) G O T O 1 1 4 00 1 1 2 K = J , J J L J = L C ( K ) • .  1 1 2 L C ( K ) - 1 D ( L J ) • ' 1 1 4 W R I T E ( 6 , 1 1 0 ) L L , L , ( L C ( L J ) , L J = J • J J ) 1 1 5 . . . . . . C O N T I N U E _ _. ; G O T O 6 5 1 2 0 I F ( K T . N E . l ) G O T O 1 3 5 N Y = N S - N G  I F ( N G . N E . O ) G O T O 1 4 5 C * * * S E T U P R E A D Y F O R D E N D O G R A M - G R O U P I N G C O M P L E T E D O 1 3 0 I. = 1 , N Y _ : L L = K S T O R E ( I ) K . G I L L ) = I 1 3 0 C O N T I N U E . G O T O 1 3 8 C * * * S E T U P R E A D Y F O R D E N D O G R A M - G R O U P I N G I N C O M P L E T E D U E T O C O N T I G U I T Y C * * * C R I T E R I O N . . . . . 145 L J =0 DO 160 I =1 ,NS . . IF (KG(I).NE.I) GO TO 160 _ _ _ _ L =0 DO 150 J =1 ,NS IF (KG(J ) . N E.I) GO TO 150  L = L+1 N J =KSTORE< J ) L C ( N J ) = j „ „ . : _ . . „ ._ ._ . v  150 CONTINUE DO 155 J = 1 , L LJ =LJ+1  LAM(LJ) = L C ( J ) 155 CONTINUE 160. .CONTINUE •. _ CALL TREE (A,N Y , L J ,NP,NH»NS, MC »L AM,CONT » KHCL D) GO TO 135 138 CALL TREE (A,NY,NS,NP,NH,NS,MC,KG,CONT,KHOLD)  135 IF (KX.NE•1) GO TO 7 STOP .... END . SUBROUTINE TREE (DATA,NY,N,NP,NH,NS,JOKE,KG,CONT,KHOLD) C ******** THIS SUBROUTINE PRINTS A DENDOGRAM ON THE LINE PRINTER C ******** DIMENSION SIZE OF CAT A MUST BE SET EQUAL TO AT LEAST (N*2)+1 C ******** FOR A MAXIMUM OUTPUT OF 120 CHARACTERS PER RECORD CHANGE THE C ******** VALUES OF LMJ AND LJR IN THE DATA STATEMENT TO 84 AND 116 DIMENSION KG(NS) ,JOKE(NS) ,CONT(NS) ,KHOLD(NH) DIMENSION DATA(NP), IMT ( 29 ) , F SP EC ( 8) INTEGER DATA,FSPEC DATA JV, JX, JY, JW,LMJ ,L JR/lH ,1H* ,1H._,1H| ,96,123/  DATA FSPEC/1H7, 1H6, 1H5, 1H4, 1H3, LH2, 1H1, 1H0/ DA TA. I MT/1H( , 1H3 ,1HI , 1 H5 ,1 H, , 1 H2 ,1 HX , 1 H, , 1 HE, 1 HI , 1 HO , 1 H. , 1H8 , IH, , 1 1H2, 1HX , IH, , 1HA , 1H1, IH , , 1H1, 1HX, IH, ,. 1HI , 1H0,1H1 ,1 HA ,1 HI ,1H ) /. 1 FORMAT ( 14 ,15(4X,14 ) ) 2 FORMAT { 5X,13HITEMS GROUPED,6X,13(4X,14)) 3 FORMAT (26X, 13 (4X, I 4) ) . 16 FORMAT (IX,4HST EP»4X,1 HI,4X »1HJ » 4X,5 HERROR,4X, 13(4X, 14)) 17 FORMAT (30X,13(4X,I4)) 20 FORMAT (3X, 129AI ). .. __ 2 5 FORMAT (2X,I 4,15(4X,14)) 26 FORMAT (16(4X,I4)> 27 FORMAT ( 2X , 16(4X, 14 ) )  32" FORMAT (29X,A1) 34 FORMAT {29X,A 1,IX, 101Ai) 55- FORMAT ( 1HI ) . ] _'.__.„;_ ; __ C *** INITIALIZE VARIABLES NJ =NY-1 L =(N*2)+1  NM =0 LX =LMJ MR J .=1 KJ =LMJ/2 KL =LJR/2 FORM =10.0 : NL =KJ IF { N. LE-. KJ) NL =N 4 WE IT F ( 6, 55 )  IF (LX.G6.L-2) GO TO 19 MJ =LX . .... GO TO 13 1 19 MJ =L 13 IF ( MR J.NE. 1) GO TO 21 KW =LMJ  C ******** PRINT OUT HEADINGS WRITE (6,2) (KG(I).rl = 1,.NL,4). .... WRITE (6,3) (KG(I),I =2,NL,4) WRITE (6,16) (KG(I),I =3,NL,4) WRITE (6,17) (KG(I),I =4,NL,4) GO TO 22  21 NL =KJ+1 KJ =KJ+KL IF (KJ.GE.N) KJ =N _ KW =LJR WRITE (6,1) (KG (I), I =NL,KJ,4) JJ =NL + 1 • WRITE 16,25) (KG(Ii,I =JJ,KJ,4) JJ =JJ+1 WRITE (6, 26) I KG ( I ), I =JJ,KJ,4) JJ =JJ+1 WRITE (6,27) (KG (I), I =JJ,KJ,4) C ******** SET UP ARRAY ORDER FOR PRINTING 22 JS =0 DA TAIL) =JX GO TO 40 •_ _. 9 JS =JS+1 K =0 L =0  DO 5 I •= 1, N J =!<G( I ) . .. L = L + 1 ., _ ; JOKE(J) =L DA TAIL) =JW L _ _ L _ _ DATA(L) =JV 5 CONTINUE L =L + 1 DA TA < L) =JX C *** OBTAIN NEXT LINE FOR PRINTING 14 K =K + 1  IF (K.GT.NJ) GO TO 4 0 . KP =NS+'K _ KM =KHOL0(K ) _ _ KK = KHOL DIKP) LL =(JGKE(KK)-JOKE(KM))-l LS = JOKE( KM) + 1 : LJ =(LS+LL)-l DO .10 I =LS,LJ 1 0 D A T A ( i ) = J Y  C ******** OBTAIN APPROPRIATE FORMAT AND PRINT LINE IF (MRJ.NE.l) GOTO 28 " IF (CONT(K ) .LT.FCRM) GO TO 80 _ . -NM =NM+1 IMTQ3) =FSPEC(NM) ' FORM =F0RM*l0.0  80 WRITE (6,IMT) K,KM,KK,CONT(K ),JX,(DATA(I ) , I •=1,MJ)-GO TO. 11 28 .WRITE (6,20) (DATA? I ) , I = MRJ,MJ) . C *** CHANGE INDEXES MODIFY LINE 11 LS =LS-1 LJ = LJ+1 . DO 12 I =LS,LJ 12 DATA(I) =JV LS =(LL/2 )+l . • _ JOKE(KM) =JOKE(KM)+LS LS =JCKE(KM) DATA(LS) =JW . GO TO 14 C ******** CHECK LINE STATUS WITH REPECT TO 80RDERS AND MODIFY ARRAYS 40 JJ = MJ-1 _ LJ =MRJ+1 IF (MRJ.NE.l) GO TO 37 IF (NL.GT.KJ) GO TO 56  DO 43 I =1,J J 43 0 AT A(I> =JV WRITE (6,34) JX , ( DAT A ( I ) , I = 1, M J ) .... .._ GO TO 65 56 WRITE (6,32) JX 65 IF (JS) 35 ,9 ,85 ; 37 IF(LX.GE.L-2)G0 TO 42 WR ITE(6, 20) JV GO TO 800 42 DO 31 I=MRJ,JJ • " 31 OATA(I)=JV WRITE (6,20) (DATA? I ) , I -=MRJ,MJ) : '  800 J F ( J S . EO.O)GO TO 9 8 5 DO 33 I=MRJ.MJ',2 33 DA TA.U>=JX .. DO 70 I =LJ,MJ,2 70 DATA(I) =JV IF ( MR J .NE. 1) GO TO 39 •  WRITE (6,34) J X, (DA T A(I),I =1,MJ) GO TO 3 3 39 WRITEI 6,20) ( DATA (I ) , 1= MR J , M J)._ 38 I F I L X . G E . L - 2 I G 0 TO 41 C * * * * * * * * SET UP I N D I C E S FOR NEXT S E C T I O N OF ARRAY MR J =MRJ + KW . . „ ' ._„. L X = L X + t J R GO TO 4 41 RETURN  END APPENDIX V An A l g o r i t h m f o r t h e G r a p h i c a l G e n e r a t i o n o f a D e n d o g r a m From a n y S t e p w i s e H i e r a r c h i c a l G r o u p i n g M e t h o d G r a p h i c a l d i s p l a y and e v e n i n t e r p r e t a t i o n o f t h e r e s u l t s o f a n y t y p e o f s t e p w i s e h i e r a r c h i c a l l i n k a g e a n a l y -s i s c a n f r e q u e n t l y be e n h a n c e d by t h e p r e s e n t a t i o n o f t h e c o m p l e t e d g r o u p i n g i n t h e f o r m o f t h e f a m i l i a r t r e e g r a p h o r d e n d o g r a m . An a l g o r i t h m f o r a c h i e v i n g t h i s g o a l i s p r e s e n t e d i n t h i s a p p e n d i x . The b a s i c o b j e c t i v e i s t o d e t e r m i n e a u n i q u e p o s i -t i o n i n g o f t h e o r d e r e d g r o u p s s u c h t h a t i n t h e d e s c e n d e n t t r e e p r o j e c t e d f r o m t h i s o r d e r i n g no h o r i z o n t a l s p a n ( l i n e ) , w h i c h b r i d g e s t h e v e r t i c a l b r a n c h e s r e p r e s e n t i n g t h e j u n c t i o n o f two g r o u p s a t some s t e p i n t h e l i n k i n g p r o c e s s , w i l l i n t e r s e c t t h e s p a n o f some o t h e r p a i r o f g r o u p s j o i n e d a t a n y p r i o r o r s u b s e q u e n t s t a g e i n t h e g e n e r a t i o n o f t h e d e n d o g r a m . T h i s g o a l w i l l be a s s i s t e d by t h e c o n v e n t i o n t h a t on t h e c o m p l e t i o n o f f ' t h e : g r o u p i ng o f any p a i r o f g r o u p s i and j i n t o a s i n g l e member g r o u p i , a l l c u r r e n t members o f t h e j * * 1 g r o u p w i l l be g i v e n t h e l a b e l i . F o r a c o m p l e t e s t e p w i s e l i n k a g e o f n s u b j e c t s some n - l s t e p s , o r p a i r s o f v g r o u p s , . w i 1 1 be g e n e r a t e d . To c l a r i f y t h e a l g o r i t h m i c p r o c e d u r e t h e f o l l o w i n g d e f i n i t i o n s w i l l be made a t t h e o u t s e t . F o r any s t e p i n t h e g r o u p i n g p r o c e d u r e a) L e t C be some v e c t o r i n w h i c h i s s t o r e d t h e f i l e o f t h e number o f members e a c h g r o u p c u r r e n t l y c o n -t a i n s . Any e l e m e n t c ^ , k = l , ••• , n w i l l t h e r e f o r e be t h e number o f s u b j e c t s i n g r o u p k. b ) L e t A be a s i n g l y d i m e n s i o n e d a r r a y w h i c h l a b e l s t h e g r o u p t o w h i c h e a c h o f t h e n s u b j e c t s b e l o n g s . Thus t h e v a l u e c o n t a i n e d i n t h e kz^ c e l l o f a r r a y A , k = l , , , , n , i s t h e g r o u p t o w h i c h t h e k s u b j e c t i s c u r r e n t l y a s s i g n e d . c ) L e t B be some a r r a y i n w h i c h e a c h o f t h e k e l e m e n t s , k = 1 , *"* , n , c o n t a i n s an i n d e x r e f e r e n c i n g t h e o r d e r i n w h i c h t h a t s u b j e c t ( t h e k s u b j e c t ) j o i n e d t h e g r o u p w h e r e i n i t i s c u r r e n t l y l o c a t e d . The a l g o r i t h m f o r o b t a i n i n g t h e r e q u i r e d u n i q u e o r d e r i n g o f a l l n - l p a i r e d g r o u p i n g s may be d e s c r i b e d i n t h e f o l l o w i n g s t e p s . 1 . I n i t i a l i z e a r r a y s A , B , and C as f o l l o w s : S e t t h e k e l e m e n t i n A e q u a l t o k . T h a t i s a - ^ k f o r a l l k - l j ' - ' v n . S e t a l l n e l e m e n t s o f a r r a y s B and C e q u a l t o 1 . 2 . L e t i and j be t h e two g r o u p s j o i n e d a t some ( b e g i n n i n g w i t h t h e f i r s t ) s t e p i n t h e g r o u p i n g a n a l y s i s . N o t e t h a t b e c a u s e o f t h e a f o r e m e n t i o n e d c o n v e n t i o n t h e l a b e l d e n o t i n g g r o u p i i s a l w a y s < j . 3 . L e t £ = c.j , t h e number o f g r o u p s c u r r e n t l y a s s i g n e d t o g r o u p i . 4 . S c a n a l l n e l e m e n t s i n A , k = 1 , • *• , n „ and f o r a n y e l e m e n t i n A w h i c h s a t i s f i e s t h e c o n d i t i o n a ^ = j ( a ) S e t a ^ i , w h i c h u p d a t e s m e m b e r s h i p i n g r o u p i a n d ( b ) a d j u s t t h e ktfn e l e m e n t o f B as f o l l o w s : L e t m = b ^ and r e s e t b ^ s u c h ' t h a t ' b^ = Z+m = m + c ^ . 5 . I n c r e m e n t t h e number o f members i n g r o u p i by t h e number o f members i n g r o u p j so t h a t c . = c . + I. 3 R e p e a t s t e p s 2 t o 5 f o r a t o t a l o f n - 1 i t e r a -t i o n s . 6 . R e d e f i n e j t o be some i n d e x , i n i t i a l l y s e t e q u a l t o z e r o , and g e n e r a t e a new a r r a y D w i t h n e l e m e n t s , w h i c h w i l l c o n t a i n t h e o r d e r e d p o s i t i o n i n g o f t h e g r o u p s as s o u g h t . 7 . I n c r e m e n t j by 1 . L e t i be t h e i n d e x s t o r e d i n t h e j * * 1 c e l l o f B , so t h a t i = b . . Now s e t d . = j . 3 ' 8 . R e p e a t s t e p 7 n t i m e s . To i l l u s t r a t e t h i s a l g o r i t h m c o n s i d e r t h e p r o b l e m o f g e n e r a t i n g a d e n d o g r a m f r o m t h e s t e p w i s e h i e r a r c h i c a l g r o u p i n g o f 6 s u b j e c t s l a b e l l e d c o n s e c u t i v e l y 1 t h r o u g h 6 . S u p p o s e f u r t h e r t h a t t h e f o l l o w i n g g r o u p s a r e p a i r e d i n e a c h o f t h e n - 1 s t e p s : i ) g r o u p s 1 and 6 , i i ) g r o u p s 3 - a n d 4 , i i i ) g r o u p s 2 and 5 , i v ) g r o u p s 1 and 2 , and v ) g r o u p s 1 and 3 . The u n i q u e p o s i t i o n i n g o f . t h e o r d e r e d p a i r s . f o r t h e g e n e r a t i o n o f t h e d e n d o g r a m f r o m t h i s s t e p -w i s e g r o u p i n g i s i l l u s t r a t e d i n F i g u r e 1 7 . F i g u r e 18 shows a 40 s u b j e c t d e n d o g r a m c o m p i l e d f r o m a s e t o f d a t a g r o u p e d by t h e w e l l known Ward ( 1 9 6 3 ) a l g o r i t h m . V S t e p 1 2 3 4 5 G r o u p s 1 and 6 3 and 4 2 and 5 1 and 2 1 and 3 1 2 3 4 5 6 1 2 3 4 5 1 1 2 3 3 5 1 1 2 3 3 2 1 1 1 3 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 2 1 1 1 2 2 2 1 3 1 2 4 2 1 3 5 6 4 2 1 6 2 5 3 4 D e n d o g r a m f o r a 6 S u b j e c t E x a m p l e 1 1 1 1 1 1 2 1 1 1 1 0 2 1 2 0 1 0 2 2 2 0 0 0 4 - 0 2 0 0 0 6 0 0 0 0 0 S t e p 1 2 3 4 5 G r o u p s 1 6 3 4 2 5 1 2 1 3 F i g u r e 1 7 : I l l u s t r a t i o n o f D e n d o g r a m A l g o r i t h m f o r a 6 S u b j e c t E x a m p l e . ITEMS GROUPED STEP I J EPROR 13 T_TTT I I l_l 24 35 31 36 14 3". 13 3T 0.0C 000100 0. 0CC0C100 1 .OOOOOOOO 1.OOOOOOOO l.6f66f633 ? .OOPOCOOO 10 11 12 2 6 22 6 e 24 7 19 29 IB 15 35 2.OOOOOOOO 2.OOOOOOOO 2.0C0000C0 2.66666603 3.00000000 3. 9999Q71 4 I I I I l_l I l _ 13 1* 15 16 17 IB 2 21 25 2 24 32 23 3 40 12 4.Of OOOOOO 4 .OOOOOOOO 4.orcooooo 5.5fOO0OCO 5.5^999180 6.CCOOOOOO 19 20 21 22 23 24 25 21 10 38 27 33 7.OOOOOOOO 7.83332634 8.00000000 8 .66666603 9.4C997711 10.0000000 25 26 27 28 29 30 23 1 24 25 2 1 1 26 20 30 39 6 23 10.0000000 11.3333330 11.3999786 13.0000000 15.2618618 18 .4999847 31 32 33 34 35 1 21 2 22 11 2 2C. 1 109619 21.3332367 33.9046936 35. 6666260 47.6425476 63.6665039 37 2 38 1 39 1 22 111.214752 11 162.280635 2 326. 645264 J • • * * * * * • • • * STOP 0 EXECUTION TERMINATED F i g u r e 17 G r a p h i c a l D i s p l a y o f a 40 S u b j e c t D e n d o g r a m G r o u p e d by t h e Ward A l g o r i t h m A P P E N D I X VI A C o m p u t e r P r o g r a m f o r E s t i m a t i n g t h e M e d i a n s o f a W e i g h t e d G r a p h S u b j e c t , i f O p t i o n e d , * t o S i d e C o n s t r a i n t s N . B . A much s h o r t e r and f a s t e r p r o g r a m o f t h i s a l g o r i t h m w h i c h i n c o r p o r a t e s o n l y t h e m o s t e f f e c t i v e o f t h e v a r i o u s h e u r i s t i c a l t e r n a t i v e s , and w h i c h d o e s n o t e v a l u a t e c o m b i n a t o r i a l e q u a l i t i e s i f t h e y o c c u r , i s n e a r i n g c o m p l e t i o n and w i l l be a v a i l a b l e o n . r e q u e s t f r o m t h e D e p a r t m e n t o f G e o g r a p h y , U n i v e r s i t y o f B r i t i s h C o l u m b i a . c ****************************************************************^^ c T H I S P R O G R A M E S T I M A T E S T H E P M E D I A N S < P L O C A T I O N S ) O F A W E I G H T E D c *** G R A P H S U B J E C T , I F O P T I O N E D , T O C E R T A I N S I D E C O N S T R A I N T S . S O L U T I O N S . c *** W I L L B E O P T I M A L O R N E A R O P T I M A L c •v M E T H O D : S E E •» -AN A L G O R I T H M F O R E S T I M A T I N G T H E M E D I A N S O F A W E I . G H T E C c *** G R A P H S U B J E C T T O S I D E C O N S T R A I N T S 1 - R • A . WHI T A K E R P H D T H E S I S c *** D E P A R T M E N T O F G E O G R A P H Y U N I V E R S I T Y O F B R I T I S H C O L U M B I A 1 9 7 1 c ** ****** **************************** ****************************** c *** N . B . N E T W O R K N O D E S S H O U L D B E L A B E L L E D . I N S E Q U E N C E 1,-...,N c ** * * * O E C K S E T U P A S F O L L O W S * * * * * c *** A ) . P R O B L E M T I T L E C A R D M A X I M U M O F 8 0 C O L U M N S A L L O W E D c *** B ) . C O N T R O L C A R D c *** 1 . . C O L S 1 - 3 N U M B E R O F N O D E S O N T H E N E T W O R K I N ) c *** 2 . . C O L S 4 - 6 N U M B E R O F P O T E N T I A L L O C A T I O N S I T E S ( M ) c *** 3 . . C O L S 7 - 9 N O . O F N O D E S T O B E F O R C E D I N T O T H E S O L U T I O N I F A N Y c *** S E E I N S T R U C T I O N D c *** 4 . . C O L 1 0 T H I S C O N T R O L S P E C I F I E S T H E M A J O R P R O G R A M O P T I O N ; c *** P U N C H 1 F O R T H E C L A S S I C A L P L A N T L O C A T I O N P R O B L E M . H E R E T H E c i»< v V A L U E O F P I S U N K N O W N A N D B Q T H P L A N T A N D T R A N S P O R T C O S T S c * * * O B T A I N . B E G I N N I N G A T A S P E C I F I E D L O W E R B O U N D ( S E E I N S T R U C T I O N c *** K - 1 ) P I S I N C R E M E N T E D A T E A C H S T E P U N T I L T H E P + l S O L U T I O N H A S c *** A T O T A L C O S T > T H A N T H A T O F T H E P S O L U T I O N c >Jc P U N C H 2 F O R S T A T I C L O C A T I O N O P T I O N W H E R E T H E V A L U E ( S ) O F P A R E c *** P R E - D E T E P M I N E O B Y T H E U S E R . ( S E E I N S T R U C T I O N K 2 ) c *** P U N C H 3 F O R D Y N A M I C L O C A T I O N B A C K W A R D R E C U R S I V E O P T I O N c *** P U N C H 4 F O R D Y N A M I C L O C A T I O N F O R W A R D R E C U R S I V E O P T I O N c *** 5 . . C O L S 1 1 - 1 2 T H I S C O N T R O L S P E C I F I E S T H E N U M B E R O F M E D I A N c *** I N D I C A T O R S T H E P R O G R A M E X P E C T S T O B E R E A D I N . I T I S D E P E N D E N T c **•* U P O N T H E V A L U E P U N C H E D I N C O L U M N 1 0 A S F O L L O W S ; c * * * I F C O L U M N 1 0 C O N T A I N S 1 T H E N P U N C H I N 1 *** I F C O L U M N 1 0 C O N T A I N S 2 T H E N P U N C H I N T H ? N U M B E R O F M E D I A N S . c . J , •v -V '4? T O B E C O M P U T E D E . G . A 3 W O U L D I N D I C A T E T H A T T H E P R O G R A M c *** E X P E C T S T O R E A D I N S. C A L C U L A T E 3 S E P E R A T E M E D I A N V A L U E S r . *** I F C O L U M N 1 0 C O N T A I N S 3 0 * 4 T H E N P U N C H I N 2 c J - ,tj 6 . . C O L 1 3 P U N C H 1 I F P L A N T S I Z E U P P E R B O U N D S A R E O P T I O N E D . {M A 5 c •*** 7 . . C O L 1 4 P U N C H 1 I F A M A X I M U M D I S T A N C E C O N S T R A I N T ( F R O M E A C H c *** N O D E T O S O M E S O U R C E ) I S O P T I O N E D ( K Y ) c •j, J , - i - "V 8 . . C O L 1 5 P U N C H 1 I F P L A N T C O S T S A R E I N C L U D E D I N T H I S R U N ( M Y ) c *** • 9 . ' . 1 6 . T H I S C O N T R O L O P T I O N S A U S E R S U P P L I E D W E I G H T I N G S Y S T E M c *** F O R P A R T I T I O N I N G T H E G R A P H I F T H E U S E R D O E S N O T W I S H T O R E L Y c *** S O L E L Y U P O N T H O S E G E N E R A T E D B Y T H E A L G O R I T H M . T H E M A X M I N c * * * P A R T I T I O N I N G C R I T E R I O N I S U S E D . P U N C H 0 I F T H I S O P T I O N I S N O T c * * * S P E C I F ! E D - P U N C H 1 I F E X T E N D E D ( R O W B Y R O W ) S E A R C H I S R E Q U I R E D c * * * P U N C H 2 F O R S H O R T S E A R C H P A T T E R N I N W H I C H A N I N I T I A L S E T I S c *** N O T D E F I N E D F O R E A C H R O W B U T O V E R A L L R O W S . . . . . . . . c .1* .A. -t* -V- -f 1 0 . . C O L 1-7 P U N C H 1 I F U S E R S U P P L I E D D A T A S U B R O U T I N E I S T O 8 E c * • * C A L L E D . S E E S E C T I O N O N U S E R S U P P L I E D S U 9 - R 0 U T T N F S ' L H W ) • c v# T - 1 1 . . C O L I B P U N C H 1 I F U S E R S U P P L I E D S U B R O U T I N E F O R R E E S T I M A T I N G c *** U N I T T R A N S P O R T C O S T S I S . C A L L E D . S E E U S E R S U P P L I E D S U B R O U T I N E S c * * * 1 2 . . C O L 1 9 F I R S T P A R T I T I O N I N G H E U R I S T I C O P T I O N - T H E S I N G E R c *** C R I T E R I O N - P U N C H 0 I F E X T E N D E D .. ( R O W B Y R O W ) S E A R C H I S c ^ ' ^ ^ R E Q U I R E D - P U N C H 1 : F O R S H O R T S E A R C H P A T T E R N - P U N C H 2 TO c aj; 4c E X C L U D E T H I S H E U R I S T I C O P T I O N c -V -.V. *V* -V ^r* . . . 1 3 . . C O L 2 0 . S E C O N D P A R T I T I O N I N G H E U R I S T I C O P T I O N P U N C H 0 IF c •v -c E X T E N D E D S E A R C H (ROW- B Y ROW) S E A C H I S R E Q U I R E D . P U N C H 1 F O R c *** S H O R T S E A R C H P A T T E R N - P U N C H 2 T O E X C L U D E T H I S H E U R I S T I C c T- -V O P T I O N c *** 1 4 . . C O L 2 1 T H I R D P A R T I T I O N I N G H E U R I S T I C O P T I O N - . W E I G H T S H I F T c sjc _fr E L I M I N A T I O N M E T H O D - P U N C H 0 T O I N C L U D E T H I S H E U R I S T I C P U N C H 1 . . .c T Y .. T O E X C L U D E I T .. ; _ .... c -A, v'-*V* 1 5 . . C O L 2 2 F O U R T H P A R T I T I O N I N G H E U R I S T I C - W E I G H T E D V E R S I O N c A. J , X O F C O O P E R S M E T H O D - P U N C H 0 TO I N C L U D E T H I S H E U R I S T I C P U N C H 1 c TO E X C L U D E T H I S H E U R I S T I C c -JU ..i, .A, IT "** S O M E C O M M E N T S O N T H E ATJOVE C H O I C E S . F R O M O N E T O A L L O F T H E c *** S P E C I F I E D M E T H O D S MAY B E U S E D T O G I V E T H E I N I T I A L S T A R T . c. T H E T H I R D H E U R I S T I C . . G I V E S . . I N G E N E R A L . T H E B E S T S T A R T I N G . . . c a. S O L U T I O N ' F O L L O W E D B Y T H E S I N G E R C R I T E R I O N . C O M P U T A T I O N T I M E S c F O R T H E P A R T I T I O N I N G P R O C E D U R E S A R E S I G N I F I C A N T L Y A F F E C T E D c *** B Y B O T H T T E N U M B E R O F P O T E N T I A L P L A N T S I T E S A N D BY T H E c *** H E U R I S T I C S C H O S E N . T H E S H O R T V E R S I O N S O F T H E F I R S T T W O c JU JU -»» -V H E U R I S T I C S A R E T H E F A S T E S T B U T T H E E X T E N D E D V E R S I O N S O F T H E S E c M E T H O D S A R E C O M P U T A T I O N A L L Y T H E M O S T E X A C T I N G . . F O R V E R Y L A R G E c _J; 3$. P R O B L E M S ( S E V E R A L H U N D R E D P O T E N T I A L P L A N T S I T E S ) T H E c ,o ' i * *T* V S H O R T V E R S I O N S O F T H E H E U R I S T I C S S H O U L D B E U S E D I F C O M P U T I N G c A- -V J P *V* *1~ I 1 T I M E IS S I G N I F I C A N T F O R T H E U S E R . I N S U C H C A S E S T H E F O U R T H c -V •** H E U R I S T I C ( A N D P E R H A P S T H E T H I R D F O R T H E B I G G E S T N E T W O R K S ) c MAY A L S O B E I G N O R E D . F O R E X A M P L E T O S O L V E A 1 0 M E D I A N " N E T W O R K c. P R O B L E M . W I T H 1 0 0 . P O T E N T I A L P L A N T S I T E S W I L L T A K E A B O U T 2 T O c A- -X/ -.V •V- 1* *>* 2 . 5 M I N U T E S ( D E P E N D I N G O N T O T A L N U M B E R O F N O D E S ) T O E X E C U T E c •A. JJV -Y- *»" I * O N A N IBM 3 6 0 / 6 7 I F A L L F O U R H E U R I S T I C S A R E U S E D . U S I N G T H E c *** •SHORT S I N G E R C R I T E R I O N S E A R C H A N D T H E T H I R D P A R T I T I O N I N G c :Js H E U R I S T I C W I L L R E D U C E T H I S T O L E S S T H A N 1 M I N U T E A S A R U L E c S U M M A R IZ I N G - - E X C E P T F O R S M A L L P R O B L E M S . U S E E X T E N D E D S I N G E R A N D . c * * * T H E T H I R D H E U R I S T I C . F O R L A RG E T O V E R Y L A R G E P R O B L E M S U S E . T H E . c A- A A -r- S H O R T V E R S I O N S O F T H E F I R S T TWO H E U R I S T I C S A N D T H E T H I R D c * * * H E U R IS T IC , . F O R E X C E P T I O N A L L Y L A R G E P R O B L E M S T H E T H I R D H E U R I S T I C c • J * -V "A. V * i f -r M A Y B E D R O P P E D I F A C O M P U T A T I O N A L L Y F A S T R E S U L T I S R E Q U I R E D c A *i» -v* 1 6 . . C O L 2 3 P U N C H 0 I F A L L C O M B I N A T I O N S O E E Q U I D I S T A N T N O D E S c 'lc ^  ( N O D E S E Q U I D I S T A N T F R O M M O R E T H A N O N E C U R R E N T M E D I A N ) A R E TO c 4- JU •T* *F B E E V A L U A T E D D U R T N G . . C O M P U T A T I O N . M A X I M U M N U M B E R C O N S I DER E D . . IS...'. c •J* A A T - -V R E S T R I C T E D . P U N C H 1 TO E X C L U D E A L L C O M B I N A T I O N S S E A R C H c A A A •v '4* -v 1 7 . . C O L 2 4 I F U S E R S U P P L I E D G U E S S A T O P T I M A L S O L U T I O N I S c A A A 1- -V O P T I O N E D P U N C H A S F O L L O W S • 1 I F T H I S I S I N A D D I T I O N T O P R O G R A M c G E N E R A T E D B E G I N N I N G S E T S OR 2 W I T H O U T T H E P R O G R A M G E N E R A T E D c A A A S E T S . N . B . T H I S O P T I O N I S O N L Y A L L O W A B L E I S C O L U M N 1 0 I S __.c. A V . A ^ i - -»» -#* P U N C H E D . WI TH . A . 2 . U S E R . M U S T . E N S U R E . P L A N T . . S I Z E C O N S T R A I N T IS c A A 1- -V N O T V I O L A T E D c - I , J . A 'i» •*«* 1 8 . . C O L 2 5 L E A V E B L A N K I F P L A N T C O S T S A R E N O T O P T I O N E D O R I F c *** T H E Y A R E A N D B O T H S E T S O F P L A N T C O S T H E U R I S T I C S . A R E T O B E c A A A A C T I V A T E D . P U N C H 1 I F T H E S E C O N D S E T O F P L A N T C O S T H E U R I S T I C S c * * * I S T O B E I G N O R E D . c. ..... 1'9 . . . . . . C O L _ . 2 6 . L E A V E . . .BL.ANK.....I F . . . .THE . . .PL AN T...S.I Z E . C O N S T R A I N T O P T I O N ... C * * * I S NOT S P E C I F I E O . I F IT IS O P T I O N E D THEN PUNCH 0 TO A C T I V A T E C * * * BOTH SET S OF P L A N T S I Z E C O N S T R A I N T H E U R I S T I C S . PUNCH 1 TO C *** A C T I V A T E THE F I R S T SET O N L Y ; PUNCH 2 TO A C T I V A T E THE SECOND C *** SET QNLY.--N.B. THE SECOND SET IS S L I G H T L Y TO BE P R E F E R R E D TO C *** THE F I R S T . C *** 20... COL 27 L E A V E BLANK IF DYNAMIC L O C A T I O N O P T I O N I S NOT-, C *** S P E C I F I E D . P U N C H 1 I F A D I F F E R E N T SET OF MARKET DEMANDS (NODE C *** W E I G H T S ) I S TO BE READ I N FOR EACH TIME PERIOD - I F NOT PUNCH ( C * * * . 2 1 . . COL 28 L E A V E BLANK I F DYNAMIC L O C A T I O N O P T I O N I S NOT C * * * S P E C I F I E D . PUNCH 1 I F A D I F F E R E N T SET OF TRANSPORT COSTS C * * * ( D I S T A N C E S ) I S TO BE READ IN FOR E A C H TIME P E R I O D - OTHERWISE C * * * L E A V E BLANK '_ . C * * * 22 COL 29 L E A V E BLANK IF DYNAMIC L O C A T I O N O P T I O N I S NOT C *** S P E C I F I E D . PUNCH 1 I F A D I F F E R E N T SET OF P L A N T C O S T S I S TO BE c *** READ I N VI A. SUBROUT I NE DATA ( S E E S E C T I O N ON USER S U P P L I E D C *** S U B R O U T I N E ) FOR EACH TIME P E R I O D - OTHERWISE L E A V E B L A N K . C *** 2 3 . . COLS 30-39' MAXIMUM COST BUDGET CONSTRAINT ON THE TOTAL C *** S Y S T E M IF O P T I O N E D . ONLY S O L U T I O N S FOUND S AT IS FY I NG • T H IS ARE C *** PR INT ED C * * * 2 4 . . COL 4 0 PUNCH T I F T H I S IS THE LAST (OR O N L Y ) J O B IN THE RUN C *** C ) . READ IN THE NODE D E S I G N A T I O N S ( L A B E L S ) OF A L L THE P O T E N T I A L C *** L O C A T I O N S I T E S - 4 COLS PER NODE. IGNORE T H I S I N S T R U C T I O N I F C * * * A L L NETWORK NODES ARE P O T E N T I A L PLANT S I T E S C * » * D ) . READ IN THE L I S T OF NODES TO BE FORCED INTO S O L U T I O N I F ANY C *** M.B. THESE NODES MIGHT L A B E L AN E X I S T I N G S Y S T E M OF L O C A T I O N S c * * * TO WHICH NEW SOURCES ARE TO BE ADDED. A L T E R N A T I V E L Y THEY W I L L c *** R E F L E C T SOME A PRIOR I .R EASON S WHICH R E Q U I R E A C E R T A I N P L A N T ( S ) C *** TO BE I N S O L U T I O N C * * * E ) . READ IN V A R I A B L E FORMAT CARD FOR P L A N T S I Z E U P PER BOUNDS I F C * * * OPTIONED E.G. ( 1 0 F 5 . 1 ) C * * * F) . READ I N PLANT S I Z E UPPER BOUNDS UNDER T H I S FORMAT I F O P T I O N E D C * * * G ) . READ IN V A R I A B L E FORMAT CARD FOR MAXIMUM D I S T A N C E C O N S T R A I N T C *** .. I F O P T I O N E D C * * * H ) . READ I N MAXIMUM D I S T A N C E V A L U E S I F O P T I O N E D UNDER T H I S FORMAT C *** I;)-. READ IN V A R I A B L E FORMAT CARD FOR USER S U P P L I E D WEIGHTING . C * * * SYSTEM FOR I N I T I A L P A R T I T I O N I N G I F OPT IGNEO C * * * J ) . READ IN THE USER S U P P L I E D WEIGHTS UNDER THE S P E C I F I E D FORMAT C *** K ) . READ IN A VECTOR ( 4 COLS PER NODE) WHICH C O N T A I N S THE V A L U E S C * * * OF. A L L L O C A T I O N ! S) . TO BE COMPUTED. T H I S WILL BE DEPENDENT UPON C *#* THE V A L U E S P R E V I O U S L Y PUNCHED UNDER I N S T R U C T I O N S E 4 ( C 0 L 1 0 ) C ANO 85 (COLS 1 1 - 1 2 ) C *** I . . IF COL 10 I S PUNCHED 1 THEN PUNCH THE LOWER BOUND S T A R T I N G C *•** POINT FOR I N C R E M E N T A T I O N OF THE V A L U E S OF P. E.G. A 2 WOULD C * * * INSRUCT THE PROGRAM TO B E G I N WITH THE 2 MEDIAN S O L U T I O N AND C * * * INCREMENT THE VALUE OF P S U C C E S S I V E L Y U N T I L THE P + l S O L U T I O N C *** EXCEEDS THE P S O L U T I O N AT WHICH T I M E E X E C U T I O N T E R M I N A T E S C * * * 2.. I F COL 10 I S PUNCHED WITH A 2 THEN THE PROGRAM EX P E C T S TO C *** COMPUTE AS MANY MEDIA N S AS ARE S P E C I F I E D I N C O L S 1 1 - 1 2 . E.G. •C *** I F T H I S I S 3 THEN 3 MEDIAN V A L U E S WILL- BE~ READ I N . SUPPOSE C *** THESE TO BE 4,7 AND 9 TH EN THE 4,7 AND 9 M E D I A N L O C A T I O N S OF c * * * THE NETWORK WILL BE. COMPUTED.. ... . ... ... ; C *** 3 . . IF COL 1 0 I S PUNCHED 3 THEN THE PROGRAM E X P E C T S TO READ IN C *** 2 MEDIAN V A L U E S ; AN UP PE R ST ART I NG AND LOWER F I N I S H I N G VALUI C * * * __. R E S P E C T I V E L Y - I N THAT ORDER. E.G. I F THESE V A L U E S WERE 7 C * * * AND 2 THEN THE PROGRAM WOULD COMPUTE THE 7 M E D I A N S O L U T I O N C *** F I R S T FOLLOWED BY THE 6 MEDIAN S O L U T I O N - (CHOSEN FROM AMON( C * * * THE P R E V I O U S 7 ) , AN. SO ON DOWN TO THE 2 M E D I A N S O L U T I O N C *** 4.. I F COL 10 IS PUNCHED 4 THEN THE PROGRAM E X P E C T S TO READ IN C *** 2 MEDIAN V A L U E S ; A LOWER S T A R T I N G AND UPPER F I N I S H I N G V A L U E C *** _. R E S P E C T I V E L Y - IN THAT ORDER. E.G. I F .THESE V A L U E S WERE.. 2 C *** AND 7 THEN THE PROGRAM WOULD COMPUTE THE 2 M E D I A N SOLUTION C *** F I R S T FOLLOWED BY THE 3 MEDIAN S O L U T I O N - ( C O N T A I N I N G THE 2 C *** MEDIANS FOUND IN T H E P R E V I O U S S T E P AS FORCED NODES) AND SO C * * * ON UP TO THE 7 MEDIAN S O L U T I O N C * * * L ) . READ IN V A R I A B L E FORMAT CARD FOR R E A D I N G IN U N I T TRANSPORT ..C. __: COSTS . (SHORT.. PATHS J FROM EACH NODE TO ALL P O T E N T I A L M E D I A N S . C *** H ) . READ I N THE SHORTEST P A T H MATRIX UNDER THE S P E C I F I E D FORMAT C *** N.B. ***WARN I N G * * * THERE WILL BE N ROWS, ONE FOR EACH NETWORK C *** NODE AND M COLUMNS, ONE FOR EACH P O T E N T I A L PLANT S I T E . THE J T E C *** COLUMN IN THE A ( I , J ) SHORT PATH M A T R I X MUST R E F E R E N C E C *** THE D I S T A N C E FROM NODE I TO THE J T H NODE S P E C I F I E D IN THE L I S T C .**.*_ OF P O T E N T I A L PLANT S I T E S ( S E E C ) . E.G. ELEMENT A l l , 5 ) MUST BE.. C * * * THE D I S T A N C E FROM THE F I R S T NODE TO THE F I F T H MEMBER OF THE C *** S U B S E T OF P O T E N T I A L PLANT S I T E S . FURTHERMORE TO SAVE STORAGE C * * * I T I S ASSUMED THAT THE SHORT PATH MATRIX I S S Y M E T R I C I . E . THE C *** D I S T A N C E FROM NODE I TO NODE J MUST BE THE SAME AS FROM NODE C *** j TO NODE I FOR EACH SUCH P A I R OF ELEMENTS IN THE M A T R I X . C **.* IF T H I S . I S NOT SO THE. PROS RAM . WI LL R E Q U I R E SOME M O D I F I C A T I O N . C *** WITH MATRIX A R E D E F I N E D AS A DOUBLY D I M E N S I O N E D A R R A Y . C *** N.B. IE DYNAMIC L O C A T I O N I S O P T I O N E D S E E I N S T R U C T I O N R F I R S T . C * * * N) . READ IN V A R I A B L E FORMAT FOR NODE ( M A R K E T ) DEMANDS ( W E I G H T S ) C *** P ) . READ I N MARKET DEMANDS (NODE W E I G H T S ) UNDER FORMAT S P E C I F I E D C *** N.B. I F DYNAMIC L O C A T I O N I S OPTIONED SEE I N S T R U C T I O N R F I R S T . C ***. Q) . READ I N PLANT COST F U N C T I O N S V I A SUBROUTINE DATA I F O P T I O N E D . . C *** N.B. I F DYNAMIC L O C A T I O N IS O P T I O N E D S E E I N S T R U C T I O N R F I R S T . C *** R) . IGNORE T H I S I N S T R U C T I O N I F 1 ) , THE DYNAMIC L O C A T I O N O P T I O N I S C *** NOT S P E C I F I E D OR 2 ) , IF IT I S BUT ONLY ONE SET OF TRANSPORT C *** C O S T S ( D I S T A N C E S ) , NODE WEI G H T S , AND PLANT COSTS ( I F O P T I O N E D ) C *** IS TO BE READ I N . OTHERWISE R E P E A T I N S T R U C T I O N S L-Q FOR EACH C. ***....• T I M E P E R I O D WHERE A P P L I C A B L E . E.G...A 3 T I M E P E R I O D S O L U T I O N C *** WITH D I F F E R E N T TRANSPORT COST ( I . E * SHORT PATH) M A T R I C E S AMD C *** NODE WEIGHTS (ONE SET FOR EACH P E R I O D ) WILL R E Q U I R E R E P I T I T . I O N C *** OF I N S T R U C T I O N S L ~ P •THREE T I M E S . FQR A S I N G L E SET OF TRANSPORT C ***. COSTS,BUT D I F F E R E N T NODE WEIGHTS FOR A L L THREE P E R I O D S CARRY C *** GUT I N S T R U C T I O N S L AND M ONCE BUT N AND P THREE T I M E S . C ***.. N.B.. ***WARNI NG*** THE ORDERING BY WHICH THE D I F F E R E N T .SET.S . . . C * * * ARE READ IN I S DEPENDENT UPON WHETHER THE FORWARDS OR C *** BACKWARDS R E C U R S I O N PROCEDURE IS O P T I O N E D . FOR A FORWARDS C *** R E C U R S I O N THE F I R S T S E T ( S ) READ I N MUST REFER TO T I M E P E R I O D C * * * .1, THE SECOND SET TO T I M E P E R I O D 2, AND THE L A S T SET TO THE C *** LAST T I M E P E R I O D . FOR THE BACKWARDS R E C U R S I O N T H I S O R D E R I N G C.*** . ... I S R E V E R S E D . THE F I R S T S E T ( S ) READ I N MUST REFER TO THE L A S T C *** TIME P E R I O D , THE SECOND SET TO THE SECOND TO L A S T TIME P E R I O D , C *** AND THE L A S T SET TO THE F I R S T T I M E P E R I O D . . C * * * S ) . REA D IN USER S U P P L I E D GIIE S S A T 0 ? TI M A L SO L U T I ON IF OPTIONED c -JL. .1, -Jy T T *J- A L L O W 4 C D L S P E R N O D E . R E A D I N A S M A N Y S E T S A S A R E S P E C I F I E D c U, „(„ - J , '1* B Y T H E M E D I A N V A L U E G I V E N I N C O L 1 1 - 1 2 . N . B . T H I S O P T I O N I S c if. A L L O W A B L E O N L Y I F C O L 1 0 I S P U N C H E D W I T H A 2 . c * * * * * * * * * * * * * * * * * U S E R S U P P L I E D S U B R O U T I N E S c •A, •A, JL, ' l * *f f- T H E F O L L O W I N G - 2 S U B R O U T I N E S M U S T B E D E F I N E D B Y T H E U S E R I F P L A N T c • C O S T S A R E I N V O K E D . c _U -V V V 1 ) . S U B R O U T I N E D A T A c J , - C -V T H I S S U B R O U T I N E I S U S E D T O R E A D I N P L A N T C O S T F U N C T I O N S F O R . c. 5jc :_C . E A C H P O T E N T I A L P L A N T . S I T E W H E R E T H E . F U N C T I O N S V A R Y A M O N G . P L A N T : c » • » v l , -V •V- 1» C O N S I D E R T H E F O L L O W I N G E X A M P L E : S U P P O S E E A C H P O T E N T I A L S I T E H A ! c -1, _v •V V V A C O S T F U N C T I O N O F T H E F O R M Y = A + 8 X W H E R E A A N D B A R E C O N S T A N T ! c -JU. -V 1- X V A R I E S A C C O R D I N G T O P L A N T S I Z E A N D T H E R E A * E 1 0 S U C H S I T E S c s^. V- V * f S U B R O U T I N E D A T A c *** C O M M O N D A T I 2 , 1 0 ) c * * * 0 0 5 I = 1 , 1 0 c *** R E A D 1 5 , 1 ) A , B c •J, ~Jr -Ji. T 'C T- 1 F O R M A T ( 2 F 5 . 2 ) c J c j / - J _ D A T ( 1 , 1 ) = A c - i , ^ 1 * D A T ( 2 , I ) =B -c *** 5 . C O N T I N U E ;. ..... c «V v*, „V , r -y . R E T U R N . . . . . . . . . . . . . . . . c x>- J»-•*** -v- -a*- E N D c «JC - A . •V ' f 1^ I N T H E F I R S T R O W O F M A T R I X D A T A L L T H E A C O N S T A N T S A R E S T O R E D c I N T H E S E C O N D R O W O F M A T R I X D A T A L L T H E 3 C O N S T A N T S A R E S T O R E D c 't* V T H E C O N T E N T S O F D A T A R E T R A N S F E R R E D T O S U B R O U T I N E C O S T B Y T H E c V V V C O M M O N S T A T E M E N T c c • t : V *t *•* — -N . B . T H I S S U B R O U T I N E . . C A N 3 E S U P P L I E D A S A D U M M Y P R O V I D I N G T H E . P L A N T C O S T F U N C T I O N S A R E I D E N T I C A L F O B . A L L P L A N T S . I N S U C H A c • J - v l , ,u ' t * *y C A S E T H E C O S T F U N C T I O N C A N B E D E F I N E D D I R E C T L Y I N S U B R O U T I N E c -J' A -V V C O S T c 2 ) . S U B R O U T I N F C O S T c J , vl* -V - i - -v* T H I S S U B R O U T I N E C A L C U L A T E S T H E C O S T F U N C T I O N S U P P L I E D B Y T H E c •A- -A- -'» T" 'i- U S E R . I T S A R G U M E N T L I S T M U S T C O N T A I N T H E ' F O L L O W I N G ; S U M . T H E . . . c Jl, -J* v*„ 't* *c C U R R E N T P L A N T S I Z E W H I C H I S C A L C U L A T E D I N T H E M A I N P R O G R A M , L R c •A* ~.«, -V v T H E I N D E X R E F E R R I N G T O S O M E P L A N T , A N D Y T H E C O S T O F P R O D U C I N G c s_ A M O U N T S U M A T S O U R C E H . - E . G . S U P P O S E T H E C O S T F U N C T I O N I S O F c »(, s»-•Y* '<~ T H E F O R M Y = A + B X , A N D T H A T T H E C O N S T A N T S A A N D B A R E N O T c •3* -V -A- I D E N T I C A L F O R E A C H P L A N T . T H E N , A S S U M I N G T H I S I N F O R M A T I O N T O c . . . H A V E B E E N P F A D I N V I A S U B R O U T I N E D A T A . A S J U S T . D E M O N S T R A T E D . .WE . c H A V E c v j , ' l - S U B R O U T I N E C O S T ( S U M , Y , L R ) c v'» -1, •V *-» -V- D O U B L E P R E C I S I O N Y c *** C O M M O N D A T ( 2 , 1 0 ) c *** Y = D A T ( 1 , L V ) + ( D A T ( 2 , L V ) * S U M ) c. R E T U R N c V- -.o V -v- '** E N D c . -a. I F P L A N T C O S T S A R E T H E S A M E O V E R A L L P L A N T S T H E N T H E I N D E : X L V ; c ,v ' I * '<* ' i » W I L L N O T B E S P E C I F I C A L L Y A C T I V A T E D B U T I S N E E D E D I N T H E c OJ. -A, '1* 1* -V* T H E S U B R O U T I N E A R G U M E N T L I S T . N • B • V A R I A B L E Y - M U S T A L W A Y S B E c ,1, v/. D O U B L E P R E C I S I O N E D . - „.. c -V -A. -X, -r* *f -V T H E . F O L L O W I N G S U B R O U T I N E . . M A Y , . B E S U P P L I E D B Y T H E U S E R . . A T H I S c vJL, "V* I " D I S C R E T I O N I F U N I T T R A N S P O R T C O S T S R E F L E C T N O N - L I N E A R I T I E S A N D c *** P L A N T S I Z E U P P E R B O U N D S O B T A I N f C *** SUBROUTINE TCOST C *** THIS SUBROUTINE MIGHT BE WRITTEN IN THE FOLLOWING CIRCUMSTANCES. C.***. IF THE PLANT SIZE CONSTRAINT IMPOSITION RESULTS IN SOME MARKET C *** BEING SUPPLIED FROM MORE THAN ONE SOURCE (PLANT) THEN THE UNIT C **,* TRANSPORT COST' OVER SOME LINK (WHICH IS AN AVERAGE VALUE BASED ON C *** COST OF SHIPPING ALL DEMAND OVER THAT LINK),MAY BE MADE REDUNDANT C •*** IF THE COST FUNCTION IS NGN LINEAR. THE USER CAN SPEC IF Y A C •*** FUNCTION IN TCOST" TO RECALCULATE NEW UNIT COSTS 8ASED ON THE C *** CURRENT AMOUNT ACTUALLY ALLOCATED OVER THAT LINK...THE ARGUMENT C *** LIST IS IDENTICAL TO THAT OF SUBROUTINE COST I.E. SUBROUTINE C *** TCOST (SUM,Y,LR) C *** A VERSION OF~THE BASIC ALGORITHM INCLUSIVE ONLY OF FORCED NODE C *** CAPABILITY AND RESTRICTICNS CN POTENTIAL S IT ES IS AVAILABLE. C *** THIS WILL PRINT. ALL COMBINATIONS OF MEDIANS IF THE OPTIMAL SETTS C *** NOT UNIQUE BUT WILL NOT EVALUATE, CURING COMPUTATION, COMBINATION; C **.* CF NODES WHICH MIGHT BE EQUIDISTANT TO MORE THAN ONE MEMBER OF T Hi C *** CURRENT MEDIAN LIST. THIS VERSION IS CONSIDERABLY FASTER FOR THE C *** BASIC ALGORITHM THAN IS THIS PROGRAM C ******************** DIMENSIONING INFORMATION *******************; C *** DIMENSION THE FOLLOWING ARRAYS TO.AT LEAST N (THE NUMBER OF C *** NETWORK NODES) STORAGE SPACES --- MC,HN,IN,ID C *** DIMENSION THE FOLLOWING ARRAYS TO AT LEAST M (THE NUMBER OF C *** POTENTIAL SITES) SPACES UP,JN,C,QS,PL, IB,KC,NZ,MZ,MXR,I X,JCt C *** NST,MT,IZ,VR,Q,MKB,V,F,W,NP,MP C *** THE FOLLOWING ARRAYS MUST BE DIMENSIONED AT LEAST EQUAL TO M7 C *** WHERE M7 IS. SOME AUDIT IONAL STORAGE SET AS IDE FOR SHARED NODES OR. C *** ALLOCATIONS. M7 IS DEFINED AT THE BEGINNING OF THE PROGRAM AS 50 C *** BUT MAY BE INCREASED FOR LARGE PROBLEMS RS,JG,JA C *** DIMENSION THE FOLLOWING ARRAYS TO AT LEAST N+M7 Lw , NC, B,, D, JO, C N 8 , KD» DN , MD,CJ C *** DIMENSION THE FOLLOWING ARRAYS TO AT LEAST M+M7 E,P,MB,K8,LC C *** DIMENSION ARRAY JQ EQUAL TO AT LEAST M+M8 ' WHERE. M8 IS FOR • C *** ADDITIONAL STORAGE AND IS DEFINED AT THE BEGINNING OF THE PROGRAM C *** AS 100 BUT MAY BE INCREASED FOR LARGER PROBLEMS WHERE THE "PLANT C *-** SIZE CONSTRAINTS ARE VERY RESTRICTIVE ' ; C *** ARRAY A MUST SE DIMENSIONED TO AT LEAST ( ( M *M )+M ) / { M *( N-M ) ) SPACES C *** DIMENSION ARRAY KG TO KR*9 STORAGE SPACES WHERE KR IS THE MAXIMUM C .**•* NUMBER .OF'.?.. MED IANS SPEC IF I ED. QUR 1 NG ANY ONE RUN „ C *** C *** OBJECT CODE = APPROXIMATELY 75000 BYTES• C »**  C * * * * * * * * * * * * * * ***** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * c * * * C *** PROGRAMMER - R .A ,WHI TAKER. DEPARTMENT OF GEOGRAPHY,UNI VERSI TY. OF.. C *** BRITISH CLUMBI A,SEPT EMBER 1970 C *** _C * * * * * * * * * * * * * * * * * * *•* * * * * * * * * * * * * * * * * »•* * * * * * * * ****** * * * * * * * * * * * * * * * DOUBLE PRECISION E , S , S 2, S3 , S 4, S6, S SS, A SUM , P DIMENSION PR0B(20) ,FMT1(20) ,FMT2(20) ,FMT3(20 )» FMT4 (20 ) , FMT 5(20) . DIMENSION LW(230),NC( 230), 81 230) ,D(230) ,MC( 130) »HN(180) ,PL(55) 5 22 2 3 138 01 MENS ION C ( 5 5 ) , I 8 ( 5 5 ) , K C | 5 5 ) , N Z ( 5 5) ,M8l 1 2 0 ) ,V(55) DIMENS I ON J N ( 5 5 ) , U P ( 5 5 ) , M Z < 5 5 ) , M X R ( 5 5 ) , J D { 2 2 0 ) , IN( 1 8 0 ) DIMENS ION I X I 5 5) ,E( 1 2 0 ) , L C ( 1 2 0 ) , K 8 ( I 2 0 ) , J C ( 5 5 ) , N B ( 2 2 0 ) CI MENS ION N S T ( 5 5 ) , M T ( 5 5 ) , I D ( 1 8 0 ) , I Z < 5 5 ) , K G ( 3 0 0 ) ,MKB( 55) DI ME NS I ON K 0 ( 2 3 0 ) , D N ( 2 3 0 ) ,RS ( 5 0 ) , J Q ( 150 ) , V R ( 5 5 ) D I M E N S I O N At 5 0 0 0 ) , J G ( 5 0 ) , J A ( 5 Q ) , J U D ( 1 0 ) , P ( 1 8 0 )  D I M E N S I O N Q S ( 5 5 ) , Q ( 5 5 ) , M 0 ( 2 2 0 ) , C J ( 2 2 0 ) D I M E N S I O N F ( 2 0 ) ,NP( 20) , M P ( 2 0 ) ,W ( 2 0 ) COMMON/COM 1/ M M , K P , M A , N M , J S , K Y , J 2 , M R J , M Y , K V P , 1 K P , I Z P , M G , I K S C0MM0N/C0M2 / AM , AP., DS , BD, SX , PM, SY E Q U I V A L E N C E ( KC ( 1) , I X ( 1) ) FORMAT ( 2 Q A 4 )  FORMAT ( 3 1 3 , I I , 1 2 , 1 7 1 1 , F 1 0 . 3 , I I ) . FORMAT ( 5X, 85H.NUMBER OF L O C A T I O N S S P E C I F I E D 1 NODES TO BE FORCED INTO S O L U T I O N ) FORMAT ( 5 X , 7 4 H N U H B E R OF L O C A T I O N S S P E C I F I E D IS L E S S THAN NUMBER Of IS GREATER THAN NUMBER 1 OF P O S S I B L E L O C A T I O N S ) FORMAT ( 4 0 H C A P A C I T Y CONSTRAINT C R I T E R I O N P R E V E N T S A , I A , 1 X, 17HLQC A 1 1 S Y S T E M H I G H L Y UN IT EACH AND C O N S T R A I N E O R E - R U N ) H O N S O L U T I O N / ) 134 FORMAT ( 5X, U O H E X E C U T I O N T E R M I N A T I N G 1 I N C R E A S E PLANT U P P E R BOUNDS BY J U S T 1 READ ( 5 , 3 ) ( P R O B ( I ) , I = 1 , 2 0 ) C ******** R E A D I N CONTROL AND T I T L E C A R D S , ' READ ( 5 , 5 ) N,M,MM,NM,K A , M A , K Y t M Y , M O , M W , J 2 , J 5 , J 6 , J 9 , J 7 , L J S , K Z P , J 1 , 1 IK- S , I 2 » I KK , J K K , BD , LF I N M7 =50 M8 =100 W. I TE ( 6 , 2) ( PROB (1 ) , I = 1 , 2 0 ) FORMAT ( 1 H 1 , 2 X , 1 9 H P R 0 B L E M NAME LH =M+-M7 , 2 0 A 4 / / ) IH =N+M7 L I = I I M *M ) +M ) / 2 -M M * ( N-I KP = C(M*M) -M)/2 SY = 1 0 . 0 * * 1 5 M) ) K3 =0 4 6 L E =d LZ =M*9 I F IBO.EQ.O.O). FORMAT ( 2 0 A 4 ) I F (N.EQ.M) GO FORMAT ( 2 0 1 4 ) BD =10. 0**15 TO 8 C *•** 8 . 61 ***** R E A D READ ( 5 , 6 ) GO TO 9 DO 6 1 I =1,N JN(1 > =1 IF (N.EQ.M) GO IN P O T E N T I A L L O C A T I O N ( J N ( I ) , i =1,M) TO 6 0 SI TE S C *•** c c * * * ***** SET UP I N D E X I N G SYSTEM IN ARRAY 10 FOR R E F E R E N C I N G SHORT ***** PATHS STORED I N ARRAY A ***** SET UP ARRAY MC A S FOLLOWS C * * * * * * * * Z E R O V A L U E S - I N D I C A T E T H I S N O D E N O T A P O T E N T I A L P L A N T S I T E C * * * * * * * * L E S S T H A N Z E R O I N D I C A T E S T H I S N O D E T O B E F O R C E D I N T O S O L U T I O ? C * * * * * * * * G R E A T E R T H A N Z E R O I N D I C A T E S . N O D E T O B E . A P O T E N T I A L L O C A T I O N C * * * * * * * * A C T U A L V A L U E S O F E L E M E N T S R E F E R E N C E T H E C O L U M N I N D I C E S O F THf C * * * * * * * * S H O R T P A T H M A T R I X A D O 4 5 I =T,N  1 0 ( 1 ) = 0 4 5 M C ( I ) = 0 D O 5 0 I = 1 , M :  L = J N ( I ) I D ( L ) =1 5 0 M C ( L ) =1 :  J = M D O 5 5 I = 1,N I F ( I D ( I ) . N E . . O . ) . . G O . _ T O . 5 5  J = J + 1 I D ( I ) = J 5 5 C O N T I N U E  G O T O 7 0 6 0 D O 6 5 I - 1 , N I D ( I ) -1 _ _. _ . „ _ _ . _ ;__ _ 6 5 M C ( I ) =1 7 0 I F { M M . E Q . 0 ) G O T O 9 0 C * * * * * * * * R E A D I N M O D E S ( I F A N Y ) W H I C H M U S T B E I N F I N A L S O L U T I O N  R E A D ( 5 , 6 ) ( N C ( I ) t I = 1 , M M ) D O 8 0 I = l , M M . J = N C ( I ) N P ( I ) = M C ( J ) M P ( I ) = M C ( J ) 8 0 M-C ( J ) = M C ( J ) * ( - - ! ) .  9 0 I F ( N - A . E Q . O ) G O T O 1 0 C * * * * * * * * R E A D I N P L A N T S I Z E U P P E R B O U N D S - I F O P T I O N E D R E A D ( 5 , 4 ) ( F M T 3 ( I ) , I = 1 , 2 0 ) . . . _ . R E A D ( 5 , F M T 3 ) ( U P ( I ) , I = 1 , M ) G O T O 1 2 1 0 D O 1-1 I = 1 , M :  1 1 U P ( I ) = S Y 1.2 I F ( K Y . E Q . O ) G O T O 1 3 C * * * * * * * * R E A D I N M A X I M U M D ( S T A N C E A N Y D E M A N D P O I N T . I S TO. B E F R O M S O M E C * * * * * * * * P L A N T - I F O P T I O N E D R E A D ( 5 , 4 ) ( F M T 4 ( I ) , I = 1 , 2 0 ) R E A D ( 5 , F M T 4 ) ( H N ( I ) , I = 1 , N )  G O T O 1 5 1 3 D O 1 4 I = 1,N 1 4 H N ( I ) = S Y _ _ : _ C * * * * * * * * R E A D I N U S E R S U P P L I E D W E I G H T I N G S Y S T E M - I F O P T I O N E D 1 5 I F ( M Q . E Q . O ) G O T O 1 8 R E A D ( 5 , 4 ) ( F M T 5 ( 1 ) , I = 1 , 2 0 ) R E A D ( 5 , F M T 5 » ( C ( I ) , I = 1 , M ) C * * * * * * * * R E A D I N V E C T O R W H I C H C O N T A I N S T H E V A L U E S O F T H E L O C A T I O N S C * * * * * * * * T H A T A R E T O B E C O M P U T E D E . G . 2 F O R A 2 M E D I A N S O L U T I O N 18 READ ( 5 , 6 ) ( J U D ( I ) , I =1,KA> IVKX =0 54 I F ( IRK.-EQ.2 ) GO TO 6 3 R E A D . ( 5 , 4 ) ( F M T 1 ( I ) , I = 1 , 2 0 ) ' C * * * * * * * * READ IN TRANSPORT CO ST , TI ME , OR , SHOR TE ST ROUTE MATRIX DO 3 5 I =1 , N . K = ID( I ). ' READ ( 5 , - F M T l ) ( D ( J ) , J =1,M) DO 3 5 J =1 »M A ( IC HE C < ( < , J , H) ) = D ( J ) ' ~ ~ 35 CONTINUE I F ( I K K . EQ.O) I K K =2 . . : . "63 IF ( 12.EQ. 2). 3(1 TO 67 '. : ~ C * * * * * * * * READ IN MODE DEMAND V A L U E S RE AD ( 5 , 4 ) ( EM"T2 ( I ) , I = 1 , 2 0 ) R E AD ( 5, EM T 2 ) ( 0( I ) , 1 =1,N) ' ' " IF ( 12 .EQ.O ) 12 =2 67 I F (Mw.EQ.O) SO TO 118 I F ( J K K .EQ .?.) GO TO 118 C A L L DATA IF ( J K K . EQ. 0) J K K =2 118 SX = 1 0 0 . 0 • IE ( I VKX. E Q . l )' GO TO 2 1 I V K X =1 ; ' . US =0.0 DO 17 I =1 ,N 17 _}S =OS + D ( I ) I J K M =0 . .. ~ A SUM =SY. ,-KRB =0 . ' a •_ 20 KR3 =KRB+1 ' . I K S =IRS IF I K R _ . S T . K A ) GO TO 5 1 5 C * * * * * * * * GET THE NEXT MEDIAN V A L U E TO BE COMPUTED KP = J U D ( K R B ) 21 IF I KP .GE .MM) GO TO 25 ' " WRITE ( 6 , 2 2 ) GO TO 5.15 25 IF (KP.LE.'M) S3 TO 24 .WRITE ( 6 , 2 3 ) " " " "&