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A land use allocation model for a lower Fraser Valley municipality Schroeter, Daniel Eric 1973

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cl  A LAND USE ALLOCATION FOR A LOWER FRASER VALLEY  MODEL MUNICIPALITY  by  DANIEL ERIC SCHROETER B.A.,  University  o f B r i t i s h Columbia,  1971  A THESIS SUBMITTED IN PARTIAL FULFILMENT  OF  THE REQUIREMENTS FOR THE DEGREE MASTER OF SCIENCE  in  t h e Department of  Agricultural  We  accept  required  this  thesis  Economics  as c o n f o r m i n g t o t h e  standard  THE UNIVERSITY OF BRITISH COLUMBIA August,  1973  i  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 of the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the 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  study.  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 copying of t h i s t h e s i s f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  be granted by  PiGiRlCULrruiZA/\  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  s h a l l not be  permission.  Department o f  AUGUST  zti~^  Department or  I t i s understood t h a t copying or  of 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 written  the Head of my  /9 73  fir^nMicS:  publication  allowed without  my  ii ABSTRACT  This in the  t h e s i s examines t h e use o f m a t h e m a t i c a l models  a s t r a c t i n g r e a l world  l a n d use systems.  The p u r p o s e o f  s t u d y was t o d e t e r m i n e what t y p e s o f d e c i s i o n m o d e l s a r e  available,  w h e t h e r t h e y c a n be a d a p t e d t o l a n d u s e a l l o c a t i o n  p r o b l e m s , a n d w h i c h i s most s u i t a b l e f o r u s e i n l a n d u s e planning.  Since  planning  was f e l t  or maximizing p u b l i c welfare,  t o be a means t o s a t i s f y i n g  the c r i t e r i o n  used  f o r model  s e l e c t i o n was t h e d e g r e e t o w h i c h t h e m o d e l c o u l d seeking  l a n d u s e p o l i c i e s w h i c h w o u l d be o p t i m a l  o f m a x i m i z i n g some measure o f s o c i a l  Using which allows  this  goals  i n the objective  today.  T h i s model s t r u c t u r e was  usefulness  of this  t y p e o f model f o r d e c i s i o n  making i n t h e l a n d use f i e l d .  be  various  aspects  incorporated  used t o f i n d of  Columbia.  p u r p o s e o f c o n s t r u c t i n g t h e m o d e l was t o i l l u s t r a t e t h e  potential  the  then  a l a n d u s e m o d e l f o r t h e C i t y and D i s t r i c t  o f L a n g l e y i n t h e Lower F r a s e r V a l l e y o f B r i t i s h The  programming  t o be t h e b e s t m o d e l s t r u c t u r e a v a i l a b l e  land use planners  used t o c o n s t r u c t  i n the sense  welfare.  a form o f l i n e a r  f o r several concurrent  f u n c t i o n was f e l t to  criterion,  aid in  of the r e a l world  l a n d use system  i n t o t h e m o d e l and how t h e m o d e l c o u l d  land use p a t t e r n s  social welfare.  possible  T h i s was done by s h o w i n g  how  could t h e n be  w h i c h w o u l d m a x i m i z e a measure  A f t e r a d i s c u s s i o n o f model  results,  f u r t h e r r e f i n e m e n t s and s t u d y were s u g g e s t e d and  iii discussed.  It suited  was  felt  to the land  that  t h e m o d e l s t r u c t u r e c h o s e n was  use p l a n n i n g  as a p o t e n t i a l p l a n n i n g  tool.  field  and o f f e r e d much  well  promise  iv  ACKNOWLEDGMENT  Without the e f f o r t , several  encouragement, and g u i d a n c e o f  p e o p l e t h i s t h e s i s w o u l d n o t have b e e n p o s s i b l e .  w i s h t o e x t e n d my s i n c e r e  Dean M i c h a e l Agriculture,  thanks t o :  Shaw, a n d t h e Canada D e p a r t m e n t o f  f o rt h e i r f i n a n c i a l support o f t h i s  Dr.  Peter L. Arcus,  f o r sharing  experience  the  a d d i t i o n a l e f f o r t he made t o make my s t u d i e s  J o h n D. Graham, f o r i n i t i a l l y  concept o f using  a multiple  format t o study problems o f land work i n t h i s a r e a n o t o n l y study, but a l s o to  further  thesis  a warm a n d  goals  linear  a d v a n c i n g t o me programming  use a l l o c a t i o n .  His i n i t i a l  s e r v e d a s an i n v a l u a b l e  a i dto this  i n this field,  including  available  support o f t h i s  G. R. W i n t e r , a n d D r . L . M. L a v k u l i c h ,  service  on t h e t h e s i s  assistance;  Schroeter,  support.  fortheir  committee; f o r h e r i n t e r e s t , encouragement,  and  my f e l l o w and  and f o r  project;  Beverly and  advisor,  l e d t o f i n a n c i a l s u p p o r t b e i n g made  research  Dr. valued  thesis  experience; Dr.  the  i n h i s r o l e as s e n i o r  project;  h i s t i m e , wisdom,  and  rewarding  I  s t u d e n t s and f r i e n d s ,  f o rtheir  advice  V  TABLE OF CONTENTS  CHAPTER  Page  1  INTRODUCTION  1  2  THE HISTORY OF LAND USE PLANNING AND CONTROL IN THE LOWER MAINLAND OF BRITISH COLUMBIA  3  Early  History  The F o r m a t i o n o f t h e Lower  3 Mainland  Regional Planning Board  5  The S i t u a t i o n T o d a y 3  6  THE USE OF MATHEMATICAL MODELS  8  Definition  8  E a r l y H i s t o r y o f Mathematical Models U s e f u l n e s s o f Mathematical Models 4  MATHEMATICAL MODELS FOR APPLICATION TO LAND USE PLANNING: A REVIEW OF THE ALTERNATIVES  8 9  12  Introduction  12  Input-Output Models  13  Simulation  17  Operational  5  ....  Games  24  M a t h e m a t i c a l Programming M o d e l s  25  Summary  33  THE USE OF LINEAR PROGRAMMING WITH MULTIPLE GOALS IN LAND RESOURCE ALLOCATION  35  The F o r m a t and M e t h o d o l o g y o f C o n v e n t i o n a l L i n e a r Programming  35  The A p p l i c a t i o n o f L i n e a r Programming t o a H y p o t h e t i c a l L a n d Use P r o b l e m  36  vi  CHAPTER  Page The F o r m a t Multiple  and M e t h o d o l o g y o f G o a l s L i n e a r Programming  A M u l t i p l e Goals Version Hypothetical  of the  L a n d Use P r o b l e m  Summary 6  7  41 47  LANGLEY MODEL STRUCTURE  48  Intoduction  48  Initial  50  Assumptions  L a n d Use A l t e r n a t i v e s  51  Technical  52  and P h y s i c a l C o n s t r a i n t s  Social Constraints  55  O u t p u t Demand C o n s t r a i n t s  56  Externality  63  Demand C o n s t r a i n t s  Accounting Constraints  65  Goal Accounting Constraints  65  Summary  70  THE LANGLEY MODEL: DATA PROBLEMS AND I N I T I A L RESULTS  71  Data Problems  71  Results  73  Further  Refinements  Discussion 8 .  39  81 82  SUMMARY AND CONCLUSIONS BIBLIOGRAPHY APPENDIX I : MATRIX PICTURE  85 87 91  APPENDIX I I : CODING EXPLANATION AND DATA DESCRIPTION FOR LANGLEY MODEL MATRIX  93  vii  L I S T OF TABLES  TABLE 1 2  A  B C  D E  F  Page LANGLEY MODEL SOLUTIONS FOR ALLOCATIONS OF LAND  75  OPTIMAL ALLOCATIONS BY THE LANGLEY MODEL UNDER VARYING POPULATION CONDITIONS  80  AREAS OF LAND S U I T A B I L I T I E S IN THE CITY AND DISTRICT OF LANGLEY  109  MINIMUM LAND REQUIREMENTS IN THE LANGLEY MODEL  USED I l l  COEFFICIENTS FOR STEP DEMAND FUNCTIONS USED IN LANGLEY MODEL  116  EMPLOYMENT DENSITY COEFFICIENTS USED IN LANGLEY MODEL  118  A C T I V I T Y CONTRIBUTIONS/ACRE TO GROSS PRODUCTION  119  A C T I V I T Y CONTRIBUTIONS/ACRE TO POLLUTION INDICES  121  viii L I S T OF  FIGURES  FIGURE  1  Page  Graphical  Illustration  S t e p Demand F u n c t i o n s  of 60  1 Chapter 1  INTRODUCTION  That government s h o u l d control  and p l a n n i n g  resources the in  o f t h e use o f s o c i e t y ' s  i s a p r i n c i p l e that  world today.  Certainly this  i n the  scarce  i s well established  t h e Lower M a i n l a n d o f B r i t i s h  regional,  a c t i v e l y intervene  land  i n most o f  p r i n c i p l e i s f r i m l y entrenched C o l u m b i a , where many  p r o v i n c i a l , and f e d e r a l government a g e n c i e s  a major r o l e i n the p l a n n i n g  municipal, play  and development o f t h e r e g i o n ' s  land a l l o c a t i o n .  With land  use p l a n n i n g  s c a l e by a m u l t i t u d e increased  being  u n d e r t a k e n on a m a j o r  o f government a g e n c i e s a n d b o a r d s ,  a t t e n t i o n has r e c e n t l y been p a i d  t o t h e development  o f new t o o l s a n d methods t o a i d i n t h i s p l a n n i n g One s u c h d e v e l o p m e n t t h a t this  area  i s the recent  models i n s t u d y i n g  This structures  that  large  process.  i s receiving particular notice i n  emergence o f t h e u s e o f m a t h e m a t i c a l land use systems.  t h e s i s e x a m i n e s t h e t y p e s o f d e c i s i o n model are presently  a v a i l a b l e , and t h e n  determines  w h e t h e r t h e y c a n be a d a p t e d t o l a n d u s e a l l o c a t i o n p r o b l e m s , and  w h i c h i s t h e most s u i t a b l e f o r u s e i n t h i s  l a n d use p l a n n i n g  c a n be i n t e r p r e t e d a s t h e d e s i g n  of  an a r e a ' s  of  t h e area) r e c e i v e s  from those  land based  land  area.  functions  so  that  society  Since  and c o n t r o l (the p o p u l a t i o n  t h e maximum a t t a i n a b l e b e n e f i t o r u t i l i t y  resources,  the c r i t e r i o n  used  for suitability  2 was  the  degree to which the  use  p o l i c i e s w h i c h w o u l d be  some measure o f  examined i n g r e a t e r  Langley  The illustrate  i s judged detail,  l a n d use  i n the  could  potential  Given t h i s superior i s used  i s chosen, i n the  City  be  usefulness  and  i t s format  r e a l world  o f model r e s u l t s ,  t h e model and patterns  social welfare.  possible  discussed.  to  t y p e o f model  o f the  l a n d use  of  Columbia.  aspects  find  the  District  T h i s was  into  maximizing  development  field.  incorporated  used to  of this  land  criterion,  l a n d use  various  m a x i m i z e a measure o f  s u g g e s t e d and  sense of  p u r p o s e o f c o n s t r u c t i n g t h e model was  the  t h e n be  i n the  model f o r t h e  d e c i s i o n making i n the  system c o u l d  and  a i d i n seeking  Lower F r a s e r V a l l e y o f B r i t i s h  the  s h o w i n g how  optimal  s o c i a l welfare.  model s t r u c t u r e t h a t  of a r a t i o n a l  model c o u l d  done  by  land  how  for  use  the  model  which would  A f t e r an  examination  f u r t h e r r e f i n e m e n t s and  study  are  3  Chapter 2  THE IN  Early  HISTORY OF LAND USE PLANNING AND CONTROL THE LOWER MAINLAND OF BRITISH COLUMBIA  History  Though many r e g a r d land  government i n t e r v e n t i o n  u s e a l l o c a t i o n a s a phenomenon f a i r l y  Fraser  recent  itself.  The f i r s t  British  tradition,  Nor  legal  that not land  a s s e t down b y p u b l i c was t h i s  land  itself,  law, c o u l d  represented  development o f t h i s notably of  be p r i v a t e l y owned. always  Early land  an a c t i v e a t t e m p t on b e h a l f  Valley.  i nthe  rights i n 1  exercised  inventories  systems, though s i m p l i s t i c  government t o d i r e c t t h e p a t t e r n Lower F r a s e r  but only  c o n t r o l over p r i v a t e property  classification  settlement  s u r v e y s o f 1858 e s t a b l i s h e d ,  p a s s i v e l y by t h e p u b l i c a u t h o r i t y . and  to the  V a l l e y , p u b l i c c o n t r o l o f t h e p r i v a t e use o f l a n d i n  the V a l l e y i s a c t u a l l y as o l d as r e s i d e n t i a l  land  into  a t best,  of the c o l o n i a l  of settlement  in--the  A l f r e d Siemens, i n h i s study o f t h e region,  reports  that  "some  townsites,  t h a t o f New W e s t m i n s t e r , were s e l e c t e d w e l l  settlement—with  due r e g a r d  to strategic  i n advance 2 considerations."  1 T h e s e s u r v e y s a r e d e s c r i b e d i n : V. J . P a r k e r , " P r o b l e m s a n d P r o g r e s s i n R a t i o n a l i z i n g t h e Use o f t h e R e s o u r c e s o f t h e Lower F r a s e r V a l l e y , " The Lower F r a s e r V a l l e y : E v o l u t i o n o f a C u l t u r a l L a n d s c a p e , e d . A. H. Siemens ( V a n c o u v e r : T a n t a l u s R e s e a r c h , 1 9 6 6 ) , p . 165. 2 A l f r e d H. S i e m e n s , "The P r o c e s s o f S e t t l e m e n t i n t h e Lower F r a s e r V a l l e y - I n - i t s P r o v i n c i a l C o n t e x t , " The Lower F r a s e r V a l l e y : E v o l u t i o n o f a C u l t u r a l L a n d s c a p e , e d . r  4  T h u s , p u b l i c l a n d use the  public interest,  of the as  uses of  e a r l y as  planning i n t o the  land,  had  i . e . the design  19th  Government c o n t r o l o v e r fairly  tenuous u n t i l  completion o f the  near the  railroad  o f new  settlers  greater put. the  t e r m i n u s on to the  already  the  land, use t h a n as  services—roads,  and  satisfy  the  the major  an  planners  o f use  numbers  joined  f o r the  suddenly  intensified provincial  on more and  more  adjudicators  allocation.  harbour f a c i l i t i e s ,  dyking  As  and  sewage m a i n s , p o l i c e and  the  l a n d use rights.  various  to  of  well,  the  e v e r i n c r e a s i n g n e e d f o r , and government b o d i e s  drainage  fire  protection.  g o v e r n m e n t s were  m a r k e t as b u y e r s and  an  Siemens  be  b r o u g h t i n c r e a s i n g demands f o r g o v e r n m e n t  t h e s e demands t h e  to enter  port  increasingly  r e s i d e n t i a l u s e s now  f i e l d , t h o u g h more as  owners o f p r o p e r t y  H.  introduced  competition  as  A.  and  West C o a s t a t t r a c t e d v a s t  region  As  p r o j e c t s , w a t e r and  various  as  remained The  R a i l w a y i n 1881  g o v e r n m e n t s were c a l l e d  growing p o p u l a t i o n  obliged  century.  e x i s t i n g primary uses i n competing  municipal  disputes,  To  Valley  however,  the Vancouver area the  Fraser  century.  turn of the  I n d u s t r i a l , c o m m e r c i a l and  enter  i n the  distribution  number o f u s e s t o w h i c h t h e V a l l e y ' s l a n d c o u l d  scarce V a l l e y land. and  spatial  l a n d use,  Canadian P a c i f i c  subsequent development of and  of the  i t s beginnings  the middle of the  intervention, in  Out  i n the  of t h i s  thus  s i t u a t i o n evolved  s u b s e q u e n t p a r t i c i p a t i o n by controlling  (Vancouver: T a n t a l u s  Research,  and  planning  1966),  p.31.  of  5 l a n d use. and  L o c a l governments e s t a b l i s h e d p l a n n i n g  c o m m i s s i o n s t o p l a n and  jurisdiction. districts,  dyking  and  drainage  The  Formation  At to bear role  on  Parks  r e g u l a t e l a n d use  boards,  school boards,  same t i m e ,  the p r o v i n c i a l  i n land planning.  t h e Lower M a i n l a n d  pressure  I n 1936  o f t h e Lower M a i n l a n d  unit  to s a t i s f y During  reacted  to these  W o r l d War  was  a comprehensive  Mainland well  first  setting  as c o n t i n u e d  of e x i s t i n g  up  demands by m u n i c i p a l  and  the p r o v i n c i a l  government t o e s t a b l i s h  the board  Planning was  various other  B o a r d on  and  local  J u n e 21,  economic  and  citizen  agency,  Lower as  town p l a n -  groups l e d  t h e Lower The  Division  undertook  i n the  councils,  Columbia  Reconstruction.  Division  this  planning  Rehabil-  Planning  l a n d use  1949.  the  a government  p a s s i n g the  Recommendations made by  ning agencies,  Regional  urged  I I t h e Government o f B r i t i s h  tasks t h a t the P l a n n i n g  i n 1945.  organized  social  which c r e a t e d a Regional  survey  brought  a more d o m i n a n t group  t h e B u r e a u o f Post-War R e h a b i l i t a t i o n  of the  t o be  Board  the need t o p l a n  as a s i n g l e  i n c r e a s i n g demands by  One  sewerage  Planning  continued  a citizens'  t h a t n e e d by  i t a t i o n A c t o f 1944, within  Regional  government t o r e c o g n i z e  agency.  and  P l a n n i n g A s s o c i a t i o n which  l a n d use and  improvement  government t o t a k e  Regional  their  up.  o f t h e Lower M a i n l a n d  the p r o v i n c i a l  within  commissions, water communities,  d i s t r i c t s were s e t  the  boards  Mainland  prime r o l e  the p r e p a r a t i o n o f a comprehensive r e g i o n a l  of  6  plan  f o r a l l of  Strait  The  the  of Georgia,  Fraser  Valley  f r o m Hope w e s t w a r d s t o  i n c l u d i n g a l l 28  has  Lower M a i n l a n d R e g i o n a l P l a n n i n g  s i n c e been r e p l a c e d  this  evolution  the  planning  has  continued  by  four  separate regional  towards i n c r e a s e d  and  to the  present.  which e x e r c i s e  authority  i n the  over various  and  control.  question  the  p r i n c i p l e (though t h e r e  question  the  degree) of  Given the least  the  planning, be  Planning,  but  a means t o  how  can  the  few  Valley's  desirability  that  of  the  over  aspects of  Valley  land  a r e many t h a t  or  satisfying  would  would planning  resources.  the  necessity,  or  l a n d use  complex s e t o f  device  at  not  be  use  plans  objectives  construed  as  are anything  or maximizing p u b l i c welfare..  planners determine p u b l i c o b j e c t i v e s  what method o r  use  public intervention i n land  should  can  t h e y d e c i d e how  interdependencies  land  four  Lower F r a s e r  public's planning  after a l l ,  l a n d use  are  p e o p l e t o d a y who  land  in  Lower M a i n l a n d ' s  i t i s reasonable to expect that so  districts,  p u b l i c p a r t i c i p a t i o n i n the  inevitability  formulated  met.  There are  of the  the  Today t h e r e  planning  t h e w i s e use  District  government i n v o l v e m e n t  r a t i o n a l i z a t i o n of  hundred d i f f e r e n t p u b l i c b o d i e s  by  region m u n i c i p a l i t i e s .  S i t u a t i o n Today  Though t h e  of  the  that  to  But  and  manipulate  exist in a  land  7  u s e s y s t e m so t h a t lies  these o b j e c t i v e s  an u n d e r s t a n d i n g o f t h e r e c e n t  o f models  and o t h e r  decisions.  simulation  By d i s t i l l i n g  are best emergence  arguments  a complex  land  land  use  use system i n t o objectives,  which o t h e r w i s e have remained a l m o s t  totally  The n e x t c h a p t e r o u t l i n e s t h e r i s i n g  t h e s e model  land  o f t h e use  o f q u a n t i f i c a t i o n c a n be i n j e c t e d i n t o t h e  qualitative. of  Herein  systems i n f o r m u l a t i n g  an o b s e r v a b l e s e t o f m a j o r r e l a t i o n s h i p s and some measure  met?  use f i e l d ,  systems i n d e c i s i o n making, but also  i n most  not only  of the s o c i a l  prominence i n the  sciences.  8 Chapter 3  THE  USE OF MATHEMATICAL MODELS  Definition  A model i s an a n a l o g u e set of relations  that exists  o r a r e p r e s e n t a t i o n o f some  i n the r e a l world.  These  relations  may be s p e c i f i e d v e r b a l l y , m a t h e m a t i c a l l y , o r v i s u a l l y . school children playing  "store",  Thus  and i n t e r s e c t o r a l i n p u t - o u t p u t  m a t r i x o f t h e Lower M a i n l a n d economy, o r a s c a l e v e r s i o n o f an internal  combustion  e n g i n e c a n a l l be c l a s s i f i e d  the purposes o f t h i s which  as models.  s t u d y , however, o n l y t h o s e e c o n o m i c  are e s s e n t i a l l y mathematically specified w i l l  E a r l y History o f Mathematical  The  increase  the p a s t t h r e e decades  For  models  be c o n s i d e r e d .  Models  i n t h e use o f m a t h e m a t i c a l models has been r e m a r k a b l e .  over  M a t h e m a t i c a l model1  l i n g was v i r t u a l l y  n o n e x i s t e n t b e f o r e 1940.  War I I and t h e s u b s e q u e n t Allied  war e f f o r t  major  The a d v e n t o f W o r l d  o p e r a t i o n a l problems  l e d r e s e a r c h e r s t o d e v e l o p what was  of the termed  1 Some examples o f m a t h e m a t i c a l m o d e l s t h a t p r e d a t e W o r l d War I I c a n be f o u n d . F o r example, Quesnay's T a b l e a u E c o n o m i q u e , p u b l i s h e d i n 1758, i s p e r h a p s t h e e a r l i e s t example o f an i n t e r s e c t o r a l i n p u t - o u t p u t m o d e l . Other n o t a b l e e a r l y works i n t h i s f i e l d a r e t h o s e o f L e o n W a l r a s i n t h e l a t e n i n e t e e n t h c e n t u r y and W. L e o n t i e f i n t h e 1930*s.  9  "operations the  research".  To a i d r e s e a r c h e r s  complex i n t e r d e p e n d e n c i e s  components o f t h e war e f f o r t ,  i n understanding  t h a t e x i s t e d between t h e v a r i o u s they  d e s c r i p t i o n o f t h e war o p e r a t i o n s  developed system.  a mathematical  Using  a t i c a l m o d e l t h e y were a b l e t o f i n d o p t i m a l  t h i s mathem-  or best solutions 2  to t h e v a r i o u s o p e r a t i o n a l problems t h a t c o n f r o n t e d From t h i s rapidly  evolved  analysis, ning, and  somewhat i g n o b l e b i r t h ,  with  i n t o what may be b r o a d l y  r e s e a r c h has  termed  systems  a p p l i c a t i o n s i n such d i v e r s e f i e l d s  international  relations,  s o c i a l psychology.  social  operations  industrial  them.  a s town  economics,  Problems i n v i r t u a l l y  every  plan-  ecology,  area  of the  s c i e n c e s have come u n d e r t h e s c r u t i n y o f s y s t e m s  analysts.  Usefulness  o f Mathematical  The  Models  widespread use o f mathematical modelling  without  its critics.  pursuit  o f such models as i n a p p r o p r i a t e , a l t h o u g h  able.  i s not  T h e r e a r e some who v i e w t h e r e c e n t  V e r n o n and H o o v e r sum up t h i s  understand-  position quite  succinctly:  One w o n d e r s , however, w h e t h e r t h e p r e o c c u p a t i o n w i t h s y s t e m s o f t h i s s o r t , s o i n g e n i o u s l y p u r s u e d by so many r e s e a r c h e r s i n t h e u r b a n f i e l d , h a s n o t come t o represent a m i s a l l o c a t i o n of scarce ingenuity. Some o f the reasons f o r the a t t r a c t i o n t o such systems a r e c l e a r enough. To t h e m i n d t h a t i n s i s t s on c l a r i t y and r i g o r ,  2 F o r a b r i e f d e c r i p t i o n o f t h i s work s e e J o s e p h L . S c h o f e r , "Systems A n a l y s i s i n T r a n s p o r t a t i o n P l a n n i n g " a t e c h n i c a l p a p e r p u t o u t b y t h e C e n t e r f o r U r b a n S t u d i e s and D e p a r t m e n t o f Systems E n g i n e e r i n g a t t h e U n i v e r s i t y o f I l l i n o i s ( C h i c a g o ) , 1970.  10 the p u r s u i t of research i n the s o c i a l sciences i s not a wholly s a t i s f y i n g a c t i v i t y . The r e s u l t s o f s u c h r e s e a r c h t e n d t o be s p o n g y , q u a l i f i e d , and e q u i v o c a l ; independent v a r i a b l e s r a r e l y prove a l t o g e t h e r independent e i t h e r o f one a n o t h e r o r o f t h e d e p e n d e n t phenomena on w h i c h t h e y a r e p r e s u m e d t o a c t ; r e s i d u a l v a r i a n c e p r o v e s more i m p o r t a n t t h a n what has b e e n " e x p l a i n e d " . The " t r u e " m o d e l l i e s v e r y d e e p , i n d e e d , i f i t e x i s t s at a l l . But the t h i r s t f o r a simple o r d e r proves h a r d t o q u e n c h ; t h e a d u m b r a t i o n s t h a t t h e o r d e r may e x i s t e x e r t an o v e r w h e l m i n g p u l l upon t h e r e s e a r c h e r . 3 A l t h o u g h most r e s e a r c h e r s w o u l d a g r e e t h a t  c a r e must be  taken not t o lose  of  s i g h t of the l i m i t a t i o n s  m a t h e m a t i c a l models  f r o m complex s o c i a l  s y s t e m s , few w o u l d  a c c e p t V e r n o n and H o o v e r ' s h y p o t h e s i s t h a t world it  social  s y s t e m s no c a u s a l i t y  does e x i s t ,  fruitless. that  One  individual  rationally  as t o r e n d e r  search  decision  system behave  units  in a social  i n accordance with c e r t a i n  does  s y s t e m and t h a t  exist within  a social  This  o f p a t t e r n s c a n be d e s c r i b e d , e x p l a i n e d A c c o r d i n g t o Meyer,  analysis  implies this  that  recogorder  order or set  and, w i t h i n  limits,  " m a t h e m a t i c a l programming  without question the best o f the t o o l s  one b e l i e v e s  or i f  o f economics i s  aims.  if  real  of the fundamental b e l i e f s  n i z a b l e e c o n o m i c and s o c i a l  regional  i n some  or order exists,  i t i s h i d d e n so d e e p l y  and p r e d i c t a b l y  forecast.  extracting  from a s t r i c t l y  employed  is  i n modern  conceptual point  i n a r e a s o n a b l y p e r v a s i v e economic  of view rationality."  E v e n i f one a c c e p t s V e r n o n and H o o v e r ' s s u g g e s t i o n t h a t  the  Raymond V e r n o n and E d g a r M. H o o v e r , "Economic A s p e c t s o f U r b a n R e s e a r c h , " The S t u d y o f U r b a n i z a t i o n e d s . P h i l i p H a u s e r and L e o S c h n o r e (New Y o r k : J o h n W i l e y and S o n s , 1965)., p . 195. 4  J o h n R. M e y e r , A m e r i c a n E c o n o m i c Review,  "Regional Economics: A Survey," L I I I , No. 1 ( J a n u a r y , 1963) p.53.  11  " t r u e " m o d e l may follow that  any  model c a n n o t be included,  be  impossible  t o u n c o v e r , does i t n e c e s s a r i l y  attempt to a b s t r a c t justified?  take a very  an  approximation of  Many r e s e a r c h e r s ,  Friedmanistic  this  the  author  approach t o model  abstraction:  The l e g i t i m a c y o f and j u s t i f i c a t i o n f o r a b s t r a c t i o n must r e s t u l t i m a t e l y . . . o n t h e l i g h t t h a t i s s h e d and t h e power t o p r e d i c t t h a t i s y i e l d e d by t h e a b s t r a c t i o n . C e r t a i n l y m a t h e m a t i c a l m o d e l s have t h e i r comings, but  u s u a l l y these short-comings are  r e q u i r e m e n t s and nature of  the  not  and  has  Despite  to regard  confronting us  inexperience  e n c o u n t e r e d w i d e s p r e a d use i n the  social  m o d e l s as  the  sciences,  and  than  understand these problems.  Models are  problems  the  only  in  likely  taken,  panacea f o r a l l of  be  mathematical  will  C a r e must be  modern c i v i l i z a t i o n .  accumulation,  acceptance  and  input  conceptual  seem t o  these d i f f i c u l t i e s  c o n t i n u e t o grow i n p o p u l a r i t y .  help  i n the  i n m o d e l s t u d i e s much more f r e q u e n t l y  almost a l l f i e l d s  not  r e l a t e d to  Problems o f data  programming  o f model methodology. modelling  a c t u a l l y inherent  models t h e m s e l v e s .  computer c a p a c i t y mentioned  are  short-  however,  social tools  In the words o f  ills  to  Claude  McMillan: M a t h e m a t i c a l programming i s a r e s p e c t e d body o f k n o w l e d g e b e c a u s e o f i t s a n a l y t i c a l power as a s u p p l e m e n t t o , r a t h e r t h a n a s u b s t i t u t e f o r human judgement i n making d e c i s i o n s about r e a l - w o r l d problems o f g r e a t c o m p l e x i t y .  5 M i l t o n Friedman, P r i c e Theory (Chicago: A l d i n e Co., 1962) p.13. 6 C l a u d e M c M i l l a n J r . , M a t h e m a t i c a l Programming: An I n t r o d u c t i o n t o t h e D e s i g n and A p p l i c a t i o n o f O p t i o n a l D e c i s i o n M a c h i n e s (New Y o r k : J o h n W i l e y and S o n s , 1 9 7 0 ) , p.v. preface. Publishing  of  12 Chapter 4  MATHEMATICAL MODELS FOR APPLICATION TO LAND USE A REVIEW OF THE ALTERNATIVES  PLANNING:  Introduction  E v e n t h o u g h t h e use o f m a t h e m a t i c a l m o d e l s a s an aid  t o d e c i s i o n making i n t h e s o c i a l  s c i e n c e s has o n l y  dev-  e l o p e d o v e r t h e l a s t two o r t h r e e d e c a d e s , t h e l i t e r a t u r e i n the  field  also  i s vast, not only  i n terms o f the almost  t h a t have been s t u d i e d this and  thesis,  i n t e r m s o f a b s o l u t e volume b u t infinite  and u s e d  number o f m e t h o d o l o g i e s 1  i n recent years.  however, t o b r i e f l y d e s c r i b e  to evaluate their  possible  p l a n n i n g and d e c i s i o n m a k i n g .  Suffice for  t h e major model groups  effectiveness  i n land  use  I t s h o u l d be u n d e r s t o o d ,  however,  t h a t no model g r o u p h a s s h a r p l y d e f i n e d b o u n d a r i e s t h a t make it  completely  distinct  models a r e f r e q u e n t l y model  from o t h e r model s t r u c t u r e s .  Actual  a c o m p o s i t e o f more t h a n one t y p e o f  structure.  T h r e e t y p e s o f m o d e l s c a n be i d e n t i f i e d literature:  i n p u t - o u t p u t models, s i m u l a t i o n models  operational  games), and programming m o d e l s .  have p o t e n t i a l  applications  from the (including  A l l o f the models  i n l a n d use p l a n n i n g .  1 F o r a r e v i e w o f t h e l i t e r a t u r e on m o d e l s u s e d i n t h e a g r i c u l t u r a l a r e a a l o n e s e e E c o n o m i c M o d e l s and Q u a n t i t a t i v e Methods f o r D e c i s i o n s and P l a n n i n g i n A g r i c u l t u r e , e d . E a r l 0. Heady (Ames: Iowa S t a t e U n i v e r s i t y P r e s s , 1 9 7 1 ) .  13 Input-Output Models  I n p u t - o u t p u t models d e s c r i b e ships  that  variables  exist within  a s y s t e m t h a t has  a c t i n g upon i t .  The  model i s i n i n t e r s e c t o r a l o r input-output  model c a n  nature of  economy, and  place  an  in that or  regions  can  various  sectors  and  within the  detail  the  and  the  To  illustrate,  economy c a n as  be  this  the  analysis.  transactions  economy b u t  the  i f there  not  only  n  final  sectors  take  between the the  demands f o r  i n the  economy  o u t p u t o f e a c h s e c t o r between  i n t e r s e c t o r a l flows  described  i n an  of The  which  a l s o between  various  are  type  complete s t r u c t u r a l  time p e r i o d ,  by  d i s t r i b u t i o n of the  (exogenous) demand and  exogenous  exogenous v a r i a b l e s a c t i n g upon  economy, u s u a l l y r e p r e s e n t e d outputs.  of  i n t e r r e g i o n a l flow  approximate the  relation-  a number o f  most common use  economy i n a g i v e n  sectors  interactivity  final  i s known, t h e n  input-output  the  accounting  array  n  Final Demand  Total Output  ln  *1  follows: Consuming S e c t o r s Producing Sectors  1  2  •  1  x  l l  x  12  x  13  •  2  x  21  x  22  x  2 3  •  3  x  31  x  32  x  33  •  *  •  *  n  X„ -i nl  V a l u e Added Total  3  Output  X  . .  *  • * «  • *  x  x  •* •  2n x  3n  <*3  x  l  x  2  x  3  •  •  •  •  •  •  •  •  •  •  o n2  X  .  0  n3  v  l  v  2  v  3  •  x  l  x  2  x  3  •  . X nn  • • «  •  v  n  x  n  d  n  X  n  14 In  this  array,  is  the value  quantity  x ^ j i s the sales of sector  added t o p r o d u c t i o n  demanded o f s e c t o r  i to sector  by s e c t o r  j , and d^ i s t h e  i by t h e " f i n a l  h o u s e h o l d s , government, p r i v a t e i n v e s t m e n t , Xi  i s the t o t a l  producing  output produced i n s e c t o r  sector  By  a process  relating  demand  sectors":  foreign  i .  Since  demand. every  i s a l s o a consuming s e c t o r , note t h a t t h e  following balances hold  it j=i  j , Vj  X j L j  true:  + d. = x  = ±L 1=1  ±  of matrix  X  I  +  J  V  J  i n v e r s i o n , an a r r a y o f l i n e a r  equations  the output o f each s e c t o r t o the f i n a l  s e c t o r c a n be c a l c u l a t e d u s i n g  the accounting  demand o f e v e r y 2 a r r a y above.  2 Knowing  that 3=1  if  an n x n a r r a y , A, o f t e c h n i c a l c o e f f i c i e n t s o f p r o d u c t i o n a r e d e f i n e d such t h a t each element of the array, a  then  1  D  —  Xj  '  n ' i = 2> L a. a,X -X . ++ d X, l j=l i  In m a t r i x  notation  this  j  j  becomes  X = AX + D , where X i s t h e o u t p u t v e c t o r i n t h e i n i t i a l i n p u t o u t p u t a r r a y , D i s t h e column v e c t o r o f f i n a l demands, and A i s t h e a r r a y d e f i n e d a b o v e . This  becomes (I - A)X = D ,  where I i s t h e n x n m u l t i p l i c a t i v e If  identity  t h e d e t e r m i n a n t o f (I - A) i s n o n - z e r o , X =  ( I - A) D 1  matrix.  then  . continued  overleaf  15  If  a set of policy  variables  e x i s t s which determines  demands t h e n o u t p u t d i s t r i b u t i o n  c a n be d e t e r m i n e d  given values o f these p o l i c y v a r i a b l e s . ulation  of this  Thus t h r o u g h  o f o u t p u t meets o r s a t i s f i e s  the values that  to a t t a i n  distribution.  This  type o f model has a p p l i c a t i o n s  although t h e nature o f i t s assumptions  input-output c o e f f i c i e n t s  theoretically valid  over time  changes i n t e c h n o l o g y ,  to the n a t i o n a l industry  i s only  p e r i o d s i n which i n p u t  h e t e r o g e n e i t y o f p r o d u c t , new 3  substitution, sources  As w e l l , t h e  o f t h e model t e n d t o l i m i t  level,  i tto studies  The m o d e l  and t h u s  i n p u t , a n d s c a l e e c o n o m i e s do n o t a r i s e .  data requirements  i n order  i n many a r e a s ,  do l i m i t  d e a l i n g with the past o r with the short run.  of  manip-  h i s objectives  t h e p o l i c y v a r i a b l e s must t a k e  this  assumes f i x e d  f o r any  t y p e o f model t h e p o l i c y maker c a n d e t e r m i n e  which d i s t r i b u t i o n and  final  applications  where a d e q u a t e i n f o r m a t i o n on i n t e r -  f l o w s has been g a t h e r e d ,  although  input-output  D e f i n i n g ( I - A ) - ^ a s M, where m^j i s an e l e m e n t o f M, t h e n X = MD r e p r e s e n t s an a r r a y o f n e q u a t i o n s o f t h e f o r m ,  x  ±  =J_t> a j=i  i j  j  ,  which r e p r e s e n t the d e s i r e d i n f o r m a t i o n . 3 F o r an e x c e l l e n t d i s c u s s i o n on t h e s e assumptions see H o l i s Chenery and P a u l C l a r k , I n t e r i n d u s t r y Economics, (New Y o r k : J o h n W i l e y a n d S o n s , 1959) p p . 33-42.  16  m o d e l s have b e e n p r e p a r e d  f o rvarious  Iowa) a n d e v e n f o r s i n g l e c i t i e s Despite  states  (St. L o u i s ,  these l i m i t a t i o n s , input-output  s u c c e s s f u l l y u s e d by many r e s e a r c h e r s . known work i n t h i s  f i e l d was p r o v i d e d  (California, Philadelphia).  a n a l y s i s has been P e r h a p s t h e most  by W a s s i l y  well  Leontief  i n h i s s t u d i e s o f t h e i n t e r i n d u s t r y s t r u c t u r e o f t h e U.S. 4 economy. C a r t e r and Heady e m p l o y e d an i n p u t - o u t p u t analysis in  their  Rice  study o f the American a g r i c u l t u r a l i n d u s t r y ,  and L e F e r n e y u s e d an i n p u t - o u t p u t  happenings i n t h e American t e x t i l e output matrices  model t o f o r e c a s t 5,6  industry.  have b e e n c o n s t r u c t e d  while  f o r many  Inputnational  7 economies.  Though l a n d  a l l o c a t i o n c a n be s t u d i e d  o u t p u t a n a l y s i s i f t h e r e l a t i o n s h i p between l a n d  v i a inputand s e c t o r a l  4 V. L e o n t i e f , S t u d i e s i n t h e S t r u c t u r e o f t h e A m e r i c a n Economy, (New Y o r k : O x f o r d U n i v e r s i t y P r e s s , 1 9 5 3 ) . 5 H. 0. C a r t e r , a n d E a r l 0. Heady, An I n p u t - O u t p u t A n a l y s i s E m p h a s i z i n g R e g i o n a l and Commodity S e c t o r s o f A g r i c u l t u r e Iowa A g r i c u l t u r a l and Home E c o n o m i c s E x p e r i m e n t a l S t a t i o n B u l l e t i n no.469, no d a t e g i v e n . 6 P h i l i p R i c e and P r e s t o n L e F e r n e y , Use o f I n p u t Output A n a l y s i s i n Studying Industry Problems: A p p l i e d t o Employment Changes i n t h e U.S. T e x t i l e I n d u s t r y . U.S. Department o f A g r i c u l t u r e , Economic Research S e r v i c e , T e c h n i c a l B u l l e t i n No. 1411. F e b r u a r y , 1970. 7 F o r a r e v i e w o f r e s e a r c h done i n t h i s a r e a s e e C h e n e r y and C l a r k , o p . c i t . , p p . 183-200.  17  o u t p u t i s known o r shown l i t t l e  generally  The  fact  long  use  of  input-  Optimizing the  than  input-output not  be  too  have t o be  m o d e l s may  have a r o l e t o p l a y  since this  issue  policy  simulation  type  exact  it  seems a p p a r e n t t h a t  certainly  as  use  o f p o l i c y c h a n g e s , and or optimal  long  be  for short not  as  in this  o f f the  since parting  use  with  searching  Input-output  development. to the  discussed  However,  case  more  models.  Nevertheless, any  role  i t i s almost  t e r m f o r e c a s t i n g and to  of  fully  a n a l y s i s has  present,  a device  field.  u n a c c e p t a b l e t o many  simulation  at the  optimizing  also  approaching or  i f input-output  planning  a device  planners.  s p e c i f i c a t i o n of  and  in this  in  problems.  developed.  models, i t w i l l  a f o l l o w i n g examination of  i n land  use  to write  i s p r o b a b l y more r e l e v a n t  in  to play  land  quick  prove p o l i t i c a l l y  p o l i c y o p t i m a may  short  f o r e c a s t i n g them,  p o l i c y - m a k e r s , a l t e r n a t e methods o f for  The  have  l i m i t e d use  model a n a l y s i s  models u s u a l l y r e q u i r e  may  planners  u s u a l l y more i n t e r e s t e d i n  o u t p u t models i n o p t i m i z i n g  information  of  of  policy-makers' u t i l i t y functions,  this  use  analysis.  term time h o r i z o n s  d i s t r i b u t i o n rather  the  type of  t h i s model s t r u c t u r e a r e  However, p l a n n e r s s h o u l d of  determined, land  t h a t planners are  industry limits  be  interest in this  term a s p e c t s o f the  can  formulate  pretesting correct  term a l l o c a t i o n .  Simulation  Though s i m u l a t i o n  i s a term t h a t  i s w i d e l y used  to  18  describe  a v a r i e t y of operational procedures,  often reserved  f o r systems o f m a t h e m a t i c a l e x p r e s s i o n s  approximate time s e q u e n t i a l and  interaction.  Simulation  of a growth p r o c e s s , dynamic e l e m e n t s . variables  i t i s most  that  r e a l world patterns  o f exchange  almost always i s a r e p l i c a t i o n  i s , most s i m u l a t i o n m o d e l s  Given a f i x e d  s e t of values  i n a system, s i m u l a t i o n  second d i s t i c t i v e  admit t o t h e h i g h r e a l world  can produce t h e time  o f many s i m u l a t i o n s  into their  formulation,  structure.  i s that  this  t y p e o f model any s t e p  will  i n v o l v e randomly choosing  f r o m an a p p r o p r i a t e  containing  this  a stochastic variable  the value  of the variable  frequency d i s t r i b u t i o n . probabilistic.  a r e s u p p r e s s e d and r e p l a c e d  Since  many r e a l w o r l d  type  In the s o l u t i o n o f  Simulation  models  I f a l l stochastic by  statistical  measures o f c e n t r a l tendency t h e n t h e model i s termed inistic.  they  assumptions o r c o n d i t i o n s  t h e same r e s u l t .  type are c a l l e d  disturbances  That i s , with  t h e same s e t o f i n i t i a l  n o t always y i e l d  this  as d e s i r e d .  d e g r e e o f v a r i a t i o n t h a t e x i s t s i n most  will  of  paths  systems by i n c o r p o r a t i n g time dependent p r o b a b i l i t y  distributions of  feature  contain  f o r t h e exogenous  o f t h e endogenous v a r i a b l e s f o r a s l o n g a p e r i o d A  that  systems c o n t a i n  determ-  some s t o c h a s t i c  elements, d e t e r m i n i s t i c models o f such systems a r e u s u a l l y not  accurate  c h a r a c t e r i z a t i o n s o f what a c t u a l l y t a k e s  although they are o f t e n  a n a l y t i c a l l y much s i m p l e r  place,  to use.  I n s e t t i n g up a s i m u l a t i o n model t h e r e s e a r c h e r begins with  an i n i t i a l  s e t o f system e q u a t i o n s  garnered  19  through the  a p p l i c a t i o n of  to a v a i l a b l e past of  and  are  the  the  Statistical  variation, regression  analysis,  and  the  of  this initial  r e s u l t s of  model i s t e s t e d  real  and  system under study. refined until  system, i t can  the If are  simulation  i n the  processing The  comparison  this initial  on  the  The  set  machinations  initial  future  model i s  approximates  i t can  are  often  These f o r e c a s t s  and  determine the efficiently  be  of  predetermine  any  v a r i a b l e s of  the  variable  form of and  interest.  forecasts  derived  in  a«  the  a distribution  their  associated  are  derived  i n one  the  s t o c h a s t i c elements i n  d i s t r i b u t i o n of p o s s i b l e then t h i s procedure i s  mathematical a n a l y s i s  variables  used to  is probabilistic,  i n the  real  values of  values which are  simulation  a mathematical a n a l y s i s  model can  the  forecast  range o f v a l u e s p o s s i b l e  probability.  easily  correlation  r e p l i c a t e s the  is deterministic,  I f the  forecasts  showing the  If  used to  form of p r e c i s e  trial.  variable  If  be  model a c c e p t a b l y  e f f e c t s o f p r o p o s e d p o l i c i e s on  single  a  i t accurately  i n t e r e s t , o r more i m p o r t a n t l y ,  the  and  system.  Once t h e  of  by  computer m a n i p u l a t i o n o f  r e a l world  restructured the  measures  o t h e r time s e r i e s  of equations with a v a i l a b l e information of  techniques  most commonly e m p l o y e d t o o l s a t t h i s s t a g e .  validity of  present data.  c e n t r a l t e n d e n c y and  s t u d i e s , Markov c h a i n  standard econometric  i s u n d u l y complex o r  p r o b a b i l i t y d i s t r i b u t i o n s are  of  two  methods. the  outcomes  followed. tedious  estimated through  then  repeated  20 trial  runs,  employing  the  same s t a r t i n g  a d e q u a t e sample d i s t r i b u t i o n type of procedure far If  i s c a l l e d Monte C a r l o a n a l y s i s  then  initial  determine  variable).  what s e t o f i n i t i a l  tests  distributions  can  be  t h e v a r i o u s g o a l s and Klein-Goldberger  thus  i s by  can  Thus t h e  be  run,  policy  goal  (endogenous i n a comb-  r e g r e s s i o n a n a l y s i s and on  initial  conditions i s required  some p o l i c y  performed  t o determine  This  sets of  I f t h e p o l i c y maker i s i n t e r e s t e d  i n a t i o n of s e v e r a l goals then  an  i n probabilistic simulation.  conditions.  r e a c h , maximize, o r minimize  statistical  and  a s e r i e s o f Monte C a r l o a n a l y s e s  f o r each s e t of  output  The  used  i t i s desirable to consider several different  maker c a n to  o f outcomes i s p r o d u c e d .  t h e most d o m i n a n t p r o c e d u r e  conditions, one  conditions, until  other  t h e Monte C a r l o  the t r a d e - o f f s a i d i n choosing  that exist optimal  output between  strategies.  s i m u l a t i o n a n a l y s i s o f t h e U.S.  economy 8  p r o v i d e s a good example o f t h e u s e  Although  a s s o c i a t e d w i t h them. themselves  with  and  model a c c u r a c y  do  collecting  data o f t e n prove  necessarily  problems  usually  l a r g e , v e r y complex r e a l a r e enormous.  procedure.  successfully  have a number o f  S i n c e s i m u l a t i o n models  systems, the data requirements bif g e n e r a t i n g and  type of  s i m u l a t i o n m o d e l s have been  a p p l i e d by many a n a l y s t s , t h e y  concern  of t h i s  The  world high  cost  prohibitive,  suffers.  8 I . Adelman and F. L . Adelman, "Dynamic-Properties o f t h e K l e i n - G o l d b e r g e r M o d e l " , E c o n o m e t r i c a 27: 596-625, October, 1959.  21 Simulation historical  models are  growth p r o c e s s e s .  assume t h a t t h e dynamism a r e  dynamic i n t h a t they  However, i n d o i n g  technical coefficients  f i x e d or  static.  By  c o e f f i c i e n t s many s i m u l a t i o n m o d e l s r u n of  studying  the  there  is little  stant  i n the  long run,  in  a s i m u l a t i o n model are  ( i n the mathematical sense),  a p p l i c a t i o n s i n the  countering  this  been t o t r e a t 9  the model, t h i s difficult  or  of  fit"  historical policy  con-  will coefficients  Howrey  only  as  to  and  stochastic var-  l e n d i n g more a c c u r a c y  to  t e n d e d t o make model s o l u t i o n  major problem encountered or refinement  b e e n t o e x a m i n e how data,  policy  i n simulation  stage.  The  i n most m o d e l s t o measure t h e i r  b a s e d on  changes o f the  than the  hold  impossible.  employed  has  the  Since  Another approach  coefficients  While probably  t h e model v a l i d a t i o n  technique  will  unless  short run.  the  a p p r o a c h has  A third during  models f a c e .  p r o b l e m , r e c e n t l y a d v a n c e d by  themselves.  same p r o b l e m s  r e g u l a r l y u p d a t e d t h e model w i l l  have v a l i d  iables  this  o r even t h a t i t s r a t e o f growth  continuous  has  i n t o the  t o assume t h a t t e c h n o l o g y  be  Kelejian,  usually  fixed technical  l o n g term t h a t i n p u t - o u t p u t reason  so t h e y  t h a t govern  employing  replicate  one  standard  "goodness  w e l l the model r e p l i c a t e s  period  f u t u r e may  changes o f the  occurs  be  forecasts.  But  since  the  substantially different  past, Naylor  argues t h a t  9 P h i l i p Howrey and H. H. K e l e j i a n , "Computer S i m u l a t i o n V e r s u s A n a l y t i c a l S o l u t i o n s " i n The D e s i g n o f Computer S i m u l a t i o n E x p e r i m e n t s , e d . Thomas H. N a y l o r (Durham, N.C.: Duke U n i v e r s i t y P r e s s , 1 9 6 9 ) .  this  10 criterion new  f o r model v a l i d a t i o n i s not  measure o f how  well  model v a l i d a t i o n w i l l  The has  use  study of  a good example o f many l a r g e  scale  land the  their  isolated studies.  structure.  use  Columbia c o n t a i n s  heavily biased  transportation a very very  u s e f u l work and,  typical  those  example o f  and  a t the  provides However,  urban  Though t h i s m o d e l i s  land uses t h a t  somewhat, l a n d  f o r the  purposes of  the  of  Simulator  University  this  simulation  to  provide  uses such  r e c r e a t i o n a l uses) i t does  use  and  a l l o c a t i o n sector  such a s e c t o r .  (neglecting,  functions  land  constructed  towards a study of  employment o r h o u s i n g  area.  Inter-Institutional Policy  (I.I.P.S.) model c u r r e n t l y b e i n g of B r i t i s h  or  systems  planning 11  o f n a t i o n a l , r e g i o n a l , and  a land The  use  Halter  t y p e o f work done i n t h i s  simulations  simulation.  land  a l l o c a t i o n in river basin  e c o n o m i e s have i n c l u d e d within  i n modelling  some  i s developed,  t o pose problems i n  simulation  r e m a i n e d l i m i t e d t o a few  Miller's  Until  a model a c t u a l l y s i m u l a t e s  continue  of  acceptable.  as  represent  exposition, study land  a use  problems. The (I.I.P.S.)  purpose of  i s not  one  of  the  land  use  finding land  o r m a x i m i z e some measure o f  sector  use  in this  patterns  s o c i a l welfare,  but  that  model optimize  i s instead  to  10 Thomas H. N a y l o r , " P o l i c y S i m u l a t i o n E x p e r i m e n t w i t h Macroeconomic Models: the S t a t e of the A r t , " American J o u r n a l o f A g r i c u l t u r a l E c o n o m i c s , V o l . 52 No. 2, May, 1970. 11 A. N. H a l t e r , and S. F. M i l l e r , R i v e r B a s i n P l a n n i n g — A Simulation Approach. Oregon A g r i c u l t u r a l E x p e r i m e n t a l S t a t i o n S p e c i a l R e p o r t No. 224, November, 1966.  22b  predict by  or  f o r e c a s t f u t u r e l a n d use  c e r t a i n changes i n the  historical  k n o w l e d g e on  requirements  population  how  the  and  l a n d use  changes i n the  past.  For  population  labour  f o r c e increases are  the  and  will The  (manufacturing,  t r a n s p o r t a t i o n and  housing  additional population  will  they  live  by  reach  their  i n and  place  of  Many o t h e r  the  u r b a n and  found which c o n t a i n  to  the  Policy  which c o n t a i n model o f t h e Indian  l a n d use  submodel f o u n d 12  Simulator.  the  which the  model,  so  then type  work on).  determine of  labourers  housing will  As  r e g i o n a l s i m u l a t i o n models s e c t o r s t h a t are very i n the  two  can  similar  Inter-Institutional  w e l l , s e v e r a l l a r g e r n a t i o n a l models  P a k i s t a n i economy and  land a l l o c a t i o n  i n t o the  and  s u c h s e c t o r s have b e e n c o n s t r u c t e d .  economy a r e  possible  what t h e i r  will  such  employment.  be  l a n d use  route  live,  on  to  determine  processing,  subsections  where t h e will  entered  work and  food  given  system r e a c t e d  i n the model  a d d i t i o n a l labourers w i l l  c o n s i s t of  force,  example, when i n f o r m a t i o n  employment l o c a t i o n s u b s e c t i o n s  where t h e  labour  necessitated  Holland's  Kresge's  simulation of  l a r g e s c a l e n a t i o a n l models which 13,14 sectors.  the  include  12 See t h e May, 1965 i s s u e ( V o l . 31, No. 2) o f t h e J o u r n a l o f the American I n s t i t u t e o f P l a n n e r s f o r examples o f u r b a n and r e g i o n a l m o d e l s w h i c h c o n t a i n l a n d a l l o c a t i o n s e c t o r s . 13 D a v i d T. K r e s g e , "A S i m u l a t i o n M o d e l f o r E c o n o m i c P l a n n i n g : A P a k i s t a n Example," E c o n o m i c D e v e l o p m e n t R e p o r t No. 81, D e v e l o p m e n t A d v i s o r y S e r v i c e , H a r v a r d U n i v e r s i t y , 1967. 14 E . P. H o l l a n d , " S i m u l a t i o n o f an Economy w i t h D e v e l o p m e n t and T r a d e P r o b l e m s , " A m e r i c a n E c o n o m i c R e v i e w, 52: 408-430, J u n e , 1962. '  Wherever s i m u l a t i o n use  planners,  studied  by  at  l e a s t i n a l l of  t h i s w r i t e r , the  f o r e c a s t i n g or estimating specific at the  set of  land  present.  finding  the  use  use  not that  an  This  simulation  pattern  socialwelfare makers a r e  function  not  t h e i r welfare  " c a n n o t be useful  use  se.  i n the  this  to the  to y i e l d  to  or  is  believes  current  quantify for  m e t h o d o l o g y b a s e d on  are  analyst,  the  consequences of p o s s i b l e p o l i c y d e c i s i o n s  to  and  simulation  a p p r o a c h d o e s a p p e a r t o be  to  Naylor,  suggest  effort  m o d e l , shows t h e  p o l i c y maker f e e l s t o be  the  particularly  N a y l o r g o e s on  the  of  policy  target values  a l t e r n a t i v e approach i s a cooperative  p o s i t i o n which the  dis-  simulation  I f the  to define  results that 16  a t t a i n i n g , v i a repeated  some  employ some s o r t 15  best  and  some  r a r e l y been  that  the  seeking  given  i s a v a i l a b l e , according  a simulation  for  satisfies  optimization.  p o l i c y maker."  using  been  However, N a y l o r  or even d e s i r e d  information  expected  fact  those that  able  literature  a l l o c a t i o n , that i s ,  p o t e n t i a l l y superior  o u t p u t v a r i a b l e s , t h e n any  premise t h a t  patterns  land  u s u a l l y those p r e v a i l i n g  to the  w i l l i n g , . o r not functions  use  o r o b j e c t i v e s , has  procedure per  models are  i n the  t h a t maximizes or  models,tespecially  optimizing  various  land  i s p r o b a b l y due  optimizing  examples  land  policies,  Optimizing  land  the  o v e r r i d i n g c o n c e r n has  future  set of public p o l i c y goals cussed.  m o d e l s h a v e b e e n e m p l o y e d by  his  that  i n which  p o l i c y maker a i d s him  trials,  in  a  welfare  maximum.  This  reasonable  i f infor-  m a t i o n on  s o c i a l w e l f a r e f u n c t i o n s i s indeed l a c k i n g , but 15 16 Thomas H. N a y l o r , op. c i t . I b i d . p. 264.  only  if  t h e number o f a l t e r n a t e p o l i c y  limited.  For  as p o l i c y  c o m b i n a t i o n s may  c o n s u m i n g t o be  tested via this  Though work by  f i n d i n g optimum s e e k i n g has  been e n c o u r a g i n g ,  as  l a n d use  simple  and  Operational  Games  Close devices  area before  problems, the probably  relatives  feel  r e l a t i o n s which they contain  then,  t o one  of  policy decisions.  A  literature game, l i k e  lies  i n the  as a  mathematical  system.  Despite  The  most o f t h e  usually these  major the  Though most games  o f gaming m o d e l s i s s p e c i f i e d v e r b a l l y , v i s u a l l y , M a t h e m a t i c a l models a r e , o f c o u r s e ,  operating,  s p e c i f i c a t i o n of  attempt to r e p r e s e n t .  specified.  except  r o l e o f s i m u l a t i o n models  some 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 ,  mathematically  progress  o f m a t h e m a t i c a l s i m u l a t i o n models  games.  two  t h a t much more  Until  model, i s a r e p r e s e n t a t i o n o f a r e a l w o r l d d i f f e r e n c e between t h e  error 18 Amiada in  remain c o n f i n e d  pretesting specific  systemic  time  and  s i m u l a t i o n can  known v a r i o u s l y i n t h e  operational or  of  d e c i s i o n r u l e s i n s i m u l a t i o n models  planning w i l l  forecasting  are  essentially trial 17Conway and Zusman and  a l e g i t i m a t e o p t i m i z i n g procedure.  for relatively in  p r o v e t o o c c o s t l y and  many r e s e a r c h e r s  work i s n e e d e d i n t h i s  reasonably  a l t e r n a t i v e s i n c r e a s e , t h e number  possible policy  approach.  decisions i s  or  structure in writing.  completely  differences in  17 R. W. Conway, "Some T a c t i c a l P r o b l e m s i n D i g i t a l S i m u l a t i o n " , Management S c i e n c e , 10: 47-61, O c t o b e r , 1963. 18 P. Zusman and A. Amiada, " S i m u l a t i o n : A T o o l o f Farm P l a n n i n g U n d e r C o n d i t i o n s o f . Weather U n c e r t a i n t y " , J o u r n a l o f Farm E c o n o m i c s , 47: 574-594, A u g u s t , 1965.  25 in  s p e c i f i c a t i o n many a n a l y s t s  According  classify  games a s s i m u l a t i o n  models.  to Feldt:  . . . an o p e r a t i o n a l game i s a t r u e s i m u l a t i o n . It isa s i m u l a t i o n i n w h i c h a l l o f most o f t h e d e c i s i o n s a s t o outcomes o f t h e e v e n t s b e i n g r e p r e s e n t e d a r e l e f t i n t h e hands o f human p l a y e r s , w i t h r e l a t i v e l y m i n o r d e c i s i o n s and a c c o u n t i n g p r o b l e m s h a n d l e d e i t h e r b y human o p e r a t o r s o r by c o m p u t e r s . ^ Though o p e r a t i o n a l the in  l a n d use f i e l d ,  e s p e c i a l l y as e d u c a t i o n a l  devices  to a i d  u n d e r s t a n d i n g t h e w o r k i n g s o f a l a n d u s e s y s t e m , t h e y do n o t  seem t o o w e l l forecasting for  games u n d o u b t e d l y have many a p p l i c a t i o n s i n  suited to contribute 20  and p l a n n i n g .  the players  t o problems o f l a n d use  B e c a u s e games must be s i m p l e  t o l e a r n and u n d e r s t a n d ,  a high  enough  degree o f  a b s t r a c t i o n n e c e s s a r i l y r e n d e r s ; gaming m o d e l s i n a c c u r a t e and unsuitable  as p r e d i c t i v e o r p l a n n i n g  M a t h e m a t i c a l Programming  A third  devices.  Models  major approach i n t h e use o f mathematical  m o d e l s a s an a i d t o d e c i s i o n m a k i n g l i e s ematical locating  programming. optimal  than simply  i n the f i e l d  o f math-  M a t h e m a t i c a l programming c o n c e r n s i t s e l f  or best  solutions to a given  system,  with  rather  f o r e c a s t i n g o r p r e d i c t i n g an e x p e c t e d s o l u t i o n .  While input-output  and s i m u l a t i o n  seek o p t i m a , t h e s e a r c h  m o d e l s c a n a l s o be u s e d t o  p r o c e d u r e s e m p l o y e d a r e n o t an  implicit  19 A l l a n F e l d t , P l a y e r ' s M a n u a l f o r CLUG; Community L a n d Use Game (New Y o r k : C o l l i e r M a c m i l l a n Canada L t d , 1 9 7 2 ) , p . 1. 20 F e l d t ' s l a n d u s e game ( s e e above f o o t n o t e ) provides a good i l l u s t r a t i o n o f t h e u s e o f o p e r a t i o n a l games a s e d u c a t i o n a l d e v i c e s i n t e a c h i n g l a n d use theory.  p a r t o f t h e model s t r u c t u r e and exhaustive  searches  Mathematical  this  implicit  rudimentary  s m a l l number o f  alternatives.  t h e o t h e r hand, a l w a y s has  in i t s structure.  specified  and  w h i c h i s an  and  goals that apply  f u n c t i o n , whose v a l u e  Besides  system under  i s t o be  a criterion  optimized,  desires  study.  t h i s mathematical statement  n o t h i n g more t h a n  of o b j e c t , which  by w h i c h t o j u d g e a l t e r n a t e  s o l u t i o n s , m a t h e m a t i c a l programming m o d e l s a l s o c o n t a i n a or array of mathematical expressions limitations real  interactivity  system being  a statement i n p u t s and into  and  an  abstracted.  which s p e c i f y  These e x p r e s s i o n s  of production p o s s i b i l i t i e s  given  When t h i s  solution, a solution  an  efficient  i f one  are  fixed  i s infeasible,  resource  i s q u i c k l y and  this  the  basically amounts o f  readily  optimal  produced.  o r unbounded, o r i f more t h a n  g l o b a l optimum i s p o s s i b l e , t h e n  in  computer  programming a l g o r i t h m , an  exists,  set  information i s fed  o p t i m i z i n g model, u s u a l l y v i a a d i g i t a l  containing  the  r e l a t i o n s h i p s which e x i s t  a given technology.  to  some  adequate r e p r e s e n t a t i o n o f r e a l world i n the  efficient  A corrollary  i s t h a t m a t h e m a t i c a l programming a l w a y s c o n t a i n s  mathematically  is  of a r e l a t i v e l y  programming, on  search procedures  usually involve  If  one  i n f o r m a t i o n i s a l s o made  available.  B e c a u s e most programming m o d e l s assume t h a t resources,  and  p r i c e s are a l l f i x e d  m a t h e m a t i c a l programming  i n any  is essentially  as  s u c h w o u l d seem e l i g i b l e  at  s i m u l a t i o n and  one  given  a static  analysis in this  situation,  analysis,  f o r much o f t h e c r i t i c i s m  input-output  technology,  and  levelled  regard.  However,  one  o f the major advantages of mathematical  ease i n which changes i n t e c h n o l o g y , be  studied.  Price  programming, and easily of  be  ranging,  a change i n one  to determine, price  ceteris  or resource  level  Thus, mathematical  does l e n d i t s e l f  can  p a r i b u s , the  effect  or, mutatis Techniques  programming, w h i l e  mutandis, have  input  coeffic-  static  to a n a l y s i s of v a r i a n c e w i t h i n the  may  parametric  analysis  i n programming w i t h v a r i a b l e  i s the  prices  hand s i d e r a n g i n g ,  s e v e r a l c o n c u r r e n t changes.  a l s o been d e v e l o p e d 21 ients.  r e s o u r c e s , and  o t h e r methods o f s e n s i t i v i t y  conducted  the e f f e c t o f  right  programming  i n nature,  systems  being  studied.  The  most commonly e m p l o y e d t y p e o f  programming u s e d programming. Lit  today  i s the Simplex  According  to  mathematical  method o f  linear  McMillan:  . . . t h e s i m p l e x m a c h i n e |Tof l i n e a r programming] i s t h e most a v a i l a b l e , t h e most f r e q u e n t l y e m p l o y e d , and t h e most w i d e l y u n d e r s t o o d o p t i m i z i n g m a c h i n e . Recurring a r t i c l e s i n p r o f e s s i o n a l and t r a d e l i t e r a t u r e a t t e s t t o t h e c o n t i n u i n g and s u c c e s s f u l e f f o r t s t o so s t r u c t u r e r e p r e s e n t a t i o n s o f r e a l - w o r l d p r o b l e m s t h a t t h e y become s u s c e p t i b l e t o s o l u t i o n v i a the s i m p l e x machine.^2  However, t h e a s s u m p t i o n s e m p l o y e d i n l i n e a r programming somewhat r e s t r i c t  the  problems of l i n e a r i t y , and  relevance of i t s a p p l i c a t i o n s . additivity,  divisibility, 23  s i n g l e - v a l u e e x p e c t a t i o n s a r e w e l l known.  static  technical  coefficients  do  The  finiteness, As w e l l ,  n e c e s s a r i l y mean c o n s t a n t  the  returns  21 F o r a d e s c r i p t i o n o f t h i s and o t h e r methods o f e x a m i n i n g p r i c e , r e s o u r c e , and t e c h n o l o g y c h a n g e s see E a r l . 0. Heady and W i l f r e d C a n d l e r , L i n e a r Programming Methods (Ames: Iowa S t a t e U n i v e r s i t y Press"^ 1958) , e s p e c i a l l y c h a p t e r s 7, 8, 16. 22 M c M i l l a n , op. c i t . , p. v i o f p r e f a c e . 23 F o r a d i s c u s s i o n o f t h e s e a s s u m p t i o n s , s e e Heady and C a n d l e r , op. c i t . , pp. 17-18.  28 to s c a l e ,  an a s s u m p t i o n  problem o f f i n i t e n e s s ,  that also,  I f n o t enough a c t i v i t i e s  c a n n o t a l w a y s be j u s t i f i e d . often  The  l e a d s t o one o f h e t e r o g e n e i t y .  are specified  then the a c t i v i t i e s can  become t o o h e t e r o g e n e o u s t o a l l o w e s t i m a t i o n o f i n t e r e s t i n g interactivity  relationships.  Many a t t e m p t s have b e e n made t o overcome t h e d i f f i culties  p o s e d by t h e s e a s s u m p t i o n s , e i t h e r  t o t h e S i m p l e x method some o t h e r  refinements  o f l i n e a r programming, o r by u s e o f  form o f programming.  programming  through  Q u a d r a t i c and o t h e r  f o r m s c a n be u s e d t o more a c c u r a t e l y  non-linear  depict  real  w o r l d r e l a t i o n s and a v o i d many o f t h e p r o b l e m s a s s o c i a t e d w i t h 24 linearity. Z e r o - o n e a n d o t h e r t y p e s o f i n t e g e r programming 25 may be u s e d t o t a c k l e d i s c o n t i n u i t i e s . Dynamic l i n e a r and d y n a m i c programming c a n be u s e d t o overcome t h e p r o b l e m s o f applying  analysis to intertemporal or sequentially 26 ordered problems. S t o c h a s t i c programming c a n i n t r o d u c e v a r 27 iation  a static  into  t h e model a n d a l l o w  However, t h e s e r e l a t i v e l y  f o r ranges o f e x p e c t a t i o n s .  new i n n o v a t i o n s  i n the f i e l d  o f math^  e m a t i c a l programming have n o t e n j o y e d w i d e s p r e a d u s e and accepatance  i n most a r e a s ,  simply because o f t h e i r  demanding t i m e and d a t a r e q u i r e m e n t s . Non-linear,  integer,  exceedingly  According to Reisch:  dynamic l i n e a r ,  24  and dynamic 25  M c M i l l a n , op. c i t . ,  p p . 173-219.  Ibid.,  p p . 312-99,  26 I b i d . , pp. 242-270, o r E a r l 0. Heady a n d L a u r e l L o f t s g a r d , " A p p l i c a t i o n o f Dynamic Programming M o d e l s f o r O p t i m a l Farm a n d Home Plans-"'"-Journal o f Farm E c o n o m i c s , X L I , F e b r u a r y , 1959. 27 See K. D. C o c k s , " D i s c r e t e S t o c h a s t i c Programming", Management S c i e n c e , V o l . 15 No. 1, 1968, p p . 72-79.  29 programming as w e l l a s s t o c h a s t i c programming have n o t p r o v e d t o be a p p l i c a b l e t h u s - f a r . . . . The main r e a s o n s a r e (a) n e c e s s a r y d a t a c a n n o t be s u p p l i e d ; (b) s e t t i n g up t h e m o d e l s t a k e s t o o much p r e p a r a t i o n t i m e ; (c) r e q u i r e ments o f c o m p u t e r t i m e and s t o r a g e c a p a c i t y f o r r e a l i s t i c m o d e l s a r e s t i l l t o o h i g h ; and (d) i n p u t s t h a t go b e y o n d t h e r e q u i r e m e n t s f o r s t a n d a r d [ l i n e a r ] programming a r e n o t w i t h i n a reasonable r e l a t i o n to the value of the a d d i t i o n a l i n f o r m a t i o n g a i n e d b y t h o s e more s o p h i s t i c a t e d methods.^8 After  r e v i e w i n g o t h e r d e c i s i o n models t h a t a r e a v a i l a b l e t o  planners  today,  programming  R e i s c h goes on t o s t a t e  c a n be c o n s i d e r e d as a w i d e l y 29  p l a n n i n g method." advantages over It  i s mathematically  set-up  matrix  time  ations. quite  very  simple  Computer  It i s a relatively  layman.  programming  programming.  f o r o t h e r more  flexible,  used  h a s many  a r e n o t as  requirements  easy  and  yet theoretically  i s much s h o r t e r t h a n  I t i s extremely  simple  linear  I t s data requirements  formulations.  demanding.  the  Certainly  linear  accepted  t h e o t h e r more i n v o l v e d f o r m s o f  a t t h e same t i m e . and  that "only  elegant severe complicated  are not o v e r l y  having widespread  t o understand  method  applic-  and i s  t o use, not o n l y f o r the a n a l y s t but a l s o f o r And p r o b a b l y most  respectability  of widespread  Nor d o e s t h e n a t u r e the a p p l i c a t i o n  of linear  f u n c t i o n s c a n be c l o s e l y  i m p o r t a n t l y , i t has g a i n e d  u s e and  acceptability.  o f i t s assumptions o v e r l y  programming. approximated  the  Non-linear by s t e p - w i s e  real  restrict world  linear  28 E r w i n R e i s c h , " P r o v e n T o o l s f o r M i c r o P l a n n i n g and D e c i s i o n s " , E c o n o m i c M o d e l s and Q u a n t i t a t i v e Methods f o r D e c i s i o n s and P l a n n i n g ~ I n A g r i c u l t u r e e d . E a r l 0. Heady (Ames: Iowa S t a t e U n i v e r s i t y P r e s s , 1971) p i 155. 29 I b i d . , p . 157.  30 f u n c t i o n s t o l e s s e n the problem o f l i n e a r i t y . divisibility  tend  A d d i t i v i t y and  t o c r e a t e problems a t t h e micro  level  and a r e n o t  u s u a l l y encountered  i n macro p l a n n i n g m o d e l s .  ness i s e s s e n t i a l l y  one o f taxonomy and measurement and d o e s n o t ,  per  se, represent a s t r u c t u r a l  Certainly with  this  deficiency i n linear  p r o b l e m i s no g r e a t e r w i t h  o t h e r mathematical models.  static,  modelling  examples o f i t s a p p l i c a t i o n  Edwin M i l l s , of  interesting  who u s e s t h i s  flexibility,  such  i n this  field  the optimal  a r e many.  L i n e a r programming  a n a l y s i s i n seeking  abounds 30 allocation.  i s presented  to not only a l l o c a t e  i s superior to the positive  input-output  area  spatial  optimal  by  amounts  distribution of  programming h o l d s Itsqualities  non-programming m o d e l s a s t h e y  mathematical models. as  device  The l i t e r a t u r e  and low i n p u t r e q u i r e m e n t s , - . d i s c u s s e d  advantage over  programming  analysis.  t o problems o f l a n d  The a d v a n t a g e s t h a t l i n e a r  model s t r u c t u r e s i n t h i s  an  today.  application  land use b u t a l s o t o i n d i c a t e 31  land uses.  inherent i n the  i n linear  sensitivity  than  programming i s t h e most d o m i n a n t f o r m o f m a t h e m a t i c a l  used i n l a n d use p l a n n i n g  A particularly  programming.  programming  The d i f f i c u l t i e s  c a n be l e s s e n e d a g r e a t d e a l t h r o u g h  with  linear  s i n g l e - v a l u e e x p e c t a t i o n s employed  Linear  The p r o b l e m o f f i n i t -  over  of simplicity,  a b o v e , a r e a s much  are over a l t e r n a t e  i s a n o r m a t i v e p r o c e s s , and  a p p r o a c h e s o f s i m u l a t i o n and  optimal  l a n d use p a t t e r n s .  P e r h a p s t h e o n l y m a j o r drawback t o t h e u s e o f c o n -  30 See t h e b i b l i o g r a p h y i n A. G. G a r d n e r , A L i n e a r P r o g r a m ming M o d e l f o r L a n d R e s o u r c e A l l o c a t i o n i n t h e Lower M a i n l a n d of" B r i t i s h C o l u m b i a ( u n p u b l i s h e d M.Sc. t h e s i s i n t h e D e p a r t m e n t o f A g r i c u l t u r a l E c o n o m i c s , 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 ) , p p . 106-108. 31 E d w i n S. M i l l s , " M a r k e t s and E f f i c i e n t R e s o u r c e A l l o c a t i o n i n U r b a n A r e a s , " S w e d i s h J o u r n a l o f E c o n o m i c s , 1972, p p . 101-113.  31  v e n t i o n a l l i n e a r programming i n land use p l a n n i n g  and other  such areas o f p u b l i c d e c i s i o n making, i s the p r o v i s i o n f o r o n l y a s i n g l e o b j e c t i v e as a c r i t e r i o n f o r o p t i m a l i t y , when i n r e a l i t y many c r i t e r i a may be r e l e v a n t . l i n e a r programs f o r optimal marginal  (value)  Traditioanlly,  land use a l l o c a t i o n have used the  p r o d u c t i v i t y o f l a n d , land r e n t s ,  production,  or some such r e l a t e d measure o f economic p r o f i t o r g a i n as the s i n g l e c r i t e r i o n a g a i n s t which a l t e r n a t e land a l l o c a t i o n s should be measured.  However, many other p o l i c y goals can be detected  i n land use d e c i s i o n s : promotion o f ^ a g r i c u l t u r a l s e l f - s u f f i c i e n c y , the p r e s e r v a t i o n municipal  o f r e c r e a t i o n a l areas and green b e l t s , i n c r e a s i n g  t a x bases, a i d i n g export i n d u s t r i e s , encouraging  employment, i s o l a t i n g land uses t o s p e c i f i c areas,  full  minimizing  p o l l u t i o n l e v e l s , encouraging c e r t a i n forms o f t r a n s p o r t a t i o n , promoting low c o s t housing and urban renewal, m a i n t a i n i n g standards and other b u i l d i n g codes, and so on.  density  These are j u s t  a few examples o f other p o l i c y o b j e c t i v e s t h a t have i n f l u e n c e d land use d e c i s i o n s .  Note t h a t d i f f e r e n t p o l i c y goals may  c o n f l i c t w i t h one another.  Encouraging the use o f the p r i v a t e  automobile, and the m i n i m i z a t i o n example o f c o n f l i c t i n g g o a l s .  o f p o l l u t i o n l e v e l s i s a prime  Promoting low d e n s i t y housing  standards, and encouraging the development o f r a p i d t r a n s i t i s another, l e s s obvious, example.  Note a l s o t h a t v i r t u a l l y a l l  of these a d d i t i o n a l goals c o n f l i c t w i t h the maximization o f land r e n t s o r some such measure o f economic g a i n , as measured i n the p r i v a t e market p l a c e .  Most o f these p o l i c i e s ^ a r e , i n f a c t ,  attempts t o a l l e v i a t e o r c o r r e c t m i s a l l o c a t i o n s o f resources r e s u l t from v a r i o u s e f f e c t s which are e x t e r n a l t o the p r i v a t e  that  32 economy.  Pollution,  f o r example, i s a n e g a t i v e  externality  w h i c h i m p o s e s enormous damages upon s o c i e t y and taken  into  account  by  p r i v a t e d e c i s i o n makers b e c a u s e i t does 32  n o t have a p r i v a t e c o s t a s s o c i a t e d w i t h on  yet i s not  t h e o t h e r hand, r e p r e s e n t s  it.  A green  belt,  a p o s i t i v e p u b l i c good e x t e r n a l -  i t y w h i c h c o n t r i b u t e s c o n s i d e r a b l e s o c i a l w e l f a r e , and 33 i s a l s o n o t a c c o u n t e d f o r by t h e p r i v a t e s e c t o r . L i n e a r programs which f i n d distributions many o t h e r  t h a t m a x i m i z e o n l y one  criteria  solutions.  exist,  are not  appropriate land policy  r e a l l y 'finding  this  problems.  and  optimal  However, t h e  r e c e n t development,  f o r m u l t i p l e g o a l s , seems t o s u c c e s s f u l l y 34 T h i s new  seeks s o l u t i o n s t h a t o p t i m i z e essentially  land  B o e h l j e , o f a f o r m o f l i n e a r programming  deficiency.  policy  when  T h i s does r e p r e s e n t a s e r i o u s d e f i c i e n c y i n t h e  allocation  allows  use  criterion,  a p p l i c a t i o n o f c o n v e n t i o n a l l i n e a r programming t o  Candler  which  form o f  linear  a criterion  The  number o f p o l i c y  goals  that  circumvent  programming  f u n c t i o n which i s  a composite of s e v e r a l d i f f e r e n t ,  goals.  by  often  conflicting,  i n c l u d e d i n the  compo-  32 See D a n i e l S c h r o e t e r , "Some E c o n o m i c A s p e c t s o f P o l l u t i o n , " ( u n p u b l i s h e d B.A. g r a d u a t i n g t h e s i s i n t h e D e p a r t m e n t o f E c o n o m i c s , 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 ) , pp. 11-13. 33 See D a n i e l S c h r o e t e r , "Green B e l t C o n s e r v a t i o n and Development R i g h t s , " ( u n p u b l i s h e d e s s a y i n the Department o f A g r i c u l t u r a l Economics, U n i v e r s i t y o f B r i t i s h Columbia, 1972), pp. 2-3. 34 W i l f r e d C a n d l e r and M i c h a e l B o e h l j e , "Use o f L i n e a r Programming i n C a p i t a l B u d g e t i n g w i t h M u l t i p l e G o a l s , " A m e r i c a n J o u r n a l o f A g r i c u l t u r a l E c o n o m i c s , V o l . 53 No. 2, May, 1971, pp. 325-330.  site  o b j e c t i v e f u n c t i o n i s not  although  use  o f much more t h a n  Though, t o my  the  i t seems e m i n e n t l y w e l l , s u i t e d  t h e a d d e d one  objectives. set-up  Nor  problems which can  requirements.  easily  Because o f the  lower  be  over  t o do  utilize  format  linear  land  resource  so. I t has a l l programming goals  unduly  Indeed, the  great e f f i c i e n c y this  of this  search procedure  computer r e q u i r e m e n t s .  o t h e r model t y p e s ,  or add  to  conventional -  method, have  substan-  B e c a u s e o f t h e many  this  c h o s e n as t h e m e t h o d o l o g y t o be planning  form o f  a p p l i e d t o problems of t h i s  advantages t h a t m u l t i p l e goals l i n e a r possess  unwieldy.  single-goal linear  does t h e m u l t i p l e g o a l s  Simplex a l g o r i t h m can  tially  prove  of allowing f o r several p o l i c y  o r computer time  nature.  a d o z e n may  s e t number,  n o t been a p p l i e d t o problems o f  advantages of c o n v e n t i o n a l  with  t o any  knowledge, t h i s m u l t i p l e g o a l s  programming has allocation,  limited  programming a p p e a r s f o r m o f programming  used i n t h i s  study  to was  of land  use  i n the F r a s e r V a l l e y .  Summary  On of can  the be  the b a s i s of t h i s  l i t e r a t u r e , three broad  somewhat a b b r e v i a t e d  sampling  c l a s s e s o f mathematical models  distinguished: input-output  analysis,  simulation  ( i n c l u d i n g gaming m o d e l s ) , and  mathematical  Though a l l o f t h e s e  have some a p p l i c a t i o n  field  of  l a n d use  have tended  techniques  planning,  input-output  t o remain r e s t r i c t e d  s h o r t t e r m f o r e c a s t i n g , and  and  programming. in  the  s i m u l a t i o n models  to studies of h i s t o r i c a l data,  p r e t e s t i n g of s p e c i f i c  policy  34  changes.  M a t h e m a t i c a l programming,  a readily  useable,  best or optimal  systematic  solutions  on t h e o t h e r hand,  format with which t o s e l e c t  from a wide v a r i e t y  A l t h o u g h t h e y a r e s h o r t t e r m and s t a t i c models a r e e x t r e m e l y  flexible  of p o s s i b l e v a r i a t i o n  linear  programming  public  d e c i s i o n making.  simplest of a l l  and r e a d i l y  seems b e s t  suited  Though  l a n d use p l a n n i n g  many d i f f e r e n t which c o n f l i c t , field,  do r e s t r i c t limiting  land resource  g o a l s and o b j e c t i v e s ,  to linear  goals objective  for application  applic-  t o problems  allocation.  in this  function, area.  Because  including  ones  i n the land  use  programming w h i c h a l l o w s seems e s p e c i a l l y  The f o l l o w i n g  methodology i n d e t a i l  a hypothetical  i t s validity  in their  suited  o p e r a t e on d e c i s i o n m a k i n g  d i s c u s s e s t h i s new examine  t o problems o f o p t i m a l  area i s quite widespread.  a recent modification  for a multiple suited  policy  system under o b s e r v a t i o n .  appears w e l l  and o p t i m a l  Indeed, i t s use i n t h i s  for analysis  models.  L i n e a r programming of  allow  programming  t h e a s s u m p t i o n s employed i n t h i s  t y p e s o f programming  t o macro d e c i s i o n  alternatives.  t y p e s o f m a t h e m a t i c a l programming,  somewhat, t h e y do n o t a p p e a r o v e r l y ation  of  i n nature,  i n the r e a l world  Of t h e s e v e r a l d i f f e r e n t  offers  l a n d use problem.  well  chapter  and u s e s i t t o  35  Chapter 5  THE USE OF LINEAR PROGRAMMING WITH MULTIPLE GOALS IN LAND RESOURCE ALLOCATION  The F o r m a t a n d M e t h o d o l o g y o f C o n v e n t i o n a l L i n e a r  The s u r v e y o f t h e l i t e r a t u r e going chapter  suggested that  for determining optimal  relatively  i n this  presented i n the forea v a i l a b l e model  land use patterns  programming w h i c h c o n t a i n s The d i s c u s s i o n  the best  a multiple  chapter w i l l  new programming method  Programming  goal  center  structure  i s a form o f l i n e a r objective  function.  on e x p l a i n i n g  and p r o v i d i n g  this  an e l e m e n t a r y  example o f i t s a p p l i c a t i o n t o l a n d u s e p r o b l e m s .  Conventional as  follows:  such  find  linear  p r o g r a m s c a n be e x p r e s s e d  the k x 1 vector  of alternative  activities  that:  Z = z(X)  a  maxxmum  subject t o : AX  5 B  and,  X > 0 where X  k x 1 vector o f alternative a c t i v i t i e s X3 . . . X]^) , a n d  Z  some  A  an m x k m a t r i x o f t e c h n i c a l c o e f f i c i e n t s w h i c h  (X]_, X2,  (linear) c r i t e r i o n function of the l e v e l s o f t h e a c t i v i t i e s , and  36 determine i n t e r a c t i v i t y l e v e l s , and  r e l a t i o n s h i p s and  B = an m x 1 v e c t o r o f r e s t r a i n t s flows and l e v e l s  Note t h a t t h i s criterion levels  format assumes t h a t :  function i s linear;  on i n t e r a c t i v i t y  (1) the o b j e c t i v e o r  (2) the r e s t r a i n t s  can be expressed as l i n e a r  inequalities;  on a c t i v i t y and (3) a  l o c a l maximum i s a g l o b a l maximum, and t h a t the d i r e c t i o n o f improvement i n the o b j e c t i v e f u n c t i o n can be determined a t any  point.  wifehfethe  The f i r s t two o f these assumptions a r e i n keeping  l i n e a r i t y p r o p e r t i e s necessary f o r the programming  m a n i p u l a t i o n o f the problem m a t r i x .  The t h i r d assumption i s  necessary i f a simple search procedure such as the Simplex Algorithm  i s t o be used t o f i n d the g l o b a l optimum.  The A p p l i c a t i o n o f L i n e a r Programming t o a H y p o t h e t i c a l Land Use Problem  To b e t t e r i l l u s t r a t e t h i s programming method, consider  the f o l l o w i n g h y p o t h e t i c a l s i t u a t i o n .  The p l a n n i n g  commission f o r a s m a l l m u n i c i p a l i t y i s c o n s i d e r i n g how t o zone the m u n i c i p a l i t y ' s 1000 acres i s a t a maximum.  so t h a t t o t a l land  value  Because land value was f e l t t o be a measure  of the p r o d u c t i v i t y o f the l a n d , the commission f e l t  that  maximizing t o t a l land value would l e a d t o a maximum o f l o c a l production. all  Since t o t a l l o c a l p r o d u c t i o n  was the source o f  income f o r the i n h a b i t a n t s o f the r e g i o n , maximizing  production  would a l s o maximize incomes.  The commission  37  has decided t h a t land may  o n l y be put to one of three  r e s i d e n t i a l , i n d u s t r i a l or a g r i c u l t u r a l unused).  Two  (or i t may  uses:  be  left  hundred and f i f t y acres o f a g r i c u l t u r a l  land  and one hundred acres of r e s i d e n t i a l land i s needed to meet the minimum food and housing requirements  of the  populace.  I t i s a l s o known t h a t an acre of r e s i d e n t i a l land employs no workers and uses up ten thousand d o l l a r s of the community's f i v e m i l l i o n d o l l a r c a p i t a l supply.  An acre of i n d u s t r i a l  Jgand employs twenty people and r e q u i r e s twenty thousand d o l l a r s of c a p i t a l , w h i l e an acre i n a g r i c u l t u r a l o n l y one person and r e q u i r e s but one starting capital.  The  employs  thousand d o l l a r s  of  r e g i o n s labour f o r c e comprises  1000  people.  The p l a n n i n g commission has decided t h a t zoning i s necessary to ensure o r d e r l y development but i s u n c e r t a i n as to the amounts of l a n d to be zoned to each category so t h a t t o t a l community land value i s a t a maximum. valued  R e s i d e n t i a l land i s  a t t h r e e thousand d o l l a r s per a c r e , w h i l e an acre  of i n d u s t r i a l land i s worth seven thousand d o l l a r s and acre o f a g r i c u l t u r e ,  one  an  thousand.  S e t t i n g up t h i s problem i n terms o f the above l i n e a r programming t a b l e a u we  get:  Land A c t i v i t i e s ( a l l ^0) RESIDENTIAL INDUSTRIAL AGRICULTURAL  (x ) x  O b j e c t i v e : Maximum of Subject to C o n s t r a i n t s on: A v a i l a b l e Land A v a i l a b l e Labour Available Capital Minimum Food Minimum Housing  (x ) 2  3000  7000  1 0 10 0 1  1 20 20 0 0  RHS  (x ) 3  1000 1 1 1 1 0  a  max. ^1000 ^1000 ^5000 > 250 * 100  38  In t h i s statement the o b j e c t i v e f u n c t i o n to be maximized i s the t o t a l community land v a l u e .  The  f i r s t c o n s t r a i n t ensures  t h a t no more land than i s a v a i l a b l e i s a l l o c a t e d . c o n s t r a i n t l i m i t s the use of labour and  to the e x i s t i n g  the t h i r d r e s t r a i n s the use o f c a p i t a l .  The  f i f t h r e s t r a i n t s ensure t h a t minimum food and requirements are met.  The  Though these l a s t two  second  supply  fourth  and  housing constraints  are  i n the form of " g r e a t e r than o r equal t o " i n e q u a l i t i e s they can e a s i l y be made to conform w i t h the r e s t of the i n e q u a l i t i e s by simply m u l t i p l y i n g each l i n e by negative t h a t the n o n - n e g a t i v i t y  unity.  Note a l s o  of each a c t i v i t y column i s an  p a r t of a l i n e a r programming t a b l e a u and  need not be  implicit included  i n the l i s t o f e x p l i c i t c o n s t r a i n t s .  When t h i s i n f o r m a t i o n  i s fed i n t o an o p t i m i z i n g  ( u s u a l l y v i a a d i g i t a l computer) the s o l u t i o n i s q u i c k l y and  r e a d i l y produced.  In t h i s case the o b j e c t i v e  function  i s a t a maximum when: RESIDENTIAL LAND USES, X i = 400 INDUSTRIAL LAND USE,  X  AGRICULTURAL LAND USE, and  = 21  2  X  t o t a l community land value has  Thus given  some i n f o r m a t i o n  3  acres,  acres,  = 579  acres,  reached $1,926,000.  about the p l a n n i n g  commissions  d e s i r e s , as w e l l as some data on t e c h n i c a l p o s s i b i l i t i e s , a l i n e a r program can be employed t o optimize makers ( s i n g l e goal) c r i t e r i o n  function.  the d e c i s i o n  routine  39  The  Format and Methodology o f M u l t i p l e Goals  Linear  Programming  However, a s h a s been p r e v i o u s l y d i s c u s s e d , many g o a l s may be p u r s u e d i n a d e c i s i o n - m a k i n g land use p l a n n i n g .  In the context  example, t h e p l a n n i n g minimizing  pollution  In t h i s  case,  c o m m i s s i o n may a l s o be c o n c e r n e d  alternate multiple  of other  with!;.maximizing t o t a l  a single  i s satisfactory.  In g e n e r a l , t h e format programming c a n be e x p r e s s e d  should  land  value.  g2(X),  compare  Instead  be u s e d .  for multiple goals  as f o l l o w s : f i n d  activities  Z = z gi(X),  community  g o a l c r i t e r i o n by which  l i n e a r programming  vector of alternative  g o a l s , many o f  l i n e a r programming o f t h e t y p e  s o l u t i o n s , no l o n g e r goals  with  l e v e l w i t h i n the m u n i c i p a l i t y , maximizing  conventional  just o u t l i n e d , with  such as  o f t h e above p r o b l e m , f o r  employment, a s w e l l a s a m u l t i p l i c i t y w h i c h may c o n f l i c t  process  linear  the k x 1  such t h a t :  . . . g (X) n  a maximum  subject t o : AX  £ B  and,  X > 0 , where X = k x 1 vector of alternative X3, . . . X^) , a n d Z = some  activities  (X]_, X , 2  ( l i n e a r ) composite c r i t e r i o n f u n c t i o n o f the l e v e l o f i n d i v i d u a l g o a l f u n c t i o n s , and  g^ (X) = t h e i t h ( l i n e a r )  goal  f u n c t i o n , and  40  A = an m x k m a t r i x o f t e c h n i c a l c o e f f i c i e n t s w h i c h d e t e r m i n e i n t e r a c t i v i t y r e l a t i o n s h i p s and l e v e l s , and B = an m x 1 v e c t o r o f r e s t r a i n t s f l o w s and l e v e l s .  While t h i s conventional the  individual  scale. case of has  linear  Often  this  an  goal goals  f o r m a t employs a l l t h e programming,  be  the  devised,  i s constant,  utility  on  some  the p o l l u t i o n  generating  generating  f o r the  between t h e s e i s an  additional  linear  other  zero  activities  extremes.  linear  appropriate  0 t o 100  with  (units) to the  and  appropriate  t o do  this,  t h a t the marginal 1  i s held constant. b a s e d on  any  cannot scale  scale for 100  (units)  the  least  continuum  a l l t h a t i s •;•  utility  of  an  T h e s e s c a l e s do  t e c h n i c a l or  What i s r e q u i r e d i s t h a t t h e y  d e c i s i o n maker's p e r c e p t i o n o f h i s u t i l i t y  be  not  financial  b a s e d on  the  function i n respect  1 In o t h e r words, s c a l e v a l u e s  doll  intermediate  t h a t l i e i n the  In o r d e r  assumption  n e c e s s a r i l y have t o be  An  utility  goal  a c t i v i t y which generates  activity,  scale unit  continuum.  of  hundred  I f the  some o t h e r  g o a l m i g h t be  g r e a t e s t amount o f p o l l u t i o n , pollution  t h a t i s , one  of ten d o l l a r s .  however a r b i t r a r y .  a p o l l u t i o n minimizing  required  expressed  a s s u m p t i o n m e r e l y demands t h a t t h e m a r g i n a l  ten times  values  assumptions  a r e m e a s u r e d i n m o n e t a r y t e r m s , i n w h i c h .:  be m e a s u r e d i n m o n e t a r y t e r m s , t h e n  to  interactivity  i t assumes i n a d d i t i o n t h a t  f u n c t i o n s can  additional dollar  must be  on  must be a d d i t i v e .  41 to each g o a l financial will  in isolation,  information  a i d him  though undoubtedly t e c h n i c a l  t h a t i s a v a i l a b l e to the  i n making u t i l i t y  although  one  acre  physical  amount o f p o l l u t i o n  p r o d u c e s , one estimation  o f a c t i v i t y A may  should  of the  activities will  not  be  be  i n the  consider  linearized only  make t o t h e  the  goal  disutility same  utility  noted  between g o a l s  enter  at a l a t e r  of the  be  d e c i s i o n making group t o  as w e l l as  the  t i m e and  u n d o u b t e d l y a c t as in  t h a t may  data  B  d e c i s i o n maker's these  i s t h a t each g o a l The  two  stage  and  activities  t o none  other.  trade-off  considerations  o f the a n a l y s i s .  format, there specified, identify  limitations  a c o n s t r a i n t on  function  d e c i s i o n maker must  as w e l l as  Under a m u l t i p l e g o a l s number o f g o a l s  the  of a c t i v i t y  c o n t r i b u t i o n s t h a t the  goals,  words,  times  p r o d u c e d by  function i n question,  to other  In o t h e r  ratio.  in isolation.  Contributions  the  acre  presume t h a t t h e  relative  d e c i s i o n maker.,  produce f i v e  t h a t one  A n o t h e r p o i n t t o be should  judgements.  and  the  i s no  although  and  limit the  to  ability  specify goals,  of doing  so,  will  number o f g o a l s  employed  the a n a l y s i s .  A M u l t i p l e Goals Version  Once t h e scaled  to the  information function,  goals  of the  have b e e n d e f i n e d and  satisfaction  can  be  H y p o t h e t i c a l L a n d Use  o f the  added t o o t h e r  interactivity  the  Problem  goal  functions  d e c i s i o n making group, information  r e l a t i o n s h i p s , and  on  the  this  objective  constraints  on  activity  levels,  and  then  examined i n a l i n e a r  framework.  To  Suppose now  t h a t , i n a d d i t i o n t o wanting t o maximize  land value,  the p l a n n i n g commission a l s o d e s i r e d t o minimize  pollution,  and  that  they  felt  t h a t an  illustrate,  t h a t these  considered  we  two  recall  programming  an  units of p o l l u t i o n was  felt  t o be  The  problem can  r e l e v a n t t o the  issue.  acre of i n d u s t r y generated disutility.  be  An  illustrated  goals  one  pollution hundred  acre of r e s i d e n t i a l u n i t s of  land disutility.  as f o l l o w s :  Activities Land  community  Suppose i t was  land created zero  i n t e r m e d i a t e , c o n t r i b u t i n g 40 now  above.  g o a l s were t h e o n l y two  acre of a g r i c u l t u r a l  d i s u t i l i t y while  t h e example u s e d  Activities  ( a l l ^0)  Accounting  Activities W  <  D  H  E-i  —  £  H  «  —  Q  D in \A X  0  0  0  1 0 10 0 1  1 20 20 0 0  1 1 1 1 0  0 0 0 0 0  0 0 0 0 0  ^1000 ^1000 ^5000 ^ 250 * 100  -3000 -7000 -1000 -10 -100 0  1 0  0 1  0 0  O b j e c t i v e : Maximum o f  Goal  EH  o  Q  H  A v a i l a b l e Land A v a i l a b l e Labour Available Capital Minimum F o o d Minimum H o u s i n g  <  53  ~  O X U ~  H  to  H  H  Q  H  to Constraints  EH  X  H X W —' D  W X  Subject  CN  P  Q  53 O  Pi  < < EH O EH  >  rtj O  •  EH PM O EH  x  2  EH  Pi  a  max.  on:  C o n t r i b u t i o n s to: T o t a l Land Value Total Pollution  In t h i s  tableau,  the objective  f u n c t i o n t o be m a x i m i z e d i s  a weighted composite o f the separate goals  o f land  m a x i m i z a t i o n and p o l l u t i o n m i n i m i z a t i o n . the weights t h a t  and \  goal  individual  pollution disutility  activities  g e n e r a t e a r e summed i n t o t h e t o t a l  and t h e p o l l u t i o n g o a l  units that  t o determine the c o n t r i b u t i o n that  The  the various  land  p o l l u t i o n (X5) to this  total pollution  function.  Since  the desire  t o m i n i m i z e p o l l u t i o n , X . 2 must t a k e on a n e g a t i v e to create  variable  c a n be a c c o m p l i s h e d by m a x i m i z i n g i t s n e g a t i v e ) .  land value  goal weight,  will  h e r e i s t o m a x i m i z e community l a n d into  the t o t a l  land value  Note t h a t land  w h i c h h a s been  that  procedure ensures that  that  i s , total  land value,  are entered  the  non-negativity  When t h i s routine,  summed  and t o t a l  procedure  non-positive tableau.  contributions,  pollution units, activity  will accounting  i s necessary t o ensure  c o n s t r a i n t on a l l a c t i v i t y  information  as  of different  i n this  the t o t a l s f o rthese  a p p e a r a s p o s i t i v e amounts i n t h e two t o t a l This  each acre  two c o n s t r a i n t l i n e s  This  c o l u m n s , X^ a n d X^.  The  c o l u m n , X4.  the contributions  i n the l a s t  a positive  be p o s i t i v e a s t h e d e s i r e  value,  u s e makes t o w a r d s t h e two g o a l s  coefficients  (minimizing  value  in order  total  a minimizing e f f e c t  e  to the  i s applied  2  makes t o t h e c o m p o s i t e o b j e c t i v e  r  respectively.  column a n d t h e p o l l u t i o n g o a l w e i g h t , ) \ ,  is  a 2  t h e d e c i s i o n making group a s s i g n s  land value  total  value  that  columns i s u p h e l d .  i s f e d i n t o an o p t i m i z i n g  l i n e a r programming-will y i e l d  anlefficient  solution  44 to (N]_  this  l a n d use  andYvj) •  desired  p r o b l e m f o r any  The  levels  p r o b l e m now  of finding  w h i c h i s p r e f e r r e d by  Candler  solution is  indicate,  i s found  That  what t h e y  in  terms of t h e i r  begins and  again.  the d i r e c t i o n  t o an  to Candler  feel and  The  a budget i s found  adjacent budgets or u n t i l  and  d e c i s i o n makers programming  they  improved  feel  A  goal  solution and  to s h i f t  the  then around  a preferred  solution  goal weights  a still  weights ("improved"  i s then the  process  solution solution  in  preferred solution  Boehlje, this  improved  an  efficient  the r e l a t i v e  the g o a l weights  identifying  continue u n t i l  initial  d e c i s i o n makers examine t h e new  i n which they  According  sequentially  lead  shift  set of r e l a t i v e  The  body.  i n which d i r e c t i o n  i s , they  solution  s e t of goal weights  preference function).  f o r t h i s new  efficient  the a i d o f parametric  think w i l l  again respecify  lies.  An  f o r some s p e c i f i e d  solution,  to  found  problem.  the  alternatively,  the p a r t i c u l a r  t o the d e c i s i o n makers.  lies.  Stated  B o e h l j e p r o p o s e what amounts t o 2  perhaps with  the o r i g i n a l solution  and  weights  of choosing  the d e c i s i o n making  approach to t h i s  presented  becomes one  f o r the g o a l weights.  t h e p r o b l e m i s one  iterative  given set of goal  solutions  process  (budgets)  of "would  t h a t i s b e t t e r than a l l  the r a t e of g a i n i n the o b j e c t i v e 3  function  i s lower  than  the per u n i t  cost of  and  B o e h l j e , op.  c i t . pp.  further  analysis."  2 Candler 3 I b i d . p.  330.  329-330.  45  The  s p e c i f i c a t i o n of i n i t i a l  pose somewhat of a problem. procedures assume t h a t any  Since  s t a r t i n g weights does  l i n e a r programming  search  l o c a l optimum found i s a g l o b a l  optimum, the a c t u a l s t a r t i n g p o i n t of the search presents d i f f i c u l t i e s t h e o r e t i c a l l y . However, p r a c t i c a l  no  considerations  of the c o s t s of a n a l y s i s , would i n d i c a t e t h a t the s t a r t i n g p o i n t should  be i n the g e n e r a l v i c i n i t y of the optimum, or as  to i t as p o s s i b l e .  To t h i s end,  close  the d e c i s i o n making group  should make an attempt to s p e c i f y the i n i t i a l i n accordance w i t h t h e i r p e r c e p t i o n  s e t of weights  of the t r a d e - o f f s t h a t  e x i s t between g o a l s .  Going back to our example problem, i f  the land use  commission f e l t t h a t f o r e g o i n g  planning  100  units  of p o l l u t i o n i s approximately worth i n c r e a s i n g land value  by  $1000, the i m p l i c a t i o n would be t h a t the i n i t i a l p o l l u t i o n goal weight c o u l d be s e t a $10 u n i t of p o l l u t i o n .  of community land value  That i s , i f w a s  s e t a t 1.0,  s t a r t i n g p o i n t would be to setYv? equal to -10.0. these two  weights are used the f o l l o w i n g i n i t i a l  per  a good If, in fact,  solution i s  obtained: the o b j e c t i v e f u n c t i o n i s a t a maximum when: X± = 100  RESIDENTIAL LAND USE, INDUSTRIAL LAND USE,  X  AGRICULTURAL LAND USE, Compare t h i s s o l u t i o n to the one data,  acres  = nil  2  X  3  = 900  obtained  acres w i t h the same set of  except w i t h t o t a l community l a n d value  c r i t e r i o n t o be maximized.  The  as the  sole  s p e c i f i c a t i o n and i n s e r t i o n  of a second c r i t e r i o n , the minimize p o l l u t i o n g o a l , has a marked e f f e c t on the s o l u t i o n v e c t o r .  Residential  and  had  industrial  acreage, both o f which generate p o l l u t i o n u n i t s ,  have d e c r e a s e d c o n s i d e r a b l y  and a g r i c u l t u r a l a c r e a g e , w h i c h  does n o t g e n e r a t e p o l l u t i o n , h a s i n c r e a s e d . of  a d d i t i o n a l goals  will  on  t h e value^o-f 1 t h e  solution vector,  any  n o t a l w a y s have s u c h a d r a m a t i c e f f e c t  model w h i c h d o e s n o t c o n t a i n  provide  solutions that  Though t h e i n c l u s i o n  i t should  be a p p a r e n t  a l l the relevant  c a n be c o n s i d e r e d  goals  cannot  t o be o p t i m a l ,  i n the  sense o f m a x i m i z i n g t h e d e c i s i o n makers' c o l l e c t i v e (preference)  that,  feels of  that  should  be n o t e d t h a t  an i n i t i a l  solution.  a more p r e f e r r e d  weight,  solution.  solution  I f t h e d e c i s i o n making  examined.  This  C a n d l e r and B o e h l j e  (near o p t i m a l )  change t h e p o l l u t i o n  that  t h i s new s o l u t i o n c o u l d again  an o p t i m a l  as o u t l i n e d  o r a t l e a s t a "good"  :  m i g h t be p o s s i b l e , however, t h a t  making group.  be  be s h i f t e d and a n o t h e r  p r o c e s s would c o n t i n u e ,  until  still  s o l u t i o n i s reached.  s o l u t i o n s encountered  exist within  group  s a y f r o m -10 t o -8, and examine t h e r e s u l t i n g  If i ti s felt  It  solution i s  s o l u t i o n would l i e i n t h e d i r e c t i o n  i m p r o v e d upon, t h e w e i g h t s c o u l d  by  t h e above i n i t i a l  a l o w e r c o s t on p o l l u t i o n , t h e y c o u l d  goal  welfare  function.  It just  that  i n this  none o f t h e  process appeal to the d e c i s i o n  T h i s would suggest t h a t e r r o r s o r d e f i c i e n c i e s t h e model s t r u c t u r e .  are  s p e c i f i e d erroneously  the  r e a l world  Alternatively,  E i t h e r some o f t h e c o n s t r a i n t s  o r some c o n s t r a i n t s  system have n o t been i n c l u d e d there  may be g o a l s  d e c i s i o n makers' w e l f a r e  function  that operate i n i n t h e model.  t h a t govern p a r t o f the t h a t have b e e n e x c l u d e d  from  the  model.  I n any o f t h e s e c a s e s ,  must be made and t h e a n a l y s i s corrections specified,  the necessary  s t a r t e d anew.  c a n be made, new c o n s t r a i n t s  corrections  The e a s e i n w h i c h  a d d e d , a n d new  goals  i s i n d e e d one o f t h e m a j o r a d v a n t a g e s o f u s i n g  t y p e o f model  this  structure.  Summary  The i n t e n t o f t h e d i s c u s s i o n to o u t l i n e goals of  t h e m e t h o d o l o g y o f l i n e a r programming w i t h  and t o p r o v i d e a simple  t h i s t y p e o f programming  The goals  i n t h i s s e c t i o n has been  following  illustration  of the a p p l i c a t i o n  to a hypothetical  land  c h a p t e r o u t l i n e s an a c t u a l  l i n e a r programming m o d e l t h a t was c o n s t r u c t e d  problems o f l a n d  use a l l o c a t i o n inhthe  multiple  use problem.  multiple to  consider  C i t y and D i s t r i c t o f  L a n g l e y i n t h e Lower M a i n l a n d o f B r i t i s h C o l u m b i a .  48 Chapter  6  LANGLEY MODEL STRUCTURE  Introduction  The  foregoing chapter  d i s c u s s e d the  application  a m u l t i p l e goals type  o f l i n e a r programming t o problems  l a n d use  A v e r y h y p o t h e t i c a l l a n d use  was  allocation.  examined.  tion  o f an  concern l a n d use  t o examine l a n d u s e  allocation  of Langley  The  City  i n t h e Lower M a i n l a n d  and  District  of B r i t i s h  is a  area of approximately  on  slopes of the F r a s e r R i v e r Basin  the  south  f r i n g e of the  urban area. the  The  District,  approximately industrial  of Langley,  on  few  found  severe  c e n t e r o f the  i n the  area.  predominantly  miles, located just  outside  area.  commercial 1  has  t h a t has  b e e n f o r c e d , by  and  exemplifies arisen  d e c a d e s as t h e M e t r o p o l i t a n V a n c o u v e r and  of  Its  Langley  urban-rural c o n f l i c t  expanded r a p i d l y  t o t h e n o r t h and  Columbia.  the western perimeter  2000 a c r e s c o n t a i n much o f t h e  activities  the past  a r e a has  City  square  and  expanding M e t r o p o l i t a n Vancouver  i s the business  the i n c r e a s i n g l y over  rapidly  was  i n the C i t y  agricultural the  125  a descrip-  model t h a t  of Langley  of  problem  i t s e l f with  actual multiple goals  constructed District  This chapter w i l l  of  the  t h e w a t e r t o t h e w e s t , t o grow o u t  urban  mountains  onto  the  1 areas  The a u t h o r u s e s " L a n g l e y " t o d e s c r i b e t h e combined o f t h e C i t y o f L a n g l e y and L a n g l e y D i s t r i c t M u n i c i p a l i t y .  flat  agricultural  Vancouver access away f r o m t h e the  reach  Langley  Langley  any  B.C.  an  previous.  which prevents  spring  42,  the  example o f an were b e i n g District of  an  Langley  urban-rural  a area  over  too  early  altered  Land Commission  Act,  provincial legislature,  intensified  farmland  interest in public  of 'this  afforded  a multi-goaled  legislation,  a readily accessible  framework, t h e  the  site  f o r the  a model f o r l a n d  Lower M a i n l a n d o f B r i t i s h  L a n g l e y model i s o n l y  i n a p r o j e c t whose end  seven  be  a r e a where p u b l i c p l a n n i n g  w o r k i n g example o f  completion of t h i s  the  Canadian  t r e n d may  result  o f L a n g l e y were c h o s e n as  d e c i s i o n making i n the  step  1971  in  authority.  t h a t a r o s e as  made w i t h i n  initial  The  21,936, up  o f the  P a r t i a l l y because of the  p a r t l y because the  within  have b e e n s e t t l i n g  further subdivision of designated  planning  minutes  well  Though i t i s y e t  sitting  w i t h o u t p r o v i n c i a l government  and  district,  noticeable deviation, this  1973  Three major  l e s s than s i x t y  area population of 2  Government's B i l l  passed a t the  l a n d use  east.  i n e v e r i n c r e a s i n g numbers.  from ten years  to detect the  highways put  s o u t h and  Vancouver c e n t r a l business  Census r e p o r t e d  by  t o the  o f V a n c o u v e r commuters who  area  thousand  lands  result will  intended be  decisions City  and  construction  allocation Columbia. t o be  a  The first  a multiple goals  land 3  use  model t o the  e n t i r e Lower F r a s e r V a l l e y o f B r i t i s h  Columbia.  2 C e n s u s o f Canada  ( O t t a w a : Queen's P r i n t e r ) ,  1961,  1971.  3 T h i s p r o j e c t i s c u r r e n t l y being undertaken i n the Department o f A g r i c u l t u r a l Economics a t 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 u n d e r t h e d i r e c t i o n o f Dr. J o h n D. Graham.  50 Initial  Assumptions  Before truction, The  entering into  a d i s c u s s i o n o f model  an e x p l a n a t i o n o f i n i t i a l  assumptions  cons-  i s necessary.  f o l l o w i n g a s s u m p t i o n s were made a t t h e o n s e t  o f model  structuring: (1) c o n v e r s i o n that is,  c o s t s from e x i s t i n g  l a n d uses t o uses  t h e model might w i s h t o a l l o c a t e t h e model s t a r t s w i t h  are n i l .  a "clean slate"  square m i l e s which i t can a l l o c a t e ,  That  o f 125  irregardless of  e x i s t i n g use. \{2)  capital  i n non-constraining.  enough c a p i t a l solution (3) L a n g l e y  t h a t t h e model might  The c o m m e r c i a l  population met  and j u s t  by l o c a l  labour  are  It the  reality  should  business  imported  population are e n t i r e l y  establishments. locally  that these  t h a t c a n be a c h i e v e d  assumptions  by t h e model.  (with the  severely  allow.  limit  However, i t  t h a t the major o b j e c t i v e o f the study  i s the  working l a n d use model, based i n  o n l y a s much a s t h e q u i t e l i m i t i n g  restraints  As w e l l , t h e  from o u t s i d e t h e a r e a .  c o n s t r u c t i o n o f an i n i t i a l reality  of the l o c a l  o f some unemployment) a n d no l a b o u r e r s  i s realized  be n o t e d  from t h e s u r r o u n d i n g  requirements  the l o c a l  o f any f e a s i b l e  examine.  isolated  f o r c e i s o n l y employed  possibility  i s , there i s  t o meet t h e r e q u i r e m e n t s  i s commercially  areas.  That  I t was f e l t  t i m e and d a t a  that to include conversion  costs,  51 a capital  c o n s t r a i n t , and  a substantive initial  effort  study.  resents  of  input  both analyst this  complication for  the  I n any to  and  of  of  this  s t u d y expand  necessity  i s that  realistic  of  the  this  the  these  author  use  practical  planning,  goals  linear  a great  programming  input  study, the  i s p e r h a p s an  writer  d e c i s i o n maker and  the  deal  body.  leaves  impossibility.  rep-  application  f r o m some p u b l i c d e c i s i o n m a k i n g  p r o b a b l y e v e n an  purposes of  a later  noted  land  theoretician, this  the  unnecessary Therefore,  assumed t h e  dual  analyst.  Alternatives  The land  t o be  resources  require  p u b l i c d e c i s i o n maker a l i k e f o r  study.  multiple  responsibility  L a n d Use  and  i s required  the  beyond the  away w i t h t h e  type of modelling  However, u n t i l realm of  t o do  further point  purposes of of t h i s  t h a t was  I t i s proposed that  model s u f f i c i e n t l y 4 assumptions.  A  i n t e r r e g i o n a l l i n k s would  t o be  matrix constructed  put  to  34  in;this  a l t e r n a t e uses,  study allows  i n c l u d i n g one  Langley  non-use  5 (vacant  land).  Three a g r i c u l t u r a l uses are  m a r k e t g a r d e n i n g , and activities  (office  f e e d l o t ) , as w e l l  services, transport  as  defined  seven  (dairying,  commercial  s e r v i c e s , urban  4 See  footnote  3 in this  chapter.  5 and  A matrix p i c t u r e , together with a coding explanation d a t a d e s c r i p t i o n , i s a v a i l a b l e i n A p p e n d i c e s I and I I .  52  shopping, urban neighbourhood suburban neighbourhood industrial  uses  retail,  suburban  reserve, golf  five  off  street parking,  uses  railroad,  single  three i n s t i t u t i o n a l ment b u i l d i n g s ) ,  uses  and one  (urban  recreational development),  road, r u r a l five  (schools, h o s p i t a l s ,  low  farm h o m e s i t e s ) ,  and o t h e r g o v e r n -  (vacant l a n d ) .  A l l of  L a n g l e y ' s a p p r o x i m a t e l y 80,000 a c r e s must be a l l o c a t e d m o d e l i n t o one  o f t h e s e 34 u s e s s u b j e c t t o 125  such t h a t a composite individual  policy  social  goals  T e c h n i c a l and P h y s i c a l  criterion  road,  residential  family dwelling,  hobby f a r m s , and  non-use  heavy  uses  recreational  and a i r p o r t ) ,  family dwelling,  four  industry,  (urban r o a d , suburban  (apartment, h i g h d e n s i t y single  retail),  recreational  c o u r s e , and c o m m e r c i a l  transportion  density  light  shopping,  park, major m e t r o p o l i t a n park o r  six  uses  suburban  and r u r a l  (food p r o c e s s i n g ,  i n d u s t r y , wood p r o c e s s i n g ) , park,  retail,  by  the  constraints  function of ten  i s maximized.  Constraints 6  The major groups. essentially  125  restraints  c a n be c l a s s i f i e d  Sixteen represent  technical or physical  into six  limiting  f a c t o r s that are 7 i n nature : population,  6 The a u t h o r u s e s t h e words " r e s t r a i n t " and "constraint" interchangeably. 7 R e f e r t o row numbers 13-22, 25-29, and 86 i n t h e m a t r i x p i c t u r e and c o d i n g e x p l a n a t i o n i n A p p e n d i c e s I a n d ' I I .  the  amount o f l a n d and l a b o u r  land uses, the labour in  the labour  i n p u t s r e q u i r e d by t h e v a r i o u s  f o r c e , the p a r t i c i p a t i o n  of the f l o o d  Most o f t h e s e A brief coding  plain,  restraint  land uses, the s i z e  t h e amount o f c l e a r e d  lines  explanation  and d a t a  an u n d e r s t a n d i n g  most o f t h e s t u d y in  while  with  d e s c r i p t i o n , s h o u l d be  of these  restraint  total  population  t o t h e 1971 p o p u l a t i o n  was  here.  the r  sufficient  lines.  However,  clarity.  constrained i n  of Langley  (as r e p o r t e d  t h e 1971 C e n s u s o f C a n a d a ) , t h e model c a n be r u n on any  given the  population,  right  and  not a constant.  t o the d e s i r e d  a d j u s t a c c o r d i n g l y as p o p u l a t i o n  requirements o f the l o c a l  increasing population.  t o cope w i t h  obtain  reliable  r u n s c a n show how population  level.  population  both  increase  t h a t l a n d u s e d e c i s i o n s must  trends,  changes.  I f the  then consecutive  l a n d u s e p a t t e r n s must  c o n d i t i o n s , and t h e p l a n n e r s  accordingly.  will  of structure allows f o r  future population  population  varies.  s e r v i c e s , and t h e minimum  T h i s type  a p r o j e c t i o n o f the d i r e c t i o n in order  value i n  V a r i a b l e s w h i c h a r e d e p e n d e n t upon  example, t h e demand f o r m e d i c a l  hospital with  by c h a n g i n g t h e p o p u l a t i o n  h a s b e e n s t r u c t u r e d i n t o t h e model a s a v a r i a b l e  population w i l l For  simply  hand s i d e o f t h e m a t r i x  Population  can  land.  p i c t u r e , together  some p o i n t s b e a r m e n t i o n i n g i n t h e i n t e r e s t s o f  Firstly,  and comp-  n e e d no f u r t h e r e x p l a n a t i o n  i n s p e c t i o n of the matrix  to reach  population  f o r c e , t h e amount o f l a n d and t h e q u a l i t y o f  land a v a i l a b l e to the d i f f e r e n t osition  of the  c h a n g e t o meet can a d j u s t t h e i r  take  planners model changing policies  54  S e c o n d l y , two allow for  options  f o r some v a r i a t i o n i n t h e  allocation.  which allow  the  cleared  land  amount o f  increase  physical  model w h i c h  resources  I n programmer's t e r m s t h e s e a r e  options  annual cost,  e x i s t i n the  or  the  model t o  i f increased  i n the  value of  increase  flood-safe  objective  "buy"  the  amount  land,  at a  s o c i a l welfare, the  available  of given  r e f l e c t e d by  function,  can  an  be  achieved.  Thirdly, of  flood protection  good. allow  In  terms of  provision  p l a i n when t h e at  i t must be  less or  protection will  constitutes the  otherwise.  t o be  Yet,  buy  because the  To  r e c o n c i l e the  of  foolish the  a per  be  acre b a s i s ,  f o r the  more  s o l u t i o n s was  f l o o d p r o t e c t i o n was  protection  remained  forced  set at  were f i n a l i z e d  p r o c e d u r e was  protection  was  indeed  that the  the  value.  i n the  first  had  goal  optimal  in  Thereafter produced  weights  repeated to ensure  s e t a t an  only  maximum and  When t h e  model  dictate  The  value that  flood  productive  zero p r o t e c t i o n .  s e t a t the  function  the  to  protected  twossolutions  identical starting conditions. two  to  flood  cost of buying  model t o t h i s f a c t ,  a higher objective  flood  of part  protection  s e c o n d s o l u t i o n i t was  flood  discontinuous  L a n g l e y m o d e l i t w o u l d be  s p e c i f i e d on  d i f f e r e n c e between t h e  the  provision  f l o o d zone, even though r e a l i t y would  were f o u n d g i v e n  solution  the  e n t i r e f l o o d p l a i n i n a c t u a l i t y can  continually only  lands i n the  that  a lumpy o r  for flood protection  equal cost. had  recognized  value.  that  55  Social Constraints  A second group o f twenty-seven c o n s t r a i n t s can be c h a r a c t e r i z e d as " s o c i a l " r e s t r i c t i o n s which impose a minimum 8 q u a l i t y of l i f e upon the model s o l u t i o n .  Minimum shopping  and o t h e r commercial requirements, minimum t r a n s p o r t a t i o n needs, minimum r e c r e a t i o n a l requirements, minimum i n s t i t u t i o n a l and r e s i d e n t i a l requirements are a l l i n c l u d e d here.  As w e l l ,  a maximum unemployment standard and a minimum m u n i c i p a l tax  base are imposed w i t h i n t h i s c a t e g o r y . Although a l l o f these c o n s t r a i n t s are q u i t e  straight  forward, and need no f u r t h e r e l u c i d a t i o n here, a d i s c u s s i o n of  the r o l e o f a minimum q u a l i t y of l i f e  optimum s e e k i n g model appears i n o r d e r .  imposed upon an The  classical  w e l f a r e economist would argue, i n P a r e t i a n terms, t h a t minimum and maximum i m p o s i t i o n s upon an economic  system  simply d i s t o r t e f f i c i e n t a l l o c a t i o n o f r e s o u r c e s and l e a d 9 away from Pareto o p t i m a l i t y .  He wouldcoontinue by  t h a t o n l y i f the system i s allowed t o f r e e l y 8  showing  allocate  Refer t o row numbers 30-36, 51-57, 68-76, and 86-87 the p i c t u r e and coding e x p l a n a t i o n i n Appendices I and I I . 9 A Pareto optimum i s one i n which the w e l f a r e o f no i n d i v i d u a l can be i n c r e a s e d f u r t h e r without d e c r e a s i n g the w e l f a r e o f some o t h e r i n d i v i d u a l o r group of i n d i v i d u a l s . in  56 resources  according  to marginal considerations  w o r t h and  utility will  any  s o c i a l welfare  of  economic  f u n c t i o n be  maximized.  However, h i s e n t i r e argument i s p r e m i s e d upon a c c e p t a n c e the  e x i s t i n g i n i t i a l d i s t r i b u t i o n of wealth,  has  not  proven acceptable  R e d i s t r i b u t i o n of wealth the  most o f t e n  a minimum q u a l i t y o f  in  the  i s very  life  was  i n mind t h a t  i m p o s e d upon t h e m o d e l  O u t p u t Demand  other  w i t h l i m i t a t i o n s i m p o s e d by the  output of the  restraints  prevent  the  amount o f  land  group are  constraints  p r o d u c e d by  imposition  protection, whatever.  s e t o f minimum  constraints  structure.  Constraints  Thirty-two  for  this  of  economic system, whether  f o r m o f minimum wages, minimum f i r e  i s with this  One  r e d i s t r i b u t i o n i s the  upon t h e  that 10  economies.  much i n e v i d e n c e .  minimum open s p a c e , minimum b u i l d i n g s p a c e o r It  assumption  t o most modern W e s t e r n  used v e h i c l e s of  of  an  of  t o any  c o n s t r a i n t s concern  themselves  economic f a c t o r s , c h i e f l y 11  various  land  activities.  m o d e l f r o m a l l o c a t i n g an p a r t i c u l a r land limiting  the  use.  These uneconomic  Included  demand f o r t h e  f e e d l o t s , a l l commercial  land  demand  in  this  output  uses, a l l r e c r e a t i o n a l  10 T h i s a s s u m p t i o n , f o r example, c o u l d l e a d t o one man's s t a r v a t i o n b e i n g P a r e t o - o p t i m a l to s o c i e t y . 11 R e f e r t o row numbers 23-24, 37-50, 58-67, and 77-82 i n t h e m a t r i x p i c t u r e and c o d i n g e x p l a n a t i o n i n A p p e n d i c e s I and I I .  57  l a n d u s e s , and a l l i n s t i t u t i o n a l  Due possible actual  uses.  t o the n a t u r e o f the model, i t d i d not prove  t o s p e c i f y t h e s e demand c o n s t r a i n t s  i n terms o f  demand s c h e d u l e s f o r t h e m u l t i p l i c i t y  s e r v i c e s p r o d u c e d by t h e v a r i o u s all  land  land  o f goods and  activities.  t h e goods and s e r v i c e s p r o d u c e d by any g i v e n  were  a l l lumped t o g e t h e r  Instead, land 12  i n t o one b a s k e t o f g o o d s ,  use and  t h e n t h e demand f o r t h e b a s k e t o f goods p r o d u c e d by any particular  land  At technical  u s e was  this point  duplicate  whereby  "industry" as  that  given  to take place  programming,  no programming  could  the u n i t return  mechanism,  be f o u n d w h i c h  E s s e n t i a l l y what was  p r o b l e m was  increased that  this  downward  n e e d e d was  or price to a  t h i s was  a  given  the type o f behaviour W h i l e no mechanism  type o f a c t i o n could  rate  observed could  be d u p l i c a t e d ,  overcome by t h e u s e o f s t e p p e d o r  which could  could  t h e amount o f b a s k e t s o f goods  i n the r e a l world.  be f o u n d whereby  functions  Firstly,  two  ( l a n d use) w o u l d d e c l i n e a t an a p p r o p r i a t e  industry  supplied,  structuring  the a c t i o n of a conventional  s l o p i n g demand c u r v e . mechanism  i n model  problems arose.  compatible to l i n e a r exactly  constrained.  the  discontinuous  be m a n i p u l a t e d t o c l o s e l y a p p r o x i m a t e  12 One b a s k e t c o n s i s t e d o f a l l goods and p r o d u c e d b y one a c r e o f t h e l a n d u s e i n q u e s t i o n , a p p r o p r i a t e amounts o f c o m p l e m e n t a r y i n p u t s .  services given  58  the  desired  action.  In terms o f model c o n s t r u c t i o n ,  this  was a c c o m p l i s h e d by h a v i n g e v e r y L a n g l e y l a n d  u s e , whose  output  split  three a  faced  separate a c t i v i t i e s .  separate  of  a downward s l o p i n g demand c u r v e ,  step  that  faced  return  value  they received  first  steps  t h e same p r o d u c t i o n  step  received  that  could  received  occur  a fourth  represented  the t o t a l  nothing  i n each.  the d i f f e r e n t  declining cular  land  artificial  T h i s column  has t h e e f f e c t  average r e t u r n  use i n  required to  steps  columns  represent  v a r i a b l e s which  T h e y do n o t r e p r e s e n t  returns,  steps  were t h e n  the various  land use a l l o c a t i o n .  lower u n i t  use under  column.  Note t h a t  p r o c e d u r e o f summing t h r e e  column o b v i o u s l y  steps  I t i s t h e columns c o n t a i n i n g  t o t a l s which represent  successively  These three  i n terms o f a c r e s  i m p l e m e n t demand l i m i t a t i o n s .  this  A l l three  demand c u r v e  more t h a n i n t e r n a l  land use a c t i v i t i e s .  and t h e f o l l o w i n g  o u t p u t p r o d u c e d by t h e l a n d  output.  i n the  a s t o t h e amount o f l a n d  separate a c t i v i t y  total  labour  product.  return  lower r e t u r n s .  function,  and  they d i f f e r e d  high  appropriately  a n d was e x p r e s s e d  produce a given containing  a fairly  successively  summed i n t o  question,  of output,  from the s a l e o f t h e i r  were t h e n c o n s t r a i n e d use  exactly  constituted  Though e a c h  i s , t h e y a l l n e e d e d t h e same amount o f l a n d  to produce a given  The  Each o f these a c t i v i t i e s  i n t h e s t e p p e d demand f u n c t i o n .  these a c t i v i t i e s  into  activities,  actual  the output  Note a l s o which  earn  into a fourth total of introducing  that  activity  a three  step  t o t h e o u t p u t p r o d u c e d by t h e p a r t i -  consideration.  59  The  second s t r u c t u r a l  area concerned  the nature  various output  producing  macroeconomic i n n a t u r e , c o u l d n o t be d i r e c t l y  problem encountered  i n this  o f t h e monetary r e t u r n t o t h e activities. output  specified  B e c a u s e t h e model i s  unit  returns or prices  i n t h e model.  T h i s meant  that  t h e demand c o n s t r a i n t s h a d t o be d e f i n e d i n t e r m s  than  conventional price  and q u a n t i t y t e r m s .  This  other  problem  was d e a l t w i t h by d e f i n i n g t h e demand c o n s t r a i n t s i n t e r m s of  declining  valued  added by p r o d u c t i o n .  w h i c h s o u g h t a maximum o f g r o s s expressed  i n value  The p o l i c y  regional production  added t e r m s and t h u s  l a n d u s e whose o u t p u t three a r t i f i c i a l contributed per  a c t i v i t i e s which represented  lowering  the value  added  production  i s t h a t as output i s  demand, o u t p u t  unit price  drops,  added component o f p r o d u c t i o n .  F o r example, each a c r e i n t h e f i r s t use  F o r each  t h a t l a n d use  amounts o f v a l u e  The a s s u m p t i o n h e r e  increased, given a fixed thus  convenient  f a c e d a downward s l o p i n g demand, t h e  s u c c e s s i v e l y lower  acre o f use.  was  provided a  b a s e upon w h i c h t o d e f i n e t h e demand c o n s t r a i n t s .  goal  step of the feedlot  land  c o n t r i b u t e s $4400 t o w a r d s t h e g o a l o f m a x i m i z i n g r e g i o n a l  production.  Each acre  $2000, and e a c h a c r e explained separate  i n the second step c o n t r i b u t e s  i n the t h i r d  a b o v e , when t h e s e activity  the e f f e c t  three  step, only  steps are t o t a l l e d  column w h i c h r e p r e s e n t s t o t a l  i s one o f i n t r o d u c i n g a t h r e e  demand c u r v e  $500.  As into a  land use,  s t e p downward s l o p i n g  f o r t h e oujtput o f t h a t l a n d u s e .  Figure  1 illustrates  t h i s mechanism  graphically,  60 Figure  1  Total Value Added  400 A c r e s o f A c t i v i t y  Total Revenue  400 A c r e s o f  C o s t o r Non V a l u e Added Component 100  200  300  of  Output  Output  400 A c r e s o f  Output  61  using  the  Figure of  1  f e e d l o t example d i s c u s s e d a b o v e . (a) shows how  feedlot  l a n d use  total  activity.  increases at a different s t e p s , and  t h a t the  maximum o f one is  feedlot to the  output, total  then  value  value  unit  Figure 1  corresponding  revenue curve  Using  can  be  this  this  approximation  Therefore  demanded a t t h a t  accuracy  the  steps  Figure 1  of output  and  inclusion  achieved  i n each f u n c t i o n .  convenience,  curve,  calculated  (c) r e p r e s e n t s  the q u a n t i t y of  at  use  i n Figure 1  of simple  a  curve (b). the  same  rudimentary  relating output  the e f f e c t  three step value  by  t h e use  Only  three  and  the  resulting  added  o f i n t r o d u c i n g rough Obviously,  greater  o f a g r e a t e r number s t e p s were u s e d  demand f u n c t i o n s i n t h i s model f o r r e a s o n s and  of  This i s  i s precisely  t o m a r k e t demand c u r v e s .  be  constant  average revenue  revenue curve  i n t h e m o d e l has  can  step  price.  Thus t h e  approximations  a  corresponding  o f a downward s l o p i n g demand c u r v e ,  the p r i c e o f a u n i t  functions  is a  revenue  c a n be  average revenue per u n i t o f output  as u n i t p r i c e .  third  i s , at various land  (c) d e t a i l s  to the t o t a l  added  different  constructed.  total  of output  l e v e l s of output,  acreages.  t h a t the  acres  added m a t e r i a l s i n e a c h u n i t  a total  (b).  value  steps are c o n s t r a i n e d to  added c u r v e  average revenue per  But  two  I f i t i s assumed t h a t t h e r e  shown i n F i g u r e 1  various  Note t h a t t o t a l  r a t e i n each of the t h r e e  first  component o f non  graph i n  added i n c r e a s e s w i t h  h u n d r e d a c r e s e a c h and  unconstrained.  cost  value  The  of  of  for a l l  simplicity  f u n c t i o n s are  probably  J  62  very  inaccurate.  given  The  point  s u f f i c i e n t accurate  t o be  information  r e a l w o r l d m a r k e t demand c u r v e c a n by  this  made t o l i n k  the  felt  (medical,  dental,  services)  in a region  land the  region.  legal,  For  that  r e a l world,  closely  any  approximated  I n e a c h c a s e an  obviously  by  of each step  to the  and  other  similar  as  for  added f u n c t i o n  to the  t o t a l population.  of  demand c u r v e s f o r t h e  was  market. services  total  market c o n d i t i o n s ,  value  this  professional  extends t o the  proportionately s h i f t i n g the  attempt  relevant  s e r v i c e s , as w e l l  i n the  of  the market f o r o f f i c e  clerical,  office  uses confronted size  model.  demand f u n c t i o n s  e x a m p l e , i t was  the  the  s i x t e e n o u t p u t demand f u n c t i o n s  t y p e were u s e d i n t h e  of  be  on  that  method.  In t o t a l ,  For  n o t e d however, i s  This  goods and  has  population other  therefore, varies the  effect  services  p r o d u c e d by size,  t h e s e a c t i v i t i e s as p o p u l a t i o n , t h a t i s , m a r k e t 13 varies. T h o s e a c t i v i t i e s whose m a r k e t s a r e d e p e n d e n t  13 The a u t h o r r e a l i z e s t h a t demand c u r v e s c a n r a r e l y be e x p e c t e d t o b e h a v e so s i m p l y . S h i f t s i n t h e demand f o r t h e p r o d u c t o f a l a n d a c t i v i t y p r o b a b l y d e p e n d on c h a n g e s i n o t h e r a c t i v i t y l e v e l s , c h a n g e s i n income d i s t r i b u t i o n , a b s o l u t e income l e v e l s , t a s t e s , and many o t h e r f a c t o r s i n a d d i t i o n to s h i f t s i n population. However, i n an e f f o r t t o k e e p t h i s i n t r o d u c t o r y m o d e l s i m p l e , and s i n c e p o p u l a t i o n i s p r o b a b l y t h e most i m p o r t a n t f a c t o r t o be c o n s i d e r e d , most o f t h e demand c u r v e s i n t h i s m o d e l a r e t i e d t o some measure of population.  63  upon a c e r t a i n p o r t i o n o f t h e urban p o p u l a t i o n  or  a l l the  market t h a t no  e f f e c t on  with  subsets, Land  A  can  no  be  Demand  f o u r t h group o f  s i m i l a r to the  last  activities.  large  assumed t o have l i t t l e  demand c o n s t r a i n t s  or  associated  constraints, eight  group j u s t d i s c u s s e d ,  Some a g r i c u l t u r a l l a n d ,  These b e n e f i t s are  i n i t s r o l e as  external  to the  constraints  s p e c i f i e s the  e x t e r n a l i t i e s have by c u r v e s f o r them.  introducing  Other than t h e i r  r e a s o n a b l e t o assume t h a t e c o n o m i c goods and,  as  ^ R e f e r t o row coding explanation  govern the  the  various  example,  a green b e l t .  p r i v a t e market p l a c e ,  t h e y c a n n o t command a p r i c e o r  group o f  i n number  for  generate a r e t u r n  p r o d u c e r e v e n t h o u g h t h e y have s o c i a l v a l u e .  and  their  Constraints  produces p o s i t i v e b e n e f i t s  that  the  activities,  demand f o r p o s i t i v e e x t e r n a l i t i e s p r o d u c e d by 14 land  have  uses, which s e r v i c e such a  activity  prices,have  s u c h as  them.  Externality  and  accordingly.  industrial  their  population,  suburban p o p u l a t i o n  demand c u r v e s c o n s t r a i n e d s u c h as  total  This  s o c i a l value conventional external  the  fourth  that  various  m a r k e t demand  nature,  these b e n e f i t s c o n s t i t u t e  such, a l l the  to  in  conventional  i t seems normal  assumptions  numbers 120-127 i n t h e m a t r i x i n A p p e n d i c e s I and I I .  picture  64  t h a t one c a n make a b o u t s u c h goods s h o u l d external  demand c u r v e s . various  production  belt  The o n l y  steps goal,  goals.  identical  o f these these  stepped  functions  Acreages  still  i n the f i r s t  into  step  the other  p o l i c y goal, while and t h i r d  step  steps  acres,  a r e summed  c o l u m n , a downward  In a s i m i l a r f a s h i o n  second  sloping effectively  demand  t h i s manner w i t h i n  such e x t e r n a l i t i e s a r e handled i n  t h e model: green b e l t b e n e f i t s  produced  large t r a c t s of natural or a g r i c u l t u r a l vegetation;  green space b e n e f i t s produced by s m a l l e r  areas o f lawns,  s h r u b b e r y , g a r d e n s , and t r e e s ; r e c r e a t i o n a l b e n e f i t s by  lands  uses which allow  a g r i c u l t u r a l lands  provided  f o r p u b l i c a c c e s s a n d u s e a t no  c h a r g e ; and a g r i c u l t u r a l s e l f - s u f f i c i e n c y by  curves  e x t e r n a l i t i e s are introduced.  In a l l , f o u r  by  f o r green  (approximately  f o r g r e e n b e l t e x t e r n a l i t i e s has been  i m p o s e d upon t h e m o d e l . for  to the regional  o f t h e g r e e n b e l t demand  somewhat l e s s ,  activity  of linking  are linked to various  When a c r e a g e s i n t h e s e t h r e e  a fourth total  demand c u r v e  functions  s o many e x t e r n a l i t y u n i t s  contribute  less.  instead  t o t h e maximize green b e l t p o l i c y  e q u a l t o d o l l a r s ) towards t h i s acres  f o r the output  F o r example, t h e demand c o n s t r a i n t s  function contribute  step  t o t h a t used  difference i s that  e x t e r n a l i t i e s are t i e d  goal.  f o r these  f o r m a t b y w h i c h t h e s e demand c u r v e s a r e i n t r o -  duced i s almost e x a c t l y  other  true  b e n e f i t s as w e l l .  The  the  hold  wYiose p r o d u c t i o n  benefits  diminishes  produced o u r dependence  65  on  "outside"  Accounting  food.  Constraints  A an  sources of  fifth  set of  constraints  a c c o u n t i n g mechanism by 15  t o one  another.  activities  t o be  This  subtotalled  included  sums t h e  acreages i n the  produce a l l types of  this  use  column.  external  As  benefits  constraints  between s e p a r a t e components o f constraint  Goal Accounting  The p r o v i d e the  for various  the  well, are  linked  land For  line  which  a g r i c u l t u r a l uses i n t o  t h i s a c c o u n t i n g mechanishm.  32  are  i n t o s e p a r a t e columns.  three  group i s a s e r i e s o f  These are  activities  i n t h i s group i s a c o n s t r a i n t  a g r i c u l t u r e land  c o l u m n s by  which v a r i o u s  mechanism a l l o w s  example,  total  e s s e n t i a l l y provides  one  acreages which  summed i n t o  Also  included  separate in  which p r o v i d e s the  various  demand  lines in total within  links  constraints.  this fifth  group.  Constraints  last  set of  same t y p e o f  The  only  difference  ten  i n d i v i d u a l goal  constraints, accounting  i s that  ten  service  i n number, as  the  fifth  t h i s set relates d i r e c t l y  functions  w h i c h make up  the  to  set. the  composite  15 and  R e f e r t o row numbers 88-119 i n t h e m a t r i x p i c t u r e c o d i n g e x p l a n a t i o n a v a i l a b l e i n A p p e n d i c e s I and I I .  66  16 social objective function. fact, at  scaled p o l i c y goal  length  These t e n c o n s t r a i n t s a r e ,  f u n c t i o n s o f the type  i n the l a t t e r h a l f o f the preceding  methodology o f m u l t i p l e goals constraints,  programming.  one f o r e a c h p o l i c y g o a l ,  discussed chapter  and p l a c e  the t o t a l s  I t i s t o these t o t a l s  sum t h e c o n t r i b u t i o n s  i n t o t e n separate  that the various  a p p l i e d t o determine the value objective  the  basis  columns.  goal weights are  f u n c t i o n a t any p a r t i c u l a r s o l u t i o n .  social As m e n t i o n e d  constraint lines  f o r the i n t r o d u c t i o n of the various  The  land use  activity  o f the composite  a b o v e , some o f t h e s e p o l i c y g o a l  on t h e  These l a s t t e n  t h a t e a c h l a n d a c t i v i t y makes t o w a r d s t h e v a r i o u s goals,  in  also  provide  demand c o n s t r a i n t s .  model a l l o c a t e s L a n g l e y l a n d among t h e t h i r t y -  four land a c t i v i t i e s ,  subject  t o a l l the c o n s t r a i n t s  discussed  above, so t h a t a composite s o c i a l o b j e c t i v e f u n c t i o n i s maximized. policy  This  goals  By  with  composite  function consists of ten  an a p p r o p r i a t e  goal weight a p p l i e d t o each.  f a r the dominant g o a l  maximize gross added a p p r o a c h .  regional production, In other  separate  i s one w h i c h s e e k s t o c a l c u l a t e d by a  w o r d s , i f an a c r e  l a n d u s e r e q u i r e s one t h o u s a n d d o l l a r s o f  value  of a particular intermediate  16 R e f e r t o row numbers 128-137 i n t h e m a t r i x p i c t u r e and c o d i n g e x p l a n a t i o n a v a i l a b l e i n A p p e n d i c e s I and I I .  67  goods per year to produce four thousand d o l l a r s of  final  output then t h a t acre w i l l have c o n t r i b u t e d a value  added  c o n t r i b u t i o n o f three thousand d o l l a r s per year to the of maximizing gross  regional production.  goal  This goal i s  a f f o r d e d a dominant p o s i t i o n i n t h i s model because the production  of a r e g i o n p r o v i d e s  a good i n d i c a t o r of  m a t e r i a l w e l l being of i t s i n h a b i t a n t s .  gross  the  I t represents  p r i v a t e market v a l u a t i o n of the t o t a l amount of goods  the and  s e r v i c e s produced i n a r e g i o n w i t h i n a given time p e r i o d ( i n thisi-case, one s i z e of the t o t a l  year) and,  as such, i s an estimate  of  the  " p i e " t h a t i s to be d i v i d e d up among the  inhabitants.  However, because o f the divergence and  p r i v a t e v a l u a t i o n s which occurs  between s o c i a l  i n a l l r e a l world economies,  t h i s s i n g l e g o a l i s not s u f f i c i e n t to maximize s o c i a l B e n e f i t s and  c o s t s which are e x t e r n a l to the p r i v a t e market  are a l s o c r e a t e d i n the p r o d u c t i o n  process  and  " p i e " to achieve  subtracted  accurate  welfare.  from the p r o d u c t i o n  estimate  of s o c i a l w e l f a r e .  and must be added a more  However, because o f  t h e i r absence from the p r i v a t e market p l a c e , no d i r e c t v a l u a t i o n of these e x t e r n a l b e n e f i t s i s a v a i l a b l e or p o s s i b l e .  Therefore,  some o t h e r v a l u a t i o n method was  the  needed to ensure t h a t  model more a c c u r a t e l y sought a s o c i a l w e l f a r e was  optimum.  accomplished i n the model by the c r e a t i o n of  a d d i t i o n a l goal f u n c t i o n s , one  This  nine  f o r each s e p a r a t e l y  identifiable  68  externality. five  F i v e of these  different  types  ground p o l l u t i o n  pollution,  and  visual  and in  agricultural the  green  of the  As  externality  four  external benefits  into  the  composite  arising  and  included  to  the  the  initial  Though i n i t i a l l y  differing  difficulty  indices,  in setting  Candler  process  determination  all  of the  g o a l s were r e s p e c i f i e d ,  and  in isolation,  of  final  e a c h g o a l was comparable, initial  i n the  on  an  index  the  In o t h e r  iterative  17 and  Boehlje,  weights  op. c i t .  that  analyst's words,  whose u n i t s were  of the  confusion  separately  p r o c e d u r e m i g h t seem t o  the purpose o f the whole concept  on  goal weights,  analyst's perception, to d o l l a r  inspection this  Candler  relative  indices that represented  specified  preceding  constructed  though s t i l l  p e r c e p t i o n of approximate d o l l a r v a l u e .  similar  Boehlje  l e d t o so much  manipulating  iterative  and  p a r t of the  e a c h g o a l was  this and  by  latter  d u r i n g the  on  were t h e n a l l  e a c h g o a l , i n a manner  approach suggested  d i s c u s s e d a t l e n g t h i n the 17  chapter.  as w e l l as  s o c i a l objective function, with  a p p r o p r i a t e w e i g h t p l a c e d on iterative  goals,  regional production,  an  On  well,  model.  combined  and  water  space, p u b l i c r e c r e a t i o n a l areas,  goal of maximizing gross  widely  pollution,  s e l f - s u f f i c i e n c y were c o n s t r u c t e d  These n i n e  and  of  (garbage o r r e f u s e ) , n o i s e  (sight) p o l l u t i o n .  seeking maximization  from green b e l t s ,  seek a m i n i m i z a t i o n  of external c o s t s — a i r  pollution,  goals  goals  roughly  units. defeat determination  69  of  r e l a t i v e weights.  units to  (dollars)  simply  I f a l l goals are s p e c i f i e d  i n t h e same  i t m i g h t seem t h a t i t w o u l d be by f a r e a s i e r  sum up a l l t h e g o a l s  into  a new s i n g l e  goal  c o u l d be u s e d as t h e s o c i a l o b j e c t i v e f u n c t i o n . a multiple goals  approach t h i s  which  I n terms o f  i s t h e same a s p l a c i n g an  e q u a l w e i g h t o f u n i t y on a l l g o a l s .  However,  there  t h a t must be made h e r e . to  i s a subtle but c r u c i a l The d i f f e r e n t  o n l y approximate d o l l a r v a l u e .  distinction  i n d i c e s were  constructed  T h e r e i s no r e a s o n t o  assume t h a t t h e y  exactly coincide with  assume t h a t t h e y  a l l d e v i a t e f r o m d o l l a r v a l u e b y t h e same  amount a n d i n t h e same d i r e c t i o n . will,  i n fact,  units.  The a t t e m p t t o r o u g h l y  weights. in  reasonably exists.  lies.  one o f s i m p l i c i t y  and m a n i p u l a t i n g  of Chapter-five this  and  initial  procedure  solution,  The v a l u e o f t h e i t e r a t i v e  search  they w i l l  they  feel  be c o r r e c t i n g  goal  simply analysis i s  i f such  an optimum  f o r t h e optimum  The d e c i s i o n makers c a n s t i l l  weights i n the d i r e c t i o n In e f f e c t ,  convenience.  points  point i n the i t e r a t i v e  c l o s e t o t h e optimum  b y no means l o s t .  initial  approximate d o l l a r value i s  I n t e r m s o f t h e d i s c u s s i o n on s t a r t i n g  that the s t a r t i n g  index  e l i m i n a t e a l o t o f c o n f u s i o n and  i n establishing  the l a t t e r h a l f  ensures  is  other than  d o e s n o t h i n g more t h a n  difficulty  I n o t h e r words, they, a l l  be c o n s t r u c t e d i n t e r m s o f d i f f e r e n t  done f o r no r e a s o n It  d o l l a r value, nor to  respecify  a preferred solution their  initial  70  r o u g h a p p r o x i m a t i o n s between g o a l value,  given  feasible  i n d e x u n i t s and d o l l a r  t h e c e r t a i n amount o f h i n d s i g h t  solutions afford  that  the i n i t i a l  them.  Summary  The i n t e n t o f t h e d i s c u s s i o n been t o o u t l i n e t h e b a s i c to discuss during  a c t u a l model d e t a i l ,  s t r u c t u r e o f t h e L a n g l e y model and  No a t t e m p t h a s b e e n made t o p r e s e n t  a s s u c h an e x e r c i s e  lengthy  and f a r t o o d r e a r y  of t h i s  thesis.  i n this  detail  would prove t o o  t o be i n c l u d e d  i n t h e main  body  R e f e r e n c e s have b e e n made t h r o u g h o u t t h e  Appendices t o t h i s greater  chapter has  c e r t a i n t e c h n i c a l and t h e o r e t i c a l problems e n c o u n t e r e d  i t s construction.  discussion  i n this  section to the relevant s t u d y where  areas i n the  t h e model c a n be e x a m i n e d i n  i f desired.  The f o l l o w i n g c h a p t e r b e g i n s w i t h an o u t l i n e o f methods o f d a t a a c q u i s i t i o n a n d p r o b l e m s o f d a t a and  then continues  results.  to a presentation  o f model  suitability  solution  71 Chapter 7  THE LANGLEY MODEL: DATA PROBLEMS AND INITIAL RESULTS  Data  Problems  The c o l l e c t i o n and g e n e r a t i o n o f data, as w e l l as the time and money needed t o c a r r y out these a c t i v i t i e s , are probably the most l i m i t i n g f a c t o r s encountered i n any k i n d o f  systems 1  modelling.  C e r t a i n l y t h i s proved t o be the case i n t h i s study.  Undoubtedly  a g r e a t d e a l o f data e x i s t s which  to the Lower F r a s e r V a l l e y land use system. of i t c o u l d be found w i t h i n the imposed  applies  However, what  little  time l i m i t s o f t h i s study  tended t o be s t r u c t u r e d so as t o negate i t s u s e f u l n e s s t o the model, o r to be s t r u c t u r e d so as t o be u s e f u l , but o n l y a f t e r much a r i t h m e t i c and a c c o u n t i n g m a n i p u l a t i o n .  One o f the most o f t e n encountered data problems one o f a g g r e g a t i o n c o m p a t i b i l i t y .  Data e x i s t e d which  was  applied  to c e r t a i n aggregates chosen, f o r v a r i o u s reasons, by the agencies or persons^ t h a t o r i g i n a l l y generated or c o l l e e t e d the d a t a . However, these aggregates were not compatible t o model aggregation.  For example, worker d e n s i t i e s were found f o r the  M e t r o p o l i t a n Vancouver j u s t Langley.  area ( i n c l u d i n g Langley) but not f o r  Costs of f l o o d p r o t e c t i o n f o r the e n t i r e Lower  1 A complete d e s c r i p t i o n o f the data used i n t h i s model i s a v a i l a b l e i n Appendix I I , T a b l e s A-F e s p e c i a l l y .  72  Fraser Valley  flood  p l a i n were a v a i l a b l e ,  of p r o t e c t i n g t h a t part o f the f l o o d were n o t .  Park p l a n n i n g standards  of Langley  but not f o r the D i s t r i c t .  but the s p e c i f i c  plain  that l i e s  c o u l d be f o u n d Often  these  type o f  the p r o d u c t i o n f u n c t i o n i n t o  found  and s o o n .  for this  i n t o one c l a s s  several classes:  However,  resulting  f o r the purposes  to l o c a l  a p p l i c a b l e d a t a c o u l d n o t be 'i cases,  some q u i t e  c o u l d n o t be f o u n d  commercial a c t i v i t i e s . of North  was a l s o u s e d  In these  cases,  t h e most i m p o r t a n t  a r e a was n o t u n s u i t a b i l i t y Even u n s u i t a b l e d a t a  of adequate time  and r e s o u r c e s t o g e n e r a t e  to suspect  f o r some standards  This  standards.  problem  encountered  of data, but rather lack of  i s b e t t e r than  a r e a s d a t a c o u l d n o t be f o u n d reason  For  A m e r i c a were assumed t o be r e l e v a n t  f o r some r e c r e a t i o n a l  However, p r o b a b l y  little  In these  c o n d i t i o n s and were i n c l u d e d i n t h e m o d e l .  procedure  data.  grouped  a s s u m p t i o n s were made and a l t e r n a t e d a t a were u s e d .  i n other areas  this  similar  o f t h e model.  however u n s u i t a b l y a g g r e g a t e d .  Lower M a i n l a n d  in  semi-  c o u l d n o t be  i n a l l labour inputs being  example, minimum p l a n n i n g s t a n d a r d s  used  skilled,  labour requirements  I n many a r e a s , d i r e c t l y  dubious  the labour input to  t y p e o f d i s a g g r e g a t i o n , o r any such  classification,  found  i n which  new d a t a , n e c e s s i t a t e d c h a n g e s i n m o d e l d e s i g n .  F o r e x a m p l e , i t was hoped t o d i s a g g r e g a t e  skilled,  i n Langley  f o r the City  a g g r e g a t i o n problems, t o g e t h e r w i t h t h e l a c k o f time to generate  costs  no d a t a , g i v e n a l a c k new d a t a .  I n many  a n d , i n some a r e a s , t h e r e was v e r y  t h a t i t even e x i s t e d .  I n f o r m a t i o n on  municipal  tax  conditions  f o r the  information land  collection  on  various  use—pollution  fall  into this  information initial  can  expeditures  outputs of  externality valuation,  added t o p r o d u c t i o n all  and  the  land  use,  demand  d i f f e r e n t land  uses,  technical information  on  r e l a t i o n s h i p s , w o r k e r d e n s i t i e s and  value  i n schools,  areas,  category.  likely  be  hospitals, recreational  G i v e n enough t i m e and  generated.  For  w o r k i n g m o d e l , however, t h i s  estimated or purely  by  the  money,  purposes of  information  was  this this  either  fabricated.  Though t h e s e p r o b l e m s o f d a t a a q u i s i t i o n u n d o u b t e d l y have r e d u c e d model r e a l i t y one  of  the  identify  primary purposes of t h i s  and  any  lesson  As  f o r any  such, t h i s future  such s i m i l a r p r o j e c t .  amount o f  s t u d y was,  be  remembered  in fact,  purposes, effort,  to t h i s  p o i n t where i t c a n land  use  future a  project,  be  or  abstracting  used  for  system r e q u i r e s  a  great  money: e f f o r t m e a s u r e d i n teams  w o r k e r s , t i m e i n m a n - y e a r s , and  money i n h u n d r e d s o f  that  to  provided  attempt a t a c c u r a t e l y  from a l a r g e t i m e , and  s i t u a t i o n has  extensions  Any  a m a t h e m a t i c a l model, t o the planning  i t should  measure t h e s e t y p e s o f p r o b l e m s f o r any  model c o n s t r u c t i o n . valuable  severely,  of  thousands  of d o l l a r s .  Results  B e c a u s e o f d a t a and constructed  in this  t o a c t u a l l a n d use  design  study obviously planning  d e f i c i e n c i e s , the has  problems.  very  little  model  application  However, model s o l u t i o n  is  still  interesting  be  carried  For t h i s  i n terms o f the type o f a n a l y s i s  o u t by a m u l t i p l e g o a l s l i n e a r  reason,  programming  a s h o r t d i s c u s s i o n on m o d e l r e s u l t s  R e c a l l t h a t a l l goals i n the o b j e c t i v e constructed  either  on  a dollar  units.  f o r t h i s were d e t a i l e d  chapter.  reasons  The  m o d e l was  then  initially  e q u a l t o p l u s o r minus u n i t y . u n i t s were o f e q u a l m a g n i t u d e r e g i o n a l production equaled tural  of  u n i t y oh  and  attempt  c a r r i e d out  Not  particularly  contribution  a one  i n column  to dairying.  uses.  p r o t e c t i o n was  an e q u a l  to  (a) o f T a b l e  weight  The  of  initial  goal  equal  1 was  residential  were a l l o c a t e d  I t was  established  ±1.0),  achieved.  uses, to  L a n g l e y ' s wooded u n i m p r o v e d a c r e a g e s .  Full  and  uses,  the  and  institutional  that  therefore a l l of Langley's  model a l s o c l e a r e d  the  which  l a n d went t o a g r i c u l t u r a l  Small acreages  protected.  gross  in  equal units  ( a l l weights  f o l l o w e d by  o p t i m a l , and  scale  agricul-  T r a n s p o r t a t i o n l a n d uses took  commerce.  weights  i s simply i n keeping with  s e t of weights  presented  i n d u s t r y and  p l a i n was  to  u n i t decrease  to avoid confusion i n setting  next biggest a l l o c a t i o n , recreational  preceding  contribution  Note t h a t p l a c i n g  u n u s u a l l y , most o f L a n g l e y * s  uses,  dollar  unit  T h i s procedure  Using t h i s  the s o l u t i o n  dollar  solved with a l l goal  to s p e c i f y a l l goals i n approximately  weights.  units  a l l t h e g o a l s does n o t mean t h a t a l l g o a l s a r e  equal importance.  was  (a one  a one  so o n ) .  i n the  were  In o t h e r words, a l l the g o a l  s e l f - s u f f i c i e n c y which equaled  water p o l l u t i o n ,  follows.  s c a l e o r on a s c a l e whose  ( i n the author's p e r c e p t i o n ) of  can  model.  function  were a p p r o x i m a t i o n s The  that  flood flood  u s e d much o f  employment  was  75 Table 1  LANGLEY MODEL SOLUTIONS FOR OPTIMAL ALLOCATIONS OF LAND (all  acreages rounded  to the nearest (a)  A c t i v i t y or A c t i v i t y Group  (b)  (c)  Objective Function C  dairy market g a r d e n i n g feedlot a l l a g r i c u l t u r a l uses  acre)  a  CLIES=5  b  (d) Used:  CGRPONLY CPOL=5 c  65558 7510 200 73268  65533 7510 200 73243  64300 7510 200 72010  65844 7510 200 73454  all  commercial  uses  11  12  12  23  all  i n d u s t r i a l uses  158  158  237  nil  59 139 88 208 494  59 164 88 208 415  59 164 88 104 415  59 139 88 208 494  urban roads r u r a l roads r a i l r o a d s and r a i l y a r d s airport o f f - s t r e e t parking a l l t r a n s p o r t a t i o n uses  61 742 169 141 336 1449  61 742 169 141 336 1449  429 729 249 141 232 1780  61 744 23 141 382 1351  apartments s i n g l e f a m i l y homes a l l r e s i d e n t i a l uses  153 915 106 8  153 915 1068  nil 1973 1973  153 908 1061  274 5 282  274 5 282  137 3 143  274 5 347  total  76730  76730  76730  76730  flood protection land clearance % employment p e r c a p i t a income  6200 11708 100 1906  6200 11683 100 1906  6200 11522 100 1915  6200 11709 100 1675  urban park recreational reserve g o l f course commercial r e c r e a t i o n a l dev. a l l r e c r e a t i o n a l uses  schools hospitals a l l i n s t i t u t i o n a l uses  ($)  a W e i g h t s on a l l g o a l s e q u a l s  ±1.0.  76 Table  1  cont'd  b W e i g h t s on p o l i c y g o a l s c o n c e r n e d w i t h r e c r e a t i o n a l b e n e f i t s , g r e e n b e l t s , and g r e e n s p a c e a l l e q u a l +5.0. A l l other g o a l weights remain s e t a t ±1.0. c W e i g h t on t h e g r o s s r e g i o n a l p r o d u c t i o n g o a l e q u a l s +1.0, w h i l e a l l o t h e r g o a l w e i g h t s a r e s e t e q u a l t o z e r o , d W e i g h t s on a l l f i v e p o l l u t i o n g o a l s e q u a l -5.0. All other goal weights are s e t equal t o +1.0.  11  achieved, the key  w h i c h i s a l s o t o be  limiting  p r i c e o f an units the  a d d i t i o n a l l a b o u r e r was  limiting  Per  labour  nature  of the  labour  initial  total  $1906.  labour  The  a d d i t i o n a l acreage,  t o 194  design  and  data  reasonable. all to  acquisition,  Land a l l o c a t i o n ,  w i t h i n a reasonable be.  In  resemble  fact,  the  ranging  on  this  The  to quite large s h i f t s  remains v i r t u a l l y other  in  by  for  t o the  which the  recreational  solution  m o d e l , as  would expect  solution,  the  goals  agricultural  g o a l s by  Compare t h e  green b e l t ,  Nor  insen-  For  self-sufficiency solution  a factor of  i n column  as  i s very  goal weights.  have b e e n s e t a t +5.0  g o a l weights remain s e t a t ±1.0.  them  notable  I n c r e a s i n g t h e w e i g h t s on  the  are  closely  as w e l l  constructed,  in relative  presented  g o a l w e i g h t s on area  land rents  s o l u t i o n brought a  changes model s o l u t i o n o n l y m a r g i n a l l y . solution  surprisingly  appears t o f a i r l y  initial  externality  acre  p r o b l e m s i n model  a f a c t o r o f t w e n t y - f i v e , model  unchanged.  three p o s i t i v e  are  for  Langley.  programming a r o u n d t h i s  is multiplied  the  d i s t a n c e o f where one  year  f o r another  w o r k e r i n c o m e s , and  example, e v e n i f t h e w e i g h t on goal  results  the model s o l u t i o n  model f e a t u r e t o l i g h t . sitive  Considering  current situation  Price parametric  u n i t s per year  these  was  a n n u a l shadow r e n t s  any  land.  worker  force)  o b j e c t i v e f u n c t i o n u n i t s per  agricultural  reflecting  Average  l a n d v a r i e d f r o m 91  the best  shadow  run,  for  of  probably  annual  constraint.  d i v i d e d by  c a p i t a income was  The  is  2533 o b j e c t i v e f u n c t i o n  "dollars") in this  (total production  $4887.  as  c o n s t r a i n t i n the model.  (social welfare  income  expected  a l l the  five initial  (b) o f T a b l e green space, while  1, and  a l l other  does e l i m i n a t i n g a l l n i n e  78 externality  g o a l s from  as was e x p e c t e d , gain  although  substantially  Column  production  pollution  s u b s t a n t i a l amounts o f p o s i t i v e e x t e r n a l -  (c) i n T a b l e  initial  solution  costs of pollution,  1 p r e s e n t s t h e s o l u t i o n when  i s most s e n s i t i v e  though even here  must be a t l e a s t d o u b l e d  weights  b e f o r e any major changes b e g i n  a c t i v i t i e s which generate  Table  from  a t +1.0.  results  from  per c a p i t a gross  goal  At this  stage,  quickly  (column d) o f  placing  a l l other goal weights  t h e low p e r c a p i t a  a h i g h p r i c e on p o l l u t i o n  remain  income t h a t  (this  i s not the  of the value of the o b j e c t i v e function but  r e g i o n a l p r o d u c t i o n d i v i d e d by p o p u l a t i o n ) .  In  trying  model s o l u t i o n ,  to visualize  dimensional  the character of t h i s  i f the s e t of a l l f e a s i b l e  a multi-dimensional concept,  to  to occur.  goal  t h e o p t i m a l s o l u t i o n when a l l t h e p o l l u t i o n  Note e s p e c i a l l y  share  to the other  The f o u r t h column  g o a l s have a w e i g h t o f -5.0, w h i l e set  t o changes i n t h e  l a r g e amounts o f p o l l u t i o n  the s o l u t i o n .  1 presents  relative  gross  for optimality.  a l l the p o l l u t i o n  weights  disappear  do  a t t h e expense o f l a n d  i n the region i s the sole c r i t e r i o n  The  a s much  generating a c t i v i t i e s  higher allocations  uses which generate ities.  t h e m o d e l change t h e s o l u t i o n  terrain,  with  initial  solutions,  itself  c a n be r e p r e s e n t e d a s a t h r e e -  the height of the t e r r a i n  corresponding  the value o f the o b j e c t i v e f u n c t i o n a t t h a t p o i n t , then the  initial almost  solution  i s l o c a t e d i n the middle  flat  plateau.  sensitivity  analysis  Unfortunately, this  of a very  large,  s i t u a t i o n makes  r a t h e r u n i n t e r e s t i n g and a l s o  renders  79 any i t e r a t i v e search f o r an optimum q u i t e f r u i t l e s s .  Social  w e l f a r e , as measured by the value of the composite o b j e c t i v e f u n c t i o n i s v i r t u a l l y the same on a l l p o i n t s of the p l a t e a u . In may  o t h e r word, the marginal i n c r e a s e s i n s o c i a l w e l f a r e t h a t be achieved through any i t e r a t i v e search are not worth the  c o s t s of s e a r c h i n g f o r them.  T h i s s i t u a t i o n should not be  construed, however, as i n d i c a t i n g t h a t i t e r a t i v e search methods, of  the type o u t l i n e d i n p r e v i o u s c h a p t e r s , have l i t t l e  while a p p l i c a t i o n . is  due  That they have l i t t l e  worth-  application in this  case  t o the nature of t h i s p a r t i c u l a r model, not t o any i n h e r e n t  d e f i c i e n c y i n the i t e r a t i v e search methods. They d o u b t l e s s w i l l have a p p l i c a t i o n s f o r o t h e r land use models.  Another i n t e r e s t i n g r e s u l t occurs when the upper l i m i t on p o p u l a t i o n i s allowed t o i n c r e a s e , c e t e r i s p a r i b u s . shows model s o l u t i o n s f o r p o p u l a t i o n increments s t a r t i n g a t a p o p u l a t i o n o f 10,000 (the 1971 Langley was  21,936 ).  The  Table 2  of 10,000,  p o p u l a t i o n of  steady d e c l i n e i n per c a p i t a income  (gross production) and i n per c a p i t a s o c i a l w e l f a r e  (measured  by v a l u e of the o b j e c t i v e f u n c t i o n ) are probably the most i n t e r e s t i n g f e a t u r e s here.  Nor  should these d e c l i n e s be unexpected.  As f a r a per c a p i t a income goes, what e s s e n t i a l l y i s being done is  t o i n c r e a s e the amount of workers i n an area t h a t has a l i m i t e d  production p o t e n t i a l .  The even more r a p i d d e c l i n e i n the per  c a p i t a share of t o t a l s o c i a l w e l f a r e occurs because the i n c r e a s i n g p o p u l a t i o n c r e a t e s much a d d i t i o n a l p o l l u t i o n but c o n t r i b u t e s very l i t t l e  i n the way  of p o s i t i v e  externalities.  Though data problems have undoubtedly  reduced  the  80  Table 2  OPTIMAL ALLOCATIONS BY THE LANGLEY MODEL UNDER VARYING POPULATION CONDITIONS  (rounded t o n e a r e s t  Activity  Group  agricultural  uses  commercial  uses  industrial  uses  recreational  uses  transportation residential  acre, a l l  uses  uses  institutional  uses  TOTAL ACREAGE  g o a l w e i g h t s = ±1.0)  Population 21936  10000  20000  30000  40000  73268  73740  73446  72526  71605  11  5  10  16  22  158  12  134  255  377  494  868  450  675  900  1449  1001  1378  1747  2117  1068  976  1054  1124  1195  282  129  256  386  515  76730  76730  76730  76730  76730  (dollar  units)  per  c a p i t a income  1906  2404  1946  1792  1715  per  capita social welfare  2134  2841  2192  1974  1854  81  applicability system and  o f t h e s e model r e s u l t s  t h a t was  t h e y do  to the a c t u a l  a b s t r a c t e d , t h e r e s u l t s do  appear  land  encouraging,  p r o v i d e some i d e a o f t h e t y p e o f r e s u l t s  type of a n a l y s i s  t h a t c a n be  produced  and  by m a t h e m a t i c a l  use m o d e l s , t o g e t h e r w i t h an u n d e r s t a n d i n g  o f the  use  the  land  potential  u s e f u l n e s s of'-'such m o d e l s .  Further  Refinements  Certainly  the type of a n a l y s i s  l i n e a r programming o f f e r s of  further  concern  study.  Several further  are suggested  Firstly,  found  collected  and  and  sorted.  Data  n e e d t o be  specified  fulfil  these  this  and  not  least  must be  constraints  by  of  and  generation  data sources d a t a needs t o  need be  be  available information  found.  data i s c o l l e c t e d ,  undertaken.  c a n be  Simplifying  r e p l a c e d by more r e a l i s t i c land uses,  Existing  satisfied  when s u f f i c i e n t  i n c r e a s e model d e t a i l .  existing  areas  c o s t methods o f g e n e r a t i n g d a t a  ments i n m o d e l d e s i g n n e e d t o be  and  and  Methods o f u p d a t i n g d a t a n e e d t o  requirements  and  deserving  study.  A l l available  documented.  requirements  Secondly,  of a c t i v i t i e s  refinements  initial  examined.  clearly  examined.  to  by  p l a n n e r s appears  the area of data c o l l e c t i o n  must be more c l o s e l y t o be  t o l a n d use  that multiple goals  improve-  A g r e a t e r amount  added t o t h e s t r u c t u r e assumptions  structuring.  c a n be  removed  Information  c o n v e r s i o n c o s t s between u s e s ,  to  on  capital  flows  82 and  c o n s t r a i n t s , and g r e a t e r  relationships Lower F r a s e r  c a n be i n c l u d e d Valley  functions  o f t h e model t o i n c l u d e This  specification of interactivity  this  i n model d e s i g n .  a s an e c o n o m i c u n i t , e n t i r e area  p r o c e s s would permit g r e a t e r  relationships  and c o n t r i b u t e  Lastly,  Since  should  expansion considered.  specification of interregional  f u r t h e r t o model  t h e whole a r e a  be  the e n t i r e  accuracy.  o f t h e p s y c h o l o g y and s o c i o l o g y  o f p o l i c y making groups needs e x a m i n a t i o n t o determine t h e e x a c t compatibility  of a multiple  goals  type o f a n a l y s i s t o d e c i s i o n  making p r o c e s s e s and t o suggest p o s s i b l e design  which would i n c r e a s e  making t o o l . is  This  needed i f t h i s  the  i s a very  improvements  i n analysis  i t s e f f e c t i v e n e s s as a d e c i s i o n important  type o f a n a l y s i s  area  o f c o n c e r n where  i s t o s u c c e s s f u l l y move  study  past  t h e o r e t i c i a n i n t o t h e r e a l world p r o c e s s e s o f land use  planning.  Discussion  On t h e b a s i s  o f model  construction  and t h e p r e c e d i n g  d i s c u s s i o n on model r e s u l t s , i t a p p e a r s t h a t m u l t i p l e linear of  programming  land  a useful  format w i t h which problems  u s e a l l o c a t i o n c a n be e x a m i n e d .  simplicity, this  provides  flexibility  and m u l t i p l e  structure particularly  well  goals  goal  I t squalities of optimization  make  s u i t e d t o t h e land use f i e l d .  By f o r c i n g s p e c i f i c a t i o n o f r e l e v a n t  p o l i c y goals  lends  t h a t has remained almost  some q u a n t i f i c a t i o n t o an a r e a  wholly q u a l i t a t i v e t o date.  this  Y e t a t t h e same t i m e , t h i s  format  structure  83  does n o t f o r c e d e c i s i o n making b o d i e s  t o completely  preference  relationships.  f u n c t i o n s and i n d i f f e r e n c e  do s o may p r o v e t o be p o l i t i c a l l y this  feature i s very  t h i s model  objectives  to at least and t h e i r  By d o i n g  systematic, optimal  such  this  type  format  examine, and As planners  i n t h e l a n d use f i e l d .  p o i n t , i t s h o u l d be n o t e d  that multiple goals  does n o t r e p r e s e n t t h e o n l y i n this  "good"  planning  area.  Though,  d e c i s i o n s i t appears the  o f t h e s t r u c t u r e s d i s c u s s e d , a l l o f t h e model  do have v e r y u s e f u l a p p l i c a t i o n s i n t h e l a n d u s e f i e l d . o t h e r c r i t e r i o n were u s e d t o measure  w o u l d be j u d g e d  most  suitable.  t h a t a c t u a l models r a r e l y  Then t o o , i t must  are, and r a r e l y  be remembered  n e e d t o b e , composed  Most o f t e n t h e y  c o n t a i n e l e m e n t s o f more t h a n  frequently  a r e a composite o f a l l t h r e e .  types  CEf some  t h a t some o t h e r s t r u c t u r e  o f o n l y one o f t h e s t r u c t u r e s d i s c u s s e d i n t h i s  they  types  the usefulness o f these  m o d e l s , i t i s p o s s i b l e and e v e n l i k e l y  different  What  an a c c e p t a b l e ,  by w h i c h t o s p e c i f y ,  o r even t h e "best" a i d a v a i l a b l e  entirely  bodies,  f o r t r a d e - o f f s between  o f a n a l y s i s represents a u s e f u l t o o l which  programming  suitable  considerations.  provides  on t h e b a s i s o f s e a r c h i n g f o r o p t i m a l most  t o some s u c h  s o l u t i o n s t o problems o f l a n d use a l l o c a t i o n .  At t h i s  tool,  Since to  i s to force the d e c i s i o n  preference patterns  use f o r d e c i s i o n making  linear  such  their  t h i n k i n t e r m s o f v a r i o u s g o a l s and  s o , t h i s model type  rational  find  can  with  s t r u c t u r e does do, however,  m a k i n g body  these.  compatible  unacceptable  bare  study.  one s t r u c t u r e and Often  o f s t r u c t u r e s a r e melded i n such  the three  a fashion that  84  each undertakes a separate it  i s best  structure  s u i t e d to providing. and  to  use  in this  are  compatible  to  land  function or  use  suggest that  field and  i s obviously a l l are  planning.  group o f  Therefore,  i t i s the folly.  only  to  functions s i n g l e out  one  A l l three  that  that one  merits  structures  c a p a b l e o f m a k i n g some c o n t r i b u t i o n  85 Chapter 8  SUMMARY AND  The  principle  CONCLUSIONS  o f the p u b l i c c o n t r o l of the p r i v a t e  use  of land  i s one w h i c h i s r e a d i l y a c c e p t e d  i n most p a r t s o f  the  world.  I n t h e Lower M a i n l a n d o f B r i t i s h  Columbia,  principle  has a s s e r t e d  settlement. the  from t h e f i r s t b e g i n n i n g s o f  Though t h e g o v e r n m e n t " s c o n t r o l o v e r l a n d u s e i n  r e g i o n was f a i r l y  century,  itself  this  tenuous u n t i l  well  i n t o the current  c o n t r o l h a s s t e a d i l y grown u n t i l ,  government, t h r o u g h i t s v a r i o u s most o f t h e p l a n n i n g in  this  agencies,  today, the  i s responsible f o r  and d e v e l o p m e n t o f t h e l a n d  resources  the region.  As  large scale  s e v e r a l new p l a n n i n g  land use p l a n n i n g  t o o l s and m e t h o d s .  d e v e l o p e d , so d i d  One m a j o r new d e v -  elopment has been t h e use o f m a t h e m a t i c a l models t o study large  land use systems.  Though t h e r e  development as a poor a l l o c a t i o n in  the f i e l d  to play  a r e t h o s e who r e g a r d  o f e f f o r t , most  this  researchers  b e l i e v e t h a t m a t h e m a t i c a l m o d e l s have some r o l e  as a p l a n n i n g  tool.  Three t y p e s o r major groups o f macroeconomic models can  be i d e n t i f i e d  from t h e l i t e r a t u r e .  examine t h e s t r u c t u r a l f l o w s in  l a n d use p l a n n i n g  within  ,for s h o r t  I n p u t - o u t p u t models  a s y s t e m and c a n be u s e d  t e r m f o r e c a s t i n g and p r e t e s t i n g  86 of p o l i c y changes. world and  Simulation  system&actions  c a n a l s o be u s e d  forecasting  (with  models c l o s e l y r e p l i c a t e r e a l  little  regard  to internal  i n l a n d use p l a n n i n g  f o r short  and p r e t e s t i n g o f p o l i c y d e c i s i o n s .  structure) term  Mathematical  programming m o d e l s examine s t r u c t u r a l c o n s t r a i n t s w i t h i n s y s t e m and c a n be u s e d t o f i n d allocations.  The i n h e r e n t  " c o r r e c t " o r "best"  optimizing  a  l a n d use  nature o f mathematical  programming makes t h i s model s t r u c t u r e much more a p p l i c a b l e to problems o f l a n d input-output  use p l a n n i n g  than e i t h e r simulation or  analysis.  A t y p e o f m a t h e m a t i c a l programming p a r t i c u l a r l y suited  t o l a n d use p l a n n i n g  form o f l i n e a r in  programming w h i c h a l l o w s  the objective  allocations posite Since use  i s the recent  function.  With t h i s  this  are relevant  m i n g m o d e l was c o n s t r u c t e d and D i s t r i c t  Columbia. severely results that  goals  type o f model, l a n d use i n respect  rather  t o a com-  than j u s t one.  t o d e c i s i o n making i n t h e l a n d  procedure i s p o t e n t i a l l y very  Using t h i s multiple  City  f o rmultiple  function of several p o l i c y goals,  field,  development o f a  c a n be f o u n d w h i c h a r e o p t i m a l  many g o a l s  goals  useful.  format, a l i n e a r  t o study land  program-  use a l l o c a t i o n i n t h e  o f L a n g l e y i n t h e Lower M a i n l a n d o f B r i t i s h  W h i l e p r o b l e m s o f d a t a a c q u i s i t i o n and l i m i t e d reduced the a c t u a l a p p l i c a b i l i t y  appear encouraging.  the m u l t i p l e  well  goals  resources  o f t h e model, model  Certainly this  type o f a n a l y s i s  structure o f f e r s to land  a p p e a r s p o t e n t i a l l y u s e f u l and i s d e s e r v i n g  use p l a n n e r s  of further  study.  87 BIBLIOGRAPHY  Adelman, I.,,.and F. 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" S i m u l a t i o n : A T o o l o f Farm P l a n n i n g Under C o n d i t i o n s o f Weather U n c e r t a i n t y , " J o u r n a l o f Farm E c o n o m i c s , X L V I I ( A u g u s t , 1 9 6 5 ) , 574-594.  91  Appendix  I  92  NOTE:  V a r i a b l e numbers g i v e n i n t h e c o d i n g e x p l a n a t i o n to t h i s matrix picture in  this  it  s h o u l d be n o t e d  in  this picture,  137  and  (see A p p e n d i x  computer o u t p u t .  columns,  numbers 138  that  For c r o s s - r e f e r e n c e purposes rows, g o i n g  correspond form  through  to  II) are not i n c l u d e d  left 260.  from  to variable to right,  top to bottom numbers 1 t o  correspond  to  variable  92a  o o _  —J L L  5T D_  _ _  fO  _  a  _  <N  3t  a.  OL  _  -4  OO  CL  or:  _  »~  vi  a  el-  __  LO  a . e-  to  a ct:  =>  ^  3  _ CC  3  r> X  QO.  Ct  _  m _  _  —* u  ca  0  LL)  < > >• X  a  H  <  1  H-  _j m  0 - <i _ J _  0 0  _J  VI  to  »~  UJ  CC _> O-  D:  *"*  ar  cc z> a 0- - J  _  CC  U, 0 U  f^l <\J  a.  - J •—  K  —<  CL  CXI  <  _j  a c c c  a  l~-  o_  n*l  CL  f\J  CL  —4  _  J -  o t  LO C J 2T 0 - n  ' to  0 2T CC O J  f _ p—<  _  00  O  0. h-  a  CL  rO  a  r\j  X 0  r> «/> x0 a Z> L_>  z>  _  r-(  CC — I  aC r<i  0  y-  _ a. CXI 3 O _ 'X—4 cf. < _ to 1 -  t-  _  _D  O  l/l  <  on CXI  i ~ o_  <  a:  <  CU.  lt  O h-  O  IL  LL  <_)  a  LL  a.  CD U- u .  u  c—1  i-  0  t-  O  -  U.  UJ UJ  I-  C_  -  to  -. m cxj  LL  UJ LU G  LL  LU LU  LL  LU  Ut  _  <T  CC _  JJ..  rxt  __  _ <"  CC. >-  t- O  < O 2  2 r-t rxi I A  o > •  —J  o! r* 11 n  -  11  u  ll  2* IT 11  _j _ _ J r, o o n cr c a _ • C C a o a u u a o o O O LJ  -J  _J  —I - J  - J -I  J  _ i p  I  o O  r-t  I T —1  I!  11  i c Of O  n- u u. > o c. 11 C  l l . ' LL> L U UJ  "Do' a t u.: U_i LLJ  -J U  U 7  cS. a  j*t ~ tr  — — _ J LL .LI.  L,"  o o o jo C O (  I  iX. CL.  ?  L;  I  _ i J - * fXl • » iu u 1  t/> o n r,- i i LL • 7_ _ r ! u . U - I  x  i  O !  i  ia  c  • t~ 1  3  T to to  7f  r\j c X' C" X L_) t O L O CO  2i  s s s sR s s s s s s ss s s sU H H HH c c cH H H H R  R  R  R  T  0  T  F  F  F  R R R U U UU  I:  E  F  F  A A  I I N N N N N N M N O O n O N N N N o o on AA A A I  C  G  T  U U UU  0  F E  E  R  E  D  D  O  O  O  C  C  C  C  S  S  S  Y  N  I  2  3  T  M  1  2  3  T  1  2  3  A  S  C  R  T  1  P 2  C R  C C R  3  T  1  P  P  .  2  P 3  P  R T  P. 1  2  R 3  T  R 1  P 2  P 3  P T  P 1  U U U  0  R  T  L 2  P. L  3  R L  T  L  N  O  D  O  T  H Y  V D  O  E  C  1  U U UU  P p pp  MM P P  R  R  R K  2  K 3  R K T  R K 1  K 2  K 3  K  K  T  R  K  K  K  K  F  1  2  3  T  1  S S H? P 2 P.URAL 1 RURAL? MNGF.BT  L  MNT»FC  L  M K J t J P R |<  -1  -1-1 -1  -C  L  -c  L S'tvi MP RK  L  MNG1JL "  L  M.MRDEV  L  UPRK1  G  -c -c  G  U°RK 2 SPRK1  G  SPRK2  G  MPRK 1  G  MPRK2  G  GOLF 1  G  GOLF 2  G  CP.DEV1  G  CR'IEV?  G  MNURI  L  -c  -c  -c  -c  -c  -c  MNSRD MNRPQ M.X'OARK "fN^A'TT" MM A I R MMHOS 0  «MS'<UL "NGOVT  HOSPl HOSP? SKUL 1 SKIJL2  GOVT i GOVT 2 HOUSAL ~ = u —  MXUNE-M  -u-u  -B  - T - 3  - U  BALA^JC TAG? I C  1-1-1  TF EE 0 "TCOM'RC  - 1 - 1 - 1  1 "1  -  TQFF- I C  =T~ - 1 - 1 - 1  TTRA'IS  1 -1-1-1  TiJCf:R  F  TUSHOP  F  VO  1 - 1 - 1 - 1  1 - 1 - 1 - 1  1  T O T A G TSCMR TSS'-inp TP. URAL TINDUS TRECRE TUPRK TSPKK TMO'.K  TGOLF. TCsO-.V TTRAM TPESID TIMSTI T!lf)S° TSKUL TGOVT TFLD'ID TFREESEC TGPRELT T G R S P ACE TAGS E L F F F R E E R EC FGR. F L T FGRS"ACE FAGSELFPEE°EC1 FR EE " E C2 GPRFLT1. GRUELT2 GR S AC F 1 GRSPAC E? AGSELF1 AGSFL = 2 WW.P MNAIR CL MMWAT FT "W-FUSE M W I S E MN S I G H T MyFRSSEC «XG"BELT MXGRSPAC "MXAGTELF" J  n  D  D A I R Y  G T D A F F F F C I F R E E E E T F F H E F E E C I E n D n O O C C N 1 2 3 T M 1  O F F I C 2  O F F I C 3  O F A I S T  T R A N S 1  T T T R R R U U A A C C C N N N N N S S R P . 2 3 T 1 2  U U U U S S S U U S S S S S S S S S S S S C H H H H C C C C . H H N N O O O O N N N N O O O R P P P P P R R R R P P 3 T 1 2 3 T 1 2 3 T 1 2 3  E E E E E p  S U H O P T  R U H A P 1  R U R A L 2  R U R A L 3  R O R A L T  T  T L H O U U U U S S S S M M M M G R T F I E W T P P P P P P P P ° P P P O I C G A CP. R R R R R R R R R R R P L L N O H V O E K K K K K K K K K K K K F " D O T Y D C 1 2 3 T 1 2 3 T 1 2 3 T 1  -1-1-11 -1-1-1 -1-1-1  1 1-1-1-1-1 1 -1-1-1  -1 1  -1 -1-1-1  -1  1 -1-1-1  c c c c c E p. p c E t: p c c p E c G G G G  1 -1  i  -1  -1  -1 -1  -1-1 -1 -1-1  -1  -1  Q G G G E E c p p c p  - c - c - r )-»-C l A A A  1  -F-E-E A A C 8 C A A 3 B  -F-E-E  -F-E-E  -F-E-E  -P-fc-t  -F-fc-fc  -F-t-fc  - L - L - C  -t  D 0  1  L  H  C C  - L - C - L  B  - C - L - L  - L  B  o  92d  < O  CO I D  <  C O L U —1 LL <-*  O  _> _ co a  <  to  _)  Q. < l  a : co  o  a  cc  K  LU r o  C J UJ (NJ  O CC l / l Q. < o  LL ( M  o LU  O CC to a . < O CC  -J  L ) LU —4  UJ  -j  _  _J  0.-0—1—1  o  Q_ CO L U  _J  — t  LL CC LLi LU _  LU  U  LL CC U J LU C L  LU  U fO  LL CC L U L U _  LU  U  LL £_ U J LU CC L U  3  K  _J  <  K  D  CD > -  LU  _J  CC  _) >  CL  C O <_  3  or  h  Q-  O  CL  O  _P _J  yu. j  D  1  U  _ O  -4  _  >• LL _ J  CO  <VJ  U  LO  ZD  C  CC CM  _  ^  O  h-  _j < _  o o o o  a > a > o >  LO  _  3  CO _  3  to _  3  • to _  3  c  — I  >  X O co a. hX O to CL f O x  a  X o  t o o_ rv> to  O-  K- C J V— •—• _  -4  CO  <- O X  o  _*:  UJ  in <  o  C_  LU  o  £_  LU  <r in  <  L,  1  CC  co  LU  _c  < _ < h- C t~ <  —  CY.  o  CC Q-  cc  CJ  t_  a: 3  o:  to  3  CC  3  a- n: n_ o  V~ Cj  o or  Ui_  r  Q-  O  or c  CC <t UJ  > t~  GIL' >  u f>: c uj  >  U  m (VI  CC O  UJ  >  -<  e o  - J LL  t~  o c-  -J U  fO  O C  -J  o >-  o m -*! II II fi to ,i _j a _ _iiJ LILJO !... n _'ir^ o' c C * - — !( o G O . —J _ J Ci o o _> u o e.t U U U  —J  —J  >-  i); CJ a  > r:  _t  -J  -a  —J  '_• c-  o u. — |_j -r  IU tl.i UJ  n to  UJ  UJ  z:  7i < to </i  ;  JY  _  u  T  LO LO LO  —I > i —i * V .—( a: O  r-1—  r,:  u.  i_» i / —  IL  1  -^r  o  _> o  92e  <S O tOU J _ J LL O J < O to  LL.  r-t  O tx  o. < o  LU  t~  O  £L  < o  LU  m  (_ Of CO CL  o  J j CM  */> Q .  < o  LU r - i  CC W  o a  LU  O Of CO u ;  _J  —1 —  O  a  J— m  —' Of. CC LU _ J  cc LU —t YCsl —1  O Of CO LU _ J L L Of LU LU  CC LU  L L Of U J LU  CC LU U m  L L (_ LU LU  CC LU  U . Of U J LU  CC L U  _  LU  _  fM  »-l  _  CL Of  -J  < CO U I  3  K  O h-Q  C a  CO  >  u  _J  Of  CC  3  >  LL _ l  3  o  (XI  CO  3  >• LL _ J  3  Q  r-l  CO  3  >• U . -J  13  O  <  _  o C3 > t -  ro  D t o _• O  o >  o o > 1— fx) o o > to  _  3  to  3  _ J r\i  3  _J  X o  to  a.  X  a  to a  X  c  t o a. <xj  _  X o <  -  ^  to  LO  K  3  O h-  _J m  — I  -  m  t o C_ t—• to  o X O _c  LU  <r o  Of U-i  —1< \ < o  Of LU  tA  I »I <r  (_) Of UJ  If*  <r  a  CO CO  o  *5 o f t -  o  Of I U t o  CC C L o f h-  < CL  K  <  ~  Of  a.  < CC  Of  3  Of  CC o  to  3  CO  CC o  3  CC, 00 Of o  O h-CC <t 2:  o or a  UJ >  t~  u cc o  t!_  >  m  o  LU  >  rxt  a n: > c _ J LL  -4  u> —  (.-*•  o o  °  _ J LL _ l L L rxi  i  I  Y— L> <XJ  . CC O  > h- 3  ,v y : (T Of ;  3 >  (  i n: a  rc  < 3  >" r y  i f •_ to O  H  <_• O  UJ .UJ _ •  LU  (X)  c io. _J _f H - t - _» cn to to 3 3 > > 3 IX 3 1 C- O  -f  x i d  v,  c o o 'o x : L3 O X ,< > I  O *3  K  LU O LL U \j  H-  t-  O  rr u> to H* 3 3 f-  92f  LU  _J  LL  rvj  OO U J  ~i  IL  -*  < o to O  CC V I CL < o O cr to a. < u O CCVO Q. < u o cr oo a.<r u  LL' LL)  hm  UJ  r\j  UJ  r-l  \—  CC. CO L U  —J  o  or co  UJ  _J f-  ct  CO  LU  CT  CO  UJ  •J _J w  u. a: U J  LU  CC U J L ) h-  o  LL  a: LU cr U J  Lt-  OCU J LU OC U J t_) r-l  LL  CC  ro  1X1  CC U J  U  LU  OC U J  L_)  =2 _J  < CO  LU  a: a  a  cr  >- ol~ C L o CC > u -j >- L L _ l o >- LL - J D o  CC  ZD  CC  3 > u_  (/I  o o o o  o O c o  vo  ^ r>  <  H- rr,  >  r - rvj  — K  > K r-l  1  < u n fc  rO <N  _1  I o Xo Xo Xc  r—  C r~  <l  oX in  1-  ^ IT  —J  LTi  CC  > >  to  r-r  a.  -J  _J o  vo  <  c/> CL  J—  to  m  LO  -  CL  a rg a. r-4 to  G  UJ  U  CC LU  cr <  O  a : LU  30 <  ^  < o rr LU  <  <r cr H  O  i i  c  r-  O  r-  cr U J  J < CD  O<  D _) < CD  i/>  < I —<  cf.CL a: t~  a <  cr. <r  ~  _J  o  CL  <t CC  cr  ^  DC  Z> cr cr Q  OO  CD o : O  Or-  o  cr:  o LU > CO  <  H  z  1  t—  o LU > r>!  o LJ  >  o  CC  rfi cn o < <s u u  O  LU  O rH ~* o  O r-l  i— n  cr <?  CC  CO  Z> cr m (X G t~ a  O  <  cr  o.u.> UJ  o o  o  -J  c: _ J  IL  _J  a  m  O C 10 o  I  111  LU V  V  LL  LI"  a.  a  IL _  c  <: is. jtyi t 0 17" >  rr rr  l-  U.J  ry.j~  c >:  r  LO  C  -j cr LL  rH !f\J  H  U-  IT' t o t  cr <  lO O •  f\J U J _J  'lil  O J O  rH fV IL  LL  U.  LL  or. cr CL O O O C <T  l i ' U.' U J LiJ I  a  a v i u.  O. —  <T * U  IL  < <f _J _ J .u a c UJ ^: r f t o |co t o _J  CJ I J H I T  c  I  U.J  h- 'LL  to  O  I LU Ul  x o —  iy,o <: 3 . a : ^ o o O O x ^^ ?: -3 ~ I T a: ^ 5 :  u rr it' a  UJ  UJ  t o |oo  92g  i O  < O  i/> I  CL W  Q- •  o LL LO  cr re  CfT.  i  •— o X  t- (  cr  u_ :  UJ  t  LLI L U U (J  C  <: <I  LS  < O  LO  UJ UJ . O  UJ  C  I  <t <r  O —'  I I I I I r-l |—  I I I I  <T. f ~ U • cr (  < — cr ;  UJ  O  _J  cr  Q  < < < <  I  I  1-1 .  UJ  ,  • -J  a.-  rj  II  n  _J  I/I 1/1 C  ! .  O O f CL  O  O c_o O O C O (J O O i  i  £  ; r.r o < . C r. ( ) u u i  -«h  77  7'  rr L_> *7T rt" i : —. •— < 2: • _> U. j'.c L : ' ! O L O 7T -X U C h D i ty. <r> rr ]LL. LL P?- Z IU. LL x 77 77  > 73  !  T  y :r x .2'  T:  :  5"  1  ' rs. a r-i 1  i/i H  r j 'n  c  :- u c: ' O J <r ^ 7* -T x. rr o o 't/i oo t_;  r c- c i- >—  777 77 >77 77 t--n (  i  y A A G G S S E 1. L F F R 3 T P c  W A AT I E RR P P R E O O S O O E L C L L L E L L C T E F  2,  T~\  SSHC°2 RURAL 1 T  VNGP^T  VNT"EC  L  MMSPRK  L  MNj'-PRK M \ ' G 3 L  C  MNRDEV UPRK1 UPRK2 SPRK'I SPRK2 MPRKI MPRK2 GOLF I GOLF 2 CR9FV1 CROEV2 MN'iJRD  L L  L G G G G G G G G G G L L L L L L L L L  MNR- 0 VN PA RK MNRA I L MM A I R MNHOSP MNSK'JL MNGOVT HOSP1 HOS°2 >j SKIJL 1 G SMJL 2 GOVT1 r GOVT 2 G HCl'JSAL E AGHOME E M X U"'! F '' G evp.-;pp E RALAMC L c TAG"IC E T FEE D TCOMRC E TOFFIC • E F TTRA ! S TUCNR E TUSHOP E 7  V  VO  92i  00  < o  O Of  o  LL LO  o  CO  CO L U CL  <  cr. I l l  LLI  i -  LO  LU  C  Cf. LLI Li. LU  <  LU  CC L U O  X t— C L  < o  o  Cr: CO L U — J  o  <  —J LL  -  _  c  _J -J  C O LLI CL  Q ' O.  o _J c- _ J  o  of n .  CO L U - J  u. H  LO  LU  _ J LL, r<1  • O O O O CD C U J ; —i (\j I CO U I  (>" • h -  r; cv v > ^  LL; CI.  CL  LO vf  C  I  U i .'Ji UJ C i / i .o n- u.i • .if a : 11^ " LU U LL  L'  O  a  LL'  [u r - . r o  L_" ;(_) L L  !i-  C  U.  C  UJ U i rU I _J ItlJ O cr ' c o c u i cr »- I L •— <j i n ;  • <f  LU u .  U I Ll >  U h  U'  I uu.  _ J <T l_l I U I— CC O III co • c _ c 'co —• i if a V , a • ' ' x o o '<r a: ~ ' LU  111 ' a;  1  EXECUTOR. SU'ri«Y SYNWl  OF  MPS/360  V2-M5  "  LESS  Y  . 0000)1  X  .000010  W  ~  V  . 000100  COUNT  THA^  .000001  THRU  _P_0_?_.  .009999  ;  .099999  ' 11 29~  1  1.000000  1. 0 0 0 0 0 0  344  A  1.000001  10.000000  84  1 0 0 . 0 0 0 0 0 0  . 55  C  10,000051  1,000,00000!  ~F. F G  '  1 00. 000001  0  i_,jOJ;O.J3GOOO_0  70  10,000.000000  33  1 0 , 0 0 0 . OOOOO1  ICO , 0 0 0 . 0 0 0 0 0 0  43~  1 0 0 , OOP. CC0001  1 , 0 0 0 , 0 0 0 . 0_0_0000  10  GREATER  IINCL.RHS)  3  .999999  "  73/222  .000999  .010030  "H  =  !  . 10000 0  '  3  .000099 :  . 00100 0  U "T  PAGE  MATRI X  RANGE  Z  :  THAN  1 , 0 0 0 , 0 0 0 . 000000  _ to  l_J.  93 Appendix I I  CODING EXPLANATION AND DATA DESCRIPTION FOR LANGLEY MODEL MATRIX  V a r i a b l e numbers 1 t h r o u g h 137 r e p r e s e n t row v e c t o r s i n t h e m a t r i x w h i l e v a r i a b l e numbers 138 t o 259 a r e a c t i v i t y c o l u m n s .  VARIABLE COMPUTER NUMBER CODE 1-12  VARIABLE NAME objective  functions  DESCRIPTION o r COMMENT  twelve a l t e r n a t e o b j e c t i v e functions. Each f u n c t i o n has a d i f f e r e n t s e t o f goal wts. a l l goals i n t h i s objective f u n c t i o n have a w e i g h t o f ± 1. T h i s does n o t mean t h a t a l l t e n goals are equally important. I t means t h a t the u n i t s used i n t h e t e n d i f f e r e n t indices are equal i n terms o f t h e i r c o n t r i b u t i o n s t o the composite objective function. Since an a t t e m p t was made t o express a l l the d i f f e r e n t u n i t s i n t e r m s o f $1 u n i t s , o b j e c t i v e f u n c t i o n "C" merely r e f l e c t s t h i s attempt and was u s e d a s a s t a r t i n g point i n determining f i n a l goal weights.  2-7, 9-10  CPOL=.8 CPOL=.9 CPOL=l.l CPOL=1.2 CPOL=1.5 CPOL=2.0 CPOL=5 CPOL=10  These e i g h t o b j e c t i v e f u n c t i o n s p l a c e w e i g h t s o f -.8, - . 9 , - 1 . 1 , -1.2, - 1 . 5 , -2.0, -5.0, and -10.0 r e s p e c t i v e l y on a l l f i v e p o l l u t i o n g o a l s . The w e i g h t s on a l l t h e o t h e r g o a l s r e m a i n s e t a t +1.0. T h e s e o b j e c t i v e f u n c t i o n s were used t o determine the e f f e c t s of varying the r e l a t i v e importance o f t h e p o l l u t i o n g o a l s i n the model.  CGRPONLY  This objective function leaves the g r o s s r e g i o n a l production g o a l w e i g h t s e t a t +1.0 b u t s e t s  P4  the w e i g h t s on a l l t h e r e m a i n i n g goals a t zero. This function was used t o determine t h e amount of i n f l u e n c e that a l l the e x t e r n a l i t y g o a l s have on t h e o p t i m a l allocation. 11-12  CLIES=5 CLIES=10  13  TOTPOP  These two o b j e c t i v e f u n c t i o n s p l a c e w e i g h t s o f +5.0 and +10.0 r e s p e c t i v e l y on t h e t h r e e posi t i v e e x t e r n a l i t y goals that are concerned w i t h m a x i m i z i n g r e c r e a t i o n a l and l i e s u r e - t i m e b e n e f i t s . The w e i g h t s on a l l the o t h e r g o a l s remain s e t a t +1.0. These two f u n c t i o n s were used t o d e t e r m i n e t h e e f f e c t s of i n c r e a s i n g the r e l a t i v e importance o f these t h r e e r e l a t e d aoals. t o t a l population (of L a n g l e y ) force  constrains t o t a l population t o t h e d e s i r e d l e v e l . TOTPOP  was i n i t i a l l y  s e t a t 21,936.  14  LABOUPv  labour  15  ALLAND  allocate a l l land  c o n s t r a i n s the a l l o c a t i o n of l a n d (between 34 a l t e r n a t e l a n d uses) t o L a n g l e y ' s 76730 a c r e s (see T a b l e A ) .  16  SDEVEL  suitable for development  c o n s t r a i n s development t o c l e a r e d , non-flood p l a i n , gently s l o p e d l a n d . E q u a l s 54663 a c r e s + BYCLIR acreage (see T a b l e A ) .  17  NEEDCLIR  needs l a n d clearance  c o n s t r a i n s those l a n d uses that require cleared land to the t o t a l amount o f c l e a r e d l a n d : 64,861 a c r e s + BYCLIR acreage (see T a b l e A ) .  NEEDFLUD needs f l o o d  protection  constrains labour force t o 39% o f t o t a l p o p u l a t i o n .  constrains that require or non-flood t o t a l amount land: 70530 acreage (see  1971 p o p u l a t i o n o f L a n g l e y as r e p o r t e d Canada (Ottawa: Oueen's P r i n t e r , 1971). 2  those l a n d uses flood protected p l a i n l a n d t o the o f t h i s tvpe o f a c r e s + BUYFLUDT Table A ) . by Census o f  P a r t i c i p a t i o n r a t e i n t h e Lower M a i n l a n d i n 1970 as r e p o r t e d i n The Lower Mainland's Economy: Trends and P r o s p e c t s (Vancouver: G r e a t e r Vancouver R e g i o n a l D i s t r i c t , 1970), p. I T .  95  19  ALLFLUD  total flood protection  20  SDAIRY  suitable for dairying  c o n s t r a i n s DAIRY a c r e a g e t o that a g r i c u l t u r a l land considered s u i t a b l e f o r d a i r y i n g ( i n c l u d i n g growing o f f e e d ) : 66030 a c r e s (see T a b l e A ) .  21  SGRDEN  suitable for market gardening  c o n s t r a i n s GARDEN a c r e a g e t o that a g r i c u l t u r a l land conside r e d s u i t a b l e f o r market garde n i n g : 5080 a c r e s + BUYFLUDl acreage (see T a b l e A ) .  22  SLNDAG  suitable f o r land based a g r i c u l t u r e  c o n s t r a i n s l a n d based a g r i c u l t u r e (DAIRY o r GARDEN) t o that land suitable f o r land based a g r i c u l t u r : 62260 a c r e s + BUYFLUDT a c r e a g e ( s e e T a b l e A ) .  23-24  FEEDl,  2 t h e f i r s t two s t e p s o f FEEDT's s t e p v a l u e added function  25  SINDUS  suitable for i n d u s t r i a l use  constrains total industry (TOTIND) t o t h e amount o f l a n d considered suitable f o r indust r i a l u s e : 2680 a c r e s ( s e e Table A ) .  26  SPARK  suitable for a major park/ recreational reserve  c o n s t r a i n s major p a r k / r e c r e a t i o n a l r e s e r v e acreage t o t h e amount o f f o r e s t e d l a n d i n L a n g l e y : 11869 a c r e s — B Y C L I R acreage (see T a b l e A ) .  27  MAXFLUD1 maximum f l o o d protection of A.R.D.A. c l a s s e s " o r g a n i c " and "excellent" land  28  MAXFLUD2 maximum f l o o d protection of A.R.D.A. d l a s s  c o n s t r a i n s t h e t o t a l amount o f f l o o d p r o t e c t i o n (BUYFLUDT) t o any d e s i r e d l e v e l . Since i t i s not economically relevant to p r o t e c t o n l y a p o r t i o n o f a ibflood p l a i n , zero a c r e s o r 6200 a c r e s (see T a b l e A) a r e the o n l y a l l o w a b l e v a l u e s f o r BUYFLUDT.  c o n s t r a i n s t h e f i r s t two s t e p s o f FEEDT's s t e p v a l u e added f u n c t i o n t o 100 a c r e s e a c h (see T a b l e C ) .  c o n s t r a i n s a buy f l o o d p r o t e c t i o n a c t i v i t y (BUYFLUDl) t o the p o r t i o n o f the f l o o d p l a i n t h a t c o n s i s t s o f o r g a n i c (peat and muck) s o i l s o r A.R.D.A. "excellent" agricultural soils (see T a b l e A ) . c o n s t r a i n s t h e s e c o n d buy I flood protection activity (BUYFLUD2) t o t h e p o r t i o n o f  96  t h e f i r s t two s t e p s o f OFFICT's demand f u n c t i o n  c o n s t r a i n s t h e f i r s t two s t e p s o f O F F I C T ' s s t e p v a l u e added f u n c t i o n (see T a b l e C ) .  t h e f i r s t two s t e p s o f TRANST's demand f u n c t i o n  c o n s t r a i n s t h e f i r s t two s t e p s o f TRANST's s t e p v a l u e added f u n c t i o n (see T a b l e C ) .  t h e f i r s t two s t e p s o f UCNRT's demand f u n c t i o n  c o n s t r a i n s t h e f i r s t two s t e p s o f UCNRT's s t e p v a l u e added f u n c t i o n (see T a b l e C ) .  t h e f i r s t two s t e p s o f USHOPT's demand f u n c t i o n  c o n s t r a i n s t h e f i r s t two s t e p s o f USHOPT's s t e p v a l u e added f u n c t i o n (see T a b l e C ) .  t h e f i r s t two s t e p s o f SCNRT's demand f u n c t i o n  c o n s t r a i n s t h e f i r s t two s t e p s o f SCNRT's s t e p v a l u e added f u n c t i o n (see T a b l e C ) .  t h e f i r s t two s t e p s o f SSHOPT's demand f u n c t i o n  c o n s t r a i n s t h e f i r s t two s t e p s o f SSHOPT's s t e p v a l u e added f u n c t i o n (see T a b l e C ) .  t h e f i r s t two s t e p s o f RURALT's demand f u n c t i o n  c o n s t r a i n s t h e f i r s t two s t e p s o f RURALT's s t e p v a l u e added f u n c t i o n (see T a b l e C ) .  minimum g r e e n b e l t acreage  e n s u r e s t h a t t h e r e w i l l be a t l e a s t 1 acre o f green b e l t f o r every 10-people i n the p o p u l a t i o n (see T a b l e B ) .  minimum t o t a l recreation  e n s u r e s t h a t t h e r e w i l l be a t l e a s t 3 a c r e s t h a t c a n be u s e d f o r r e c r e a t i o n f o r e v e r y 100 p e o p l e i n t h e p o p u l a t i o n (see Table B).  minimum u r b a n  minimum park  park  suburban  minimum m a j o r park/recreational reserve  e n s u r e s t h a t t h e r e w i l l be a t l e a s t 1 a c r e o f urban park (UPRKT) f o r e v e r y 395 p e o p l e t h a t l i v e i n urban h o u s i n g (see Table B). e n s u r e s t h a t t h e r e w i l l be a t l e a s t 1 a c r e o f suburban park (SPRKT) f o r e v e r y 395 p e o p l e t h a t l i v e i n suburban housing (see T a b l e B ) . e n s u r e s t h a t t h e r e w i l l be a t l e a s t 1 a c r e o f major park/ r e c r e a t i o n a l r e s e r v e (MPRKT) f o r e v e r y 3 33 p e o p l e i n t h e t o t a l p o p u l a t i o n (see T a b l e B ) .  98  99  100  85  MXUNEM  maximum unemployment  s e t s a n u p p e r l i m i t on unemployment o f t h e l a b o u r f o r c e a t 10%  86  EMPOPP  employment opportunities  s e t s t h e employment d e n s i t i e s ( i n workers p e r a c r e ) o f any a c t i v i t y t h a t p r o v i d e s employment (see T a b l e D ) .  budget  e n s u r e s t h a t t h e y e a r l y munic i p a l e x p e n d i t u r e s on r e s i d e n t i a l , commercial and i n d u s t r i a l l a n d u s e s do n o t e x c e e d t h e t a x r e v e n u e s g e n e r a t e d by these a c t i v i t i e s . Municipal e x p e n d i t u r e s on r e s i d e n t i a l land u s e s a r e assumed t o i n c l u d e t h e costs of providing theresidents with schools, parks, h o s p i t a l s , roads, e t c . The n e t m u n i c i p a l r e v e n u e ( t a x r e v e n u e minus expenditures) f o r these a c t i v i t i e s are s e t as f o l l o w s : ACTIVITY NET REVENUE (?/ACRJD  87  balancer  TOTCOM TOTIND APART 5/ACRE 1AGRE SACRE AGHOME 88  TAGRIC  total agriculture land  +500.00 +1000.OO -25.OO -10.OO 0.00 +50.00 +200.OO J  3  4 4  4  4  4  sums DAIRY, GARDEN & FEEDT i n t o TOTAG  These r e p r e s e n t p e r s o n a l e s t i m a t e s . 4 Based on d a t a i n Economic A s p e c t s o f Urban Sprawl (New W e s t m i n s t e r : Lower M a i n l a n d R e g i o n a l P l a n n i n g B o a r d , 1 9 5 6 ) , p . 42.  101  t o t a l feed l o t acreage  sums FEEDl, 2 and 3 i n t o FEEDT.  t o t a l commercial acreage  sums UCNRT, USHOPT, SCNRT, SSHOPT and RURALT i n t o TCOMRC,  total office s e r v i c e s acreage  sums OFFICl, 2 and 3 i n t o OFFICT  t o t a l transport s e r v i c e s acreage  sums TRANS1, 2 and 3 i n t o TRANST.  t o t a l urban neighbourhood r e t a i l services  sums UCNRl, 2 and 3 i n t o UCNRT.  t o t a l urban shopping c e n t e r  sums USHOP1, 2 and 3 i n t o USHOPT.  t o t a l suburban neighbourhood r e t a i l services  sums SCNRl, 2 and 3 i n t o SCNRT.  t o t a l suburban shopping c e n t e r  sums SSH0P1,2 and 3 i n t o SSHOPT.  total rural r e t a i l services  sums RURALl, 2 and 3 i n t o RURALT.  total industrial acreage  sums FOOD, LIGHT, HEAVY, and WOOD i n t o TOTIND.  total recreational acreage  sums UPRKT, SPRKT, MPRKT, GOLFT, and CRDEVT i n t o TOTREC.  t o t a l urban acreage  sums UPRKl, 2 and 3 i n t o UPRKT.  park  t o t a l suburban park acreage  sums SPRK1, 2 and 3 i n t o SPRKT  t o t a l major park acreage  sums MPRKl, 2 and 3 i n t o MPRKT  t o t a l g o l f course acreage  sums GOLF1, 2 and 3 i n t o GOLFT  t o t a l commercial recreationl development  sums CRDEV1, 2 and 3 i n t o CRDEVT  t o t a l transportasums URBRD, SUBRD, RURRD, t i o n , u t i l i t i e s and PARK, RAILRD, and AIRPRT i n t o communications TOTRAN acreage  102 106  TRESID  total residential acreage  sums APART, 5/ACRE, 1ACRE 5ACRE, a n d AGHOME i n t o TOTRES,  107  TINSTI  total institut i o n a l acreage  sums HOSPT, SKULT, and GOVTT i n t o TOTINS.  108  THOSP  total hospital land acreage  sums HOSP1, 2 a n d 3 i n t o HOSPT.  109  TSKUL  t o t a l school acreage  sums SKUL1, 2 and 3 SKULT.  110  TGOUT  t o t a l other institutional acreage  sums GOVT1, 2 and 3 i n t o GOVTT.  111  TFLOOD  t o t a l flood protection bought  sums BUYFLUDl a n d 2 BUYFLUDT.  112  TFREEREC t o t a l " f r e e " recreational acreage  113  TGRBELT  114  TGRSPACE t o t a l g r e e n acreage  115  TAGSELF  116  FFREEREC " f r e e " recreation coefficients  t o t a l green acreage  into  into  sums FREEREC1, 2 and 3 i n t o FREERECT. belt  sums GRBELT1, 2 and 3 i n t o GRBELTT.  space  sums GRSPACEl, 2 and 3 i n t o GRSPACET.  t o t a l acreage contributing to agricultural self sufficiency  sums AGSELF1, 2 and 3 i n t o AGSELFT.  sums t h e t o t a l a c r e s i n l a n d a c t i v i t i e s 1-89 w h i c h p r o v i d e f r e e p u b l i c r e c r e a t i o n u s e and p l a c e s t h e sum i n t o FREERECT. An a c r e o f u r b a n p a r k , s u b u r b a n park o r major park a l l c o n t r i b u t e 1 a c r e t o FREERECT, w h i l e an a c r e o f s c h o o l l a n d c o n t r i b u t e s o.6 a c r e s t o F R E E R E C T . 5  F r a c t i o n o f t o t a l s c h o o l a c r e a g e t h a t i s gym a r e a , p l a y i n g a r e a s o r p l a y i n g f i e l d s a s r e p o r t e d i n R e p o r t On S c h o o l and P a r k S t a n d a r d s ( V a n c o u v e r : C i t y o f V a n c o u v e r P l a n n i n g D e p a r t m e n t , 1957) , p. 3.  103  FRGRBELT g r e e n b e l t coefficients  sums t h e t o t a l a r e a i n l a n d a c t i v i t i e s 1-89 w h i c h c o n s t i t u t e s a green b e l t . The sum i s p l a c e d i n GRBELTT. A l l o f DAIRY, GARDEN MPRKT, a n d GOLFT make up t o t a l green b e l t acreage.  FGRSPACE g r e e n s p a c e coefficients  sums t h e t o t a l a r e a i n l a n d a c t i v i t i e s 1-89 w h i c h c o n s t i t u t e s green space (lawn). The sum i s p l a c e d i n GRSPACET. A l l of u r b a n p a r k , s u b u r b a n p a r k and hobby f a r m s p l u s o n e - q u a r t e r o f 1-acre r e s i d e n t i a l l o t s , farm h o m e s i t e s , and h o s p i t a l g r o u n d s p l u s one-tenth o f apartment l o t s , 5 / a c r e r e s i d e n t i a l l o t s , and other i n s t i t u t i o n a l acreage a r e assumed t o be g r e e n s p a c e .  FAGSELF  sums t h e t o t a l a r e a i n l a n d a c t i v i t i e s 1-89 w h i c h c o n t r i b u t e s to a g r i c u l t u r a l s e l f s u f f i c i e n c y The sum i s p l a c e d i n AGSELF. A l l o f DAIRY, GARDEN, and FEEDT contribute to agricultural self sufficiency.  agricultural self sufficiency coefficients  F R E E R E C l t h e f i r s t two and 2 steps of FREERECT's demand f u n c t i o n  s e t s t h e f i r s t two s t e p s o f FREERECT's s t e p demand f u n c t i o n e a c h a t 1 a c r e p e r 250 p e o p l e i n t h e t o t a l p o p u l a t i o n (see Table C ) .  GRBELT1 and 2  s e t s t h e f i r s t two s t e p s o f GRBELTT's s t e p demand f u n c t i o n e a c h at> 1 a c r e p e r p e r s o n i n t h e total population.  t h e f i r s t two steps of GRBELTT's demand f u n c t i o n  GRSPACEl t h e f i r s t two s t e p s s e t s t h e f i r s t two s t e p s o f GRSPACET's s t e p demand f u n c t i o n and 2 o f GRSPACET's e a c h a t 1 a c r e p e r 2 50 p e o p l e demand f u n c t i o n i n t h e t o t a l p o p u l a t i o n (see Table C ) . AGSELF1 and 2  t h e f i r s t two s t e p s s e t s t h e f i r s t two s t e p s o f o f AGSELFT's AGSELFT's s t e p demand f u n c t i o n demand f u n c t i o n each a t 1 acre p e r person i n the t o t a l p o p u l a t i o n (see T a b l e C ) .  MXGRP  gross r e g i o n a l product maximization  sums t h e y e a r l y c o n t r i b u t i o n s t h a t t h e v a r i o u s a c t i v i t i e s make towards gross r e g i o n a l product. T h i s sum i s p l a c e d i n GRP. A l l the c o e f f i c i e n t s i n t h i s l i n e are i n t e r m s o f v a l u e added d o l l a r s per acre o f a c t i v i t y . Activities  104a  t h a t f a c e a s t e p v a l u e added f u n c t i o n w i l l have t h r e e c o e f ficients i n this restraint line, one f o r e a c h s t e p o f t h e i r v a l u e added f u n c t i o n (see T a b l e E ) . MNAIRPOL m i n i m i z e a i r pollution MNWATER minimize water pollution MNREFUSE m i n i m i z e g r o u n d pollution MNNOISE minimize noise pollution MNSIGHT minimize s i g h t pollution  T h e s e f i v e a c c o u n t i n g rows sum the y e a r l y c o n t r i b u t i o n s t h a t the v a r i o u s a c t i v i t i e s make t o w a r d s i n d i c e s o f a i r p o l l u t i o n , water p o l l u t i o n , ground wastes, noise p o l l u t i o n and s i g h t p o l l u t i o n r e s p e c t i v e l y (see T a b l e F ) .  MXFREREC m a x i m i z e " f r e e " recreational acreage  T h i s a c c o u n t i n g row i n t r o d u c e s a s t e p demand f u n c t i o n i n t o FREERECT b y s e t t i n g d i f f e r e n t yearly contribution coefficients f o r a c r e i n FREERECl c o n t r i b u t e s 10000 u n i t s t o MXFREREC,while an a c r e i n FREEREC2 c o n t r i b u t e s o n l y 10 u n i t s . An a t t e m p t was made t o e q u a t e t h e s e i n d e x u n i t s with d o l l a r s of b e n e f i t received (see T a b l e C ) .  MXGRBELT m a x i m i z e g r e e n b e l t acreage  T h i s a c c o u n t i n g row i n t r o d u c e s a s t e p demand f u n c t i o n i n t o GRBELTT by s e t t i n g d i f f e r e n t steps. E v e r y a c r e i n GRBELTl c o n t r i b u t e s 100 u n i t s t o MXGRBELT w h i l e an a c r e i n GRBELT2 c o n t r i b u t e s 5 0 u n i t s a n d one a c r e i n GRBELT 3 c o n t r i b u t e s o n l y 25 units. An a t t e m p t was made t o equate these index u n i t s with d o l l a r s o f b e n e f i t r e c e i v e d (see Table C ) .  MXGRSPAC m a x i m i z e g r e e n space acreage  T h i s a c c o u n t i n g row i n t r o d u c e s a s t e p demand f u n c t i o n i n t o GRSPACET b y s e t t i n g d i f f e r e n t yearly contribution coefficients f o r GRSPACET's t h r e e d i f f e r e n t steps. E v e r y a c r e i n GRSPACEl c o n t r i b u t e s 10000 u n i t s t o MXGRSPACE, w h i l e an a c r e i n MXGRSPACE2 c o n t r i b u t e s 5000 u n i t s and an a c r e i n GRSPACE c o n t r i b u t e o n l y 10 u n i t s . An a t t e m p t was made t o e q u a t e t h e s e i n d e x u n i t s with d o l l a r s of benefit received (see T a b l e C ) .  104b  137  MXAGSELF m a x i m i z e a g r i cultural selfsufficiency  138  TOTAG  total agriculture acreage  139  DAIRY  d a i r y farm acreage  140  GARDEN  acreage i n market gardens  FEEDl, 2, 3, and T  t o t a l beef acreage  145  TOTCOM  t o t a l acreage i n commerce  146149  OFFIC1, 2, 3, and T  office services acreage  includes medical, dental, l e g a l and o t h e r p r o f e s s i o n a l o r business services. O F F I C l and OFFIC2 a n d OFFIC3= O F F I C T .  150153  TRANS1, 2, 3, and T  transport acreage  i n c l u d e s auto and a l l i e d s a l e s and s e r v i c e s as w e l l a s t h r o u g h highway b u s i n e s s . TRANS1, and TRANS2, and TRANS3= TRANST.  154157  UCNRl, 2, 3, and T  urban neighbourany r e t a i l o u t l e t t h a t c a t e r s hood r e t a i l a c r e a g e t o a s p e c i f i c u r b a n n e i g h b o u r hood. UCNRl, and UCNR2, a n d UCNR3= UCNRT.  158161  USHOP1, 2, 3, and T  urban shopping center acreage  USHOP1, a n d USHOP2, USHOPT  162165  SCNR1, 2, 3, and T  suburban n e i g h bourhood r e t a i l acreage  SCNR1, and SCNR2, and SCNR3= SCNRT.  141144  T h i s a c c o u n t i n g row i n t r o d u c e s a s t e p demand f u n c t i o n i n t o AGSELFT b y s e t t i n g d i f f e r e n t yearly contribution coefficients f o r AGSELFT's t h r e e d i f f e r e n t steps. Every a c r e i n AGSELFl c o n t r i b u t e s 100 u n i t s t o MXAGSELF w h i l e an a c r e i n AGSELF2 c o n t r i b u t e s 50 u n i t s and one a c r e i n AGSELF3 c o n t r i b u t e s o n l y 25 u n i t s An a t t e m p t was made t o e q u a t e these index u n i t s with d o l l a r s o f b e n e f i t r e c e i v e d ( s e e T a b l e C)  includes  feedlot  services  feed  crop  acreage  F E E D l and FEED2 and FEED3= FEEDT ground  floor  space o n l y  and USHOP3=  105  166169  SSHOPL, 2,  3,  suburban shopping center acreage  SSHOPl, and SSHOP2, and SSHOP3= SSHOPT. RURALl, and RURAL2, and RURAL3= RURALT.  and T 170173  RURAL1, 2, 3, and T  rural retail o u t l e t acreage  174  TOTIMD  total industrial acreage  175  FOOD  acreage i n food processing  176  LIGHT  a c r e a g e i n manufacturing  177  HEAVY  acreage i n heavy industry  178  WOOD  a c r e a g e i n wood products  179  TOTREC  total recreational acreage  180183  UPRK1, 2, 3, and T  urban park  SPRKl, 2, 3, and T  suburban acreage  188191  MPRK1, 2, 3, and T  major park/ recreational reserve acreage  MPRK1, and MPRK2, a n d MPRK3= MPRKT  192195  GOLFl, 2, 3, and T  g o l f course acreage  GOLFl, GOLFT  CRDEVl, 2, 3, and T  commercial r e c r e a t i o n a l development acreage  sports stadiums, p l a y l a n d s , drive-in theaters, e t c .  TOTRAN  t o t a l acreage i n transportation, c o m m u n i c a t i o n s , and utilities  184187  196199  200  acreage  park  includes  construction  UPRK1, and UPRK2, and UERK3= UPRKT  , SBRK1, and SPRK2, and SPRK3= SPRKT  and GOLF2, a n d GOLF3=  106 road  i n c l u d e s communication u t i l i t y r i g h t o f way.  and  i n c l u d e s communication u t i l i t y r i g h t o f way.  and  acreage  i n c l u d e s communication u t i l i t y r i g h t o f way.  and  parking  does n o t i n c l u d e r o a d - s i d e parking.  201  URBRD  urban  acreage  202  SUBRD  suburban acreage  203  RURRD  rural  road  204  PARK  off-street acreage  205  RAILRD  r a i l r o a d r i g h t of way and r a i l y a r d acreage  206  AIRPRT  airport  207  TOTRES  total residential acreage  208  APART  acreage of apartments  209  5/ACRE  acreage i n h i g h 5 l o t s / a c r e , e x c l u d i n g roads d e n s i t y s i n g l e fam- and l a n e s . C o n s t i t u t e s urban i l y dwellings housing.  210  1ACRE  1 l o t / a c r e , e x c l u d i n g roads a c r e a g e i n lowd e n s i t y s i n g l e fam- and l a n e s . C o n s t i t u t e s sub~^_ i l y dwellings urban housing.  211  5 AC RE  hobby f a r m  212  AGHOME  f a r m home and homesite  213  TOTINS  total institutional acreage  214217  HOSP1, 2, 3, and T  hospital  218221  SKUL1, 2, 3, and T  school  222-, 225  GOVT1, 2, 3, and T  other i n s t i t u t i o n a l acreages  road acre-  acreage  low-rise  acreage  acreages  acreages  3 s t o r y , 35-unit apartments, c o n s t i t u t e s urban housing.  I n c l u d e s hobby f a r m s , a v e r a g e 1 l o t / 5 a c r e s , e x c l u d i n g roads and l a n e s . Constitutes rural housing. 1 acre area. r u r a l housing.  Constitutes  Includes grounds. HOSPLl + HOSP2 + HOSP3 = HOSPT. Includes grounds. SKULl SKUL2 + SKUL3 = SKULT.  +  GOVTl + GOVT2 + GOVT3 = GOVTT.  107 LDSLAK  vacant land  BUYFLUDT buy f l o o d p r o t e c t 1, and 2 i o n a c t i v i t i e s  Since f l o o d p r o t e c t i o n i s i n d i v i s i b l e , BUYFLUDT must come i n at zero or maximum. BUYFLUDT equals BUYFLUDl + BUYFLUD2.  BYCLIR  buy land ance  Only a p p l i e s t o t h a t land which i s p r e s e n t l y wooded.  TOTPOP  t o t a l Langley population  LABOUR  labour f o r c e available  UNEMP  t o t a l unemployed  clear-  FREEREC1 " f r e e " r e c r e a t i o n a l T o t a l acres i n l a n d a c t i v 2, 3, acreage i t i e s 138-226 which c o n f e r an and T e x t e r n a l i t y by v i r t u e of being open to the p u b l i c f o r r e c r e a t i o n a l use at no c o s t . FREERECl + FREEREC2 + FREEREC 3 = FREERECT. GRBELT1, green b e l t acreage 2, 3, and T  Large t r a c t s (greater than 50 acres) of v e g e t a t i o n covered l a n d . Equals the t o t a l acres i n l a n d a c t i v i t i e s 138-226 which c o n f e r an e x t e r n a l i t y by c o n s t i t u t i n g a green b e l t . GRBELT1 + GRBELT2 + GRBELT3 = GRBELTT.  GRSPACEl green space acreage Small p l o t s of lawn and kept 2, 3, grounds. Equals the t o t a l acres and T i n land a c t i v i t i e s 138-226 which c o n f e r an e x t e r n a l i t y by v i r t u e of being green spaces. GRSPACEl + GRSPACE2 + GRSPACE3 = GRSPACET. AGSELF1, t o t a l acreage which T o t a l acreage i n land a c t i v 2, 3, c o n t r i b u t e s towards i t i e s 138-226 which c o n f e r an and T a g r i c u l t u r a l s e l f - e x t e r n a l i t y by v i r t u e of c o n t r i b u t i n g towards a g r i c u l t u r a l sufficiency. self-sufficiency. GRP  t o t a l gross regiona l product (valueadded)  Goal accounting  column,  AIRPOL  total air pollution (index u n i t s )  Goal accounting  column.  108 253  WATERPOL t o t a l w a t e r p o l lution (index)  Goal  accounting  column.  254  REFUSE  Goal  accounting  column.  255  NOISEPOL t o t a l n o i s e p o l u t i o n (index)  Goal  accounting  column.  256  SIGHTPOL t o t a l v i s u a l pollution (index)  Goal  accounting  column.  257  FREEREC  Goal  accounting  column.  258  GRBELT  t o t a l worth o f f r e e r e c r e a t i o a n l acreage t o t a l worth o f green b e l t acreage  Goal  accounting  column.  259  GRSPACE  total space  Goal  accounting  column.  260  AGSELF  t o t a l worth o f Goal c o n t r i b u t i o n s towards agricultural selfsufficiency  accounting  column.  t o t a l ground pollution (index)  worth o f green acreage  109 Table  A  AREAS OF LAND S U I T A B I L I T I E S IN THE CITY AND DISTRICT OF LANGLEY (acres)  t o t a l area area unsuited f o r any use (marsh, water, rock outcrops etc.) t o t a l area t o be a l l o c a t e d : upper l i m i t o f ALLAND. (1) - (2) = (3) area o f "excellent" s o i l s (ARDA classes 1, 2, organic) i n flood p l a i n : upper l i m i t of MXFLUD1 area o f "good" s o i l s (ARDA classes 3, 4) i n f l o o d p l a i n : upper l i m i t o f MXFLUD2. . . t o t a l area o f 1948 flood p l a i n : upper l i m i t o f ATJJFLUD. (4) + (5) = (6) area with greater than 15% slope area o f uncleared land: upper l i m i t o f MXCLIR. . t o t a l area suitable f o r development: upper l i m i t o f SDEVEL. (1) - (6) - (7) - (8) = (9) t o t a l area suitable f o r any land use requiring cleared land: upper l i m i t o f NEEDCLIR. (1) - (8) + BYCLIR = (10) . . . BYCLIR t o t a l area suitable f o r any land use requiring f l o o d protection: upper l i m i t o f NEEDFLUD. (1) - (6) + BUYFLUDT = (11) . . BUYFLUDT ARDA land c l a s s i f i c a t i o n acreages: ARDA 1 and 2 ("excellent") ARDA 3 and 4 ("good") ARDA 5 and 6 ("fair") ARDA 7 ("poor") ARDA organic ("peat and muck s o i l s " ) . . t o t a l ARDA acreage t o t a l suitable f o r d a i r y i n g : upper l i m i t o f SDAIRY. (12) + (13) = (16) t o t a l suitable f o r market gardening: upper l i m i t o f SGRDEN. (12) + (13) = (17) . . t o t a l suitable f o r land based a g r i c u l t u r e : upper l i m i t o f SLNDAG. (12) + (13) + (15) = (18) t o t a l suitable f o r major park/recreational reserve: upper l i m i t o f SPARK. (8) - BYCLIR = (19) . . . . -BYCLIR t o t a l area suitable f o r i n d u s t r i a l use: upper l i m i t o f SESTDUS. .  82460  a  (1)  5730  a  (2)  76730  (3)  , 243CT , . 377CT , 620(T 2688° . 11869  (6) (7) (8)  54663  (9)  + 64861  (10)  + 70530  (11)  5080 60950 8370 5730 . 2430 82460  a a 3 a a  (4) (5)  (12) (13) (14) (2) (15) (1)  66030  (16)  7510  (17)  68460  (18)  + 11869  (19)  2680  (20)  .  ^ . L . Lee, Regional Farmland Study (Abbotsford, B.C.: Central Fraser V a l l e y Regional D i s t r i c t Planning Department, 1972), p. 5.  110 Table A  cont'd  b Ibid., planimetric analysis of map of Langley flood p l a i n . c The Lower Mainland Looks Ahead (New Westminster, B.C.: Lower Mainland Regional Planning Board, 1963), p. 26. d Equals area o f unimproved farm land as reported by the LVstunion Bureau o f S t a t i s t i c s , A g r i c u l t u r a l Census of Canada (Ottawa: Queen's P r i n t e r , 1966) e Space f o r Jjndustry: Summary Report (Vancouver: Greater Vancouver Regional D i s t r i c t Planning Department, 1971), p. 8, 20.  Ill Table B MINIMUM LAND REQUIREMENTS USED IN THE LANGLEY MODEL  MNOFIC: minimum o f f i c e services land requirements medical-dental o f f i c e requirements: 2897 sq. ft./lOOO population (46% f o r buildings) business and other professional o f f i c e requirements: 3392 sq. ft./lOOO population (49% f o r buildings) combined o f f i c e services requirements: 2994 sq. ft./lOOO papulation (buildings only) o r 1 acre/14925 pop. MNTRAN: minimum transport services land requirements through highway business requirements: 17302 sq. ft./lOOO population (10% f o r buildings) l o c a l highway business requirements: 2429 sq. ft./lOOO population (8% f o r b u i l d i n g s ) auto and a l l i e d sales requirements: 10313 sq. ft./lOOO population (10% f o r buildings) auto and a l l i e d service requirements: 9825 sq. ft./lOOO population (18% f o r b u i l d i n g s ) 3  3  combined transport services requirements: 4256 sq. ft./lOOO population (buildings only) o r 1 acre/10638 pop. MNUCNR: minimum urban neighbourhood r e t a i l land requirements: from a personal estimate that the average market o f an urban o r suburban neighbourhood r e t a i l o u t l e t (approximately 400 sq. f t . ground space) i s approximately 400 persons o r 115 households: 1 acre/43478 population MNUSHP: ntinimum urban shopping center land requirements: town planning standards: 12 sq.j_ft. o f urban o r suburban shopping center o r subcenter per person. 1 acre/3663 population MNSCNR: minimum suburban neighbourhood r e t a i l land requirements: see MNUCNR. MNSSHP: minimum suburban shopping center land requirements: see MNUSHP. MNRURL: minimum r u r a l r e t a i l land requirements: from a personal estimate that r e t a i l area o f 400 square f t . can serve a r u r a l population o f 200 persons o r approximately 60 households: 1 acre/22222 population MNGRBT: miriimum green b e l t acreage requirements: set a approximately the r a t i o o f green b e l t t o people that e x i s t s i n the Lower Eraser V a l l e y today: 1 acre/10 population  112 Table  __  B  cont'd  develpment 1 acre per: h acre o f t o t a l conroercial land use, o r 175 dwelling u n i t s , o r 175 persons i n the labour force, o r 1 acre o f corrmercial r e c r e a t i o n a l development.  MNRAIL: irdnimum r a i l r o a d and r a i l y a r d land requirements: a r b i t r a r i l y s e t a t 1 acre/1 acre i n a corrmercial o r i n d u s t r i a l land use. MNAIR: minimum a i r p o r t land requirements: ^ from a report on land requirements f o r minor a i r p o r t s : h sq. mile/25000 population, o r 1 acre/156 population. MNHOSP: minimum h o s p i t a l land requirements: town planning standards: 4.5 beds/1000 population" and from personal estimate: 200 bed h o s p i t a l requires a minimum o f 4 acres o f land.  __  1 acre/10000 population. MNSKUL: minimum school land requirements: from a report on rrdnimum Langley school needs i n 1961: space needed f o r 1650 secondary p u p i l s and 2544 elementary p u p i l s , and from town planning standards: minimum 4 acres per,250 p u p i l s , and from 1961 Langley population:3 14585 people. n  x  __  1 acre/200 population. MNGOVT: minimum other i n s t i t u t i o n a l land set a r b i t r a r i l y a t : 1 acre/10000 population.  requirements:  HOUSAL: minimum r e s i d e n t i a l land requirements t o ensure that a l l the population i s housed: f i v e average housing types were a r b i t r a r i l y set: APART (low-rise apartment): 35 units/1 acre (including grounds) 5/AGRE (high density single family housing): 5 units/1 acre (including grounds) 1ACRE (low density s i n g l e family housing): 1 unit/1 acre (including grounds) 5ACRE (hobby farm): 1 unit/5 acre AGHCME (farm home and hcmesite): 1 unit/1 acre. a l l o f these housing types have an average density o f 3.5 people/unit. k  AGHCME: land requirements f o r farm homes and homesites: a r b i t r a r i l y provides 1 u n i t o f AGHCME (1 acre) f o r each 80 acres o f dairy farm, 100 acres o f market garden, and 10 acres o f feedlot.  113 Table B cont'd MNTREC: minimum total recreational acreage requirements: set arbitrarily at 1 acre/33 population MNUPRK: minimum urban neighbourhood park land requirements: town planning standards: a irdnimum 4 acre park within 1/3 mile and a minimum 25 acre park within 1 mile (where housing density averages 12 people/acre). This averages to a combined neighbourhood park requirement of 1 acre/395 population. c  MNSPRK: minimum suburaban neighbourhood park land requirements: see MNUPRK. MNMPRK: minimum major park or recreational reserve land requirements: town planning standards: a minimum of 4 acres of major metro park or recreational reserve for every 1000 people: 1 acre/250 population d  MNGOLF: nujiimum golf course land requirements: town planning standards: a f u l l size golf course (150 acres) for every 50,000 people, or 3 acres per 1000 people:^ 1 acre/333 population MNRDEV: irdnimum commercial recreational development land requirements: intensive commercial recreational land requirements: 3650 sq. ft./lOOO population (23% for buildings) extensive ccimmercial recreational land requirements: 58731 sq. ft./lOOO population (28% for buildings) commercial recreational land requirements: 17079 sq. ft./lOOO population (buildings only) or 1 acre/264 pop. MNURD: minimum urban road land requirements: from an estimate made by transportation economist W.G. Waters II (University of British Columbia, Department of Economics) that transportation and communications land uses account to approximately 30% of total urban area. Since residential land use accounts for the bulk of the remaining area, the minimum urban road requirement was set at 1 acre/2.5 acres urban housing. MNSRD: rrdjiimum suburban road land requirements: estimate by W.G. Waters (see MNURD above): 1: acre/4 acres suburban housing. MNRRD: minimum rural road land requirements: personal estimate: 1 acre/100 acres of agricultural land or rural housing. MNPRK: minimum parking land requirements: frcm U.S.A. nationally recognized standard: 4 sq. f t . / l sq. f t . commercial ground floor space, 1 parking space per employee, and from Langley Zoning Bylaw (off-street parking reo^rements) : 1 parking space per dwelling unit, and from personal estimates: 1 parking space = 250 sq. f t . , 1 sq. f t . parking required/1 sq. f t . ccmmercial recreational 3  e  114 Table B cont'd  1975 Metropolitan Tulsa Ccrnmunity Land Needs (Tulsa: Tulsa Metropolitan Area Planning Commission, 1959), pp. 47, 75-82. b Commercial Development i n (3oquitlam (New Westminster: Lower Mainland Regional Planning Board, 1967), p. 2. c Land f o r L i v i n g (New Westminster: Lower Mainland Regional Planning Board, 1963), p. 16. d Land f o r Liesure (New Tfestminster: Lower Mainland Regional Planning Board, 1961), pp. 6, 15. e Implementation, Langley Plan (New Westminster: Lower Mainland Regional Planning Board, 1957), pp. 65a-65b. f A i r p o r t s f o r the Lower F i n l a n d (New Westiriinster: Lower Mainland Regional Planning Board, 1953), p. 7. g Figure advanced by Richmond town planner, Bob Reynolds, on a Vancouver h o t - l i n e radio show i n the summer of 1972. h School Needs f o r Langley (New Westminster: Lower Mainland Regional Planning Board, 1961), pp. 17, 28. i Report on School and Park Standards (Vancouver:Technical Planning Branch o f the Planning Department o f the C i t y of Vancouver, 1957), p. 3. j Census o f Canada (Ottawa: Queen's P r i n t e r , 1961). k Average number o f persons per household f o r Metropolitan Vancouver fringe areas as reported i n the 1966 Census o f Canada (Ottawa: Queen's P r i n t e r , 1966).  Table C COEFFICIENTS FOR STEP DEMAND FUNCTIONS USED IN LANGLEY MODEL  Land Use o r  1st Step o f Function  Upper L i m i t (acres)  3  ,  Contribution to Policy Goal (units/acre)  2nd Step o f Function  Upper Limit (acres)  3  ,  3rd Step o f Function  Contribution to < Upper L i m i t Policy Goal (acres) (units/acre)  3  Contribution to Policy Goal (units/acre)  FEEDT  100  $4400 to MXGRP  OFFICT  TOTPOP/12048  $340000 to MXGRP TOTPOP/12048  $100000 to MXGRP  rain  Externality Activity (Code)  $50000 to MXGRP  TRANST  TOTPOP/8475  $340000 to MXGRP TOTPOP/8475  $100000 to MXGRP  -P Ul  $50000 to MXGRP  UCNRT^  urban urban population/34483 $340000 to MXGRP population/34483 $100000 to MXGRP urban $340000 to MXGRP population/29 33 $100000 to MXGRP  USHOPT  urban population/2933  SCNRT  suburban suburban population/34483 $340000 to MXGRP population/34483 $100000 to MXGRP  d  e  SSHOPT  6  suburban population/2933  suburban $340000 to MXGRP population/2933  $100000 to MXGRP  PJURALT  rural rural population/17857 $340000 to MXGRP pppulation/17857 $100000 to MXGRP  UPRKT  urban population/316  $1000 to MXGRP  urban population/316  $750 to MXGRP  SPRKT  suburban population/316  $1000 to MXGRP  suburban population/316  $750 to MXGRP  MPRKT  TOTPOP/267  $200 to MXGRP  TOTPOP/267  $175 to MXGRP  GOLFT  TOTPOP/200  $133 to MXGRP  TOTPOP/200  $100 to MXGRP  f  5  6  <D  activities in this (solum n are uncor  $2000 to MXGRP  s;tep  100  all  C  $500 to MXGRP  $50000 to MXGRP $50000 to MXGRP $50000 to MXGRP $50000 to MXGRP $50000 to MXGRP $500 to MXGRP $500 to MXGRP $125 to MXGRP $75 to MXGRP  Table C cont'd  CRDEVT  TOTPOP/211  $20000 t o MXGRP TOTPOP/211  $15000 t o MXGRP  $7500 t o MXGRP  HOSPT  TOTPOP/8000  $300000 t o MXGRP TOTPOP/8000  $150000 t o MXGRP  $50000 t o MXGRP  SKULT  TOTPOP/160  $35000 t o MXGRP TOTPOP/160  $15000 t o MXGRP  $5000 t o MXGRP  GOVTT  TOTPOP/8000  $200000 t o MXGRP TOTPOP/8000  $100000 t o MXGRP  T3  $50000 t o MXGRP  FREERECT^  TGTPOP/250  10000 units t o MXFREREC  5000 units t o MXFREREC  C •H (0 S-l  10 u n i t s t o MXFREREC  GRBELTT  TOTPOP/250  TOTPOP/1  100 units t o MXGRBELT  TOTPOP/1  50 units t o MXGRBELT  GRSPACET  TOTPOP/250  2000 units t o MXGRSPAC  TOTPOP/250  1000 u n i t s t o MXGRSPAC  AGSELFT  TOTBOP/1  100 units t o MXAGSELF  TOTPOP/1  50 u n i t s t o MXAGSELF  G  G  G  <D  -P W  c: o o c 3  25 u n i t s t o MXGRBELT 10 u n i t s t o MXGRSPAC 25 u n i t s t o MXAGSELF  The f i r s t two steps o f a l l o f the land use a c t i v i t y demand functions (except FEEDT's) are both a r b i t r a r i l y s e t a t 125% o f the corresponding minimum land requirements f o r that a c t i v i t y (see Table B). The t h i r d steps are a l l l e f t unconstrained (to avoid forcing land into slack). a  The sources of data f o r those functions i n t h i s column that contribute t o the MXGRP goal are given i n Table D. Data sources f o r a l l other functions are explained i n footnote (g) below. °The f i r s t two steps o f FEEDT's demand function are set a t 100 acres each and the t h i r d step i s unconstrained. This function i s not linked t o any Langley population because i t was f e l t that the market f o r l o c a l fed beef was e s s e n t i a l l y the population o f the entire Lower Mainland. Since FEEDT produces 150 animals per acre per year, and since the t o t a l market f o r beef animals i n the Lower Mainland i s somewhat i n excess o f 100,000 animals per year, the output o f FEEDT would begin t o a f f e c t market p r i c e a t quite low l e v e l s o f FEEDT. FEED1 and 2 (the two most p r o f i t a b l e steps i n the FEEDT demand function) were therefore constrained t o 100 acres each. ^Urban population = 122.5 people/acre x APART acres +17.5 people/acre x 5/ACRE acres. Suburban population = 3,5 people/acre x 1ACKE acres.  Table C cont'd  ^Rural population = 0.7 people/acre x 5ACRE acres + 3.5 people/acre x AGHCME acres. ^ A l l the data given on these four externality a c t i v i t y functions i s e n t i r e l y fabricated. The demand functions are based on the author's perception o f the amount o f u t i l i t y society derives from these four types of externalities.  118  Table  D  EMPLOYMENT DENSITY COEFFICIENTS USED IN LANGLEY MODEL  Activity  Enrployment Density (workers/acre)  DAIRY  0.025  GARDEN  0.050  FEEDT  0.400  TOTCCM  40.000  Derivation o f C o e f f i c i e n t  estimate: 2 f u l l - t i m e workers per 80 acre d a i r y farm, estimate: 5 f u l l - t i m e workers per 100 acre market garden, estimate: 4 f u l l - t i m e workers per 1 0 acre feedlot. average worker density on commercial land uses f o r several U.S.A. c i t i e s . average worker density on i n d u s t r i a l land uses i n the Lower Mainland.b estimate: 1 f u l l - t i m e caretaker per 4 acre urban park, estimate: 1 f u l l - t i m e caretaker per 4 acre suburban park, estimate: 5 f u l l - t i m e caretakers per 100 acre major park/recreational reserve, estimate: 5 f u l l - t i m e caretakers per 150 acre g o l f course, estimate: 5 workers per acre of commercial r e c r e a t i o n a l development, estimate: 1 f u l l - t i m e caretaker per 35 u n i t apartment (1 acre). estimate: 200 f u l l - t i m e workers per 200 bed h o s p i t a l (4 acres). estimate: 20 f u l l - t i m e employees per 250 p u p i l elementary school (4 acres) and 40 f u l l time employees per 500 p u p i l secondary school (8 acres). estimate: assumed t o be the same as TCTCOM. one unemployed person "occupies" one person i n the labour force. 3  TOTIND  19.200  UPRKT  0.250  SPRKT  0.250  MPRKT  0.050  GOLFT  0.030  CRDEVT  5.000  APART  1.000  HOSPT  50.000  SKULT  5.000  GOVTT UNEMP  40.000 1.000  Southeastern Wisconsin Regional Planning Cctrcnission Land UseTransportation Study (Milwaukee: Southeastern Wisconsin Regional Planning Ccmrtission, 1965) Planning Report No. 7, V o l . l pp. 3 2 , 3 4 , 84. 3D Space f o r Industry: Technical Report (Vancouver: Greater Vancouver Regional D i s t r i c t , 1 9 7 1 ) , p. 74. f  119 Table  E  A C T I V I T Y CONTRIBUTIONS/ACRE TO  Activity Code  DAIRY  Contributicn t o MXGRP ($/acre) 200  GROSS PRODUCTION  Derivation o f C o e f f i c i e n t ( c o e f f i c i e n t s f o r which no explanation appears were e i t h e r estimated o r set a r b i t r a r i l y . ) (net farm income + h i r e d labour expense)/total operated farm acres f o r Eraser V a l l e y d a i r y farms i n 1970. 2/3 x (gross value of production)/total acreage of vegetable production i n the Lower Mainland i n 1971. (net farm income + h i r e d labour expense) f o r a 1500 animal/year commercial f e e d l o t . T o t a l acreage assumed t o be 10 a c r e s . a  GARDEN  450  b  FEEDl  4400  c  FEED2 2000 500 FEED3 a l l TOTCCM a c t i v i t i e s : 340000 1st step  (non-service r e t a i l trade p a y r o l l + 1% of nonservice r e t a i l sales + service trade p a y r o l l + 10000/year per working p r o p r i e t o r ) ^ / t o t a l acres of commercial f l o o r space f o r a l l Vancouver C i t y and Metropolitan area commercial a c t i v i t i e s . e  a l l TOTCCM a c t i v i t i e s : 100000 2nd step a l l TOTCCM 50000 activities: 3rd step TOTIND 60000  value added by a l l i n d u s t r i a l land u s e s / t o t a l i n d u s t r i a l acreage f o r Metropolitan Vancouver. estimate: 1 worker/4 acres @ $4000/year. e  UPRK1 and SPRK1 UPRK2 and SPRK2 UPRK3 and SPRK3 MPRKl MPRK2 MPRK3 GOLF1 GOLF2 G0LF3 CRDEVl CRDEV2 CRDEV3 APART 5ACRE  1000 750 500 200 175 125 133 100 75 20000 15000 7500 2000 50  estimate: 5 workers/100 acres @ $4000/year.  estimate: 5 workers/150 acre g o l f course @ $4000/year.  estimate: 5 workers/acre @ $4000/year. \  estimate: 1 caretaker per apartment (1 acre) @ $2000/year. estimate: $250 of value added farm product per 5 acre hobby farm per year.'  120 Table  HOSPl  300000  HOSP2 HOSP3 SKULl SKUL2 SKUL3 GOVT1 GOVT2 GOVT3 BUYFLUDT  150000 50000 35000  E cont'd  estimate: 200 workers per 200 bed h o s p i t a l (4 acres) @ $6000/year.  estimate: 5 workers/acre @ $7000/year.  15000 5000 200000  100000 50000 -82  BYCLIR  -75  estimate: 40 workers/acre @ $5000/year.  (annual maintenance costs + 10% x i n i t i a l c a p i t a l c o s t s ) / t o t a l acres protected: flood protecting e n t i r e Lower Fraser V a l l e y flood p l a i n t o 1948 flood l e v e l s . 9 This annual charge i s subtracted from MXGRP because i t represents an intermediate good i n the production o f output on flood protected land. estimate: costs o f servicing c a p i t a l costs o f c l e a r i n g land. This annual charge i s subtracted from MXGRP because i t represents an intermediate good i n the production o f output on cleared land.  1970 Fraser V a l l e y Dairy Farms by Municipality ( V i c t o r i a : B.C. Department o f Agriculture, 1971). fo 1971 Production o f Vegetable Crops Together with an Estimate of Farm Value ( V i c t o r i a : B.C. Department o f A g r i c u l t u r e , 1972). °E. T. Osborne. The Fed-Beef Industry i n the Fraser V a l l e y Region of B.C. (unpublished M.B.A. Thesis i n the Department o f Commerce a t the University o f B r i t i s h Columbia, 1968), pp. 153-154. a  ^Census o f Canada (Ottawa: Queen's P r i n t e r , 1966). CCTrrrtercial Floor Space (Vancouver: Greater Vancouver Regional D i s t r i c t , 1970). ^Space f o r Industry: Summary Report (Vancouver: Greater Vancouver Regional D i s t r i c t , 1971) pp. 8, 44. ^Preliminary Report on Flood Control and Hydro-Electric Power i n the Fraser River Basin ( V i c t o r i a : Colonist P r i n t e r ' s Ltd., 1958), pp. 49, 61.  121  Table F A C T I V I T Y CONTRIBUTIONS/ACRE TO POLLUTION INDICES  Activity Code  Contribution/Acre t o :  DAIRY GARDEN FEEDT TOTCCM TOTIND  MNAIRPOL  MNWATER  MNREFUSE  MNNOISE  MNSIGHT  (units)  (units)  (units)  (units)  (units)  1  10  0  2  0  0 10 2  5 500 50 4000  0 0 200 1000 20  1 10 10  0 50 20  750 50  250 0 0 0  4000 0  UPRKT SPRKT JMPRKT GOLFT URBRD SUBRD RURRD PARK RAILRD APART 5/ACRE 1ACRE 5ACRE AGHCME HOSPT SKULT GOVTT LDSLAK  1  0 1  1 1 1 10  4000 400  5 1  80 800 200  0 1 2  10 4 1 0 1 20 4 2 0  1500 200 40 18 40 1000 500 50 0  0 0  CRDEVT  a  20  50  20 20  0 0  20 5 1  5 750 100  50  0 1 0 1000  50 500 750 5  125 40 8 40 100 50 50 20  5 5 2 5 0 100 5 0  10 250 50 100 20 0  0 0 250  i,0 0 0 0 0 0  A l l o f these indices wre e n t i r e l y fabricated by the author and lack any s t a t i s t i c a l backing. An attempt was made t o define the indices i n terms o f $ 1 p o l l u t i o n d i s u t i l i t y u n i t s . a  

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