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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 MODEL FOR A LOWER FRASER VALLEY MUNICIPALITY by DANIEL ERIC SCHROETER B.A., U n i v e r s i t y 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 i n the Department of A g r i c u l t u r a l Economics We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1973 i In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of PiGiRlCULrruiZA/\ fir^nMicS: The University of B r i t i s h Columbia Vancouver 8, Canada Date AUGUST zti~^ /9 73 i i ABSTRACT T h i s t h e s i s examines the use of mathematical models i n a s t r a c t i n g r e a l world la n d use systems. The purpose of the study was to determine what types o f d e c i s i o n models are a v a i l a b l e , whether they can be adapted t o land use a l l o c a t i o n problems, and which i s most s u i t a b l e f o r use i n land use p l a n n i n g . Since p l a n n i n g was f e l t to be a means t o s a t i s f y i n g or maximizing p u b l i c w e l f a r e , 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 the degree t o which the model c o u l d a i d i n seeking l a n d use p o l i c i e s which would be o p t i m a l i n the sense of maximizing some measure of s o c i a l w e l f a r e . Using t h i s c r i t e r i o n , a form of l i n e a r programming which allows f o r s e v e r a l concurrent goals i n the o b j e c t i v e f u n c t i o n was f e l t t o be the bes t model s t r u c t u r e a v a i l a b l e to l a n d use planners today. T h i s model s t r u c t u r e was then used t o c o n s t r u c t a land use model f o r the C i t y and D i s t r i c t o f Langley i n the Lower F r a s e r V a l l e y of B r i t i s h Columbia. The purpose of c o n s t r u c t i n g the model was to i l l u s t r a t e the p o t e n t i a l u s e f u l n e s s of t h i s type o f model f o r d e c i s i o n making i n the l a n d use f i e l d . T h i s was done by showing how the v a r i o u s aspects o f the r e a l world la n d use system c o u l d be i n c o r p o r a t e d i n t o the model and how the model c o u l d then be used to f i n d l a n d use p a t t e r n s which would maximize a measure of s o c i a l w e l f a r e . A f t e r a d i s c u s s i o n o f model r e s u l t s , p o s s i b l e f u r t h e r refinements and study were suggested and i i i d i s c u s s e d . I t was f e l t t h a t the model s t r u c t u r e chosen was w e l l s u i t e d t o the l a n d use p l a n n i n g f i e l d and o f f e r e d much promise as a p o t e n t i a l p l a n n i n g t o o l . i v ACKNOWLEDGMENT Without the e f f o r t , encouragement, and guidance o f s e v e r a l people t h i s t h e s i s would not have been p o s s i b l e . I wish t o extend my s i n c e r e thanks t o : Dean M i c h a e l Shaw, and the Canada Department of A g r i c u l t u r e , f o r t h e i r f i n a n c i a l support o f t h i s p r o j e c t ; Dr. P eter L. Arcus, f o r s h a r i n g h i s time, wisdom, and experience i n h i s r o l e as s e n i o r t h e s i s a d v i s o r , and f o r the a d d i t i o n a l e f f o r t he made to make my s t u d i e s a warm and rewarding experience; Dr. John D. Graham, f o r i n i t i a l l y advancing to me the concept o f u s i n g a m u l t i p l e g o als l i n e a r programming format to study problems of la n d use a l l o c a t i o n . His i n i t i a l work i n t h i s area not o n l y served as an i n v a l u a b l e a i d to t h i s study, but a l s o l e d to f i n a n c i a l support being made a v a i l a b l e to f u r t h e r r e s e a r c h i n t h i s f i e l d , i n c l u d i n g support o f t h i s t h e s i s p r o j e c t ; Dr. G. R. Winter, and Dr. L. M. L a v k u l i c h , f o r t h e i r v a l u e d s e r v i c e on the t h e s i s committee; B e v e r l y Schroeter, f o r her i n t e r e s t , encouragement, and a s s i s t a n c e ; and my f e l l o w students and f r i e n d s , f o r t h e i r advice and support. 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 E a r l y H i s t o r y 3 The Formation of the Lower Mainland Regio n a l P l a n n i n g Board 5 The S i t u a t i o n Today 6 3 THE USE OF MATHEMATICAL MODELS 8 D e f i n i t i o n 8 E a r l y H i s t o r y o f Mathematical Models .... 8 Use f u l n e s s of Mathematical Models 9 4 MATHEMATICAL MODELS FOR APPLICATION TO LAND USE PLANNING: A REVIEW OF THE ALTERNATIVES 12 I n t r o d u c t i o n 12 Input-Output Models 13 Si m u l a t i o n 17 O p e r a t i o n a l Games 24 Mathematical Programming Models 25 Summary 33 5 THE USE OF LINEAR PROGRAMMING WITH MULTIPLE GOALS IN LAND RESOURCE ALLOCATION 35 The Format and Methodology of Conventional 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 to a H y p o t h e t i c a l Land Use Problem 36 v i CHAPTER Page The Format and Methodology of M u l t i p l e Goals L i n e a r Programming 39 A M u l t i p l e Goals V e r s i o n o f the H y p o t h e t i c a l Land Use Problem 41 Summary 47 6 LANGLEY MODEL STRUCTURE 48 In t o d u c t i o n 48 I n i t i a l Assumptions 50 Land Use A l t e r n a t i v e s 51 T e c h n i c a l and P h y s i c a l C o n s t r a i n t s 52 S o c i a l C o n s t r a i n t s 55 Output Demand C o n s t r a i n t s 56 E x t e r n a l i t y Demand C o n s t r a i n t s 6 3 Accounting C o n s t r a i n t s 65 Goal Accounting C o n s t r a i n t s 65 Summary 70 7 THE LANGLEY MODEL: DATA PROBLEMS AND INITIAL RESULTS 71 Data Problems 71 Re s u l t s 73 F u r t h e r Refinements 81 D i s c u s s i o n 82 8 . SUMMARY AND CONCLUSIONS 85 BIBLIOGRAPHY 87 APPENDIX I: MATRIX PICTURE 91 APPENDIX I I : CODING EXPLANATION AND DATA DESCRIPTION FOR LANGLEY MODEL MATRIX 93 v i i LIST OF TABLES TABLE Page 1 LANGLEY MODEL SOLUTIONS FOR ALLOCATIONS OF LAND 75 2 OPTIMAL ALLOCATIONS BY THE LANGLEY MODEL UNDER VARYING POPULATION CONDITIONS 80 A AREAS OF LAND SUITABILITIES IN THE CITY AND DISTRICT OF LANGLEY 109 B MINIMUM LAND REQUIREMENTS USED IN THE LANGLEY MODEL I l l C COEFFICIENTS FOR STEP DEMAND FUNCTIONS USED IN LANGLEY MODEL 116 D EMPLOYMENT DENSITY COEFFICIENTS USED IN LANGLEY MODEL 118 E ACTIVITY CONTRIBUTIONS/ACRE TO GROSS PRODUCTION 119 F ACTIVITY CONTRIBUTIONS/ACRE TO POLLUTION INDICES 121 v i i i LIST OF FIGURES FIGURE Page 1 G r a p h i c a l I l l u s t r a t i o n of Step Demand Fun c t i o n s 60 1 Chapter 1  INTRODUCTION That government should a c t i v e l y i n t e r v e n e i n the c o n t r o l and p l a n n i n g o f the use of s o c i e t y ' s s c a r c e l a n d resources i s a p r i n c i p l e t h a t i s w e l l e s t a b l i s h e d i n most of the world today. C e r t a i n l y t h i s p r i n c i p l e i s f r i m l y entrenched i n the Lower Mainland o f B r i t i s h Columbia, where many m u n i c i p a l , r e g i o n a l , p r o v i n c i a l , and f e d e r a l government agencies p l a y a major r o l e i n the p l a n n i n g and development of the re g i o n ' s land a l l o c a t i o n . With l a n d use p l a n n i n g being undertaken on a major s c a l e by a m u l t i t u d e of government agencies and boards, i n c r e a s e d 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 the development of new t o o l s and methods t o a i d i n t h i s p l a n n i n g p r o c e s s . One such development t h a t i s r e c e i v i n g p a r t i c u l a r n o t i c e i n t h i s area i s the re c e n t emergence of the use of mathematical models i n s t u d y i n g l a r g e l a n d use systems. T h i s t h e s i s examines the types o f d e c i s i o n model s t r u c t u r e s t h a t are p r e s e n t l y a v a i l a b l e , and then determines whether they can be adapted t o land use a l l o c a t i o n problems, and which i s the most s u i t a b l e f o r use i n t h i s a rea. Since l a n d use p l a n n i n g can be i n t e r p r e t e d as the design and c o n t r o l of an area's lan d based f u n c t i o n s so t h a t s o c i e t y (the p o p u l a t i o n of the area) r e c e i v e s the maximum a t t a i n a b l e b e n e f i t or u t i l i t y from those lan d r e s o u r c e s , the c r i t e r i o n used f o r s u i t a b i l i t y 2 was the degree to which the model c o u l d a i d i n seeking l a n d use p o l i c i e s which would be o p t i m a l i n the sense of maximizing some measure of s o c i a l w e l f a r e . Given t h i s c r i t e r i o n , the model s t r u c t u r e t h a t i s judged s u p e r i o r i s chosen, i t s format examined i n g r e a t e r d e t a i l , and i s used i n the development of a r a t i o n a l l a n d use model f o r the C i t y and D i s t r i c t of Langley i n the Lower F r a s e r V a l l e y of B r i t i s h Columbia. The purpose of c o n s t r u c t i n g the model was t o i l l u s t r a t e the p o t e n t i a l u s e f u l n e s s o f t h i s type of model f o r d e c i s i o n making i n the l a n d use f i e l d . T h i s was done by showing how the v a r i o u s aspects o f the r e a l world l a n d use system c o u l d be i n c o r p o r a t e d i n t o the model and how the model c o u l d then be used to f i n d l a n d use p a t t e r n s which would maximize a measure of s o c i a l w e l f a r e . A f t e r an examination o f model r e s u l t s , p o s s i b l e f u r t h e r refinements and study are suggested and d i s c u s s e d . 3 Chapter 2 THE HISTORY OF LAND USE PLANNING AND CONTROL IN THE LOWER MAINLAND OF BRITISH COLUMBIA E a r l y H i s t o r y Though many regard government i n t e r v e n t i o n i n t o land use a l l o c a t i o n as a phenomenon f a i r l y r e c e n t t o the F r a s e r V a l l e y , p u b l i c c o n t r o l o f the p r i v a t e use of 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 settlement i t s e l f . The f i r s t l e g a l surveys of 1858 e s t a b l i s h e d , i n the B r i t i s h t r a d i t i o n , t h a t not l a n d i t s e l f , but o n l y r i g h t s i n 1 l a n d as s e t down by p u b l i c law, c o u l d be p r i v a t e l y owned. Nor was t h i s c o n t r o l over p r i v a t e p r o p e r t y always e x e r c i s e d p a s s i v e l y by the p u b l i c a u t h o r i t y . E a r l y l a n d i n v e n t o r i e s and l a n d c l a s s i f i c a t i o n systems, though s i m p l i s t i c a t best , r e p r e s e n t e d an a c t i v e attempt on b e h a l f of the c o l o n i a l government t o d i r e c t the p a t t e r n o f sett l e m e n t in--the Lower F r a s e r V a l l e y . A l f r e d Siemens, i n h i s study o f the development o f t h i s r e g i o n , r e p o r t s t h a t "some to w n s i t e s , n o t a b l y t h a t o f New Westminster, were s e l e c t e d w e l l i n advance 2 of s e t t l e m e n t — w i t h due regard to s t r a t e g i c c o n s i d e r a t i o n s . " 1 These surveys are d e s c r i b e d i n : V. J . Parker, "Problems and Progress i n R a t i o n a l i z i n g the Use of the Resources of the 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  of a C u l t u r a l Landscape, ed. A. H. Siemens (Vancouver: rTantalus Research, 1966), p. 165. 2 A l f r e d H. Siemens, "The Process o f Settlement i n the 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 Context," 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 Landscape, ed. 4 Thus, p u b l i c l a n d use p l a n n i n g i . e . the i n t e r v e n t i o n , i n the p u b l i c i n t e r e s t , i n t o the d e s i g n of the s p a t i a l d i s t r i b u t i o n of the uses of l a n d , had i t s beginnings i n the F r a s e r V a l l e y as e a r l y as the middle of the 19th c e n t u r y . Government c o n t r o l over la n d use, however, remained f a i r l y tenuous u n t i l near the t u r n of the c e n t u r y . The completion o f the Canadian P a c i f i c Railway i n 1881 and subsequent development of the Vancouver area as the major p o r t and r a i l r o a d terminus on the West Coast a t t r a c t e d v a s t numbers of new s e t t l e r s t o the r e g i o n and i n t r o d u c e d an i n c r e a s i n g l y g r e a t e r number of uses t o which the V a l l e y ' s l a n d c o u l d be put. I n d u s t r i a l , commercial and r e s i d e n t i a l uses now j o i n e d the a l r e a d y e x i s t i n g primary uses i n competing f o r the suddenly s c a r c e V a l l e y l a n d . As the c o m p e t i t i o n i n t e n s i f i e d p r o v i n c i a l and m u n i c i p a l governments were c a l l e d on more and more to e n t e r the land, use f i e l d , though more as a d j u d i c a t o r s of d i s p u t e s , than as p l a n n e r s o f use a l l o c a t i o n . As w e l l , the growing p o p u l a t i o n brought i n c r e a s i n g demands f o r government s e r v i c e s — r o a d s , harbour f a c i l i t i e s , dyking and drainage p r o j e c t s , water and sewage mains, p o l i c e and f i r e p r o t e c t i o n . To s a t i s f y these demands the v a r i o u s governments were o b l i g e d to e n t e r the l a n d use market as buyers and thus as owners of p r o p e r t y r i g h t s . Out of t h i s s i t u a t i o n evolved an ever i n c r e a s i n g need f o r , and subsequent p a r t i c i p a t i o n by v a r i o u s government bodies i n the c o n t r o l l i n g and p l a n n i n g of A. H. Siemens (Vancouver: T a n t a l u s Research, 1966), p.31. 5 l a n d use. 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 boards and commissions to p l a n and r e g u l a t e land use w i t h i n t h e i r j u r i s d i c t i o n . Parks boards, s c h o o l boards, improvement d i s t r i c t s , dyking commissions, water communities, and sewerage and drainage d i s t r i c t s were s e t up. The Formation of the Lower Mainland Regional P l a n n i n g Board At the same time, pressure continued to be brought to bear on the p r o v i n c i a l government to take a more dominant r o l e i n l a n d p l a n n i n g . In 1936 a c i t i z e n s ' group o r g a n i z e d the Lower Mainland R e g i o n a l P l a n n i n g A s s o c i a t i o n which urged the p r o v i n c i a l government to r e c o g n i z e the need to p l a n the l a n d use of the Lower Mainland as a s i n g l e s o c i a l and economic u n i t and to s a t i s f y t h a t need by s e t t i n g up a government p l a n n i n g agency. During World War I I the Government of B r i t i s h Columbia r e a c t e d to these i n c r e a s i n g demands by p a s s i n g the R e h a b i l -i t a t i o n A c t of 1944, which c r e a t e d a Regional P l a n n i n g D i v i s i o n w i t h i n the Bureau of Post-War R e h a b i l i t a t i o n and R e c o n s t r u c t i o n . One of the f i r s t t a sks t h a t the P l a n n i n g D i v i s i o n undertook was a comprehensive survey of e x i s t i n g l a n d use i n the Lower Mainland i n 1945. Recommendations made by t h i s agency, as w e l l as continued demands by m u n i c i p a l c o u n c i l s , town p l a n -n i n g agencies, and v a r i o u s other l o c a l c i t i z e n groups l e d the p r o v i n c i a l government to e s t a b l i s h the Lower Mainland Regional P l a n n i n g Board on June 21, 1949. The prime r o l e of the board was the p r e p a r a t i o n of a comprehensive r e g i o n a l 6 p l a n f o r a l l of the F r a s e r V a l l e y from Hope westwards to the S t r a i t o f Georgia, i n c l u d i n g a l l 28 r e g i o n m u n i c i p a l i t i e s . The S i t u a t i o n Today Though the Lower Mainland Regional P l a n n i n g D i s t r i c t has s i n c e been r e p l a c e d by f o u r separate r e g i o n a l d i s t r i c t s , t h i s e v o l u t i o n towards i n c r e a s e d government involvement i n the p l a n n i n g and r a t i o n a l i z a t i o n of the Lower Mainland's l a n d has c o n t i n u e d to the p r e s e n t . Today there are over f o u r hundred d i f f e r e n t p u b l i c bodies i n the Lower F r a s e r V a l l e y which e x e r c i s e a u t h o r i t y over v a r i o u s aspects o f l a n d use p l a n n i n g and c o n t r o l . There are few people today who would q u e s t i o n the p r i n c i p l e (though t h e r e are many t h a t would q u e s t i o n the degree) of p u b l i c p a r t i c i p a t i o n i n the p l a n n i n g of the wise use of the V a l l e y ' s l a n d r e s o u r c e s . Given the d e s i r a b i l i t y o r the n e c e s s i t y , or a t l e a s t the i n e v i t a b i l i t y of p u b l i c i n t e r v e n t i o n i n l a n d use p l a n n i n g , i t i s reasonable to expect t h a t l a n d use plans be formulated so t h a t the p u b l i c ' s p l a n n i n g o b j e c t i v e s are met. P l a n n i n g , a f t e r a l l , should not be construed as anything but a means t o s a t i s f y i n g or maximizing p u b l i c w e l f a r e . . But how can l a n d use p l a n n e r s determine p u b l i c o b j e c t i v e s and by what method or d e v i c e can they decide how t o manipulate the complex s e t of i n t e r d e p e n d e n c i e s t h a t e x i s t i n a l a n d 7 use system so t h a t these o b j e c t i v e s are be s t met? Herein l i e s an understanding o f the r e c e n t emergence of the use of models and o t h e r s i m u l a t i o n systems i n f o r m u l a t i n g l a n d use d e c i s i o n s . By d i s t i l l i n g a complex land use system i n to an o b s e r v a b l e s e t of major r e l a t i o n s h i p s and o b j e c t i v e s , some measure o f q u a n t i f i c a t i o n can be i n j e c t e d i n t o the arguments which otherwise have remained almost t o t a l l y q u a l i t a t i v e . The next chapter o u t l i n e s the r i s i n g prominence of these model systems i n d e c i s i o n making, not o n l y i n the l a n d use f i e l d , but a l s o i n most of the s o c i a l s c i e n c e s . 8 Chapter 3  THE USE OF MATHEMATICAL MODELS D e f i n i t i o n A model i s an analogue or a r e p r e s e n t a t i o n of some s e t o f r e l a t i o n s t h a t e x i s t s i n the r e a l world. These r e l a t i o n s may be s p e c i f i e d v e r b a l l y , mathematically, or v i s u a l l y . Thus sch o o l c h i l d r e n p l a y i n g " s t o r e " , 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 matrix o f the Lower Mainland economy, or a s c a l e v e r s i o n o f an i n t e r n a l combustion engine can a l l be c l a s s i f i e d as models. For the purposes of t h i s study, however, on l y those economic models which are e s s e n t i a l l y m athematically s p e c i f i e d w i l l be c o n s i d e r e d . E a r l y H i s t o r y o f Mathematical Models The i n c r e a s e i n the use of mathematical models over the p a s t three decades has been remarkable. Mathematical model-1 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. The advent of World War I I and the subsequent major o p e r a t i o n a l problems of the A l l i e d war e f f o r t l e d r e s e a r c h e r s t o develop what was termed 1 Some examples of mathematical models t h a t predate World War I I can be found. For example, Quesnay's Tableau  Economique, p u b l i s h e d i n 1758, i s perhaps the e a r l i e s t example of 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 model. 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 are those of Leon Walras i n the l a t e n i n e t e e n t h century and W. L e o n t i e f i n the 1930*s. 9 "ope r a t i o n s r e s e a r c h " . To a i d r e s e a r c h e r s i n understanding the complex in t e r d e p e n d e n c i e s t h a t e x i s t e d between the v a r i o u s components of the war e f f o r t , they developed a mathematical d e s c r i p t i o n o f the war o p e r a t i o n s system. Using t h i s mathem-a t i c a l model they were ab l e t o f i n d o p t i m a l o r bes t s o l u t i o n s 2 to the 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 them. From t h i s somewhat i g n o b l e b i r t h , o p e r a t i o n s r e s e a r c h has r a p i d l y e v o l v e d i n t o what may be b r o a d l y termed systems a n a l y s i s , w i t h 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 as town p l a n -n i n g , i n t e r n a t i o n a l r e l a t i o n s , i n d u s t r i a l economics, ecology, and s o c i a l psychology. Problems i n v i r t u a l l y every area of the s o c i a l s c i e n c e s have come under the s c r u t i n y o f systems a n a l y s t s . U s e f u l n e s s o f Mathematical Models The widespread use of mathematical m o d e l l i n g i s not without i t s c r i t i c s . There are some who view the r e c e n t p u r s u i t o f such models as i n a p p r o p r i a t e , although understand-a b l e . Vernon and Hoover sum up t h i s p o s i t i o n q u i t e s u c c i n c t l y : One wonders, however, whether the pr e o c c u p a t i o n w i t h systems of t h i s s o r t , so i n g e n i o u s l y pursued by so many r e s e a r c h e r s i n the urban f i e l d , has not come t o re p r e s e n t a m i s a l l o c a t i o n of sca r c e i n g e n u i t y . Some of the reasons f o r the a t t r a c t i o n to such systems are c l e a r enough. To the mind 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 For a b r i e f d e c r i p t i o n o f t h i s work see Joseph L. Schofer, "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 paper put out by the Center f o r Urban S t u d i e s and Department o f Systems E n g i n e e r i n g a t the U n i v e r s i t y o f I l l i n o i s (Chicago), 1970. 10 the p u r s u i t of r e s e a r c h i n the s o c i a l s c i e n c e s 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 such r e -search tend t o be spongy, 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 of one another or of the dependent phenomena on which they are presumed to a c t ; r e s i d u a l v a r i a n c e proves more important than what has been " e x p l a i n e d " . The " t r u e " model l i e s very deep, indeed, i f i t e x i s t s a t a l l . But the t h i r s t f o r a simple o r d e r proves hard to quench; the adumbrations t h a t the order may e x i s t e x e r t an overwhelming p u l l upon the r e s e a r c h e r . 3 Although most r e s e a r c h e r s would agree t h a t care must be taken not t o l o s e s i g h t of the l i m i t a t i o n s o f e x t r a c t i n g mathematical models from complex s o c i a l systems, few would accept Vernon and Hoover's h y p o t h e s i s t h a t i n some r e a l world s o c i a l systems no c a u s a l i t y o r o r d e r e x i s t s , o r i f i t does e x i s t , i t i s hidden so deeply as to render search f r u i t l e s s . One of the fundamental b e l i e f s o f economics i s t h a t i n d i v i d u a l d e c i s i o n u n i t s i n a s o c i a l system behave r a t i o n a l l y and p r e d i c t a b l y i n accordance w i t h c e r t a i n r ecog-n i z a b l e economic and s o c i a l aims. T h i s i m p l i e s t h a t o r d e r does e x i s t w i t h i n a s o c i a l system and t h a t t h i s o r d e r or s e t of p a t t e r n s can be d e s c r i b e d , e x p l a i n e d and, w i t h i n l i m i t s , f o r e c a s t . A c c o r d i n g t o Meyer, "mathematical programming i s without q u e s t i o n the b e s t o f the t o o l s employed i n modern r e g i o n a l a n a l y s i s from a s t r i c t l y c o nceptual p o i n t of view i f one b e l i e v e s i n a reasonably p e r v a s i v e economic r a t i o n a l i t y . " Even i f one accepts Vernon and Hoover's s u g g e s t i o n t h a t the Raymond Vernon and Edgar M. Hoover, "Economic Aspects of Urban Research," The Study of U r b a n i z a t i o n eds. P h i l i p Hauser and Leo Schnore (New York: John Wiley and Sons, 1965)., p. 195. 4 John R. Meyer, "Regional Economics: A Survey," American Economic Review, L I I I , No. 1 (January, 1963) p.53. 11 " t r u e " model may be i m p o s s i b l e to uncover, does i t n e c e s s a r i l y f o l l o w t h a t any attempt to a b s t r a c t an approximation of the model cannot be j u s t i f i e d ? Many r e s e a r c h e r s , t h i s author i n c l u d e d , take a very F r i e d m a n i s t i c approach t o model a b s t r a c t i o n : The l e g i t i m a c y of 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 the l i g h t t h a t i s shed and the power t o p r e d i c t t h a t i s y i e l d e d by the a b s t r a c t i o n . C e r t a i n l y mathematical models have t h e i r s h o r t -comings, but u s u a l l y these short-comings are r e l a t e d to i n p u t requirements and are not a c t u a l l y i n h e r e n t i n the c o n c e p t u a l nature of the models themselves. Problems o f data accumulation, computer c a p a c i t y and programming i n e x p e r i e n c e seem t o be mentioned i n model s t u d i e s much more f r e q u e n t l y than problems of model methodology. Des p i t e these d i f f i c u l t i e s mathematical m o d e l l i n g has encountered widespread use and acceptance i n almost a l l f i e l d s i n the s o c i a l s c i e n c e s , and w i l l l i k e l y c o n tinue t o grow i n p o p u l a r i t y . Care must be taken, however, not to r e g a r d models as the panacea f o r a l l of the s o c i a l i l l s c o n f r o n t i n g modern c i v i l i z a t i o n . Models are o n l y t o o l s to h e l p us understand these problems. In the words of Claude M c M i l l a n : Mathematical programming i s a r e s p e c t e d body of knowledge because of i t s a n a l y t i c a l power as a supplement t o , r a t h e r than 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 of g r e a t complexity. 5 M i l t o n Friedman, P r i c e Theory (Chicago: A l d i n e P u b l i s h i n g Co., 1962) p.13. 6 Claude M c M i l l a n J r . , Mathematical Programming: An I n t r o d u c t i o n t o the Design 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 Machines (New York: John Wiley and Sons, 1970), p.v. of p r e f a c e . -12 Chapter 4 MATHEMATICAL MODELS FOR APPLICATION TO LAND USE PLANNING: A REVIEW OF THE ALTERNATIVES I n t r o d u c t i o n Even though the use of mathematical models as an a i d t o d e c i s i o n making i n the s o c i a l s c i e n c e s has o n l y dev-eloped over the l a s t two or three decades, the l i t e r a t u r e i n the f i e l d i s v a s t , not o n l y i n terms of a b s o l u t e volume but a l s o i n terms of the almost i n f i n i t e number of methodologies 1 t h a t have been s t u d i e d and used i n r e c e n t y e a r s . S u f f i c e f o r t h i s t h e s i s , however, to b r i e f l y d e s c r i b e the major model groups and t o e v a l u a t e t h e i r p o s s i b l e e f f e c t i v e n e s s i n lan d use p l a n n i n g and d e c i s i o n making. I t should be understood, however, t h a t no model group has s h a r p l y d e f i n e d boundaries t h a t make i t completely d i s t i n c t from o t h e r model s t r u c t u r e s . A c t u a l models are f r e q u e n t l y a composite of more than one type of model s t r u c t u r e . Three types of models can be i d e n t i f i e d from the l i t e r a t u r e : i n p u t - o u t p u t models, s i m u l a t i o n models ( i n c l u d i n g o p e r a t i o n a l games), and programming models. A l l of the models have p o t e n t i a l a p p l i c a t i o n s i n l a n d use p l a n n i n g . 1 For a review o f the l i t e r a t u r e on models used i n the a g r i c u l t u r a l area alone see Economic Models 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 , ed. 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). 13 Input-Output Models Input-output models d e s c r i b 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 t h a t e x i s t w i t h i n a system t h a t has a number of exogenous v a r i a b l e s a c t i n g upon i t . The most common use of t h i s type of model i s i n i n t e r s e c t o r a l or i n t e r r e g i o n a l flow a n a l y s i s . The i n p u t - o u t p u t model can approximate the complete s t r u c t u r a l nature o f an economy, and can d e t a i l the t r a n s a c t i o n s which take p l a c e i n t h a t economy i n a given time p e r i o d , not o n l y between s e c t o r s or r e g i o n s w i t h i n the economy but a l s o between the v a r i o u s s e c t o r s and the exogenous v a r i a b l e s a c t i n g upon the economy, u s u a l l y r e p r e s e n t e d by the v a r i o u s f i n a l demands f o r o u t puts. To i l l u s t r a t e , i f there are n s e c t o r s i n the economy and the d i s t r i b u t i o n of the output o f each s e c t o r between f i n a l (exogenous) demand and i n t e r s e c t o r a l flows i s known, then the economy can be d e s c r i b e d i n an i n p u t - o u t p u t a c c o u n t i n g a r r a y as f o l l o w s : Consuming Sectors Producing F i n a l T o t a l S e c t o r s 1 2 3 • * . . n Demand Output 1 x l l x12 x13 • • * x l n *1 x l 2 x 2 1 x22 x2 3 • « • * x 2 n x2 3 * x 3 1 x32 • x 3 3 • • • * • x 3 n • <*3 • x 3 • • n * X„ -i • X o • X 0 • . . X • d • X n l n2 n3 nn n n Value Added v l v 2 v 3 • • • v n T o t a l Output x l x 2 x 3 • « • x n 14 In t h i s a r r a y , x ^ j i s the s a l e s o f s e c t o r i to s e c t o r j , Vj i s the va l u e added t o p r o d u c t i o n by s e c t o r j , and d^ i s the q u a n t i t y demanded of s e c t o r i by the " f i n a l demand s e c t o r s " : households, government, p r i v a t e investment, f o r e i g n demand. X i i s the t o t a l output produced i n s e c t o r i . Since every producing s e c t o r i s a l s o a consuming s e c t o r , note t h a t the f o l l o w i n g balances h o l d t r u e : it X j L j + d. = x± = ±L X I J + V J j = i 1=1 By a process o f m a t r i x i n v e r s i o n , an a r r a y o f l i n e a r equations r e l a t i n g the output of each s e c t o r t o the f i n a l demand of every 2 s e c t o r can be c a l c u l a t e d u s i n g the acc o u n t i n g a r r a y above. 2 Knowing t h a t 3=1 i f an n x n a r r a y , A, of 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 are d e f i n e d such t h a t each element of the a r r a y , a 1D — Xj ' then n X, = > . a, -X . + d ' i 2 L a i j X j + j = l l In matrix n o t a t i o n t h i s becomes X = AX + D , where X i s the output v e c t o r i n the i n i t i a l i n p u t -output a r r a y , D i s the column v e c t o r o f f i n a l demands, and A i s the a r r a y d e f i n e d above. T h i s becomes (I - A)X = D , where I i s the n x n m u l t i p l i c a t i v e i d e n t i t y m a t r i x . I f the determinant o f (I - A) i s non-zero, then X = (I - A) 1D . continued o v e r l e a f 15 I f a s e t of p o l i c y v a r i a b l e s e x i s t s which determines f i n a l demands then output d i s t r i b u t i o n can be determined f o r any give n v a l u e s o f these p o l i c y v a r i a b l e s . Thus through manip-u l a t i o n o f t h i s type o f model the p o l i c y maker can determine which d i s t r i b u t i o n o f output meets or s a t i s f i e s h i s o b j e c t i v e s and the va l u e s t h a t the p o l i c y v a r i a b l e s must take i n order to a t t a i n t h i s d i s t r i b u t i o n . T h i s type o f model has a p p l i c a t i o n s i n many areas, although the nature o f i t s assumptions do l i m i t i t t o s t u d i e s d e a l i n g w i t h the past o r with the s h o r t run. The model assumes f i x e d i n p u t - o u t p u t c o e f f i c i e n t s and thus i s o n l y t h e o r e t i c a l l y v a l i d over time p e r i o d s i n which i n p u t s u b s t i t u t i o n , changes i n technology, h e t e r o g e n e i t y o f product, new sources 3 of i n p u t , and s c a l e economies do not a r i s e . As w e l l , the data requirements o f the model tend t o l i m i t a p p l i c a t i o n s to the n a t i o n a l l e v e l , where adequate i n f o r m a t i o n on i n t e r -i n d u s t r y flows has been gathered, although i n p u t - o u t p u t D e f i n i n g (I - A ) - ^ as M, where m^j i s an element of M, then X = MD re p r e s e n t s an a r r a y o f n equations of the form, x ± =J_ t> i j a j , j = i 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 For an e x c e l l e n t d i s c u s s i o n on these assumptions see H o l i s Chenery and Paul C l a r k , I n t e r i n d u s t r y Economics, (New York: John Wiley and Sons, 1959) pp. 33-42. 16 models have been prepared f o r v a r i o u s s t a t e s ( C a l i f o r n i a , Iowa) and even f o r s i n g l e c i t i e s (St. L o u i s , P h i l a d e l p h i a ) . Despite these l i m i t a t i o n s , i n p u t - o u t p u t a n a l y s i s has been s u c c e s s f u l l y used by many r e s e a r c h e r s . Perhaps the most w e l l known work i n t h i s f i e l d was pr o v i d e d by Wa s s i l y L e o n t i e f i n h i s s t u d i e s o f the i n t e r i n d u s t r y s t r u c t u r e o f the U.S. 4 economy. C a r t e r and Heady employed an inp u t - o u t p u t a n a l y s i s i n t h e i r 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 , w h i l e Rice and LeFerney used an in p u t - o u t p u t model to f o r e c a s t 5,6 happenings i n the American t e x t i l e i n d u s t r y . Input-output m a t r i c e s have been c o n s t r u c t e d f o r many n a t i o n a l 7 economies. Though land a l l o c a t i o n can be s t u d i e d v i a i n p u t -output a n a l y s i s i f the r e l a t i o n s h i p between l a n d and 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 the S t r u c t u r e o f the American  Economy, (New York: Oxford U n i v e r s i t y P r e s s , 1953). 5 H. 0. C a r t e r , and E a r l 0. Heady, An Input-Output  A n a l y s i s Emphasizing Regional and Commodity Se 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 Economics Experimental S t a t i o n B u l l e t i n no.469, no date g i v e n . 6 P h i l i p Rice and Pres t o n LeFerney, Use of Input- Output A n a l y s i s i n Studying Ind u s t r y Problems: A p p l i e d t o  Employment Changes i n the U.S. T e x t i l e I n d u s t r y . U.S. Department of 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. February, 1970. 7 For a review o f r e s e a r c h done i n t h i s area see Chenery and C l a r k , op. c i t . , pp. 183-200. 17 output i s known or can be determined, l a n d use p l a n n e r s have shown l i t t l e i n t e r e s t i n t h i s type o f a n a l y s i s . The s h o r t term aspects o f t h i s model s t r u c t u r e are of l i m i t e d use i n the g e n e r a l l y long term time h o r i z o n s of l a n d use p l a n n e r s . The f a c t t h a t planners are u s u a l l y more i n t e r e s t e d i n o p t i m i z i n g i n d u s t r y d i s t r i b u t i o n r a t h e r than f o r e c a s t i n g them, a l s o l i m i t s the use of i n p u t - o u t p u t model a n a l y s i s i n t h i s f i e l d . However, pla n n e r s should not be too quick to w r i t e o f f the use of i n p u t - output models i n o p t i m i z i n g type problems. O p t i m i z i n g models u s u a l l y r e q u i r e exact s p e c i f i c a t i o n of the policy-makers' u t i l i t y f u n c t i o n s , and s i n c e p a r t i n g w i t h t h i s i n f o r m a t i o n may prove p o l i t i c a l l y unacceptable to many policy-makers, a l t e r n a t e methods of approaching or s e a r c h i n g f o r p o l i c y optima may have t o be developed. Input-output models may have a r o l e to p l a y i n t h i s development. However, s i n c e t h i s i s s u e i s probably more r e l e v a n t to the case of p o l i c y s i m u l a t i o n models, i t w i l l be d i s c u s s e d more f u l l y i n a f o l l o w i n g examination of s i m u l a t i o n models. N e v e r t h e l e s s , i t seems apparent t h a t i f i n p u t - o u t p u t a n a l y s i s has any r o l e to p l a y i n l a n d use p l a n n i n g a t the present, i t i s almost c e r t a i n l y as a d e v i c e f o r s h o r t term f o r e c a s t i n g and p r e t e s t i n g of p o l i c y changes, and not as a d e v i c e to formulate c o r r e c t o r o p t i m a l long term a l l o c a t i o n . S i m u l a t i o n 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 d e s c r i b e a v a r i e t y o f o p e r a t i o n a l procedures, i t i s most o f t e n r e s e r v e d f o r systems of mathematical exp r e s s i o n s t h a t approximate time s e q u e n t i a l r e a l world p a t t e r n s o f exchange and i n t e r a c t i o n . S i m u l a t i o n almost always i s a r e p l i c a t i o n of a growth p r o c e s s , t h a t i s , most s i m u l a t i o n models c o n t a i n dynamic elements. Given a f i x e d s e t of va l u e s f o r the exogenous v a r i a b l e s i n a system, s i m u l a t i o n can produce the time paths o f the endogenous v a r i a b l e s f o r as long a p e r i o d as d e s i r e d . A second d i s t i c t i v e f e a t u r e o f many s i m u l a t i o n s i s t h a t they admit t o the h i g h degree of v a r i a t i o n t h a t e x i s t s i n most r e a l world 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 d i s t r i b u t i o n s i n t o t h e i r s t r u c t u r e . That i s , w i t h t h i s type of f o r m u l a t i o n , the same s e t of i n i t i a l assumptions or c o n d i t i o n s w i l l not always y i e l d the same r e s u l t . In the s o l u t i o n o f t h i s type o f model any step c o n t a i n i n g a s t o c h a s t i c v a r i a b l e w i l l i n v o l v e randomly choosing the va l u e o f the v a r i a b l e from an a p p r o p r i a t e frequency d i s t r i b u t i o n . S i m u l a t i o n models of t h i s type are c a l l e d p r o b a b i l i s t i c . I f a l l s t o c h a s t i c d i s t u r b a n c e s are suppressed and r e p l a c e d by s t a t i s t i c a l measures of c e n t r a l tendency then the model i s termed determ-i n i s t i c . Since many r e a l world systems c o n t a i n 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 of such systems are u s u a l l y not a c c u r a t e 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 takes p l a c e , although they are o f t e n a n a l y t i c a l l y much si m p l e r t o use. In s e t t i n g up a s i m u l a t i o n model the r e s e a r c h e r begins w i t h an i n i t i a l s e t o f system equations garnered 19 through the a p p l i c a t i o n of standard econometric techniques to a v a i l a b l e p a s t and p r e s e n t data. S t a t i s t i c a l measures of c e n t r a l tendency and v a r i a t i o n , r e g r e s s i o n and c o r r e l a t i o n s t u d i e s , Markov c h a i n a n a l y s i s , and o t h e r time s e r i e s p r o c e s s i n g are the most commonly employed t o o l s at t h i s stage. The v a l i d i t y of t h i s i n i t i a l model i s t e s t e d by a comparison of the r e s u l t s o f computer m a n i p u l a t i o n of t h i s i n i t i a l s e t of equations with a v a i l a b l e i n f o r m a t i o n on the machinations of the r e a l world system under study. The i n i t i a l model i s r e s t r u c t u r e d and r e f i n e d u n t i l i t a c c u r a t e l y approximates the r e a l system. Once the model a c c e p t a b l y r e p l i c a t e s the r e a l system, i t can be used t o f o r e c a s t f u t u r e v a l u e s of v a r i a b l e s of i n t e r e s t , or more i m p o r t a n t l y , i t can be used to predetermine the e f f e c t s of proposed p o l i c i e s on any v a r i a b l e s of i n t e r e s t . I f the s i m u l a t i o n i s d e t e r m i n i s t i c , the v a r i a b l e f o r e c a s t s are i n the form of p r e c i s e v a l u e s which are d e r i v e d i n a« s i n g l e t r i a l . I f the s i m u l a t i o n i s p r o b a b i l i s t i c , the v a r i a b l e f o r e c a s t s are o f t e n i n the form of a d i s t r i b u t i o n showing the range o f v a l u e s p o s s i b l e and t h e i r a s s o c i a t e d p r o b a b i l i t y . These f o r e c a s t s are d e r i v e d i n one o f two methods. I f a mathematical a n a l y s i s of the s t o c h a s t i c elements i n the model can determine the d i s t r i b u t i o n of p o s s i b l e outcomes e a s i l y and e f f i c i e n t l y then t h i s procedure i s f o l l o w e d . I f mathematical a n a l y s i s i s unduly complex or t e d i o u s then the 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 e s timated through repeated 20 t r i a l runs, employing the same s t a r t i n g c o n d i t i o n s , u n t i l an adequate sample d i s t r i b u t i o n of outcomes i s produced. T h i s type of procedure i s c a l l e d Monte C a r l o a n a l y s i s and i s by f a r the most dominant procedure used i n p r o b a b i l i s t i c s i m u l a t i o n . I f i t i s d e s i r a b l e t o c o n s i d e r s e v e r a l d i f f e r e n t s e t s of i n i t i a l c o n d i t i o n s , then a s e r i e s of Monte C a r l o analyses can be run, one f o r each s e t of i n i t i a l c o n d i t i o n s . Thus the p o l i c y maker can determine what s e t o f i n i t i a l c o n d i t i o n s i s r e q u i r e d to reach, maximize, or minimize some p o l i c y g o a l (endogenous output v a r i a b l e ) . I f the p o l i c y maker i s i n t e r e s t e d i n a comb-i n a t i o n o f s e v e r a l goals then r e g r e s s i o n a n a l y s i s and o t h e r s t a t i s t i c a l t e s t s can be performed on the Monte C a r l o output d i s t r i b u t i o n s t o determine the t r a d e - o f f s t h a t e x i s t between the v a r i o u s goals and thus a i d i n choosing o p t i m a l s t r a t e g i e s . The K l e i n - G o l d b e r g e r s i m u l a t i o n a n a l y s i s of the U.S. economy 8 p r o v i d e s a good example of the use of t h i s type of procedure. Although s i m u l a t i o n models have been s u c c e s s f u l l y a p p l i e d by many a n a l y s t s , they do have a number of problems a s s o c i a t e d w i t h them. Since s i m u l a t i o n models u s u a l l y concern themselves w i t h l a r g e , v e r y complex r e a l world systems, the data requirements are enormous. The h i g h c o s t bif g e n e r a t i n g and c o l l e c t i n g data o f t e n prove p r o h i b i t i v e , and model accuracy n e c e s s a r i l y s u f f e r s . 8 I. Adelman and F. L. Adelman, "Dynamic-Properties of the K l e i n - G o l d b e r g e r Model", Econometrica 27: 596-625, October, 1959. 21 S i m u l a t i o n models are dynamic i n t h a t they r e p l i c a t e h i s t o r i c a l growth p r o c e s s e s . However, i n doing so they u s u a l l y assume t h a t the t e c h n i c a l c o e f f i c i e n t s t h a t govern t h i s dynamism are f i x e d or s t a t i c . By employing f i x e d t e c h n i c a l c o e f f i c i e n t s many s i m u l a t i o n models run i n t o the same problems of s t u d y i n g the l o n g term t h a t i n p u t - o u t p u t models f a c e . Since there i s l i t t l e reason to assume t h a t technology w i l l h o l d con-s t a n t i n the long run, or even t h a t i t s r a t e of growth w i l l be continuous ( i n the mathematical sense), u n l e s s the c o e f f i c i e n t s i n a s i m u l a t i o n model are r e g u l a r l y updated the model w i l l o n l y have v a l i d a p p l i c a t i o n s i n the s h o r t run. Another approach to c o u n t e r i n g t h i s problem, r e c e n t l y advanced by Howrey and K e l e j i a n , has been t o t r e a t the c o e f f i c i e n t s as s t o c h a s t i c v a r -9 i a b l e s themselves. While probably l e n d i n g more accuracy t o the model, t h i s approach has tended t o make model s o l u t i o n d i f f i c u l t o r i m p o s s i b l e . A t h i r d major problem encountered i n s i m u l a t i o n occurs d u r i n g the model v a l i d a t i o n or refinement stage. The standard technique employed i n most models to measure t h e i r "goodness of f i t " has been t o examine how w e l l the model r e p l i c a t e s h i s t o r i c a l data, based on one p e r i o d f o r e c a s t s . But s i n c e the p o l i c y changes of the f u t u r e may be s u b s t a n t i a l l y d i f f e r e n t than the p o l i c y changes of the p a s t , Naylor argues t h a t t h i s 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 Versus A n a l y t i c a l S o l u t i o n s " i n The Design of  Computer S i m u l a t i o n Experiments, ed. Thomas H. Naylor (Durham, N.C.: Duke U n i v e r s i t y P r e s s , 1969). 10 c r i t e r i o n f o r model v a l i d a t i o n i s not a c c e p t a b l e . U n t i l some new measure of how w e l l a model a c t u a l l y simulates i s developed, model v a l i d a t i o n w i l l continue to pose problems i n s i m u l a t i o n . The use of s i m u l a t i o n i n m o d e l l i n g l a n d use systems has remained l i m i t e d to a few i s o l a t e d s t u d i e s . H a l t e r and M i l l e r ' s study of l a n d a l l o c a t i o n i n r i v e r b a s i n p l a n n i n g p r o v i d e s 11 a good example of the type of work done i n t h i s a r ea. However, many l a r g e s c a l e s i m u l a t i o n s of n a t i o n a l , r e g i o n a l , and urban economies have i n c l u d e d a l a n d use or l a n d a l l o c a t i o n s e c t o r w i t h i n t h e i r s t r u c t u r e . The I n t e r - I n s t i t u t i o n a l P o l i c y Simulator (I.I.P.S.) model c u r r e n t l y being c o n s t r u c t e d a t the U n i v e r s i t y o f B r i t i s h Columbia c o n t a i n s such a s e c t o r . Though t h i s model i s h e a v i l y b i a s e d towards a study of those lan d uses t h a t p r o v i d e employment or housing ( n e g l e c t i n g , somewhat, l a n d uses such as t r a n s p o r t a t i o n f u n c t i o n s and r e c r e a t i o n a l uses) i t does r e p r e s e n t a v e r y u s e f u l work and, f o r the purposes of t h i s e x p o s i t i o n , a very t y p i c a l example of the use of s i m u l a t i o n to study l a n d use problems. The purpose of the l a n d use s e c t o r i n t h i s model (I.I.P.S.) i s not one of f i n d i n g l a n d use p a t t e r n s t h a t o p t i m i z e or maximize some measure of s o c i a l w e l f a r e , but i s i n s t e a d to 10 Thomas H. Naylor, " P o l i c y S i m u l a t i o n Experiment 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 Economics, 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 S i m u l a t i o n Approach. Oregon A g r i c u l t u r a l Experimental S t a t i o n S p e c i a l Report No. 224, November, 1966. 22b p r e d i c t or f o r e c a s t f u t u r e l a n d use requirements n e c e s s i t a t e d by c e r t a i n changes i n the p o p u l a t i o n and labour f o r c e , g i v e n h i s t o r i c a l knowledge on how the l a n d use system r e a c t e d to such changes i n the p a s t . For example, when i n f o r m a t i o n on p o s s i b l e p o p u l a t i o n and labour f o r c e i n c r e a s e s are entered i n t o the model, the employment l o c a t i o n s u b s e c t i o n s i n the model determine where the a d d i t i o n a l l a b o u r e r s w i l l work and what t h e i r work w i l l c o n s i s t of (manufacturing, food p r o c e s s i n g , and so on). The t r a n s p o r t a t i o n and housing s u b s e c t i o n s w i l l then determine where the a d d i t i o n a l p o p u l a t i o n w i l l l i v e , the type of housing they w i l l l i v e i n and the route by which the l a b o u r e r s w i l l r each t h e i r p l a c e of employment. Many oth e r urban and r e g i o n a l s i m u l a t i o n models can be found which c o n t a i n l a n d use s e c t o r s t h a t are very s i m i l a r to the l a n d use submodel found i n the I n t e r - I n s t i t u t i o n a l 12 P o l i c y S i m u l a t o r . As 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 which c o n t a i n such s e c t o r s have been c o n s t r u c t e d . Kresge's model of the P a k i s t a n i economy and Holland's s i m u l a t i o n of the Indian economy are two l a r g e s c a l e n a t i o a n l models which i n c l u d e 13,14 l a n d a l l o c a t i o n s e c t o r s . 12 See the May, 1965 i s s u e ( V o l . 31, No. 2) of the J o u r n a l o f the American I n s t i t u t e of Planners f o r examples of urban and r e g i o n a l models which 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 David T. Kresge, "A S i m u l a t i o n Model f o r Economic P l a n n i n g : A P a k i s t a n Example," Economic Development Report No. 81, Development A d v i s o r y S e r v i c e , Harvard 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 of an Economy wit h Development and Trade Problems," American Economic Review, 52: 408-430, June, 1962. ' Wherever s i m u l a t i o n models have been employed by l a n d use p l a n n e r s , a t l e a s t i n a l l of the examples i n the l i t e r a t u r e s t u d i e d by t h i s w r i t e r , the o v e r r i d i n g concern has been f o r f o r e c a s t i n g or e s t i m a t i n g f u t u r e l a n d use p a t t e r n s given some s p e c i f i c s e t of l a n d use p o l i c i e s , u s u a l l y those p r e v a i l i n g a t the p r e s e n t . O p t i m i z i n g l a n d use a l l o c a t i o n , t h a t i s , f i n d i n g the l a n d use p a t t e r n t h a t maximizes or s a t i s f i e s some se t o f p u b l i c p o l i c y g o a l s or o b j e c t i v e s , has r a r e l y been d i s -cussed. T h i s i s probably due to the f a c t t h a t s i m u l a t i o n i s not an o p t i m i z i n g procedure per se. However, Naylor b e l i e v e s t h a t s i m u l a t i o n models are p o t e n t i a l l y s u p e r i o r to c u r r e n t o p t i m i z i n g m o d e l s , t e s p e c i a l l y those t h a t employ some s o r t of 15 s o c i a l w e l f a r e f u n c t i o n i n the o p t i m i z a t i o n . I f the p o l i c y makers are not w i l l i n g , . o r not a b l e to d e f i n e or q u a n t i f y t h e i r w e l f a r e f u n c t i o n s or even d e s i r e d t a r g e t v a l u e s f o r v a r i o u s output v a r i a b l e s , then any methodology based on the premise t h a t t h i s i n f o r m a t i o n i s a v a i l a b l e , a c c o r d i n g to Naylor, "cannot be expected to y i e l d r e s u l t s t h a t are p a r t i c u l a r l y 16 u s e f u l to the p o l i c y maker." Naylor goes on to suggest t h a t the b e s t a l t e r n a t i v e approach i s a c o o p e r a t i v e e f f o r t i n which the a n a l y s t , u s i n g a s i m u l a t i o n model, shows the p o l i c y maker 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 and a i d s him i n seeking and a t t a i n i n g , v i a repeated s i m u l a t i o n t r i a l s , a p o s i t i o n which the p o l i c y maker f e e l s to be h i s w e l f a r e maximum. T h i s approach does appear to be reasonable i f i n f o r -mation 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 o n l y 15 16 Thomas H. N a y l o r , op. c i t . I b i d . p. 264. i f the number of a l t e r n a t e p o l i c y d e c i s i o n s i s reasonably l i m i t e d . For as p o l i c y a l t e r n a t i v e s i n c r e a s e , the number of p o s s i b l e p o l i c y combinations may prove t o o c c o s t l y and time consuming t o be t e s t e d v i a t h i s e s s e n t i a l l y t r i a l and e r r o r 17- 18 approach. Though work by Conway and Zusman and Amiada i n f i n d i n g optimum seeking 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 has been encouraging, many r e s e a r c h e r s f e e l t h a t much more work i s needed i n t h i s area b e f o r e s i m u l a t i o n can progress as a l e g i t i m a t e o p t i m i z i n g procedure. U n t i l then, except f o r r e l a t i v e l y simple problems, the r o l e o f s i m u l a t i o n models i n l a n d use p l a n n i n g w i l l probably remain c o n f i n e d to one of f o r e c a s t i n g and p r e t e s t i n g s p e c i f i c p o l i c y d e c i s i o n s . O p e r a t i o n a l Games Close r e l a t i v e s of mathematical s i m u l a t i o n models are d e v i c e s known v a r i o u s l y i n the l i t e r a t u r e as o p e r a t i n g , o p e r a t i o n a l or systemic games. A game, l i k e a mathematical model, i s a r e p r e s e n t a t i o n of a r e a l world system. The major d i f f e r e n c e between the two l i e s i n the s p e c i f i c a t i o n o f the r e l a t i o n s which they attempt to r e p r e s e n t . Though most games c o n t a i n some mathematical s p e c i f i c a t i o n , most of the s t r u c t u r e of gaming models 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 , or i n w r i t i n g . Mathematical models are, o f course, u s u a l l y completely mathematically s p e c i f i e d . D e s p i t e these d i f f e r e n c e s i n 17 R. W. Conway, "Some T a c t i c a l Problems i n D i g i t a l S i m u l a t i o n " , Management Sc i e n c e , 10: 47-61, October, 1963. 18 P. Zusman and A. Amiada, " S i m u l a t i o n : A T o o l of Farm Pl a n n i n g Under C o n d i t i o n s of. Weather U n c e r t a i n t y " , J o u r n a l of  Farm Economics, 47: 574-594, August, 1965. 25 i n s p e c i f i c a t i o n many a n a l y s t s c l a s s i f y games as s i m u l a t i o n models. Ac c o r d i n g t o F e l d t : . . . 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 . I t i s a s i m u l a t i o n i n which a l l of most of the d e c i s i o n s as t o outcomes of the events being r e p r e s e n t e d are l e f t i n the hands of human p l a y e r s , w i t h r e l a t i v e l y minor d e c i s i o n s and a c c o u n t i n g problems handled e i t h e r by human op e r a t o r s or by c o m p u t e r s . ^ Though o p e r a t i o n a l games undoubtedly have many a p p l i c a t i o n s i n the 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 d e v i c e s to a i d i n understanding the workings of a l a n d use system, they do not seem too w e l l s u i t e d to c o n t r i b u t e to problems of l a n d use 20 f o r e c a s t i n g and p l a n n i n g . Because games must be simple enough f o r the p l a y e r s t o l e a r n and understand, a hig h degree of a b s t r a c t i o n n e c e s s a r i l y renders; gaming models i n a c c u r a t e and u n s u i t a b l e as p r e d i c t i v e o r p l a n n i n g d e v i c e s . Mathematical Programming Models A t h i r d major approach i n the use o f mathematical models as an a i d to d e c i s i o n making l i e s i n the f i e l d o f math-e m a t i c a l programming. Mathematical programming concerns i t s e l f with l o c a t i n g o p t i m a l o r bes t s o l u t i o n s t o a giv e n system, r a t h e r than simply f o r e c a s t i n g o r p r e d i c t i n g an expected s o l u t i o n . While i n p u t - o u t p u t and s i m u l a t i o n models can a l s o be used to seek optima, the search procedures employed are not an i m p l i c i t 19 A l l a n F e l d t , P l a y e r ' s Manual f o r CLUG; Community  Land Use Game (New York: C o l l i e r Macmillan Canada L t d , 1972), p. 1. 20 F e l d t ' s l a n d use game (see above footnote) p r o v i d e s a good i l l u s t r a t i o n o f the use of o p e r a t i o n a l games as educ-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 of the model s t r u c t u r e and u s u a l l y i n v o l v e rudimentary exhaustive searches of a r e l a t i v e l y s m a l l number of a l t e r n a t i v e s . Mathematical programming, on the o t h e r hand, always has e f f i c i e n t s earch procedures i m p l i c i t i n i t s s t r u c t u r e . A c o r r o l l a r y t o t h i s i s t h a t mathematical programming always c o n t a i n s some mathematically s p e c i f i e d f u n c t i o n , whose va l u e i s to be o p t i m i z e d , and which i s an adequate r e p r e s e n t a t i o n o f r e a l world d e s i r e s and g o a l s t h a t apply i n the system under study. Besides t h i s mathematical statement of o b j e c t , which i s n o t h i n g more than a c r i t e r i o n by which to judge a l t e r n a t e s o l u t i o n s , mathematical programming models a l s o c o n t a i n a s e t or a r r a y of mathematical e x p r e s s i o n s which s p e c i f y the resource l i m i t a t i o n s and 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 which e x i s t i n the r e a l system being a b s t r a c t e d . These e x p r e s s i o n s are b a s i c a l l y a statement of p r o d u c t i o n p o s s i b i l i t i e s g i v e n f i x e d amounts of i n p u t s and a g i v e n technology. When t h i s i n f o r m a t i o n i s f e d i n t o an 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 computer c o n t a i n i n g an e f f i c i e n t programming a l g o r i t h m , an o p t i m a l s o l u t i o n , i f one e x i s t s , i s q u i c k l y and r e a d i l y produced. I f a s o l u t i o n i s i n f e a s i b l e , or unbounded, or i f more than one g l o b a l optimum i s p o s s i b l e , then t h i s i n f o r m a t i o n i s a l s o made a v a i l a b l e . Because most programming models assume t h a t technology, r e s o u r c e s , and p r i c e s are a l l f i x e d i n any one g i v e n s i t u a t i o n , mathematical programming i s e s s e n t i a l l y a s t a t i c a n a l y s i s , and as such would seem e l i g i b l e f o r much o f the c r i t i c i s m l e v e l l e d a t s i m u l a t i o n and i n p u t - o u t p u t a n a l y s i s i n t h i s r e g a r d . However, one of the major advantages of mathematical programming i s the ease i n which changes i n technology, r e s o u r c e s , and p r i c e s may be s t u d i e d . P r i c e ranging, r i g h t hand s i d e ranging, parametric programming, and o t h e r methods of s e n s i t i v i t y a n a l y s i s can e a s i l y be conducted to determine, c e t e r i s p a r i b u s , the e f f e c t of a change i n one p r i c e or resource l e v e l o r , mutatis mutandis, the e f f e c t of s e v e r a l concurrent changes. Techniques have a l s o been developed i n programming with v a r i a b l e i n p u t c o e f f i c -21 i e n t s . Thus, mathematical programming, while s t a t i c i n nature, does l e n d i t s e l f 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 systems being s t u d i e d . The most commonly employed type of mathematical programming used today i s the Simplex method of l i n e a r programming. A c c o r d i n g to M c M i l l a n : . . . the simplex machine |Tof l i n e a r programming] i s the Lit most a v a i l a b l e , the most f r e q u e n t l y employed, and the most w i d e l y understood o p t i m i z i n g machine. R e c u r r i n g a r t i c l e s i n p r o f e s s i o n a l and trade l i t e r a t u r e a t t e s t to the 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 of r e a l - w o r l d problems t h a t they 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 simplex machine.^2 However, the assumptions employed i n l i n e a r programming do somewhat r e s t r i c t the r e l e v a n c e of i t s a p p l i c a t i o n s . The problems of l i n e a r i t y , a d d i t i v i t y , d i v i s i b i l i t y , f i n i t e n e s s , 23 and s i n g l e - v a l u e e x p e c t a t i o n s are w e l l known. As w e l l , the s t a t i c t e c h n i c a l c o e f f i c i e n t s n e c e s s a r i l y mean constant r e t u r n s 21 For a d e s c r i p t i o n of t h i s and o t h e r methods of examining p r i c e , r e s o u r c e , and technology changes see E a r l . 0. Heady and W i l f r e d Candler, L i n e a r Programming Methods (Ames: Iowa St 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 chapters 7, 8, 16. 22 M c M i l l a n , op. c i t . , p. v i of p r e f a c e . 23 For a d i s c u s s i o n of these assumptions, see Heady and Candler, op. c i t . , pp. 17-18. 28 to s c a l e , an assumption t h a t cannot always be j u s t i f i e d . The problem o f f i n i t e n e s s , a l s o , o f t e n l e a d s to one of h e t e r o g e n e i t y . I f not enough a c t i v i t i e s are s p e c i f i e d then the a c t i v i t i e s can become too heterogeneous t o al l o w e s t i m a t i o n of i n t e r e s t i n g 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 . Many attempts have been made to overcome the d i f f i -c u l t i e s posed by these assumptions, e i t h e r through refinements to the Simplex method of l i n e a r programming, or by use of some oth e r form of programming. Quad r a t i c and oth e r n o n - l i n e a r programming forms can be used t o more a c c u r a t e l y d e p i c t r e a l w orld r e l a t i o n s and a v o i d many o f the problems a s s o c i a t e d with 24 l i n e a r i t y . Zero-one and o t h e r types o f i n t e g e r programming 25 may be used 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 dynamic programming can be used t o overcome the problems of a p p l y i n g a s t a t i c a n a l y s i s t o i n t e r t e m p o r a l or s e q u e n t i a l l y 26 ordered problems. S t o c h a s t i c programming can i n t r o d u c e v a r -27 i a t i o n i n t o the model and a l l o w f o r ranges o f e x p e c t a t i o n s . However, these r e l a t i v e l y 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 not enjoyed widespread use and accepatance i n most areas, simply because o f t h e i r e x c e e d i n g l y demanding time and data requirements. A c c o r d i n g t o Re i s c h : N o n - l i n e a r , i n t e g e r , dynamic l i n e a r , and dynamic 24 25 McM i l l a n , op. c i t . , pp. 173-219. I b i d . , pp. 312-99, 26 I b i d . , pp. 242-270, or E a r l 0. Heady and 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 Models f o r Optimal Farm and Home Plans-"'"-Journal o f Farm Economics, XLI, February, 1959. 27 See K. D. Cocks, " D i s c r e t e S t o c h a s t i c Programming", Management Science, V o l . 15 No. 1, 1968, pp. 72-79. 29 programming as w e l l as s t o c h a s t i c programming have not proved t o be a p p l i c a b l e t h u s - f a r . . . . The main reasons are (a) necessary data cannot be s u p p l i e d ; (b) s e t t i n g up the models takes too much p r e p a r a t i o n time; (c) r e q u i r e -ments of computer time and storage c a p a c i t y f o r r e a l i s t i c models are s t i l l too high ; and (d) i n p u t s t h a t go beyond the requirements f o r standard [ l i n e a r ] programming are not w i t h i n a reasonable r e l a t i o n t o 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 gained by those more s o p h i s t i c a t e d methods.^8 A f t e r 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 are a v a i l a b l e t o plan n e r s today, R e i s c h goes on to s t a t e t h a t "only l i n e a r programming can be c o n s i d e r e d as a w i d e l y accepted and used 29 p l a n n i n g method." C e r t a i n l y l i n e a r programming has many advantages over the ot h e r more i n v o l v e d forms o f programming. I t i s mathematically very simple y e t t h e o r e t i c a l l y e l e g a n t a t the same time. I t s data requirements are not as severe and set-up time i s much s h o r t e r than f o r other more complicated m a t r i x f o r m u l a t i o n s . Computer requirements are not o v e r l y demanding. I t i s extremely f l e x i b l e , having widespread a p p l i c -a t i o n s . I t i s a r e l a t i v e l y easy t o understand method and i s q u i t e simple 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 the layman. And probably most i m p o r t a n t l y , i t has gained the r e s p e c t a b i l i t y of widespread use and a c c e p t a b i l i t y . Nor does the nature of i t s assumptions o v e r l y r e s t r i c t the a p p l i c a t i o n o f l i n e a r programming. N o n - l i n e a r r e a l world f u n c t i o n s can be c l o s e l y approximated by step-wise l i n e a r 28 Erwin R e i s c h , "Proven Tools f o r Micro P l a n n i n g and D e c i s i o n s " , Economic Models 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 ed. 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 of l i n e a r i t y . A d d i t i v i t y and d i v i s i b i l i t y tend to c r e a t e problems a t the micro l e v e l and are not u s u a l l y encountered i n macro p l a n n i n g models. The problem of f i n i t -ness i s e s s e n t i a l l y one of taxonomy and measurement and does not, per se, r e p r e s e n t a s t r u c t u r a l d e f i c i e n c y i n l i n e a r programming. C e r t a i n l y t h i s problem i s no g r e a t e r w i t h l i n e a r programming than w i t h o t h e r mathematical models. The d i f f i c u l t i e s i n h e r e n t i n the s t a t i c , s i n g l e - v a l u e e x p e c t a t i o n s employed i n l i n e a r programming can be l e s s e n e d a gre a t d e a l through s e n s i t i v i t y a n a l y s i s . L i n e a r programming i s the most dominant form o f mathematical m o d e l l i n g used i n l a n d use p l a n n i n g today. The l i t e r a t u r e abounds 30 w i t h examples of i t s a p p l i c a t i o n t o problems of land a l l o c a t i o n . A p a r t i c u l a r l y i n t e r e s t i n g a p p l i c a t i o n i n t h i s area i s presented by Edwin M i l l s , who uses t h i s d e v i c e t o not o n l y a l l o c a t e o p t i m a l amounts of l a n d use but a l s o t o i n d i c a t e the op t i m a l s p a t i a l d i s t r i b u t i o n of 31 lan d uses. The advantages t h a t l i n e a r programming holds over model s t r u c t u r e s i n t h i s f i e l d are many. I t s q u a l i t i e s o f s i m p l i c i t y , f l e x i b i l i t y , and low i n p u t requirements,-.discussed above, are as much an advantage over non-programming models as they are over a l t e r n a t e mathematical models. L i n e a r programming i s a normative p r o c e s s , and as such i s s u p e r i o r t o the p o s i t i v e approaches of s i m u l a t i o n and inp u t - o u t p u t a n a l y s i s i n seeking o p t i m a l l a n d use p a t t e r n s . Perhaps the o n l y major drawback t o the use of con-30 See the b i b l i o g r a p h y i n A. G. Gardner, A L i n e a r Program- ming Model f o r Land Resource A l l o c a t i o n i n the Lower Mainland of"  B r i t i s h Columbia (unpublished M.Sc. t h e s i s i n the Department of 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, 1971), pp. 106-108. 31 Edwin S. M i l l s , "Markets and E f f i c i e n t Resource A l l o c a t i o n i n Urban Areas," Swedish J o u r n a l o f Economics, 1972, pp. 101-113. 31 ventional l i n e a r programming i n land use planning and other such areas of public decision making, i s the provision for only a single objective as a c r i t e r i o n for optimality, when in r e a l i t y many c r i t e r i a may be relevant. T r a d i t i o a n l l y , l i n e a r programs for optimal land use a l l o c a t i o n have used the marginal (value) productivity of land, land rents, production, or some such related measure of economic p r o f i t or gain as the single c r i t e r i o n against which alternate 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 in land use decisions: 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 preservation of recreational areas and green b e l t s , increasing municipal tax bases, aiding export industries, encouraging f u l l employment, i s o l a t i n g land uses to s p e c i f i c areas, minimizing p o l l u t i o n l e v e l s , encouraging certain forms of transportation, promoting low cost housing and urban renewal, maintaining density standards and other building codes, and so on. These are just a few examples of other p o l i c y objectives that have influenced land use decisions. Note that 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 with one another. Encouraging the use of the private automobile, and the minimization of p o l l u t i o n l e v e l s i s a prime example of c o n f l i c t i n g goals. Promoting low density housing standards, and encouraging the development of rapid t r a n s i t i s another, less obvious, example. Note also that v i r t u a l l y a l l of these additional goals c o n f l i c t with the maximization of land rents or some such measure of economic gain, as measured i n the private market place. Most of these p o l i c i e s ^ a r e , i n fact, attempts to a l l e v i a t e or correct misallocations of resources that r e s u l t from various e f f e c t s which are external to the private 32 economy. P o l l u t i o n , f o r example, i s a n e g a t i v e e x t e r n a l i t y which imposes enormous damages upon s o c i e t y and y e t i s not taken i n t o account by p r i v a t e d e c i s i o n makers because i t does 32 not 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 i t . A green b e l t , on the o t h e r hand, r e p r e s e n t s 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 which 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 which 33 i s a l s o not accounted f o r by the 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 a p p r o p r i a t e l a n d use d i s t r i b u t i o n s t h a t maximize o n l y one p o l i c y c r i t e r i o n , when many oth e r c r i t e r i a e x i s t , are not r e a l l y ' f i n d i n g o p t i m a l s o l u t i o n s . 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 the a p p l i c a t i o n of c o n v e n t i o n a l l i n e a r programming to l a n d a l l o c a t i o n problems. However, the r e c e n t development, by Candler and B o e h l j e , of a form of l i n e a r programming t h a t allows f o r m u l t i p l e g o a l s , seems to s u c c e s s f u l l y circumvent 34 t h i s d e f i c i e n c y . T h i s new form of l i n e a r programming seeks s o l u t i o n s t h a t o p t i m i z e a c r i t e r i o n f u n c t i o n which i s e s s e n t i a l l y a composite of s e v e r a l d i f f e r e n t , o f t e n c o n f l i c t i n g , p o l i c y g o a l s . The number of p o l i c y g o als 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 Economic Aspects of P o l l u t i o n , " (unpublished B.A. g r a d u a t i n g t h e s i s i n the Department of Economics, U n i v e r s i t y of B r i t i s h Columbia, 1971), 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 , " (unpublished essay i n the Department of A g r i c u l t u r a l Economics, U n i v e r s i t y of B r i t i s h Columbia, 1972), pp. 2-3. 34 W i l f r e d Candler and M i c h a e l B o e h l j e , "Use of L i n e a r Programming i n C a p i t a l Budgeting w i t h M u l t i p l e Goals," American  J o u r n a l of A g r i c u l t u r a l Economics, V o l . 53 No. 2, May, 1971, pp. 325-330. s i t e o b j e c t i v e f u n c t i o n i s not l i m i t e d t o any s e t number, although use of much more than a dozen may prove unwieldy. Though, t o my knowledge, t h i s m u l t i p l e goals form of l i n e a r programming has not been a p p l i e d to problems of l a n d resource a l l o c a t i o n , i t seems eminently well, s u i t e d to do so. I t has a l l the advantages of c o n v e n t i o n a l s i n g l e - g o a l l i n e a r programming wit h the added one of a l l o w i n g f o r s e v e r a l p o l i c y g o als or o b j e c t i v e s . Nor does the m u l t i p l e goals format unduly add to set-up or computer time requirements. Indeed, the c o n v e n t i o n a l Simplex a l g o r i t h m can e a s i l y be a p p l i e d to problems of t h i s -n a t u r e . Because o f the g r e a t e f f i c i e n c y o f t h i s method, problems which can u t i l i z e t h i s search procedure have substan-t i a l l y lower computer requirements. Because of the many advantages t h a t m u l t i p l e goals l i n e a r programming appears t o possess over o t h e r model types, t h i s form of programming was chosen as the methodology to be used i n t h i s study o f l a n d use p l a n n i n g i n the F r a s e r V a l l e y . Summary On the b a s i s of t h i s somewhat a b b r e v i a t e d sampling of the l i t e r a t u r e , t h r e e broad c l a s s e s of mathematical models can be d i s t i n g u i s h e d : i n p u t - o u t p u t a n a l y s i s , s i m u l a t i o n ( i n c l u d i n g gaming models), and mathematical programming. Though a l l of these techniques have some a p p l i c a t i o n i n the f i e l d o f l a n d use p l a n n i n g , i n p u t - o u t p u t and s i m u l a t i o n models have tended to remain r e s t r i c t e d to s t u d i e s of h i s t o r i c a l data, s h o r t term f o r e c a s t i n g , and p r e t e s t i n g of s p e c i f i c p o l i c y 34 changes. Mathematical programming, on the ot h e r hand, o f f e r s a r e a d i l y useable, s y s t e m a t i c format with which to s e l e c t best or o p t i m a l s o l u t i o n s from a wide v a r i e t y o f a l t e r n a t i v e s . Although they are s h o r t term and s t a t i c i n nature, programming models are extremely f l e x i b l e and r e a d i l y a l l o w f o r a n a l y s i s of p o s s i b l e v a r i a t i o n i n the r e a l world system under o b s e r v a t i o n . Of the s e v e r a l d i f f e r e n t types of mathematical programming, l i n e a r programming seems bes t s u i t e d to problems of o p t i m a l p u b l i c d e c i s i o n making. Though the assumptions employed i n t h i s s i m p l e s t of a l l types of programming do r e s t r i c t i t s v a l i d i t y somewhat, they do not appear o v e r l y l i m i t i n g i n t h e i r a p p l i c -a t i o n t o macro d e c i s i o n models. L i n e a r programming appears w e l l s u i t e d to problems of l a n d use p l a n n i n g and o p t i m a l l a n d resource a l l o c a t i o n . Indeed, i t s use i n t h i s area i s q u i t e widespread. Because many d i f f e r e n t p o l i c y goals and o b j e c t i v e s , i n c l u d i n g ones which c o n f l i c t , operate on d e c i s i o n making i n the lan d use f i e l d , a r e c e n t m o d i f i c a t i o n t o l i n e a r programming which allows f o r a m u l t i p l e g oals o b j e c t i v e f u n c t i o n , seems e s p e c i a l l y w e l l s u i t e d f o r a p p l i c a t i o n i n t h i s area. The f o l l o w i n g chapter d i s c u s s e s t h i s new methodology i n d e t a i l and uses i t to examine a h y p o t h e t i c a l l a n d use problem. 35 Chapter 5 THE USE OF LINEAR PROGRAMMING WITH MULTIPLE GOALS IN LAND RESOURCE ALLOCATION The Format and Methodology of Conv e n t i o n a l L i n e a r Programming The survey o f the l i t e r a t u r e presented i n the f o r e -going chapter suggested t h a t the b e s t a v a i l a b l e model s t r u c t u r e f o r d etermining o p t i m a l l a n d use p a t t e r n s i s a form of l i n e a r programming which c o n t a i n s a m u l t i p l e g o a l o b j e c t i v e f u n c t i o n . The d i s c u s s i o n i n t h i s chapter w i l l c e n t e r on e x p l a i n i n g t h i s r e l a t i v e l y new programming method and p r o v i d i n g an elementary example o f i t s a p p l i c a t i o n t o l a n d use problems. Co n v e n t i o n a l l i n e a r programs can be expressed as f o l l o w s : f i n d the k x 1 v e c t o r o f a l t e r n a t i v e a c t i v i t i e s such t h a t : Z = z(X) a maxxmum su b j e c t t o : AX 5 B and, X > 0 where X k x 1 v e c t o r o f a l t e r n a t i v e a c t i v i t i e s (X]_, X2, X3 . . . X]^) , and Z some ( l i n e a r ) 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 s of the a c t i v i t i e s , and A an m x k matr i x of t e c h n i c a l c o e f f i c i e n t s which 36 determine i n t e r a c t i v i t y relationships and le v e l s , and B = an m x 1 vector of r e s t r a i n t s on i n t e r a c t i v i t y flows and level s Note that t h i s format assumes that: (1) the objective or c r i t e r i o n function i s li n e a r ; (2) the re s t r a i n t s on a c t i v i t y l e v e l s can be expressed as l i n e a r i n e q u a l i t i e s ; and (3) a l o c a l maximum i s a global maximum, and that the di r e c t i o n of improvement i n the objective function can be determined at any point. The f i r s t two of these assumptions are i n keeping wifehfethe l i n e a r i t y properties necessary for the programming manipulation of the problem matrix. The t h i r d assumption i s necessary i f a simple search procedure such as the Simplex Algorithm i s to be used to f i n d the global optimum. The Application of Linear Programming to a Hypothetical Land Use Problem To better i l l u s t r a t e t h i s programming method, consider the following hypothetical s i t u a t i o n . The planning commission for a small municipality i s considering how to zone the municipality's 1000 acres so that t o t a l land value i s at a maximum. Because land value was f e l t to be a measure of the productivity of the land, the commission f e l t that maximizing t o t a l land value would lead to a maximum of l o c a l production. Since t o t a l l o c a l production was the source of a l l income for the inhabitants of the region, maximizing production would also maximize incomes. The commission 37 has decided that land may only be put to one of three uses: 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 (or i t may be l e f t unused). Two hundred and f i f t y acres of 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. It i s also known that 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 requires twenty thousand do l l a r s of c a p i t a l , while an acre i n a g r i c u l t u r a l employs only one person and requires but one thousand d o l l a r s of s t a r t i n g c a p i t a l . The regions labour force comprises 1000 people. The planning commission has decided that zoning i s necessary to ensure orderly development but i s uncertain as to the amounts of land to be zoned to each category so that t o t a l community land value i s at a maximum. Residential land i s valued at three thousand d o l l a r s per acre, while 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 an acre of agriculture, one thousand. Setting up t h i s problem i n terms of the above l i n e a r programming tableau we get: Land A c t i v i t i e s ( a l l ^0) RESIDENTIAL INDUSTRIAL AGRICULTURAL RHS (xx) (x2) (x3) Objective: Maximum of 3000 7000 1000 a max. Subject to Constraints on: Available Land Available Labour Available Capital Minimum Food Minimum Housing 1 0 10 0 1 1 20 20 0 0 1 1 1 1 0 ^1000 ^1000 ^5000 > 250 * 100 38 In t h i s statement the objective function to be maximized i s the t o t a l community land value. The f i r s t constraint ensures that no more land than i s available i s allocated. The second constraint l i m i t s the use of labour to the e x i s t i n g supply and the t h i r d restrains the use of c a p i t a l . The fourth and f i f t h r e s t r a i n t s ensure that minimum food and housing requirements are met. Though these l a s t two constraints are i n the form of "greater than or equal to" i n e q u a l i t i e s they can e a s i l y be made to conform with the rest of the i n e q u a l i t i e s by simply multiplying each l i n e by negative unity. Note also that the non-negativity of each a c t i v i t y column i s an i m p l i c i t part of a l i n e a r programming tableau and need not be included in the l i s t of e x p l i c i t constraints. When t h i s information i s fed into an optimizing routine (usually v i a a d i g i t a l computer) the solution i s quickly and r e a d i l y produced. In t h i s case the objective function i s at a maximum when: RESIDENTIAL LAND USES, Xi = 400 acres, INDUSTRIAL LAND USE, X 2 = 21 acres, AGRICULTURAL LAND USE, X 3 = 579 acres, and t o t a l community land value has reached $1,926,000. Thus given some information about the planning commissions desires, as well as some data on technical p o s s i b i l i t i e s , a l i n e a r program can be employed to optimize the decision makers (single goal) c r i t e r i o n function. 39 The Format and Methodology o f M u l t i p l e Goals L i n e a r Programming However, as has been p r e v i o u s l y d i s c u s s e d , many goals may be pursued i n a decision-making process such as land use p l a n n i n g . In the context o f the above problem, f o r example, the p l a n n i n g commission may a l s o be concerned w i t h m i n i m i z i n g p o l l u t i o n 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 employment, as w e l l as a m u l t i p l i c i t y o f oth e r g o a l s , many of which may c o n f l i c t with!;.maximizing t o t a l community land v a l u e . In t h i s case, c o n v e n t i o n a l l i n e a r programming of the type j u s t o u t l i n e d , w i t h a s i n g l e g o a l c r i t e r i o n by which compare a l t e r n a t e s o l u t i o n s , no lon g e r i s s a t i s f a c t o r y . Instead m u l t i p l e goals l i n e a r programming should be used. In g e n e r a l , the format f o r m u l t i p l e goals l i n e a r programming can be expressed as f o l l o w s : f i n d the k x 1 v e c t o r o f a l t e r n a t i v e a c t i v i t i e s such t h a t : Z = z g i ( X ) , g2(X), . . . g n(X) a maximum s u b j e c t t o : AX £ B and, X > 0 , where X = k x 1 v e c t o r of a l t e r n a t i v e a c t i v i t i e s (X]_, X 2, X3, . . . X^) , and Z = some ( 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 of 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) = the i t h ( l i n e a r ) g o a l f u n c t i o n , and 40 A = an m x k matrix of t e c h n i c a l c o e f f i c i e n t s which determine 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 of r e s t r a i n t s on i n t e r a c t i v i t y flows and l e v e l s . While t h i s format employs a l l the assumptions of c o n v e n t i o n a l l i n e a r programming, i t assumes i n a d d i t i o n t h a t the i n d i v i d u a l g o a l f u n c t i o n s can be expressed on some l i n e a r s c a l e . Often goals are measured i n monetary terms, i n which .: case t h i s assumption merely demands t h a t the marginal u t i l i t y of an a d d i t i o n a l d o l l a r i s constant, t h a t i s , one hundred d o l l has ten times the u t i l i t y o f ten d o l l a r s . I f the g o a l cannot be measured i n monetary terms, then some oth e r l i n e a r s c a l e must be d e v i s e d , however a r b i t r a r y . An a p p r o p r i a t e s c a l e f o r a p o l l u t i o n m i n i m i z i n g g o a l might be 0 to 100 w i t h 100 (units) to the p o l l u t i o n g e n e r a t i n g a c t i v i t y which generates the g r e a t e s t amount of p o l l u t i o n , zero (units) to the l e a s t p o l l u t i o n g e n e r a t i n g a c t i v i t y , and a p p r o p r i a t e i n t e r m e d i a t e v a l u e s f o r the other a c t i v i t i e s t h a t l i e i n the continuum between these extremes. In o r d e r to do t h i s , a l l t h a t i s •;• r e q u i r e d i s an assumption t h a t the marginal u t i l i t y o f an 1 a d d i t i o n a l s c a l e u n i t i s h e l d constant. These s c a l e s do not n e c e s s a r i l y have to be based on any t e c h n i c a l or f i n a n c i a l continuum. What i s r e q u i r e d i s t h a t they be based on the d e c i s i o n maker's p e r c e p t i o n of h i s u t i l i t y f u n c t i o n i n r e s p e c t 1 In o t h e r words, s c a l e v a l u e s must be a d d i t i v e . 41 to each g o a l i n i s o l a t i o n , though undoubtedly t e c h n i c a l and f i n a n c i a l i n f o r m a t i o n t h a t i s a v a i l a b l e to the d e c i s i o n maker., w i l l a i d him i n making u t i l i t y judgements. In o t h e r words, although one acre of a c t i v i t y A may produce f i v e times the p h y s i c a l amount of p o l l u t i o n t h a t one acre of a c t i v i t y B produces, one should not presume t h a t the d e c i s i o n maker's e s t i m a t i o n of the r e l a t i v e d i s u t i l i t y produced by these two a c t i v i t i e s w i l l be i n the same r a t i o . Another p o i n t to be noted i s t h a t each g o a l f u n c t i o n should be l i n e a r i z e d i n i s o l a t i o n . The d e c i s i o n maker must c o n s i d e r o n l y the u t i l i t y c o n t r i b u t i o n s t h a t the a c t i v i t i e s make to the g o a l f u n c t i o n i n q u e s t i o n , and to none ot h e r . C o n t r i b u t i o n s to o t h e r g o a l s , as w e l l as t r a d e - o f f c o n s i d e r a t i o n s between goals e n t e r a t a l a t e r stage o f the a n a l y s i s . Under a m u l t i p l e goals format, there i s no l i m i t t o the number of goals t h a t may be s p e c i f i e d , although the a b i l i t y o f the d e c i s i o n making group t o i d e n t i f y and s p e c i f y g o a l s , as w e l l as the time and data l i m i t a t i o n s of doing so, w i l l undoubtedly a c t as a c o n s t r a i n t on the number of g o a l s employed i n the a n a l y s i s . A M u l t i p l e Goals V e r s i o n of the H y p o t h e t i c a l Land Use Problem Once the goals have been d e f i n e d and the g o a l f u n c t i o n s s c a l e d t o the s a t i s f a c t i o n of the d e c i s i o n making group, t h i s i n f o r m a t i o n can be added to o t h e r i n f o r m a t i o n on the o b j e c t i v e f u n c t i o n , 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 c o n s t r a i n t s on a c t i v i t y l e v e l s , and then examined i n a l i n e a r programming framework. To i l l u s t r a t e , we r e c a l l the example used above. Suppose now t h a t , i n a d d i t i o n to wanting to maximize community land v a l u e , the p l a n n i n g commission a l s o d e s i r e d to minimize p o l l u t i o n , and t h a t these two goals were the o n l y two goals t h a t they c o n s i d e r e d r e l e v a n t to the i s s u e . Suppose i t was f e l t t h a t an acre of a g r i c u l t u r a l l a n d c r e a t e d zero p o l l u t i o n d i s u t i l i t y w h i l e an acre of i n d u s t r y generated one hundred u n i t s of p o l l u t i o n d i s u t i l i t y . An acre of r e s i d e n t i a l l a n d was f e l t t o be 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 u n i t s of d i s u t i l i t y . The problem can now be i l l u s t r a t e d as f o l l o w s : A c t i v i t i e s ( a l l ^0) Land A c t i v i t i e s A ccounting A c t i v i t i e s < H E-i — £ H W X Q ~ H « CN H X W —' D Q 53 H D EH O X U ~ H Pi o W Q 53 H P X O to H < EH — Q D in \A X < < EH > rtj O EH PM • O O EH EH EH Pi O b j e c t i v e : Maximum of 0 0 0 x2 a max. Subject t o C o n s t r a i n t s on: A v a i l a b l e Land 1 1 1 0 0 ^1000 A v a i l a b l e Labour 0 20 1 0 0 ^1000 A v a i l a b l e C a p i t a l 10 20 1 0 0 ^5000 Minimum Food 0 0 1 0 0 ^ 250 Minimum Housing 1 0 0 0 0 * 100 Goal C o n t r i b u t i o n s to: T o t a l Land Value T o t a l P o l l u t i o n -3000 -7000 -1000 -10 -100 0 1 0 0 1 0 0 In t h i s t a b l e a u , the o b j e c t i v e f u n c t i o n t o be maximized i s a weighted composite of the separate goals o f land v a l u e maximization and p o l l u t i o n m i n i m i z a t i o n . and \ 2 a r e the weights t h a t the d e c i s i o n making group a s s i g n s t o the land v a l u e g o a l and the p o l l u t i o n g o a l r e s p e c t i v e l y . The i n d i v i d u a l p o l l u t i o n d i s u t i l i t y u n i t s t h a t the v a r i o u s l a n d a c t i v i t i e s generate are summed i n t o the t o t a l p o l l u t i o n (X5) column and the p o l l u t i o n g o a l w e i g h t , ) \ 2 , i s a p p l i e d to t h i s t o t a l t o determine the c o n t r i b u t i o n t h a t t o t a l p o l l u t i o n makes t o the composite o b j e c t i v e f u n c t i o n . Since the d e s i r e i s t o minimize p o l l u t i o n , X . 2 must take on a negative v a l u e i n o r d e r to c r e a t e a m i n i m i z i n g e f f e c t (minimizing a p o s i t i v e v a r i a b l e can be accomplished by maximizing i t s n e g a t i v e ) . The t o t a l l a n d v a l u e g o a l weight, w i l l be p o s i t i v e as the d e s i r e here i s to maximize community l a n d v a l u e , which has been summed i n t o the t o t a l l a n d value column, X4. Note t h a t the c o n t r i b u t i o n s t h a t each acre o f d i f f e r e n t land use makes towards the two goals are entered as n o n - p o s i t i v e c o e f f i c i e n t s i n the l a s t two c o n s t r a i n t l i n e s i n t h i s t a b l e a u . T h i s procedure ensures t h a t the t o t a l s f o r these c o n t r i b u t i o n s , t h a t i s , t o t a l l a n d v a l u e , and t o t a l p o l l u t i o n u n i t s , w i l l appear as p o s i t i v e amounts i n the two t o t a l a c t i v i t y accounting columns, X^ and X^. T h i s procedure i s necessary t o ensure t h a t the n o n - n e g a t i v i t y c o n s t r a i n t on a l l a c t i v i t y columns i s upheld. When t h i s i n f o r m a t i o n i s f e d i n t o an o p t i m i z i n g r o u t i n e , l i n e a r programming-will y i e l d a n l e f f i c i e n t s o l u t i o n 44 to t h i s l a n d use problem f o r any given s e t o f g o a l weights (N]_ a n d Y v j ) • The problem now becomes one of choosing the d e s i r e d l e v e l s f o r the g o a l weights. S t a t e d a l t e r n a t i v e l y , the problem i s one of f i n d i n g the p a r t i c u l a r e f f i c i e n t s o l u t i o n which i s p r e f e r r e d by the d e c i s i o n making body. Candler and B o e h l j e propose what amounts t o an 2 i t e r a t i v e approach to t h i s problem. An i n i t i a l e f f i c i e n t s o l u t i o n i s found f o r some s p e c i f i e d s e t of g o a l weights and i s presented t o the d e c i s i o n makers. The d e c i s i o n makers then i n d i c a t e , perhaps w i t h the a i d o f parametric programming around the o r i g i n a l s o l u t i o n , i n which d i r e c t i o n they f e e l a p r e f e r r e d s o l u t i o n l i e s . That i s , they s h i f t the r e l a t i v e g o a l weights to what they t h i n k w i l l l e a d t o an improved s o l u t i o n ("improved" i n terms of t h e i r p r e f e r e n c e f u n c t i o n ) . A s o l u t i o n i s then found f o r t h i s new s e t of r e l a t i v e g o a l weights and the process begins a g a i n . The d e c i s i o n makers examine the new s o l u t i o n and again r e s p e c i f y the g o a l weights to s h i f t the s o l u t i o n i n the d i r e c t i o n i n which they f e e l a s t i l l p r e f e r r e d s o l u t i o n l i e s . A c c o r d i n g to Candler and B o e h l j e , t h i s process of s e q u e n t i a l l y i d e n t i f y i n g improved s o l u t i o n s (budgets) "would continue u n t i l a budget i s found t h a t i s b e t t e r than a l l adjacent budgets or u n t i l the r a t e of g a i n i n the o b j e c t i v e 3 f u n c t i o n i s lower than the per u n i t c o s t of f u r t h e r a n a l y s i s . " 2 Candler and B o e h l j e , op. c i t . pp. 329-330. 3 I b i d . p. 330. 4 5 The s p e c i f i c a t i o n of i n i t i a l s t a r t i n g weights does pose somewhat of a problem. Since l i n e a r programming search procedures assume that any l o c a l optimum found i s a global optimum, the actual s t a r t i n g point of the search presents no 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 considerations of the costs of analysis, would indicate that the s t a r t i n g point should be i n the general v i c i n i t y of the optimum, or as close to i t as possible. To t h i s end, the decision making group should make an attempt to specify the i n i t i a l set of weights in accordance with t h e i r perception of the trade-offs that e x i s t between goals. Going back to our example problem, i f the land use planning commission f e l t that foregoing 100 units of p o l l u t i o n i s approximately worth increasing land value by $1000, the implication would be that the i n i t i a l p o l l u t i o n goal weight could be set a $10 of community land value per unit of p o l l u t i o n . That i s , i f w a s set at 1.0, a good s t a r t i n g point would be to setYv? equal to -10.0. I f , i n f a c t , these two weights are used the following i n i t i a l solution i s obtained: the objective function i s at a maximum when: RESIDENTIAL LAND USE, X± = 100 acres INDUSTRIAL LAND USE, X 2 = n i l AGRICULTURAL LAND USE, X 3 = 900 acres Compare t h i s solution to the one obtained with the same set of data, except with t o t a l community land value as the sole c r i t e r i o n to be maximized. The 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 goal, has had a marked e f f e c t on the solution vector. Residential and i n d u s t r i a l acreage, both o f which generate p o l l u t i o n u n i t s , have decreased 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 acreage, which does not generate p o l l u t i o n , has i n c r e a s e d . Though the i n c l u s i o n of a d d i t i o n a l goals w i l l not always have such a dramatic e f f e c t on the value^o-f 1 the s o l u t i o n v e c t o r , i t should be apparent t h a t any model which does not c o n t a i n a l l the r e l e v a n t goals cannot p r o v i d e s o l u t i o n s t h a t can be c o n s i d e r e d t o be o p t i m a l , i n the sense of maximizing the d e c i s i o n makers' c o l l e c t i v e w e l f a r e (preference) f u n c t i o n . I t should be noted t h a t the above i n i t i a l s o l u t i o n i s j u s t t h a t , an i n i t i a l s o l u t i o n . I f the d e c i s i o n making group f e e l s t h a t a more p r e f e r r e d s o l u t i o n would l i e i n the d i r e c t i o n of a lower c o s t on p o l l u t i o n , they c o u l d change the p o l l u t i o n g o a l weight, say from -10 t o -8, and examine the r e s u l t i n g s o l u t i o n . I f i t i s f e l t t h a t t h i s new s o l u t i o n c o u l d s t i l l be improved upon, the weights c o u l d again be s h i f t e d and another s o l u t i o n examined. T h i s process would co n t i n u e , as o u t l i n e d by Candler and Bo e h l j e u n t i l an op t i m a l o r a t l e a s t a "good" (near optimal) s o l u t i o n : i s reached. I t might be p o s s i b l e , however, t h a t none of the s o l u t i o n s encountered i n t h i s process appeal to the d e c i s i o n making group. 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 e x i s t w i t h i n the model s t r u c t u r e . E i t h e r some of the c o n s t r a i n t s are s p e c i f i e d e r r o n e o u s l y o r some c o n s t r a i n t s t h a t operate i n the r e a l world system have not been i n c l u d e d i n the model. A l t e r n a t i v e l y , there may be goals t h a t govern p a r t o f the d e c i s i o n makers' w e l f a r e f u n c t i o n t h a t have been excluded from the model. In any of these cases, the necessary c o r r e c t i o n s must be made and the a n a l y s i s s t a r t e d anew. The ease i n which c o r r e c t i o n s can be made, new c o n s t r a i n t s added, and new goals s p e c i f i e d , i s indeed one of the major advantages of u s i n g t h i s type of model s t r u c t u r e . Summary The i n t e n t o f the d i s c u s s i o n i n t h i s s e c t i o n has been to o u t l i n e the methodology of l i n e a r programming wi t h m u l t i p l e g o a l s and to p r o v i d e a simple i l l u s t r a t i o n o f the a p p l i c a t i o n of t h i s type o f programming t o a h y p o t h e t i c a l l a n d use problem. The f o l l o w i n g chapter o u t l i n e s an a c t u a l m u l t i p l e goals l i n e a r programming model t h a t was c o n s t r u c t e d to c o n s i d e r problems of l a n d use a l l o c a t i o n inhthe C i t y and D i s t r i c t o f Langley i n the Lower Mainland o f B r i t i s h Columbia. 48 Chapter 6  LANGLEY MODEL STRUCTURE I n t r o d u c t i o n The f o r e g o i n g chapter d i s c u s s e d the a p p l i c a t i o n of a m u l t i p l e goals type of l i n e a r programming t o problems of l a n d use a l l o c a t i o n . A very h y p o t h e t i c a l l a n d use problem was examined. T h i s chapter w i l l concern i t s e l f w i t h a d e s c r i p -t i o n of an a c t u a l m u l t i p l e g o a l s l a n d use model t h a t was c o n s t r u c t e d t o examine lan d use a l l o c a t i o n i n the C i t y and D i s t r i c t o f Langley i n the Lower Mainland o f B r i t i s h Columbia. The C i t y and D i s t r i c t of Langley i s a predominantly a g r i c u l t u r a l area of approximately 125 square m i l e s , l o c a t e d on the south s l o p e s of the F r a s e r R i v e r B a s i n j u s t o u t s i d e the f r i n g e of the r a p i d l y expanding M e t r o p o l i t a n Vancouver urban area. The C i t y of Langley, on the western perimeter of the D i s t r i c t , i s the business c e n t e r o f the area. I t s approximately 2000 acres c o n t a i n much o f the commercial and 1 i n d u s t r i a l a c t i v i t i e s found i n the a r e a . Langley e x e m p l i f i e s the i n c r e a s i n g l y severe u r b a n - r u r a l c o n f l i c t t h a t has a r i s e n over the p a s t few decades as the M e t r o p o l i t a n Vancouver urban area has expanded r a p i d l y and has been f o r c e d , by the mountains t o the n o r t h and the water to the west, to grow out onto the 1 The author uses "Langley" to d e s c r i b e the combined areas o f the C i t y of Langley and Langley D i s t r i c t M u n i c i p a l i t y . f l a t a g r i c u l t u r a l lands t o the south and e a s t . Three major Vancouver access highways put Langley l e s s than s i x t y minutes away from the Vancouver c e n t r a l business d i s t r i c t , w e l l w i t h i n the reach of Vancouver commuters who have been s e t t l i n g i n the Langley area i n ever i n c r e a s i n g numbers. The 1971 Canadian Census r e p o r t e d an area p o p u l a t i o n o f 21,936, up over seven 2 thousand from ten years p r e v i o u s . Though i t i s y e t too e a r l y to d e t e c t any n o t i c e a b l e d e v i a t i o n , t h i s t r e n d may be a l t e r e d by the B.C. Government's B i l l 42, the Land Commission A c t , passed a t the 1973 s p r i n g s i t t i n g o f the p r o v i n c i a l l e g i s l a t u r e , which prevents f u r t h e r s u b d i v i s i o n o f d e s i g n a t e d farmland without p r o v i n c i a l government a u t h o r i t y . P a r t i a l l y because of the i n t e n s i f i e d i n t e r e s t i n p u b l i c l a n d use p l a n n i n g t h a t arose as a r e s u l t of 'this l e g i s l a t i o n , and p a r t l y because the Langley area a f f o r d e d a r e a d i l y a c c e s s i b l e example of an u r b a n - r u r a l area where p u b l i c p l a n n i n g d e c i s i o n s were being made w i t h i n a m u l t i - g o a l e d framework, the C i t y and D i s t r i c t o f Langley were chosen as the s i t e f o r the c o n s t r u c t i o n of an i n i t i a l working example of a model f o r l a n d a l l o c a t i o n d e c i s i o n making i n the Lower Mainland of B r i t i s h Columbia. The completion of t h i s Langley model i s o n l y intended t o be a f i r s t step i n a p r o j e c t whose end r e s u l t w i l l be a m u l t i p l e goals l a n d 3 use model to 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 Census of Canada (Ottawa: 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 at the U n i v e r s i t y of B r i t i s h Columbia under the d i r e c t i o n of Dr. John D. Graham. 50 I n i t i a l Assumptions Before e n t e r i n g i n t o a d i s c u s s i o n o f model cons-t r u c t i o n , an e x p l a n a t i o n o f i n i t i a l assumptions i s necessary. The f o l l o w i n g assumptions were made a t the onset o f model s t r u c t u r i n g : (1) c o n v e r s i o n c o s t s from e x i s t i n g l a n d uses t o uses t h a t the model might wish to a l l o c a t e are n i l . That i s , the model s t a r t s with a " c l e a n s l a t e " o f 125 square m i l e s which i t can a l l o c a t e , i r r e g a r d l e s s of e x i s t i n g use. \{2) c a p i t a l i n n o n - c o n s t r a i n i n g . That i s , t h e r e i s enough c a p i t a l t o meet the requirements o f any f e a s i b l e s o l u t i o n t h a t the model might examine. (3) Langley i s commercially i s o l a t e d from the surrounding areas. The commercial requirements o f the l o c a l p o p u l a t i o n and j u s t the l o c a l p o p u l a t i o n are e n t i r e l y met by l o c a l b u s i n e s s e s t a b l i s h m e n t s . As w e l l , the labour f o r c e i s o n l y employed l o c a l l y (with the p o s s i b i l i t y of some unemployment) and no l a b o u r e r s are imported from o u t s i d e the area. I t i s r e a l i z e d t h a t these assumptions s e v e r e l y l i m i t the r e a l i t y t h a t can be achieved by the model. However, i t should be noted t h a t the major o b j e c t i v e o f the study i s the c o n s t r u c t i o n o f an i n i t i a l working l a n d use model, based i n r e a l i t y o n l y as much as the q u i t e l i m i t i n g time and data r e s t r a i n t s a l l o w . I t was f e l t t h a t t o i n c l u d e c o n v e r s i o n c o s t s , 51 a c a p i t a l c o n s t r a i n t , and i n t e r r e g i o n a l l i n k s would r e q u i r e a s u b s t a n t i v e e f f o r t t h a t was beyond the r e s o u r c e s of t h i s i n i t i a l study. I t i s proposed t h a t a l a t e r study expand the model s u f f i c i e n t l y t o do away wit h the n e c e s s i t y of these 4 assumptions. A f u r t h e r p o i n t to be noted i s t h a t the author rep-r e s e n t s both a n a l y s t and p u b l i c d e c i s i o n maker a l i k e f o r the purposes o f t h i s study. In any r e a l i s t i c p r a c t i c a l a p p l i c a t i o n of t h i s type o f m o d e l l i n g to l a n d use p l a n n i n g , a g r e a t d e a l of i n p u t i s r e q u i r e d from some p u b l i c d e c i s i o n making body. However, u n t i l m u l t i p l e goals l i n e a r programming leaves the realm of the t h e o r e t i c i a n , t h i s i n p u t i s perhaps an unnecessary c o m p l i c a t i o n and probably even an i m p o s s i b i l i t y . T h e r e f o r e , f o r the purposes of t h i s study, the w r i t e r assumed the d u a l r e s p o n s i b i l i t y of d e c i s i o n maker and a n a l y s t . Land Use A l t e r n a t i v e s The m a t r i x c o n s t r u c t e d i n ; t h i s study allows Langley l a n d to be put to 34 a l t e r n a t e uses, i n c l u d i n g one non-use 5 (vacant l a n d ) . Three a g r i c u l t u r a l uses are d e f i n e d ( d a i r y i n g , market gardening, and f e e d l o t ) , as w e l l as seven commercial a c t i v i t i e s ( o f f i c e s e r v i c e s , t r a n s p o r t s e r v i c e s , urban 4 See f o o t n o t e 3 i n t h i s chapter. 5 A m a t r i x p i c t u r e , t o g e t h e r w i t h a coding e x p l a n a t i o n and data d e s c r i p t i o n , i s a v a i l a b l e i n Appendices I and I I . 52 shopping, urban neighbourhood r e t a i l , suburban shopping, suburban neighbourhood r e t a i l , and r u r a l r e t a i l ) , f o u r i n d u s t r i a l uses (food p r o c e s s i n g , l i g h t i n d u s t r y , heavy i n d u s t r y , wood p r o c e s s i n g ) , f i v e r e c r e a t i o n a l uses (urban park, suburban park, major m e t r o p o l i t a n park or r e c r e a t i o n a l r e s e r v e , g o l f course, and commercial r e c r e a t i o n a l development), s i x t r a n s p o r t i o n uses (urban road, suburban road, r u r a l road, o f f s t r e e t p a r k i n g , r a i l r o a d , and a i r p o r t ) , f i v e r e s i d e n t i a l uses (apartment, h i g h d e n s i t y s i n g l e f a m i l y d w e l l i n g , low d e n s i t y s i n g l e f a m i l y d w e l l i n g , hobby farms, and farm homesites), t h r e e i n s t i t u t i o n a l uses (s c h o o l s , h o s p i t a l s , and o t h e r govern-ment b u i l d i n g s ) , and one non-use (vacant l a n d ) . A l l of Langley's approximately 80,000 ac r e s must be a l l o c a t e d by the model i n t o one of these 34 uses s u b j e c t t o 125 c o n s t r a i n t s such t h a t a composite s o c i a l c r i t e r i o n f u n c t i o n o f ten i n d i v i d u a l p o l i c y g o als i s maximized. T e c h n i c a l and P h y s i c a l C o n s t r a i n t s 6 The 125 r e s t r a i n t s can be c l a s s i f i e d i n t o s i x major groups. S i x t e e n r e p r e s e n t l i m i t i n g f a c t o r s t h a t are 7 e s s e n t i a l l y t e c h n i c a l or p h y s i c a l i n nature : p o p u l a t i o n , 6 The author uses the words " r e s t r a i n t " and " c o n s t r a i n t " i n t e r c h a n g e a b l y . 7 Refer t o row numbers 13-22, 25-29, and 86 i n the matrix 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'II. the amount o f l a n d and labour i n p u t s r e q u i r e d by the v a r i o u s land uses, the labour f o r c e , the p a r t i c i p a t i o n of the p o p u l a t i o n i n the labour f o r c e , the amount of l a n d and the q u a l i t y of land a v a i l a b l e t o the d i f f e r e n t l a n d uses, the s i z e and comp-o s i t i o n of the f l o o d p l a i n , the amount of c l e a r e d l a n d . Most of these r e s t r a i n t l i n e s need no f u r t h e r e x p l a n a t i o n here. A b r i e f i n s p e c t i o n of the mat r i x p i c t u r e , together w i t h the r coding e x p l a n a t i o n and data d e s c r i p t i o n , s h o u l d be s u f f i c i e n t to reach an understanding of these r e s t r a i n t l i n e s . However, some p o i n t s bear mentioning i n the i n t e r e s t s of c l a r i t y . F i r s t l y , w h i l e t o t a l p o p u l a t i o n was c o n s t r a i n e d i n most of the study t o the 1971 p o p u l a t i o n of Langley (as r e p o r t e d i n the 1971 Census of Canada), the model can be run on any giv e n p o p u l a t i o n , simply by changing the p o p u l a t i o n value i n the r i g h t hand s i d e of the m a t r i x t o the d e s i r e d l e v e l . P o p u l a t i o n has been s t r u c t u r e d i n t o the model as a v a r i a b l e and not a c o n s t a n t . V a r i a b l e s which are dependent upon p o p u l a t i o n w i l l 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 v a r i e s . For example, the demand f o r medical s e r v i c e s , and the minimum h o s p i t a l requirements o f the l o c a l p o p u l a t i o n w i l l both i n c r e a s e w i t h i n c r e a s i n g p o p u l a t i o n . T h i s type of s t r u c t u r e allows f o r a p r o j e c t i o n o f the d i r e c t i o n t h a t l a n d use d e c i s i o n s must take i n o r d e r t o cope wi t h f u t u r e p o p u l a t i o n changes. I f the planners can o b t a i n r e l i a b l e p o p u l a t i o n t r e n d s , then c o n s e c u t i v e model runs can show how l a n d use p a t t e r n s must change to meet changing p o p u l a t i o n c o n d i t i o n s , and the planners can a d j u s t t h e i r p o l i c i e s a c c o r d i n g l y . 54 Secondly, two o p t i o n s e x i s t i n the model which all o w f o r some v a r i a t i o n i n the p h y s i c a l r e s o u r c e s a v a i l a b l e f o r a l l o c a t i o n . In programmer's terms these are "buy" o p t i o n s which a l l o w the model to i n c r e a s e the amount of c l e a r e d l a n d or the amount of f l o o d - s a f e l a n d , a t a g i v e n annual c o s t , i f i n c r e a s e d s o c i a l w e l f a r e , r e f l e c t e d by an i n c r e a s e i n the v a l u e of the o b j e c t i v e f u n c t i o n , can be a c hieved. T h i r d l y , i t must be r e c o g n i z e d t h a t the p r o v i s i o n of f l o o d p r o t e c t i o n c o n s t i t u t e s a lumpy or d i s c o n t i n u o u s good. In terms of the Langley model i t would be f o o l i s h t o a l l o w p r o v i s i o n f o r f l o o d p r o t e c t i o n of p a r t o f the f l o o d p l a i n when 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 be p r o t e c t e d a t l e s s o r equal c o s t . Yet, because the c o s t of buying f l o o d p r o t e c t i o n had to be s p e c i f i e d on a per acre b a s i s , the model w i l l c o n t i n u a l l y o n l y buy p r o t e c t i o n f o r the more p r o d u c t i v e lands i n the f l o o d zone, even though r e a l i t y would d i c t a t e o therwise. To r e c o n c i l e the model t o t h i s f a c t , t w o s s o l u t i o n s were found g i v e n i d e n t i c a l s t a r t i n g c o n d i t i o n s . The o n l y d i f f e r e n c e between the two s o l u t i o n s was t h a t i n the f i r s t s o l u t i o n f l o o d p r o t e c t i o n was f o r c e d to the maximum and i n the second s o l u t i o n i t was s e t a t zero p r o t e c t i o n . T h e r e a f t e r f l o o d p r o t e c t i o n remained s e t a t the v a l u e t h a t had produced a h i g h e r o b j e c t i v e f u n c t i o n v a l u e . When the g o a l weights were f i n a l i z e d the procedure was repeated to ensure t h a t f l o o d p r o t e c t i o n was indeed s e t a t an o p t i m a l v a l u e . 55 Social Constraints A second group of twenty-seven constraints can be characterized 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 solution. Minimum shopping and other commercial requirements, minimum transportation needs, minimum recreational 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 included here. As well, a maximum unemployment standard and a minimum municipal tax base are imposed within t h i s category. Although a l l of these constraints are quite straight forward, and need no further elucidation here, a discussion of the role of a minimum qu a l i t y of l i f e imposed upon an optimum seeking model appears i n order. The c l a s s i c a l welfare economist would argue, i n Paretian terms, that minimum and maximum impositions 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 of resources and lead 9 away from Pareto optimality. He wouldcoontinue by showing that only i f the system i s allowed to f r e e l y a l l o c a t e 8 Refer to row numbers 30-36, 51-57, 68-76, and 86-87 in the picture and coding explanation i n Appendices I and I I . 9 A Pareto optimum i s one i n which the welfare of no in d i v i d u a l can be increased further without decreasing the welfare of some other i n d i v i d u a l or group of i n d i v i d u a l s . 56 r e s o u r c e s a c c o r d i n g to marginal c o n s i d e r a t i o n s of economic worth and u t i l i t y w i l l any s o c i a l w e l f a r e f u n c t i o n be maximized. However, h i s e n t i r e argument i s premised upon acceptance o f 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 o f wealth, an assumption t h a t 10 has not proven a c c e p t a b l e t o most modern Western economies. R e d i s t r i b u t i o n of wealth i s very much i n evidence. One o f the most o f t e n used v e h i c l e s of r e d i s t r i b u t i o n i s the i m p o s i t i o n of a minimum q u a l i t y of l i f e upon the economic system, whether i n the form of minimum wages, minimum f i r e p r o t e c t i o n , minimum open space, minimum b u i l d i n g space or whatever. I t i s w i t h t h i s i n mind t h a t t h i s s e t of minimum c o n s t r a i n t s was imposed upon the model s t r u c t u r e . Output Demand C o n s t r a i n t s T h i r t y - t w o o t h e r c o n s t r a i n t s concern themselves wi t h l i m i t a t i o n s imposed by economic f a c t o r s , c h i e f l y demand 11 f o r the output o f the v a r i o u s l a n d a c t i v i t i e s . These r e s t r a i n t s prevent the model from a l l o c a t i n g an uneconomic amount of l a n d t o any p a r t i c u l a r land use. Included i n t h i s group are c o n s t r a i n t s l i m i t i n g the demand f o r the output produced by f e e d l o t s , a l l commercial l a n d uses, a l l r e c r e a t i o n a l 10 T h i s assumption, 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 being P a r e t o - o p t i m a l to s o c i e t y . 11 Refer t o row numbers 23-24, 37-50, 58-67, and 77-82 i n the m a t r i x 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 . 57 la n d uses, and a l l i n s t i t u t i o n a l l a n d uses. Due t o the nature of the model, i t d i d not prove p o s s i b l e t o s p e c i f y these demand c o n s t r a i n t s i n terms of a c t u a l demand schedules f o r the m u l t i p l i c i t y o f goods and s e r v i c e s produced by the v a r i o u s l a n d a c t i v i t i e s . Instead, a l l the goods and s e r v i c e s produced by any g i v e n l a n d use 12 were a l l lumped t o g e t h e r i n t o one basket of goods, and then the demand f o r the basket of goods produced by any p a r t i c u l a r l a n d use was c o n s t r a i n e d . At t h i s p o i n t i n model s t r u c t u r i n g two t e c h n i c a l problems arose. F i r s t l y , no programming mechanism, compatible t o l i n e a r programming, c o u l d be found which c o u l d e x a c t l y d u p l i c a t e the a c t i o n of a c o n v e n t i o n a l downward s l o p i n g demand curve. E s s e n t i a l l y what was needed was a mechanism whereby the u n i t r e t u r n o r p r i c e t o a gi v e n " i n d u s t r y " (land use) would d e c l i n e a t an a p p r o p r i a t e r a t e as t h a t i n d u s t r y i n c r e a s e d the amount of baskets of goods s u p p l i e d , g i v e n t h a t t h i s was the type of behaviour observed to take p l a c e i n the r e a l world. While no mechanism c o u l d be found whereby t h i s type o f a c t i o n c o u l d be d u p l i c a t e d , the problem was overcome by the use of stepped or d i s c o n t i n u o u s f u n c t i o n s which c o u l d be manipulated t o c l o s e l y approximate 12 One basket c o n s i s t e d of a l l goods and s e r v i c e s produced by one acre of the l a n d use i n q u e s t i o n , g i v e n a p p r o p r i a t e amounts of complementary i n p u t s . 58 the d e s i r e d a c t i o n . In terms of model c o n s t r u c t i o n , t h i s was accomplished by having every Langley l a n d use, whose output f a c e d a downward s l o p i n g demand curve, s p l i t i n t o three separate a c t i v i t i e s . Each of these a c t i v i t i e s c o n s t i t u t e d a separate step i n the stepped demand f u n c t i o n . Though each of these a c t i v i t i e s f aced e x a c t l y the same p r o d u c t i o n f u n c t i o n , t h a t i s , they a l l needed the same amount o f l a n d and labour to produce a giv e n value of output, they d i f f e r e d i n the r e t u r n they r e c e i v e d from the s a l e o f t h e i r product. The f i r s t s tep r e c e i v e d a f a i r l y h i g h r e t u r n and the f o l l o w i n g steps r e c e i v e d s u c c e s s i v e l y lower r e t u r n s . These three steps were then c o n s t r a i n e d a p p r o p r i a t e l y as to the amount o f land use t h a t c o u l d occur i n each. A l l t h r e e steps were then summed i n t o a f o u r t h separate a c t i v i t y column. T h i s column repr e s e n t e d the t o t a l output produced by the l a n d use i n q u e s t i o n , and was expressed i n terms of acres r e q u i r e d t o produce a giv e n t o t a l output. Note t h a t the v a r i o u s columns c o n t a i n i n g the d i f f e r e n t demand curve steps r e p r e s e n t n o t h i n g more than i n t e r n a l a r t i f i c i a l v a r i a b l e s which implement demand l i m i t a t i o n s . They do not r e p r e s e n t a c t u a l l a n d use a c t i v i t i e s . I t i s the columns c o n t a i n i n g the output t o t a l s which r e p r e s e n t l a n d use a l l o c a t i o n . Note a l s o t h a t t h i s procedure of summing th r e e a c t i v i t i e s , which earn s u c c e s s i v e l y lower u n i t r e t u r n s , i n t o a f o u r t h t o t a l a c t i v i t y column o b v i o u s l y has the e f f e c t o f i n t r o d u c i n g a t h r e e step d e c l i n i n g average r e t u r n t o the output produced by the p a r t i -c u l a r l a n d use under c o n s i d e r a t i o n . 59 The second s t r u c t u r a l problem encountered i n t h i s area concerned the nature o f the monetary r e t u r n to the v a r i o u s output producing a c t i v i t i e s . Because the model i s macroeconomic i n nature, output u n i t r e t u r n s o r p r i c e s c o u l d not be d i r e c t l y s p e c i f i e d i n the model. T h i s meant t h a t the demand c o n s t r a i n t s had to be d e f i n e d i n terms ot h e r than c o n v e n t i o n a l p r i c e and q u a n t i t y terms. T h i s problem was d e a l t w i t h by d e f i n i n g the demand c o n s t r a i n t s i n terms of d e c l i n i n g v a l u e d added by p r o d u c t i o n . The p o l i c y g o a l which sought a maximum of gross r e g i o n a l p r o d u c t i o n was expressed i n va l u e added terms and thus p r o v i d e d a convenient base upon which t o d e f i n e the demand c o n s t r a i n t s . For each l a n d use whose output faced a downward s l o p i n g demand, the thr e e a r t i f i c i a l a c t i v i t i e s which re p r e s e n t e d t h a t l a n d use c o n t r i b u t e d s u c c e s s i v e l y lower amounts o f va l u e added p r o d u c t i o n per acre o f use. The assumption here i s t h a t as output i s i n c r e a s e d , g i v e n a f i x e d demand, output u n i t p r i c e drops, thus lowering the va l u e added component of p r o d u c t i o n . For example, each acre i n the f i r s t step of the f e e d l o t l a n d use c o n t r i b u t e s $4400 towards the g o a l o f maximizing r e g i o n a l p r o d u c t i o n . Each acre i n the second step c o n t r i b u t e s $2000, and each acre i n the t h i r d s t ep, o n l y $500. As e x p l a i n e d above, when these t h r e e steps are t o t a l l e d i n t o a separate a c t i v i t y column which r e p r e s e n t s t o t a l l a n d use, the e f f e c t i s one of i n t r o d u c i n g a th r e e step downward s l o p i n g demand curve f o r the oujtput o f t h a t l a n d use. F i g u r e 1 i l l u s t r a t e s t h i s mechanism g r a p h i c a l l y , 60 F i g u r e 1 T o t a l Value Added T o t a l Revenue 400 Acres of A c t i v i t y 400 Acres of Output Cost or Non Value Added Component of Output 100 200 300 400 Acres of Output 61 u s i n g the f e e d l o t example d i s c u s s e d above. The graph i n F i g u r e 1 (a) shows how t o t a l v alue added i n c r e a s e s with acres of f e e d l o t l a n d use a c t i v i t y . Note t h a t t o t a l v alue added i n c r e a s e s a t a d i f f e r e n t r a t e i n each of the t h r e e d i f f e r e n t s t e p s , and t h a t the f i r s t two steps are c o n s t r a i n e d to a maximum of one hundred acres each and t h a t the t h i r d step i s u n c o n s t r a i n e d . I f i t i s assumed t h a t there i s a constant c o s t component of non value added m a t e r i a l s i n each u n i t of f e e d l o t output, then a t o t a l revenue curve corresponding to the t o t a l v alue added curve can be c o n s t r u c t e d . T h i s i s shown i n F i g u r e 1 (b). Using t h i s t o t a l revenue curve, average revenue per u n i t of output can be c a l c u l a t e d a t v a r i o u s l e v e l s of output, t h i s i s , a t v a r i o u s l a n d use acreages. F i g u r e 1 (c) d e t a i l s the average revenue curve corresponding to the t o t a l revenue curve i n F i g u r e 1 (b). But average revenue per u n i t of output i s p r e c i s e l y the same as u n i t p r i c e . T h e r e f o r e F i g u r e 1 (c) r e p r e s e n t s a rudimentary J approximation o f a downward s l o p i n g demand curve, r e l a t i n g the p r i c e o f a u n i t of output and the q u a n t i t y of output demanded a t t h a t p r i c e . Thus the i n c l u s i o n of simple t h r e e step v a l u e added f u n c t i o n s i n the model has the e f f e c t of i n t r o d u c i n g rough approximations to market demand curves. O b v i o u s l y , g r e a t e r accuracy can be achieved by the use of a g r e a t e r number of steps i n each f u n c t i o n . Only t h r e e steps were used f o r a l l demand f u n c t i o n s i n t h i s model f o r reasons of s i m p l i c i t y and convenience, and the r e s u l t i n g f u n c t i o n s are probably 62 very i n a c c u r a t e . The p o i n t to be noted however, i s t h a t given s u f f i c i e n t a c c u r a t e i n f o r m a t i o n on the r e a l world, any r e a l world market demand curve can be c l o s e l y approximated by t h i s method. In t o t a l , s i x t e e n output demand f u n c t i o n s of t h i s type were used i n the model. In each case an attempt was made to l i n k the demand f u n c t i o n s to the r e l e v a n t market. For example, i t was f e l t t h a t the market f o r o f f i c e s e r v i c e s (medical, d e n t a l , l e g a l , c l e r i c a l , and o t h e r p r o f e s s i o n a l s e r v i c e s ) i n a r e g i o n o b v i o u s l y extends t o the t o t a l p o p u l a t i o n of the r e g i o n . For o f f i c e s e r v i c e s , as w e l l as f o r other l a n d uses c o n f r o n t e d by s i m i l a r market c o n d i t i o n s , t h e r e f o r e , the s i z e of each step i n the value added f u n c t i o n v a r i e s p r o p o r t i o n a t e l y to the t o t a l p o p u l a t i o n . T h i s has the e f f e c t o f s h i f t i n g the demand curves f o r the goods and s e r v i c e s produced by these 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 , market 13 s i z e , v a r i e s . Those a c t i v i t i e s whose markets are dependent 13 The author r e a l i z e s t h a t demand curves can r a r e l y be expected to behave so simply. S h i f t s i n the demand f o r the product of a l a n d a c t i v i t y probably depend on changes i n o t h e r a c t i v i t y l e v e l s , changes 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 other 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 p o p u l a t i o n . However, i n an e f f o r t to keep t h i s i n t r o d u c t o r y model simple, and s i n c e p o p u l a t i o n i s probably the most important f a c t o r to be c o n s i d e r e d , most of the demand curves i n t h i s model are t i e d t o some measure of p o p u l a t i o n . 63 upon a c e r t a i n p o r t i o n o f the t o t a l p o p u l a t i o n , such as the urban p o p u l a t i o n or suburban p o p u l a t i o n subsets, have t h e i r demand curves c o n s t r a i n e d a c c o r d i n g l y . Land a c t i v i t i e s , such as a l l the i n d u s t r i a l uses, which s e r v i c e such a l a r g e market t h a t t h e i r a c t i v i t y can be assumed to have l i t t l e o r no e f f e c t on p r i c e s , h a v e no demand c o n s t r a i n t s a s s o c i a t e d w i t h them. E x t e r n a l i t y Demand C o n s t r a i n t s A f o u r t h group of c o n s t r a i n t s , e i g h t i n number and s i m i l a r to the l a s t group j u s t d i s c u s s e d , govern the 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 produced by the v a r i o u s 14 l a n d a c t i v i t i e s . Some a g r i c u l t u r a l l a n d , f o r example, produces p o s i t i v e b e n e f i t s i n i t s r o l e as a green b e l t . These b e n e f i t s are e x t e r n a l to the p r i v a t e market p l a c e , i n t h a t they cannot command a p r i c e or generate a r e t u r n to the producer even though they have s o c i a l v a l u e . T h i s f o u r t h group o f c o n s t r a i n t s s p e c i f i e s the s o c i a l value t h a t v a r i o u s e x t e r n a l i t i e s have by i n t r o d u c i n g c o n v e n t i o n a l market demand curves f o r them. Other than t h e i r e x t e r n a l nature, i t seems reasonable to assume t h a t these b e n e f i t s c o n s t i t u t e normal economic goods and, as such, a l l the c o n v e n t i o n a l assumptions ^ R e f e r t o row numbers 120-127 i n the matrix 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 . 6 4 t h a t one can make about such goods should h o l d t r u e f o r these e x t e r n a l b e n e f i t s as w e l l . The format by which these demand curves are i n t r o -duced i s almost e x a c t l y i d e n t i c a l to t h a t used f o r the output demand c u r v e s . The o n l y d i f f e r e n c e i s t h a t i n s t e a d o f l i n k i n g the v a r i o u s steps o f these stepped f u n c t i o n s to the r e g i o n a l p r o d u c t i o n g o a l , these f u n c t i o n s are l i n k e d t o v a r i o u s o t h e r g o a l s . For example, the demand c o n s t r a i n t s f o r green b e l t e x t e r n a l i t i e s are t i e d t o the maximize green b e l t p o l i c y g o a l . Acreages i n the f i r s t s tep of the green b e l t demand f u n c t i o n c o n t r i b u t e so many e x t e r n a l i t y u n i t s (approximately equal t o d o l l a r s ) towards t h i s p o l i c y g o a l , w h i l e second step acres c o n t r i b u t e somewhat l e s s , and t h i r d step a c r e s , s t i l l l e s s . When acreages i n these t h r e e steps are summed i n t o a f o u r t h t o t a l a c t i v i t y column, a downward s l o p i n g demand curve f o r green b e l t e x t e r n a l i t i e s has been e f f e c t i v e l y imposed upon the model. In a s i m i l a r f a s h i o n demand curves f o r the ot h e r e x t e r n a l i t i e s are i n t r o d u c e d . In a l l , f o u r such e x t e r n a l i t i e s are handled i n t h i s manner w i t h i n the model: green b e l t b e n e f i t s produced by l a r g e t r a c t s o f n a t u r a l o r a g r i c u l t u r a l v e g e t a t i o n ; green space b e n e f i t s produced by s m a l l e r areas o f lawns, shrubbery, gardens, 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 p r o v i d e d by lands uses which al l o w f o r p u b l i c access and use a t no charge; 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 b e n e f i t s produced by a g r i c u l t u r a l lands wYiose p r o d u c t i o n d i m i n i s h e s our dependence 65 on " o u t s i d e " sources of food. Accounting C o n s t r a i n t s A f i f t h s e t of c o n s t r a i n t s e s s e n t i a l l y p r o v i d e s an a c c o u n t i n g mechanism by which v a r i o u s a c t i v i t i e s are l i n k e d 15 to one another. T h i s mechanism allows f o r v a r i o u s land a c t i v i t i e s to be s u b t o t a l l e d i n t o separate columns. For example, i n c l u d e d i n t h i s group i s a c o n s t r a i n t l i n e which sums the acreages i n the t h r e e a g r i c u l t u r a l uses i n t o one t o t a l a g r i c u l t u r e l a n d use column. As w e l l , acreages which produce a l l types of e x t e r n a l b e n e f i t s are summed i n t o separate columns by t h i s a c c o u n t i n g mechanishm. A l s o i n c l u d e d i n t h i s group i s a s e r i e s o f c o n s t r a i n t s which p r o v i d e s the l i n k s between separate components of the v a r i o u s demand c o n s t r a i n t s . These are 32 c o n s t r a i n t l i n e s i n t o t a l w i t h i n t h i s f i f t h group. Goal Accounting C o n s t r a i n t s The l a s t s e t o f c o n s t r a i n t s , ten i n number, pro v i d e the same type o f a ccounting s e r v i c e as the f i f t h s e t . The o n l y d i f f e r e n c e i s t h a t t h i s s e t r e l a t e s d i r e c t l y t o the ten i n d i v i d u a l g o a l f u n c t i o n s which make up the composite 15 Refer to row numbers 88-119 i n the m a t r i x p i c t u r e and coding e x p l a n a t i o n a v a i l a b l e i n Appendices I and I I . 66 16 s o c i a l o b j e c t i v e f u n c t i o n . These ten c o n s t r a i n t s a r e , i n f a c t , s c a l e d p o l i c y g o a l f u n c t i o n s o f the type d i s c u s s e d a t l e n g t h i n the l a t t e r h a l f o f the pre c e d i n g chapter on the methodology of m u l t i p l e goals programming. These l a s t ten c o n s t r a i n t s , one f o r each p o l i c y g o a l , sum the c o n t r i b u t i o n s t h a t each l a n d a c t i v i t y makes towards the v a r i o u s l a n d use g o a l s , and p l a c e the t o t a l s i n t o t en separate a c t i v i t y columns. I t i s t o these t o t a l s t h a t the v a r i o u s g o a l weights are a p p l i e d t o determine the value o f the composite s o c i a l o b j e c t i v e 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 . As mentioned above, some of these p o l i c y g o a l c o n s t r a i n t l i n e s a l s o p r o v i d e the b a s i s f o r the i n t r o d u c t i o n o f the v a r i o u s demand c o n s t r a i n t s . The model a l l o c a t e s Langley l a n d among the t h i r t y -f o u r l a n d a c t i v i t i e s , s u b j e c t t o a l l the c o n s t r a i n t s d i s c u s s e d 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. T h i s composite f u n c t i o n c o n s i s t s o f ten separate p o l i c y g oals w i t h an a p p r o p r i a t e g o a l weight a p p l i e d t o each. By f a r the dominant g o a l i s one which seeks t o maximize gross r e g i o n a l p r o d u c t i o n , c a l c u l a t e d by a va l u e added approach. In o t h e r words, i f an acre o f a p a r t i c u l a r l a n d use r e q u i r e s one thousand d o l l a r s of i n t e r m e d i a t e 16 Refer t o row numbers 128-137 i n the matrix p i c t u r e and coding e x p l a n a t i o n a v a i l a b l e i n Appendices I and I I . 67 goods per year to produce four thousand d o l l a r s of f i n a l output then that acre w i l l have contributed a value added contribution of three thousand d o l l a r s per year to the goal of maximizing gross regional production. This goal i s afforded a dominant position i n t h i s model because the gross production of a region provides a good indicator of the material well being of i t s inhabitants. I t represents the private market valuation of the t o t a l amount of goods and services produced i n a region within a given time period (in thisi-case, one year) and, as such, i s an estimate of the size of the t o t a l "pie" that i s to be divided up among the inhabitants. However, because of the divergence between s o c i a l and private valuations which occurs i n a l l r e a l world economies, t h i s single goal i s not s u f f i c i e n t to maximize s o c i a l welfare. Benefits and costs which are external to the private market are also created i n the production process and must be added and subtracted from the production "pie" to achieve a more accurate estimate of s o c i a l welfare. However, because of t h e i r absence from the private market place, no d i r e c t valuation of these external benefits i s available or possible. Therefore, some other valuation method was needed to ensure that the model more accurately sought a s o c i a l welfare optimum. This was accomplished i n the model by the creation of nine additional goal functions, one for each separately i d e n t i f i a b l e 68 e x t e r n a l i t y . F i v e of these goals seek a m i n i m i z a t i o n of f i v e d i f f e r e n t types of e x t e r n a l c o s t s — a i r p o l l u t i o n , water p o l l u t i o n , ground p o l l u t i o n (garbage or r e f u s e ) , n o i s e p o l l u t i o n , and v i s u a l ( s i g h t ) p o l l u t i o n . As w e l l , f o u r goals seeking maximization of the e x t e r n a l b e n e f i t s a r i s i n g from green b e l t s , green space, p u b l i c r e c r e a t i o n a l areas, 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 were c o n s t r u c t e d and i n c l u d e d i n the model. These nine e x t e r n a l i t y g o a l s , as w e l l as the i n i t i a l g o a l of maximizing gross r e g i o n a l p r o d u c t i o n , were then a l l combined i n t o the composite s o c i a l o b j e c t i v e f u n c t i o n , w i t h an a p p r o p r i a t e weight p l a c e d on each g o a l , i n a manner s i m i l a r t o the i t e r a t i v e approach suggested by Candler and B o e h l j e and d i s c u s s e d a t l e n g t h i n the l a t t e r p a r t of the p r e c e d i n g 17 chapter. Though i n i t i a l l y each g o a l was c o n s t r u c t e d on w i d e l y d i f f e r i n g i n d i c e s , t h i s process l e d to so much c o n f u s i o n and d i f f i c u l t y i n s e t t i n g and m a n i p u l a t i n g r e l a t i v e weights d u r i n g the i t e r a t i v e d e t e r m i n a t i o n of f i n a l g o a l weights, t h a t a l l o f the goals were r e s p e c i f i e d , though s t i l l s e p a r a t e l y and i n i s o l a t i o n , on i n d i c e s t h a t r e p r e s e n t e d the a n a l y s t ' s p e r c e p t i o n of approximate d o l l a r v a l u e . In o t h e r words, each g o a l was s p e c i f i e d on an index whose u n i t s were roughly comparable, i n the a n a l y s t ' s p e r c e p t i o n , t o d o l l a r u n i t s . On i n i t i a l i n s p e c t i o n t h i s procedure might seem to d e f e a t the purpose of the whole concept of the i t e r a t i v e d e t e r m i n a t i o n 17 Candler and B o e h l j e , op. c i t . 69 of r e l a t i v e weights. I f a l l goals are s p e c i f i e d i n the same u n i t s ( d o l l a r s ) i t might seem t h a t i t would be by f a r e a s i e r to simply sum up a l l the go a l s i n t o a new s i n g l e g o a l which c o u l d be used as the s o c i a l o b j e c t i v e f u n c t i o n . In terms of a m u l t i p l e g o a l s approach t h i s i s the same as p l a c i n g an equal weight of u n i t y on a l l g o a l s . However, there i s a s u b t l e but c r u c i a l d i s t i n c t i o n t h a t must be made here. The d i f f e r e n t i n d i c e s were c o n s t r u c t e d to o n l y approximate d o l l a r v a l u e . There i s no reason t o assume t h a t they e x a c t l y c o i n c i d e w i t h d o l l a r v a l u e , nor to assume t h a t they a l l d e v i a t e from d o l l a r v a l u e by the same amount and i n the same d i r e c t i o n . In o t h e r words, they, a l l w i l l , i n f a c t , be c o n s t r u c t e d i n terms of d i f f e r e n t index u n i t s . The attempt t o roughly approximate d o l l a r v alue i s done f o r no reason o t h e r than one of s i m p l i c i t y and convenience. I t does n o t h i n g more than 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 d i f f i c u l t y i n e s t a b l i s h i n g and ma n i p u l a t i n g i n i t i a l g o a l weights. In terms of the d i s c u s s i o n on s t a r t i n g p o i n t s i n the l a t t e r h a l f o f C h a p t e r - f i v e t h i s procedure simply ensures t h a t the s t a r t i n g p o i n t i n the i t e r a t i v e a n a l y s i s i s reasonably c l o s e t o the optimum s o l u t i o n , i f such an optimum e x i s t s . The va l u e o f the i t e r a t i v e search f o r the optimum i s by no means l o s t . The d e c i s i o n makers can s t i l l r e s p e c i f y i n i t i a l weights i n the d i r e c t i o n they f e e l a p r e f e r r e d s o l u t i o n l i e s . In e f f e c t , they w i l l be c o r r e c t i n g t h e i r i n i t i a l 7 0 rough approximations between g o a l index u n i t s and d o l l a r v a l u e , given the c e r t a i n amount o f h i n d s i g h t t h a t the i n i t i a l f e a s i b l e s o l u t i o n s a f f o r d them. Summary The i n t e n t o f the d i s c u s s i o n i n t h i s chapter has been t o o u t l i n e the b a s i c s t r u c t u r e o f the Langley model and to d i s c u s s 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 encountered d u r i n g i t s c o n s t r u c t i o n . No attempt has been made to present a c t u a l model d e t a i l , as such an e x e r c i s e would prove too lengthy and f a r too dr e a r y t o be i n c l u d e d i n the main body of t h i s t h e s i s . References have been made throughout the d i s c u s s i o n i n t h i s s e c t i o n t o the r e l e v a n t areas i n the Appendices t o t h i s study where the model can be examined i n g r e a t e r d e t a i l i f d e s i r e d . The f o l l o w i n g chapter begins w i t h an o u t l i n e o f methods of data a c q u i s i t i o n and problems of data s u i t a b i l i t y and then continues t o a p r e s e n t a t i o n of model s o l u t i o n r e s u l t s . 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 generation of data, as well as the time and money needed to carry out these a c t i v i t i e s , are probably the most l i m i t i n g factors encountered i n any kind of systems 1 modelling. Certainly t h i s proved to be the case i n t h i s study. Undoubtedly a great deal of data exists which applies to the Lower Fraser Valley land use system. However, what l i t t l e of i t could be found within the imposed time l i m i t s of t h i s study tended to be structured so as to negate i t s usefulness to the model, or to be structured so as to be useful, but only aft e r much arithmetic and accounting manipulation. One of the most often encountered data problems was one of aggregation compatibility. Data existed which applied to cer t a i n aggregates chosen, for various reasons, by the agencies or persons^ that o r i g i n a l l y generated or colleeted the data. However, these aggregates were not compatible to model aggre-gation. For example, worker densities were found for the Metropolitan Vancouver area (including Langley) but not for just Langley. Costs of flood protection for the entire Lower 1 A complete description of the data used i n t h i s model i s available i n Appendix I I , Tables A-F e s p e c i a l l y . 7 2 F r a s e r V a l l e y f l o o d p l a i n were a v a i l a b l e , but the s p e c i f i c c o s t s of p r o t e c t i n g t h a t p a r t o f the f l o o d p l a i n t h a t l i e s i n Langley were not. Park p l a n n i n g standards c o u l d be found f o r the C i t y of Langley but not f o r the D i s t r i c t . Often these type o f aggregation problems, t o g e t h e r w i t h the l a c k o f time i n which to generate new data, n e c e s s i t a t e d changes i n model d e s i g n . For example, i t was hoped t o disaggregate the labour i n p u t t o the p r o d u c t i o n f u n c t i o n i n t o s e v e r a l c l a s s e s : s k i l l e d , semi-s k i l l e d , and so on. However, labour requirements c o u l d not be found f o r t h i s type o f d i s a g g r e g a t i o n , or any such s i m i l a r c l a s s i f i c a t i o n , r e s u l t i n g i n a l l labour i n p u t s being grouped i n t o one c l a s s f o r the purposes o f the model. In many areas, d i r e c t l y a p p l i c a b l e data c o u l d not be 'i found however u n s u i t a b l y aggregated. In these cases, some q u i t e dubious assumptions were made and a l t e r n a t e data were used. For example, minimum p l a n n i n g standards c o u l d not be found f o r some Lower Mainland commercial a c t i v i t i e s . In these cases, standards used i n other areas of North America were assumed t o be r e l e v a n t to l o c a l c o n d i t i o n s and were i n c l u d e d i n the model. T h i s procedure was a l s o used f o r some r e c r e a t i o n a l standards. However, probably the most important problem encountered i n t h i s area was not u n s u i t a b i l i t y o f data, but r a t h e r l a c k of data. Even u n s u i t a b l e data i s b e t t e r than no data, given a l a c k of adequate time and reso u r c e s to generate new data. In many areas data c o u l d not be found and, i n some areas, there was very l i t t l e reason to suspect t h a t i t even e x i s t e d . Information on m u n i c i p a l tax c o l l e c t i o n and e x p e d i t u r e s by l a n d use, demand c o n d i t i o n s f o r the v a r i o u s outputs of the d i f f e r e n t l a n d uses, i n f o r m a t i o n on e x t e r n a l i t y v a l u a t i o n , t e c h n i c a l i n f o r m a t i o n on land u s e — p o l l u t i o n r e l a t i o n s h i p s , worker d e n s i t i e s and v a l u e added to p r o d u c t i o n i n s c h o o l s , h o s p i t a l s , r e c r e a t i o n a l areas, a l l f a l l i n t o t h i s category. Given enough time and money, t h i s i n f o r m a t i o n can l i k e l y be generated. For the purposes of t h i s i n i t i a l working model, however, t h i s i n f o r m a t i o n was e i t h e r e s t imated or p u r e l y f a b r i c a t e d . Though these problems of data a q u i s i t i o n undoubtedly have reduced model r e a l i t y s e v e r e l y , i t should be remembered t h a t one of the primary purposes of t h i s study was, i n f a c t , t o i d e n t i f y and measure these types of problems f o r any f u t u r e model c o n s t r u c t i o n . As such, t h i s s i t u a t i o n has p r o v i d e d a v a l u a b l e l e s s o n f o r any f u t u r e extensions to t h i s p r o j e c t , or any such s i m i l a r p r o j e c t . Any attempt a t a c c u r a t e l y a b s t r a c t i n g a mathematical model, to the p o i n t where i t can be used f o r p l a n n i n g purposes, from a l a r g e l a n d use system r e q u i r e s a g r e a t amount of e f f o r t , time, and money: e f f o r t measured i n teams of workers, time i n man-years, and money i n hundreds of thousands o f d o l l a r s . R e s u l t s Because of data and d e s i g n d e f i c i e n c i e s , the model c o n s t r u c t e d i n t h i s study o b v i o u s l y has very l i t t l e a p p l i c a t i o n to a c t u a l l a n d use p l a n n i n g problems. However, model s o l u t i o n i s s t i l l i n t e r e s t i n g i n terms of the type of a n a l y s i s t h a t can be c a r r i e d out by a m u l t i p l e goals l i n e a r programming model. For t h i s reason, a s h o r t d i s c u s s i o n on model r e s u l t s f o l l o w 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 f u n c t i o n were c o n s t r u c t e d e i t h e r on a d o l l a r s c a l e or on a s c a l e whose u n i t s were approximations ( i n the author's p e r c e p t i o n ) of d o l l a r u n i t s . The reasons f o r t h i s were d e t a i l e d i n the p r e c e d i n g chapter. The model was then i n i t i a l l y s o l v e d w i t h a l l g o a l weights equal t o p l u s or minus u n i t y . In o t h e r words, a l l the g o a l s c a l e u n i t s were of equal magnitude (a one d o l l a r c o n t r i b u t i o n t o gross r e g i o n a l p r o d u c t i o n equaled a one u n i t c o n t r i b u t i o n 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 which equaled a one u n i t decrease i n water p o l l u t i o n , and so on). Note t h a t p l a c i n g an equal weight of u n i t y oh a l l the goals does not mean t h a t a l l goals are of equal importance. T h i s procedure i s simply i n keeping w i t h the attempt t o s p e c i f y a l l goals i n approximately equal u n i t s which was c a r r i e d out to a v o i d c o n f u s i o n i n s e t t i n g i n i t i a l g o a l weights. Using t h i s s e t of weights ( a l l weights equal ±1.0), the s o l u t i o n presented i n column (a) of Table 1 was achieved. Not u n u s u a l l y , most o f Langley*s l a n d went to a g r i c u l t u r a l uses, p a r t i c u l a r l y t o d a i r y i n g . T r a n s p o r t a t i o n l a n d uses took the next b i g g e s t a l l o c a t i o n , f o l l o w e d by r e s i d e n t i a l uses, and r e c r e a t i o n a l uses. Small acreages were a l l o c a t e d t o i n s t i t u t i o n a l uses, i n d u s t r y and commerce. I t was e s t a b l i s h e d t h a t f l o o d p r o t e c t i o n was o p t i m a l , and t h e r e f o r e a l l o f Langley's f l o o d p l a i n was p r o t e c t e d . The model a l s o c l e a r e d and used much of Langley's wooded unimproved acreages. F u l l employment was 75 Table 1 LANGLEY MODEL SOLUTIONS FOR OPTIMAL ALLOCATIONS OF LAND ( a l l acreages rounded to the n e a r e s t acre) (a) (b) (c) (d) A c t i v i t y or O b j e c t i v e F u n c t i o n Used: A c t i v i t y Group C a CLIES=5 b CGRPONLYcCPOL=5 d a i r y 65558 65533 64300 65844 market gardening 7510 7510 7510 7510 f e e d l o t 200 200 200 200 a l l a g r i c u l t u r a l uses 73268 73243 72010 73454 a l l commercial uses 11 12 12 23 a l l i n d u s t r i a l uses 158 158 237 n i l urban park 59 59 59 59 r e c r e a t i o n a l r e s e r v e 139 164 164 139 g o l f course 88 88 88 88 commercial r e c r e a t i o n a l dev. 208 208 104 208 a l l r e c r e a t i o n a l uses 494 415 415 494 urban roads 61 61 429 61 r u r a l roads 742 742 729 744 r a i l r o a d s and r a i l y a r d s 169 169 249 23 a i r p o r t 141 141 141 141 o f f - s t r e e t p a r k i n g 336 336 232 382 a l l t r a n s p o r t a t i o n uses 1449 1449 1780 1351 apartments 153 153 n i l 153 s i n g l e f a m i l y homes 915 915 1973 908 a l l r e s i d e n t i a l uses 106 8 1068 1973 1061 schools 274 274 137 274 h o s p i t a l s 5 5 3 5 a l l i n s t i t u t i o n a l uses 282 282 143 347 t o t a l 76730 76730 76730 76730 f l o o d p r o t e c t i o n 6200 6200 6200 6200 land c l e a r a n c e 11708 11683 11522 11709 % employment 100 100 100 100 per c a p i t a income ($) 1906 1906 1915 1675 a Weights on a l l g o a l s equals ±1.0. 76 Table 1 cont'd b Weights on p o l i c y g o als concerned w i t h r e c r e a t i o n a l b e n e f i t s , green b e l t s , and green space a l l equal +5.0. A l l o t h e r g o a l weights remain s e t a t ±1.0. c Weight on the gross r e g i o n a l p r o d u c t i o n g o a l equals +1.0, w h i l e a l l o t h e r g o a l weights are s e t equal t o zero, d Weights on a l l f i v e p o l l u t i o n g o als equal -5.0. A l l o t h e r g o a l weights are s e t equal t o +1.0. 11 achieved, which i s a l s o to be expected as l a b o u r i s probably the key l i m i t i n g c o n s t r a i n t i n the model. The annual shadow p r i c e of an a d d i t i o n a l l a b o u r e r was 2533 o b j e c t i v e f u n c t i o n u n i t s ( s o c i a l w e l f a r e " d o l l a r s " ) i n t h i s i n i t i a l run, r e f l e c t i n g the l i m i t i n g nature of the labour c o n s t r a i n t . Average worker income ( t o t a l p r o d u c t i o n d i v i d e d by t o t a l l a b o u r f o r c e ) was $4887. Per c a p i t a income was $1906. The annual shadow r e n t s f o r l a n d v a r i e d from 91 o b j e c t i v e f u n c t i o n u n i t s per year f o r any a d d i t i o n a l acreage, t o 194 u n i t s per year f o r another acre of the b e s t a g r i c u l t u r a l l a n d . C o n s i d e r i n g the problems i n model de s i g n and data a c q u i s i t i o n , these r e s u l t s are s u r p r i s i n g l y r e a sonable. Land a l l o c a t i o n , worker incomes, and l a n d r e n t s are a l l w i t h i n a reasonable d i s t a n c e of where one would expect them to be. In f a c t , the model s o l u t i o n appears t o f a i r l y c l o s e l y resemble the c u r r e n t s i t u a t i o n i n Langley. P r i c e ranging on t h i s i n i t i a l s o l u t i o n , as w e l l as parametric programming around t h i s s o l u t i o n brought a n o t a b l e model f e a t u r e t o l i g h t . The model, as c o n s t r u c t e d , i s very i n s e n -s i t i v e t o q u i t e l a r g e s h i f t s i n r e l a t i v e g o a l weights. For example, even i f the weight on the 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 g o a l i s m u l t i p l i e d by a f a c t o r of t w e n t y - f i v e , model s o l u t i o n remains v i r t u a l l y unchanged. I n c r e a s i n g the weights on a l l the o t h e r t h r e e p o s i t i v e e x t e r n a l i t y g o a l s by a f a c t o r of f i v e changes model s o l u t i o n o n l y m a r g i n a l l y . Compare the i n i t i a l s o l u t i o n t o the s o l u t i o n presented i n column (b) of Table 1, f o r which the g o a l weights on the green b e l t , green space, and r e c r e a t i o n a l area goals have been s e t a t +5.0 while a l l o t h e r g o a l weights remain s e t a t ±1.0. Nor does e l i m i n a t i n g a l l nine 78 e x t e r n a l i t y g o als from the model change the s o l u t i o n as much as was expected, although p o l l u t i o n g e n e r a t i n g a c t i v i t i e s do ga i n s u b s t a n t i a l l y h i g h e r a l l o c a t i o n s a t the expense of l a n d uses which generate s u b s t a n t i a l amounts of p o s i t i v e e x t e r n a l -i t i e s . Column (c) i n Table 1 presents the s o l u t i o n when gross p r o d u c t i o n i n the r e g i o n i s the s o l e c r i t e r i o n f o r o p t i m a l i t y . The i n i t i a l s o l u t i o n i s most s e n s i t i v e to changes i n the c o s t s of p o l l u t i o n , though even here a l l the p o l l u t i o n g o a l weights must be a t l e a s t doubled r e l a t i v e to the oth e r g o a l weights b e f o r e any major changes begin t o occur. At t h i s stage, a c t i v i t i e s which generate l a r g e amounts of p o l l u t i o n q u i c k l y d i s a p p e a r from the s o l u t i o n . The f o u r t h column (column d) of Table 1 prese n t s the o p t i m a l s o l u t i o n when a l l the p o l l u t i o n g o a ls have a weight of -5.0, while a l l other g o a l weights remain s e t a t +1.0. Note e s p e c i a l l y the low per c a p i t a income t h a t r e s u l t s from p l a c i n g a hi g h p r i c e on p o l l u t i o n ( t h i s i s not the per c a p i t a share of the value o f the o b j e c t i v e f u n c t i o n but gross 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 t r y i n g t o v i s u a l i z e the c h a r a c t e r of t h i s i n i t i a l model s o l u t i o n , i f the s e t of a l l f e a s i b l e s o l u t i o n s , i t s e l f a m u l t i - d i m e n s i o n a l concept, can be re p r e s e n t e d as a t h r e e -dimensional t e r r a i n , w i t h the h e i g h t o f the t e r r a i n corresponding to the va l u e 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 i n i t i a l s o l u t i o n i s l o c a t e d i n the middle o f a very l a r g e , almost f l a t p l a t e a u . U n f o r t u n a t e l y , t h i s s i t u a t i o n makes s e n s i t i v i t y a n a l y s i s 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 for an optimum quite f r u i t l e s s . Social welfare, as measured by the value of the composite objective function i s v i r t u a l l y the same on a l l points of the plateau. In other word, the marginal increases i n s o c i a l welfare that may be achieved through any i t e r a t i v e search are not worth the costs of searching for them. This s i t u a t i o n should not be construed, however, as in d i c a t i n g that i t e r a t i v e search methods, of the type outlined i n previous chapters, have l i t t l e worth-while application. That they have l i t t l e a pplication i n t h i s case i s due to the nature of t h i s p a r t i c u l a r model, not to any inherent deficiency i n the i t e r a t i v e search methods. They doubtless w i l l have applications for other 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 population i s allowed to increase, c e t e r i s paribus. Table 2 shows model solutions for population increments of 10,000, st a r t i n g at a population of 10,000 (the 1971 population of Langley was 21,936 ). The steady decline i n per capita income (gross production) and i n per capita s o c i a l welfare (measured by value of the objective function) are probably the most in t e r e s t i n g features here. Nor should these declines be unexpected. As far a per capita income goes, what e s s e n t i a l l y i s being done i s to increase the amount of workers i n an area that has a limited production p o t e n t i a l . The even more rapid decline i n the per capita share of t o t a l s o c i a l welfare occurs because the increasing population creates much additional p o l l u t i o n but contributes very l i t t l e i n the way of p o s i t i v e e x t e r n a l i t i e s . 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 a c r e , a l l g o a l weights = ±1.0) A c t i v i t y Group P o p u l a t i o n 21936 10000 20000 30000 40000 a g r i c u l t u r a l uses 73268 73740 73446 72526 71605 commercial uses 11 5 10 16 22 i n d u s t r i a l uses 158 12 134 255 377 r e c r e a t i o n a l uses 494 868 450 675 900 t r a n s p o r t a t i o n uses 1449 1001 1378 1747 2117 r e s i d e n t i a l uses 1068 976 1054 1124 1195 i n s t i t u t i o n a l uses 282 129 256 386 515 TOTAL ACREAGE 76730 76730 76730 76730 76730 ( d o l l a r u n i t s ) per c a p i t a income 1906 2404 1946 1792 1715 per c a p i t a s o c i a l 2134 2841 2192 1974 1854 w e l f a r e 81 a p p l i c a b i l i t y of these model r e s u l t s t o the a c t u a l l a n d use system t h a t was a b s t r a c t e d , the r e s u l t s do appear encouraging, and they do p r o v i d e some i d e a of the type of r e s u l t s and the type of a n a l y s i s t h a t can be produced by mathematical l a n d use models, t o g e t h e r w i t h an understanding of the p o t e n t i a l u s e f u l n e s s of'-'such models. F u r t h e r Refinements C e r t a i n l y the type of a n a l y s i s t h a t m u l t i p l e goals l i n e a r programming o f f e r s to l a n d use p l a n n e r s appears d e s e r v i n g of f u r t h e r study. S e v e r a l f u r t h e r refinements and areas of concern are suggested by t h i s i n i t i a l study. F i r s t l y , the area of data c o l l e c t i o n and g e n e r a t i o n must be more c l o s e l y examined. A l l a v a i l a b l e data sources need to be found and c l e a r l y documented. E x i s t i n g data needs t o be c o l l e c t e d and s o r t e d . Methods o f updating data need to be examined. Data requirements not s a t i s f i e d by a v a i l a b l e i n f o r m a t i o n need to be s p e c i f i e d and l e a s t c o s t methods of g e n e r a t i n g data to f u l f i l these requirements must be found. Secondly, when s u f f i c i e n t data i s c o l l e c t e d , improve-ments i n model d e s i g n need to be undertaken. A g r e a t e r amount of a c t i v i t i e s and c o n s t r a i n t s can be added to the s t r u c t u r e t o i n c r e a s e model d e t a i l . S i m p l i f y i n g assumptions can be removed and r e p l a c e d by more r e a l i s t i c s t r u c t u r i n g . Information on e x i s t i n g l a n d uses, c o n v e r s i o n c o s t s between uses, c a p i t a l flows 82 and c o n s t r a i n t s , and g r e a t e r s p e c i f i c a t i o n o f 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 can be i n c l u d e d i n model d e s i g n . Since the e n t i r e Lower F r a s e r V a l l e y f u n c t i o n s as an economic u n i t , expansion of the model t o i n c l u d e t h i s e n t i r e area should be c o n s i d e r e d . T h i s process would permit g r e a t e r s p e c i f i c a t i o n o f i n t e r r e g i o n a l r e l a t i o n s h i p s and c o n t r i b u t e f u r t h e r t o model accuracy. L a s t l y , the whole area of the psychology and s o c i o l o g y of p o l i c y making groups needs examination t o determine the exact c o m p a t i b i l i t y o f a m u l t i p l e goals type o f a n a l y s i s t o d e c i s i o n making processes and to suggest p o s s i b l e improvements i n a n a l y s i s design which would i n c r e a s e 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 making t o o l . T h i s i s a very important area o f concern where study i s needed i f t h i s type o f a n a l y s i s i s to s u c c e s s f u l l y move past the t h e o r e t i c i a n i n t o the r e a l world processes o f land use pl a n n i n g . D i s c u s s i o n On the b a s i s of model c o n s t r u c t i o n and the pre 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 appears t h a t m u l t i p l e goals l i n e a r programming p r o v i d e s a u s e f u l format w i t h which problems of l a n d use a l l o c a t i o n can be examined. I t s q u a l i t i e s of s i m p l i c i t y , f l e x i b i l i t y and m u l t i p l e g o a l o p t i m i z a t i o n make t h i s s t r u c t u r e p a r t i c u l a r l y w e l l s u i t e d t o the 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 t h i s format lends some q u a n t i f i c a t i o n t o an area t h a t has remained almost wholly q u a l i t a t i v e t o date. Yet a t the same time, t h i s s t r u c t u r e 83 does not f o r c e d e c i s i o n making bodies t o completely bare t h e i r p r e f e r e n c e f u n c t i o n s and i n d i f f e r e n c e r e l a t i o n s h i p s . Since t o do so may prove t o be p o l i t i c a l l y unacceptable to some such b o d i e s , t h i s f e a t u r e i s very compatible w i t h such c o n s i d e r a t i o n s . What t h i s model s t r u c t u r e does do, however, i s to f o r c e the d e c i s i o n making body to a t l e a s t t h i n k i n terms of v a r i o u s goals and o b j e c t i v e s and t h e i r p r e f e r e n c e p a t t e r n s f o r t r a d e - o f f s between these. By doing so, t h i s model type p r o v i d e s an a c c e p t a b l e , s y s t e m a t i c , r a t i o n a l format by which to s p e c i f y , examine, and f i n d o p t i m a l s o l u t i o n s t o problems of l a n d use a l l o c a t i o n . As such t h i s type o f a n a l y s i s r e p r e s e n t s a u s e f u l t o o l which pla n n e r s can use f o r d e c i s i o n making i n the l a n d use f i e l d . A t t h i s p o i n t , i t should be noted t h a t m u l t i p l e goals l i n e a r programming does not r e p r e s e n t the o n l y "good" p l a n n i n g t o o l , or even the "best" a i d a v a i l a b l e i n t h i s a r ea. Though, on the b a s i s o f s e a r c h i n g f o r op t i m a l d e c i s i o n s i t appears the most s u i t a b l e o f the s t r u c t u r e s d i s c u s s e d , a l l of the model types do have very u s e f u l a p p l i c a t i o n s i n the land use f i e l d . CEf some othe r c r i t e r i o n were used t o measure the u s e f u l n e s s of these models, i t i s p o s s i b l e and even l i k e l y t h a t some oth e r s t r u c t u r e would be judged most s u i t a b l e . Then too, i t must be remembered t h a t a c t u a l models r a r e l y are, and r a r e l y need to be, composed e n t i r e l y o f o n l y one of the 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 study. Most o f t e n they c o n t a i n elements of more than one s t r u c t u r e and f r e q u e n t l y they are a composite of a l l t h r e e . Often the three d i f f e r e n t types o f s t r u c t u r e s are melded i n such a f a s h i o n t h a t 84 each undertakes a separate f u n c t i o n or group of f u n c t i o n s t h a t i t i s b e s t s u i t e d to p r o v i d i n g . T h e r e f o r e , to s i n g l e out one s t r u c t u r e and to suggest t h a t i t i s the o n l y one t h a t m e r i t s use i n t h i s f i e l d i s o b v i o u s l y f o l l y . A l l three s t r u c t u r e s are compatible and a l l are capable of making some c o n t r i b u t i o n to l a n d use p l a n n i n g . 85 Chapter 8 SUMMARY AND CONCLUSIONS The p r i n c i p l e o f the p u b l i c c o n t r o l o f the p r i v a t e use o f l a n d i s one which i s r e a d i l y accepted i n most p a r t s of the world. In the Lower Mainland o f B r i t i s h Columbia, t h i s p r i n c i p l e has a s s e r t e d i t s e l f from the f i r s t beginnings of set t l e m e n t . Though the government"s c o n t r o l over la n d use i n the r e g i o n was f a i r l y tenuous u n t i l w e l l i n t o the c u r r e n t century, t h i s c o n t r o l has s t e a d i l y grown u n t i l , today, the government, through i t s v a r i o u s agencies, i s r e s p o n s i b l e f o r most of the p l a n n i n g and development of the land r e s o u r c e s i n the r e g i o n . As l a r g e s c a l e l a n d use p l a n n i n g developed, so d i d s e v e r a l new p l a n n i n g t o o l s and methods. One major new dev-elopment has been the use of mathematical models t o study l a r g e l a n d use systems. Though there are those who regard t h i s development as a poor a l l o c a t i o n o f e f f o r t , most r e s e a r c h e r s i n the f i e l d b e l i e v e t h a t mathematical models have some r o l e to p l a y as a p l a n n i n g t o o l . Three types o r major groups of macroeconomic models can be i d e n t i f i e d from the l i t e r a t u r e . Input-output models examine the s t r u c t u r a l flows w i t h i n a system and can be used i n l a n d use p l a n n i n g ,for s h o r t term 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. S i m u l a t i o n models c l o s e l y r e p l i c a t e r e a l world system&actions (with l i t t l e r egard t o i n t e r n a l s t r u c t u r e ) and can a l s o be used i n land use p l a n n i n g f o r s h o r t term f o r e c a s t i n g and p r e t e s t i n g of p o l i c y d e c i s i o n s . Mathematical programming models 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 a system and can be used t o f i n d " c o r r e c t " or "best" l a n d use a l l o c a t i o n s . The i n h e r e n t o p t i m i z i n g 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 of land use p l a n n i n g than e i t h e r s i m u l a t i o n or inp u t - o u t p u t a n a l y s i s . A type o f mathematical programming p a r t i c u l a r l y w e l l s u i t e d t o land use p l a n n i n g i s the r e c e n t development of a form of l i n e a r programming which allows f o r m u l t i p l e goals i n the o b j e c t i v e f u n c t i o n . With t h i s type o f model, land use a l l o c a t i o n s can be found which are o p t i m a l i n r e s p e c t to a com-p o s i t e f u n c t i o n o f s e v e r a l p o l i c y g o a l s , r a t h e r than j u s t one. Since many go a l s are r e l e v a n t t o d e c i s i o n making i n the land use f i e l d , t h i s procedure i s p o t e n t i a l l y very u s e f u l . Using t h i s m u l t i p l e goals format, a l i n e a r program-ming model was c o n s t r u c t e d t o study l a n d use a l l o c a t i o n i n the C i t y and D i s t r i c t o f Langley i n the Lower Mainland o f B r i t i s h Columbia. While problems of data a c q u i s i t i o n and l i m i t e d resources s e v e r e l y reduced the a c t u a l a p p l i c a b i l i t y of the model, model r e s u l t s appear encouraging. C e r t a i n l y t h i s type o f a n a l y s i s t h a t the m u l t i p l e goals s t r u c t u r e o f f e r s to l a n d use planners appears 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 o f f u r t h e r study. 87 BIBLIOGRAPHY Adelman, I.,,.and F. L. Adelman. "Dynamic Properties of the Klein-Goldberger Model," Econometrica, XXVII (October, 1959), 596-625. B r i t i s h Columbia Department of Agriculture. 1971 Production  of Vegetable Crops Together with an Estimate of Farm  Value. V i c t o r i a , B. C , 1972. B r i t i s h Columbia Department of Agriculture. 1970 Fraser Valley  Dairy Farms by Municipality. V i c t o r i a , B. C , 1971. Canada, Dominion Bureau of S t a t i s t i c s (now S t a t i s t i c s Canada). Census of Canada. 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South-E a s t e r n Wisconsin Regional P l a n n i n g Commission Land Use- T r a n s p o r t a t i o n Study" P l a n n i n g Report No. V, V o l . T~, Milwaukee, 1965. T u l s a M e t r o p o l i t a n Area P l a n n i n g Commission. 1975 M e t r o p o l i t a n  T u l s a Community Land Needs. T u l s a , 1959. Zusman, P., and A. Amiada. " S i m u l a t i o n : A T o o l of Farm Pl a n n i n g Under C o n d i t i o n s of Weather U n c e r t a i n t y , " J o u r n a l o f Farm  Economics, XLVII (August, 1965), 574-594. 91 Appendix I 92 NOTE: V a r i a b l e numbers given i n the coding e x p l a n a t i o n to t h i s m atrix p i c t u r e (see Appendix II) are not i n c l u d e d i n t h i s computer output. For c r o s s - r e f e r e n c e purposes i t should be noted t h a t rows, going from top to bottom i n t h i s p i c t u r e , correspond to v a r i a b l e numbers 1 to 137 and columns, form l e f t t o r i g h t , correspond to v a r i a b l e numbers 138 through to 260. 9 2 a o o —J L L —< _ CL 5T D_ _ _ fO _ a _ <N 3t a . OL _ - 4 OO CL or: _ »~ v i a el- _ _ f^l L O a . e- <\J t o a ct: _ *"* => ^ a. »~ 3 _ CC m 3 Q- Ct _ r> O . CC _ —* ar U J u c a 0 X LL) < > >• - J •— X U , 0 a 0 K U H 0 cc z> a _ J H -CC _> O- < _j m 0 - - J 0 - < i _ J CXI D : _ _ < _j V I a a l ~ -t o 1 c o_ n*l c CL f \J c CL —4 _ J -LO C J 2T 0 - n ' to 0 2T CC O J f_ p—< O 0 . h -a CL r O _ 00 X 0 a r\j r> «/> x 0 a r-( Z> L_> _ CC I— z> 0 aC r<i _D O _ a. CXI 3 O _ 'X —4 y- cf. < _ to 1 -t - _ < l / l i ~ o_ < on CXI I - a : < - to -. C U . l t O h-O I L L L <_) m a L L a . cxj CD U- u. u c—1 i - 0 t - O -U . U J U J L L U J L U G L L LU L U rxt L L L U U t _ C_ <T CC _ JJ.. __ _ <" CC. >-t - O < O 2 2 11 l l a _ u u u r-t rxi I A r, o o • C C a o o 2* o > • —J o! r*- IT 11 n 11 _j _ _ J n cr c a o a O O L J o O I T —1 r-t I! 11 i c Of O n- u u. > o c. 11 l l . ' LL> C LU UJ a t u.: U_i LLJ - J _ J —I - J - J "Do' U 7 cS. a - J — — j*t ~ tr U _J LL .LI. L," o t -I J _ i p O ! I I iX. CL. _ i J - * fXl • ? I » i u u 1 L ; t/> on r,- i i LL • 7_ _ r ! u . U - I x i i a c 1 o o o jo C O ( • t~ 3 T 7f to to r\j c X' C" X L_) tO LO CO i F F I: E E E C G 0 0 F F F F T T R R A A R U U U U A C C C C U U U U s s s s H H H H s s s c c c I I N N N N N N M N O O n O N N N N s s s s s s s s H H H H o o o n R R R R T U U U U 0 P. A R R A A R T A I U U U U P p p p R R R R M M P P R R R E D D O O O C C C C S S S S R P R R P P P P R P. R R P P P P L L L L N O H V O E - K K K K K K K K Y N I 2 3 T M 1 2 3 T 1 2 3 T 1 2 3 T 1 . 2 3 T 1 2 3 T 1 2 3 T 1 2 3 T D O T Y D C 1 2 3 T 1 2 3 T K K K K F 1 2 3 T 1 S S H? P 2 P.URAL 1 R U R A L ? 2i M N G F . B T L - 1 - 1 - 1 M N T » F C L - 1 M K J t J P R |< L -C L -c S'tvi MP RK L -c MNG1JL " L M . M R D E V L U P R K 1 G -c U ° R K 2 G -c S P R K 1 G -c S P R K 2 G -c MPRK 1 G -c M P R K 2 G -c G O L F 1 G -c G O L F 2 G C P . D E V 1 G C R ' I E V ? G M N U R I L MNSRD MNRPQ M.X'OARK "fN^A'TT" MM A I R MMHOS 0«MS'<UL " N G O V T HOSPl H O S P ? SKUL 1 S K I J L 2 GOVT i GOVT 2 HOUSAL MXUNE-M -u-u ~ = u — - T - 3 -B - U B A L A ^ J C T A G ? I C T F E E 0 " T C O M ' R C -TQFF- I C TTRA'IS 1 - 1 - 1 - 1 - 1 - 1 1 "1 = T ~ - 1 - 1 - 1 1 - 1 - 1 - 1 1 TiJCf:R F - 1 - 1 - 1 1 T U S H O P F - 1 - 1 - 1 1 VO G T D O O O T T T T U U U U S S S S R R R R T T T D A F F F F C I F F F F R R R R U U U U S S S S S S S S S S S S U U U U O L H O U U U U S S S S M M M M G O A R E E E E T F F F F A A A A C C C C H H H H C C C C . H H H H R R R R T F I E W T P P P P P P P P ° P P P O T I H E F E E C I I I I N N N N N N N N O O O O N N N N O O O O A A A A I C G A C P . R R R R R R R R R R R P L A R E n D n O O C C C C S S S S R P . R P P P P P R R R R P P P P L L L L N O H V O E K K K K K K K K K K K K F " G Y N 1 2 3 T M 1 2 3 T 1 2 3 T 1 2 3 T 1 2 3 T 1 2 3 T 1 2 3 T 1 2 3 T D O T Y D C 1 2 3 T 1 2 3 T 1 2 3 T 1 TSCMR E - 1 - 1 - 1 1 TSS' - inp E - 1 - 1 - 1 TP. URAL E - 1 - 1 - 1 1 T I N D U S T R E C R E E E 1 - 1 - 1 - 1 - 1 1 - 1 - 1 - 1 TUPRK p - 1 - 1 - 1 1 TSPKK T M O ' . K - 1 - 1 - 1 1 - 1 - 1 - 1 1 T G O L F . c - 1 T C s O - . V c TTRAM c T P E S I D c T I M S T I c T!lf)S° E T S K U L p. TGOVT p i T F L D ' I D c T F R E E S E C E T G P R E L T t: T G R S P ACE p TAGS E L F c F F R E E R EC c - 1 - 1 - 1 F G R J. F L T p - 1 - 1 - 1 F G R S " A C E E - 1 - 1 F A G S E L - c - 1 - 1 - 1 FPEE°EC1 G F R E E " E C 2 G G P R F L T 1 . G G R U E L T 2 G GR S n AC F1 Q G R S P A C E? G A G S E L F 1 G AGSFL = 2 G W W . P E - c - c - r )-»-C -F-E-E -F-E-E -F-E-E -F-E-E - P - f c - t -F - f c-fc -F-t-fc - t - L - L - C - L - C - L - C - L - L - L M N A I R D C L E l A A D MMWAT FT c A A C 8 0 1 " W - F U S E p C L H M W I S E p A 1 A A C B B MN S I G H T c 3 B C M y F R S S E C p «XG"BELT M X G R S P A C " M X A G T E L F " o 92d < < _> _ O CC O CC O CC o o o LL CC LL CC LL CC LL £_ CO _ J CC _) O C O I D - J O CO L U —1 co a < o to a . < _) to Q. < l C J l / l Q. < L ) a : co U J - j a cc _ _ J 0 . - 0 — 1 — 1 Q_ CO L U _ J LLi LU _ LU U J LU C L L U L U L U _ LU U J LU CC L U 3 K L U _ J < CO <_ K D h Q-CD >- O _ P >• LL _ J _ J y u . j D > U 1 ZD _ O L O _j o a > o a > o o > o c > L O _ 3 CO _ 3 to _ 3 • to _ 3 X O co X O to x a to X o to K- C J V— •—• <- O X o i n < o <r o i n < L , 1 < _ < h- C t~ CC < — CY. Q -a : 3 o: to 3 CC 3 a- n: V~ Cj CC o or r U J U i _ G I L ' u f>: c uj U CC O U J e o - J o c- - J O C - J LL (M LL <-* LU K LU r o UJ (NJ LU —4 U t— U fO U <VJ U - 4 _ C L 3 or O CL I— CC C CM _ ^ O h -< _ a. h-C L f O o_ rv> O- - 4 _ CO _*: U J C_ L U £ _ L U CC L U cc CJ t_ Q - O or c n_ o <t > t~ > m > (VI > -< LL t~ U fO o >-_t _j a _ n _'ir^ o' c o G o o _> u o e.t o m -*! II II fi ii LO to ,i _ J ILJ !... !( C * - — - J O . —J _ J Ci -a —J U U U —J —J >-i); CJ a '_• c-> r: o u. — IU tl.i UJ |_j -r n U J U J to z: 7i < to </i co _ c ;JY _ —I i>—i *V .—( T a : O i_» i / 1 u LO r-1— — L O L O r,: u. IL -^ r o o _> o 9 2 e <S O tO U J _ J LL O J < O t o L U _ J LL. r-t O tx o. < o LU t~ O CC W £L < o LU m (_ Of CO CL o J j CM o a */> Q . < o LU r - i O Of CO u; —1 — J— —' Of. CC LU _ J m O a cc LU —t Y-Csl O Of CO LU _ J —1 LL Of LU LU CC LU LL Of U J LU CC LU U m LL (_ LU LU CC LU _ fM U . Of U J LU CC L U »-l _ LU _ CL - J < CO UI 3 Of 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 . 1 3 O - J D to _• < _ O o > o C3 > t - r o o o > 1— fx) o o > ^ to _ 3 -to 3 _ J m to 3 _ J r\i LO _ 3 _ J X o t o a. I— X a to a m X c to a. <xj X o to C_ t—• K O h- - to < o X O _c LU tA <r o Of U-i —1 < (_) Of UJ If* \ < o Of LU <r a *5 o f t -Of I U to < CC CL o f h-CL < ~ Of a . < CC Of 3 Of CC o t o 3 CO CC o 3 CC, 00 Of o K O h-CC <t 2: o or a U J > t~ u cc o t!_ > m u> — o LU > rxt a n: > - 4 (.-*• c _ J LL o o _ J LL ° _ l L L rxi CO CO o o I »-I <r i <_• O UJ .UJ _ • L U I Y— L> . CC O > h- 3 <XJ ( ,v y : (T Of ; H (X) i n: a rc < 3 >" r y 3 > i f •_ c io. _J _f t o O to to 3 3 C - O - f x i d v, H- t - _» c- n > > 3 IX 3 1 c o o 'o x : L3 O X ,< > I O L U O LL rr *3 U \j O H* K H- t - f -u> to 3 3 9 2 f < o to LU _J LL rvj O OO U J ~i I L -* CC V I CL < o LL' h-O cr to a. < u LL) m O CC VO Q. < u U J r\j o cr oo a. <r u U J r-l O CC. CO L U —J \— r-o or co U J _ J f -ct CO LU •J o CT CO U J _J w u. a: U J L U CC U J L ) h-L L a: LU 1X1 CC U J U ro L L cr U J LU OC U J L_) Lt- OC U J LU OC U J t_) r-l =2 LU a: a _ J < CO a cr >- o l~ CL o a. CC > u -j CC CC >- LL _ l o CC ZD >- LL - J D o ^  CC 3 > u_ - J _J o (/I < I T o o > K— o O > H- rr, o c > r - rvj o o > K r-l vo ^ r> 1 vo rO —J <N to _1 I o c/> CL J— X o to CL m X o LO a rg X c a. r-4 r— C r~ - to <l o X G U J in U CC LU r-r < O a : LU LTi ^ < o rr LU < <r cr H 1 - O r - cr U J i/> <I —< cf. CL a: t~ cr. <r ~ _ J cr CL <t CC ^ DC Z> cr cr Q OO CD o : O Z> cr m (X G t~ a i— cr <? z1 O Or- n LU > t— o cr: o LU > CO o o LU > r>! L J cr o. U J > o o - J u. c: _ J I L m o _ J a i c i < u n fc O < D a < o < O rH CO O r-l o < cr < 30 < J < CD _) < CD O CC ~* o o CC < H rfi cn <s u u 111 U.J I LU V V I L LL LI" _ a . a c <: is. jtyi t 0 17" rr rr U.J c >: l- r y . j ~ r L O C > C J I J H I T IT' t o t -j cr cr < L L U - lO O • LL U . L L O C 10 o c I rH !f\J H f\J U J 'lil rH fV - O J O I L I L _ J _ J < <f _J _ J . u a c U J ^: r f t o |co t o iy, or. cr CL O O O O O C <T -3 l i ' U.' U J LiJ I a a v i u. t o x O . h- ' LL o — <T * U O — o <: 3 . a : ^ oo x ^ ^ ?: ~ IT a: ^ 5 : I LU Ul U J U J u r r t o |oo it' a 92g i U J U J U J . O U J C I < O i/> I O CL W Q- • o cr re i LL CfT. LLI L U ( L O •— o X t- ( cr U J u_ : <T. f~ U • cr ( < — cr ; < <I LS L O 1-1 . < O U J , U J O C — ' O _ J : cr Q t <t <r I I I I I r-l |— I I I I < < < < I I a. - r j II n _ J ! . O O f CL O £ ; r.r o < . C r. ( ) u u i I/I 1/1 C 77 7' - « h T > 73 rr L_> *7T rt" i : —. •— < 2: • _> U. j'.c L : ' X U C h D x 77 77 y :r x .2' T : : • - J O c_o O i ! O LO 7T !-- 1 i ty. <r> rr ]LL. LL P?- Z IU. LL 5" r c- c i-O C O (J O O i ' rs.1 i / i H r j 'n a r-i :- u c: ' O J c <r ^ 7* -T x. rr o o 't/i oo t_; >— 777 77 >77 77 t--n ( i y A A G G S S E c 1. L W A A T I E R R P P 2, T~\ R E F F R O O S O O E L C L 3 T P L L E L L C T E F SSHC°2 RURAL 1 VNGP^T T VNT"EC L MMSPRK L MNj'-PRK L M \ ' G 3 L C L MNRDEV L UPRK1 G UPRK2 G SPRK'I G SPRK2 G MPRKI G MPRK2 G GOLF I G GOLF 2 G CR9FV1 G CROEV2 G MN'iJRD L L M N R - 0 L V N P A R K L MNRA I L L MM A I R L MNHOSP L MNSK'JL L MNGOVT L HOSP1 HOS°2 SKIJL 1 >j SMJL 2 G GOVT1 r7 GOVT 2 G HCl'JSAL E AGHOME E M X U"'! F '' G evp.-;pp E RALAMC L TAG"IC c T FEE D E TCOMRC E TOFFIC • E TTRAV! S F TUCNR E TUSHOP E VO 92i 00 < o CO L U — J L L O Of CO CL < o L U o Cr: CO L U — J L L cr. I l l LLI CC L U O L O o X t— C L c _J o i - L O L U C - J Cf. LLI Li . CO LLI LU _ CL o _ J < - Q' O. c- _ J o of n . < o CO L U - J u. H < LO L U _ J LL, r<1 I CO U I r; cv v > ^ LL; CI. • O O O O CD C ; —i (\j LL' [u r - . ro • h - L_" ;(_) LL U . CL C I LO vf i UJ C i / i • . i f a : 11^  LL L ' O U i .'J- (>" .o n- u.i " LU U a ! i - C C U J UJ U i rU UI Ll > I _ J ItlJ O cr 'co IU I— c u i co • cr »- IL —• i •— <j i n ; a • • <f a : ~ ' LU u . U ' I U h u u . L U _ J <T l_ l CC 111 O III ' a; c _ c 1 'co if a V , ' ' x o o '<r E X E C U T O R . M P S / 3 6 0 V 2 - M 5 : P A G E 3 = 7 3 / 2 2 2 S U ' r i « Y OF MATRI X S Y N W l RANGE COUNT I I N C L . R H S ) Z " L E S S T H A ^ . 0 0 0 0 0 1 Y . 0 0 0 0 ) 1 THRU _ P _ 0 _ ? _ . ! X . 0 0 0 0 1 0 . 0 0 0 0 9 9 W ~ . 0 0 0 1 0 0 : . 0 0 0 9 9 9 V . 0 0 1 0 0 0 . 0 0 9 9 9 9 ; 3 U . 0 1 0 0 3 0 . 0 9 9 9 9 9 ' 11 "T ' . 1 0 0 0 0 0 . 9 9 9 9 9 9 29~ 1 1 . 0 0 0 0 0 0 1 . 0 0 0 0 0 0 3 4 4 A 1 . 0 0 0 0 0 1 1 0 . 0 0 0 0 0 0 84 "H " 1 0 , 0 0 0 0 5 1 ' 1 0 0 . 0 0 0 0 0 0 . 5 5 C 1 0 0 . 0 0 0 0 0 1 i_,jOJ;O.J3GOOO_0 70 0 1 , 0 0 0 , 0 0 0 0 0 ! 1 0 , 0 0 0 . 0 0 0 0 0 0 33 ~F. 1 0 , 0 0 0 . OOOOO1 ICO , 0 0 0 . 0 0 0 0 0 0 43~ F 1 0 0 , O O P . C C 0 0 0 1 1 , 0 0 0 , 0 0 0 . 0_0_0000 10 G G R E A T E R T H A N 1 , 0 0 0 , 0 0 0 . 0 0 0 0 0 0 _ 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 through 137 r e p r e s e n t row v e c t o r s i n the matr i x w h i l e v a r i a b l e numbers 138 to 259 are a c t i v i t y columns. VARIABLE COMPUTER VARIABLE NUMBER CODE NAME DESCRIPTION or COMMENT 1-12 2-7, 9-10 CPOL=.8 CPOL=.9 CPOL=l.l CPOL=1.2 CPOL=1.5 CPOL=2.0 CPOL=5 CPOL=10 CGRPONLY o b j e c t i v e f u n c t i o n s twelve a l t e r n a t e o b j e c t i v e f u n c t i o n s . Each f u n c t i o n has a d i f f e r e n t s e t of go a l wts. a l l g o a ls i n t h i s o b j e c t i v e f u n c t i o n have a weight o f ± 1. T h i s does not mean t h a t a l l t e n goals are e q u a l l y important. I t means t h a t the u n i t s used i n the ten d i f f e r e n t i n d i c e s are equal i n terms of t h e i r c o n t r i b u -t i o n s t o the composite o b j e c t i v e f u n c t i o n . Since an attempt was made to express a l l the d i f f e r e n t u n i t s i n terms of $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 used as a s t a r t i n g p o i n t i n determining f i n a l g o a l weights. 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 weights of -.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 weights on a l l the other goals remain s e t a t +1.0. These 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 v a r y i n g the r e l a t i v e import-ance of the p o l l u t i o n goals i n the model. T h i s o b j e c t i v e f u n c t i o n l e a v e s the gross r e g i o n a l p r o d u c t i o n g o a l weight s e t at +1.0 but s e t s P4 the weights on a l l the remaining goals at zero. This f u n c t i o n was used to determine the amount of i n f l u e n c e t h a t a l l the e x t e r -n a l i t y goals have on the optimal a l l o c a t i o n . 11-12 CLIES=5 CLIES=10 These two o b j e c t i v e functions place weights of +5.0 and +10.0 r e s p e c t i v e l y on the three pos-i t i v e e x t e r n a l i t y goals t h a t are concerned w i t h maximizing 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 weights on a l l the other goals remain set at +1.0. These two fu n c t i o n s were used to determine the 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 of these three r e l a t e d a o a l s . 13 TOTPOP t o t a l population c o n s t r a i n s t o t a l population (of Langley) to the d e s i r e d l e v e l . TOTPOP was i n i t i a l l y set at 21,936. 14 LABOUPv labour force c o n s t r a i n s labour f o r c e to 39% of t o t a l p o p u l a t i o n . 15 ALLAND a l l o c a t e a l l c o n s t r a i n s the a l l o c a t i o n of land land (between 34 a l t e r n a t e land uses) t o Langley's 76730 acres (see Table A ) . 16 SDEVEL s u i t a b l e f o r development 17 NEEDCLIR needs land clearance NEEDFLUD needs f l o o d p r o t e c t i o n c o n s t r a i n s development to c l e a r e d , non-flood p l a i n , gently sloped land. Equals 54663 acres + BYCLIR acreage (see Table A ) . c o n s t r a i n s those land uses tha t r e q u i r e c l e a r e d land to the t o t a l amount of c l e a r e d land: 64,861 acres + BYCLIR acreage (see Table A). co n s t r a i n s those land uses that r e q u i r e f l o o d protected or non-flood p l a i n land to the t o t a l amount of t h i s tvpe of land: 70530 acres + BUYFLUDT acreage (see Table A). 1971 population of Langley as reported by Census of Canada (Ottawa: Oueen's P r i n t e r , 1971). 2 P a r t i c i p a t i o n rate i n the Lower Mainland i n 1970 as reported i n The Lower Mainland's Economy: Trends and Prospects (Vancouver: Greater Vancouver Regional D i s t r i c t , 1970), p. I T . 95 19 ALLFLUD t o t a l f l o o d p r o t e c t i o n 20 SDAIRY s u i t a b l e f o r d a i r y i n g 21 SGRDEN s u i t a b l e f o r market gardening 22 SLNDAG s u i t a b l e f o r l a n d based a g r i c u l t u r e 23-24 FEEDl, 2 the f i r s t two steps of FEEDT's step v a l u e added f u n c t i o n 25 SINDUS s u i t a b l e f o r i n d u s t r i a l use 26 SPARK s u i t a b l e f o r a major park/ r e c r e a t i o n a l r e s e r v e 27 MAXFLUD1 maximum f l o o d p r o t e c t i o n o f A.R.D.A. c l a s s e s " o r g a n i c " and " e x c e l l e n t " l a n d 28 MAXFLUD2 maximum f l o o d p r o t e c t i o n o f A.R.D.A. d l a s s c o n s t r a i n s the t o t a l amount of f l o o d p r o t e c t i o n (BUYFLUDT) to any d e s i r e d l e v e l . Since i t i s not ec o n o m i c a l l y r e l e v a n t to p r o t e c t o n l y a p o r t i o n of a ibflood p l a i n , zero a c r e s o r 6200 acres (see Table A) are 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 DAIRY acreage t o t h a t a g r i c u l t u r a l l a n d c o n s i d -ered 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 of f e e d ) : 66030 acres (see Table A ) . c o n s t r a i n s GARDEN acreage t o t h a t a g r i c u l t u r a l l a n d c o n s i d -ered s u i t a b l e f o r market gard-ening: 5080 acres + BUYFLUDl acreage (see Table A ) . 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 or GARDEN) to t h a t l a n d s u i t a b l e f o r land based a g r i c u l t u r : 62260 acres + BUYFLUDT acreage (see Table A ) . c o n s t r a i n s the f i r s t two steps of FEEDT's step v a l u e added f u n c t i o n t o 100 acres each (see Table C ) . c o n s t r a i n s t o t a l i n d u s t r y (TOTIND) t o the amount of land c o n s i d e r e d s u i t a b l e f o r ind u s -t r i a l use: 2680 ac r e s (see Table A ) . 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 to the amount of f o r e s t e d l a n d i n Langley: 11869 a c r e s — B Y C L I R acreage (see Table A ) . c o n s t r a i n s a buy f l o o d pro-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 of the f l o o d p l a i n t h a t c o n s i s t s of o r g a n i c (peat and muck) s o i l s o r A.R.D.A. " e x c e l l e n t " a g r i c u l t u r a l s o i l s (see Table A ) . c o n s t r a i n s the second buy I f l o o d p r o t e c t i o n a c t i v i t y (BUYFLUD2) to the p o r t i o n of 96 the f i r s t two steps o f OFFICT's demand f u n c t i o n the f i r s t two steps o f TRANST's demand f u n c t i o n the f i r s t two steps o f UCNRT's demand f u n c t i o n the f i r s t two steps o f USHOPT's demand f u n c t i o n the f i r s t two steps o f SCNRT's demand f u n c t i o n the f i r s t two steps o f SSHOPT's demand f u n c t i o n the f i r s t two steps o f RURALT's demand f u n c t i o n minimum green b e l t acreage minimum t o t a l r e c r e a t i o n minimum urban park minimum suburban park minimum major p a r k / r e c r e a t i o n a l r e s e r v e c o n s t r a i n s the f i r s t two steps of OFFICT's step v a l u e added f u n c t i o n (see Table C ) . c o n s t r a i n s the f i r s t two steps of TRANST's step v a l u e added f u n c t i o n (see Table C ) . c o n s t r a i n s the f i r s t two steps o f UCNRT's step value added f u n c t i o n (see Table C ) . c o n s t r a i n s the f i r s t two steps of USHOPT's step v a l u e added f u n c t i o n (see Table C ) . c o n s t r a i n s the f i r s t two steps o f SCNRT's step v a l u e added f u n c t i o n (see Table C ) . c o n s t r a i n s the f i r s t two steps o f SSHOPT's step v a l u e added f u n c t i o n (see Table C ) . c o n s t r a i n s the f i r s t two steps of RURALT's step v a l u e added f u n c t i o n (see Table C ) . ensures 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 ) . ensures 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 can be used f o r r e c r e a t i o n f o r every 100 people i n the p o p u l a t i o n (see Table B ) . ensures 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 urban park (UPRKT) f o r every 395 people t h a t l i v e i n urban housing (see Table B). ensures t h a t there w i l l be a t l e a s t 1 acre o f suburban park (SPRKT) f o r every 395 people t h a t l i v e i n suburban housing (see T a b l e B ) . ensures t h a t there w i l l be a t l e a s t 1 acre 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 every 3 33 people i n the t o t a l p o p u l a t i o n (see Table B). 98 99 100 85 MXUNEM maximum unemploy-ment se t s an upper l i m i t on unemployment of the labour f o r c e a t 10% 86 EMPOPP employment o p p o r t u n i t i e s s e t s the employment d e n s i t i e s ( i n workers per acre) o f any a c t i v i t y t h a t p r o v i d e s employ-ment (see Table D). 87 budget b a l a n c e r ensures t h a t the y e a r l y muni-c 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 indu s -t r i a l l a n d uses do not exceed the tax revenues generated by these a c t i v i t i e s . M u n i c i p a l expenditures on r e s i d e n t i a l land uses are assumed t o i n c l u d e the c o s t s o f p r o v i d i n g the r e s i d e n t s w i t h s c h o o l s , parks, h o s p i t a l s , roads, e t c . The net m u n i c i p a l revenue (tax revenue 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 TOTCOM TOTIND APART 5/ACRE 1AGRE SACRE AGHOME +500.00 J +1000.OO3 -25.OO4 -10.OO4 0.00 4 +50.00 4 +200.OO4 88 TAGRIC t o t a l a g r i c u l -t u r e l a n d 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 data i n Economic Aspects o f Urban Sprawl (New Westminster : Lower Mainland Regional P l a n n i n g Board, 1956), p. 42. 1 0 1 t o t a l feed l o t acreage t o t a l commercial acreage t o t a l o f f i c e services acreage t o t a l transport services acreage t o t a l urban neighbourhood r e t a i l services sums FEEDl, 2 and 3 into FEEDT. sums UCNRT, USHOPT, SCNRT, SSHOPT and RURALT into TCOMRC, sums OFFICl, 2 and 3 into OFFICT sums TRANS1, 2 and 3 into TRANST. sums UCNRl, 2 and 3 into UCNRT. t o t a l urban shopping center t o t a l suburban neighbourhood r e t a i l services sums USHOP1, 2 and 3 into USHOPT. sums SCNRl, 2 and 3 into SCNRT. t o t a l suburban shopping center t o t a l r u r a l r e t a i l services sums SSH0P1,2 and 3 into SSHOPT. sums RURALl, 2 and 3 into RURALT. t o t a l i n d u s t r i a l acreage t o t a l recreational acreage t o t a l urban park acreage t o t a l suburban park acreage t o t a l major park acreage t o t a l golf course acreage t o t a l commercial r e c r e a t i o n l development sums FOOD, LIGHT, HEAVY, and WOOD into TOTIND. sums UPRKT, SPRKT, MPRKT, GOLFT, and CRDEVT into TOTREC. sums UPRKl, 2 and 3 into UPRKT. sums SPRK1, 2 and 3 into SPRKT sums MPRKl, 2 and 3 into MPRKT sums GOLF1, 2 and 3 into GOLFT sums CRDEV1, 2 and 3 into CRDEVT t o t a l transporta-t i o n , u t i l i t i e s and communications acreage sums URBRD, SUBRD, RURRD, PARK, RAILRD, and AIRPRT into TOTRAN 102 106 TRESID t o t a l r e s i d e n t i a l acreage 107 TINSTI t o t a l i n s t i t u -t i o n a l acreage 108 THOSP t o t a l h o s p i t a l l a n d acreage 109 TSKUL t o t a l s c h o o l acreage 110 TGOUT t o t a l o t h e r i n s t i t u t i o n a l acreage 111 TFLOOD t o t a l f l o o d p r o t e c t i o n bought 112 TFREEREC t o t a l " f r e e " r e c r e a t i o n a l acreage 113 TGRBELT t o t a l green b e l t acreage 114 TGRSPACE t o t a l green space acreage 115 TAGSELF t o t a l acreage c o n t r i b u t i n g 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 116 FFREEREC " f r e e " r e c r e a t i o n c o e f f i c i e n t s sums APART, 5/ACRE, 1ACRE 5ACRE, and AGHOME i n t o TOTRES, sums HOSPT, SKULT, and GOVTT i n t o TOTINS. sums HOSP1, 2 and 3 i n t o HOSPT. sums SKUL1, 2 and 3 i n t o SKULT. sums GOVT1, 2 and 3 i n t o GOVTT. sums BUYFLUDl and 2 i n t o BUYFLUDT. sums FREEREC1, 2 and 3 i n t o FREERECT. sums GRBELT1, 2 and 3 i n t o GRBELTT. sums GRSPACEl, 2 and 3 i n t o GRSPACET. sums AGSELF1, 2 and 3 i n t o AGSELFT. sums the t o t a l a cres i n land a c t i v i t i e s 1-89 which 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 use and p l a c e s the sum i n t o FREERECT. An acre of urban park, suburban park o r major park a l l c o n t r i -bute 1 acre to FREERECT, while an acre 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 acres t o FREERECT. 5 F r a c t i o n o f t o t a l s c h o o l acreage t h a t i s gym ar e a , p l a y i n g areas or p l a y i n g f i e l d s as r e p o r t e d i n Report On School and Park  Standards ( Vancouver: C i t y o f Vancouver P l a n n i n g Department, 1957) , p. 3. 103 FRGRBELT green b e l t c o e f f i c i e n t s FGRSPACE green space c o e f f i c i e n t s FAGSELF 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 c o e f f i c i e n t s sums the t o t a l area i n land a c t i v i t i e s 1-89 which 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, and GOLFT make up t o t a l green b e l t acreage. sums the t o t a l area i n l a n d a c t i v i t i e s 1-89 which 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 o f urban park, suburban park and hobby farms p l u s one-quarter o f 1-acre r e s i d e n t i a l l o t s , farm homesites, and h o s p i t a l grounds p l u s one-tenth o f apartment l o t s , 5/acre r e s i d e n t i a l l o t s , and othe r i n s t i t u t i o n a l acreage are assumed to be green space. sums the t o t a l area i n land a c t i v i t i e s 1-89 which 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 of DAIRY, GARDEN, and FEEDT c o n t r i b u t e t o 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 . FREERECl the 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 the f i r s t two steps o f FREERECT's step demand f u n c t i o n each a t 1 acre per 250 people i n the t o t a l p o p u l a t i o n (see Table C ) . GRBELT1 the f i r s t two and 2 steps o f GRBELTT's demand f u n c t i o n GRSPACEl the f i r s t two steps and 2 of GRSPACET's demand f u n c t i o n s e t s the f i r s t two steps o f GRBELTT's step demand f u n c t i o n each at> 1 acre per person i n the t o t a l p o p u l a t i o n . s e t s the f i r s t two steps o f GRSPACET's step demand f u n c t i o n each a t 1 acre per 2 50 people i n the t o t a l p o p u l a t i o n (see Table C ) . AGSELF1 the f i r s t two steps and 2 o f AGSELFT's demand f u n c t i o n s e t s the f i r s t two steps of AGSELFT's step demand f u n c t i o n each a t 1 acre per person i n the t o t a l p o p u l a t i o n (see Table C ) . MXGRP gross r e g i o n a l sums the y e a r l y c o n t r i b u t i o n s product maximiz- t h a t the v a r i o u s a c t i v i t i e s make a t i o n 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 terms of value added d o l l a r s per acre o f a c t i v i t y . A c t i v i t i e s 104a MNAIRPOL MNWATER MNREFUSE MNNOISE MNSIGHT minimize a i r p o l l u t i o n minimize water p o l l u t i o n minimize ground p o l l u t i o n minimize n o i s e p o l l u t i o n minimize s i g h t p o l l u t i o n MXFREREC maximize "f r e e " r e c r e a t i o n a l acreage MXGRBELT maximize green b e l t acreage t h a t f a c e a step value added f u n c t i o n w i l l have three c o e f -f i c i e n t s i n t h i s r e s t r a i n t l i n e , one f o r each step of t h e i r v a l u e -added f u n c t i o n (see Table E ) . These 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 towards 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, noi s e 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 Table F ) . 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 step demand f u n c t i o n i n t o FREERECT by s e t t i n g d i f f e r e n t y e a r l y c o n t r i b u t i o n c o e f f i c i e n t s f o r acre 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 acre 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 attempt was made t o equate these index u n i t s w i t h 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 ) . 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 step 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 s t e p s . Every acre i n GRBELTl c o n t r i b u t e s 100 u n i t s t o MXGRBELT whi l e an acre i n GRBELT2 c o n t r i -butes 5 0 u n i t s and one acre i n GRBELT 3 c o n t r i b u t e s o n l y 25 u n i t s . An attempt was made to equate these index u n i t s w i t h 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 maximize green 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 step demand f u n c t i o n i n t o GRSPACET by s e t t i n g d i f f e r e n t y e a r l y c o n t r i b u t i o n c o e f f i c i e n t s f o r GRSPACET's thr e e d i f f e r e n t s t e p s . Every acre i n GRSPACEl c o n t r i b u t e s 10000 u n i t s to MXGRSPACE, wh i l e an acre i n MXGRSPACE2 c o n t r i b u t e s 5000 u n i t s and an acre i n GRSPACE c o n t r i b u t e o n l y 10 u n i t s . An attempt was made t o equate these index u n i t s w i t h 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 ) . 104b 137 MXAGSELF maximize 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 138 139 140 141-144 TOTAG t o t a l a g r i c u l -t u r e acreage DAIRY d a i r y farm acreage GARDEN FEEDl, 2, 3, and T acreage i n market gardens t o t a l beef f e e d l o 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 step demand f u n c t i o n i n t o AGSELFT by s e t t i n g d i f f e r e n t y e a r l y c o n t r i b u t i o n c o e f f i c i e n t s f o r AGSELFT's t h r e e d i f f e r e n t s t e p s . Every acre i n AGSELFl c o n t r i b u t e s 100 u n i t s to MXAGSELF whi l e an acre i n AGSELF2 c o n t r i -butes 50 u n i t s and one acre 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 attempt was made to equate these index u n i t s w i t h d o l l a r s of b e n e f i t r e c e i v e d (see Table C) i n c l u d e s feed crop acreage FEEDl and FEED2 and FEED3= FEEDT 145 TOTCOM t o t a l acreage i n commerce ground f l o o r space o n l y 146-149 OFFIC1, 2, 3, and T o f f i c e s e r v i c e s acreage i n c l u d e s m e d i c a l , d e n t a l , l e g a l and ot h e r p r o f e s s i o n a l or bus i n e s s s e r v i c e s . OFFICl and OFFIC2 and OFFIC3= OFFICT. 150-153 TRANS1, 2, 3, and T t r a n s p o r t s e r v i c e s 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 as through highway b u s i n e s s . TRANS1, and TRANS2, and TRANS3= TRANST. 154- UCNRl, urban neighbour- any r e t a i l o u t l e t t h a t c a t e r s 157 2, 3, hood r e t a i l acreage t o a s p e c i f i c urban neighbour-and T hood. UCNRl, and UCNR2, and UCNR3= UCNRT. 158-161 USHOP1, 2, 3, and T urban shopping c e n t e r acreage USHOP1, and USHOP2, and USHOP3= USHOPT 162- SCNR1, suburban n e i g h -165 2, 3, bourhood r e t a i l and T acreage SCNR1, and SCNR2, and SCNR3= SCNRT. 105 166-169 170-173 174 175 176 177 178 179 180-183 184-187 188-191 192-195 196-199 200 SSHOPL, 2 , 3 , and T RURAL1, 2, 3, and T suburban shop- SSHOPl, and SSHOP2, and pi n g c e n t e r acreage SSHOP3= SSHOPT. r u r a l r e t a i l o u t l e t acreage TOTIMD t o t a l i n d u s t r i a l acreage FOOD acreage i n food p r o c e s s i n g LIGHT HEAVY WOOD TOTREC UPRK1, 2, 3, and T SPRKl, 2, 3, and T MPRK1, 2, 3, and T GOLFl, 2, 3, and T acreage i n manu-f a c t u r i n g acreage i n heavy i n d u s t r y acreage i n wood products t o t a l r e c r e a t i o n a l acreage urban park acreage suburban park acreage major park/ r e c r e a t i o n a l r e s e r v e acreage g o l f course acreage RURALl, and RURAL2, and RURAL3= RURALT. i n c l u d e s c o n s t r u c t i o n UPRK1, and UPRK2, and UERK3= UPRKT , SBRK1, and SPRK2, and SPRK3= SPRKT MPRK1, and MPRK2, and MPRK3= MPRKT GOLFl, and GOLF2, and GOLF3= GOLFT CRDEVl, commercial r e c r e a - s p o r t s stadiums, p l a y l a n d s , 2, 3, t i o n a l development d r i v e - i n t h e a t e r s , e t c . and T acreage TOTRAN t o t a l acreage i n t r a n s p o r t a t i o n , communications, and u t i l i t i e s 106 201 202 203 204 205 206 207 208 209 210 211 212 213 214-217 URBRD SUBRD RURRD PARK RAILRD AIRPRT TOTRES APART 5/ACRE 1ACRE 5 AC RE AGHOME TOTINS HOSP1, 2, 3, and T urban road acreage suburban road a c r e -acreage r u r a l road acreage o f f - s t r e e t p a r k i n g acreage r a i l r o a d r i g h t of way and r a i l y ard acreage a i r p o r t acreage t o t a l r e s i d e n t i a l acreage acreage of l o w - r i s e apartments acreage i n h i g h -d e n s i t y s i n g l e fam-i l y d w e l l i n g s acreage i n low-d e n s i t y s i n g l e fam-i l y d w e l l i n g s hobby farm acreage farm home and homesite t o t a l i n s t i t u t i o n a l acreage h o s p i t a l acreages i n c l u d e s communication and u t i l i t y r i g h t of way. i n c l u d e s communication and u t i l i t y r i g h t of way. i n c l u d e s communication and u t i l i t y r i g h t of way. does not i n c l u d e r o a d - s i d e p a r k i n g . 3 s t o r y , 35-unit apartments, c o n s t i t u t e s urban housing. 5 l o t s / a c r e , e x c l u d i n g roads and l a n e s . C o n s t i t u t e s urban housing. 1 l o t / a c r e , e x c l u d i n g roads and l a n e s . C o n s t i t u t e s sub~^_ urban housing. Includes hobby farms, average 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 . C o n s t i t u t e s r u r a l housing. 1 acre area. C o n s t i t u t e s r u r a l housing. Includes grounds. HOSPLl + HOSP2 + HOSP3 = HOSPT. 218-221 SKUL1, 2, 3, and T s c h o o l acreages Includes grounds. SKULl + SKUL2 + SKUL3 = SKULT. 222-, 225 GOVT1, 2, 3, and T o t h e r i n s t i t u t i o n a l GOVTl + GOVT2 + GOVT3 = GOVTT. acreages 107 LDSLAK vacant land BUYFLUDT buy flood protect-1, and 2 ion a c t i v i t i e s BYCLIR buy land c l e a r -ance TOTPOP t o t a l Langley population LABOUR labour force available UNEMP t o t a l unemployed Since flood protection i s i n d i v i s i b l e , BUYFLUDT must come in at zero or maximum. BUYFLUDT equals BUYFLUDl + BUYFLUD2. Only applies to that land which i s presently wooded. FREEREC1 "free" recreational 2, 3, acreage and T Total acres i n land a c t i v -i t i e s 138-226 which confer an externality by vir t u e of being open to the public for recrea-t i o n a l use at no cost. FREERECl + FREEREC2 + FREEREC 3 = FREE-RECT. GRBELT1, green b e l t acreage 2, 3, and T Large tracts (greater than 50 acres) of vegetation covered land. Equals the t o t a l acres in land a c t i v i t i e s 138-226 which confer an externality by constituting a green b e l t . GRBELT1 + GRBELT2 + GRBELT3 = GRBELTT. GRSPACEl green space acreage 2, 3, and T Small plots of lawn and kept grounds. Equals the t o t a l acres in land a c t i v i t i e s 138-226 which confer an externality by v i r t u e of being green spaces. GRSPACEl + GRSPACE2 + GRSPACE3 = GRSPACET. AGSELF1, t o t a l acreage which 2, 3, contributes towards and T 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 . Total acreage i n land a c t i v -i t i e s 138-226 which confer an externality by vir t u e of c o n t r i -buting towards 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 . GRP t o t a l gross region-a l product (value-added) Goal accounting column, AIRPOL t o t a l a i r p o l l u t i o n (index units) Goal accounting column. 108 253 254 255 256 257 258 259 260 WATERPOL REFUSE NOISEPOL t o t a l water p o l -l u t i o n (index) t o t a l ground p o l l u t i o n (index) t o t a l n o i s e p o l -u t i o n (index) SIGHTPOL t o t a l v i s u a l p o l l u t i o n (index) FREEREC 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 GRBELT t o t a l worth of green b e l t acreage GRSPACE t o t a l worth of green space acreage AGSELF t o t a l worth of c o n t r i b u t i o n s towards 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 Goal a c c o u n t i n g column. Goal accounting column. Goal accounting column. Goal accounting column. Goal a c c o u n t i n g column. Goal accounting column. Goal a c c o u n t i n g column. Goal accounting column. 109 Table A AREAS OF LAND SUITABILITIES IN THE CITY AND DISTRICT OF LANGLEY (acres) t o t a l area 82460a (1) area unsuited for any use (marsh, water, rock outcrops etc.) 5730a (2) tota l area to be allocated: upper li m i t of ALLAND. (1) - (2) = (3) 76730 (3) area of "excellent" s o i l s (ARDA classes 1, 2, organic) i n flood plain: upper li m i t , of MXFLUD1 243CT (4) area of "good" s o i l s (ARDA classes 3, 4) i n , flood plain: upper li m i t of MXFLUD2. . . . 377CT (5) tota l area of 1948 flood plain: upper limit , of ATJJFLUD. (4) + (5) = (6) 620(T (6) area with greater than 15% slope 2688° (7) area of uncleared land: upper limit of MXCLIR. . . 11869 (8) tota l area suitable for development: upper lim i t of SDEVEL. (1) - (6) - (7) - (8) = (9) 54663 (9) tota l area suitable for any land use requiring cleared land: upper limit of NEEDCLIR. (1) - (8) + BYCLIR = (10) . . . BYCLIR + 64861 (10) to t a l area suitable for any land use requiring flood protection: upper li m i t of NEEDFLUD. (1) - (6) + BUYFLUDT = (11) . . BUYFLUDT + 70530 (11) ARDA land classification acreages: ARDA 1 and 2 ("excellent") 5080 (12) ARDA 3 and 4 ("good") 60950a (13) ARDA 5 and 6 ("fair") 8370a (14) ARDA 7 ("poor") 57303 (2) ARDA organic ("peat and muck soils") . . . 2430a (15) tota l ARDA acreage 82460a (1) to t a l suitable for dairying: upper li m i t of SDAIRY. (12) + (13) = (16) 66030 (16) to t a l suitable for market gardening: upper lim i t of SGRDEN. (12) + (13) = (17) . . . 7510 (17) tota l suitable for land based agriculture: upper limit of SLNDAG. (12) + (13) + (15) = (18) 68460 (18) total suitable for major park/recreational reserve: upper li m i t of SPARK. (8) - BYCLIR = (19) . . . . -BYCLIR + 11869 (19) to t a l area suitable for industrial use: upper limit of SESTDUS. . 2680 (20) ^ . L . Lee, Regional Farmland Study (Abbotsford, B.C.: Central Fraser Valley 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 plain. c The Lower Mainland Looks Ahead (New Westminster, B.C.: Lower Mainland Regional Planning Board, 1963), p. 26. d Equals area of unimproved farm land as reported by the LVstunion Bureau of Statistics, Agricultural Census of Canada (Ottawa: Queen's Printer, 1966) e Space for Jjndustry: Summary Report (Vancouver: Greater Vancouver Regional D i s t r i c t Planning Department, 1971), p. 8, 20. I l l T able B MINIMUM LAND REQUIREMENTS USED IN THE LANGLEY MODEL MNOFIC: minimum office services land requirements medical-dental office requirements: 2897 sq. ft./lOOO population (46% for buildings) business and other professional office requirements: 3392 sq. ft./lOOO population (49% for buildings) combined office services requirements: 2994 sq. ft./lOOO papulation (buildings only) or 1 acre/14925 pop. MNTRAN: minimum transport services land requirements through highway business requirements: 17302 sq. ft./lOOO population (10% for buildings) local highway business requirements: 2429 sq. ft./lOOO population (8% for buildings) 3 auto and a l l i e d sales requirements: 10313 sq. ft./lOOO population (10% for buildings) auto and a l l i e d service requirements: 9825 sq. ft./lOOO population (18% for buildings) 3 combined transport services requirements: 4256 sq. ft./lOOO population (buildings only) or 1 acre/10638 pop. MNUCNR: minimum urban neighbourhood r e t a i l land requirements: from a personal estimate that the average market of an urban or suburban neighbourhood r e t a i l outlet (approximately 400 sq. f t . ground space) i s approximately 400 persons or 115 households: 1 acre/43478 population MNUSHP: ntinimum urban shopping center land requirements: town planning standards: 12 sq.j_ft. of urban or suburban shopping center or 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 rural r e t a i l land requirements: from a personal estimate that r e t a i l area of 400 square f t . can serve a rural population of 200 persons or approximately 60 households: 1 acre/22222 population MNGRBT: miriimum green belt acreage requirements: set a approximately the ratio of green belt to people that exists i n the Lower Eraser Valley today: 1 acre/10 population Table B cont'd 112 develpment __ 1 acre per: h acre of tot a l conroercial land use, or 175 dwelling units, or 175 persons i n the labour force, or 1 acre of corrmercial recreational development. MNRAIL: irdnimum railroad and railyard land requirements: a r b i t r a r i l y set at 1 acre/1 acre i n a corrmercial or industrial land use. MNAIR: minimum airport land requirements: ^ from a report on land requirements for minor airports: h sq. mile/25000 population, or 1 acre/156 population. MNHOSP: minimum hospital land requirements: town planning standards: 4.5 beds/1000 population" and from personal estimate: 200 bed hospital requires a minimum of 4 acres of land. __ 1 acre/10000 population. MNSKUL: minimum school land requirements: n from a report on rrdnimum Langley school needs i n 1961: space needed for 1650 secondary pupils and 2544 elementary pupils, and from town planning standards: x minimum 4 acres per,250 pupils, and from 1961 Langley population:3 14585 people. __ 1 acre/200 population. MNGOVT: minimum other institutional land requirements: set a r b i t r a r i l y at: 1 acre/10000 population. HOUSAL: minimum residential land requirements to ensure that a l l the population i s housed: five average housing types were ar 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 single family housing): 1 unit/1 acre (including grounds) 5ACRE (hobby farm): 1 unit/5 acre AGHCME (farm home and hcmesite): 1 unit/1 acre. k a l l of these housing types have an average density of 3.5 people/unit. AGHCME: land requirements for farm homes and homesites: ar b i t r a r i l y provides 1 unit of AGHCME (1 acre) for each 80 acres of dairy farm, 100 acres of market garden, and 10 acres of 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).c This averages to a combined neighbourhood park requirement of 1 acre/395 population. 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:d 1 acre/250 population 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:3 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) : e 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 Table B cont'd 114 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 for Living (New Westminster: Lower Mainland Regional Planning Board, 1963), p. 16. d Land for 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 Airports for 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 hot-line radio show i n the summer of 1972. h School Needs for Langley (New Westminster: Lower Mainland Regional Planning Board, 1961), pp. 17, 28. i Report on School and Park Standards (Vancouver:Technical Planning Branch of the Planning Department of the City of Vancouver, 1957), p. 3. j Census of Canada (Ottawa: Queen's Printer, 1961). k Average number of persons per household for Metropolitan Vancouver fringe areas as reported i n the 1966 Census of Canada (Ottawa: Queen's Printer, 1966). Table C COEFFICIENTS FOR STEP DEMAND FUNCTIONS USED IN LANGLEY MODEL Land Use or 1st Step of Function , 2nd Step of Function , 3rd Step of Function Externality Activity (Code) Upper Limit 3 (acres) Contribution to Policy Goal (units/acre) Upper Limit 3 (acres) Contribution to Policy Goal (units/acre) < Upper Limit 3 (acres) Contribution to Policy Goal (units/acre) FEEDTC 100 $4400 to MXGRP 100 $2000 to MXGRP <D $500 to MXGRP OFFICT TOTPOP/12048 $340000 to MXGRP TOTPOP/12048 $100000 to MXGRP rain $50000 to MXGRP TRANST TOTPOP/8475 $340000 to MXGRP TOTPOP/8475 $100000 to MXGRP -P Ul $50000 to MXGRP UCNRT^  urban population/34483 $340000 to MXGRP urban population/34483 $100000 to MXGRP uncor $50000 to MXGRP USHOPTd urban population/2933 $340000 to MXGRP urban population/29 33 $100000 to MXGRP n are $50000 to MXGRP SCNRTe suburban population/34483 $340000 to MXGRP suburban population/34483 $100000 to MXGRP solum $50000 to MXGRP SSHOPT6 suburban population/2933 $340000 to MXGRP suburban population/2933 $100000 to MXGRP this ( $50000 to MXGRP PJURALTf rural population/17857 $340000 to MXGRP rural pppulation/17857 $100000 to MXGRP vities in $50000 to MXGRP UPRKT5 urban population/316 $1000 to MXGRP urban population/316 $750 to MXGRP vities in $500 to MXGRP SPRKT6 suburban population/316 $1000 to MXGRP suburban population/316 $750 to MXGRP acti $500 to MXGRP MPRKT TOTPOP/267 $200 to MXGRP TOTPOP/267 $175 to MXGRP ;tep $125 to MXGRP GOLFT TOTPOP/200 $133 to MXGRP TOTPOP/200 $100 to MXGRP all s $75 to MXGRP Table C cont'd CRDEVT TOTPOP/211 $20000 to MXGRP TOTPOP/211 $15000 to MXGRP HOSPT TOTPOP/8000 $300000 to MXGRP TOTPOP/8000 $150000 to MXGRP SKULT TOTPOP/160 $35000 to MXGRP TOTPOP/160 $15000 to MXGRP GOVTT TOTPOP/8000 $200000 to MXGRP TOTPOP/8000 $100000 to MXGRP FREERECT^ TGTPOP/250 10000 units to MXFREREC TOTPOP/250 5000 units to MXFREREC GRBELTTG TOTPOP/1 100 units to MXGRBELT TOTPOP/1 50 units to MXGRBELT GRSPACETG TOTPOP/250 2000 units to MXGRSPAC TOTPOP/250 1000 units to MXGRSPAC AGSELFTG TOTBOP/1 100 units to MXAGSELF TOTPOP/1 50 units to MXAGSELF T3 <D C •H (0 S-l -P W c: o o c 3 $7500 to MXGRP $50000 to MXGRP $5000 to MXGRP $50000 to MXGRP 10 units to MXFREREC 25 units to MXGRBELT 10 units to MXGRSPAC 25 units to MXAGSELF aThe f i r s t two steps of a l l of the land use activity demand functions (except FEEDT's) are both arbit-r a r i l y set at 125% of the corresponding minimum land requirements for that ac t i v i t y (see Table B). The third steps are a l l l e f t unconstrained (to avoid forcing land into slack). The sources of data for those functions i n this column that contribute to the MXGRP goal are given i n Table D. Data sources for a l l other functions are explained i n footnote (g) below. °The f i r s t two steps of FEEDT's demand function are set at 100 acres each and the thi r d step i s unconstrained. This function i s not linked to any Langley population because i t was f e l t that the market for local fed beef was essentially the population of the entire Lower Mainland. Since FEEDT produces 150 animals per acre per year, and since the tot a l market for beef animals in the Lower Mainland i s somewhat i n excess of 100,000 animals per year, the output of FEEDT would begin to affect market price at quite low levels of FEEDT. FEED1 and 2 (the two most profitable steps i n the FEEDT demand function) were therefore constrained to 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 activity functions i s entirely fabricated. The demand functions are based on the author's perception of the amount of u t i l i t y society derives from these four types of externalities. 1 1 8 Table D EMPLOYMENT DENSITY COEFFICIENTS USED IN LANGLEY MODEL Activity Enrployment Density (workers/acre) Derivation of Coefficient DAIRY GARDEN FEEDT TOTCCM TOTIND UPRKT SPRKT MPRKT GOLFT CRDEVT APART HOSPT SKULT GOVTT UNEMP 0.025 0.050 0.400 40.000 19.200 0.250 0.250 0.050 0.030 5.000 1.000 50.000 5.000 40.000 1.000 acre acre estimate: 2 full-time workers per 80 acre dairy farm, estimate: 5 full-time workers per 100 acre market garden, estimate: 4 full-time workers per 10 acre feedlot. average worker density on commercial land uses for several U.S.A. c i t i e s . 3 average worker density on industrial land uses i n the Lower Mainland.b estimate: 1 full-time caretaker per 4 urban park, estimate: 1 full-time caretaker per 4 suburban park, estimate: 5 full-time caretakers per 100 acre major park/recreational reserve, estimate: 5 full-time caretakers per 150 acre golf course, estimate: 5 workers per acre of commercial recreational development, estimate: 1 full-time caretaker per 35 unit apartment (1 acre). estimate: 200 full-time workers per 200 bed hospital (4 acres). estimate: 20 full-time employees per 250 pupil elementary school (4 acres) and 40 f u l l -time employees per 500 pupil secondary school (8 acres). estimate: assumed to be the same as TCTCOM. one unemployed person "occupies" one person i n the labour force. Southeastern Wisconsin Regional Planning Cctrcnission Land Use- Transportation Study (Milwaukee: Southeastern Wisconsin Regional Planning Ccmrtission, 1965) Planning Report No. 7, Vol. l f pp. 32,34, 84. 3D Space for Industry: Technical Report (Vancouver: Greater Vancouver Regional D i s t r i c t , 1971), p. 74. Table E 119 ACTIVITY CONTRIBUTIONS/ACRE TO GROSS PRODUCTION Activity Contributicn Derivation of Coefficient Code to MXGRP (coefficients for which no explanation appears ($/acre) were either estimated or set arbitrarily.) DAIRY 200 (net farm income + hired labour expense)/total operated farm acres for Eraser Valley dairy farms i n 1970.a GARDEN 450 2/3 x (gross value of production)/total acreage of vegetable production i n the Lower Mainland i n 1971.b FEEDl FEED2 4400 2000 (net farm income + hired labour expense) for a 1500 animal/year commercial feedlot. Total acreage assumed to be 10 acres. c FEED3 500 a l l TOTCCM ac t i v i t i e s : 1st step a l l TOTCCM 3 4 0 0 0 0 (non-service r e t a i l trade payroll + 1% of non-service r e t a i l sales + service trade payroll + 10000/year per working proprietor)^/total acres of commercial floor space e for a l l Vancouver City and Metropolitan area commercial a c t i v i t i e s . a c t i v i t i e s : 100000 2nd step a l l TOTCCM ac t i v i t i e s : 50000 3rd step TOTIND UPRK1 and SPRK1 60000 1000 value added by a l l industrial land uses/total industrial acreage for Metropolitan Vancouver.e estimate: 1 worker/4 acres @ $4000/year. UPRK2 and 750 SPRK2 UPRK3 and 500 SPRK3 MPRKl MPRK2 200 175 estimate: 5 workers/100 acres @ $4000/year. MPRK3 125 GOLF1 GOLF2 G0LF3 133 100 75 estimate: 5 workers/150 acre golf course @ $4000/year. CRDEVl CRDEV2 20000 15000 estimate: 5 workers/acre @ $4000/year. \ CRDEV3 7500 APART 2000 estimate: 1 caretaker per apartment (1 acre) @ $2000/year. 5ACRE 50 estimate: $250 of value added farm product per 5 acre hobby farm per year.' 120 Table E cont'd HOSPl HOSP2 HOSP3 300000 150000 50000 estimate: 200 workers per 200 bed hospital (4 acres) @ $6000/year. SKULl SKUL2 SKUL3 35000 15000 5000 estimate: 5 workers/acre @ $7000/year. GOVT1 GOVT2 GOVT3 200000 100000 50000 estimate: 40 workers/acre @ $5000/year. BUYFLUDT -82 (annual maintenance costs + 10% x i n i t i a l capital costs)/total acres protected: flood protecting entire Lower Fraser Valley flood plain to 1948 flood levels.9 This annual charge i s subtracted from MXGRP because i t represents an intermediate good in the production of output on flood protected land. BYCLIR - 7 5 estimate: costs of servicing capital costs of clearing land. This annual charge i s subtracted from MXGRP because i t represents an intermediate good i n the production of output on cleared land. a1970 Fraser Valley Dairy Farms by Municipality (Victoria: B.C. Department of Agriculture, 1971). fo 1971 Production of Vegetable Crops Together with an Estimate  of Farm Value (Victoria: B.C. Department of Agriculture, 1972). °E. T. Osborne. The Fed-Beef Industry i n the Fraser Valley Region  of B.C. (unpublished M.B.A. Thesis i n the Department of Commerce at the University of Brit i s h Columbia, 1968), pp. 153-154. ^Census of Canada (Ottawa: Queen's Printer, 1966). CCTrrrtercial Floor Space (Vancouver: Greater Vancouver Regional D i s t r i c t , 1970). ^Space for 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 in the Fraser River Basin (Victoria: Colonist Printer's Ltd., 1958), pp. 49, 61. Table F 1 2 1 ACTIVITY CONTRIBUTIONS/ACRE TO POLLUTION INDICES Activity Contribution/Acre t o : a Code MNAIRPOL MNWATER MNREFUSE MNNOISE MNSIGHT (units) (units) (units) (units) (units) DAIRY 1 10 0 2 0 GARDEN 0 5 0 1 0 FEEDT 10 500 0 10 50 TOTCCM 2 50 200 10 20 TOTIND 4000 4000 1000 750 250 UPRKT 0 1 20 50 0 SPRKT 0 1 20 50 0 JMPRKT 0 1 20 0 0 GOLFT 0 1 20 0 0 CRDEVT 1 10 20 5 0 URBRD 4000 5 5 750 250 SUBRD 400 1 1 100 50 RURRD 80 0 0 50 10 PARK 800 1 1 500 250 RAILRD 200 2 0 750 50 APART 10 1500 1000 5 100 5/ACRE 4 200 125 5 20 1ACRE 1 40 40 5 0 5ACRE 0 18 8 2 i,0 AGHCME 1 40 40 5 0 HOSPT 20 1000 100 0 0 SKULT 4 500 50 100 0 GOVTT 2 50 50 5 0 LDSLAK 0 0 20 0 0 a A l l of these indices wre entirely fabricated by the author and lack any s t a t i s t i c a l backing. An attempt was made to define the indices i n terms of $1 pollution d i s u t i l i t y units. 

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