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An application of linear programming to log allocation in the forest industry of British Columbia Sydneysmith, Sam 1964

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AN APPLICATION OF LINEAR PROGRAMMING TO LOG ALLOCATION IN THE FOREST INDUSTRY OF BRITISH COLUMBIA by Sam S y d n e y s m i t h B . A . S c , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1956 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Ma s t e r o f A r t s i n t he Department o f Economics We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1964 In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of • B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and study* I further agree that per mission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that, copying or publi cation of this thesis for financial gain s h a l l not be allowed without my written permission^. S. Sydneysmith. Department of ranpftMIC-fi — The University of B r i t i s h Columbia, Vancouver 8, Canada D a * e April 24th. 1964. i i ABSTRACT This t h e s i s presents an a p p l i c a t i o n of l i n e a r program ming to the question of e f f i c i e n t l o g a l l o c a t i o n i n the f o r  est i n d u s t r y of B r i t i s h Columbia. Current procedures f o r a l l o c a t i n g logs among a l t e r n a t i v e u t i l i z a t i o n processes are discussed and i t i s suggested t h a t a more e f f i c i e n t a l l o c a t i o n might be obtained through a systematic approach to the pro blem. The economic n e c e s s i t y of improving net returns t o the l o g supply i s emphasized. A l i n e a r programme l o g - a l l o c a t i o n model i s presented, based on an i n t e g r a t e d - i n d u s t r y i n the c o a s t a l r e g i o n of B r i t i s h Columbia.. The model encompasses three main cate g o r i e s of log-use, namely s a w m i l l i n g , plywood production and pulp production, and demonstrates how a given supply of logs may be o p t i m a l l y d i s t r i b u t e d among these s t r u c t u r a l l y d i f  f e r e n t log-conversion processes. Emphasis throughout t h i s study i s on the s t r u c t u r e of the l i n e a r programme model, although considerable e f f o r t was d i r e c t e d to obt a i n i n g r e a l i s t i c data. S o l u t i o n s of the model, obtained through the s e r v i c e s of the Computing Cen t r e at the U n i v e r s i t y of B r i t i s h Columbia, are dis c u s s e d , and a s u p e r f i c i a l comparison i s made wit h a c t u a l l o g a l l o  c a t i o n i n the i n d u s t r y . M o d i f i c a t i o n s of the model to s u i t the l o g - a l l o c a t i o n problem faced by an i n d i v i d u a l f i r m i n the short-run are discussed and normal c o m p a r a t i v e - s t a t i c s a p p l i c a t i o n s are considered. i i i I t i s p o i n t e d out t h a t many of the s i m p l i f y i n g assump t i o n s i n the model may be r e l a x e d . However, the main l i m i t a t i o n t o i t s p r a c t i c a l a p p l i c a t i o n by i n d u s t r y and government l i e s i n the q u a l i t y and type o f d a t a a v a i l a b l e . I n t h i s r e s p e c t i t i s suggested t h a t the l i n e a r programme model of t h i s t h e s i s p r o v i d e s a v a l u a b l e guide t o the p r o  d u c t i o n d a t a r e q u i r e d t o improve economic e f f i c i e n c y i n the f o r e s t i n d u s t r y . i v ACKNOWLEDGEMENTS The encouragement and advice of Dr. P.H.Pearse, as d i r e c t o r of t h i s study, and Dr. G.Rosenbluth, as methodological c o n s u l t a n t , i s g r a t e f u l l y acknowledged. In a d d i t i o n , the author wishes to thank a l l of those persons who supplied data and t e c h n i c a l knowledge on the various phases of the f o r e s t i n d u s t r y i n B r i t i s h Columbia. V TABLE OF CONTENTS Page L i s t of Tables . v i i L i s t of Figures v i i i CHAPTER I : INTRODUCTION 1 I . L o g - A l l o c a t i o n i n the Forest Industry 2 I I . Scope of the Study 6 CHAPTER I I : THE CASE FOR IMPROVED LOG ALLOCATION . . . 12 I. The Nature of the Forest Industry . . 12 I I . Problems of the Industry 16 I I I . Industry Trends 21 IV. Conclusions 26 CHAPTER I I I : EXPOSITION OF THE LOG-ALLOCATION PROBLEM . 2 8 CHAPTER IV: CONSTRUCTION OF A LOG-ALLOCATION LINEAR PROGRAMME MODEL 3^ I . Basic O u t l i n e of the Model 34- I I . S t r u c t u r e of the Model 36 I I I . Assumptions of the Model 4-4- CHAPTER V: DATA AND SOLUTION OF THE MODEL 4-9 r. A c t i v i t y C o e f f i c i e n t s 4-9 I I . U n i t - L e v e l Returns of the A c t i v i t i e s 60 111. R e s t r a i n t s 66 v i Page IV. Method of S o l u t i o n 6 7 V. S o l u t i o n s of the Lin e a r Programme . . 72 Appendix to Chapter F i v e 7 4 CHAPTER V I : RESULTS AND DISCUSSION 38 I . R e s u l t s 88 I I . A p p l i c a t i o n s of the Model 95 Appendix to Chapter S i x 1 0 4 CHAPTER V I I : SUMMARY AND CONCLUSION 109 v i i LIST OF TABLES gage The Value of Forest Production i n B r i t i s h Columbia by Indust r ia l Categories i n Selected Years 14 The Log Supply i n the Coastal Region of B r i t i s h Columbia, 1962 . 74 Log Sawing A c t i v i t y Technological Coe f f i c i en t s . . . 75 Log Peel ing A c t i v i t y Technologica l Coe f f i c i en t s . . . 76 Plywood Production A c t i v i t y Coe f f i c i en t s 77 Pulp Production A c t i v i t y Coe f f i c i en t s ',78 Plywood Manufacturing Costs as Percent of Sales Pr ice 79 Plywood Production A c t i v i t y Unit-Level Returns . . . 80 Pulp Production A c t i v i t y Unit-Level Returns 81 Log Se l l i ng A c t i v i t y Unit-Level Returns . 82 Lumber Se l l i ng A c t i v i t y Unit-Level Ret irns . . . . . 8 3 Veneer S e l l i ng Ac t i v i t y Unit-Level Returns 8 4 Pulp Chip S e l l i ng A c t i v i t y Unit-Level Returns . . . . 85 Modified Coe f f i c i en t s of A c t i v i t y No. 9 ; for Case II 86 Volume of Log Supply by Species and Grade . . . . . . 87 Tota l Net Returns, i n Lumber, Plywood, and Pulp Production 104 Optimum Log A l l oca t i on 105 Lumber Sales by Species and Grade . . . . . . . . . .106 Plywood and Veneer Sales . , 107 Pulp Sales 108 v i i i LIST OF FIGURES F i g u r e I Schematic P r e s e n t a t i o n o f Log A l l o c a t i o n L i n e a r Programme Model. ( a t t a c h e d i n s i d e r e a r c o v e r ) CHAPTER I INTRODUCTION The forest resource of B r i t i s h Columbia i s pre dominately owned by the Crown, but exploited e n t i r e l y by private enterprise. A question of ' e f f i c i e n c y of resource a l l o c a t i o n ' arises therefore at two l e v e l s : at the forest management l e v e l which i s i n the hands of the P r o v i n c i a l Government, and at the entrepreneurial or private l e v e l of firms consuming timber as a raw material. The conditions for optimal resource a l l o c a t i o n at each l e v e l d i f f e r , and give r i s e to d i f f e r e n t maximizing c r i t e r i a . As owner of most of the forest resource the Pro v i n c i a l Government controls the a l l o c a t i o n of timber to the forest industry.'1' To e f f i c i e n t l y discharge t h i s function i t must be guided by the s o c i a l aspects of resource ex p l o i t a t i o n as well as by the demands of the private forest industry, for the ultimate objective of resource manage ment i s to maximize the value of the resource to society. The e f f i c i e n t owner i s concerned therefore with estimat ing future benefits and costs associated with the forest resource, and with establishing time rates of preference to be applied i n discounting these future benefits to present 1 The term 'forest industry' as used i n t h i s thesis refers c o l l e c t i v e l y to the private firms engaged i n extraction and processing of logs. Government administration of the forests i s therefore not considered a part of the forest industry. 2 values. On the st r e n g t h of these evaluations the owner i s able to develop s u i t a b l e forest-management p o l i c i e s to guide p r i v a t e i n d u s t r y i n i t s e x p l o i t a t i o n of the resource. At the i n d u s t r y l e v e l e f f i c i e n t resource a l l o c a t i o n r e s u l t s from the p r o f i t - m a x i m i z i n g behaviour of each p r i  vate f i r m . The supply of timber to each f i r m i s one of s e v e r a l input f a c t o r s which i t seeks to a l l o c a t e i n a man ner that y i e l d s i t a maximum o v e r - a l l net r e t u r n . E f f i c i e n t a l l o c a t i o n of timber i n the f o r e s t i n d u s t r y as a whole th e r e f o r e depends upon the aggregate p r o f i t - m a x i m i z i n g be haviour of a l l firms i n the i n d u s t r y , each one o p t i m i z i n g i t s own production f u n c t i o n . The problem i s removed from any welfare i m p l i c a t i o n s other than those imposed by the government. This t h e s i s considers the e f f i c i e n c y of l o g a l l o  c a t i o n at the second of these two l e v e l s , namely, at the l e v e l of i n d u s t r i a l use. The main o b j e c t i v e i s t o present a systematic approach to the problem of a t t a i n i n g an o p t i  mal a l l o c a t i o n of logs among various u t i l i z a t i o n processes, such t h a t the r e t u r n to a given supply of logs i s maximized. I t w i l l be u s e f u l t h e r e f o r e to elaborate upon current l o g - a l l o c a t i o n procedures i n the i n d u s t r y , before c o n s i d e r i n g the l o g - a l l o c a t i o n problem faced by i n d i v i d u a l f i r m s . 1. LOG ALLOCATION IN THE FOREST INDUSTRY Two forms of l o g a l l o c a t i o n i n the f o r e s t i n d u s t r y may be d i s t i n g u i s h e d . F i r s t , a l l o c a t i o n of logs among competitive f i r m s . Second, a l l o c a t i o n of logs w i t h i n each 3 f i r m , between i t s v a r i o u s a l t e r n a t i v e u t i l i z a t i o n p r o c e s s e s . Log A l l o c a t i o n Among F i r m s . T h i s p r o blem i s s o l v e d ( t h e o r e t i c a l l y ) t h r o u g h the open market f o r l o g s , where p r i c e s a re e s t a b l i s h e d by the u s u a l f o r c e s o f s u p p l y and demand and the p r i c e system r a  t i o n s t h e a v a i l a b l e s u p p l y t o c o m p e t i t i v e f i r m s i n the market. Not a l l o f the t i m b e r i n the c o a s t a l r e g i o n goes t h r o u g h t h e open-log market, but p r i c e s e s t a b l i s h e d t h e r e a r e used as a b a s i s f o r c o m p l e t i n g most " c l o s e d " market t r a n s a c t i o n s . I n t h e o r y t h e p r i c e system s h o u l d r e s u l t i n maximum economic e f f i c i e n c y i n l o g a l l o c a t i o n f o r the econ omy as a whole, p r o v i d i n g t h a t p e r f e c t c o m p e t i t i o n e x i s t s t h r o u g h o u t (many b u y e r s , many s e l l e r s , homogeneous p r o d u c t , f r e e e x i t and e n t r y o f f i r m s , e t c . ) . I n p r a c t i c e i t i s q u e s t i o n a b l e whether the market i s 2 a c t u a l l y "open and c o m p e t i t i v e " . F o r example, i n the case o f i n t e g r a t e d f i r m s w h i c h have t h e i r own l o g g i n g o p e r  a t i o n s i t might be argued t h a t " i n t e r n a l " l o g t r a n s a c t i o n s o c c u r r i n g w i t h i n t h e f i r m a r e not a p a r t o f the open mar k e t . I n a d d i t i o n t h e r e i s an element o f d i r e c t p r i c e d i s  t o r t i o n a r i s i n g f r o m an i n t e r p l a y between the op e n - l o g market and the market f o r c h i p s and small-wood r e s i d u e s . 2 "The c o n d i t i o n and o p e r a t i o n o f the o p e n - l o g market and c h i p and small-wood m a r k e t i n g i n the "Vancouver F o r e s t D i s t r i c t " , R e p o r t t o the S e l e c t S t a n d i n g Committee o f  F o r e s t r y o f the L e g i s l a t i v e Assembly o f B r i t i s h C o l u m b i a . B. C. F o r e s t S e r v i c e , V i c t o r i a , ( J a n . 1962) - s u p p l i e d i n c o n f i d e n c e t o the a u t h o r . 1+ A t t h e p r e s e n t time the i n d u s t r y a c c e p t s t h a t the p r i c e p a i d under c o n t r a c t f o r wood c h i p s can he r e l a t e d d i r e c t l y t o t h e p r i c e e s t a b l i s h e d on the Vancouver l o g market f o r 3 number t h r e e grade hemlock, and v i c e v e r s a . Any f i r m i n a p o s i t i o n t o c o n t r o l p r i c e s i n e i t h e r o f t h e s e markets t h e r e f o r e c o u l d q u i t e f e a s i b l y m a n i p u l a t e p r i c e s i n the o t h e r market. I n the s h o r t r u n , w i t h g i v e n u t i l i z a t i o n c a p a c i t i e s and g i v e n l o g s u p p l i e s f o r each f i r m , the a ggregate a l l o c a t i o n o f l o g s t o d i f f e r e n t u t i l i z a t i o n p r o c e s s e s may not c o i n c i d e w i t h the o p t i m a l a l l o c a t i o n p l a n f o r the i n d u s t r y as a whole. Such m i s a l l o c a t i o n r e f l e c t s i m p e r f e c t i o n s i n the l o g - a l l o  c a t i o n system. k Log A l l o c a t i o n W i t h i n I n t e g r a t e d F i r m s . The i n t e g r a t e d f o r e s t p r o d u c t s f i r m i s u s u a l l y con f r o n t e d w i t h a g i v e n o r known l o g s u p p l y w h i c h i t seeks t o a l l o c a t e among i t s v a r i o u s c o n v e r s i o n p r o c e s s e s i n a manner t h a t maximizes t o t a l n et r e t u r n t o t h e e n t e r p r i s e . P r o f i t - m a x i m i z i n g b e h a v i o u r of the i n t e g r a t e d f i r m t h e r e f o r e i n  v o l v e s s e e k i n g an optimum a l l o c a t i o n o f i t s l o g s u p p l y . The 3 "The c o n d i t i o n and o p e r a t i o n o f the o p e n - l o g market and c h i p and small-wood m a r k e t i n g i n t h e Vancouver F o r e s t D i s t r i c t " , R e p o r t t o t h e S e l e c t S t a n d i n g Committee o f F o r e s  t r y o f the L e g i s l a t i v e Assembly o f B r i t i s h C olumbia. B. C. F o r e s t S e r v i c e , V i c t o r i a , ( J a n . 1 9 6 2 ) - s u p p l i e d i n c o n f i d e n c e t o the a u t h o r . k I n t h i s t h e s i s a l l r e f e r e n c e s t o the f i r m w i l l assume an i n t e g r a t e d f o r e s t - p r o d u c t s e n t e r p r i s e . Most o f the d i s  c u s s i o n however i s e q u a l l y a p p l i c a b l e t o s m a l l e r f i r m s s p e c i a l i z i n g i n a s i n g l e u t i l i z a t i o n p r o c e s s , such as saw- m i l l i n g . 5 g i v e n l o g s u p p l y i s u s u a l l y d e t e r m i n e d by the f i r m ' s own h o l d i n g s o f t i m b e r - c u t t i n g r i g h t s , b u t i t may be augmented by t r a d i n g o r p u r c h a s i n g l o g s on t h e open market. C u r r e n t P r o c e d u r e s . The a l l o c a t i o n o f l o g s among v a r i o u s c o n v e r s i o n p r o  c e s s e s i n an i n t e g r a t e d f o r e s t p r o d u c t s e n t e r p r i s e i s p r e s e n t l y a haphazard p r o c e d u r e conducted m o s t l y by " r u l e s o f thumb". Each i n t e g r a t e d f i r m g e n e r a l l y has a c e n t r a l  i z e d c o - o r d i n a t o r r e s p o n s i b l e f o r c o n t r o l l i n g l o g a l l o c a t i o n on a b a s i s o f e x p e r i e n c e , s p e c i a l knowledge, and a t a l e n t f o r a s s e s s i n g the l o g demand s u b m i t t e d by each m i l l . I n t h i s r e s p e c t the c e n t r a l l o g s u p p l y c o n t r o l o f f i c e p e r f orms a c o - o r d i n a t i o n r a t h e r t h a n an a l l o c a t i o n f u n c t i o n , and i t i s d i f f i c u l t t o c o n c e i v e o f any h i g h degree o f economic e f  f i c i e n c y r e s u l t i n g from t h e forme r approach. I t i s t r u e t h a t market s i t u a t i o n s may change r a p i d  l y and u n p r e d i c t a b l y , r e q u i r i n g d e c i s i o n s based on e x p e r i e n c e and i n d i v i d u a l 'judgement. N e v e r t h e l e s s , the l a c k o f a s y s t e m a t i c and r i g o r o u s a n a l y s i s o f normal l o g r e q u i r e m e n t s , and t h e absence of a t e c h n i q u e f o r q u i c k l y d e t e r m i n i n g the most e f f i c i e n t ( p r o f i t - m a x i m i z i n g ) a l l o c a t i o n o f l o g s , a l  most c e r t a i n l y r e s u l t s i n a h i g h , and u n n e c e s s a r y , degree o f i n e f f i c i e n c y . Some i m p o r t a n t o b s t a c l e s t o d e v e l o p i n g an e f f e c t i v e l o g - a l l o c a t i o n p r o c e d u r e a r e the l a c k o f s u f f i c i e n t spe c i f i c d a t a on l o g y i e l d s and p r o d u c t i o n c o s t s , and the c o m p l e x i t y o f the c a l c u l a t i o n s i n v o l v e d even when the 6 required data are available. It may be argued that the industry has hitherto lacked an incentive to develop efficient log-allocation procedures, for profits have been high and exploitation characterized by an "extensive" use of a relatively cheap and ubiquitous resource. It w i l l be pointed out below though that these conditions are changing rapidly and that trends in the value of the resource are already such that efficiency in al l o  cating timber among alternate uses demands a more system atic and rigorous approach. The development of such an approach comprises the subject of this study. II. SCOPE OF THE STUDY This study examines the problem of efficient a l l o  cation of logs in the forest industry. Specifically, the purpose is to develop a solution technique for arriving at the optimal allocation plan for a given supply of logs. The form of the problem therefore i s similar to that faced by an integrated forest products enterprise in allocating i t s logs among alternative u t i l i z a t i o n plants. The log-allocation problem lends i t s e l f to mathemat i c a l solution by linear programming techniques. Thus we shall discuss the elements of "choice", "optimization", and "restraint", and their mathematical analogues in a linear programme model of the allocation problem, before proceeding to a solution and discussion of practical applications. The emphasis throughout is on the structure of the model. Although considerable effort has been devoted to 7 s e e k i n g o u t t h e m o s t a c c u r a t e d a t a a v a i l a b l e f o r t h e p u r  p o s e s o f t h e a n a l y s i s , n o c l a i m i s made f o r a h i g h d e g r e e o f s t a t i s t i c a l a c c u r a c y . F r e q u e n t l y t h e d a t a a v a i l a b l e h a s b e e n i n s u f f i c i e n t o r i n a f o r m p o o r l y s u i t e d f o r t h e a n a l  y s i s . I n t h e s e i n s t a n c e s ( a s n o t e d b e l o w ) , e x t r a p o l a t i o n s a n d a s s u m p t i o n s h a v e b e e n m a d e . N e v e r t h e l e s s , i t i s h o p e d t h a t t h e b r o a d l i n e s o f t h e r e s u l t s a c h i e v e d w i l l p r o v i d e a u s e f u l i n s i g h t t o t h e e f f i c i e n c y o f t h e l o g - a l l o c a t i o n p r o c e s s . T h u s , w h i l e t h e p r i m a r y o b j e c t i v e i s t o p r e s e n t a r i g o r o u s m e t h o d o l o g i c a l a p p r o a c h t o t h e p r o b l e m , m o r e a c c u r  a t e e m p i r i c a l d a t a a r e r e q u i r e d b e f o r e m u c h r e l i a n c e c a n be p l a c e d o n t h e n u m e r i c a l r e s u l t s . T h e s t u d y p o i n t s o u t t h e n a t u r e o f t h e t e c h n i c a l a n d e c o n o m i c d a t a t h a t a r e r e q u i r e d f o r t h e p r a c t i c a l a p p l i c a t i o n o f t h e s o l u t i o n d e v e l o p e d . F r a m e w o r k o f A n a l y s i s . " L i n e a r p r o g r a m m i n g i s c o n c e r n e d w i t h t h e p r o b l e m o f p l a n n i n g a c o m p l e x o f i n t e r d e p e n d e n t a c t i v i t i e s i n t h e b e s t p o s s i b l e ( o p t i m a l ) f a s h i o n . " T h e v a l i d i t y o f r e s u l t s h o w  e v e r d e p e n d s e n t i r e l y u p o n t h e q u a l i t y o f d a t a s u p p l i e d t o t h e p r o b l e m . T h r e e d i f f e r e n t a p p r o a c h e s w e r e c o n s i d e r e d t o t h e t a s k o f a s s e m b l i n g d a t a f o r t h e p r o p o s e d m o d e l ; t h e s e w e r e 5 A . C h a r n e s , W. W. C o o p e r a n d A . H e n d e r s o n , A n I n t r o  d u c t i o n t o L i n e a r P r o g r a m m i n g , W i l e y & S o n s I n c . , N . Y . , c h . 1. 8 to base the model on either, (i) imaginary data, ( i i ) case-study data, ( i i i ) industry-average data. The f i r s t alternative was discarded on grounds that i t would convey l i t t l e of the practical significance of a linear programme approach to log allocation, and would be of limited interest to those concerned with industry pro blems. Nor would i t specify the shortcomings of existing data. It would be l i t t l e more than an exercise in linear programming. The second alternative was discarded for two reasons. F i r s t l y , i t was considered unlikely that any firm would be prepared to supply sufficiently detailed data, much of which would probably be regarded as confidential. Secondly, a case study would inevitably reflect special characteristics of the case-study firm. The third alternative was selected since i t would avoid most of the above objections and would be of practi cal interest to both government and industry. Although the purpose of this study is mainly to demonstrate the appli cation of linear programming to log allocation i t was con sidered important to keep the demonstration as r e a l i s t i c and as general as possible. It w i l l be shown later how both government and industry might modify the "industry-wide" model to analyze their own special problems. 9 I n c o n j u n c t i o n w i t h the d e c i s i o n t o use i n d u s t r y - average d a t a i t was d e c i d e d t o base the model on the concept o f an " i n t e g r a t e d - i n d u s t r y " . The g e n e r a l approach was t o t r e a t a l l the t i m b e r - u s i n g i n d u s t r i e s i n a g i v e n l o g - m a r k e t a r e a , drawing on a g i v e n s u p p l y o f l o g s , as a s i n g l e com p l e t e l y - i n t e g r a t e d e n t i t y w i t h i n w h i c h t i m b e r c o u l d be moved about w i t h o u t c o s t t o i t s p l a c e o f h i g h e s t v a l u e - i n - u s e . Having d e c i d e d t o base the l i n e a r programme model on an " i n t e g r a t e d - i n d u s t r y " u s i n g i n d u s t r y - a v e r a g e d a t a i t r e  mained t o s e l e c t an a r e a i n w h i c h t h e s e p r o p o s a l s c o u l d be e f f e c t i v e l y implemented. The a r e a chosen was t h a t p a r t o f the B r i t i s h Columbia f o r e s t i n d u s t r y w h i c h draws i t s s u p p l y o f l o g s f rom B r i t i s h C olumbia F o r e s t S e r v i c e I n v e n t o r y Zone 1 and p a r t of I n v e n  t o r y Zone 2. I n v e n t o r y Zone 1 c o v e r s the l o w e r c o a s t a l main l a n d and Vancouver I s l a n d , and the r e l e v a n t p a r t o f I n v e n t o r y Zone 2 comprises the c o a s t a l s e c t o r o f the P r i n c e R u p e r t F o r e s t D i s t r i c t . T h i s a r e a w i l l h e n c e f o r t h be r e f e r r e d t o c o l l e c t i v e l y as the " c o a s t a l r e g i o n . " G u t h r i e and Armstrong have u n d e r l i n e d t h e g e o g r a p h i  c a l " i s o l a t i o n " o f the c o a s t a l r e g i o n , bounded on the e a s t by the Coast Range, on t h e west by the P a c i f i c Ocean and t o 7 t h e n o r t h and s o u t h by f r o n t i e r s w i t h t h e U n i t e d S t a t e s . 6 C o n t i n u o u s F o r e s t I n v e n t o r y o f B r i t i s h C o l u m b i a . B.C. Dept. o f Lands & F o r e s t s , V i c t o r i a , B.C., (1958), T a b l e s A - I . 7 J . A. G u t h r i e , G. R. A r m s t r o n g , Western F o r e s t I n d u s t r y - An Economic O u t l o o k , The John Hopkins P r e s s , B a l t i m o r e , (1961), c h . l . 1 0 Rankin has pointed out the low cost of water transportation o f o r forest products within the region. We can summarize the p r i n c i p a l reasons for s e l e c t  ing the coastal region as our study area therefore, as f o l  lows: (i) An isolated economic u n i t . The geographic " i s o l a t i o n " and high mobility of logs within the coastal region make i t free from any ex ternal complications such as s i g n i f i c a n t log movements into and out of the region. The major portion of conversion f a c i l i t i e s i n the region i s concentrated i n the Gulf of Georgia area, into which most of the coastal log supply flows at low cost. The assump tions of a completely "integrated-industry" are thus not u n r e a l i s t i c , ( i i ) A consistent log supply. The commercial tim ber of the coastal region i s r e l a t i v e l y uni form with respect to species, size and grades. There are d e f i n i t e b i o l o g i c a l trends within the region, such as a higher percentage of spruce i n the northern sector, but these are sub s t a n t i a l l y o f f s e t by cheap transportation 8 A. G. Rankin, "Cost-Price Relationships i n the Forest Industry", The Forestry Chronicle, v. 3 9 , (Mar. 1 9 6 3 ) , PP. 6 9 - 7 9 . 11 and by the a c t i v e open market f o r l o g s i n the G u l f o f G e o r g i a , ( i i i ) A s i n g l e a d m i n i s t r a t i v e a r e a . Management o f the t i m b e r r e s o u r c e i n t h i s r e g i o n i s under the s o l e j u r i s d i c t i o n o f the B r i t i s h Columbia F o r e s t S e r v i c e . T h i s has the d e s i r a b l e r e  s u l t ( f o r our purposes) o f u n i f o r m and c o n s i s t e n t d a t a r e p o r t i n g . I n a d d i t i o n t h i s s i n g l e a u t h o r i t y r e p o r t s t o t h e government o f B r i t i s h Columbia o n l y , so t h a t p o l i t i c a l i n  f l u e n c e s on management o f the f o r e s t s a r e u n i f o r m and c o n s i s t e n t , ( i v ) A s u b s t a n t i a l p o r t i o n o f the i n d u s t r y . L a t e s t o f f i c i a l f i g u r e s f o r B r i t i s h Columbia show t h a t i n 1962 a p p r o x i m a t e l y 5 8 p e r c e n t o f t h e t o t a l l o g - s c a l e f o r B r i t i s h Columbia was r e  corded i n the c o a s t a l r e g i o n . 9 Thus a l t h o u g h the c o a s t a l r e g i o n c o m p r i s e s a r e l a t i v e l y s m a l l p o r t i o n o f the t o t a l a r e a o f B r i t i s h C o l u m b i a , i t a c c o u n t s f o r a v e r y s i g n i f i c a n t p o r t i o n o f the c u r r e n t l o g - s c a l e . T h i s p e r m i t s the model t o be employed w i t h more c o n f i d e n c e i n d e m o n s t r a t i n g the i m p l i c a t i o n s o f a change i n f o r e s t management p o l i c y on the f o r e s t i n d u s t r y ' s p r o f i t s and consumption p a t t e r n . 9 Annual R e p o r t , B. C. F o r e s t S e r v i c e , V i c t o r i a , B.C., (19627: 1 2 CHAPTER I I THE CASE FOR IMPROVED LOG ALLOCATION The main r e a s o n s f o r i n e f f i c i e n c y i n l o g a l l o c a t i o n have a l r e a d y been r e f e r r e d t o ; i n t h i s c h a p t e r the case f o r improved l o g a l l o c a t i o n i n the i n d u s t r y w i l l be examined. I t w i l l be shown t h a t improvement i n l o g - a l l o c a t i o n o f f e r s one o f t h e most p r o m i s i n g avenues toward g r e a t e r economic e f f i c i e n c y i n t h e i n d u s t r y as a whole. We s h a l l b e g i n by b r i e f l y r e v i e w i n g t h e f o r e s t i n  d u s t r y i n the c o a s t a l r e g i o n . The t r e n d s i n i n d u s t r i a l be h a v i o u r aimed a t overcoming some of the i n c r e a s i n g d i f f i  c u l t i e s f a c e d by t h e i n d u s t r y w i l l be i n d i c a t e d , and the most l i k e l y f u t u r e c o u r s e s o f t e c h n o l o g i c a l development and r e s e a r c h w i l l be o u t l i n e d . I . THE NATURE OF THE FOREST INDUSTRY S i z e . The f o r e s t i n d u s t r y i n the c o a s t a l r e g i o n was founded on saw-timber p r o d u c t i o n , w h i c h enj o y e d a u n i q u e advantage d e r i v e d from t h e p a r t i c u l a r l y l a r g e h i g h - q u a l i t y t r e e s t h a t grew i n abundance a l o n g the c o a s t . T e c h n o l o g i c a l p r o g r e s s and expanding markets i n Canada and abroad l e d t o i n c r e a s  i n g r a t e s o f e x p l o i t a t i o n o f the f o r e s t s , and t o a more i n t e n  s i v e use o f the l o g s h a r v e s t e d . A t p r e s e n t most o f the f o r  e s t r e s o u r c e i n t h e c o a s t a l r e g i o n , and i n d e e d t h r o u g h o u t B r i t i s h C o l u m b i a , i s under some f o r m o f f o r e s t management. 1 3 The i n d u s t r i e s w h i c h consume l o g s d i r e c t l y as a raw m a t e r i a l a r e u s u a l l y d i v i d e d i n t o t h r e e c a t e g o r i e s ; saw m i l l s , plywood m i l l s , and p u l p m i l l s . The r e l a t i v e s i z e and growth r a t e f o r each o f t h e s e i n d u s t r i e s , f o r s e l e c t e d y e a r s from 1 9 5 4 t o 1 9 6 2 , i s shown i n T a b l e 2 . 1 on the f o l  l o w i n g page. I t i s ap p a r e n t t h a t lumber p r o d u c t i o n i s the major i n d u s t r y i n terms o f t o t a l v a l u e of p r o d u c t i o n . The slow per c a p i t a growth r a t e i n the lumber i n d u s  t r y has been d i s c u s s e d by many w r i t e r s 1 1 ' " ' " 2 ' ' 1 " ^ who note the low l e v e l o f r e s e a r c h and development e x p e n d i t u r e i n t h i s i n d u s t r y . I n a d d i t i o n , s u b s t i t u t e p r o d u c t s have undoubted l y i n f l u e n c e d the demand f o r lumber a d v e r s e l y , and reduced i t s r a t e o f growth. 3y c o n t r a s t , the plywood and p u l p i n d u s  t r i e s have e x p e r i e n c e d s u b s t a n t i a l growth over the p a s t t e n y e a r s . Much o f the growth i n plywood demand can be t r a c e d t o a more g e n e r a l t r e n d toward the use o f components and sheet m a t e r i a l s t o reduce o n - s i t e l a b o u r i n c o n s t r u c t i o n . Z i n u v - ska has s t r e s s e d the importance o f c o n s t r u c t i o n l a b o u r c o s t s -I h. as an e x t e r n a l i n f l u e n c e on plywood demand. 1 1 J . A. Z i n u v s k a , "The F o r e s t P r o d u c t s M i x i n a Changing Economy," P r o c e e d i n g s , Soc. o f American F o r e s t e r s , ( I 9 6 0 ) , PP. 5 9 - 6 3 . 1 2 J o s e p h Zaremba, Economics o f American Lumber I n d u s t r y , R o b e r t S p e l l e r P u b l i s h e r s , New Y o r k , ( 1 9 6 3 ) , ch. 5 . 1 3 J . A. G u t h r i e and G. R. A r m s t r o n g , Western F o r e s t I n d u s t r y - An Economic O u t l o o k , The John Hopkins P r e s s , B a l t i  more, ( 1 9 6 1 ) , ch. 2 and 3 . 14 J . ik. Z i n u v s k a , op. c i t . TABLE 2.1 THE VALUE OF FOREST PRODUCTION IN BRITISH COLUMBIA BY INDUSTRIAL CATEGORIES IN SELECTED YEARS ( m i l l i o n s o f d o l l a r s ) * 1954 1957 1959 1961 1962 Lumber 270.1 275.4 323.9 359.0 388.3 Plywood 53.0 71.8 75.5 79.8 89.6 P u l p & Paper 156.2 177.4 23^.5 258.4 292.0 S o u r c e : A n n u a l R e p o r t , B r i t i s h C o l u m b i a F o r e s t S e r v i c e , V i c t o r i a , B. C., (1962). * V a l u e s i n c l u d e l o a d i n g and f r e i g h t charges w i t h i n t h e P r o v i n c e . 15 I t s h o u l d be noted however t h a t t h i s growth i n plywood con sumption has o c c u r r e d l a r g e l y a t the expense o f lumber con sumption i n t r a d i t i o n a l lumber m a r k e t s . Growth i n the p u l p i n d u s t r y has been c o n s i s t e n t l y s t r o n g , f o l l o w i n g t h e s t e a d y i n c r e a s e i n paper and paper- p r o d u c t s consumption, and augmented by an i n c r e a s i n g use o f h i g h - g r a d e p u l p or c e l l u l o s e as a raw m a t e r i a l f o r the chem i c a l i n d u s t r i e s . O r g a n i z a t i o n o f the I n d u s t r y . The lumber i n d u s t r y i s c h a r a c t e r i z e d by a l a r g e num ber o f i n d i v i d u a l e n t e r p r i s e s consuming a w i d e l y a v a i l a b l e raw m a t e r i a l w i t h r e l a t i v e l y l o w - c a p i t a l r e q u i r e m e n t s and r e l a t i v e l y easy e n t r y . Some w r i t e r s have s u g g e s t e d t h a t t h i s fragmented s t r u c t u r e o f the lumber i n d u s t r y has been l a r g e l y r e s p o n s i b l e f o r i t s l a c k o f t e c h n o l o g i c a l and mar k e t r e s e a r c h and t h u s i t s s l u g g i s h r a t e of growth. To quote Z i n u v s k a , " . . . t h i s i s an i n d u s t r y s t r u c t u r e w h i c h might warm the h e a r t o f Adam S m i t h , but i t would break the 15 h e a r t o f any i n d u s t r i a l r e s e a r c h d i r e c t o r . " N e v e r t h e l e s s , w h i l e t h e r e i s a l a r g e number o f s m a l l s a w m i l l s on the P a c i f i c C o a s t t h e r e a r e a l s o s e v e r a l v e r y l a r g e lumber p r o  d u c i n g p l a n t s . I n c o n t r a s t , the number o f i n d i v i d u a l plywood and p u l p p r o d u c t i o n p l a n t s i n the c o a s t a l r e g i o n i s r e l a t i v e l y s m a l l . L a t e s t f i g u r e s i n d i c a t e t h e r e were s i x t e e n plywood r? J . A. Z i n u v s k a , "The F u t u r e f o r Wood i n a C o m p e t i t i v e M a r k e t , " paper f o r p r e s e n t a t i o n a t j o i n t m e e t i n g Puget Sound and Columbia R i v e r s e c t i o n s , S o c i e t y o f American F o r  e s t e r , Longview, Wash., May 4, 1963. 16 16 17 m i l l s and t h i r t e e n p u l p m i l l s i n o p e r a t i o n i n the coa s t a l r e g i o n d u r i n g 1961. C o n c e n t r a t i o n i n the plywood and p u l p i n d u s t r i e s i s c l o s e l y a s s o c i a t e d w i t h l a r g e economies o f s c a l e , and the h i g h l e v e l of t e c h n o l o g i c a l i n n o v a t i o n n e c e s s a r y t o expand i n t o w o r l d m a r k e t s . C o s t c o n t r o l i n the f o r e s t i n d u s t r y seems t o v a r y i n v e r s e l y w i t h the age o f the i n d u s t r y . Thus t h e r e i s l e s s knowledge o f u n i t - c o s t s i n the l o g g i n g and lumber i n d u s t r i e s t h a n t h e r e i s i n the younger plywood and p u l p i n d u s t r i e s . Inadequate c o s t c o n t r o l here r e f e r s t o an i n s u f f i c i e n t know l e d g e of u n i t - c o s t s f o r each stage of the p r o d u c t i o n p r o c e s s . The problem may e x i s t f o r two r e a s o n s : (a) i n s u f f i c i e n t d a t a or (b) r i s k and u n c e r t a i n t y . An i n s u f f i c i e n c y of d a t a may a r i s e m e r e l y because th e r e q u i r e d d a t a i s d i f f i c u l t t o c o l l e c t . F o r example, th e l o g g i n g i n d u s t r y i s f a c e d w i t h a heterogeneous s u p p l y o f raw m a t e r i a l ( t r e e s ) from w h i c h i t produces an e q u a l l y heterogeneous p r o d u c t ( l o g s ) . G r e a t d i f f i c u l t y i s e x p e r i  enced t h e r e f o r e i n d e t e r m i n i n g the c o s t o f p r o d u c t i o n 16 Veneer and Plywood M i l l s . Dominion Bureau o f S t a t i s t i c s , C a t a l o g u e No. 35-206, Queen's P r i n t e r s , Ottawa, (July, 1963). 17 P u l p and Paper M i l l s . Dominion Bureau o f S t a t i s t i c s , C a t a l o g u e No. 36-204, Queen's P r i n t e r s , Ottawa, ( O c t . 1963). I I . PROBLEMS OF THE INDUSTRY 17 a t t r i b u t a b l e t o each l o g , p a r t i c u l a r l y s i n c e l o g s o f d i f  f e r e n t grade a r e t y p i c a l l y produced from t h e same f e l l e d t r e e . A s i m i l a r problem a r i s e s i n the lumber i n d u s t r y where a v a r i e t y o f d i f f e r e n t lumber grades and s i z e s a r e f r e q u e n t l y produced from the same l o g . Plywood and p u l p m i l l o p e r a t i o n s a r e more amenable t o measurement of p r o  d u c t i o n d a t a and show a r e a s o n a b l y h i g h degree o f c o s t con t r o l . I n a d d i t i o n t o t h e d i f f i c u l t y o f c o l l e c t i n g u n i t - c o s t d a t a t h e r e has so f a r been l i t t l e i n c e n t i v e t o t a c k l e t h e problem. I n the l o g g i n g and lumber i n d u s t r i e s o f the p a s t t h e small-owner approach has p r e v a i l e d , w i t h l i t t l e o r no a t t e n t i o n d i r e c t e d t o u n i t - c o s t s as l o n g as the b a l a n c e sheet showed a p r o f i t . O n l y r e c e n t l y w i t h the emergence of l a r g e i n t e g r a t e d f i r m s and a growing awareness by the i n d u s t r y o f t h e r e a l l i m i t s t o i t s r e s o u r c e b a s e, has t h e r e been any attempt t o measure u n i t - c o s t s o f p r o d u c t i o n , (b) R i s k and U n c e r t a i n t y . Inadequate c o s t - c o n t r o l i n the f o r e s t i n d u s t r y due t o r i s k and u n c e r t a i n t y i s i n e v i t a b l e . F or example, l o g  g i n g c o s t s v a r y w i t h s e a s o n a l weather c o n d i t i o n s w h i c h i n  f l u e n c e t h e f i r e h a z a r d and t h e p h y s i c a l d i f f i c u l t i e s o f l o g g i n g . Some measure of the p r e d i c t a b i l i t y o f t h e s e r i s k s i s p o s s i b l e but i t depends upon d a t a w h i c h i s l a c k i n g i n IB most p a r t s o f the p r o v i n c e . The u n c e r t a i n t y o f f u t u r e stumpage v a l u e s p r e s e n t s a problem t h r o u g h o u t the f o r e s t i n d u s t r y . Stumpage r e  f l e c t s t h e t r u e r e s i d u a l o f market p r i c e l e s s e s t i m a t e d c o s t o f p r o d u c t i o n . S i n c e ownership o f the f o r e s t r e s o u r c e i n most p a r t s o f the p r o v i n c e remains i n the hands o f the Crown, and s i n c e the government changes stumpage w i t h h a r v e s t , t h e s e c o s t s t o t h e f o r e s t i n d u s t r y a r e h i g h l y u n p r e d i c t a b l e . 1 I n summary, the l o g g i n g and l o g - c o n v e r s i o n i n d u s t r i e s s u f f e r v a r y i n g degrees o f in a d e q u a t e p r o d u c t i o n - c o s t c o n t r o l . The l a c k o f u n i t - c o s t s c u r t a i l s t he measurement o f co n v e r  s i o n e f f i c i e n c y and p r o d u c t i v i t y w h i c h i n t u r n d i r e c t l y a f  f e c t s e a r n i n g s . The l a c k o f c o n t r o l over s u p p l y c o s t s i s r e f l e c t e d i n the i n d u s t r y ' s c u r r e n t haphazard a l l o c a t i o n o f l o g s among v a r i o u s c o n v e r s i o n p r o c e s s e s . C l e a r l y the i n d u s t r y must overcome t h e s e c o s t - c o n t r o l d e f i c i e n c i e s i f i t i s t o expand and meet the i n c r e a s i n g c o m p e t i t i o n f o r i t s m arkets. Market Problems. The lumber i n d u s t r y o f B r i t i s h C olumbia f a c e s s t r o n g c o m p e t i t i o n from s u b s t i t u t e p r o d u c t s , n o t a b l y plywood, p l a s t i c s , c o n c r e t e , a l u m i n i u m , and s t e e l . Most o f t h i s c o m p e t i t i o n has o c c u r r e d i n t h e t r a d i t i o n a l markets f o r lum ber where the i n d u s t r y has been f o r c e d t o e i t h e r reduce c o s t s , reduce p r o f i t s , or concede the market. The plywood 18 A. M i l t o n Moore, F o r e s t r y Tenures and Taxes i n Canada, C a n a d i a n Tax F o u n d a t i o n , T o r o n t o , (1957), ch. 2. 19 and p u l p i n d u s t r i e s , u n l i k e lumber, have n ot f a c e d the same degree of c o m p e t i t i o n from s u b s t i t u t e s . I n f a c t , t h e y have made i n r o a d s i n t o a r e a s p r e v i o u s l y dominated by lumber, 19 c o t t o n , j u t e , e t c . The lumber and p u l p i n d u s t r i e s o f B r i t i s h Columbia s e l l a major p o r t i o n o f t h e i r o u t p u t i n f o r e i g n m a r k e t s ; 20 a p p r o x i m a t e l y 77 p e r c e n t of the lumber o u t p u t and 89 p e r  c e n t o f the p u l p o u t p u t from t h e c o a s t a l r e g i o n i s e x p o r t - 21 ed. S i n c e c o a s t a l B r i t i s h C o l u m b i a a c c o u n t s f o r a s m a l l f r a c t i o n o f t h e t o t a l w o r l d s u p p l y o f t h e s e p r o d u c t s t h i s r e g i o n must, e s s e n t i a l l y , t a k e w o r l d market p r i c e s f o r t h e s e p r o d u c t s as g i v e n . The c o a s t m i l l s , w h i c h a r e o f t e n c o n s t r u c t e d s p e c i f  i c a l l y t o s e r v e f o r e i g n m a r k e t s , s h i p a h i g h e r p e r c e n t a g e of t h e i r o u t p u t t o w o r l d markets t h a n does the average m i l l i n t he p r o v i n c e . However, i t i s g e n e r a l l y agreed t h a t the f o r e s t i n d u s t r y o f the i n t e r i o r r e g i o n of B r i t i s h Columbia w i l l a ccount f o r an i n c r e a s i n g s hare o f the t o t a l f o r e s t i n d u s t r y i n the p r o v i n c e , as s t o c k s o f l a r g e - d i a m e t e r mature t i m b e r on the c o a s t a r e d e p l e t e d , and improved t e c h n o l o g y makes p o s s i b l e the h a r v e s t i n g o f p r e v i o u s l y non-commercial 22 s t a n d s o f s m a l l d i a m e t e r s p r u c e , balsam, p i n e s and f i r . 19 J . A. G u t h r i e and G. R. Arm s t r o n g , op. c i t . c h . 6. 20 P r o d u c t i o n , Lumber M i l l s , Dominion Bureau o f S t a - t i s t i c s , C a t a l o g u e No. 35-003, Queen's P r i n t e r s , Ottawa, (accumu l a t e d monthly i s s u e s 1963). 21 L. Read, Economist & S t a t i s t i c i a n , C o u n c i l o f F o r e s t I n d u s t r i e s , Vancouver, B.C. (1964) - u n o f f i c i a l e s t i m a t e , s i n c e p r i m a r y source not a v a i l a b l e . 22 J . A. G u t h r i e and G. R. Ar m s t r o n g , op. c i t . c h . 5. 20 As the lumber i n d u s t r y i s f o r c e d i n t o t i m b e r o f de c l i n i n g s i z e and q u a l i t y i t s p o s i t i o n i n w o r l d markets w i l l become weaker. The i n d u s t r y w i l l be f o r c e d t o compete i n a market f o r p r o d u c t s produced f r o m p r e d o m i n a n t l y second- growth f o r e s t s , not u n l i k e t h o s e found i n o t h e r p a r t s o f the w o r l d , and w i l l t h e r e b y l o s e i t s f o r m e r n a t u r a l a d v a n t  age i n l a r g e - d i m e n s i o n c l e a r lumber. Raw M a t e r i a l S u p p l y . A p e r v a s i v e problem f a c i n g the f o r e s t i n d u s t r y i s the d e c l i n i n g q u a l i t y of t i m b e r . The problem a f f e c t s t h e lumber and plywood i n d u s t r i e s most, f o r no m a t t e r how e f  f i c i e n t t he l o g g i n g and l o g - c o n v e r s i o n t e c h n i q u e s may be come, l o g q u a l i t y u l t i m a t e l y governs lumber and plywood q u a l i t y . The t r a n s i t i o n t o l o w e r q u a l i t y raw m a t e r i a l i s has t e n e d by the f a c t t h a t lumber and plywood a r e c o m p e t i  t o r s f o r the l i m i t e d s t o c k o f h i g h q u a l i t y l o g s . The p l y  wood i n d u s t r y i n p a r t i c u l a r p r e f e r s l a r g e - d i a m e t e r l o g s , c o n t a i n i n g a h i g h p e r c e n t a g e of c l e a r wood. The p r i c e o f t h e s e has r i s e n , i n the f a c e o f l i m i t e d s u p p l y , t o the p o i n t where l o g c o s t s now a c c o u n t f o r up t o 60 p e r c e n t o f 2~\ 2k t o t a l plywood c o s t s . J ' 23 W. E. Mayhew, "A New Method of A l l o c a t i n g C o s t s t o Veneer by Grades," F o r . P r o d . J o u r n . , v. 8, (Aug. 1958), pp. 27A-30A. 2k A. G. R a n k i n , " C o s t - P r i c e R e l a t i o n s h i p s i n the F o r e s t I n d u s t r y , " The F o r e s t r y C h r o n i c l e , v. 39, (Mar. 1963), PP. 69-79. 21 While timber quality is declining the annual har- 25 vest i s increasing steadily, 7 and the logging industry is being forced to log In more inaccessible areas and at progressively higher elevations. These trends are being accompanied however by technological advances in logging methods, and by more intensive u t i l i z a t i o n of the raw material on the part of the log-conversion industries. III. INDUSTRY TRENDS The forest industry is f u l l y aware of i t s production and market problems, and has taken steps to overcome them. ' ' Research and development in technology and mar keting have been strongly supported in recent years. The plywood and pulp industry have always been strong in re search and development, but only in recent years have the logging and lumber industries made important advances in these field s . Structural changes in the industry have also been significant. These trends are discussed in the f o l  lowing sections. Research and Development. (a) Technological Research and Development. Recent technological advances in a l l phases of the forest industry have been directed towards lowering unit- production costs, raising over-all u t i l i z a t i o n of the raw 25 "Progress to September, 1962 and Future Prospects of the Br i t i s h Columbia Sustained Yield Forestry Program." B.C. F.S., submitted to Tariff Commission of the United States of America, Washington, D.C. October, 1962. 26 J. A. Zinuvska, op. c i t . 27 A. G. Rankin, op. c i t . 2 2 m a t e r i a l , and d e v e l o p i n g new p r o d u c t s . F r e q u e n t l y t h e s e advances o c c u r c o n c u r r e n t l y . A g r e a t d e a l o f the r e s e a r c h e f f o r t i s c a r r i e d out by government a g e n c i e s and the u n i  v e r s i t i e s , a s i d e f r o m i n d u s t r i a l r e s e a r c h performed by t h e l a r g e r p r i v a t e e n t e r p r i s e s . ( i ) R e d u c t i o n i n u n i t - c o s t s . These g a i n s a r e made t h r o u g h an improvement i n p r o c e s s i n g e f f i c i e n c y , e i t h e r by r a i s i n g o u t p u t p e r d o l l a r o f o p e r a t i n g c o s t o r r e d u c i n g c o s t p e r u n i t of t h r o u g h p u t . Some examples a r e p o r t a b l e - s p a r l o g g i n g , band saws and automated g r e e n - c h a i n s i n saw m i l l s , t h i n n e r v e n e e r s i n plywood p r o d u c t i o n , and c o n t i n u o u s d i g e s t e r s i n p u l p i n g . ( i i ) I n c r e a s e d u t i l i z a t i o n . These a r e b a s i c a l l y p h y s i c a l g a i n s aimed a t r a i s i n g o v e r - a l l c o n v e r s i o n e f f i c i e n c y where h i g h e r p r o d u c t y i e l d s p e r u n i t of raw m a t e r i a l l e a d t o h i g h e r n e t r e t u r n s . T y p i c a l examples a r e c h i p p i n g of s a w m i l l w a s t e , p a t c h i n g veneer s h e e t s t o r a i s e t h e i r g r a d e , and r e c o v e r i n g c h e m i c a l s ( s u c h as l i g n o s u l p h o n a t e s ) from p u l p - m i l l e f f l u e n t s . ( i i i ) New p r o d u c t s . The development o f new p r o d u c t s may be s t i m u l a t e d by r e s e a r c h d i s c o v e r i e s or by changing mar k e t demands, o r some c o m b i n a t i o n of t h e s e f o r c e s . New p r o  d u c t s may r e s u l t f r om the improvement of e x i s t i n g p r o c e s s e s o r t h e y may r e q u i r e an e n t i r e l y new p r o d u c t i o n f a c i l i t y . Some examples a r e g l u e d - l a m i n a t e d s t r u c t u r a l beams, mixed s p e c i e plywoods, s p e c i a l t y p u l p s ( h i g h a l p h a c e l  l u l o s e ) and w o o d - e x t r a c t a d h e s i v e s . 23 (b) Market R e s e a r c h and Development. B o t h i n d i v i d u a l f i r m s and i n d u s t r y o r g a n i z a t i o n s have made e f f o r t s i n r e c e n t y e a r s t o improve t h e market a c c e p t a n c e o f wood p r o d u c t s , t h r o u g h h i g h e r s t a n d a r d s o f market s e r v i c e and by m a i n t a i n i n g c o m p e t i t i v e p r i c e s and q u a l i t y . I n a d d i t i o n , the i n d u s t r y has sought new mar k e t s e i t h e r t h r o u g h p r o d u c t d i f f e r e n t i a t i o n or t h r o u g h d i r e c t p r o m o t i o n o f new uses f o r wood. ( i ) Improved market a c c e p t a n c e . B e t t e r s e r v i c e t o lum ber customers has been a c h i e v e d t h r o u g h b u n d l i n g o f lumber, where each bundle c l a s s has a f i x e d p r o p o r t i o n of s p e c i  f i e d grades o f lumber s e l e c t e d t o conform w i t h a t y p i c a l customer's needs. Improved s e r v i c e t h r o u g h the e s t a b l i s h  ment o f a u t h o r i z e d lumber d e a l e r s has a l s o promoted b e t t e r a c c e p t a n c e . ( i i ) New M a r k e t s . I n d u s t r y a s s o c i a t i o n s have been p a r t i c  u l a r l y a c t i v e i n p r o m o t i n g new uses f o r wood p r o d u c t s t h r o u g h p u b l i c a t i o n s and a d v e r t i s e m e n t s . F o r example, drawings and p l a n s f o r houses, b o a t s , t r a i l e r s , e t c . , are o f f e r e d a t s u b s i d i z e d ( o f t e n z e r o ) p r i c e s as an i n d u c e  ment t o use wood p r o d u c t s . P r o m o t i o n o f o v e r s e a s markets has been a t t e m p t e d t h r o u g h o r g a n i z e d t r a d e m i s s i o n s t o f o r e i g n markets by prominent B r i t i s h Columbia wood-product m a n u f a c t u r e r s . S t r u c t u r a l A d j u s t m e n t s . There i s a marked tendency i n t h e f o r e s t i n d u s t r y of B r i t i s h Columbia towards i n t e g r a t i o n , b o t h v e r t i c a l and 2k h o r i z o n t a l . I t w i l l he w o r t h w h i l e t o c o n s i d e r here the im p l i c a t i o n s o f t h e s e two forms o f s t r u c t u r a l a d j u s t m e n t . (a) H o r i z o n t a l I n t e g r a t i o n . We have noted the c o m p e t i t i o n between t h e lumber and plywood i n d u s t r i e s f o r h i g h - q u a l i t y l o g s . I n a d d i t i o n , t h e lumber i n d u s t r y f a c e s c o m p e t i t i o n f rom t h e p u l p i n d u s  t r y f o r p o o r e r - g r a d e l o g s . T h i s c o n f l i c t c a n be r e s o l v e d , a t l e a s t i n p a r t , t h r o u g h h o r i z o n t a l i n t e g r a t i o n . Thus the l a r g e f i r m w i t h s u b s t a n t i a l t i m b e r h o l d i n g s seeks t o a c h i e v e maximum u t i l i z a t i o n o f i t s t i m b e r by i n t e g r a t i n g the lumber, plywood and p u l p p r o d u c t i o n p r o c e s s e s w i t h i n i t s own p l a n t . The h i g h - g r a d e p e e l e r l o g s a r e used as p l y  wood s t o c k ; the b u l k o f the b e t t e r grade l o g s goes t o the s a w m i l l s ; and the r e m a i n i n g l o g s p l u s waste from lumber and p l y w o o d - m i l l o p e r a t i o n s go t o the p u l p m i l l . T h i s f a c i l i t a t e s t he a l l o c a t i o n o f l o g s t o t h e i r most p r o f i t a b l e use. However, i n t e g r a t i o n o f t h i s s o r t i s p r a c t i c a l o n l y w i t h i n the l a r g e r companies. Some measure o f p a r t i a l i n t e g r a t i o n i s se c u r e d by the exchange o f l o g s between f i r m s o r by the s a l e of p u l p c h i p s . F o r example, a p u l p m i l l may exchange Douglas f i r p e e l e r l o g s f r o m i t s own h o l d i n g s , f o r hemlock or s m a l l l o g s t h a t a plywood m i l l w i s h e s t o d i s p o s e o f , o r a saw m i l l may s e l l i t s c h i p s t o a nearby p u l p m i l l . (b) V e r t i c a l I n t e g r a t i o n . The development of the f o r e s t i n d u s t r y c o n c u r s w i t h S c o t t ' s s t a g e 11 i n a l l a s p e c t s , w i t h the p o s s i b l e e x c e p t i o n o f i n d u s t r i a l o r g a n i z a t i o n . As Z i n u v s k a has p o i n t e d out however, the inducement t o i n v e s t i n the f o r e s t i n d u s t r y d e r i v e s from m a n u f a c t u r i n g p r o f i t s i n consuming t i m b e r ( o r wood) as a raw m a t e r i a l r a t h e r t h a n from the r e - 2Q t u r n s t o growing t i m b e r . ' Thus f o r e s t i n d u s t r y f i r m s t e n d t o merge w i t h ( i n v e s t i n ) more h i g h l y r e f i n e d wood-manu f a c t u r i n g p r o c e s s e s . T h i s i s s u b s t a n t i a t e d by the r e c e n t t r e n d t o v e r t i c a l i n t e g r a t i o n f o r w a r d s by many o f the l a r g e r f o r e s t p r o d u c t e n t e r p r i s e s . V e r t i c a l i n t e g r a t i o n f o r l a r g e i n t e g r a t e d f i r m s p r o  v i d e s a degree of s t a b i l i z a t i o n i n p r o d u c t p r i c e s and s a l e s volumes, t h r o u g h t h e mutual a s s o c i a t i o n o f a g i v e n p r o d u c t i v e c a p a c i t y w i t h a known and c a p t i v e market. A l s o , v e r t i c a l i n t e g r a t i o n f r e q u e n t l y r e s u l t s i n a b r o a d e r f i n a n c i a l base f o r t h e i n t e g r a t e d u n i t , e n a b l i n g economies o f s c a l e t h r o u g h i n v e s t m e n t i n l a r g e - c a p a c i t y p l a n t s , w h i c h i n t u r n a r e sup p o r t e d by t h e guaranteed s u p p l i e s and ma r k e t s , ( c ) Summary. S t r u c t u r a l a d j u s t m e n t s i n the f o r e s t i n d u s t r y i n r e c e n t y e a r s have r e s u l t e d i n the f o r m a t i o n of l a r g e i n t e  g r a t e d f o r e s t - p r o d u c t e n t e r p r i s e s . A l t h o u g h some s o c i a l b e n e f i t s from more complete u t i l i z a t i o n o f the raw m a t e r i a l may r e s u l t , t h i s need not n e c e s s a r i l y c o i n c i d e w i t h g r e a t e r 28 A. D. S c o t t , "The Development o f the E x t r a c t i v e I n d u s t r i e s , " The Can. J o u r . Econ. and P o l . S c i e n c e , v. 28, (Feb. 1962), pp. 70-87. 29 J . A. Z i n u v s k a , op. c i t . 26 economic efficiency. A discussion of the ramifications of government forest policy on structural adjustments with in the forest industry, however, i s outside the scope of this thesis. Nevertheless i t is apparent that the form ation of large integrated firms creates a managerial problem for each, in finding the optimal distribution of i t s available log supply among the various u t i l i z a t i o n processes. This was discussed in the previous chapter. IV. CONCLUSIONS To quote Zinuvska again, "...the challenge of competition in the forest industry today i s the challenge 3 0 of technology and markets." ~* The problem is basically a question of achieving maximum economic efficiency in the use of the forest resource. With strong competition in world markets and in many cases from substitute products, the forest industry as a whole is impelled to lower i t s 31 32 unit-costs. In addition, as several writers suggest, ' the forest industry is under considerable pressure to achieve f u l l e r u t i l i z a t i o n of the raw material, which may not always coincide with attaining maximum economic ef ficiency. Nevertheless, i t is generally agreed that both questions can be solved only through greater expenditures on research and development, in both technology and marketing. 30. J. A. Zinuvska, op. c i t . 31. op. c i t . 32. J. A. Guthrie & G. R. Armstrong, op. c i t . ch. 6. 27 As a prerequis i te to r a i s i n g the economic e f f i c i enc y of log-conversion however, the industry must ensure that i t uses i t s raw mater ia l i n an optimum manner. S p e c i f i  c a l l y , i t i s important that each log be a l located to the process where i t can earn i t s highest re turn . It i s c lear that th is condi t ion i s r e f l ec ted i n the question of e f f i c i  ent log a l l o c a t i o n discussed i n the previous chapter. In add i t ion to observing the problem of log a l l o  cat ion we have now discussed i t s r e l a t i v e importance to the future expansion of the fores t industry . As a part of the research and development e f f o r t ca l l ed f o r , improved log a l l o ca t i on of fers one of the more promising and f r u i t  f u l avenues toward greater economic e f f i c i ency i n the i n  dustry as a whole. 2 8 CHAPTER I I I EXPOSITION OF THE LOG-ALLOCATION PROBLEM The problem o f e f f i c i e n t l y a l l o c a t i n g l o g s among a l  t e r n a t i v e u t i l i z a t i o n p r o c e s s e s has been d e s c r i b e d and i t s i mportance emphasized. I n t h i s c h a p t e r the elements o f t h e problem w i l l be d i s c u s s e d , w i t h a view toward s y n t h e s i z i n g a l i n e a r programming s o l u t i o n i n the f o l l o w i n g c h a p t e r . Three c o n d i t i o n s u n d e r l y the problem o f l o g a l l o  c a t i o n i n the f o r e s t i n d u s t r y , namely: (1) The o b j e c t i v e o f p r o f i t m a x i m i z a t i o n on the p a r t o f the owners of l o g - u t i l i z a t i o n p l a n t s . ( 2 ) A l i m i t e d l o g s u p p l y . (3) Interdependence among the v a r i o u s l o g - c o n v e r s i o n p r o c e s s e s . These c o n d i t i o n s w h i c h determine the n a t u r e o f the l i n e a r programme model w a r r a n t e x a m i n a t i o n i n f u r t h e r d e t a i l . P r o f i t M a x i m i z a t i o n . I t i s not u n r e a s o n a b l e t o assume t h a t each f i r m i n t h e f o r e s t i n d u s t r y a t t e m p t s t o maximize p r o f i t s and hence seeks t o maximize t h e r e t u r n t o a l l i t s i m p u t s , i n c l u d i n g i t s l o g s u p p l y . I f the l o g s u p p l y i s g i v e n , a t z e r o c o s t , t h e n maxi mum r e t u r n , n e t o f m a n u f a c t u r i n g and c o n v e r s i o n c o s t s , con s t i t u t e s the h i g h e s t p o s s i b l e sum of e n t r e p r e n e u r i a l p r o f i t 29 on l o g u t i l i z a t i o n and economic r e n t t o the l o g s u p p l y . On the o t h e r hand, i f l o g c o s t s a re deducted from t o t a l r e t u r n s , t h e maximum r e t u r n r e p r e s e n t s maximum e n t r e p r e n e u r i a l p r o f i t ( o n l y ) t h a t can be earned i n l o g u t i l i z a t i o n . L i m i t e d Log S u p p l y . F o l l o w i n g the assumptions d i s c u s s e d e a r l i e r , we w i l l c o n s i d e r a s i n g l e i n t e g r a t e d - i n d u s t r y w i t h a t o t a l l o g con sumption e q u a l t o the a n n u a l l o g s c a l e o f the c o a s t a l r e g i o n . The l o g s u p p l y i s l i m i t e d i n a g i v e n y e a r by the a c t u a l h a r  v e s t of t i m b e r i n the r e g i o n , p l u s any c a r r y o v e r of i n v e n t o r y f r om p r e v i o u s y e a r s . I n the c o a s t a l r e g i o n t h i s h a r v e s t i s based a l m o s t e n t i r e l y on n a t u r a l " o l d - g r o w t h " f o r e s t s so t h a t t h e l o g s u p p l y i s v e r y heterogeneous w i t h r e s p e c t t o s i z e , q u a l i t y and s p e c i e . Government r e g u l a t i o n s r e q u i r i n g , among o t h e r t h i n g s , t h e complete removal o f a l l u s e a b l e t i m b e r from each s t a n d l o g g e d , tend t o i n c r e a s e the h e t e r o g e n e i t y o f the h a r v e s t by f o r c i n g the i n c l u s i o n o f m a t e r i a l t h a t would o t h e r  w i s e be l e f t u n h a r v e s t e d . I n p r a c t i c e , each f i r m i s f r e e t o purchase o r t r a d e l o g s on t h e open market and t h e r e b y attempt t o assemble the b e s t s u p p l y o f l o g s f o r i t s p a r t i c u l a r o p e r a t i o n s . I n t e r d e p e n d e n t U t i l i z a t i o n P r o c e s s e s . I t w i l l be e x p e d i e n t t o c a t e g o r i z e t h e u t i l i z a t i o n p r o c e s s e s under t h r e e h e a d i n g s : lumber p r o d u c t i o n , p l y  wood p r o d u c t i o n , and p u l p p r o d u c t i o n . We s h a l l d i s r e g a r d l e s s i m p o r t a n t s p e c i a l t y p r o c e s s e s w h i c h i n v o l v e the p r o d u c t i o n 30 o f such p r o d u c t s as s h i n g l e s , p a r t i c l e - b o a r d , e t c . (a) Lumber P r o d u c t i o n . I n broad terms we may d e s c r i b e t h i s p r o c e s s as the c o n v e r s i o n o f l o g s i n t o wood p r o d u c t s whose shape and d i  mension a r e l i m i t e d by t h e s i z e and q u a l i t y o f t h e l o g im p u t e . (b) Plywood P r o d u c t i o n . T h i s p r o c e s s may be d e s c r i b e d as t h e c o n v e r s i o n o f l o g s i n t o wood p r o d u c t s whose shape and d i m e n s i o n a r e not g o v e r n e d , e n t i r e l y , by the i n i t i a l l o g d i m e n s i o n . (c) P u l p P r o d u c t i o n . P u l p p r o d u c t i o n i n v o l v e s decomposing l o g s i n t o e l e  m e n t a l wood f i b r e s , e i t h e r c h e m i c a l l y or m e c h a n i c a l l y , and t h e subsequent r e c o n s t i t u t i o n o f t h e s e f i b r e s i n t o p u l p s h e e t s . I n t h e model t o f o l l o w we s h a l l n o t c o n s i d e r p r o  c e s s i n g beyond the p u l p s t a g e , i . e . , i n t o paper p r o d u c t s , c h e m i c a l s , t e x t i l e s , e t c . P u l p can c o n v e n i e n t l y be r e g a r d  ed as a f i n a l p r o d u c t s i n c e i t a c t u a l l y e n t e r s the market i n t h i s form. P r o c e s s I n t e r d e p e n d e n c i e s . There a r e two t y p e s o f i n t e r d e p e n d e n c e t o be c o n s i d e r  ed i n the l o g - a l l o c a t i o n problem. I n the f i r s t i n s t a n c e each l o g - u t i l i z a t i o n p r o c e s s o r a c t i v i t y draws raw m a t e r i a l d i r e c t l y f r om the l o g g i n g i n d u s t r y and consumes a p o r t i o n o f t h e l i m i t e d l o g s u p p l y . T h e r e f o r e any n e t i n c r e a s e i n consumption i n one u t i l i z a t i o n p r o c e s s i n a g i v e n y e a r r e  duces the s u p p l y o f l o g s a v a i l a b l e t o the o t h e r u t i l i z a t i o n 31 processes. In an optimizing problem this relationship per  se i s t r i v i a l , for logs would be allocated solely to the process which yields the maximum return per log. Neverthe less, this interdependence is important and of practical significance to the industry, for optimal log allocation in this instance depends on the relative prices of the f i n a l products. A second type of interdependence arises through the physical or technological relationships between u t i l i z a t i o n processes. For example, the pulp industry has become in  creasingly dependent upon sawmill residues for i t s raw 33 material (chip) supply. Similarly, the cores remaining after a log has been peeled for veneer are often sawn into lumber, and waste from this operation converted into chips for pulping. Technological interdependencies arise through 34 the production of "composite-products". Typical examples are plywood doors (a combination of plywood and lumber) and laminated beams (a combination of different types of lumber). It should be observed in particular that plywood and pulp are also composite-products, of veneer and chips, respective l y . It is clear that a change in the production level of a composite-product w i l l influence the production levels of 33 J. A. Guthrie & G. R. Armstrong, Western Forest  Industry - An Economic Outlook, The John Hopkins Press, Baltimore, (1961), ch. 3. 34 We shall define composite-products as products which involve combining two or more wood products produced earlier in the u t i l i z a t i o n system to form a new product. 32 i t s components, and the production l e v e l s of other compo site-products using any of the same components. Technological or physical interdependence - that i s where the actual l e v e l of one process depends on the act ual l e v e l of some other process(es) - i s the key to a l i n e a r programme formulation of the l o g - a l l o c a t i o n problem. The objective i s to f i n d that optimum combination of processes and process l e v e l s which w i l l y i e l d a maximum return to the log supply as a whole. Summary. A l o g - a l l o c a t i o n problem arises out of three charac t e r i s t i c s of the forest industry: p r o f i t maximization, a limi t e d log supply,and technological interdependence among di f f e r e n t u t i l i z a t i o n processes. This i s not to suggest that these are the only relevant considerations,for we have overlooked questions of limited process capacity, im mobility of factors of production, homogeneity of produc t i o n functions, etc. Some of these considerations w i l l be discussed l a t e r . It should be noted that each u t i l i z a t i o n category, lumber, plywood and pulp, presents i t s own programming problem r e s u l t i n g from a limited supply of raw material with technologically interdependent a c t i v i t i e s and a p r o f i t -33 OcT "if. "i<"J maximizing objective. ' ' ' In addition, however, plywood and pulp production involve a recombination of in termediate products (i.e., veneer and chips) to form f i n a l products (i. e . , plywood and pulp), whereas sawmilling results directly in a f i n a l product plus some intermediate products (e.g. chips) for use in other technologically interdependent a c t i v i t i e s . The contribution of this thesis l i e s in the construction of a linear programme model which combines these structurally different activities within the confines of a single problem or model. 35 F. H. Curtis, "Linear Programming the Management of a Forest Property," Journal of Forestry, v. 60, (Sept. 1962), pp. 611-616. 36 N. D. Jackson and G. W. Swinton, "Linear Programming in Lumber Production," For. Prod. Journ. v. XI, (June, 1961), pp. 272-274. 37 E. Koenigsberg, "Linear Programming Applied to the Plywood Industry," For. Prod. Journ. v. XI, (Sept. i960), pp. 481-486. 38 A. E. Paull, "Linear Programming - A Key to Optimum Newsprint Production," Pulp & Paper Mag, of Can., v. 57, ( 1 9 5 6 ) , pp. 145-151. 34 CHAPTER IV CONSTRUCTION OF A LOG-ALLOCATION LINEAR PROGRAMME MODEL The previous chapter d e a l t w i t h the nature of the l o g - a l l o c a t i o n problem. In t h i s chapter we s h a l l proceed to a more d e t a i l e d d i s c u s s i o n of i t s main c h a r a c t e r i s t i c s and consider t h e i r mathematical i n t e r p r e t a t i o n i n a l o g - a l l o c a t i o n l i n e a r programme model. A la r g e part of the d i s  c u s s i o n w i l l r e l a t e to the assumptions made concerning the r e a l world and i t s mathematical analogue. We s h a l l post pone to a l a t e r chapter a l l d i s c u s s i o n of the i m p l i c a t i o n s of r e l a x i n g these assumptions. I. BASIC OUTLINE OF THE MODEL Lin e a r programming problems have three b a s i c compo nents : (1) An o b j e c t i v e . ( 2 ) Input r e s t r a i n t s . (3) A l t e r n a t e interdependent a c t i v i t i e s . I t has been shown that the l o g - a l l o c a t i o n problem under con s i d e r a t i o n here contains each of these three elements. Thus the o b j e c t i v e i s to maximize t o t a l net r e t u r n to an i n t e  grated i n d u s t r y which consumes, i n a s p e c i f i c p e r i o d , a given supply of logs i n the production of lumber, plywood and pulp. The a v a i l a b l e supply of logs comprises the input r e s t r a i n t s , and the various wood-conversion processes, saw m i l l s , plywood m i l l s , and pulp m i l l s , correspond to 35 a l t e r n a t e i n t e r d e p e n d e n t a c t i v i t i e s . G i v e n t h e s e t h r e e b a s i c components we may d e s c r i b e a l i n e a r programme as f o l l o w s : L e t a ^ = the amount o f i n p u t i r e q u i r e d p e r u n i t o f o u t p u t f r o m a c t i v  i t y j . 39 p. = the r e t u r n p e r u n i t - l e v e l J o f a c t i v i t y j b. = the a v a i l a b l e s u p p l y o f i n p u t i X . = the l e v e l o f a c t i v i t y j , 3 e x p r e s s e d as a m u l t i p l e o f i t s u n i t - l e v e l . Then, i n a m a x i m i z i n g problem t h e o b j e c t i v e i s t o : n Maximize \ X . . p. J = 1 m S u b j e c t t o : (1) 2 X j • a i j < b i 5 J = 1»2, ....n i = 1 (2) X > 0 J F i g u r e I ( i n s i d e back c o v e r ) i s a s c h e m a t i c p r e s e n  t a t i o n o f the l o g - a l l o c a t i o n l i n e a r programme model 39 The u n i t - l e v e l o f each a c t i v i t y may be a r b i t r a r i l y d e f i n e d t o s u i t t h e problem. F o r example, t h e u n i t - l e v e l o f t h e l o g - c o n v e r s i o n a c t i v i t i e s i n the model i s d e f i n e d as 100 c u b i c o f l o g consumed. I n c h a p t e r f i v e we s h a l l system a t i c a l l y d e f i n e the u n i t - l e v e l s o f a l l a c t i v i t i e s i n the model. 36 d e s c r i b e d i n t h i s t h e s i s . Each column i n the f i g u r e con s i s t i n g o f a l i s t o f a c t i v i t y c o e f f i c i e n t s ( a - j j ) , r e p r e  s e n t s an a c t i v i t y or p r o c e s s such as sawing a l o g or m a n u f a c t u r i n g a plywood; th e column headings d e s c r i b e the v a r i o u s a c t i v i t i e s c o n s i d e r e d . Each a c t i v i t y i s a s s i g n e d a number from one to 93. The f i n a l column on the r i g h t - hand s i d e l i s t s the i n p u t s or s u p p l y r e s t r a i n t s t o the model ( b ^ ) . The f i r s t t w e l v e f i g u r e s c o r r e s p o n d t o the l o g - s u p p l y r e s t r a i n t s ; the remainder r e f e r t o i n t e r m e d i a t e i n  p u t r e s t r a i n t s , w h i c h we s h a l l d e s c r i b e l a t e r on i n t h i s c h a p t e r . There are a t o t a l of 37 r e s t r a i n t s i n the model. The bottom two rows o f the f i g u r e c o n t a i n t h e u n i t - l e v e l r e t u r n s f o r each a c t i v i t y ( p . ) , and the c o r r e s p o n d i n g a c t i v i t y l e v e l s ( X j ) . I I . STRUCTURE OF THE MODEL A l l o c a t i o n O b j e c t i v e . The o b j e c t i v e f u n c t i o n , or m a t h e m a t i c a l e x p r e s s i o n r e p r e s e n t i n g t o t a l n et r e t u r n t o the e n t e r p r i s e , i s a sum m a t i o n of the r e t u r n s from a l l a c t i v i t i e s i n c l u d e d i n t h e l o g - a l l o c a t i o n problem. A s o l u t i o n t o the problem con s i s t s of f i n d i n g a s e t o f n o n - n e g a t i v e v a l u e s f o r the X j a c t i v i t y l e v e l s w h i c h w i l l s a t i s f y the o b j e c t i v e f u n c t i o n , s u b j e c t t o t h e s u p p l y r e s t r a i n t s . I t s h o u l d be noted t h a t any a c t i v i t y may be d e f i n e d t o s u i t the problem. The o n l y r e q u i r e m e n t i s t h a t an a c  t i v i t y must be i n t e r d e p e n d e n t w i t h a t l e a s t one o t h e r 37 a c t i v i t y i n the model. I n o t h e r words, the f i n a l l e v e l o f an a c t i v i t y may not be a r b i t r a r i l y d e t e r m i n e d a t the s t a r t . N o r m a l l y we a r e i n t e r e s t e d i n m a x i m i z i n g t o t a l net r e t u r n , as d e s c r i b e d i n the p r e v i o u s c h a p t e r . I f t h e u n i t - l e v e l r e t u r n o f each a c t i v i t y i n the model i s e x p r e s s e d as a n e t - r e t u r n f i g u r e , t h e n c l e a r l y the summation o f a c  t i v i t y r e t u r n s i n the o b j e c t i v e f u n c t i o n w i l l y i e l d a t o t a l net r e t u r n f i g u r e . I n t h e model p r e s e n t e d i n t h i s t h e s i s i t w i l l he n e c e s s a r y t o c o n s i d e r p o s i t i v e r e t u r n s and n e g a t i v e r e t u r n s , c o r r e s p o n d i n g t o revenue a c t i v i t i e s and c o s t a c t i v i t i e s , r e s p e c t i v e l y . The summation o f t h e s e i n the o b j e c t i v e f u n c t i o n however w i l l s t i l l y i e l d a t o t a l n e t r e t u r n f i g u r e . T h i s w i l l be demonstrated l a t e r , i n c h a p t e r f i v e . A c t i v i t i e s . C o m p i l a t i o n o f the p e r t i n e n t a c t i v i t i e s i s an im p o r t a n t s t e p i n t h e c o n s t r u c t i o n o f a l i n e a r programme model. I n t h i s l o g - a l l o c a t i o n problem we have r e c o g n i z e d lumber, plywood and p u l p - p r o d u c t i o n a c t i v i t i e s as the b a s i c components o f the model, but i t i s n e c e s s a r y t o b reak t h e s e down more t h o r o u g h l y . F o r example, l o g s may be e i t h e r sawn i n t o lumber, p e e l e d f o r v e n e e r , or c h i p p e d , and the veneer and c h i p s may i n t u r n be s o l d d i r e c t l y o r recombined i n t o plywood or p u l p , r e s p e c t i v e l y . T e c h n o l o g i c a l i n t e r d e p e n d e n c e between a c t i v i t i e s was noted i n the p r e v i o u s c h a p t e r as one o f t h e main s o u r c e s 38 o f t he a l l o c a t i o n "problem". F u r t h e r m o r e , a l a r g e measure o f t h i s i n t e r d e p e n d e n c e came from the e x i s t e n c e o f i n t e r  m e diate p r o d u c t s such as veneer and c h i p s , w h i c h c o u l d be combined i n t o composite p r o d u c t s such as plywood and p u l p . I t i s l o g i c a l t h e r e f o r e t h a t a s u c c e s s f u l l o g - a l l o c a t i o n model s h o u l d i n c l u d e s e p a r a t e a c t i v i t i e s f o r each method o f p r o d u c t i o n and consumption o f t h e s e i n t e r m e d i a t e p r o  d u c t s . The a c t i v i t i e s p r o d u c i n g i n t e r m e d i a t e p r o d u c t s a r e r e f e r r e d t o as " i n t e r m e d i a t e - a c t i v i t i e s " , as opposed t o " f i n a l - a c t i v i t i e s " w h i c h produce a f i n a l p r o d u c t consumed o u t s i d e the model. A t an optimum s o l u t i o n t h e " i n t e r  m e d i a t e - a c t i v i t y " l e v e l s would i n d i c a t e how the g i v e n t o t a l l o g s u p p l y s h o u l d be d i s t r i b u t e d among the v a r i o u s l o g - c o n v e r s i o n p r o c e s s e s ( s a w i n g , p e e l i n g and c h i p p i n g ) ; w h i l e t h e " f i n a l - a c t i v i t y " l e v e l s would show how the i n t e r m e d i a t e p r o d u c t s ( l u m b e r , veneer and c h i p s ) s h o u l d be d i s p o s e d o f e i t h e r d i r e c t l y , w i t h o u t f u r t h e r m a n u f a c t u r i n g , o r i n com b i n a t i o n as p a r t o f a f i n a l p r o d u c t such as plywood o r p u l p . I n t e r m e d i a t e A c t i v i t i e s . These a c t i v i t i e s d e s c r i b e the consumption o f l o g s t o produce p r o d u c t s such as lumber, veneer and c h i p s . They might t h e r e f o r e be termed c u t t i n g a c t i v i t i e s . I n t h e model t h e s e a r e s e g r e g a t e d i n t o t h r e e n a t u r a l c a t e g o r i e s , sawing, p e e l i n g , and c h i p p i n g . A d e t a i l e d d i s c u s s i o n o f the a c t i v i t y c o e f f i c i e n t s i n each a c t i v i t y i s g i v e n i n the next 39 c h a p t e r , but i t s h o u l d be n o t e d here t h a t c u t t i n g a c t i v i  t i e s a r e the o n l y a c t i v i t i e s w i t h c o e f f i c i e n t s i n t h e P r i m a r y I n p u t rows of the model, i n d i c a t i n g t h a t t h e y are the o n l y a c t i v i t i e s w h i c h consume l o g s as i n p u t , see F i g u r e I . N e g a t i v e s i g n s are g i v e n t o the a c t i v i t y c o e f f i c i  e n t s and u n i t - l e v e l r e t u r n s o f each i n t e r m e d i a t e a c t i v i t y , ( i . e . , s a w i n g , p e e l i n g and c h i p p i n g ) , see F i g u r e I . T h i s f o l l o w s f r om the n a t u r e o f the i n t e r m e d i a t e r e s t r a i n t s t o t h e problem, d i s c u s s e d i n a l a t e r s e c t i o n o f t h i s c h a p t e r . A t t h i s p o i n t we may note t h a t i n t e r m e d i a t e a c t i v i t i e s produce i n p u t s f o r o t h e r a c t i v i t i e s i n the model, ( i . e . , t h e y produce n e g a t i v e o u t p u t s from the model) and i n c u r c o s t s of p r o d u c t i o n w h i c h must be deducted from g r o s s revenue t o y i e l d a net r e t u r n f i g u r e ( i . e . , t h e y i n  c u r n e g a t i v e r e t u r n s ) . F i n a l A c t i v i t i e s . The f i n a l a c t i v i t i e s i n c l u d e : (a) r e c o m b i n a t i o n o f i n t e r m e d i a t e p r o d u c t s i n t o f i n a l p r o d u c t s f o r s a l e , and (b) d i r e c t s e l l i n g o f i n t e r m e d i a t e i n p u t s , and p r i m a r y i n p u t s . (a) R e c o m b i n a t i o n A c t i v i t i e s . These a c t i v i t i e s d e s c r i b e the r e c o m b i n a t i o n o f i n t e r  m e diate i n p u t s i n t o new p r o d u c t s . Each a c t i v i t y r e p r e s e n t s kO See Page No. k l . 40 a unique c o m b i n a t i o n o f i n t e r m e d i a t e p r o d u c t s and the a c t i v i t y c o e f f i c i e n t s d e s c r i b e t h e r e l a t i v e p r o p o r t i o n s o f i n t e r m e d i a t e i n p u t s t a k e n i n each ca s e . R e c o m b i n a t i o n a c  t i v i t i e s r e s u l t i n " c o m p o s i t e - p r o d u c t s " , d e s c r i b e d i n the p r e v i o u s c h a p t e r . Thus, composite p r o d u c t s such as p l y  wood and p u l p a r e produced by r e c o m b i n i n g o u t p u t s from c u t t i n g a c t i v i t i e s (veneer p e e l i n g , c h i p c h i p p i n g ) , (b) S e l l i n g A c t i v i t i e s . These a c t i v i t i e s a r i s e from an imbalance o c c u r r i n g between p r o d u c t i o n and consumption o f i n t e r m e d i a t e p r o  d u c t s . F o r example, the p r o p o r t i o n s i n w h i c h veneers o f d i f f e r e n t grade a r e r e q u i r e d i n the manufacture o f p l y  wood a r e u n l i k e l y t o c o r r e s p o n d c l o s e l y t o t h e p r o p o r t i o n s i n w hich the veneers a r e produced from the l o g , so t h a t t h e r e w i l l be some excess veneer o f some gra d e s . To ac count f o r t h i s s u r p l u s we must i n c l u d e a c t i v i t i e s o f s e l l  i n g or d i s p o s i n g o f veneer d i r e c t l y , w i t h o u t f u r t h e r manufacture. A s i m i l a r s i t u a t i o n e x i s t s f o r mixed s p e c i e p u l p s , where we might e x p e c t t h a t the t e c h n i c a l r e q u i r e m e n t s d e t e r m i n i n g the p r o p o r t i o n s i n w h i c h d i f f e r e n t s p e c i e s o f c h i p s must be mixed w i l l d i f f e r from the p r o p o r t i o n s i n w h i c h t h e v a r i o u s s p e c i e s a r e a v a i l a b l e . T h e r e f o r e , c h i p - s e l l i n g a c t i v i t i e s w i l l be r e q u i r e d t o d i s p o s e of any c h i p s u r p l u s e s . F i n a l l y t h e r e i s the p o s s i b i l i t y t h a t the optimum p a t t e r n of a l l o c a t i o n f o r the g i v e n l o g s u p p l y may not c a l l kl f o r ' u t i l i z a t i o n ' o f a l l l o g s . Hence the model a l s o p r o  v i d e s f o r d i r e c t s e l l i n g o f whole l o g s . These s e l l i n g a c t i v i t i e s , i n e f f e c t , guarantee t h a t a l l l o g s w i l l be " d i s p o s e d o f " i n one way o r a n o t h e r and t h a t a l l wood p r o d u c t s or u n c u t l o g s w i l l be s o l d . These s e l l i n g a c t i v i t i e s c o r r e s p o n d t o " d i s p o s a l " a c t i v i t i e s r e  f e r r e d t o i n the g e n e r a l l i t e r a t u r e on l i n e a r program- kl ming, and a r e e s s e n t i a l t o g e n e r a t i n g e q u a l i t y r e s t r a i n t s i n t he model so t h a t a unique m a t h e m a t i c a l s o l u t i o n i s p o s s i b l e . R e s t r a i n t s . The r e s t r a i n t s t o t h e l o g - a l l o c a t i o n l i n e a r program me f a l l i n t o two c a t e g o r i e s : p r i m a r y i n p u t r e s t r a i n t s and i n t e r m e d i a t e i n p u t r e s t r a i n t s . We s h a l l c o n s i d e r t h e s e s e p a r a t e l y . P r i m a r y I n p u t R e s t r a i n t s . The p r i m a r y i n p u t r e s t r a i n t s c o r r e s p o n d t o the ' a v a i l a b l e l o g s u p p l y ' , w h i c h i s ' g i v e n ' i n t h i s revenue- m a x i m i z i n g problem and c o n s i s t s of a known m i x t u r e of d i f  f e r e n t s p e c i e s and d i f f e r e n t grades of l o g s . S i n c e the o b j e c t i v e f u n c t i o n i s t o maximize net r e t u r n , t h e s e r e  s t r a i n t s t a k e the form t h a t no more t h a n the g i v e n s u p p l y o f l o g s may be consumed. I n t e r m e d i a t e I n p u t R e s t r a i n t s . The i n t e r m e d i a t e a c t i v i t i e s produce i n t e r m e d i a t e kl R. Dorfman, P. A. Samuelson and R. M. Solow, L i n e a r  Programming and Economic A n a l y s i s , McGraw H i l l , New Y o r k , (1958), c h . 3. \2 p r o d u c t s w h i c h may be consumed i n b o t h r e c o m b i n a t i o n and s e l l i n g f i n a l a c t i v i t i e s . T h i s f e a t u r e g i v e s r i s e t o i n t e r  m e diate i n p u t r e s t r a i n t s i n the model. We s h a l l assume t h a t t h e r e i s t o be no a c c u m u l a t i o n o f i n t e r m e d i a t e p r o d u c t s . S u b s e q u e n t l y , s i n c e i n t e r m e d i a t e p r o d u c t s are b o t h produced and consumed i n the model the i n t e r m e d i a t e i n p u t r e s t r a i n t s w i l l be z e r o . I n t e r m e d i a t e i n p u t s w i t h z e r o - v a l u e c o n s t r a i n t s a r e t h e d i s t i n g u i s h i n g f e a t u r e o f t h i s model, and d e s e r v e f u r t h e r comment. We have seen t h a t the f i r s t s t a g e of any l o g - c o n v e r s i o n p r o c e s s i n v o l v e s a c u t t i n g a c t i v i t y w h i c h y i e l d s a p r o d u c t (lumber, veneer or c h i p s ) t h a t becomes an i n p u t to e i t h e r a s e l l i n g a c t i v i t y o r a p r o d u c t i o n a c t i v i t y , o r b o t h . S i n c e l i n e a r programming t h e o r y r e q u i r e s t h a t a l l a c t i v i t y l e v e l s be n o n - n e g a t i v e t h e r e i s a danger t h a t "wood" may be c ounted t w i c e i n any g i v e n r e s t r a i n t e q u a t i o n (row i n F i g u r e I ) , once i n an i n t e r m e d i a t e a c t i v i t y and a g a i n i n a f i n a l a c t i v i t y . T h i s d i f f i c u l t y i s overcome by a s s i g n i n g n e g a t i v e v a l u e s t o the t e c h n i c a l c o e f f i c i e n t s o f each i n t e r m e d i a t e a c t i v i t y . These n e g a t i v e t e c h n i c a l c o e f f i c i e n t s d e s c r i b e the o u t t u r n of i n t e r m e d i a t e p r o d u c t s w h i c h a r e consumed i n f i n a l a c t i v i t i e s ( r e c o m b i n a t i o n or d i r e c t s e l l i n g ) where the a c t i v i t y c o e f f i c i e n t s are p o s i t i v e . Thus, n e g a t i v e a c t i v i t y c o e f f i c i e n t s are a s s o c i a t e d w i t h t h e p r o d u c t i o n of i n t e r m e d i a t e p r o d u c t s , and p o s i t i v e a c t i v i t y c o e f f i c i e n t s are a s s o c i a t e d w i t h the consumption 43 o f i n t e r m e d i a t e p r o d u c t s . The c o n d i t i o n o f "no accumula t i o n " means t h a t each r e s t r a i n t e x p r e s s i o n must sum t o z e r o . M a t h e m a t i c a l l y , f o r the model r e p r e s e n t e d i n F i g u r e I , t h i s c o n d i t i o n may be e x p r e s s e d as f o l l o w s : 37 i = 13 Xj . a±. = 0 ; j = 1 , 2 , • 93 . D o u b l e - c o u n t i n g c o u l d o c c u r a l s o i n the o b j e c t i v e f u n c t i o n , b u t i t i s a v o i d e d i n the same way as above. Thus, n e g a t i v e v a l u e s a re a s s i g n e d t o t h e u n i t - l e v e l r e t u r n s o f each i n t e r m e d i a t e a c t i v i t y , and p o s i t i v e v a l u e s t o the u n i t - l e v e l ^ r e t u r n s o f each f i n a l ( r e c o m b i n a t i o n o r s e l l i n g ) a c t i v i t y . N e g a t i v e r e t u r n s i n the o b j e c t i v e f u n c t i o n can be r e g a r d e d as c o s t s ( n e g a t i v e p r i c e s ) w h i c h s u b t r a c t f rom t o t a l g r o s s revenues t o g i v e T o t a l Net R e t u r n . The d i s t i n c t i o n between p r i m a r y and i n t e r m e d i a t e i n  p u t s demands emphasis. P r i m a r y i n p u t s a r e ' g i v e n ' t o the model. T o t a l n e t r e t u r n i s t o be maximized w i t h r e s p e c t t o t h e s e p r i m a r y i n p u t s and the p r i m a r y i n p u t r e s t r a i n t c o n d i t i o n s t a k e the form of i n e q u a l i t i e s . On t h e o t h e r hand, i n t e r m e d i a t e i n p u t s a r e v a r i a b l e s w i t h i n the o p t i m i z  i n g problem. They a r e the l i n k i n the model between p r i m a r y i n p u t and f i n a l p r o d u c t but t h e y a r e i n no way a c o n d i t i o n  a l p a r t o f the problem. kk I t may help to r e a l i z e that the intermediate or cut t i n g a c t i v i t i e s take primary inputs (logs) as inputs and produce intermediate products (lumber, veneer, chips) as outputs, whereas the f i n a l a c t i v i t i e s , both recombination and s e l l i n g (excepting l o g s e l l i n g ) , take intermediate pro ducts (lumber, veneer, chips) as inputs and produce f i n a l products (plywood, pulp, and lumber; veneer and chips) as out puts . I I I . ASSUMPTIONS OF THE MODEL Two types of assumption may be d i s t i n g u i s h e d i n the model. F i r s t l y , Fundamental assumptions a s s o c i a t e d w i t h the mathematical procedures of l i n e a r programming. Secondly, C o r r e l a t i o n assumptions required to reduce the a c t u a l l o g - a l l o c a t i o n problem under c o n s i d e r a t i o n to a form amenable to l i n e a r programming. Fundamental Assumptions. These assumptions are unavoidable as long as we are c o n s i d e r i n g a l i n e a r programme model. They have s i g n i f i  cant i n t e r p r e t a t i o n s from an economic viewpoint, as we s h a l l d i s c u s s below. (1) The term ' l i n e a r programming' r e f e r s to the a p p l i  c a t i o n of a p a r t i c u l a r mathematical procedure, based on ' l i n e a r r e l a t i o n s h i p s ' and ' l i n e a r i n e q u a l i t i e s ' , to pro blems i n v o l v i n g choice. Mathematically, the term l i n e a r r e f e r s to the existence of a l l v a r i a b l e s i n the expressions as homogeneous and of degree one. That i s there can be no 45 2 3 s q u a r e d o r higher-power terms ( s u c h as X or X ) nor any l o g a r i t h m i c o r r o o t terms ( s u c h as l o g X or X 2 ) . To the economist t h i s i n d i c a t e s c o n s t a n t r e t u r n s t o s c a l e i n each p r o d u c t i o n p r o c e s s o r a c t i v i t y . ( i i ) The o b j e c t i v e f u n c t i o n was shown p r e v i o u s l y t o be e q u a l t o the summation o f the r e t u r n s from each a c t i v i t y ; m v i z : \ X p Z , j j The X - v a r i a b l e i n t h i s f u n c t i o n must be homogeneous and o f degree one. T h e r e f o r e , f o r the o b j e c t i v e f u n c t i o n t o r e p r e s e n t a l i n e a r r e l a t i o n s h i p , p.. must be a c o n s t a n t ; and c o n s t a n t v a l u e s o f p. i n t h i s model i m p l y b o t h i n - f i n i t e e l a s t i c i t i e s of demand f o r the f i n a l - p r o d u c t a c t i v i t y o u t p u t s , and c o n s t a n t c o s t s f o r the i n t e r m e d i a t e a c t i v i t y o u t p u t s . ( i i i ) F i n a l l y , the l i n e a r r e l a t i o n s h i p s c o n d i t i o n a l upon t h e model mean t h a t , f o r a g i v e n a c t i v i t y , each i n p u t i s t a k e n i n a f i x e d r e l a t i v e p r o p o r t i o n t o the o t h e r i n p u t s t o t h e a c t i v i t y . I n economic terms t h i s means f i x e d f a c t o r  p r o p o r t i o n s . The a c t u a l r e l a t i v e p r o p o r t i o n s t a k e n a r e g i v e n by t h e t e c h n i c a l c o e f f i c i e n t s o f each a c t i v i t y . These c o e f f i c i e n t s a r e c o n s t a n t s d e t e r m i n e d a t t h e o u t s e t , w h i c h t o g e t h e r make up t h e i n t e r n a l s t r u c t u r e o f the v a r i o u s a c t i v i t i e s and form the s k e l e t o n o f the model. The p r e  c i s i o n o f the a c t i v i t y c o e f f i c i e n t s i s c r i t i c a l t o the r e l i a b i l i t y and a c c u r a c y o f a l i n e a r programme s o l u t i o n . k6 C o r r e l a t i o n A s s u mptions, These assumptions a re m a i n l y t o keep the model as s i m p l e as p o s s i b l e . We have a l r e a d y d i s c u s s e d the concept o f an i n t e g r a t e d i n d u s t r y , on w h i c h the model i n t h i s t h e s i s i s based. We s h a l l c o n s i d e r some a d d i t i o n a l s i m p l i f y i n g a ssumptions below. ( i ) We s h a l l assume z e r o t r a n s p o r t a t i o n c o s t s between p r o c e s s e s so t h a t , f o r example, c h i p s produced from s a w m i l l waste a re i d e n t i c a l i n v a l u e w i t h t h o s e produced from veneer waste o r from whole l o g s c h i p p e d a t the p u l p m i l l . The low- c o s t water t r a n s p o r t a t i o n between u t i l i z a t i o n s i t e s on the c o a s t , and the t r e n d toward i n t e g r a t e d u t i l i z a t i o n p l a n t s , (where s a w m i l l i n g , p e e l i n g and p u l p i n g f a c i l i t i e s a r e l o c a t e d t o g e t h e r a t a c e n t r a l s i t e ) make t h i s a s s u m p t i o n q u i t e p l a u s i b l e . ( i i ) We s h a l l assume l o n g - r u n c o n d i t i o n s w h e r e i n t h e r e i s s u f f i c i e n t time f o r p l a n t c a p a c i t y t o a l t e r t o accommodate whatever p a t t e r n o f l o g u t i l i z a t i o n emerges from the optimum a l l o c a t i o n p l a n . G i v e n a f i x e d l o g s u p p l y , and i n f i n i t e p r i c e e l a s t i c i t y o f demand, the l i n e a r programme s o l u t i o n would s p e c i f y the l e v e l a t which each a c t i v i t y s h o u l d be o p e r a t e d t o a c h i e v e maximum t o t a l net r e t u r n . ( i i i ) We s h a l l assume t h a t the sawing and p e e l i n g a c t i v i t i e s i n c l u d e d f o r each type o f l o g i n the model y i e l d t he maximum amount o f h i g h - g r a d e p r o d u c t p o s s i b l e . T h i s a s s u m p t i o n r e  q u i r e s some e x p l a n a t i o n . W i t h i n f a i r l y narrow l i m i t s t h e grade o f the p r o d u c t produced can be r a i s e d by expending more i n the p r o d u c t i o n p r o c e s s ( e . g . , p a t c h i n g veneers t o remove k n o t s , or 4 7 m e t i c u l o u s l y c a r v i n g out the highest p r i c e d pieces of lumber obta i n a b l e from a l o g ) . The assumption here does not i n v o l v e such t e c h n i c a l p e r f e c t i o n . Rather, i t presumes that m i l l s are c u r r e n t l y o p t i m i z i n g t h e i r economic returns to process ing each type of l o g and that to e x t r a c t h i g h e r - q u a l i t y m a t e r i a l would i n v o l v e c o s t s (both opportunity and operating costs) i n excess of the r e t u r n s that would be gained. This assumption i s required i n order to avoid i n c l u d i n g a l l tech n i c a l l y conceivable methods of c u t t i n g each l o g - a s i t u a t i o n t h a t would g r e a t l y complicate the model while adding l i t t l e to i t s u s e f u l n e s s . ( i v ) We s h a l l assume that the i n t e g r a t e d i n d u s t r y on which t h i s model i s based i s comprised of average u t i l i z a t i o n p l a n t s f o r the c o a s t a l r e g i o n , r a t h e r than one s i n g l e complex. In t h i s manner the industry-average data employed i n the model bears some measure of r e a l i s m to the a c t u a l world. The manipulation of t h i s industry-average data and any imputed conclusions from i t are of course subject to the assumptions b u i l t i n t o the model. Summary. These assumptions are n e c e s s a r i l y r e s t r i c t i v e i n the l i g h t of c o n s t r u c t i n g a model of such a complex e n t i t y as the f o r e s t i n d u s t r y . Some of the assumptions are d i c t a t e d by the nature of l i n e a r programming techniques; some are s i m p l i f i c a t i o n s to make the demonstration as c l e a r as pos s i b l e . The assumptions discussed above are considered the more important q u a l i f i c a t i o n s of the model. Other 48 assumptions such as those i m p l i c i t i n the production f u n c t i o n s of the model ( i . e . , m o b i l i t y and homogeneity of f a c t o r inputs) should not be overlooked i n assessing the model's usefulness. In chapter s i x we s h a l l consider the i m p l i c a t i o n s of r e l a x i n g some of the C o r r e l a t i o n assumptions by a l t e r i n g the s t r u c t u r e of the model, but the Fundamental assumptions w i l l remain. 49 CHAPTER V DATA AND SOLUTION OF THE MODEL The p r i n c i p a l o b j e c t i v e i n t h i s study i s to demon s t r a t e an a p p l i c a t i o n of l i n e a r programming to the l o g - a l l o c a t i o n problem. An i n t e g r a t e d i n d u s t r y has been assum ed, along w i t h other assumptions, to keep the model reason a b l y simple. Nevertheless, to render the r e s u l t s as r e a l i s  t i c as p o s s i b l e considerable e f f o r t has been made to ob t a i n accurate and r e p r e s e n t a t i v e data. Data used are from published research s t u d i e s wherever p o s s i b l e . Where usable published data were not a v a i l a b l e estimates were obtained from experts i n the i n d u s t r y . A d i s c u s s i o n of the q u a l i t y of the data f o l l o w s i n chapter s i x . The data f a l l s i n t o three c a t e g o r i e s ; a c t i v i t y co e f f i c i e n t s , a c t i v i t y u n i t - l e v e l r e t u r n s , and input (supply) r e s t r a i n t s . Tables of data r e f e r r e d to i n t h i s chapter are c o l l e c t e d together i n an appendix at the end of t h i s chapter. I . ACTIVITY COEFFICIENTS A d i s c u s s i o n of the a c t i v i t y c o e f f i c i e n t s and t h e i r c o m p i l a t i o n f o r the l o g - a l l o c a t i o n model f a l l s i n t o two p a r t s . The f i r s t deals w i t h the p r i m a r y - i n p u t / l o g - a l l o  c a t i o n sector or l o g - a l l o c a t i o n c o e f f i c i e n t s , the second p a r t covers the i n t e r m e d i a t e - i n p u t / f i n a l - p r o d u c t sector or t e c h n o l o g i c a l c o e f f i c i e n t s , see Figure I ( i n s i d e back c o v e r ) . 50 1. L o g - A l l o c a t i o n C o e f f i c i e n t s . The p r i m a r y - i n p u t l o g - a l l o c a t i o n s e c t o r of t h e model d e a l s o n l y w i t h a l l o c a t i n g a g i v e n s u p p l y o f l o g s among v a r  i o u s i n t e r m e d i a t e a c t i v i t i e s . There i s no consumption o r p r o c e s s i n g o f the l o g i n v o l v e d . A l l the l o g - a l l o c a t i o n co e f f i c i e n t s a r e t h e r e f o r e u n i t y , and t h e r e i s o n l y one l o g - a l l o c a t i o n c o e f f i c i e n t p e r i n t e r m e d i a t e a c t i v i t y . I n t h i s model we have c o n s i d e r e d t w e l v e d i f f e r e n t t y p e s o f l o g , as shown i n T a b l e 5 . 1 . The s u p p l y of each t y p e o f l o g may be a l l o c a t e d to any s a wing, p e e l i n g or c h i p  p i n g ( i . e . , i n t e r m e d i a t e ) a c t i v i t y or a c o m b i n a t i o n o f t h e s e . 2. T e c h n o l o g i c a l C o e f f i c i e n t s . The t e c h n o l o g i c a l c o e f f i c i e n t s d e s c r i b e the t e c h n i  c a l r e l a t i o n s h i p s i m p l i c i t i n each a c t i v i t y . I n the i n t e r  m e d i a t e a c t i v i t i e s t h e y d e s c r i b e the p r o p o r t i o n s o f the s e v e r a l i n t e r m e d i a t e p r o d u c t s produced. I n the f i n a l r e  c o m b i n a t i o n a c t i v i t i e s t h e y d e s c r i b e the p r o p o r t i o n s i n w h i c h i n t e r m e d i a t e p r o d u c t s a r e recombined t o form f i n a l p r o d u c t s . I n t h e s e l l i n g a c t i v i t i e s t h e r e i s no t r a n s f o r m a t i o n o f the i t e m so t h a t a l l s e l l i n g a c t i v i t y c o e f f i c i e n t s a r e u n i t y . T e c h n o l o g i c a l C o e f f i c i e n t s - I n t e r m e d i a t e A c t i v i t i e s . The i n t e r m e d i a t e a c t i v i t i e s a r e s a wing, p e e l i n g and c h i p p i n g o f l o g s . The u n i t - l e v e l o f each i s d e f i n e d as one hundred c u b i c f e e t of l o g s consumed i n the a c t i v i t y . An e x p l a n a t i o n o f the d a t a used i n c o m p i l i n g t h e s e 51 a c t i v i t y c o e f f i c i e n t s f o l l o w s , (a) Log-sawing A c t i v i t i e s . There a r e t w e l v e l o g - s a w i n g a c t i v i t i e s c o r r e s p o n d  i n g t o the t w e l v e d i f f e r e n t l o g c a t e g o r i e s c o n s i d e r e d i n the problem. W h i l e the i m p l i c a t i o n i s t h a t any l o g can be sawn, t h i s may not i n f a c t be t r u e due t o t e c h n i c a l l i m i  t a t i o n s o f s i z e and shape. However, any l o g s w h i c h a r e t e c h n o l o g i c a l l y unsawable can be bypassed i n the s e l e c t i o n o f l o g - s a w i n g a c t i v i t i e s . I t was assumed here t h a t any t e c h n i c a l l y sawable l o g would be d i r e c t e d t o an a p p r o p r i a t e l o g - s a w i n g a c t i v i t y , and t h a t e c o n o m i c a l l y sawable l o g s would be d e t e r m i n e d from t h e s e by t h e l i n e a r programming model, i n the l i g h t o f lumber y i e l d s , lumber p r i c e s and s a w i n g - c o s t d a t a . The lumber y i e l d from any l o g i s d i v i d e d i n t o f o u r c a t e g o r i e s , based on lumber grades approved by the B r i t i s h C olumbia Lumber M a n u f a c t u r e r s ' A s s o c i a t i o n , v i z . 1. C l e a r . 2. S e l e c t / M e r c h a n t a b l e . 3. C o n s t r u c t i o n / S t a n d a r d . 4. U t i l i t y / E c o n o m y . I n a d d i t i o n t o the lumber y i e l d f r o m each l o g - s a w i n g a c t i v i t y , a ccount was t a k e n o f the p r o d u c t i o n o f waste- wood s u i t a b l e f o r c h i p p i n g , and o f h o g - f u e l ( i n c l u d i n g saw d u s t ) s u i t a b l e o n l y f o r b u r n i n g . 42 S t a n d a r d G r a d i n g and D r e s s i n g R u l e s No. 59. B r i t i s h C o l u m b i a Lumber M a n u f a c t u r e r s ' A s s o c i a t i o n , Vancouver, (Feb. 1961). 52 The y i e l d s o f i n t e r m e d i a t e p r o d u c t ( l u m b e r , c h i p s and hog f u e l ) r e s u l t i n g from each l o g - s a w i n g a c t i v i t y a re shown i n T a b l e 5.2, e x p r e s s e d as a pe r c e n t a g e o f the volume o f l o g consumed (sawn). T h i s d a t a was co m p i l e d as averages f r o m numerous d i f f e r e n t s o u r c e s , d i s c u s s e d below. The a c t i v i t y c o e f f i c i e n t s o f each l o g - s a w i n g a c t i v i t y i n F i g u r e I a r e based on the d a t a i n Table 5.2, but e x p r e s s e d as d e c i m a l s . T h i s i s so t h a t the c o e f f i c i e n t s w i l l be r e a d i l y a p p l i c a b l e t o any u n i t - v o l u m e o f l o g s consumed; and the u n i t - v o l u m e may be c o n v e n i e n t l y d e f i n e d t o s u i t any p a r t i c  u l a r s c a l e o f problem and d a t a . Douglas f i r lumber y i e l d s were o b t a i n e d p r i n c i p a l l y f r o m d a t a p u b l i s h e d by the U n i t e d S t a t e s Department o f A g r i  c u l t u r e , based on stu d i e s , conducted i n U n i t e d S t a t e s saw m i l l s west o f the Cascade Mountains i n Oregon and Wash- 1+3 kk k^  kg k 7 i n g t o n . J ' » y ' ' I t was assumed t h a t t h i s N o r t h w e s t e r n L 3 E. E. Matson, "Lumber Grade Recovery from Oregon Co a s t Type Douglas F i r , " U.S. Dept. o f A g r i c u l t u r e , Pac. N.W. F o r  e s t and Range E x p t . S t a . R e s e a r c h Paper No. 3, P o r t l a n d , Oregon, (May, 1952). M+ E. E. Matson, "Lumber Grades from Young-Growth Douglas F i r , " U.S. Dept. of A g r i . , Pac. N.W. F o r e s t & Range E x p t . S t a . R e s e a r c h Note No. 79, P o r t l a n d , Oregon, ( S e p t . 1952). K 5 E. E. Matson, "Lumber Grades from Douglas F i r P e e l e r L ogs," U.S. Dept. o f A g r i c , Pac. N.W. F o r e s t and Range E x p t . S t a . , R e s e a r c h Note No. 83, P o r t l a n d , Oregon, (Nov. 1952). k6 E. E. Matson, "Lumber Grades from Old-Growth Douglas F i r S a w m i l l Logs," U.S. Dept. o f A g r i c , Pac. N.W. F o r e s t and Range E x p t . S t a . , R e s e a r c h Note No. 125, ( J a n . 1956). L 7 E. H. C l a r k e , "Lumber Grade Recovery from Old-Growth D o u g l a s F i r a t a N o r t h w e s t e r n Oregon S a w m i l l , " U.S. Dept. o f A g r i c . , Pac. N.W. F o r e s t and Range E x p t . S t a . , R e s e a r c h Note No. 191, P o r t l a n d , Oregon, (Oct. I960). 53 U n i t e d S t a t e s d a t a was a p p l i c a b l e t o the B r i t i s h Columbia c o a s t a l a r e a , a f t e r a p p r o p r i a t e a d j u s t m e n t s f o r d i f f e r e n c e s i n t he grade r u l e s . Some Ca n a d i a n Douglas f i r lumber y i e l d d a t a was o b t a i n e d f r o m the F o r e s t P r o d u c t s L a b o r a t o r y , Van c o u v e r , B. C . k Q ^ Hemlock lumber y i e l d s a r e based on Ca n a d i a n d a t a pub- 50 l i s h e d by the F o r e s t P r o d u c t s L a b o r a t o r y i n Vancouver and on d i s c u s s i o n s w i t h m i l l managers o f l o c a l f o r e s t p r o  d u c t s companies. Spruce lumber y i e l d s a r e based s o l e l y upon e s t i m a t e s s u p p l i e d by m i l l managers i n the Vancouver a r e a . The c h i p y i e l d d a t a f o r s a w m i l l s was i n t e r p o l a t e d from s e v e r a l s o u r c e s . I n the absence o f s p e c i f i c d a t a on c h i p y i e l d s w i t h r e s p e c t t o l o g grade on the B. C. c o a s t , a c o r r e l a t i o n between w a s t e - c h i p p r o d u c t i o n and l o g diam e t e r was a d o p t e d . ^ 2 An average d i a m e t e r f o r each l o g 4 8 C. F. McBride and J . M. K i n g h o r n , "Lumber Degrade caused by Ambrosia B e e t l e s , " B r i t i s h Columbia Lumberman, v. 44, ( J u l y , I 9 6 0 ) . 49 J . D o b i e , "A M i l l i n g S t u d y of 150-year o l d Douglas F i r , " P u b l i c a t i o n No. 1032, F o r e s t P r o d u c t s R e s e a r c h B r a n c h , Canada, Dept. o f F o r e s t r y , Ottawa, 1962. 50 C. F. M c B r i d e , J . M. K i n g h o r n , op. c i t . 51 B. Dowdle and R. B a i n , "Lumber o r C h i p s - A Compari son o f S m a l l - l o g U t i l i z a t i o n A l t e r n a t i v e s , " U.S. Dept. o f A g r i c , N o r t h e a s t e r n F o r . E x p t . S t a . , ( I 9 6 0 ) . 52 M. E. H a m l i n , " E x p e r i e n c e R e p o r t - Wastewood and C h i p Volume Measurement," Amer. Pulpwood A s s o c . , N o r t h e a s t Tech. Comm. M i n u t e s , (1956). 5k grade was de t e r m i n e d from the lumber y i e l d d a t a and a c a l  c u l a t i o n made o f the c h i p p a b l e waste from sawing each grade o f l o g . The weighted average d i a m e t e r s f o r each l o g grade v a r y w i t h s o u r c e s o f l o g s u p p l y but the e s t i m a t e d c h i p - y i e l d d a t a i s c o n s i d e r e d r e p r e s e n t a t i v e f o r the r e g i o n as a whole. (b) L o g - p e e l i n g A c t i v i t i e s . Some grades o f l o g i n p u t a r e not exposed t o l o g - p e e l i n g a c t i v i t i e s due t o the l i m i t a t i o n s o f l o g - p e e l i n g t e c h n o l o g y . I n g e n e r a l , the l o w e s t grades o f any s p e c i e have v e r y poor veneer y i e l d s and h i g h p e e l i n g and p a t c h i n g c o s t s . The p e e l i n g o f t h e s e l o g s i s t h e r e f o r e u n l i k e l y t o be i n c l u d e d i n any p r o f i t - m a x i m i z i n g s o l u t i o n t o t h e problem. A c c o r d i n g l y we have e x c l u d e d p e e l i n g a c t i v i t i e s f o r t he No. 3 grade saw-logs o f each s p e c i e i n the model. As w i t h the l o g - s a w i n g a c t i v i t i e s , f i g u r e s i n each l o g - p e e l i n g a c t i v i t y r e p r e s e n t the y i e l d s o f ve n e e r , c h i p s and h o g - f u e l , e x p r e s s e d as a p e r c e n t a g e o f the volume of l o g consumed ( p e e l e d ) . A l l veneer y i e l d s a r e on a gre e n - veneer b a s i s ; a p p r o x i m a t e l y e i g h t p e r c e n t volume s h r i n k  age o c c u r s d u r i n g t h e veneer d r y i n g s t a g e b u t t h i s l o s s i s a c c o u n t e d f o r l a t e r i n the plywood p r o d u c t i o n a c t i v i t y c o e f f i c i e n t s . The y i e l d s o f gr e e n veneer from Douglas f i r l o g s , shown i n T a b l e 5.3 were c o m p i l e d from p u b l i s h e d r e s u l t s o f an e x t e n s i v e s t u d y c a r r i e d out by the U.S. Department o f A g r i c u l t u r e , P a c i f i c Northwest F o r e s t and Range E x p e r i m e n t a l 55 53 S t a t i o n , P o r t l a n d , Oregon, and from d i s c u s s i o n s between t h e a u t h o r and t e c h n i c a l e x p e r t s i n the plywood i n d u s t r y i n Vancouver. The p u b l i s h e d A m e r i c a n s t u d y c o v e r e d a s u r  vey o f 18 plywood p l a n t s i n Oregon and Wa s h i n g t o n , and was based on the veneer y i e l d from more t h a n f i v e m i l l i o n board f e e t of Douglas F i r l o g s . However, t h i s s t u d y was c a r r i e d o u t between 1950 and 1955 and the i n d u s t r y ' s t e c h n o l o g y has advanced a p p r e c i a b l y i n the p a s t t e n y e a r s , so t h a t con s i d e r a b l e w e i g h t was g i v e n t o the c u r r e n t ( b u t e s t i m a t e d and u n o f f i c i a l ) d a t a o b t a i n e d from l o c a l e x p e r t s . Un p u b l i s h e d d a t a by McBride c o n c u r s w i t h t h e s e u n o f f i c i a l 54 e s t i m a t e s f o r t h e c o a s t a l r e g i o n . The hemlock and spruce veneer y i e l d s a r e based s o l e l y on e s t i m a t e s by l o c a l ex p e r t s . The d a t a used i n the model and shown i n Ta b l e 5.3 a r e c o n s i d e r e d t o be a r e a l i s t i c o v e r - a l l average f o r c o a s t a l B. C. The p r o d u c t i o n o f c h i p p a b l e waste a s s o c i a t e d w i t h l o g - p e e l i n g a c t i v i t i e s i s based on d a t a quoted by G u t h r i e 55 and Armstrong f o r Western Oregon, from d a t a c o m p i l e d by 56 Guernsey f o r B r i t i s h C o l u m b i a , and from i n f o r m a t i o n 53 E. H. C l a r k e and A. C. Knauss, "Veneer R e c o v e r y f r o m Douglas F i r Logs," U.S. Dept. o f A g r i c . Pac. N.W. F o r e s t and Range E x p t . S t a . R e s e a r c h Paper No. 23, P o r t l a n d , Oregon, (Aug. 1957). 54 C. F. McBride - u n p u b l i s h e d r e s u l t s . 55 J . A. G u t h r i e and G. R. Armstrong, W e s t e r n F o r e s t  I n d u s t r y - An Economic O u t l o o k , The John Hopkins P r e s s , B a l t i - more, (1961), p. 138. 56 F. W. Guernsey, "Some C o n v e r s i o n F a c t o r s f o r B r i t i s h C o l u m b i a F o r e s t P r o d u c t s , " F o r e s t P r o d u c t s L a b o r a t o r y o f Can ada p u b l i c a t i o n , V - 1027, Dept. o f N o r t h e r n A f f a i r s and N a t i o n a l R e s o u r c e s , Ottawa, (1959). 56 57 p r e p a r e d by the I n s t i t u t e of F o r e s t P r o d u c t s , W a s h i n g t o n , J ' H o g - f u e l p r o d u c t i o n i s based on d a t a from the same s o u r c e s . The a p p l i c a b i l i t y o f a l l d a t a t o c o n d i t i o n s i n c o a s t a l B r i t i s h Columbia was c o n f i r m e d by l o c a l e x p e r t s , ( c ) L o g - c h i p p i n g A c t i v i t i e s . These a c t i v i t i e s r e p r e s e n t the s i m p l e p r o c e s s o f c o n v e r t i n g an e n t i r e l o g i n t o p u l p c h i p s . Any l o g can be c h i p p e d . A c c o r d i n g l y , t h e r e a r e 12 l o g - c h i p p i n g a c t i v i t i e s t o meet the 12 d i f f e r e n t l o g c a t e g o r i e s i n t h e model. I t i s assumed t h a t t h e r e i s 100 p e r c e n t r e c o v e r y o f the l o g volume, as c h i p s . I n p r a c t i c e t h i s r e p r e s e n t s an over e s t i m a t e , f o r f i n e p a r t i c l e s r e c o v e r e d from s c r e e n i n g the c h i p s a r e n o r m a l l y d i v e r t e d t o hog f u e l . T e c h n o l o g i c a l C o e f f i c i e n t s - F i n a l A c t i v i t i e s . The f i n a l a c t i v i t i e s i n c l u d e b o t h r e c o m b i n a t i o n a c t i v i t i e s and s e l l i n g a c t i v i t i e s , as d i s c u s s e d e a r l i e r . We s h a l l d i s c u s s each of t h e s e s e p a r a t e l y , below, (a) Plywood P r o d u c t i o n A c t i v i t i e s . There a r e n i n e d i f f e r e n t veneer grades produced from p e e l i n g a c t i v i t i e s i n t h e model. The number of t h e o r e t i c a l l y p o s s i b l e plywoods w h i c h c o u l d be manufactured f r o m t h e s e veneers depends upon the v a r i o u s veneer t h i c k n e s s e s , p a n e l t h i c k n e s s (number o f l a y e r s o f v e n e e r ) , and the s p e c i f i c m i x t u r e s o f veneer f o r each plywood grade. The number of m a t h e m a t i c a l p o s s i b i l i t i e s i s v e r y l a r g e , but many can be r e a d i l y e l i m i n a t e d i n the model as i m p r a c t i c a b l e a l t e r n a t i v e s . 57 Anon. C o n v e r s i o n F a c t o r s f o r P a c i f i c N o rthwest F o r e s t P r o d u c t s , I n s t i t u t e o f F o r e s t P r o d u c t s , S e a t t l e , W a s hington, ( J u n e , 1957). 57 The D o u g l a s - f i r plywoods i n t h i s model a r e based upon C a n a d i a n S t a n d a r d s A s s o c i a t i o n s p e c i f i c a t i o n s w i t h r e g a r d t o the p a r t i c u l a r veneer grades i n each plywood. The a c t u a l p e r c e n t a g e o f each veneer grade i n a p a r t i c u l a r plywood depends upon veneer t h i c k n e s s and p a n e l t h i c k n e s s . F o r sim p l i c i t y , a s t a n d a r d p a n e l t h i c k n e s s o f 3 / 8 i n c h i s assumed and the r e l a t i v e p r o p o r t i o n s o f each grade o f veneer i n t h e v a r i o u s plywoods i s based on d a t a p r o v i d e d by l o c a l manu f a c t u r e r s . The r e m a i n i n g plywood p r o d u c t i o n a c t i v i t i e s c o n c e r n m i x e d - s p e c i e and n o n - D o u g l a s - f i r , s i n g l e - s p e c i e plywoods. These are not i n f a c t produced t o any l a r g e e x t e n t on t h e c o a s t and m a n u f a c t u r i n g s p e c i f i c a t i o n s a r e not r e a d i l y a v a i l a b l e . E s t i m a t e s were o b t a i n e d however based on Doug l a s - f i r s p e c i f i c a t i o n s . The p l y w o o d - p r o d u c t i o n a c t i v i t y c o e f f i c i e n t s a r e shown i n T a b l e 5*k. These a r e c a l c u l a t e d on a d r y - v e n e e r b a s i s , a c c o u n t i n g f o r an assumed volume s h r i n k a g e o f 8 p e r  c e n t f o r a l l veneers i n t h e d r y i n g s tage a f t e r p e e l i n g . S a n d i n g l o s s e s o f Ik p e r c e n t o f t h e green-veneer volume f o r a . 3 / 8 i n c h p a n e l are i n c l u d e d f o r a l l plywoods ex c e p t D o u g l a s - f i r s h e a t h i n g grade ( " s h e a t h i n g grade" i n f e r s unsanded plyw o o d ) . These a d j u s t m e n t s l e a d t o a r e q u i r e  ment o f a p p r o x i m a t e l y 1 . 2 8 c u b i c f e e t g r e e n veneer t o p r o  duce one c u b i c f o o t o f sanded plywood, and 1 . 0 9 c u b i c f e e t 5 8 Douglas F i r Plywood - CSA/0121/1961, C a n a d i a n S t a n d a r d s A s s o c i a t i o n , Queen's P r i n t e r s , Ottawa, ( 1 9 6 1 ) . 58 g r e e n veneer t o produce one c u b i c f o o t o f unsanded or s h e a t h  i n g plywood. The plywood p r o d u c t i o n c o e f f i c i e n t s f o r green veneer are e x p r e s s e d on a b a s i s o f the c u b i c f e e t green veneer r e q u i r e d per c u b i c f o o t o f plywood produced. (b) P u l p P r o d u c t i o n A c t i v i t i e s . Three o f the seven p u l p p r o d u c t i o n a c t i v i t i e s i n t h i s model a r e based r o u g h l y on the p r i n c i p a l t y p e s o f p u l p c u r  r e n t l y produced i n B r i t i s h C olumbia. The remainder a r e h y p o t h e t i c a l . The a c t i v i t y c o e f f i c i e n t s a r e based on p r o d u c t i o n o f one a i r - d r i e d t o n o f p u l p i n each a c t i v i t y . S i n c e the p u l p c h i p i n p u t s a r e i n v o l u m e - u n i t s and a c t i v i t y o u t p u t s i n w e i ght u n i t s , a c o n v e r s i o n f a c t o r i s b u i l t - i n t o t h e c o e f f i c i e n t s t o account f o r t h i s change i n u n i t s . The c a l c u l a t e d a c t i v i t y c o e f f i c i e n t s a r e shown i n T a b l e 5 . 5 . (c) L o g - s e l l i n g A c t i v i t i e s . One u n i t of l o g i n p u t i s o b v i o u s l y n e c e s s a r y per u n i t o f o u t p u t from each l o g - s e l l i n g a c t i v i t y (as shown i n F i g u r e I ) . The u n i t - l e v e l o f each l o g - s e l l i n g a c t i v i t y t h e r e f o r e depends upon the u n i t - v o l u m e s e l e c t e d i n the i n t e r m e d i a t e a c t i v i t i e s . (d) L u m b e r - s e l l i n g A c t i v i t i e s . The lumber produced from the l o g s i n the model may be c o n s i d e r e d as an i n p u t t o the l u m b e r - s e l l i n g a c t i v i t i e s . I t i s assumed t h a t no p h y s i c a l t r a n s f o r m a t i o n o c c u r s i n the s e l l i n g a c t i v i t i e s and hence each c u b i c f o o t of i n t e r m e d i a t e i n p u t (lumber) r e s u l t s i n e x a c t l y one c u b i c f o o t o f lumber s o l d , so t h a t the l u m b e r - s e l l i n g a c t i v i t y c o e f f i c i e n t s each 59 have a v a l u e o f one. There i s a s e p a r a t e and d i s t i n c t s e l l i n g a c t i v i t y f o r each i n t e r m e d i a t e lumber i n p u t , g i v - . i n g a t o t a l o f t w e l v e . (e) V e n e e r - s e l l i n g A c t i v i t i e s . The veneer s e l l i n g a c t i v i t i e s a re c o n s t r u c t e d i n a manner s i m i l a r t o the l u m b e r - s e l l i n g a c t i v i t i e s . I t i s assumed t h a t no p h y s i c a l t r a n s f o r m a t i o n o c c u r s between p e e l i n g and s e l l i n g veneer ( t h e market i s f o r green un- sanded v e n e e r ) . Each c u b i c f o o t o f i n t e r m e d i a t e i n p u t t o a v e n e e r - s e l l i n g a c t i v i t y c o r r e s p o n d s t o e x a c t l y one c u b i c f o o t o f veneer s o l d , so t h a t the v e n e e r - s e l l i n g a c t i v i t y c o e f f i c i e n t s have a v a l u e o f one. There i s a s e p a r a t e a c t i v i t y f o r each d i f f e r e n t i n t e r m e d i a t e veneer i n p u t g i v i n g a t o t a l o f n i n e v e n e e r - s e l l i n g a c t i v i t i e s . ( f ) C h i p - s e l l i n g A c t i v i t i e s . There a r e o n l y t h r e e c h i p - s e l l i n g a c t i v i t i e s , c o r  r e s p o n d i n g t o the t h r e e l o g s p e c i e s . The t o t a l c h i p s u p p l y i s produced by the i n t e r m e d i a t e a c t i v i t i e s , s awing, p e e l  i n g and c h i p p i n g , each o f w h i c h c o n t r i b u t e s t o one o f the i n t e r m e d i a t e c h i p - s u p p l y i n p u t s . L i k e the o t h e r s e l l i n g a c t i v i t i e s , each c h i p - s e l l i n g a c t i v i t y has a t e c h n i c a l co e f f i c i e n t o f one, t h e r e b e i n g no p h y s i c a l changes between c h i p p i n g and s e l l i n g , and one c u b i c f o o t o f c h i p s s o l d r e q u i r e s one c u b i c f o o t of c h i p s i n p u t . (g) H o g - f u e l S e l l i n g A c t i v i t y . An a c t i v i t y t o account f o r the d i s p o s a l of hog f u e l produced i n t h e sawing and p e e l i n g a c t i v i t i e s i s r e q u i r e d 60 t o make t h e model complete. L i k e the o t h e r s e l l i n g a c t i v i  t i e s , one c u b i c f o o t o f hog f u e l s o l d r e q u i r e s one c u b i c f o o t i n p u t . The t e c h n i c a l c o e f f i c i e n t i s t h e r e f o r e one. I I . UNIT-LEVEL RETURNS OF THE ACTIVITIES The u n i t - l e v e l r e t u r n s o f the v a r i o u s a c t i v i t i e s w i l l be d i s c u s s e d i n the same o r d e r as t h e t e c h n i c a l c o e f f i c i e n t s d i s c u s s e d i n the p r e c e d i n g s e c t i o n , b e g i n n i n g w i t h the i n t e r  mediate a c t i v i t i e s w h i c h produce o u t p u t s f o r consumption by the r e m a i n i n g f i n a l a c t i v i t i e s . The l o g s u p p l y i n the model i s assumed g i v e n , a t z e r o c o s t . The o b j e c t i v e f u n c t i o n t h e r e f o r e c o n s i s t s of e n t r e p r e n e u r i a l p r o f i t p l u s economic r e n t t o the l o g sup p l y . A d i s c u s s i o n o f the p u r e l y p r o f i t - m a x i m i z i n g case w i l l be found i n c h a p t e r s i x , d e a l i n g w i t h an a p p l i c a t i o n o f t h i s model t o the l o g - a l l o c a t i o n problem of an i n t e g r a t e d f i r m . I n t e r m e d i a t e A c t i v i t y U n i t - l e v e l R e t u r n s , (a) Sawing A c t i v i t y R e t u r n s . When a v a r i e t y o f l o g s a re p r o c e s s e d i n a s a w m i l l no attempt i s made t o determine the lumber o u t p u t per l o g . Indeed, the c o l l e c t i o n of such d a t a would be p r a c t i c a l l y im p o s s i b l e i n most s a w m i l l s w i t h o u t s e r i o u s l y d i s r u p t i n g p r o  d u c t i o n . Hence most p r o d u c t i o n d a t a i s r e p o r t e d on an a v e r a g e - l o g b a s i s . R a t h e r t h a n a ttempt t o a s s e s s the sawing c o s t f o r each type and grade of l o g i n v o l v e d i n t h e model i t w i l l be assumed t h a t sawing c o s t s per u n i t volume of l o g are c o n s t a n t . 6 1 I n t e r - i n d u s t r y comparisons of l o g - c o n v e r s i o n c o s t s by R a n k i n ^ g i v e an average f i g u r e o f $ 2 6 . 0 0 per thousand f e e t b o a r d measure ( a b b r e v i a t e d MBM) o f lumber produced, f o r saw i n g c o s t s i n t h e c o a s t a l r e g i o n o f B r i t i s h C o l u m b i a . T h i s f i g u r e i n c l u d e s s e l l i n g , a d m i n i s t r a t i o n , d e p r e c i a t i o n and a l l m a n u f a c t u r i n g c o s t s , b u t i s e x c l u s i v e o f l o g c o s t s . To c o n v e r t t h i s f i g u r e t o a log-volume b a s i s an a v e r  age lumber y i e l d f i g u r e f o r a l l l o g s i n c l u d e d i n the model was c a l c u l a t e d f rom t h e d a t a o f Ta b l e 5 . 2 . T h i s , m u l t i p l i e d by R a n k i n ' s average s a w i n g - c o s t f i g u r e gave an average l o g - sawing c o s t o f $ 2 0 . 0 0 per hundred c u b i c f e e t of l o g , f o r each t y p e o f l o g i n the model. See F i g u r e I . (b) P e e l i n g - a c t i v i t y R e t u r n s . C o s t d a t a f o r plywood p r o d u c t i o n a r e a l s o o n l y a v a i l  a b l e on an a v e r a g e - l o g b a s i s . The method o u t l i n e d above t o c a l c u l a t e l o g - s a w i n g c o s t s on a log-volume b a s i s was used t o e s t i m a t e p e e l i n g c o s t s p e r u n i t o f each s p e c i e o f l o g a l  l o c a t e d t o the p e e l i n g a c t i v i t i e s . The c a l c u l a t i o n gave an average p e e l i n g c o s t o f $ 6 . 0 0 p e r hundred c u b i c f e e t o f l o g p e e l e d f o r a l l p e e l i n g a c t i v i t i e s i n t h e model. See F i g u r e I . (c) C h i p p i n g - a c t i v i t y R e t u r n s . C h i p p i n g c o s t s i n the model a r e based on r e c e n t d a t a g i v e n by M c B r i d e ^ f o r c h i p p i n g i n the i n t e r i o r r e g i o n o f 5 9 A. G. R a n k i n , " C o s t - P r i c e R e l a t i o n s h i p s i n the F o r e s t I n d u s t r y , " The F o r e s t r y C h r o n i c l e , v. 3 9 , (Mar. 1 9 6 3 ) , PP. 6 9 - 7 7 . 6 0 C. F. M c B r i d e , " B a r k i n g and C h i p p i n g i n the I n t e r i o r o f B r i t i s h C o l u m b i a , " Canada Lumberman, v.83 , ( J u l y , 1 9 6 3 ) , pp. 53-55 . 62 Br i t i s h Columbia, and on the estimates of personnel in the industry. A cost of $2.00 per 100 cubic feet of log chipped was adopted for a l l logs included in the model, see Figure I . Recombination Activity Unit-level Returns. These activities represent plywood production and pulp production,and use the intermediate inputs, veneer and chips. The unit-level returns of these activities are based on recent market prices, (a) Plywood Production Activity Returns. Unit-level returns for the pure Douglas-fir plywoods are based on November, 1963 prices quoted by a major Van couver producer for 3/8 inch panels. The prices were con verted to a 100 cubic foot plywood basis to correspond to the units used in the veneer-peeling a c t i v i t i e s . See Figure I . Table 5.7 shows the price data used for calculating unit-level returns for the Douglas f i r , hemlock, spruce and mixed-specie plywoods in the model. The hemlock, spruce and mixed-specie price data are quite arbitrary. The unit-level return of each plywood-production activity was calculated by subtracting plywood manufactur ing cost from market price. Manufacturing cost is the cost of a l l stages of plywood production beyond the veneer-sel ling stage (i. e . , after the veneer has been peeled and chipped into standard sizes suitable for shipping or further manu facture) . The veneer-peeling a c t i v i t i e s , as explained ear l i e r , incorporate manufacturing costs up to the green-veneer 63 selling stage. Plywood-production costs are taken from published data (which unfortunately was several years old) and from 61 the estimates of experts. Costs incurred in the various stages of plywood production are expressed as percentages of sales price in Table 5 .7 . From these, the unit-level return values in Table 5.6 are calculated by deducting plywood manufacturing costs comprising drying, lay-up and pressing, trimming, patching, sanding (where applicable) and shipping costs from the market price, (b) Pulp Production Activity Returns. Pulp production unit-level returns in the model are calculated by deducting pulp production costs, exclud ing the cost of chips, from recent market prices. The pulp-manufacturing cost adopted in the calculation is 14-3.00 per air-dry short ton of pulp, for a l l pulp-pro duction activities in the model. This is an average figure for typical kraft pulp mills in the coastal region, based on 62 estimates by an expert in the f i e l d . The market prices for pulp are similarily based on well-informed estimates. Since there is no large open market for this product, these f i g  ures are no more than estimates of recent average 61 to'. E. Mayhew, "A New Method of Allocating Costs to Veneer by Grades," For. Prod. Journ., v. 8 , (Aug. 1 9 5 8 ) , pp. 27A-30A. 62 Ian Hudson, Senior Development Engineer, Sandwell & Co. Ltd., Consulting Engineers, Vancouver, B.C. 6 4 p r i c e s . ^ Most p u l p s produced i n N o r t h America are s o l d i n the U n i t e d S t a t e s a t a u n i f o r m d e l i v e r e d - p r i c e anywhere i n t h e c o u n t r y . P u l p p r i c e s a t the m i l l a re t h e r e f o r e g r e a t l y 6 U i n f l u e n c e d by t h e p r o x i m i t y of the m i l l s ' customers. F o r the purpose o f t h i s a n a l y s i s an average m i l l - n e t p r i c e of $115.00 per a i r - d r y s h o r t - t o n o f p u l p i s adopted. U s i n g t h i s v a l u e as a base, market p r i c e s a r e e s t i m a t e d f o r each of the s e v e r a l p u l p s i n the model. See T a b l e 5.8. S e l l i n g A c t i v i t y U n i t - l e v e l R e t u r n s . The s e l l i n g a c t i v i t y u n i t - l e v e l r e t u r n s are s i m p l y t a k e n from r e c e n t market p r i c e s , or e s t i m a t e s where market p r i c e s are not a v a i l a b l e . U n l i k e the u n i t - l e v e l r e t u r n s o f plywood and p u l p - p r o d u c t i o n a c t i v i t i e s d i s c u s s e d i n the p r e v i o u s s e c t i o n , a c t u a l market d a t a a r e much more r e a d i l y a v a i l a b l e i n p u b l i s h e d form, (a) L o g - s e l l i n g A c t i v i t y R e t u r n s . The u n i t - l e v e l r e t u r n s f o r l o g - s e l l i n g a c t i v i t i e s a r e based on c u r r e n t l o g p r i c e s p u b l i s h e d i n the B r i t i s h Colum- 6 5 b i a Lumberman. S i n c e most l o g p r i c e s i n t h i s r e f e r e n c e 6 3 Most wood p u l p i n N o r t h America i s s o l d under c o n t r a c t a t p r i c e s announced q u a r t e r l y by the p r o d u c e r s . Most s a l e s a r e conducted t h r o u g h agents whose s t a n d a r d f e e i s t h r e e p e r  cent of s e l l i n g p r i c e , l e s s f r e i g h t , u n l e s s s a l e s a r e t o a c a p t i v e market i n w h i c h case commissions are not p a i d . 6 4 F o r example, a p u l p m i l l i n the c o a s t a l r e g i o n f a c e s about a t e n d o l l a r d i f f e r e n t i a l i n i t s m i l l - n e t p r i c e s be tween E a s t e r n and Western U n i t e d S t a t e s m a r k e t s ; and up t o twenty d o l l a r s d i f f e r e n t i a l between E a s t e r n U n i t e d S t a t e s and o t h e r w o r l d markets where f r e i g h t c o s t s , p a i d by the p r o  d u c e r , c u t more s e v e r e l y i n t o m i l l - n e t p r i c e . 65 R o b e r t S c h u l t z & Co. L t d . , Log P r i c e R e p o r t , B r i t i s h  C olumbia Lumberman, v. 47 ( O c t o b e r , 1963)? p. 80. show a $10,00 spread between h i g h e s t and l o w e s t p r i c e s p a i d , the average or median o f the p r i c e s shown was s e l e c  t e d f o r the model. See T a b l e 5.9. A c o n v e r s i o n f a c t o r o f s i x feet-board-measure l o g s c a l e e q u a l s one c u b i c f o o t , i s used t o c o n v e r t l o g v a l u e s t o the u n i t s used i n the m o d e l . ^ (b) L u m b e r - s e l l i n g A c t i v i t y R e t u r n s . U n i t - l e v e l r e t u r n s f o r l u m b e r - s e l l i n g a c t i v i t i e s a r e based on r e t a i l p r i c e s quoted by a major p r o d u c e r i n t h e c o a s t a l r e g i o n . See Ta b l e 5.10. ( c ) V e n e e r - s e l l i n g A c t i v i t y R e t u r n s . M a r k e t - p r i c e d a t a f o r veneers i s d i f f i c u l t t o o b t a i n a l t h o u g h t h i s p r o d u c t i s marketed t o a l i m i t e d e x t e n t i n t h e U n i t e d S t a t e s . The u s e f u l n e s s of the a v a i l a b l e d a t a i s f r e q u e n t l y l i m i t e d by the g r o u p i n g o f veneer grades under s i n g l e average p r i c e s . Moreover, hemlock and spruce veneers are v e r y r a r e l y s o l d as such. Most o f t h e d a t a i n T a b l e 5.11 a r e e s t i m a t e s , p a r t i c u l a r l y t h e v a l u e s f o r hemlock and s p r u c e . N e i t h e r o f t h e s e l a t t e r s p e c i e s i s used f o r f a c e - veneer a t p r e s e n t , so t h a t t h e &. and B grade p r i c e s a re assumed t o be e q u i v a l e n t t o Douglas f i r C and D g r a d e s , w h i c h a r e used e x c l u s i v e l y as plywood " c o r e " m a t e r i a l . 66 6 FBM l o g s c a l e = 1 c u b i c f o o t i s the g e n e r a l l y a c c e p t e d c o n v e r s i o n f a c t o r f o r t h e c o a s t a l r e g i o n . I n the i n t e r i o r o f B.C., an o f f i c i a l c o n v e r s i o n f a c t o r o f 5.75 FBM l o g s c a l e = 1 c u b i c f o o t i s used. See C o n v e r s i o n  F a c t o r s f o r Pac. N.W. F o r . P r o d u c t s , I n s t i t u t e o f F o r e s t P r o d u c t s , S t a t e o f Washington, j O j Andersen H a l l , S e a t t l e 5, Washington, 1957. 66 (d) Chip-selling Activity Returns. The unit-level returns of chip-selling activities are based on chip prices recently paid in the Vancouver chip market. These prices are available on a "unit" 67 basis, and a conversion factor of 'one unit equals 67 cubic feet solid wood' is used to convert the date to cub ic feet, to meet the requirements of the model. See Table 5.12. III. RESTRAINTS A distinction has already been drawn between the primary input restraints, which comprise the available sup ply of logs, and the intermediate input restraints, which stipulate that a l l intermediate products (lumber, veneer and chips) are to be completely consumed as input to f i n a l a c t i v i t i e s . Since intermediate products are entirely con sumed by f i n a l activities i t follows that the values of the intermediate restraints must be zero. It remains there fore to discuss the sources of data for the primary input restraints. 67 A "unit" of chips is defined as 200 cubic feet gross, volume of uncorapacted chips. The stipulation 'uncompacted' is indefinite, for the several methods of storing and trans porting chips lead to various degrees of compaction. A more rigorous quantification that is finding increasing ac ceptance is the "bone-dry unit," defined as the quantity of pulp chips which w i l l weigh 24-00 pounds in an oven-dry condition. 67 \ The log supply data are based on the annual log scale of the British Columbia coastal region forest industry. Log scale data for each specie in the model are available in the 1962 Annual Report of the British Columbia Forest Service. In addition, a breakdown of the log scale by log-grades was obtained from industry personnel and statisticians. The log supply data are summarized in Table 5.1. IV. METHOD OF SOLUTION Elementary linear programmes involving only a few activities and restraints can be solved by hand using the 68 simplex method. More complicated programmes require ma chines to handle the very large quantities of data involved. A machine solution of the log-allocation model presented in this thesis was carried out on the IBM 1620 d i g i t a l com puter in the computing centre of the University of B r i t i s h Columbia. In this chapter we shall b r i e f l y describe the format and special features of the machine programme employed in obtaining a solution to the model. Three variations of the model that were solved to demonstrate i t s practical appli cation w i l l then be outlined. Results of these solutions w i l l be presented and discussed later in chapter six. IBM 1620 Computer Programme. The machine programme used is identified as: 68 M. J. Baumol, Economic Theory and Operations Analysis, Prentice Hall Inc., Cal i f . , (June, 1962) , ch. !?. 68 "1620 Linear Programming Code for Card Input/Output," by- N icho l s , Nicke l and Davis. Procedural de ta i l s and mathematical elements of the programme are f u l l y discussed in the above reference, and need not be repeated here. However, some important propert ies and cha rac te r i s t i c s of the programme which are of in te res t to th i s pa r t i cu l a r app l i c a t i on , fo l low. Spec ia l Features of the Machine Programme. The programme i s designed fo r input from cards. All output i s on the typewriter although an opt ional f i n a l - matrix punch-out on cards i s ava i l ab l e . The s ize of problem which can be handled by th i s programme i s l imi ted by memory- capacity of the computer according to the fo l lowing r e l a t i o n  sh ip . (m + 2 ) (n + 3) * memory - 3920 10 where: m i s the number of r e s t r i c t i o n s , n i s the number of ' r e a l ' a c t i v i t i e s , and 'memory' i s equal to 40,000 for the par  t i c u l a r machine i n the Un ivers i t y comput ing centre. The term ' r e a l ' a c t i v i t i e s refers to those other than the ' d i sposa l ' a c t i v i t i e s . The l a t t e r are a c t i v i t i e s de signed to convert the inequa l i t y r es t r a in t expressions into e q u a l i t i e s . For the intermediate input r e s t r a i n t s , s e l l i n g a c t i v i t i e s were included as the corresponding d isposa l ac  t i v i t i e s and for computational purposes these s e l l i n g 69 a c t i v i t i e s cannot t h e r e f o r e be c o n s i d e r e d as ' r e a l ' , i n the m a t h e m a t i c a l sense. F o r the l o g - a l l o c a t i o n problem shown i n F i g u r e I t h e r e a r e m = 37 r e s t r a i n t s and n = 57 ' r e a l ' a c t i v i t i e s , t o g i v e a v a l u e o f 2340 f o r t h e l e f t - h a n d s i d e o f the c a p a c i t y e x p r e s s i o n above. The v a l u e o f the r i g h t - hand s i d e o f the e x p r e s s i o n i s 3908. There are t h r e e s p e c i a l f e a t u r e s o f t h i s computer programme wh i c h make i t q u i t e v e r s a t i l e , and p a r t i c u l a r l y u s e f u l t o the l o g - a l l o c a t i o n problem under d i s c u s s i o n . ( i ) C o s t o r p r i c e changes f o r t h e v a r i o u s a c t i v i t i e s , i n terms o f the o r i g i n a l d a t a , can be made w i t h o u t h a v i n g t o r e l o a d or r e - s o l v e the o r i g i n a l m a t r i x . These changes are i n s t i t u t e d by i n s e r t i n g a s p e c i a l deck of c a r d s , r e f e r r e d t o as the 'Cost Changer' deck, i n t o the o v e r - a l l programme, u s u a l l y behind the o r i g i n a l - d a t a c a r d s . When e n t e r i n g the programme i n t o the machine a sto p i s ex e c u t e d a f t e r t h e C o s t Changer e n t r y and the r e q u i r e d c o s t change ( p r i c e change) d a t a can th e n be e n t e r e d e i t h e r by c a r d i n p u t or by th e t y p e w r i t e r . An u n l i m i t e d number o f c o s t changes may be made i n any s i n g l e r u n . ( i i ) The r e s t r a i n t s of the problem can a l s o be v a r i e d w i t h  out r e l o a d i n g o r r e - s o l v i n g t h e o r i g i n a l m a t r i x . These changes a r e a c h i e v e d i n e x a c t l y t h e same manner as c o s t changes, by i n s e r t i n g a s p e c i a l deck of c a r d s r e f e r r e d t o as the ' R H 3 Changer', ( r i g h t - h a n d s i d e changer) deck. Data f o r RHS changes may be e n t e r e d by c a r d s o r by t y p e w r i t e r f o r an u n l i m i t e d number of changes i n any s i n g l e r u n . B o t h 70 c o s t and RHS Changer decks can be i n c l u d e d i n the programme, ( i i i ) The o v e r - a l l programme c o n s i s t s o f s e v e r a l r e l a t i v e  l y independent sub-programmes w h i c h can be d e l e t e d or a l t e r  ed i n many r e s p e c t s w i t h o u t i n t e r f e r e n c e w i t h o t h e r sub- programmes. S e q u e n t i a l l o a d i n g of the sub-programmes i s a u t o m a t i c . T h i s f e a t u r e e n a b l e s a s k i l l e d computor p r o  grammer t o m a n i p u l a t e the v a r i o u s sub-programmes t o make ad j u s t m e n t s i n the a c t u a l s o l u t i o n p r o c e d u r e . Programme Output-Format. I f d e s i r e d , the r e s u l t s of each i t e r a t i o n i n the s o l u t i o n of the problem may be t y p e d out as p a r t o f the o u t  p u t . A l t e r n a t e l y , the c o u r s e o f the s o l u t i o n may be moni t o r e d by s e t t i n g an a p p r o p r i a t e s w i t c h on and o f f p e r i o d  i c a l l y . The i n f o r m a t i o n m o n i t o r e d f o r each i t e r a t i o n con s i s t s o f the i t e r a t i o n number, v a l u e o f t h e o b j e c t i v e f u n c t i o n , t h e p a r t i c u l a r v a r i a b l e ( a c t i v i t y ) removed from the p r e v i o u s s o l u t i o n , and the p a r t i c u l a r v a r i a b l e i n s e r t e d i n t o the l a t e s t s o l u t i o n . The v a l u e o f the o b j e c t i v e f u n c t i o n i n c r e a s e s ( i n a ma x i m i z i n g problem) w i t h each s u c c e s s i v e i t e r a t i o n up t o th e f i n a l or o p t i m a l v a l u e . When the o p t i m a l s o l u t i o n has been o b t a i n e d a F i n a l B a s i s Output i s e i t h e r punched on c a r d s or typed o u t , c o n t a i n i n g the f o l l o w i n g i n f o r m a t i o n : ( i ) The f i n a l v a l u e o f the o b j e c t i v e f u n c t i o n . ( i i ) The i d e n t i f i c a t i o n number and i n p u t p r i c e ( c o s t ) of each a c t i v i t y i n t h e F i n a l B a s i s . (These a c t i v i t i e s c o r  respond t o the ones t o be employed t o r e a c h an optimum 71 v a l u e of the o b j e c t i v e f u n c t i o n ) . ( i i i ) The " l e v e l " a t wh i c h each a c t i v i t y i n the F i n a l B a s i s i s t o be o p e r a t e d , e x p r e s s e d as a m u l t i p l e o f the a c t i v i t y ' s " u n i t - l e v e l " d e f i n e d i n the o r i g i n a l problem. ( i v ) The upper and lower l i m i t s t o the l e v e l o f each a c t i v  i t y i n the F i n a l B a s i s , beyond w h i c h the a c t i v i t y would be dropped from the F i n a l B a s i s and r e p l a c e d w i t h a F i n a l Non- B a s i s a c t i v i t y . (v) The F i n a l Non-Basis a c t i v i t i e s w h i c h would e n t e r t h e F i n a l B a s i s i n the event t h a t one o f the F i n a l B a s i s a c t i v i t i e s exceeded one of i t s l i m i t s . The remainder o f the o p t i m a l s o l u t i o n c o n s i s t s o f a F i n a l Non-Basis Output c o n t a i n i n g the f o l l o w i n g i n f o r m a t i o n . ( v i ) The i d e n t i f i c a t i o n number and i n p u t p r i c e ( c o s t ) o f each a c t i v i t y i n the F i n a l N o n - B a s i s . ( v i i ) The "shadow p r i c e " o f each n o n - b a s i s a c t i v i t y . T h i s v a l u e r e p r e s e n t s t h e p e n a l t y t o t h e whole system i f a u n i t o f t h i s a c t i v i t y i s f o r c e d i n t o the f i n a l s o l u t i o n . I n o t h e r words, i f a n o n - b a s i s a c t i v i t y ( e x c l u d e d from the o p t i m a l p l a n ) i s i n f a c t used or f o r c e d i n t o the f i n a l p l a n o f a c t i v i t i e s ( c o n t r a r y t o o p t i m a l c o n d i t i o n s ) i t w i l l i n c u r a l o s s t o the system e q u a l t o i t s shadow p r i c e per u n i t o f the a c t i v i t y used. ( v i i i ) The upper and l o w e r l i m i t s t o the l e v e l a t which t h i s a c t i v i t y c o u l d be f o r c e d i n t o t h e s o l u t i o n . As more and more o f t h i s a c t i v i t y i s used the l o s s i n c u r r e d goes up, u n t i l e v e n t u a l l y some o t h e r a c t i v i t y i n the F i n a l B a s i s i s f o r c e d o u t . 72 ( i x ) The F i n a l B a s i s v a r i a b l e s w h i c h would l e a v e the s o l u  t i o n (be f o r c e d out) f o l l o w i n g the events of ( v i i i ) above. A punch-out on c a r d s o f the complete f i n a l m a t r i x i s o p t i o n a l . When o b t a i n e d , the m a t r i x , deck punch-out comes complete w i t h c o n t r o l c a r d s , i n a form a t s u i t a b l e f o r d i r e c t r e l o a d i n g by the normal Data Loader r o u t i n e . T r a n s f o r m a t i o n o f R e s u l t s . The output format o f t h e s o l u t i o n programme, o u t  l i n e d above, i s d i f f i c u l t t o i n t e r p r e t . An a u x i l i a r y p r o  gramme was w r i t t e n t h e r e f o r e t o t r a n s f o r m the output d a t a i n t o a form more c o n s i s t e n t w i t h t h e l o g - a l l o c a t i o n model and more e a s i l y u n d e r s t o o d by t h e i n t e r e s t e d r e a d e r . U s i n g the f i n a l s o l u t i o n as i n p u t , on c a r d s , t h i s programme a d j u s t s a c t i v i t y l e v e l s t o a h u n d r e d - c u b i c - f o o t basis,computes t o t a l r e t u r n f o r each a c t i v i t y and d i s c a r d s t h e F i n a l Mom-Basis v a r i a b l e - l i m i t s . The s o l u t i o n d a t a i s c o l l e c t e d and r e a r r a n g e d i n t o a more r e a d a b l e form. M a x i  mum v a l u e o f the o b j e c t i v e f u n c t i o n and number of i t e r a t i o n s r e q u i r e d t o r e a c h a s o l u t i o n a r e shown, and each column o f f i g u r e s i s g i v e n a c l e a r l y u n d e r s t o o d heading i n c l u d i n g the a p p r o p r i a t e u n i t s . A sample o f the t r a n s f o r m a t i o n p r o  gramme output i s shown i n Appendix I t o t h i s t h e s i s . V. SOLUTION OF LINEAR PROGRAMME Three v a r i a t i o n s o f the l o g - a l l o c a t i o n l i n e a r p r o  gramme model p r e s e n t e d i n t h i s t h e s i s were s o l v e d , as f o l l o w s : A P P E N D I X T 0 H A P T E R F I V E 73 Case I . Using the data presented i n chapter f i v e , as shown i n Figure I . Case I I . Using the same data as i n Case I but with modified a c t i v i t y c o e f f i c i e n t s f o r the No. 3 hemlock log-sawing a c t i v i t y . Case I I I . Using the same data as i n Case I I but wi t h a modified l o g supply. D e t a i l s of the m o d i f i c a t i o n s i n Case I I and Case I I I are shown i n Tables 5.13 and 5.14 r e s p e c t i v e l y . Apart from demonstrating t h a t the Cost Changer sub- programme works e f f e c t i v e l y , these m o d i f i c a t i o n s i n Case I I and Case I I I provide a u s e f u l i n s i g h t i n t o p r a c t i c a l a p p l i  c a t i o n s of the model, which we s h a l l discuss i n the next chapter. TABLE 5 . 1 7 4 THE LOG SUPPLY IN THE COASTAL REGION OF BRITISH COLUMBIA 1962 PERCENTAGE DISTRIBUTION BY GRADES AND AVAILABLE TOTAL SUPPLY IN MILLIONS OF CUBIC FEET Log Description * Distribution (percent) Available Supply (million cubic feet) Douglas F i r # 1 peeler 3 6 . 1 .t u # 2 " 7 1 4 . 2 .i # 3 14 2 8 . 4 II it #k II 9 1 8 . 3 " 11 # 2 sawlog 2 3 46 . 7 .. ., # 3 it 4 4 8 9 . 5 Total Douglas F i r : 1 0 0 • 2 0 3 . 2 Hemlock # 1 sawlog 5 1 3 . 8 it # 2 " 1 8 5 0 . 0 # 3 7 7 2 1 3 . 6 Total Hemlock: 1 0 0 2 7 7 . 4 Spruce # 1 sawlog 4 1 . 3 II # 2 " 3 5 1 1 . 4 " # 3 " 61 1 9 . 8 Total Spruce 1 0 0 3 2 . 5 Sources: Distribution based on data by industry statis ticians. See text pp. **Available supply on data from: Annual Report, BCFS. TABLE 5.2 LOG SAWING ACTIVITY TECHNOLOGICAL COEFFICIENTS PERCENT YIELD OF LUMBER, CHIPS AND HOG FUEL Douglas F i r Hemlock Spruce Log peelers sawlogs sawlogs sawlogs Output \^ # 1 #2 #3 % *2 # 2 #3 *2 # 3 Lumber: Clear 38 32 18 11 11 k 22 11 5 33 16 7 Sel/Merch. 13 14 21 17 17 19 16 16 10 14 17 12 Const/std. 15 15 23 29 29 32 26 27 26 12 26 28 Util/Econ. 6 10 8 9 9 9 6 9 13 8 6 8 Chips: 13 lk 16 22 20 22 15 23 32 18 20 25 Hog Fuel: 15 15 lk 13 lk 14 15 14 14 15 15 14 Source: See Text pp. 51-54. TABLE 5 . 3 LOG-PEELING ACTIVITY COEFFICIENTS PERCENT YIELDS OF GREEN VENEER, CHIPS AND HOG FUEL Douglas F i r Hemlock Spruce \ L o g p e e l e r s sawlogs sawlogs sawlogs Output \ v # 1 # 2 # 3 # k # 2 # 3 # 1 # # 2 3 # 1 # # 2 3 A 3 0 2 4 1 7 1 0 6 - 1 2 6 1 2 6 B 8 8 9 1 0 1 2 - 1 2 1 2 1 2 1 2 C 2 3 2 8 3 3 3 8 3 9 - 2 9 3 4 2 9 3>+ C h i p s 2 2 2 1 2 0 1 9 1 8 - 2 0 2 0 2 0 2 0 Hog F u e l 1 7 1 9 2 1 2 3 2 5 - 2 7 2 8 2 7 2 8 S h r i n k a g e 8 8 8 8 8 - 8 8 8 8 S o u r c e : See T e x t pp. 5 4 - 5 6 . TABLE 5.4 PLYWOOD PRODUCTION ACTIVITY COEFFICIENTS CUBIC FEET OF GREEN VENEER REQUIRED PER CUBIC FOOT OF FINISHED PLYWOOD \ P i y - \woods Ve- \ ne e r s \ ^ G2S G/Sol G1S S2S SIS Sh'g 1 2 1 2 F/H/S F/H F/S H/S S/H Douglas F i r - A " B 11 C .776 .388 .388 - .388 - .776 .388 - .504 .504 .892 .504 .892 1.090 .388 - .776 .776 - Hem l o c k A " B " C .388 .388 .776 .504 .504 - .388 - .388 - .504 - - .892 Spruce A " B C .388 - .388 .776 .504 .504 - .388 .504 - - - - - .504 .892 F i g u r e s on a g r e e n veneer b a s i s a l l o w i n g average lh% sanding l o s s (where a p p r o p r i a t e ) and 8% s h r i n k a g e l o s s i n d r y i n g . See t e x t pp. 56-58. S o u r c e : Douglas f i r plywoods - C a n a d i a n S t a n d a r d s A s s o c i a t i o n P u b l i c a t i o n , CSA 0121 - 1961. TABLE 5 - 5 PULP PRODUCTION ACTIVITY COEFFICIENTS CUBIC FEET OF CHIPS (SOLID VOLUME) PER AIR-DRY TON OF PULP Chip Specie Yield of Pulp (*) Density (lb/CF)* Wood Conspt 1n (CF/ADT) " Chip Compositions for each Pulp F H 40/60 F H 5 0 / 5 0 F H S 3 0 / 5 0 / 2 0 F H S 4 5 / 4 5 / 1 0 F 100 H 100 S 100 Douglas F i r 4 2 . 3 28.0 1 5 2 6 3 . 5 79.0 49.0 7 2 . 0 1 5 2 — — Hemlock 41.4 2 6 . 5 164 9 5 . 5 79.0 8 1 . 5 7 2 . 0 - 164 - Spruce 41.4 24.0 181 - - 3 2 . 8 16.0 - - 181 Moisture-free weight, green volume. ADT = Air Dry Ton Sources: 1. Wood Properties - Canadian Woods - Their Properties and Uses, Appendix Tables, Queen's Printers, Ottawa. 2. Chip Compositions - data supplied by industry personnel. (See text pp. 58 TABLE 5.6 PLYWOOD MANUFACTURING COSTS AS PERCENT OF SALES PRICE P r o d u c t i o n Step C o s t % 1. Logs 47 2. P e e l i n g 10 3. D r y i n g 9 4. Lay-up and P r e s s i n g 15 5. Trimming 2 6. P a n e l P a t c h i n g 7 7. P a n e l Sanding 6 8. Warehouse & S h i p p i n g 4 100% S o u r c e : See Text pp. 62-63. TABLE 5.7 PLYWOOD PRODUCTION ACTIVITY UNIT-LEVEL RETURNS CALCULATION OF UNIT-LEVEL RETURNS FROM MARKET PRICE & MANUFACTURING COST DATA Plywood D e s c r i p t i o n Market P r i c e Manufacturing Costs $/cu.ft. 3/8" plywood U n i t - l e v e l Return $/cu.ft. 3/8" plywood Species L Grade $ per c u . f t . Doug.fir i 1 G2S ^ 5.00 2.40 2.60 I I u G/Sol 4 . 5 0 2.40 2.10 it I I G13 4.00 1.80 2.20 II it S2S 4.40 2.00 2.40 tt I I SIS 3.90 1.80 2.10 it it Sheath ing 2.30 1.00 1 .30 Hemlock - 3.00 2.40 0.60 u - 2.50 2.00 0.50 Spruce - 3.00 2.40 0.60 it - 2 .50 2.00 0.50 Fir/Hem /Spruce - 4.00 1.80 2.20 Fir/Hem - 4.30 2.00 2.30 F i r / S p r c - 4 . 2 5 2.00 2.25 Hem/Sprc - 2.80 1 .80 1.00 Sprc/Hem - 2 .90 1.80 1.10 See t e x t pp. 6 2 - 6 3 . 8 1 TABLE 5 . 8 PULP PRODUCTION ACTIVITY UNIT-LEVEL RETURNS CALCULATION OF UNIT-LEVEL RETURNS FROM MARKET PRICES AND MANUFACTURING COST DATA Ch i p M i x Manu A c t i v i t y ( c u . f t . p e r Assumed f a c t u r i n g U n i t - l e v e l ADT p u l p * ) P r i c e C o s t R e t u r n P u l p D.F. Hem. Spruce $/ADT p u l p $/ADT p u l p $/ADT p u l p a 64 9 5 1 1 5 . 0 0 4 3 . 0 0 7 2 . 0 0 b 7 9 7 9 - 1 1 5 . 0 0 4 3 . 0 0 7 2 . 0 0 c 4 9 81 3 3 ' 1 2 5 . 0 0 4 3 . 0 0 8 2 . 0 0 d 7 2 7 2 16 1 2 5 . 0 0 4 3 . 0 0 8 2 . 0 0 e 1 5 2 - - 1 1 0 . 0 0 4 3 . 0 0 6 7 . 0 0 f - 164 - 1 1 0 . 0 0 4 3 . 0 0 6 7 . 0 0 g - - 1 8 1 1 1 0 . 0 0 4 3 . 0 0 6 7 . 0 0 ADT = a i r d r y t o n S o u r c e ; See t e x t pp. 63-64. TABLE 5 . 9 LOG SELLING ACTIVITY UNIT-LEVEL RETURNS Log D e s c r i p t i o n Market P r i c e Dec. 1 9 6 3 $/MBM Log S c a l e A c t i v i t y U n i t - L e v e l R e t u r n $ / 1 0 0 c u . f t . Douglas F i r # 1 p e e l e r 1 2 0 . 0 0 7 2 . 0 0 II it # 2 • » 1 1 0 . 0 0 6 6 . 0 0 ii ii # 3 " 1 0 0 . 0 0 6 0 . 0 0 ti II # 4 " 9 0 . 0 0 5 4 . 0 0 ti II # 2 sawlog 7 0 . 0 0 4 2 . 0 0 it II # 3 " 6 0 . 0 0 3 6 . 0 0 Hemlock; # 1 sawlog 5 5 . 0 0 3 3 . 0 0 # 2 n 5 0 . 0 0 3 0 . 0 0 # 3 II 4 5 . 0 0 2 7 . 0 0 Spruce # 1 sawlog 7 0 . 0 0 4 2 . 0 0 " # 2 it 5 5 . o o 3 3 . 0 0 " # 3 ii 4 5 . o o 2 7 . 0 0 S o u r c e : See t e x t pp. 64-65. TABLE 5.10 LUMBER SELLING ACTIVITY UNIT-LEVEL RETURNS S p e c i e Grade Market P r i c e $/MBM A c t i v i t y U n i t - L e v e l R e t u r n $/CCF lumber Douglas F i r C l e a r 195.00 235.00 Sel/Mer 110.00 132.00 Cons/Std 95.00 11H - .00 U t i l / E c o n 35.00 42.00 Hemlock C l e a r 120.00 144.00 Sel/Mer 85.00 102.00 Cons/Std 75.00 90.00 U t i l / E c o n 35.00 42.00 Spruce C l e a r 225.00 270.00 Sel/Mer 80.00 96.00 Cons/Std 70.00 84.00 U t i l / E c o n 35.00 42.00 S o u r c e : See Text p. 65. 84 TABLE 5.11 VENEER SELLING ACTIVITY UNIT-LEVEL RETURNS S p e c i e Veneer Grade Market P r i c e , ($/M/l6ths)-* • A c t i v i t y U n i t - L e v e l R e t u r n $/100 c u . f t . v e n e e r Douglas F i r A , B 9.00 173.00 ti II C 5.75 110.00 Hemlock A , B 5.75 110.00 u C 4.50 86.50 S p r u c e A ,B 5.75 110.00 it C 4.50 86.50 * 1 1 M/I6ths = 1000 x l 6 x 12 c u b i c f e e t . S o u r c e : See t e x t pp. 65-66. TABLE 5.12 PULP CHIP SELLING ACTIVITY UNIT-LEVEL RETURNS Specie Market Price 1/200 cu.ft."unit" Activity Unit-Level Return 1/100 cu.ft.of chips Douglas F i r 12.00 18.00 Hemlock 14.00 21.00 Spruce 14.00 21.00 Source: See Text p. 66. TABLE 5.13 MODIFIED COEFFICIENTS OF ACTIVITY NO. 9; FOR CASE II Modified sawing activity coefficients for No. 3 hem lock log. VOLUME OF PRODUCT PER CUBIC FOOT OF LOG CONSUMED Original ^ Coefficients (Case I) Modified ^ Coefficients (Case II and Case III) Lumber: Clear 0.05 0.02 Sel/mer 0.10 0.05 Coast/std 0.26 0.14 util/econ 0.13 0.12 Chips 0.32 0.40 Hog Fuel 0.14 0.27 * Industry-average data. **Arbitrarily assumed coefficients. TABLE 5.1k VOLUME OF LOGS IN MILLION CUBIC FEET, BY SPECIES AND GRADE Log Descr ip t ion Or ig ina l Log Supply (Case I and Case II) Modif ied * Log Supply (Case III) Douglas F i r # 1 peeler 61 60 II # 2 » 142 140 ii # 3 .. 2 8 4 3 0 0 .. #k II 1 8 3 2 5 0 " " # 2 sawlog 4 6 7 5 5 0 II # 3 a 8 9 5 1 0 0 0 Hemlock # 1 sawlog 1 3 8 1 5 0 " # 2 " 5 0 0 7 5 0 " # 3 11 2 1 3 6 1 3 5 0 Spruce # 1 sawlog 1 3 5 0 # 2 » 1 1 4 2 0 0 # 3 " 1 9 8 3 0 0 Coasta l region data , 1962. (See text pp. 67 * * A r b i t r a r i l y assumed log supply. 88 CHAPTER VI RESULTS AND DISCUSSION I. RESULTS The IBM 1620 computer programme outlined in chapter five performed smoothly and no d i f f i c u l t y was encountered in obtain ing a solution. Case I required 28 iterations to reach an opti mal solution; running time for the main solution programme was approximately twelve minutes with iteration times ranging from ten seconds to 45 seconds per iteration. The solution and f i n a l matrix for Case I were punched out on cards. Solutions for Case II and Case III were obtained more easily, since i t was quite possible to start with the Case I f i n a l matrix, in sert the required modifications via the appropriate changer- programme, and reach an optimal solution in two or three i t e r  ations. As described in chapter five the f i n a l solution for each case was obtained on cards, rather than typed-out, so that i t could be used as input to the solution-transformation pro gramme. A sample of the transformed data is presented in Appendix I to this thesis. For each case, an optimum solution to the model comprised a maximum value for the objective function and a l i s t i n g of the f i n a l basis and non-basis variables (or a c t i v i t i e s ) . The f i  nal basis variables specify optimum activities and activity levels, necessary to achieve a maximum value for the objective function. The non-basis variables represent activities ex cluded from an optimal solution. 39 V a r i a b l e s numbered from one t o 37 i n c l u s i v e , c o r r e s p o n d t o i n t e r m e d i a t e a c t i v i t i e s i n the model, see F i g u r e I . Those s p e c i f i e d among the f i n a l b a s i s v a r i a b l e s comprise the o p t i  mum l o g - a l l o c a t i o n p l a n f o r the p a r t i c u l a r c a s e . The r emain i n g v a r i a b l e s , numbered from 33 t o 9 L i n c l u s i v e , c o r r e s p o n d t o f i n a l a c t i v i t i e s , and those s p e c i f i e d among the f i n a l b a s i s v a r i a b l e s comprise the optimum mix o f f i n a l p r o d u c t s produced i n the p a r t i c u l a r c a s e . As e x p l a i n e d i n c h a p t e r f i v e , the programme s o l u t i o n i n c l u d e s upper and l o w e r l i m i t s t o the u n i t - l e v e l r e t u r n f o r each f i n a l b a s i s v a r i a b l e , and s p e c i f i e s t h e f i n a l n o n - b a s i s v a r i a b l e t h a t w i l l e n t e r the s o l u t i o n when e i t h e r l i m i t i s exceeded. C o m p i l e d R e s u l t s To e v a l u a t e the r e s u l t s from each case i t w i l l be u s e  f u l t o c o m p i l e the s o l u t i o n d a t a i n t a b l e s , shown below, s e t t i n g f o r t h comparisons between the optimum l o g a l l o c a t i o n s , f i n a l p r o d u c t m i x e s , s a l e s volumes, e t c . T a b l e 6.1 shows the t o t a l net r e t u r n f o r cases I , I I and I I I by each u t i l i z a t i o n c a t e g o r y , namely lumber, plywood and p u l p p r o d u c t i o n . These net r e t u r n f i g u r e s r e p r e s e n t g r o s s r e t u r n s , l e s s m a n u f a c t u r i n g c o s t s , e x c l u d i n g the c o s t o f l o g s . I t s h o u l d be noted t h a t the p u l p p r o d u c t i o n f i g u r e s do not i n c l u d e a d e d u c t i o n f o r the c o s t o f c h i p s from s a w m i l l and p e e l i n g a c t i v i t i e s . S i m i l a r l y , t h e lumber p r o d u c t i o n and p l y  wood p r o d u c t i o n r e t u r n s do not i n c l u d e r e t u r n s from the s a l e of c h i p s t o p u l p p r o d u c t i o n a c t i v i t i e s . The optimum l o g - a l l o c a t i o n p l a n s f o r each case are shown 90 i n Table 6.2, as the percentage of the supply of each type of l o g that i s consumed i n each u t i l i z a t i o n category. F i n a l l y , Tables 6.3, 6.4 and 6.5 show the f i n a l out put of lumber, plywood and pulp, r e s p e c t i v e l y , f o r each case. These f i g u r e s i n d i c a t e the type and q u a n t i t y of each f i n a l product (lumber, plywood, veneer, pulp, and chips) that would be produced w i t h an optimum a l l o c a t i o n of l o g s , i . e . , at a maximum value f o r the o b j e c t i v e f u n c t i o n . D i s c u s s i o n of R e s u l t s . The p r a c t i c a l s i g n i f i c a n c e of these r e s u l t s i s l i m i t e d by the s i m p l i f y i n g assumptions a p p l i e d i n the c o n s t r u c t i o n of the model. I t i s p o s s i b l e nevertheless to observe some r e l e v a n c i e s to the r e a l world. For example, according to the model ( a l l cases) no logs should be s o l d ; i t i s more p r o f i t a b l e to convert the t o t a l given log supply than s e l l any part of i t . S i m i l a r l y , no chips should be s o l d , i n c l u d i n g those produced from sawmill and plywood m i l l waste. Since the model i s based on the e n t i r e l o g supply of the c o a s t a l r e g i o n we must i n f e r that any s a l e s would be to markets outside the c o a s t a l r e g i o n . Both of these no-sale c o n d i t i o n s are s u b s t a n t i a l l y met i n p r a c t i c e 70 at .the present time. 70 Log exports i n 1962, from the whole province, were ap proximately 1.5% of the c o a s t a l r e g i o n l o g s c a l e . Chip exports from the province were approximately 0.5% of the c o a s t a l r e g i o n l o g s c a l e . Annual Report, B r i t i s h Columbia Forest S e r v i c e . Province of B r i t i s h Columbia, Dept. of Lands and F o r e s t s , V i c  t o r i a , B.C., (1962). 91 With respect to log allocation, the figures for Case I in Table 6.2 show almost two-thirds of the log supply going to sawmills. In practice approximately 82 percent of the combined f i r , hemlock and spruce supply for the coastal region in 1962 71 was converted to lumber. On the other hand, the percentage of logs allocated to plywood production in Case I of the model is significantly higher than industry practice, (the increase 72 is equal to the difference in lumber production noted above). The percentage of logs allocated to chipping in the model (Case I) i s the same as industry practice for 1962. Insofar as the technological information and simplifying assumptions in the model are reasonable, the results of case I indicate that the coastal region forest industry in 1962 misallocated logs to lumber production^which might otherv/ise have been more profitably utilized in the manufacture of plywood. In case II, a slight variation in technological coef ficients of the No. 3 hemlock log sawing activity results in a significantly modified log-allocation plan. Chipping is now the main activity accounting for almost half of the log input. This pattern of log allocation clearly does not correspond 71 Annual Report, 1963, British Columbia Lumber Manufac- turers Association, Vancouver, (1964). 72 L. Read, Economist and Statistician, Council of Forest Industries, Vancouver, (1964) - personal correspondence. 73 Ibid. 92 c l o s e l y w i t h t h a t a c t u a l l y observed i n the r e g i o n . The s i g  n i f i c a n t p o i n t however i s t o observe t h a t the l o g - a l l o c a t i o n p roblem i s dominated by a r e l a t i v e l y l a r g e volume o f No. 3 hemlock l o g s . Thus, i f the q u a l i t y o f No. 3 hemlock l o g s d e c l i n e s as p o s t u l a t e d i n the m o d i f i e d a c t i v i t y c o e f f i c i e n t s o f case I I , the e n t i r e a l l o c a t i o n o f t h e s e l o g s may be s h i f t  ed from lumber t o c h i p s . I n p r a c t i c e o f c o u r s e such a w h o l e s a l e t r a n s f e r o f l o g s from one u t i l i z a t i o n p r o c e s s t o a n o t h e r i s l i m i t e d t o the l o n g - r u n , where u t i l i z a t i o n c a p a c i t i e s f o r each p r o c e s s may be v a r i e d . F u r t h e r m o r e , i t i s p o s s i b l e t h a t a g i v e n l o g s u p p l y may be o p t i m a l l y d i v i d e d between two or more u t i l i z a t i o n c a t e  g o r i e s . T h i s i s e v i d e n t i n case I I I where the s u p p l y of No. 3 s p r u c e i s a l l o c a t e d between sawing, 10 p e r c e n t , and c h i p p i n g , 9 0 p e r c e n t . The o u t p u t o f Douglas f i r lumber s p e c i f i e d i n Case I , ( T a b l e 6 . 3 ) i s 2 5 p e r c e n t below the a c t u a l output i n 1962 i n 7k the c o a s t a l r e g i o n . T h i s r e f l e c t s the l a r g e r p o r t i o n of l o g s a l l o c a t e d t o plywood p r o d u c t i o n i n t h e case I s o l u t i o n as d i s  c u s sed above. Hemlock and spruce lumber p r o d u c t i o n i s moder a t e l y h i g h e r i n case I t h a n a c t u a l e x p e r i e n c e would i n d i c a t e , 7 5 b u t the p r o p o r t i o n s are r e a s o n a b l y c l o s e . Comparisons be tween case I and case I I show t h a t Douglas f i r and spruce lum b e r o u t p u t i s u n a f f e c t e d by the m o d i f i e d a c t i v i t y c o e f f i c i e n t s 7k Annual R e p o r t 1 9 6 3 , B r i t i s h Columbia Lumber Manu- f a c t u r e r s A s s o c i a t i o n , Vancouver ( 1 9 6 4 ) . 7 5 I b i d . 93 of No. 3 hemlock logs. In plywood production each case of the model speci f i e d two Douglas f i r plywoods and one mixed-specie plywood. As discussed e a r l i e r , mixed-specie plywoods have only a l i m i  ted market i n B r i t i s h Columbia. In addition, however, the model specified d i r e c t sales of each grade of hemlock veneer. As f a r as can be ascertained from available data there i s no market for hemlock veneer, per se, i n the coastal region. We must conclude therefore that the veneer and plywood pro duction data i s inaccurate r e l a t i v e to the lumber and pulp- production data. This i s not surprising i n view of the divergent sources of data involved i n the model. I t i s of in t e r e s t however to observe that i n each case hemlock ve neer i s involved i n two separate f i n a l basis a c t i v i t i e s ; hemlock veneer.sales and hemlock - Douglas f i r mixed-specie plywood. This i s an i l l u s t r a t i o n of the interdependence be tween these two a c t i v i t i e s . The solution data on upper and lower l i m i t s to u n i t - l e v e l returns, and the corresponding l i m i t i n g variables for each f i n a l basis variable, are a useful guide to the r e l a t i v e importance of each a c t i v i t y i n the optimum solution. For ex ample, a c t i v i t y number s i x (sawing No. 3> Douglas f i r sawlogs) i s a f i n a l basis variable with a u n i t - l e v e l return (cost) of -$20.00; see Appendix I. The lower l i m i t to t h i s u n i t - l e v e l return figure i s -$31.22, beyond which i t would be more p r o f i t  able, from the point of view of t o t a l net return to the log supply, 94 i •  "i t o employ a c t i v i t y number 29 ( c h i p p i n g No. 3 , Douglas f i r s a w l o g s ) . There i s a " c u s h i o n " here o f more t h a n $11.00. C o n v e r s e l y , a c  t i v i t y number 16 w i t h a u n i t - l e v e l r e t u r n of -$6.00 has a lo w e r l i m i t o f -$6.99, below w h i c h i t would be r e p l a c e d by a c t i v i t y number 14. The " c u s h i o n " here i s $0.99. C l e a r l y , i t w i l l t a k e a r e l a t i v e l y l a r g e (56 p e r c e n t ) d e c r e a s e i n u n i t - l e v e l r e t u r n ( i n c r e a s e i n c o s t ) o f a c t i v i t y number s i x b e f o r e i t i s r e p l a c e d , whereas a c t i v i t y number s i x t e e n w i l l be r e p l a c e d f o l l o w i n g a s i m i l a r change o f o n l y 16.5 p e r c e n t . I t s h o u l d be noted t h a t t h e r e i s no u p p e r - l i m i t i n g v a r i  a b l e t o e i t h e r of t h e s e i n t e r m e d i a t e a c t i v i t i e s . I n o t h e r words, i f t h e i r u n i t - l e v e l r e t u r n s a r e any h i g h e r ( l o w e r c o s t ) t h e r e i s no b e t t e r a l t e r n a t i v e . N o r m a l l y , each a c t i v i t y has b o t h upper and l o w e r l i m i t i n g v a r i a b l e s as e v i d e n t i n a c t i v i t y number 3 8 , 76 and o t h e r f i n a l a c t i v i t i e s w i t h p o s i t i v e u n i t - l e v e l r e t u r n s . Two c o n c l u s i o n s f o l l o w from t h e s e o b s e r v a t i o n s on t h e d i f f e r e n t c a s e s o f the model. F i r s t , t h e v a l i d i t y o f any p r a c t i  c a l c o n c l u s i o n s drawn from the s o l u t i o n o f the model depends h e a v i l y on the q u a l i t y of the d a t a used. The f u l l v a l u e of t h e model cannot be r e a l i z e d u n l e s s a c c u r a t e d a t a on y i e l d s , p r i c e s and c o s t s a r e a v a i l a b l e . Second, g i v e n c o n f i d e n c e i n the q u a l i t y of the d a t a , s i g  n i f i c a n t changes i n t o t a l net r e t u r n and optimum use o f the l o g s may r e s u l t f r o m a r e l a t i v e l y minor change i n y i e l d s , p r i c e s o r c o s t s . I n an example, a de c r e a s e i n t h e lumber y i e l d f r om a No. 3 hemlock l o g r e s u l t e d i n a $3.2 m i l l i o n d e c r ease i n t o t a l n e t r e t u r n (see T a b l e 6.1, cases I and I I ) . I t a l s o r e s u l t e d i n a 76 Some a c t i v i t i e s , such as No. 11, have an u p p e r - l i m i t i n g v a r i a b l e but t h e programme f a i l s t o s p e c i f y a m e a n i n g f u l upper u n i t - l e v e l r e t u r n f i g u r e , due t o i n s u f f i c i e n t space f o r l a r g e r numbers i n t h e machine programme f o r m a t . 9 5 major adjustment i n the l o g - a l l o c a t i o n plan (see Table 6 . 2 ) . I I . APPLICATIONS OF THE MODEL In t h i s study the a c q u i s i t i o n of accurate data and the i n  c l u s i o n of a l l possible variables i n the model was considered of secondary importance to presenting a clear demonstration of the application of l i n e a r programming to l o g - a l l o c a t i o n pro blems. Many simplifying assumptions were included to make th i s demonstration as simple as possible. In t h i s section we s h a l l consider relaxing some of these simplifying assump tions to make the model more v a l i d . In chapter four we distinguished between fundamental and c o r r e l a t i o n assumptions i n the model. The fundamental assumptions are basic to l i n e a r programming and cannot be re laxed without destroying the l i n e a r i t y of the model. The cor r e l a t i o n assumptions however serve merely to simplify the ex p o s i t i o n . We may conveniently relax some of the l a t t e r assumptions, by considering an a p p l i c a t i o n of the model to the l o g - a l l o c a t i o n problem faced by an i n d i v i d u a l firm. Log-Allocation Linear Programme for an Integrated Firm. We s h a l l take the approach of modifying the s i m p l i f i e d l i n e a r programme model presented i n t h i s thesis to conform more with the l o g - a l l o c a t i o n problem of an integrated firm. This w i l l involve re-defining the objective function and relaxing some o r i g i n a l assumptions, (a) A P r o f i t Objective-Function. The f i r s t step i s to deduct log costs from the t o t a l 9 6 n e t r e t u r n o b j e c t i v e f u n c t i o n . An o p t i m a l s o l u t i o n t h e n c o r  r esponds t o maximum e n t r e p r e n e u r i a l p r o f i t , e x c l u d i n g economic r e n t t o the s u p p l y o f l o g s . T h i s c o r r e c t i o n may be made e i t h e r by a d d i n g l o g c o s t s t o the u n i t - l e v e l r e t u r n o f each i n t e r  m e diate ( i . e . , l o g c u t t i n g ) a c t i v i t y , or by d e d u c t i n g an a v e r  age l o g c o s t from the u n i t - l e v e l r e t u r n of each f i n a l a c t i v i t y . The former p r o c e d u r e i s the more a c c u r a t e a l t h o u g h the l a t t e r i s p r o b a b l y s i m p l e r t o a p p l y . I n e i t h e r case the u n i t - l e v e l r e t u r n o f l o g - s e l l i n g a c t i v i t i e s would have t o be a d j u s t e d t o z e r o ( o r a n e g a t i v e v a l u e i f ex p e c t e d s a l v a g e v a l u e s f o r l o g s were l e s s t h a n purchase p r i c e ) t o a v o i d d o u b l e - c o s t i n g o f l o g s . (b) U t i l i z a t i o n C a p a c i t y R e s t r a i n t s . The next b a s i c m o d i f i c a t i o n i s t o account f o r the l i m i  t e d u t i l i z a t i o n c a p a c i t y o f a s i n g l e f i r m i n the s h o r t - r u n . P r e v i o u s l y , the l o n g - r u n c o n d i t i o n o f u n l i m i t e d c a p a c i t y was assumed. I n p r a c t i c e i t i s the s h o r t - r u n l o g - a l l o c a t i o n p r o b l e m t h a t i s of i n t e r e s t t o t h e f i r m ' s c u r r e n t o p e r a t i o n s , w i t h l o n g - r u n c o n s i d e r a t i o n s a p p l i c a b l e more t o i n v e s t m e n t and p o l i c y d e c i s i o n s , w h i c h a re d i s c u s s e d l a t e r . U t i l i z a t i o n c a p a c i t y r e s t r a i n t s can be r e a d i l y i n c o r  p o r a t e d i n t o the l o g - a l l o c a t i o n model i f no more t h a n the t o t a l c a p a c i t y o f each u t i l i z a t i o n c a t e g o r y ( s a w i n g , p e e l i n g and c h i p p i n g ) need be s p e c i f i e d . Each c a p a c i t y r e s t r a i n t r e q u i r e s an a d d i t i o n a l row i n the m a t r i x of the model, w i t h a c t i v i t y c o e f f i c i e n t s o f u n i t y f o r each i n t e r m e d i a t e a c t i v i t y a f f e c t e d by the r e s t r a i n t . Thus, the s a w m i l l i n g c a p a c i t y 97 r e s t r a i n t (row) would have a 'one' i n each log-sawing i n t e r  mediate a c t i v i t y column. T o t a l a v a i l a b l e c a p a c i t y f o r each u t i l i z a t i o n category, expressed i n the same u n i t s as the p r i  mary inputs i s entered i n the r e s t r a i n t column. The r e s t r a i n t expressions ( i n e q u a l i t i e s ) f o r each l i m i t e d u t i l i z a t i o n ca p a c i t y take the same form as the primary input r e s t r a i n t ex pre s s i o n s ; i . e . , t o t a l consumption of each u t i l i z a t i o n - c a p a c i t y must be l e s s - t h a n or equal-to the a v a i l a b l e u t i l i z a t i o n capa c i t y . D i s p o s a l a c t i v i t i e s corresponding to the "use" of i d l e c a p a c i t y are required f o r each u t i l i z a t i o n - c a p a c i t y r e s t r a i n t . The u n i t - l e v e l , returns f o r these c a p a c i t y - d i s p o s a l or i d l e - c a p a c i t y a c t i v i t i e s are given negative values corresponding to the cost of i d l e c a p a c i t y , which su b t r a c t s from t o t a l p r o f i t . A more exact model would i n c l u d e separate c a p a c i t y r e  s t r a i n t s f o r some or a l l of the intermediate a c t i v i t i e s i n the model, but t h i s would make i t considerably more cumber some. To s p e c i f y c a p a c i t y r e s t r a i n t s on each o r i g i n a l i n t e r  mediate a c t i v i t y would add as many new r e s t r a i n t s , and as many new (d i s p o s a l ) a c t i v i t i e s , as there were o r i g i n a l i n t e r  mediate a c t i v i t i e s , (c) Log Costs. In the s i m p l i f i e d i n t e g r a t e d - i n d u s t r y model of t h i s study we assumed that l o g costs were e s t a b l i s h e d on the open market. This i s a reasonable assumption f o r any f i r m operat ing w i t h i n the Vancouver l o g market. In the event however that a f i r m does have access to logs (or intermediate products 98 such as c h i p s ) of the same grade and q u a l i t y but f r om two or more d i f f e r e n t s o u r c e s a t two or more d i f f e r e n t p r i c e s , t h e n t h e s e must be t r e a t e d as s e p a r a t e i n p u t s t o the model. Each one w i t h i t s own c o r r e s p o n d i n g s a w i n g , p e e l i n g , and c h i p p i n g a c t i v i t i e s . A p p l i c a t i o n s o f the Model t o the Problems o f a F i r m . A f i r m can u s e f u l l y a p p l y a l o g - a l l o c a t i o n model such as the one o u t l i n e d above, i n two s e p a r a t e ways: (a) i n the s h o r t - r u n , (b) i n the l o n g - r u n . (a) S h o r t - r u n A p p l i c a t i o n s . I n the s h o r t - r u n the f i r m seeks t o maximize i t s ( s h o r t - run) p r o f i t s s u b j e c t t o g i v e n p r i c e s , c o s t s , p r o d u c t i o n capa c i t i e s and l e v e l of t e c h n o l o g y . I n t h i s c o n t e x t , the f i r m , l a c k i n g any monopoly or monopsony i n f l u e n c e on the market, can t a k e c e r t a i n d a t a as g i v e n , namely: 1. Log S u p p l y : ( i ) Volume of a v a i l a b l e l o g s , by s p e c i e s and grade. ( i i ) Log p r i c e s ( d e l i v e r e d t o the f i r m ) . 2 . A c t i v i t i e s : ( i i i ) A v a i l a b l e u t i l i z a t i o n p r o c e s s e s and c a p a c i t i e s , ( i v ) T e c h n o l o g i c a l c o e f f i c i e n t s of each u t i l i z a t i o n p r o c e s s , (v) U n i t - c o s t s f o r each u t i l i z a t i o n p r o c e s s . 3 . P r o d u c t s ( v i ) Demand p r i c e s f o r i n t e r m e d i a t e and f i n a l p r o d u c t s . G i v e n t h i s i n f o r m a t i o n the f i r m can c o n s t r u c t and s o l v e a l o g - a l l o c a t i o n programme s i m i l a r t o the one d e s c r i b e d i n t h i s t h e s i s , w i t h the a p p r o p r i a t e m o d i f i c a t i o n s o u t l i n e d 99 e a r l i e r . T h i s model would be used t o determine t h e maximum p r o f i t t he f i r m c o u l d e a r n and the l o g - a l l o c a t i o n p l a n t o be f o l l o w e d i n a t t a i n i n g t h i s p r o f i t . These two r e l a t e d p i e c e s of i n f o r m a t i o n , maximum p r o  f i t and optimum l o g - a l l o c a t i o n p l a n , would se r v e as u s e f u l g u i d e s t o c u r r e n t o p e r a t i o n s o f the f i r m , i n d i c a t i n g whether, and where, a d d i t i o n a l p r o f i t s might be earned t h r o u g h more e f f i c i e n t l o g a l l o c a t i o n . The e f f e c t o f a change i n c o m p o s i t i o n o f the l o g sup p l y o r a change i n p r o d u c t p r i c e s c o u l d be i n v e s t i g a t e d by s o l v i n g the model a second time w i t h the new d a t a , (as we demonstrated w i t h c a s e s I I and I I I i n t h i s s t u d y ) . Changes i n maximum p r o f i t and optimum l o g - a l l o c a t i o n p l a n c o u l d be noted and e v a l u a t e d w i t h r e s p e c t t o the o r i g i n a l s o l u t i o n , (b) Long-run A p p l i c a t i o n s . Long-run changes may be a n a l y z e d t h r o u g h the same p r o  cedure o u t l i n e d above, namely, by o b s e r v i n g changes i n maxi mum p r o f i t and optimum l o g a l l o c a t i o n . I n the l o n g r u n , how e v e r , u t i l i z a t i o n c a p a c i t i e s and t e c h n o l o g i c a l c o e f f i c i e n t s may v a r y a l s o , r e s u l t i n g i n a change i n the b a s i c m a t r i x of t h e model. I f u t i l i z a t i o n - c a p a c i t y r e s t r a i n t s a r e e x c l u d e d from th e model e n t i r e l y , as i n the model p r e s e n t e d i n t h i s t h e s i s , t h e n the optimum l o n g - r u n l o g - a l l o c a t i o n p l a n c o r r e s p o n d s t o an i d e a l c o m b i n a t i o n o f u t i l i z a t i o n c a p a c i t i e s o f t h e v a r i o u s p r o c e s s e s . T h i s s o l u t i o n , c e t e r u s p a r i b u s , would p r o v i d e the f i r m w i t h a u s e f u l guide f o r in v e s t m e n t p l a n s i n f u t u r e e x p a n s i o n . 100 A l t e r n a t e l y , t h e f i r m may i n v e s t i g a t e d i f f e r e n t u t i l  i z a t i o n c a p a c i t y c o m b i n a t i o n s r e s u l t i n g f r om a g i v e n amount o f new i n v e s t m e n t i n p l a n t s , f o l l o w i n g a b e n e f i t - c o s t a n a l y s i s p r o c e d - 77 u r e f o r each proposed i n v e s t m e n t p l a n . The model a l l o w s a d i r e c t c omparison between b e n e f i t - c o s t r a t i o s f o r d i f f e r e n t i n v e s t m e n t ( e x p a n s i o n ) schemes, e i  t h e r on a c e t e r u s p a r i b u s b a s i s or i n c l u d i n g f o r e c a s t changes i n o t h e r c o n d i t i o n s ( s u c h as p r i c e s or l o g s u p p l y ) . F i n a l l y , the f i r m c o u l d employ a l i n e a r programme model such as t h i s t o e v a l u a t e b e n e f i t s from t e c h n o l o g i c a l i n  n o v a t i o n . The i n d u s t r i a l r e s e a r c h group ( o r i t s e q u i v a l e n t ) c o n s t a n t l y f a c e s a p r o blem of j u s t i f y i n g v a r i o u s avenues of r e s e a r c h endeavour. An e s t i m a t e o f the e f f e c t t h a t f u t u r e developments i n wood c o n v e r s i o n t e c h n i q u e s , e t c . , might have on t o t a l net r e t u r n and optimum l o g a l l o c a t i o n would be of g r e a t v a l u e . The L o g - A l l o c a t i o n L i n e a r Programme D u a l . R e f e r e n c e was made e a r l i e r t o the " m a t h e m a t i c a l d u a l " o f a l i n e a r programme model and i t s p a r t i c u l a r i n t e r p r e t a t i o n i n the l o g - a l l o c a t i o n problem. To i l l u s t r a t e the n a t u r e of the l o g - a l l o c a t i o n d u a l we s h a l l c o n s i d e r a s i m p l e problem i n w h i c h l o g s are consumed d i r e c t l y i n the p r o d u c t i o n o f b a s i c wood p r o d u c t s ; c o m p l i c a t i o n s o f i n t e r m e d i a t e p r o d u c t s , e t c . , w i l l be i g n o r e d . 77 J . D a v i s , D. W. R o s s , A. D. S c o t t and W. R. D. S e w e l l , B e n e f i t - C o s t A n a l y s i s Handbook, Queen's P r i n t e r , Ottawa, 1962. 101 I n the p r i m a r y problem, s i m i l a r i n form t o the model i n t h i s s t u d y , the o b j e c t i v e i s t o maximize the r e t u r n t o a g i v e n s u p p l y o f l o g s . P r o d u c t p r i c e s a r e known and assumed c o n s t a n t ; the l o g s u p p l y i s f i x e d . The v a r i a b l e s i n the p r o  blem a r e t h e l e v e l s o f p r o d u c t i o n o f v a r i o u s l o g - c o n v e r s i o n p r o c e s s e s or a c t i v i t i e s , k s o l u t i o n t o the p r i m a r y problem c o m p r i s e s s e l e c t i n g the most p r o f i t a b l e c o m b i n a t i o n o f l o g - c o n v e r s i o n p r o c e s s e s , and s p e c i f y i n g t h e i r c o r r e s p o n d i n g l e v e l s of p r o d u c t i o n , s u b j e c t t o the l i m i t a t i o n s imposed by a r e s t r i c t e d , f i x e d , s u p p l y o f l o g s . Suppose now t h a t i t i s r e q u i r e d t o f i n d what p r o p o r  t i o n o f i t s p r o f i t s the r e s o u r c e owner ( e . g . f i r m ) owes t o each type o f l o g i n the g i v e n s u p p l y . To do t h i s the f i r m w i l l s e t up a c c o u n t i n g p r i c e s f o r t h e s e l o g s w h i c h a r e j u s t h i g h enough t o g i v e t o t h e s e i n p u t s a combined v a l u e e q u a l t o the t o t a l p r o f i t of t h e f i r m . That i s , the f i r m ' s p r o f i t s a f t e r p a y i n g t h e s e imputed p r i c e s f o r l o g s would be z e r o . I n programming te r m s , the problem i s t o f i n d imputed p r i c e s f o r the l o g s w h i c h m i n i m i z e the t o t a l a c c o u n t i n g c o s t o f t h e s e r e s o u r c e s , and y e t i n v o l v e an imputed l o g - c o s t per u n i t o f each wood-product no l e s s t h a n the p r o f i t p e r u n i t o f each wood-product. The d i s t i n c t i o n may be c l a r i f i e d by c o n s i d e r i n g two l i n e a r programme o u t l i n e s as f o l l o w s : 1 0 2 The Primary Problem. Maximize p r o f i t : P = p X + p X + p X I 1 2 2 3 3 Subject to resource l i m i t a t i o n s : a X + a X + a X ^ b I I 1 1 2 2 1 3 3 1 a 2 1 X l + a 2 2 X 2 + a 2 3 X 3 * b 2 and the requirement that no a c t i v i t y l e v e l s be negative. X l - °> X 2 ~ °' X 3 - 0 The Dual Problem. Minimize Account Costs: (Imputed p r i c e s ) . C = b 1 Y1 + b p Y 2 Subject to the requirement that a l l p r o f i t s are imputed: a Y + a Y — p 1 1 1 2 1 2 1 a Y + a Y — p 1 2 1 2 2 2 P2 a Y + a Y ^ p 1 3 1 2 3 3 3 and the requirement that no accounting p r i c e s be negative: Y ^ 0 , Y — 0 , Y — 0 . 1 ' 2 ' 3 I t should be noted that the same matrix elements apply i n the primary and the dual model. The only changes are that rows i n the primary become columns i n the dual (and c o n v e r s e l y ) ; the i n e q u a l i t y signs i n the r e s t r a i n t expressions are reversed, and the o b j e c t i v e f u n c t i o n becomes one of minimizing r a t h e r 1 0 3 than maximizing. There i s no d i f f e r e n c e i n the s o l u t i o n tech nique . A p p l i c a t i o n s of the Dual. The dual l o g - a l l o c a t i o n problem described above i s probably of more value to a resource owner than to a resource e x p l o i t e r , since i t c o n s i d e r s l o g cost as the o p t i m i z i n g v a r i  able r a t h e r than product output. In t h i s sense, the term resource owner may encompass three c l a s s e s of owner: the P r o v i n c i a l Government which owns almost the e n t i r e resource 78 i n B r i t i s h Columbia; the l a r g e i n t e g r a t e d f i r m with i t s own holdings of timber; the log-buyer f o r a f i r m purchasing logs on the open-market. The log-buyer i n t e r p r e t a t i o n i s the most rigorous ap p l i c a t i o n of the l o g - a l l o c a t i o n d u a l , f o r two reasons. F i r s t l y , the log-buyer i s concerned only w i t h logs (not t r e e s or timber). Secondly, the log-buyer i s predominantly c o s t - o r i e n t e d . The l i s t of imputed log p r i c e s generated i n the dual s o l u t i o n would i n d i c a t e to the resource owner the r e l a t i v e value of logs to h i s p a r t i c u l a r l o g - u t i l i z a t i o n complex. 78 This i s our o r i g i n a l i n t e r p r e t a t i o n of the term resource owner, used throughout the e a r l i e r part of t h i s t h e s i s . A P P E N D I X T 0 C H A P T E R S I X TABLE 6.1 TOTAL NET RETURNS, IN LUMBER, PLYWOOD, AND PULP PRODUCTION Ca l c u l a t e d as t o t a l gross returns l e s s manufacturing c o s t s , excluding the cost of l o g s . FIGURES IN MILLIONS OF DOLLARS U t i l i z a t i o n Category- CASE I CASE I I CASE I I I Lumber Production 120.0 64.7 46.3 Plywood Production and Veneer Sales 134.4 134 A 160.3 Pulp Production and Chip Sales 7 0 . 5 125.6 110.5 Hog-Fuel 3.6 5.6 8.2 T o t a l Net Return • 333.5 330.3 325.3 TABLE 6.2 OPTIMUM LOG ALLOCATION Showing percentage of available log supply allocated to various u t i l i z a t i o n processes, - by species and grade - CASE I CASE II CASE III Saw' g Peel'g Chip'g Sav/' g Peel'g Chip'g Saw' g Peel'g Chip'g Doug. F i r #1 peeler 100 100 100 u # 2 " - 100 — — 100 _ — 100 _ " " #3 " - 100 — - 100 — — 100 _ II II #1+ II - 100 - — 100 — 100 _ " " #2 sawl'g - 100 - — 100 100 " #3 " 100 - - 100 - - 100 - - Hemlock #1 sawl'g 100 100 _ a 100 _ '* #2 1 1 - 100 — — 100 — — 100 _ #3 " - - 100 - - 100 - - 100 Spruce #1 sawl'g 100 100 — 100 _ II II 100 - — 100 — 100 — — " #3 " 100 100 9 .7 — 9 0 . 3 Total Los Consumptn. . m i l l i o n c u . f t . 329 .6 1 6 3 . 7 1 9 . 8 116.0 1 6 3 . 7 233 . h 142.9 2 0 5 . 0 162 .1 percentage 64 .0 3 2 . 0 4 . 0 2 2 . 5 3 2 . 0 k5.5 2 8 . 0 40 .0 3 2 . 0 ACTUAL CONSUMPTION 8 2 . 0 1 4 . 0 4 . 0 . 1962 - B. C. COAST TABLE 6 . 3 LUMBER SALES BY SPECIES AND GRADE Volume of lumber s a l e s by four grade c a t e g o r i e s , i n MILLION cubic f e e t , and value of lumber s a l e s by sp e c i e , i n MILLIONS of d o l l a r s CASE I CASE I I CASE I I I FIR HEM SPRU FIR HEM SPRU FIR HEM SPRU Cle a r 3 . 5 8 0 13.716 2.253 3 . 5 8 0 3 . 0 3 6 2.253 1.945 1.358 1.743 Sel/Mer 1 7 . 0 0 5 23.568 2.120 1 7 . 0 0 5 2 . 2 0 8 2.120 14.290 2.133 1 .245 Cons/Std 28.64-0 59.124 3.120 28.640 3 . 5 8 8 3.120 2 3 . 6 2 0 7.842 1.079 U t i l / E c o n 8 . 0 5 5 2 8 . 5 9 6 0 . 7 8 8 8 . 0 5 5 0 . 8 2 8 0.788 6.670 8 . 0 1 0 0 . 2 0 5 T o t a l s M i l l i o n c u . f t . 57.280 125.004 8 . 2 8 1 57.280 9.660 8.281 46 . 525 19.843 4.272 Percent 30 65 5 76 13 11 75 21 4 M i l l i o n D o l l a r s 6 6 . 6 7 8 103.541 11.064 66.678 10.171 11.064 52.997 15.093 1 0 . 0 8 7 ACTUAL 1962 PRODN. 40 57 3 * See t e x t p. 51 f o r lumber grades. 1 0 7 TABLE 6 . 4 PLYW00D AND VENEER SALES Volumes of finished plywood and green veneer sold, in hundreds of cubic feet Only activities (products) specified in the solution are shown. Product Description CASE I CASE II CASE III (Hundreds of Cubic Feet) Plywoods Douglas F i r G 1 S 3 7 8 , 8 1 4 3 7 8 , 8 1 4 413,917 " » 8 1 S 72,116 7 2 , 1 1 6 1 0 6 , 8 2 2 Fir/Hem mix 1 1 3 , 6 0 7 1 1 3 , 6 0 7 1 1 9 , 2 6 9 Veneers Hemlock A • 3 0 , 0 0 0 3 0 , 0 0 0 4 , 9 5 0 " B 6 0 , 0 0 0 6 0 , 0 0 0 9 0 , 0 0 0 " C 112,740 112,740 1 9 4 , 8 3 8 1 0 8 TABLE 6 . 5 PULP SALES Volumes of pulp produced and so ld , i n a i r-dry tons Pulps i den t i f i ed by the i r chip-mixtures. Pulp Produced CASE I CASE II (a i r-dry tons) CASE III 3 0 / F i r / 5 0 H e m / 2 0 Spruce 4 5 F i r / 4 5 H e m / 1 0 Spruce 1 0 0 Hemlock 5 9 1 , 3 1 6 1 7 5 , 0 3 5 1 2 1 , 4 8 1 5 9 1 , 3 1 6 1 7 5 , 0 3 5 l , 0 0 7 , l k 0 8 4 1 , 8 5 7 6 6 8 , 1 0 7 109 CHAPTER VII SUMMARY AND CONCLUSION . The forest industry of B r i t i s h Columbia faces strong competition in i t s t r ad i t i ona l markets from other world pro  ducers and from subst i tute products. To meet th i s compe t i t i o n i t is essent ia l that the industry reduce i t s uni t-costs and ra ise the economic e f f i c i ency of i t s log-conversion pro  cesses. One aspect of th is required e f f i c i ency i s to ensure that each log i s consumed in a manner that maximizes t o t a l net return to the log supply. As far as can be ascer ta ined, current log-a l loca t ion procedures in the industry resu l t i n considerable i n e f f i c i e n c y . This ar ises from two causes; i n s u f f i c i e n t data on log costs and y i e l d s , and the absence of an e f f ec t i ve technique for handl  ing the data ava i l ab l e . The problem of determining an optimal d i s t r i b u t i o n of logs among a l ternate conversion processes i s amenable to a l i nea r programme so lu t ion . Appl icat ions of l i nea r programming to var  ious opt imizat ion problems in each sector of the industry have 79 SO 8x been described i n the l i t e r a t u r e / " ' ' The cont r ibut ion of 79 F. H. C u r t i s , "L inear Programming the Management of a Forest Property , " Journal of Fores t ry , v. 60, (Sept. 1962), pp. 611-616. 80 N. D. Jackson & G. ¥ . Swinton, "L inear Programming i n Lumber Product ion, " For . Prod. Jour, v. XI, (June, 1961), pp. 272-274. 81 E. Koenigsberg, "L inear Programming Applied to the P ly  wood Industry, " For . Prod. Jour, v. XI, (Sept. I960), Dp. 481- 486. ~ " n o ' t h i s t h e s i s i s a d e m o n s t r a t i o n o f the a p p l i c a b i l i t y o f l i n e a r programming t o the l o g - a l l o c a t i o n problem f a c e d by the i n d u s  t r y as a whole, encompassing a l l o f the p r i n c i p a l u t i l i z a  t i o n a l t e r n a t i v e s . Fundamental d i f f e r e n c e s i n p r o d u c t i o n p r o c e s s e s e x i s t between v a r i o u s l o g - u t i l i z a t i o n s e c t o r s of the i n d u s t r y . P r i n c i p a l l y , lumber p r o d u c t i o n i n v o l v e s a s i n g l e b r e a k i n g - down o p e r a t i o n (sawing) t o produce a f i n a l p r o d u c t , whereas plywood and p u l p p r o d u c t i o n r e q u i r e a breaking-down p r o c e s s ( p e e l i n g and c h i p p i n g ) f o l l o w e d by a r e c o m b i n a t i o n p r o c e s s , ( g l u e i n g and p u l p i n g ) t o produce f i n a l p r o d u c t s . R e c o n c i l i n g t h e s e d i f f e r e n c e s i n a l i n e a r programme model r e q u i r e d i n t r o  d u c i n g the concept o f i n t e r m e d i a t e a c t i v i t i e s . I n t e r m e d i a t e a c t i v i t i e s were viewed as p r o c e s s e s w h i c h produced i n t e r m e d i a t e p r o d u c t s t o be consumed as i n p u t s t o o t h e r a c t i v i t i e s i n the model. They were i n c l u d e d i n the model w i t h n e g a t i v e a c t i v i t y c o e f f i c i e n t s and n e g a t i v e a c t i v i t y r e t u r n s , ( i . e . , c o s t s ) . I t was assumed t h a t a l l i n t e r m e d i a t e p r o d u c t s produced would be consumed; i . e . , t h e r e would be no a c c u m u l a t i o n o f i n t e r m e d i a t e p r o d u c t s . The scope o f the model was l i m i t e d t o the c o a s t a l r e g i o n o f B r i t i s h Columbia. Data were assembled from r e s e a r c h j o u r  n a l s , t r a d e j o u r n a l s and from p e r s o n a l d i s c u s s i o n s between the a u t h o r and p e r s o n n e l i n the i n d u s t r y w i t h e x p e r t knowledge of c u r r e n t o p e r a t i o n s i n the f o r e s t i n d u s t r y . S o l u t i o n s o f the model were o b t a i n e d t h r o u g h the s e r v i c e s o f the computing c e n t r e a t the U n i v e r s i t y o f B r i t i s h C olumbia. V a r i a t i o n s i n the model demonstrated i t s p r a c t i c a l v a l u e i n I l l a n a l y z i n g the e f f e c t s of changing l o g s u p p l i e s and changing t e c h n o l o g i c a l c o e f f i c i e n t s , on t o t a l net r e t u r n and optimum l o g a l l o c a t i o n . I t was shown t h a t t h e model i s a p o w e r f u l t o o l f o r i n v e s t i g a t i n g s h o r t - r u n and l o n g - r u n changes i n the i n t e  g r a t e d f o r e s t p r o d u c t s f i r m . I t can be c o n c l u d e d t h a t , as a c o m p a r a t i v e s t a t i c s economic model, the l o g - a l l o c a t i o n l i n e a r programme p r o v i d e s an e f f i c i e n t means o f e v a l u a t i n g the consequences of changing eco nomic v a r i a b l e s i n the i n d u s t r y or f i r m . Once c o n s t r u c t e d , the model may be s o l v e d an i n d e f i n i t e number o f ti m e s t o i n v e s t i  gate v a r i o u s c o m b i n a t i o n s o f v a r i a b l e s of i n t e r e s t to the a n a l y s t . The many s i m p l i f y i n g assumptions on which the model has been c o n s t r u c t e d can be removed, though t o do so would r e q u i r e more d a t a and i n v o l v e i n c r e a s i n g c o m p l e x i t y and c o s t i n t h e s o l u t i o n . Indeed, some o f the fundamental assumptions o f l i n  e a r programming may u l t i m a t e l y be removed by c o n s t r u c t i n g a n o n - l i n e a r l o g - a l l o c a t i o n problem, but t h i s degree o f complex i t y i s beyond the scope of t h i s t h e s i s . The p r i n c i p a l l i m i t a t i o n t o the l i n e a r programme d e s c r i b e d i n t h i s t h e s i s has been the q u a l i t y o f the d a t a a v a i l a b l e . The v a l i d i t y o f the r e s u l t s , w hich i s governed s u b s t a n t i a l l y by the q u a l i t y o f the d a t a employed, d e t e r m i n e s the u s e f u l n e s s o f the model as a p r a c t i c a l economic t o o l of management. Thus, one of the p r i n c i p a l c o n t r i b u t i o n s of t h i s s t u d y has been t o empha s i z e the v a l u e o f c o l l e c t i n g a c c u r a t e d a t a i n the f o r e s t p r o  d u c t s i n d u s t r y . T h i s q u e s t i o n has a l r e a d y r e c e i v e d c o n s i d e r a b l e at tent ion within the industry and e f fo r t s are being d i r  ected towards solv ing the fundamental problem of adequately descr ib ing and grading logs and log products (notably l u m b e r ) . 8 1 > 8 2 , 8 3 However, much of the data co l l ec ted i s of the wrong type for th i s kind of ana l y s i s , and several new kinds of data must be co l l ec ted in order to exp lo i t the f u l l po tent i a l of linear-programming ana l ys i s . The foundation of a l l o ca t i on models such as the one described i n th i s thes is provide a guide to the production data required. 81 P. H. Lane, "Evaluat ing Log and Tree Qual i ty fo r Wood Products , " For . Prod. Jour . , v. XIV, (Mar. 1963), pp. 89-93. 82 C. k. Newport, C. R. Lockard and C. L. Vaughan, "Log and Tree Grading as a Means of Measuring Qua l i t y , " Rept. Working Group as approved by The Nat ional Log Grade Committee, Madison, May, 1958, Forest Serv ice , U.S. Dept. of Ag r i c . Wash., D .C . , (May, 1959). 83 J . H. Jenkins, "Grade Marking of Canadian Lumber," Report No. I89, For . Prod. Res. Branch,.Canada Dept. of F o r . , Ottawa, (Sept. 1962). A P P E N D I X I A * 0 L O G A L L O C A T I O N L I N E A R P R O G R A M M E N U M b E R O F I T E R A T I O N S = 2 9 - R E S U L T S M A X I M U M V A L U E O F O B J E C T I V E F U N C T I O N = 3> 3 3 0 > 3 5 9 > 3 0 0 P E R Y E A R O P T I M U M L O G A L L O C A T I O N P L A N i 1 U N I T V A R I A b L E ' L E V E L A C T I V I T Y . T O T A L L O W E R L I 1 I T . 1 N G ' U P P E R ' L I M I T I N G 1 R E T U R N L E V E L R E T U R N L I M I T V A R I A b L E L I M I T V A R I A B L E 1 ( i / C . C . F o ) ( C . C . F . ) ( i X l . O O O ) ( W C . C . F . ) ( i / C . C . F . ) : 6 - 2 0 . 0 0 0 8 9 5 0 0 0 . 0 - 1 7 9 0 0 . 0 0 0 - 3 1 . 2 1 8 5 2 9 . 9 9 9 9 . 9 0 0 . 0 . i • 7 - 2 0 . 0 0 0 1 3 8 0 0 0 . 0 - 2 7 6 0 . 0 0 0 - 2 5 . 1 1 7 3 2 0 9 9 9 9 . 9 0 0 0 i 1 0 - 2 0 . 0 0 0 1 3 0 0 0 . 0 - 2 6 0 . 0 0 0 - 6 3 . 6 5 6 0 2 2 9 9 9 9 . 9 0 0 0 • . r i 1 1 - 2 0 . 0 0 0 1 1 4 0 0 0 . 0 - 2 2 6 0 . 0 0 0 - 3 5 . 0 9 2 8 2 3- 9 9 9 9 . 9 0 0 0 5 2 " • i 1 5 - 6 . 0 0 0 6 1 0 0 0 . 1 - 3 6 6 . 0 0 1 - 1 3 . 5 3 2 3 1 3 9 9 9 9 . 9 0 0 0 4 8 ' — r 1 6 - 6 . 0 0 0 1 4 2 0 0 0 . 0 - 8 5 2 . 0 0 0 - 6 . 9 9 2 3 1 4 9 9 9 9 . 9 0 0 0 I 1 7 . - 6 . 0 0 0 2 8 4 0 0 0 . 0 - 1 7 0 4 . 0 0 0 - 3 3 . 9 2 8 1 3 9 9 9 9 . 9 0 0 0 I 1 8 - 6 . 0 0 0 1 8 3 0 0 0 . 0 - 1 0 9 8 . 0 0 0 - 3 8 . 9 9 2 4 4 9 9 9 9 . 9 0 0 0 ' 1 1 9 - 6 . 0 0 0 4 6 7 0 0 0 . 4 - 2 8 0 2 . 0 0 2 - 3 0 . 8 5 7 2 5 9 9 9 9 . 9 0 0 0 1 2 2 1 - 6 . 0 0 0 5 0 0 0 0 0 . 0 - 3 0 0 0 . 0 0 0 - 9 . 3 6 4 3 8 9 9 9 9 . 9 0 0 0 2 0 ' 3 2 - 2 . 0 0 0 2 1 3 6 0 0 0 . 0 - 4 2 7 2 . 0 0 0 - 1 3 . 8 5 3 6 6 6 9 9 9 9 . 9 0 0 0 5 1 | • 3 b . - 2 . 0 0 0 ' 1 9 8 0 0 0 . 0 - 3 9 6 . 0 0 0 - 5 . 8 0 6 8 1 2 9 9 9 9 . 9 0 0 0 1 ! 3 8 2 2 0 . 0 0 0 3 7 0 a 1 4 . 5 8 3 3 3 9 . 1 9 0 2 1 3 . 7 1 0 9 3 6 . 2 3 8 . 5 5 2 0 8 9 1 4 0 . 2 1 0 . 0 0 0 7 2 1 1 6 . 0 • 1 5 1 4 4 . 3 6 0 1 9 9 . 7 1 3 4 4 1 2 1 8 . 7 6 5 4 3 6 1 4 6 2 2 0 . 0 . 0 0 . 0 0 0 . 0 . 0 0 0 2 0 1 . 4 4 8 0 8 9 ' - 2 8 3 . 3 8 9 9 2 3 4 7 2 3 0 . 0 0 0 1 1 3 6 0 6 . 9 2 6 1 2 9 . 5 8 7 2 2 5 . 0 0 0 1 4 8 2 5 0 . 5 7 3 3 4 1 ! 5 3 8 2 . 0 0 0 5 9 1 3 1 6 . 6 4 8 4 8 7 . 9 6 1 . 8 0 . 8 7 7 7 1 2 8 7 . 4 9 3 5 3 4 j 5 4 8 2 . 0 0 0 1 7 5 0 3 4 . 6 . 1 4 3 5 2 . 8 3 7 7 8 o 1 4 5 4 5 1 8 3 . 6 4 9 1 1 2 J 5 6 6 7 . 0 0 0 1 0 0 7 1 3 9 . ' 5 . 6 7 4 7 8 . 3 4 7 5 0 . 7 4 1 8 5 1 7 2 . 7 5 . 1 8 1 2 ' - 1 [ 7 0 2 3 5 . 0 0 0 3 5 8 0 0 . 0 8 4 1 3 . 0 0 0 - 4 5 . 4 6 3 9 2 9 " " 3 0 9 . 1 2 4 9 2 7 1 1 3 0 . 0 0 0 1 7 0 0 5 0 . 0 2 2 1 0 6 . 5 0 0 7 0 . 9 5 5 0 2 9 . 2 6 2 . 9 9 1 1 3 1 7 2 1 1 5 . 0 0 0 2 8 6 4 0 0 . 0 3 2 9 3 6 . 0 0 0 7 9 . 9 4 2 . 1 2 9 2 0 0 . 7 1 4 5 5 • i 7 3 4 0 . 0 0 0 8 0 5 5 0 . 0 3 2 2 2 . 0 0 0 - 8 4 . 6 5 0 6 2 9 ' 5 9 . 8 4 7 9 1 4 • 7 4 . 1 4 5 . 0 0 0 3 0 3 6 0 . 0 4 4 0 2 . 2 0 0 1 2 1 . 7 3 9 5 . 2 0 1 7 5 . 5 8 5 3 • 8 " '' ' 7 5 1 0 0 . 0 0 0 2 2 0 8 0 . 0 2 2 0 8 . 0 0 0 6 8 . 0 1 6 8 . 2 0 1 2 1 . 0 2 7 4 . 8 | 7 6 9 0 . 0 0 0 3 5 8 8 0 . 0 3 2 2 9 . 2 0 0 7 0 . 3 1 8 1 2 0 1 0 2 . 4 6 0 7 8 ! 7 7 4 0 . 0 0 0 8 2 8 0 . 0 3 3 1 . 2 0 0 - 4 5 . 2 8 8 5 2 0 7 7 . 3 8 2 1 8 7 8 2 7 0 . 0 0 0 2 2 5 3 0 . 0 ' 6 0 8 3 . 1 0 0 1 7 5 . 6 6 9 7 2 3 3 2 4 . 3 8 3 1 1 2 !, 7 9 9 5 . 0 0 0 2 1 2 0 0 . 0 2 0 1 4 . 0 0 0 6 . 2 1 8 5 2 3 1 2 6 . 7 2 3 5 1 2 ! 8 0 8 5 . 0 0 0 3 1 2 0 0 . 0 2 6 5 2 . 0 0 0 2 6 . 9 5 0 6 2 3 9 8 . 5 9 5 7 1 2 1 I 8 1 4 0 . 0 0 0 . 7 8 7 9 . 9 3 1 5 . 1 9 6 - 2 1 1 . 5 4 7 6 2 3 • 8 7 . 5 8 5 2 1 2 12 ho 9 8 >7 6 5 CA» £^ <ji a ca «D O O 00 CM CM <f CM CM \0 <t- <r o <t" <t" CM vO O ! CM CM in m c n c j > [ O CO CO I un co o m <t CM v£) o O O rH CO CO CM CM CM - - i Is- O rH <f CA CT> i n (<i H rH o r^ -%o <ri cn m rH o co co i n <r m <t-. co I*"- O rH O . I s - CO ON O O vO co r- loo r- O O rH. o o CM o o CO o o n o o CA j cn vO o o o o o o o o o O O •£> <—i <—I CO m v o r - CO CO CO o o o o o o o o CO o o <—i O o o o o XI o o o o o <J- o o CO o o Is- o o CO o o ~M o m —1 —1 o o o o o o o o o O \0 Oi rH CO rH CO O <f CO O CA ' T " I I CO r** tO LO "af CO NON- bAb I S V A R I A D L E S VARIAbLE UN IT LEVEL Kc TURN SHADOW PRICE • . 1 • 2 -20.000 -20.000 29.4954 23.7199 3 4 5 -20.000 -20.000 -20.000 2 7.9281 32.9924 24.8572 - " 8 . 9 12 -20.000 -20.000 -20.000 3.3643 14.5122 3.8068 13 14 20 -6.000 -6.000 -6.000 7.5323 .9923 - 5.1173 "22 23 '24 -6.000 • -6.000 -2.000 43.6560 15.0928 86.7777 2 5 26 27 -2.000 -2.000 -2.000 70.4006 60.3058 50.2109 - 28 29 30 -2.000 -2.000 -2.000 45.2788 11.2185 22.4743 • '• 31 33 34 -2.000 -2.000 : ' -2.000 15.1570 50.6392 19.8760 36 37 39 260.000 210.000 240.000 12.5782 52.5782 - 12.5782 41 42 43 13 0.000 60.000 50.000 12.5700 68.7040 78.7040 44 45 48 60.000 50.000 225.000 82.9860 107.2681 4.9999; '•• • 49 50 51 100.000 110.000 72.000 19.3920 9.73.60 5.1137 52 . 55 57 72.000 67.000 67.000 7.5543 . 23.9689 4 0.4097 4 3 58 72.000 72.6257 59 66.000 62.2486 60 I 60.000 58 . 1538 61 54.000 54.0589 62. 42.000 61.1268 63 36.000 33. 0665 64 33.000 28.328Q 65 30.000 ' 24.0107 66 27. 000 ' 1 1 .8536 67, 42.000 65.9816 68 33.000 44.2184 69 2 7. 000 30_. 3424_ 82 ' 173.000 " ' 93."30 9 2 83 . -. 173. 000 67. 5360 84 110.000 20.7982 89 110.000 36.8095 91 18.000 41.8480 92 21. 000 _ _ 1 L » JL5?Jk. 93 21.000 38.3424 BIBLIOGRAPHY BOOKS Baumol, W. J. Economic Theory and Operations Analysis. Prentice H a l l , C a l i f . (1962). Charnes, A. ¥., Cooper, W. N., and Henderson, A. An Introduction to Linear Programming. Wiley & Sons Inc. New York, (I960). Dorfman, R., Samuelson, P.A., and Solow, R.M. Linear  Programming and Economic Analysis. McGraw H i l l , New York, (1958). Davis, J . , Ross, D. W., Scott, A. D. and Sewell, W. R. D. Benefit-Cost Analysis Handbook. Queen's Prin t e r , Ottawa, (1962). Guthrie, John A., and Armstrong, George, R. Western  Forest Industry - An Economic Outlook. The John Hopkins Press, Baltimore, (1961). Moore, A. Milton. Forestry Tenures and Taxes i n Canada. Canadian Tax Foundation, Toronto, (1957). Scott, Anthony. Natural Resources - The Economics of  Conservation. Univ. Toronto Press, (1955). Zaremba, Joseph. Economics of American Lumber Industry. JOURNAL ARTICLES C u r t i s , F. H. "Linear Programming the Management of a Forest Property." Journal of Forestry, v. 60, Sept. 1962), pp. 611-616. Hamlin, M. E. "Experience Report - Wastewood and Chip Volume Measurement." Amer. Pulpwood Assoc. North east Tech. Comm. Minutes, (1956;, pp. 12-14. Jackson, N. D., and Swinton, G. W. "Linear Program ming i n Lumber Production." For. Prod. Jour, v. XI, (June, 1961). Koenigsberg, E., "Linear Programming Applied to the Ply wood Industry." For. Prod. Journ., v. XI, (Sept. I960), pp. 481-486. Lane, Paul. "Evaluating Log and True Quality for Wood Products." For. Prod. Journ., v. XIX, (Mar. 1963), PP. 89-93. Mayhew, W. E. "A New Method of A l l o c a t i n g Costs to Veneer by Grades." For. Prod. Jour, v. 3 , (Aug. 1 9 5 8 ) , pp. 27A-30A. McBride, C. F., and Kinghorn, J . M. "Lumber Degrade Caused by jAmbrosia B e a t l e s . " B r i t i s h Columbia  Lumberman, v. 4 4 , ( J u l y , i 9 6 0 ) , pp. 80-83. McBride, C. F. "Barking and Chipping i n the I n t e r i o r of B r i t i s h Columbia." Canada Lumberman, v. 8 3 , ( J u l y , 1 9 6 3 ) , PP. 5 3 - 5 5 . P a u l l , A. E. "Li n e a r Programming - A Key to Optimum Newsprint Production." Pulp and Paper Mag, of Canada, v. 5 7 , ( 1 9 5 6 ) , pp. 1 4 5 - 1 5 1 . " Rankin, A. G. "C o s t - P r i c e R e l a t i o n s h i p s i n the Forest Industry." The F o r e s t r y C h r o n i c l e , v. 3 9 , (Mar. 1 9 6 3 ) , PP. 6 9 - 7 9 . S c o t t , Anthony, D. "The Development of the E x t r a c t i v e I n d u s t r i e s . " The Can. Journ. Econ. and P o l . Science, v. 2 8 , (Feb. 1 9 6 2 ) . GOVERNMENT AND INSTITUTIONAL PUBLICATIONS Anon. Conversion Factors per P a c i f i c Northwest Forest Products. I n s t i t u t e of Forest Products, S e a t t l e , Wash., ( 1 9 5 7 ) . Annual Report 1962. B r i t i s h Columbia Forest S e r v i c e , B.C. Dept. of Lands and F o r e s t s , V i c t o r i a , (1963). B r i t i s h Columbia Dept. of Lands and F o r e s t s . Continuous Fo r e s t Inventory of B r i t i s h Columbia, V i c t o r i a , (195^8). B r i t i s h Columbia Forest S e r v i c e . "Progress to September, 1962, and fu t u r e Prospects of the B r i t i s h Columbia Sustained Y i e l d F o r e s t r y Program." Submitt a l to Tar iff Commission of the United S t a t e s of America, Wash., D.C , v. I. B r i t i s h Columbia Forest S e r v i c e . "The C o n d i t i o n and Operation of the Open Log-Market and Chip and Small- Wood Marketing i n the Vancouver Forest D i s t r i c t . " Rept. to the S e l e c t Standing Committee of F o r e s t r y of  the L e g i s l a t i v e Assembly of B r i t i s h Columbia, V i c t o r i a , (Jan. 1962) - supplied i n confidence to the author. B r i t i s h Columbia Lumber Manufacturers A s s o c i a t i o n . Annual  Report, 1963. Vancouver, (1964). Clarke, E. H. "Lumber Grade Recovery from Old-Growth Douglas F i r at a Northwestern Oregon Sawmill^" U.S. Dept. of A g r i c , P a c N.W. Forest and Range Expt. Sta. Research Note No. 1 9 1 , Portland, Oregon,(Oct. I960). Clarke, E. H., and Knauss, A. C. "Veneer Recovery from Douglas F i r Logs." U.S. Dept. of A g r i c , Pac. N.W. Forest and Range Expt. Sta. Research Paper No. 2 3 , Portland, Oregon, ( 1 9 5 7 ) . Canadian Standards Association. Douglas F i r Plywood - C S A / 0 1 2 1 / 1 9 . Queen's Pri n t e r s , Ottawa, ( 1 9 6 1 ) . Dominion Bureau of S t a t i s t i c s . Catl'g. No. 3 5 - 2 0 6 . Ve neer and Plywood M i l l s . Queen's P r i n t e r s , Ottawa, ( 1 9 6 3 ) . Dominion Bureau of S t a t i s t i c s . Catl'g. No. 3 6 - 2 0 4 . Pulp and Paper M i l l s . Queen's P r i n t e r s , Ottawa, ( 1 9 6 3 ) . Dominion Bureau of S t a t i s t i c s . Catl'g. No. 3 5 - 0 0 3 . Production, Lumber M i l l s . Queen's P r i n t e r s , Ottawa. Dobie, J. "A M i l l i n g Study of 1 5 0 - y e a r Old Douglas F i r . " Forest Products Research Branch Publication No. 1 0 3 2 . Canada Dept. of Forestry, Ottawa, ( 1 9 6 2 ) . Dowdle, Barney, and Bain, Robert. "Lumber or Chips - A Comparison of Small-log U t i l i z a t i o n A l t e r n a t i v e s . " V.S. Dept. of A g r i c , Northeastern For. Expt. Sta. (I960). Guernsey, F. W. "Some Conversion Factors for B r i t i s h Columbia Forest Products." Forest Products Labor atory of Canada Publication No. V - 1 0 2 7 , Dept. Nth. A f f a i r s , Ottawa, ( 1 9 5 9 ) . Jenkins, J. H. "Grade Marking of Canadian Lumber." Report No. 1 8 9 . For. Prod. Research Branch, Can. Dept. of Forestry, Ottawa, (Sept. 1 9 6 2 ) . Matson, E. E. "Lumber Grade Recovery from Oregon Coast Type Douglas F i r . " U.S. Dept. of Agriculture, P a c N.W. Forest and Range Expt. Sta. Research Paper No. 3 , Portland, Oregon, (May, 1 9 5 2 ) . Matson, E. E. '.'-Lumber Grades from Young-Growth Douglas Fir.'" U.S. Dept. of A g r i c , Pac. N.W. Forest and Range Expt. Sta. Research Note No. 7 9 , Portland, Oregon, (Sept. 1 9 5 2 ) . Matson, E. E. "Lumber Grades from Douglas F i r Peeler Logs." U.S. Dept. of A g r i c , Pac. N.tf. Forest and Range Expt. S t a . , Research Note No. 8 3 , P o r t l a n d , Oregon, (Nov. 1 9 5 2 ) . Matson, E. E. "Lumber Grades from Old-Growth Douglas F i r Sawmill Logs." U.S. Dept. of A g r i c , Pac. N.W. For est and Range Expt. S t a . , Research Note No. 1 2 5 , (Jan. 1 9 5 6 ) . Newport, C a r l , A., Lockard, C. R., and Vaughan, C. L. Log and Tree Grading as a Means of Measuring Q u a l i t y . " Rept. of Working Group as approved by the N a t i o n a l Log Grade Committee, Madison, Wis. May, 1 9 5 8 . - Forest S e r v i c e , U.S. Dept. of A g r i c , Wash. D.C. (May, 1 9 5 9 ) . UNPUBLISHED PAPERS McBride, C. F. "Conversion Factors f o r B. C. Coast Forest Industry." - Mimeo Notes f o r Lectu r e , ( 1 9 6 2 ) . Zinuvska, John, A. "The Future f o r Wood i n a Competitive Market." Paper f o r p r e s e n t a t i o n at j o i n t meeting Puget Sound and Columbia R i v e r s e c t i o n s , S o c i e t y of American F o r e s t e r s , Longview, Wash. (May 4 , 1 9 6 3 ) . X >< JX X x X - x X B X sx X X X X X >< X * x x „x J ,4 I I I N T E R M E D I A T E N P U T5 •3> i— 3 .1 .i K .1 .1 ro .1 .1 o> O J O O .1 T--i C D CO O O .1 .1 O J O O . 1 ro -1 c s C D 1 O O -A . 1 O * O J . 1 £ J i l J_ Co o O . 1 O J .1 <=> ON .1 .J_ £ S >^ M il i i -» cr» 8 « Hi a CP m .1 .1 CO CO . , O J — Ca co *n c» .1 .1 •=> »^» =r 1!° 0~-» 0~> r-o .1 r% c, A, — — J* T .i .i — rv. — <—> O J <r> o 2 t o — J — I .1 J _ ! <-C H5 J- "J- i i O-i <-» Oo .1 .1 ;l —. OJ 4> r o .i J .i P R I M A R Y I N P U T S 33 * * * * * •» * * * * * O j r \ J — O J NI OJ I N r v O 0 c a- ? > cn. m O o "D m m o Q cn o =>• T J - T J c n o T J 33 > o > o o > cz ux X x r- i * X It * x x _s X <X IfX Jx >>< .rx *~ ? x to CD Ci s § g -a 1— e i ° O T J 3D o r-i t nil "a T l I a r r-O cn rt CD r <n>Tj HOO-FUEL > > < —I rn cn O O o o 0 0 0 0 0 0 0 0 0 O O O O O O O O O O O O — to — c o - ( ^ ~ r s j — o i f u j o > c > a i o i s j u j H i . h j - > 33 > 2 o D m r 

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