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UBC Theses and Dissertations

Log allocation by dynamic programming Bailey, Gordon Raymond 1970

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LOG ALLOCATION by DYNAMIC PROGRAMMING by GORDON R. BAILEY B . S c , U n i v e r s i t y of Aberdeen, 1952 M . F . , U n i v e r s i t y of B r i t i s h Co lumbia , 1964 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department o f FORESTRY We accept t h i s t h e s i s as conforming to the requ i red s tandard THE UNIVERSITY OF BRITISH COLUMBIA September, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f f~OA£^Tr\y The U n i v e r s i t y o f B r i t i s h Co lumbia V a n c o u v e r 8, Canada D a t e ? U0\/EHh£r/l I97Q ABSTRACT This t h e s i s desc r ibes an o p t i m i z a t i o n model f o r the a l -l o c a t i o n of logs from a l t e r n a t i v e sources f o r a s e r i e s of s u c -c e s s i v e t ime p e r i o d s . The model was formula ted as a m u l t i - s t a g e d e c i s i o n process and i s so lved by dynamic programming. The a n a l y t i c framework c o n s i s t s of two connected recur rence equa-t i o n s , each i n c o r p o r a t i n g two d e c i s i o n - v a r i a b l e s . These two r e -l a t i o n s h i p s , together w i th the f e a s i b i l i t y reg ions de f ined f o r a h y p o t h e t i c a l prob lem, desc r i be the o p t i m i z a t i o n p rocess . The hypo the t i ca l a l l o c a t i o n problem, used as a v e h i c l e f o r model development, requ i res m i l l demands f o r q u a n t i t i e s of pu lp logs and sawlogs to be s a t i s f i e d f o r three t ime p e r i o d s . Logs are d e l i v e r e d from f i v e a v a i l a b l e s o u r c e s , four l o g -producing areas and an open log market , and temporary log s u r -p luses are a l l owed . Only a l i m i t e d quan t i t y of logs i s a v a i l a b l e from each source i n each p e r i o d , two o f the f ou r f o r e s t areas supply on ly pu lp logs and a t h i r d area i s i n a c c e s s i b l e i n one p e r i o d . - The v a r i a b l e u n i t cos ts of d e l i v e r e d logs d i f f e r not on ly between each source and per iod but are a l s o dependent upon •the magnitude of an a l l o c a t i o n . In a d d i t i o n to s a t i s f y i n g m i l l demands f o r s p e c i f i e d q u a n t i t i e s of l o g s , there i s a f u r t h e r requirement w i th regard i v to sawlogs . Average lumber p r i c e s are assumed to be dependent upon the t ree spec ies processed and v a r i a b l e log convers ion cos ts are assumed to decrease w i th i nc rease i n l og d iameter . Consequent ly , the comparison of a l t e r n a t i v e a l l o c a t i o n p o l i c i e s . i n vo l ves not only the sum of the v a r i a b l e d e l i v e r e d log cos ts but a l s o a measure of the va lue of d e l i v e r e d sawlogs. In the t h e s i s "sawlog net wor th" i s eva lua ted and combined w i th log p roduc t ion and log t r a n s p o r t a t i o n cos ts to g i ve a composite c o s t te rm, "ne t d e l i v e r e d log c o s t . " Th is i s the measure used to eva lua te each a l l o c a t i o n and i s i nco rpo ra ted i n the f i r s t o f the two recur rence equa t i ons . Th is equat ion i s used to de-r i v e minimum c o s t a l l o c a t i o n p o l i c i e s f o r a l l p o s s i b l e q u a n t i -t i e s tha t may be a l l o c a t e d from each p e r i o d . In the second equat ion the minimum cos ts de r i ved from the f i r s t a l l o c a t i o n process are combined w i th a second term t o , e v a l u a t e a l t e r n a t i v e a l l o c a t i o n s between p e r i o d s . Th is second c o s t component i s i ncu r red when su rp lus logs are " c o l d -decked" f o r subsequent m i l l c o n v e r s i o n . . To ensure a s u f f i c i e n t f low of l o g s , and to take advantage of seasonal d i f f e r e n c e s i n "net d e l i v e r e d log c o s t s , " log su rp luses are pe rm i t t ed . When log su rp luses are " co ld -decked " - a d d i t i o n a l l og hand l ing cos ts a r e , i n c u r r e d which must be cons idered when a l t e r n a t i v e a l l o c a -t i ons between per iods are e v a l u a t e d . Th is requirement i s s a t -i s f i e d by i n t r oduc ing the a d d i t i o n a l cos t component i n t o the second equa t i on . V The d i f f e r e n t d e r i v a t i o n s g iven f o r the "net d e l i v e r e d log c o s t s " f o r f i v e sources of logs demonstrate an important f ea tu re of the f o r m u l a t i o n : there i s no requirement t ha t cos ts must be l i n e a r l y r e l a t e d to the quan t i t y a l l o c a t e d . Th is freedom i s we l l i l l u s t r a t e d by the i n t r o d u c t i o n of a d d i t i o n a l " f i x e d " cos ts which are dependent upon the magnitudes of the q u a n t i t i e s a l l o c a t e d . In a demonstrat ion of the f l e x i b i l i t y of the f o rmu la t i on a complex log p roduc t ion system was assumed f o r source two. For each a l l o c a t i o n from t h i s source two opt imal q u a n t i t i e s were d e r i v e d . The f i r s t was the quan t i t y o f sawlogs s e l e c t e d from s p e c i f i e d log c l a s s e s , the second was the opt imal p o r t i o n o f the quan t i t y of pee le r logs developed tha t should be t r aded . An exchange cou ld be made e i t h e r f o r saw logs , or f o r p u l p l o g s , or bo th . With the development of t h i s log a l l o c a t i o n model a f r esh approach to log p roduc t ion p lann ing i s now p o s s i b l e . The a n a l y t i c framework i s capable of ex tens i ve adap ta t ion and the model i t s e l f can be r e a d i l y mod i f ied to s u i t a v a r i e t y of con-d i t i o n s . Whether used as d e s c r i b e d , or as pa r t of a l a r g e r a n a l y t i c a l sys tem, the computat ional advantages of dynamic p ro -gramming are now a v a i l a b l e to the p lanne r . TABLE OF CONTENTS Chapter Page I. INTRODUCTION . < . . . . . . , . . . . . . . . . 1 Pas t Work . . . . . . . . . . . . . . . . 1 The Thes is Model 6 The Hypo the t i ca l Problem . . . • . . . . . . . . 7 Convers ion P lan ts . . . . . . . 8 Log Supply 11 Log Storage . . . 12 Economic Ob jec t i ve 13 I I . THE FORMULATION AND DEVELOPMENT OF THE MODEL 16 Be l lman 's A p p l i c a t i o n of Dynamic Programming to a Resource A l l o c a t i o n Problem . . . 19 Ex tens ion of the B a s i c Recurs ive Equat ion 22 A p p l i c a t i o n of the Extended Recurrence Equat ions to Log A l l o c a t i o n 27 A l l o c a t i o n from Log Sources 27 A l l o c a t i o n between Per iods . . . . . . . . . . 31 I I I . DERIVATION OF NET DELIVERED LOG COSTS . . . . . . . • 40 I n t r oduc t i on . . . . . . . . . . . . . . . 40 Return per Un i t of De l i ve red Sawlogs . . . . . . 42 Net D e l i v e r e d Log Costs . . . . . . . . . . 44 v i i Chapter " Page Log Costs f o r Each Source 46 Log Costs f o r Source One . . . . . . . . . . 47 Net D e l i v e r e d Log Costs . . . . . . . . . 48 Log Costs f o r Source Two . . . . . . . . . . 48 Log S o r t i n g and Trad ing 51 Costs t ha t are Independent of Log S o r t i n g and Trad ing P o l i c i e s . . . . . . . . . 55 S t a r t - u p Costs - . . 55 V a r i a b l e U n i t Costs to Woods Landings . . . 56 Costs t ha t are Dependent upon Log S o r t i n g and Trad ing P o l i c i e s . . . . . . . . . 58 D i f f e r e n t i a l Sawlog Net Worth 59 Optimal Log S o r t i n g and Trad ing P o l i c i e s . . . 60 Net D e l i v e r e d Log Costs . 64 Log Costs f o r Source Three . . . • . • 64 Log Produc t ion and T ranspo r ta t i on Costs . . . . 65 Optimal Cos t ing P o l i c i e s 66 D i f f e r e n t i a l Sawlog Net Worth 70 Net De l i ve red Log Costs 72 . Log Costs f o r Source Four . . . . . . . . . 72 Net D e l i v e r e d Log Costs 74 Log Costs f o r Source F ive 75 Net De l i ve red Log Costs 76 v i i i Chapter Page IV. RESULTS . . . . . . . . . . 78 D i s c u s s i o n o f the S o l u t i o n to the Problem . . . . . . 78 Optimal A l l o c a t i o n P o l i c y . . . 79 Log A l l o c a t i o n from A l t e r n a t i v e Sources . . . . . 83 Optimal A l l o c a t i o n among Sources . . . . . . . 90 Log A l l o c a t i o n between Per iods . . . . . . . . 93 Optimal A l l o c a t i o n between Per iods 94 D i s c u s s i o n of the Formulated Model . . . . . . . • 94 Mandatory Assumptions . . . . 97 M o d i f i c a t i o n of the Model 100 S e n s i t i v i t y and E r r o r Analyses .- 102 V. CONCLUSIONS . . . . . . . . . . . . . . . . 108 LITERATURE CITED . 112 APPENDICES . • . . . . 115 A. COSTS OF LOG SORTING, LOADING AND TRANSPORTING AND DIFFERENTIAL SAWLOG NET WORTH FOR SOURCE TWO . . . . 116 S o r t i n g and Loading Costs . . . . . • 117 Log T ranspo r t a t i on Costs . . . . . . . . . . . 122 D i f f e r e n t i a l Sawlog Net Worth . . . . . . . . . 123 i x Chapter Page B: LISTING OF THE COMPUTER-CODED PROGRAMME "DYALOG" . • . . • . 128 Coded Terms used i n DYALOG '. . . . . . • . . . . • . 129 L i s t i n g of DYALOG . . . . . . . . . • . . . • . 136 C. OUTPUT TABLES GENERATED BY "DYALOG" . . . . . 147 Symbol Guide f o r Table Se r i es . - . . • 148 L i s t i n g of Output Tables Generated by DYALOG . . . . 150 INDEX 205 LIST OF TABLES Table Page I. F i r s t Optimal A l l o c a t i o n P o l i c y . . . . 80 I I . Second Optimal A l l o c a t i o n P o l i c y . . 82 I I I . A l l o c a t i o n s from Source 1 f o r Per iod 1 84 IV. A l l o c a t i o n s from Source 2 f o r Pe r i od 1 85 V. A l l o c a t i o n s from Source 3 f o r Pe r i od 1 86 V I . A l l o c a t i o n s from Source 4 f o r Pe r i od 1 87 V I I . A l l o c a t i o n s from Source 5 f o r Pe r i od 1 88 V I I I . Optimal A l l o c a t i o n w i t h i n Pe r iod 1 . . 91 IX. Optimal A l l o c a t i o n between Pe r iod 1 and a l l Pas t Per iods . . . . . . . . . . . . . . . 95 X. Optimal A l l o c a t i o n between Pe r iod 2 and a l l Past Per iods . . . . . . . . 96 X I . P r e c i s i o n of the Magnitude or Value Computed by "DYALOG" f o r Se lec ted V a r i a b l e s A f f e c t e d by Log S o r t i n g and Trad ing P o l i c i e s f o r Source 2 . . . . . 106 LIST OF FIGURES F igure Page 1. Boundar ies of the Log A l l o c a t i o n Problem . 9 2. Log Flow Diagram . . . . . - . . . . . . . . . . . 10 3. F lowchar t of the Log A l l o c a t i o n Model ' . . . . . . . . 17 4 . R e l a t i o n s h i p between. f t k ( X t J Y t ) and F t ( X , Y \ 26 5. The E f f e c t of Log Diameter and Species on Sawlog Value . . 45 6. D e l i v e r e d Cost of Pulpwood from Source One . . . . . . 49 7. F lowchar t of the D e r i v a t i o n of Net D e l i v e r e d Log Costs f o r Source Two . • . 50 8. Volume Components of Q u a n t i t i e s A l l o c a t e d from Source Two . . . . 54 9 . V a r i a b l e Log Costs to Woods Landings f o r Source Two . . . 57 10. Response Sur faces f o r Log S o r t i n g and Trad ing P o l i c i e s f o r = 3 , = 1 • • • • 62 11. Response Sur face f o r Log S o r t i n g and Trad ing P o l i c i e s f o r x - ^ = 3 , y - ^ = 2 . . . . . . . . . . 63 12. Log Cost Components f o r Source Three . . . . . . . . 67 13. F lowchar t of the D e r i v a t i o n of Net De l i ve red Log Costs f o r Source Three 73 14. D e l i v e r e d Cost o f Pulpwood from Source F ive 77 15. Combined.Cost of Loading Pee le r and Non-Pee ler Logs . . . 118 16. Cost of S e l e c t i n g the Quant i t y of Pee le r Logs requ i red f o r Trad ing . . . . • . - . . . . . . . . • 120 17'.. Cost per Thousand Cun i ts of S o r t i n g Non-Traded Logs . '• . . 121 ACKNOWLEDGMENTS The author wishes to acknowledge the help and use fu l c r i t i c i s m g iven by h i s t h e s i s s u p e r v i s o r , Dr . C. W. Boyd, Facu l t y of F o r e s t r y , dur ing the two-year term of the s tudy . His con t i nu ing and e n t h u s i a s t i c reviews a t a l l stages dur ing the development of the t h e s i s i s g r e a t l y v a l u e d . The general d i r e c t i o n and guidance g iven to the author over many years by Dr . J . H. G. Sm i th , Facu l t y of F o r e s t r y , i s deeply a p p r e c i a t e d . As pro-tern research s u p e r v i s o r h i s adv ice and encouragement, i n l a rge measure, s t imu la ted the a u t h o r ' s i n t e r e s t i n mathematical-model b u i l d i n g . The c r i t i c a l reviews by the remaining Committee members, Dr . L. G. M i t t e n , F a c u l t y of Commerce, Dr . M. J . D o r l i n g , Facu l t y of A g r i c u l t u r e , and D rs . D. Haley and A. Kozak of the Facu l t y of Fo res t r y were of g rea t b e n e f i t . The author i s g r a t e f u l f o r t h e i r c o n s t r u c t i v e c r i t i c i s m and comments. The f i n a n c i a l a s s i s t a n c e from the Department of F i s h e r i e s and Fo res t r y i n the form of educa t iona l l e a v e from the Vancouver Fo res t Products Laboratory and the r e c e i p t of two U n i v e r s i t y Fo res t Fe l l owsh ips from the Facu l t y of F o r e s t r y , U n i v e r s i t y o f B r i t i s h Co lumbia , i s very much a p p r e c i a t e d . GLOSSARY OF TERMS "Backward pass" = a computat ional p rocedure , i n which the sequence of s tages examined i n the " forward pass" i s r e v e r s e d , from which a s e t of opt imal a l l o c a t i o n s i s i d e n t i f i e d from the a l l o c a t i o n s made i n the " forward p a s s . " c t k ^ x t k ' ^ t k ^ = " n e t d e l i v e r e d log c o s t " of a l l o c a t i n g the q u a n t i t i e s x ^ and from source k i n pe r iod t . c c ' t k ^ x t k ' ^ t k ^ = s u m ° ^ t ' i e v a r i a ' 3 ^ e 1°9 p roduc t ion and log t r a n s p o r t a -t i o n cos ts i n c u r r e d when the q u a n t i t i e s x . ^ and are d e l i v e r e d to the p u l p m i l l and sawmi l l y a r d s , r e s p e c t i v e l y . ch = u n i t v a r i a b l e log hand l ing cos t per thousands of c u n i t s when su rp lus logs are p laced i n , and taken out o f , " c o l d -deck" s to rage areas i n m i l l y a r d s . c f = cos t of en l a rg i ng land ings and c o n s t r u c t i n g a d d i t i o n a l spur roads f o r source one. c f l = s t a r t - u p cos ts when logg ing i s i n i t i a t e d i n any pe r iod on source two. c f2 = an a d d i t i o n a l f i x e d c o s t - i n c u r r e d when the t o t a l volume x i v cut from source two i n any pe r iod i s g rea te r than the quan t i t y vn^ . c ^ t ^ x t 2 ' ^ t 2 ^ = P o r t i ° n ° f the net d e l i v e r e d log cos t f o r source two i n pe r i od t which i s independent of log s o r t i n g and t rad ing p o l i c i e s , expressed as a f u n c t i o n of the cut volumes x t 2 and y t 2 . c l l ^ . = v a r i a b l e c o s t . p e r thousand c u n i t s of logs yarded to woods land ings when the volume cut on source two i n pe r iod t does not exceed the q u a n t i t y vn^ . c l 2 , = v a r i a b l e c o s t per thousand c u n i t s of logs yarded to woods land ings f o r the p o r t i o n of the volume cu t on source two i n pe r iod t which exceeds the q u a n t i t y v n ^ . cm = time requ i red to produce one thousand cub ic f e e t ( ten c u n i t s ) of lumber, expressed as number o f .minutes x log d i a m e t e r - i n c h . cn = v a r i a b l e cos t per thousand c u n i t s of l oad ing non-pee le r 1ogs. -" c o l d - d e c k " = an exp ress ion commonly i n use i n the f o r e s t i ndus t r y which r e f e r s to a quan t i t y o f logs s to red f o r a pe r iod c o r r e s -ponding to a planned de lay i n log p r o c e s s i n g . cp = v a r i a b l e c o s t per thousand c u n i t s of l oad ing pee le r l o g s . XV c s t ^ x t 2 ' ^ t 2 ' ^ c ' t r ^ = s u m ° ^ ^ n e v a n a D 1 e c o s t s of l og s o r t i n g and l o a d -ing when the q u a n t i t i e s x ^ and are a l l o c a t e d from source two i n per iod t f o r a log s o r t i n g and t r ad ing p o l i c y denoted by the quan t i t y yc -and the r a t i o t r . ess = v a r i a b l e cos t per thousand c u n i t s of log s e l e c t i o n and temporary s t o c k - p i l i n g a t woods l a n d i n g s . c t t ( x ^ 2 > y ^ - 2 » y c ' t r ) = v a r i a b l e net t r a n s p o r t a t i o n cos t i ncu r red when the q u a n t i t i e s x ^ and y ^ a r e d e l i v e r e d to m i l l yards from source two i n pe r iod t f o r a log t r ad ing and s o r t i n g p o l i c y denoted by the quan t i t y yc and the r a t i o t r . c t t ^ = v a r i a b l e c o s t per thousand c u n i t s i ncu r red i n pe r iod t when logs are t ranspor ted from source two to the log market and , a f t e r exchange, an equal l og volume i s d e l i v -ered to the p u l p m i l l or s a w m i l l . c t x ^ = v a r i a b l e cos t per thousand c u n i t s of t r a n s p o r t i n g logs d i r e c t l y from source two to the p u l p m i l l i n pe r iod t . cty^. = v a r i a b l e cos t per thousand c u n i t s of t r a n s p o r t i n g logs d i r e c t l y from source two to the sawmil1 i n pe r iod t . " c u n i t " = wood volume of one hundred cub ic f e e t ; C w t k ^ x t k ' y t k ^ = " d i f f e r e n t i a l sawlog net wor th" m u l t i p l i e d by the quan t i t y y t 2 . xv i c w t ^ x t 2 ' V t 2 ' V c ' t . r O = " d i f f e r e n t i a l sawlog net wor th" of the d e l i v e r e d volume, y^2» when the q u a n t i t i e s x ^ and y ^ are a l l o c a t e d from source two i n pe r iod t f o r a - l o g s o r t i n g and t r ad i ng p o l i c y denoted by the quan t i t y yc and the r a t i o t r . c X j . = cos t of purchased logs per thousand c u n i t s d e l i v e r e d to the p u l p m i l l i n per iod t . c x ^ t k = v a r " ' a ' 3 l e c o s t per thousand c u n i t s of pu lp logs d e l i v e r e d from source k i n per iod t.when the cut volume does not exceed the quan t i t y vm^ f o r k = 1 , 3 , 5 . 0 x 2 ^ = v a r i a b l e c o s t per thousand c u n i t s of pu lp logs d e l i v e r e d from source k i n pe r iod t f o r the p o r t i o n of the cut volume which exceeds the quan t i t y vm^ f o r k = 1 ,3 ,5 . cy^ = c o s t of purchased logs per thousand c u n i t s d e l i v e r e d to the sawmi l l i n pe r iod t . c y l ^ g = v a r i a b l e c o s t . p e r thousand c u n i t s of sawlogs d e l i v e r e d from source three i n pe r iod t when the cut volume does not exceed the quan t i t y vm^. ' c y 2 ^ = v a r i a b l e cos t per thousand cun i t s of sawlogs d e l i v e r e d from source three i n pe r iod t f o r the p o r t i o n of the cut volume which exceeds the quan t i t y vnig.. D ^ = average log d iameter i n inches of the quan t i t y y ^ . " d i f f e r e n t i a l sawlog net wor th" = d i f f e r e n c e i n d o l l a r s between the sawlog net worth of the quan t i t y y ^ conver ted i n t o lumber and the sawlog net worth of an equal quan t i t y of sawlogs having an average log diameter of Ds inches and con ta in i ng Ps.100 per cent by volume of D o u g l a s - f i r l o g s . d l = average log d iameter o f western hemlock logs i n the l a rge log c l a s s . d ip = average log diameter of D o u g l a s - f i r logs i n the l a rge log c l a s s . Ds = average log d iameter i n inches of the quan t i t y y ^ s e l -ected from the t o t a l q u a n t i t y , x^3+y^3> a l l o c a t e d from source three when y ^ ± ^- xt3* ds = average log diameter of western hemlock logs i n the smal l log c l a s s . dsp = average log diameter of D o u g l a s - f i r logs i n the smal l l og c l a s s . dp- = average log d iameter i n inches of pee le r l o g s . dt^. = average log d iameter i n inches of the quan t i t y of logs a v a i l a b l e from the l o c a l log market i n per iod t . x v i i i Dx^. = demand f o r p u l p l o g s , i n thousands of c u n i t s , i n pe r iod t . Dy^.. = demand f o r saw logs , i n thousands of c u n i t s , i n per iod t . . c ' - y t ^ x t 2 ' ^ t 2 ' ^ c > t r ^ = a v e r a 9 e d iameter i n inches of the quan t i t y yc when the q u a n t i t i e s x ^ and y^ are a l l o c a t e d from source two i n pe r iod t f o r a log s o r t i n g and t r ad ing p o l i c y de^ noted by the quan t i t y y c a n d the r a t i o t r . f t ^ ( X t , Y t ) = minimum net d e l i v e r e d log c o s t of a l l o c a t i n g the q u a n t i -t i e s Xj. and Yj. from sources 1 to k i n per iod t . F ^ X ^ . Y ^ ) = minimum cos t o f a l l o c a t i n g the q u a n t i t i e s X and Y from a l l a v a i l a b l e sources i n 1 to t p e r i o d s . " forward pass" = a computat ional process f o r d e r i v i n g opt imal a l l o c a -t i ons from a s e r i e s of stages by comparing a l l o c a t i o n s from each success i ve stage w i th a l l o c a t i o n s from a l l pre^ v i o u s l y examined s t a g e s . g ^ X ^ j Y j . ) = sum of the v a r i a b l e log hand l ing cos ts i ncu r red i n per iod t by " c o l d - d e c k i n g " su rp lus logs from an a l l o c a t i o n of the q u a n t i t i e s X^ and Y ^ . INDEX t (x 1 .2>y t 3) = numerical i ndex , 0 , 1 , 2 , 3 , which r e f e r s to the opt imal c o s t i n g p o l i c y f o r source three where 0 = no a p p l i c a b l e p o l i c y , 1 = l e a s t cos t p r i o r i t y to p u l p l o g s , . 2 = l e a s t x i x c o s t p r i o r i t y to saw logs , and 3 = s i n g l e log product o n l y . " l a r g e log c l a s s " = merchantable length non-pee ler logs having a m i n i -mum top d iameter of e i gh t i n c h e s . mc = v a r i a b l e m i l l convers ion c o s t per 100..minutes. ms = minimum amount to be pa id to the c o n t r a c t o r f o r d e l i v e r e d l o g s . "net d e l i v e r e d log c o s t " = sum of the v a r i a b l e log p roduc t ion and log t r a n s p o r t a t i o n cos ts combined w i th the " d i f f e r e n t i a l saw-log net wor th" of the quan t i t y y ^ . = r a t i o of the volume of D o u g l a s - f i r logs to the quan t i t y y t k -" p o l i c y t a b l e " = t a b u l a t i o n s , f o r each s tage examined i n a " forward p a s s , " of the opt imal a l l o c a t i o n s f o r tha t p a r t i c u l a r s t a g e , and the eva luated c o s t f u n c t i o n . Ps = r a t i o o f the volume of D o u g l a s - f i r to the quan t i t y y ^ s e l e c t e d from the t o t a l q u a n t i t y , x t 3 + y t 3 » a l l o c a t e d from source three when y ^ 1 ^ - x t 3 ' pc = r a t i o of the volume of pee le r logs to the a v a i l a b l e log supply f o r source two. XX pi = r a t i o o f the volume of D o u g l a s - f i r logs i n the l a rge log c l a s s to the t o t a l volume of the l a rge log c l a s s . pp = r a t i o o f the volume of D o u g l a s - f i r logs i n the pee le r log c l a s s to the t o t a l volume of pee le r l o g s . ps = r a t i o of the volume of D o u g l a s - f i r logs i n . t h e smal l log c l a s s to the t o t a l volume of the smal l log c l a s s . p t^ = r a t i o i n pe r iod t of the quan t i t y of D o u g l a s - f i r logs to the volume of logs a v a i l a b l e from the l o c a l log market . p y t ( x t 2 » y t 2 ' y c ' t r ) = r a t i o of the volume of D o u g l a s - f i r logs to the q u a n t i t y yc when the q u a n t i t i e s x ^ a n d y t 2 are a l l o c a t e d from source two i n per iod t f o r a log s o r t i n g and t r ad ing p o l i c y denoted by the quan t i t y yc and the r a t i o t r . Q-| and Q 2 = volume of D o u g l a s - f i r and western hemlock, r e s p e c t i v e l y , conta ined i n the l a rge log c l a s s . and = volume of D b u g l a s - f i r and western hemlock, r e s p e c t i v e l y , conta ined i n the smal l log c l a s s . rs = r a t i o of the volume of the smal l log c l a s s to the t o t a l volume of non-pee le r c l a s s e s of l o g s . "sawlog net wor th" = d i f f e r e n c e i n d o l l a r s per thousand c u n i t s o f l o g s ' between the f . o . b . p r i c e of manufactured lumber and xx i v a r i a b l e m i l l convers ion c o s t s . "smal l log c l a s s " = merchantable lengths of logs having a minimum top d iameter o f four inches a f t e r pee le r logs and la rge logs i n the non-pee le r log c l a s s have been manufactured. t c = f i x e d p r i c e d i f f e r e n c e per thousand c u n i t s between.t raded pee le r logs and an equal volume of replacement l o g s . t r = r a t i o of the volume of pee le r logs t raded f o r sawlogs to the quan t i t y of logs cut from source two. v ^ = quan t i t y of logs * i n thousands of c u n i t s , a v a i l a b l e from source k i n pe r iod t . vm^ = quan t i t y of l o g s , i n thousands of c u n i t s , which can be cut i n each per iod w i thout changing u n i t v a r i a b l e cos ts f o r k = 1 , 3 , 5 . v s . ^ = q u a n t i t y , i n thousands of c u n i t s , of sawlogs a v a i l a b l e from source k i n pe r iod t . w n t and w p t = average f . o . b . m i l l p r i c e s per thousand cun i t s i n pe r iod t f o r a l l lumber manufactured from western hemlock and D o u g l a s - f i r , r e s p e c t i v e l y , l e s s m i l l yard and a l l v a r i a b l e convers ion cos ts tha t are independent of l o g . d i a m e t e r ; X • = t o t a l quan t i t y of p u l p l o g s , i n thousands of c u n i t s , x x i i a l l o c a t e d dur ing the p lann ing c y c l e . X^ = quan t i t y of p u l p l o g s , i n thousands o f c u n i t s , a l l o c a t e d from a l l a v a i l a b l e sources i n pe r iod t . x ^ = quan t i t y of p u l p l o g s , i n thousands of c u n i t s , a l l o c a t e d from source k i n per iod t . X t (X^.,Y^.) = quan t i t y of pu lp logs a l l o c a t e d from per iod t when the q u a n t i t i e s X and Y are o p t i m a l l y a l l o c a t e d from 1 to t per iods i n a " forward p a s s . " * X t ( X t , Y t ) = opt imal a l l o c a t i o n of pu lp logs from per iod t , i d e n t i f i e d from a "backward p a s s . " x t k ^ t ' ^ = Quant i t y ° f pu lp l ogs . a l l oca ted i n pe r iod t from source k i n the " forward p a s s . " x t k ^ t * ^ = 0 P t l ' r T , a ^ a l l o c a t i o n of pu lp logs from source k i n each pe r iod t , i d e n t i f i e d from a "backward p a s s . " o x ^ X j Y ) = quan t i t y of pu lp logs a l l o c a t e d from source k i n each, pe r iod t when the q u a n t i t i e s X and Y are o p t i m a l l y a l l o c a t e d from a l l per iods i n the p lann ing c y c l e . x l = pu lp log volume component, i n thousands of c u n i t s , of the a l l o c a t i o n x^ j and y ^ which together w i th the quan t i t y y l does not exceed the quan t i t y vm^. x x i i i x2 = pu lp log volume component, i n thousands of c u n i t s , of the p o r t i o n o f the a l l o c a t i o n x ^ and y ^ which exceeds the quan t i t y vm^. Y = t o t a l quan t i t y of saw logs , i n thousands of c u n i t s , a l l o -cated dur ing the p lann ing c y c l e . Y t = quan t i t y o f saw logs , i n thousands of c u n i t s , a l l o c a t e d from a l l a v a i l a b l e sources i n pe r iod t . y ^ = quan t i t y of saw logs , i n thousands of c u n i t s , a l l o c a t e d from source k i n pe r iod t . Y t ( X t , Y t ) •= quan t i t y of sawlogs a l l o c a t e d from per iod t when the quan-t i t i e s X and Y are o p t i m a l l y a l l o c a t e d from 1 to t per iods i n a " forward p a s s . " Y t ( X t , Y t ) = opt imal a l l o c a t i o n o f sawlogs from per iod t , i d e n t i f i e d from a "backward p a s s . " y t k ^ t ' ^ t ^ = Quant i ty ° f sawlogs a l l o c a t e d i n pe r iod t from source k i n the " forward p a s s . " y t ( < ( X t , Y t ) = opt imal a l l o c a t i o n o f sawlogs from source k i n each pe r i od t , i d e n t i f i e d from a "backward p a s s . " o y ^ ( X , Y ) = quan t i t y o f sawlogs a l l oca ted from source k. i n each per iod t when the q u a n t i t i e s X and Y are o p t i m a l l y a l l o c a t e d x x i v from a l l per iods i n the p lann ing c y c l e . y l = sawlog volume component, i n thousands of c u n i t s , of the a l l o c a t i o n x . ^ and y ^ which together w i th the quan t i t y x l does not exceed the quan t i t y virig. y2 = sawlog volume component, i n thousands of c u n i t s , of the p o r t i o n o f the a l l o c a t i o n x ^ and y ^ which exceeds the q u a n t i t y vm^. yc = p o r t i o n , i n thousands of c u n i t s , of the quan t i t y y ^ tha t i s s e l e c t e d from the non-pee le r log c l a s s e s of source two. ^ t^ xt2'^t2^ = m ' ' n i ' n i u n i va lue i n pe r iod t of the sum of a l l v a r i a b l e log p roduc t ion and t r a n s p o r t a t i o n cos ts tha t are dependent upon the log s o r t i n g and t r ad ing p o l i c i e s of source two l e s s the " d i f f e r e n t i a l sawlog net wor th" de r i ved f o r each p o l i c y . LOG ALLOCATION BY DYNAMIC PROGRAMMING CHAPTER I INTRODUCTION What i s an a l l o c a t i o n problem? What c rea tes the problem of log a l l o c a t i o n ? And what i s requ i red to so l ve problems of t h i s type? A b a s i c economic d e c i s i o n f a c i n g any e n t e r p r i s e i s how to d i s -t r i b u t e a sca rce resource among competing uses i n the most e f f e c t i v e way. A s i m i l a r s i t u a t i o n a r i s e s when the resources requ i red by a se t of demands may be ass igned from a l t e r n a t i v e s o u r c e s . In both cases management i s requ i red to s e l e c t an a l l o c a t i o n tha t s a t i s f i e s some c r i t e r i o n ; c o n v e n t i o n a l l y a measure of t a n g i b l e cos ts or b e n e f i t s . For an i n t eg ra ted company i n the pr imary f o r e s t products s e c t o r the problem of log a l l o c a t i o n f r e q u e n t l y i nc ludes both s i t u a t i o n s : a quan t i t y of logs must be d i s t r i b u t e d among many types of convers ion p lan ts and numerous p roduc t i on -s to rage schedules may each s a t i s f y log demands f o r one or more p e r i o d s . A s o l u t i o n to t h i s many-sided ques t ion requ i res t an examinat ion of a l l wood-cu t t ing opt ions i n a p lann ing c y c l e , an a p p r a i s a l of a l t e r n a t i v e end-uses and the development of a s u i t a b l e measure f o r comparing d i f f e r e n t a l l o c a t i o n s . A. PAST WORK Few pub l i shed s tud ies have examined the combined l og a l l o c a t i o n problem d e s c r i b e d . Q u a n t i t a t i v e models developed f o r comparing a l t e r -na t i ve wood-cu t t ing p lans have been main ly used f o r f o r e s t r e g u l a t i o n 2 wh i l e the opt imal d i s t r i b u t i o n of logs among a l t e r n a t i v e pr imary conver -s i on p lan ts has r a r e l y i nc luded a l t e r n a t i v e sources of l o g s . The s e l e c t i o n of a f o r e s t management p o l i c y to a t t a i n s p e c i f i e d o b j e c t i v e s i nc l udes the r e g u l a t i o n of a s e r i e s of p e r i o d i c c u t s . Mathematical models have been developed from which opt imal c u t t i n g schedules f o r va r ious p lann ing c r i t e r i a have been d e r i v e d . Arimuzu (1958) , i n an e a r l y a p p l i c a t i o n of dynamic programming to f o r e s t r y prob-lems, maximized the t o t a l volume cut over a ten year p e r i o d . His model f o rmu la t i on d i d not r equ i r e the a l l o c a t i o n of d i f f e r e n t log p roduc ts . C u r t i s (1962) showed how h i s l i n e a r programming model , designed f o r a p a r t i c u l a r company, was used to d e r i v e two c u t t i n g plans f o r a f o r e s t t r a c t over a twenty-year p e r i o d . One p lan maximized wood tonnage, the o t h e r , which requ i red a d d i t i o n a l p r e - c a l c u l a t i o n s , maximized present wor th . Loucks (1964) a l s o s e l e c t e d the methods of l i n e a r programming to develop s u s t a i n e d - y i e l d c u t t i n g schedu les . He presented two c u t t i n g p l a n s , each designed f o r a l i m i t e d p e r i o d . His f i r s t schedule max i -mized the t o t a l volume cut wh i l e the second minimized the number of acres to be c u t , w i th each o b j e c t i v e being sub jec t to var ious con-s t r a i n t s . In a t h i r d study i n which l i n e a r programming was used , K i d d , Thompson and Hoepner (1966) avoided the problem of an unce r t a i n f u tu re i n t h e i r net worth max imizat ion model. They assumed a l l c o s t s , p r i c e s , machine and labour p r o d u c t i v i t i e s to remain unchanged from e x i s t i n g l e v e l s f o r a f i f t y - y e a r p e r i o d . Fo l l ow ing a d i f f e r e n t approach, Hool (1966) developed a f o r e s t p r o d u c t i o n - c o n t r o l system i n which the uncer -t a i n t y of f u tu re t ree growth f o l l o w i n g stand t reatment was expressed by 3 p r o b a b i l i t y m a t r i c e s . He formulated h is dynamic programming-Markov cha in model to maximize t o t a l merchantable c u b i c - f o o t y i e l d over the p lann ing p e r i o d . - Nau t i ya l and Pearse (1967) , w i th the a id of l i n e a r programming, i n v e s t i g a t e d the interdependence between r o t a t i o n l e n g t h , the per iod of convers ion of an i r r e g u l a r f o r e s t to a s u s t a i n e d - y i e l d proper ty and the present worth of the f o r e s t . In t h e i r mathematical model they assumed stumpage va lues per acre were a known f u n c t i o n of stand age. Schreuder (1968) used average u n i t volume log p r i c e s to r e f l e c t d i f f e r e n c e s i n spec ies compos i t i on , s i z e c l a s s e s and pre-p lanned u t i l i z a t i o n . These p r i c e s , ass igned to i n d i v i d u a l f o r e s t s t a n d s , were embodied i n h i s dynamic programming model of the f o r e s t p roduc t ion p rocess . By i n c l u d i n g one or more pr imary convers ion p lan ts and p e r m i t t i n g , i f b e n e f i c i a l , t r a n s a c t i o n s on an open log market he r e l a t e d c u t t i n g p lans to p l a n t de-mands. In h i s f o rmu la t i on an assumed o v e r a l l o b j e c t i v e was to maximize the present worth of the q u a n t i t i e s of wood p rocessed . However, because log demands from the convers ion p lan ts were amalgamated i n t o one compo-s i t e q u a n t i t y , h i s model d i d not cons ide r the opt imal a l l o c a t i o n of l og v a r i e t i e s to each p l a n t . The opt imal d i s t r i b u t i o n of a g iven quan t i t y of logs among a l t e r n a t i v e convers ion p lan ts has been examined i n severa l s t u d i e s . Model f o rmu la t i on has u s u a l l y emphasized i n d u s t r i a l r a t h e r than f o r e s t management problems and has r e l i e d upon the techniques of l i n e a r program-ming to de r i ve opt imal s o l u t i o n s . The mathematical model designed by Done l l y (1965) f o r the plywood manufactur ing f a c i l i t i e s of a m u l t i - p r o d u c t f i r m extended from the 4 a v a i l a b l e wood supply to the s a l e s f o r e c a s t s f o r a wide range of p l y -wood p roduc ts . His pr imary purpose was to op t im ize the u t i l i z a t i o n of the p l a n t ' s resources by improved p roduc t ion s c h e d u l i n g . The p o r t i o n of the f i r m ' s t imber supply which was ass igned to the plywood p l a n t , s a l e s f o r e c a s t s and cos t es t imates formed the bas i s f o r p r e l i m i n a r y p roduc t ion p l a n s . I f p ro jec ted p r o f i t s were i n s u f f i c i e n t , wood supply a v a i l a b i l i -t i e s were re-examined together w i th s a l e s and cos ts to i d e n t i f y oppor-t u n i t i e s f o r improved p r o f i t e x p e c t a t i o n s . Pearse and Sydneysmith (1966) , i n t h e i r study of the u t i l i z a t i o n o f the coas ta l supply of wood i n B r i t i s h Columbia i n 1962, developed an a l l o c a t i o n model f o r a pr imary product u t i l i z a t i o n complex. I n i t i a l l y , they assumed the supply of wood may be a l l o c a t e d among p lan ts producing un l im i t ed q u a n t i t i e s of lumber, veneer , p lywood, wood ch ips and hog f u e l . L a t e r , they d i scussed the m o d i f i c a t i o n s necessary to adapt t h e i r b a s i c model to l ong - te rm , or a l t e r n a t i v e l y , sho r t - t e rm opt imal l og a l l o c a t i o n f o r an i n d i v i d u a l m u l t i - p r o d u c t company. The a l l o c a t i o n of d i f f e r e n t types of wood was i nc luded by Wolfe and Bates (1968) - in t h e i r a n a l y s i s of the manufactur-ing of the q u a n t i t i e s of pulp demanded f o r the p roduc t ion of d i f f e r e n t grades of paper . The p roduc t ion elements cons idered f o r wood a l l o c a t i o n d e c i s i o n s were the p r o d u c t i v i t y i n .each wood type of the a v a i l a b l e . m a n -hours of w o o d - c u t t e r s , the maximum s to rage l e v e l s permi t ted f o r each wood t y p e , wood t r a n s p o r t a t i o n cos ts and the cos ts of conve r t i ng wood i n t o p u l p . Wood procurement cos ts f o r a v e r t i c a l l y i n t eg ra ted f o r e s t products company were minimized by a log a l l o c a t i o n model developed by Thompson and Richards (1969). They eva lua ted f u tu re wood ha rves t i ng and 5 market purchase a l t e r n a t i v e s over a twenty-year p lann ing p e r i o d . By comparing cos ts f o r a l l acceptab le mixes of the d i f f e r e n t v a r i e t i e s of raw ma te r i a l developed by each c u t t i n g p o l i c y , they i d e n t i f i e d optimum wood supp ly schedu les . For r eg iona l wood a l l o c a t i o n and p roduc t ion p lann ing Lb'nner (1968) and Car l sson (1968) o f the Swedish Logging Research Foundat ion have together developed models cor responding approx imate ly to f i v e - y e a r , one-year and one-month p lann ing c y c l e s . The i r sys tem, designed to improve f i n a l d e c i s i o n - m a k i n g , generates s o l u t i o n s grounded on both f o rma l i z ed d e c i s i o n - r u l e s and s u b - o p t i m i z a t i o n s . P lann ing i s i n i t i a t e d by matching a n t i c i p a t e d log demands f o r a f i v e - y e a r pe r iod w i th groups of f o r e s t h o l d i n g s . The bas i s f o r s e l e c t i o n i s a compromise between cont inued adherence to c u t t i n g schedules p resc r i bed by f o r e s t management p o l i c i e s and the d e s i r e to concent ra te ha rves t i ng o p e r a t i o n s . Pa r t o f t h i s process i nc ludes the p r e l i m i n a r y c l a s s i f i c a t i o n of log products to prov ide a bas i s f o r a l l o c a t i o n s made i n the more d e t a i l e d p lann ing s t a g e s . The one-year and one-month log p roduc t ion models , which i nc lude wood t r a n s p o r t a t i o n and s torage c a p a b i l i t i e s , assess the f e a s i b i l i t y of proposed annual cuts i n terms of the a v a i l a b i l i t y of wood-cut ters and ha r ves t i ng equipment. In the one-year logg ing p l a n , a l t e r n a t i v e log a l l o c a t i o n p o l i c i e s are compared w i th ordered se ts of c r i t e r i a known to a f f e c t l ogg ing c o s t s . (These c r i t e r i a are r e l a t e d to the u t i l i z a t i o n of ha r ves t i ng machines, the s u i t a b i l i t y of season f o r logg ing p a r t i c u l a r areas and the degree of concen t ra t i on i n time and space of logg ing o p e r a t i o n s . ) For each f e a s i b l e a l l o c a t i o n p o l i c y optimum hau l i ng and storage schedules are der i ved from l i n e a r programming models. Al though t h e i r p lann ing procedure does not p rov ide opt imal p o l i c i e s , the authors c l a i m an i n d i c a t e d improvement i n log a l l o c a t i o n i s more r e a d i l y accep-tab le to the f i n a l dec is ion-maker by e x p l i c i t l y r e q u i r i n g the r e -examinat ion of i n i t i a l d e c i s i o n - r u l e s . B. THE-THESIS MODEL I t i s the purpose of t h i s t h e s i s to desc r i be an ope ra t i ona l p l a n -ning model f o r i d e n t i f y i n g opt imal log p roduc t ion schedules f o r a m u l t i p l e - p r o d u c t f o r e s t e n t e r p r i s e . P lann ing o b j e c t i v e s requ i re the a l l o c a t i o n of the q u a n t i t i e s of wood demanded i n success i ve time per iods from a l t e r n a t i v e log s o u r c e s . The model , based on the technique of dynamic programming, desc r i bes an o p t i m i z i n g procedure designed f o r i d e n t i f y i n g sho r t - t e rm log a l l o c a t i o n p o l i c i e s . The method by which a s o l u t i o n to a dec i s i on -p rob lem i s obta ined by dynamic programming requ i res each s i t u a t i o n to be c h a r a c t e r i z e d by a recur rence r e l a t i o n -s h i p . Because i n each case t h i s r e l a t i o n s h i p i s un ique , a hypo the t i ca l log a l l o c a t i o n problem i s used as a v e h i c l e f o r model f o r m u l a t i o n . For d e s c r i p t i v e purposes an annual p lann ing c y c l e i s d i v i d e d i n t o three log p roduc t ion per iods each approx imate ly the same l e n g t h . The maximum number of p lann ing per iods tha t may be i nc luded i n the o p t i m i z a t i o n process i s not r e s t r i c t e d by the model and there i s no r e -quirement tha t a l l s tages must be of the same l e n g t h . The number of per iods s e l e c t e d represents the minimum number of s tages needed to i l l u s t r a t e the computat ional p rocess . A l s o , three p lann ing seasons are f r e q u e n t l y used f o r annual log p roduc t ion schedu l ing by the f o r e s t i ndus t r y i n B r i t i s h Columbia. Th is i s p a r t i c u l a r l y t rue i n the i n t e r i o r o f the Prov ince where seasonal d i f f e r e n c e s commonly f o r ce c o n s i d e r a t i o n of "w in te r " and "summer" logg ing seasons separated by a sp r i ng "b reak-up" p e r i o d . Dur ing "b reak-up" l og p roduc t ion i s e i t h e r non -ex i s t en t or i s s e v e r e l y c u r t a i l e d . I t may be h e l p f u l to view the hypo the t i ca l a l l o c a -t i o n prob lem, desc r ibed i n the f o l l o w i n g s e c t i o n i n genera l te rms, as due i n p a r t to these w e l l - r e c o g n i z e d seasonal d i f f e r e n c e s . A sho r t - t e rm p lann ing c y c l e was s e l e c t e d f o r two reasons . One, i n the sho r t - t e rm an e r r o r i n an assumption that v a r i a b l e log p roduc t ion cos ts are a l l l i n e a r l y r e l a t e d to the quan t i t y of logs cu t i s more p ro -nounced. The f o rmu la t i on w i l l demonstrate the computat ional ease w i th which a l l i d e n t i f i e d v a r i a b l e p roduc t ion components, i n c l u d i n g those not l i n e a r l y r e l a t e d to cut volume, are i nco rpo ra ted i n the model. Two, by c o n s i d e r i n g on ly sho r t - t e rm log p roduc t ion s u f f i c i e n t working c a p i t a l may be assumed to be a v a i l a b l e as a p r e - r e q u i s i t e o f being i n b u s i n e s s . The cos t of t h i s c a p i t a l i s assumed to have no e f f e c t on log a l l o c a t i o n p o l i c i e s w i t h i n the p lann ing c y c l e . C. THE HYPOTHETICAL PROBLEM The log a l l o c a t i o n problem i s viewed from the pe rspec t i ve of the management group r e s p o n s i b l e f o r the log supply o f a sawmi l l and f o r the d e l i v e r y of s p e c i f i e d q u a n t i t i e s of the logs requ i red by a p u l p m i l l . 8 The ex ten t of the group 's a u t h o r i t y and r e s p o n s i b i l i t i e s i s shown by the boundaries i n d i c a t e d i n F igure 1. Log p roduc t ion i s planned f o r an annual c y c l e of three time per iods each of which i s a s s o c i a t e d wi th seasonal d i f f e r e n c e s i n the quan t i t y and d e l i v e r e d cos t of a v a i l a b l e l o g s . In each pe r i od the known m i l l demands f o r pu lp logs and sawlogs must be s a t i s f i e d . Un l i ke the demand f o r p u l p l o g s , sawmi l l demands i n -c lude a p re fe rence f o r c e r t a i n types of l o g s . This q u a n t i f i a b l e p r e f e r -ence r e f l e c t s a premium p r i c e a v a i l a b l e to the sawmi l l f o r lumber manufactured from c e r t a i n t ree spec ies and a reduced u n i t v a r i a b l e m i l l convers ion c o s t when the average log d iameter exceeds a s t i p u l a t e d s tandard s i z e . M i l l demands may be s a t i s f i e d from f i v e sources of l o g s : two pu lp log -p roduc ing a r e a s , i d e n t i f i e d as sources one and f i v e ; two areas from which both sawlogs and pu lp logs can be c u t , sources two and t h r e e ; and a l o c a l l og market , source f o u r ; To ensure demands w i l l always be f u l f i l l e d , and to b e n e f i t from favourab le cos t d i f f e r e n c e s between p e r i o d s , d e l i v e r e d l o g s , su rp lus to cu r ren t m i l l m i l l demands, may be s to red i n " c o l d - d e c k s " u n t i l needed. These log supply a l t e r n a -t i v e s are shown i n F igure 2 . Minimum " s a f e t y s t o c k s " are assumed to be mainta ined a t a cons tant l e v e l between p e r i o d s . The major g u i d e l i n e s cons idered when each a l l o c a t i o n p o l i c y i s examined are grouped i n t o fou r c a t e g o r i e s . 1. Convers ion P lan ts The a p p r a i s a l of competing uses f o r d e l i v e r e d logs must p rov ide f o r the planned p roduc t ion runs of each convers ion p l a n t and s p e c i f y i f PRODUCTION STAGES CONSIDERED FOREST FOREST _ T YPE & ALLOW-M A N A G E - ABLE MENT CUT FROM ALTERNA- 5 AREAS TIVES FOR 3 PERIODS 1 LOG HARVEST-ING SAWLOG SAWMILL SAWLOG V A L U E / UNIT i . MILL TRANSPORT YARD COSTS 1 MARKET r ~ 1 REQUIRE-DECI- PULPWOOD PULPMILL I PULPWOOD V A L U E / UNIT MENTS DEMANDS SIONS TRANSPORT YARD COSTS OS ft) F i g . 1 Boundaries of the Log A l l o c a t i o n Problem to PRODUCTIVE A R E A S L O G PRODUCTION L O G S O R T i N G MILL YARD I N V E N T O R I E S MILL REQUIREMENTS 11 p a r t i c u l a r log grades are p r e f e r r e d . The e f f e c t of v a r i a t i o n i n the a l l o c a t e d grades of logs on log convers ion costs and the f i n a l re tu rn from marketed products must be pa r t of the rank ing c r i t e r i o n . For the s p e c i f i c problem examined sawlog grades are based on spec ies and log d iameter . Q u a n t i t i e s of p e e l e r l o g s , developed from source two, may, i f b e n e f i c i a l to the e n t e r p r i s e , be t raded f o r equal volumes of wood. A p r i c e d i f f e r e n c e between t raded pee le r logs and replacement logs r e s u l t s i n a d d i t i o n a l revenue acc ru ing to the e n t e r p r i s e . No pu lp log grades are recogn ized and the u n i t re tu rn from convers ion of p u l p l o g s , assumed c o n s t a n t , i s not a f f e c t e d by log a l l o c a t i o n d e c i s i o n s . 2. Log Supply The number of wood-cu t t ing a l t e r n a t i v e s i s l i m i t e d by ex te rna l c o n s t r a i n t s . Fo res t r e g u l a t i o n by l e g i s l a t i o n , or by enforced f o r e s t management p o l i c i e s , may s p e c i f y l i m i t s w i t h i n which the quan t i t y of logs cu t on des ignated areas must be ma in ta ined . Because logs purchased on the open market p rov ide pa r t of the log supply the t o t a l quan t i t y of a v a i l a b l e logs may f l u c t u a t e s e a s o n a l l y and be i n f l u e n c e d by the a c t i v i -t i e s o f competing o r g a n i z a t i o n s . Fur ther r e s t r i c t i o n s may be due to seasonal d i f f e r e n c e s i n the a c c e s s i b i l i t y o f i n d i v i d u a l p roduc t i ve areas when e x t r a c t i o n cos ts may tempora r i l y i nc rease to a l e v e l banning f u r t h e r economic c o n s i d e r a t i o n . • In the problem analysed i n t h i s t h e s i s the maximum q u a n t i t y of logs a v a i l a b l e from each s o u r c e , i n c l u d i n g market l o g s , i s assumed to be pre-determined f o r each time per iod i n the annual p lann ing c y c l e . 12 To compare a l t e r n a t i v e c u t t i n g schedules d i f f e r e n c e s between log p roduc t ion cos ts f o r each per iod must be i d e n t i f i e d f o r the range of log q u a n t i t i e s and grades a v a i l a b l e from each a r e a . Only those cos t items a f f e c t e d when log a l l o c a t i o n s vary need be cons i de red . One cos t component f r e q u e n t l y inc luded a r i s e s from log s o r t i n g . Log s e l e c t i o n p o l i c i e s range from a s s i g n i n g a l l logs produced from a p a r t i c u l a r f o r e s t type to a p a r t i c u l a r p l an t to a log grad ing p o l i c y respons ive to cu r ren t p l an t requ i rements . The cos t components s p e c i f i e d f o r each of the f i v e log sources i d e n t i f i e d i n the formulated model have been s e l e c t e d to i l l u s t r a t e a v a r i e t y of r e p r e s e n t a t i v e log p roduc t ion systems. 3. Log Storage To reduce the adverse e f f e c t s of seasonal d i f f e r e n c e s i n log produc t ion c o s t s , d e l i v e r e d logs su rp lus to immediate demands are s to red f o r l a t e r use . To avo id conges t ion i n the log r e c e i v i n g area su rp lus logs are " c o l d - d e c k e d " f o r l a t e r use . The v a r i a b l e hand l ing cos ts i ncu r red f o r b u i l d - u p and breakdown of these s torage decks are par t of the cos ts of a l l o c a t i o n f o r a p a r t i c u l a r t ime p e r i o d . Al though d e t a i l e d examinat ion o f l og yard inven to ry management i s seldom neces-s a r y , a l i m i t on s torage c a p a c i t y may r e s t r i c t l og d e l i v e r i e s . In t h i s study an excess of c u r r e n t l og d e l i v e r i e s over the m i l l demands of each per iod i s l i m i t e d to a volume equal to the m i l l demands of the next p e r i o d . 13 4. Economic Ob jec t i ve The i n t e r - r e l a t i o n s h i p s between log p roduc t ion c o s t s , the cos ts of " c o l d - d e c k i n g , " and the e f f e c t i v e n e s s w i th which d e l i v e r e d logs s a t i s f y demands are the key determinants i n rank ing a l t e r n a t i v e a l l o c a -t i o n s . A s u i t a b l e rank ing index must combine these components i n t o a s i n g l e measure f o r each a l l o c a t i o n p o l i c y . An i n d i c a t e d opt imal p o l i c y may vary w i th the type of index deve loped. For example, an a l l o c a t i o n tha t min imizes cos ts does not n e c e s s a r i l y maximize p r o f i t s . Consequent ly , the s e l e c t i o n of an app rop r ia te c r i t e r i o n must r e f l e c t the s ta ted o b j e c t i v e s of management and conform to company p o l i c i e s . For the log a l l o c a t i o n problem being examined, the o b j e c t i v e of management i s to maximize the d i f f e r e n c e between the re tu rn from log convers ion and the sum of the d e l i v e r e d log cos ts , and the cos t of " c o l d - d e c k i n g " su rp lus l o g s . Th is o b j e c t i v e , expressed i n a d i f f e r e n t manner, i s used as the economic c r i t e r i o n to i d e n t i f y an opt imal p o l i c y f o r the hypo the t i ca l log a l l o c a t i o n problem d e s c r i b e d . The o b j e c t i v e f u n c t i o n inco rpora ted i n the recur rence r e l a t i o n s h i p s , developed f o r the problem, i s s p e c i f i e d as the m in im iza t i on of the combined c o s t of d e l i v e r e d logs and the cos t of " c o l d - d e c k i n g " l ess the re tu rn from log c o n v e r s i o n . A c h a r a c t e r i s t i c of a l l numer i ca l l y -based s o l u t i o n s i s t h e i r dependency upon the r e l i a b i l i t y of b a s i c i n f o r m a t i o n . Est imates of f u tu re q u a n t i t i e s , cos ts and p r i c e s are g e n e r a l l y based upon past exper -ience and cond i t i oned i n some way by f u tu re e x p e c t a t i o n s . Because of the u n c e r t a i n t y inheren t i n f o recas ted v a l u e s , p r o v i s i o n must be made to v e r i f y s o l u t i o n s based on such e s t i m a t e s . Two groups of methods by 14 which unce r t a i n t y may be cons idered are a v a i l a b l e . In a " d e t e r m i n i s t i c " group, i n which s i n g l e - v a l u e d f o r e c a s t s are used to eva lua te a l t e r n a t i v e s o l u t i o n s , the v a l i d i t y o f an i n d i c a t e d opt imal s o l u t i o n i s t es ted by s e n s i t i v i t y a n a l y s e s . By determin ing the range over which f o recas ted va lues may vary w i thou t a f f e c t i n g the s o l u t i o n the " robus tness " of the s o l u t i o n i s examined. In a second group of methods, which use a " s t o c h a s t i c " approach , each s i n g l e - v a l u e d es t imate i s rep laced by a range of p o s s i b l e outcomes, w i th each outcome ass igned a p r o b a b i l i t y of i t s r e a l i z a t i o n . The advantage of the f i r s t approach i s i t s r e l a t i v e s i m -p l i c i t y ; of the second , i t s completeness. E i t h e r approach may be i n c o r -porated i n a dynamic programming f o r m u l a t i o n . For the log 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 a "de te rmin -i s t i c " o p t i m i z a t i o n model i s p r e f e r r e d . Even i n the s h o r t - t e r m , the need f o r exac t p r o b a b i l i t y statements i s d i f f i c u l t to s a t i s f y i n a h i g h l y c y c l i c a l i n d u s t r y . A l s o , when the f i n a l p o l i c y implemented i s n e c e s s a r i l y a composite of both numerical ev idence and s u b j e c t i v e e v a l -ua t i on by the r e s p o n s i b l e dec i s i on -make r , the need f o r s i m p l i c i t y i s paramount. A c c o r d i n g l y , ques t ions of unce r t a i n t y are not i nc luded i n the a l l o c a t i o n model desc r ibed i n t h i s t h e s i s . The method of p resen ta t i on fo l l owed i s to de fe r d i s c u s s i o n of the d e t a i l e d mathematical development of l og p roduc t ion cos t f u n c t i o n s , a v a i l a b l e log q u a n t i t i e s and a l t e r n a t i v e l og grad ing and s o r t i n g p o l i c i e s u n t i l the a l l o c a t i o n model has been d e s c r i b e d . A c c o r d i n g l y , the development of the recur rence r e l a t i o n s h i p s of the formulated model i s d i scussed i n Chapter I I . The development i nc l udes c o n s i d e r a t i o n of the log hand l ing cos ts i n c u r r e d by " c o l d - d e c k i n g " logs and the manner i n which log cos ts are compared. The d e r i v a t i o n o f these log p roduc t ion cos ts f o r each source i n each of three time per iods i s examined i n d e t a i l i n Chapter I I I . Fo l l ow ing t h i s f u l l d e s c r i p t i o n , the computer-coded programme, DYALOG (Appendix B ) , i s used wi th manufactured inpu t data to demonstrate the computat ional f e a s i b i l i t y of the f o r m u l a t i o n . The t r i a l r e s u l t s are presented and d i scussed i n Chapter IV. The d i s -cuss ion concludes w i th observa t ions and comments regard ing the u t i l i t y of the model (Chapter V ) . CHAPTER II THE FORMULATION AND DEVELOPMENT OF THE MODEL To so l ve the hypo the t i ca l log a l l o c a t i o n problem desc r ibed i n Chapter I by the computat ion process of dynamic programming requ i res the development o f recur rence equa t i ons . The r e l a t i o n s h i p s incorpora ted i n these e q u a t i o n s , the d e c i s i o n - v a r i a b l e s s e l e c t e d and the l i m i t s w i t h -i n which the r e l a t i o n s h i p s are v a l i d , together de f i ne the mathematical model. In t h i s chapter the methodology of dynamic programming i s exam-i n e d , the necessary a n a l y t i c a l framework f o r s o l v i n g the log a l l o c a t i o n problem developed and the formulated model d e s c r i b e d . A d e s c r i p t i v e f low cha r t i s in t roduced to f a c i l i t a t e d i s c u s s i o n (F igu re 3). Computa-t i o n a l d e t a i l s , necessary to eva lua te the c o s t and re tu rn components d e f i n e d , are desc r ibed i n Chapter I I I . The e s s e n t i a l c h a r a c t e r i s t i c s of the problem desc r ibed a r e : ( i ) q u a n t i t i e s of pu lp logs and sawlogs are demanded i n each of th ree t ime per iods and must be s a t i s f i e d by log d e l i v e r i e s from a maximum of f i v e s o u r c e s , ( i i ) the maximum quan t i t y of logs tha t may be d e l i v e r e d from each source i s r e s t r i c t e d f o r each p e r i o d , ( i i i ) d e l i v e r e d logs tha t are su rp lus to immediate m i l l demands may be used i n a l a t e r p e r i o d , ( i v ) the v a r i a t i o n between time per iods of u n i t c o s t s , p r i c e s , the a v a i l a b l e log supply and m i l l demands i s known, F,{X,Y) = 0 I t = 2 T - 5 f t o ' V V • 0 k = 1 Y t - C y = max i s \ yes k = 1 x u - max[ ° ; X t * Y t - 1 ^ 1 wtm-ytk] xtk * Xt yes C a l c : ctk ( xtk' ytk> 8 - ^k^tk^tk l^ tk - l^ t -^k^t^ tk ' no a = 0 * 3 X , tk ytk \ *tk ' "tk + l yes x u ( X t , Y t ) - X yes k = k+1 yes XT = min r t + l K K n << = XT "t = YT k = K x t k ( X T , Y T ) - x u ( X t . Y t ) y t k ( X T , Y T ) - y t k ( X t , Y t ) X t • X t - x t k ( X T . Y T ) Yt " V * t k ( X T ' Y T > k = k-1 For computat ional purposes the index t i s increased by one, and Dx 1 = Dy 1 = Dx 5 = 0 y 5 = 0. 1. = m a x f o ; Y - £ D y . 1 L j-1 J ' t-1 K j-1 m=l j m t-1 K t-1 n -i - E D Y j - E V S J j= l J m=I J Y t - Y f t t-1 K X t » max 0 ; X - £ D x . ; X + Y - Y - £ £ t j=l 1 j=- • 1 m=l t-1 m-vJ>t-im-£<DyDv] g t ( x t . Y t ) - ( X t + Y t ) . e h yes no 1 t 1^= Y. 1 t i V t = T-1 4 X j=2 J 4 Y j=2 J XT - X t ( X , Y ) YT • Y t ( X , Y ) x°tk(J4DxJ,J4Dyj) 3 x * " ( x t , y t ) 4 4 V t v ( E O x . E D y . ) - y t k ( X T , Y T ) tk j = 2 3 j = 2 J tk X - X-XT(X,Y) Y' = Y-YT(X,Y) X « X Y = Y t .- t-1 ( END ^ Figure 3 Flowchart of the Log A l l o c a t i o n Model 18 (v) the re tu rn per u n i t of d e l i v e r e d sawlogs i s assumed to be dependent upon t ree spec ies and log d iamete r , and a constant per u n i t r e tu rn i s assumed f o r pu lp log c o n v e r s i o n , ( v i ) the o b j e c t i v e of management i s to min imize the sum of the c o s t of d e l i v e r e d logs and the cos t of " c o l d - d e c k i n g " su rp lus logs l e s s the re tu rn from log c o n v e r s i o n . Th is m in im iz ing problem requ i r es two mutua l ly dependent se ts of d e c i s -ions to be made: the opt imal s e l e c t i o n of q u a n t i t i e s of logs f o r d e l i v e r y i n each time per iod and the opt imal p a r t i t i o n i n g of these q u a n t i t i e s among the sources a v a i l a b l e i n each p e r i o d . • In the fo rmu la -t i o n to be d e s c r i b e d , t h e a l l o c a t i o n of l og d e l i v e r i e s w i t h i n each time per iod i s regarded as a m u l t i - s t a g e d e c i s i o n process i n which each stage represents an a v a i l a b l e source of l o g s . - The s e t of opt imal p o l i c i e s f o r the range of q u a n t i t i e s tha t cou ld be ass igned to each t ime per iod i s i d e n t i f i e d . A l l o c a t i o n between time per iods i s s i m i l a r l y regarded as a r e l a t e d m u l t i - s t a g e d e c i s i o n process i n which time per iods prov ide an ordered sequence of s t a g e s . The computat ional advantages of f o rmu la t i ng c e r t a i n types of problems i n sequen t i a l form have been demonstrated by Bel lman i n h i s development of dynamic programming. He showed how, prov ided c e r t a i n c o n d i t i o n s are s a t i s f i e d , the r e c u r s i v e equat ion technique of dynamic programming may be used to op t im ize the t o t a l r e tu rn from a l l o c a t i n g a l i m i t e d resource among competing a c t i v i t i e s . Because the mathematical model developed i n t h i s t h e s i s i nco rpo ra tes f u n c t i o n a l e q u a t i o n s , the 19 adapta t ion of the g e n e r a l i z e d a l l o c a t i o n problem o u t l i n e d by Bel lman w i l l be d i scussed i n d e t a i l . Subsequent ex tens ions and a d d i t i o n a l modi-f i c a t i o n s requ i red l a t e r may thereby be more r e a d i l y r e l a t e d to the bas i c concept . • A. BELLMAN'S APPLICATION OF DYNAMIC PROGRAM-MING TO A RESOURCE ALLOCATION PROBLEM. Bel lman (1957) d i scussed the p a r t i t i o n i n g of a v a i l a b l e resources among competing a c t i v i t i e s to maximize t o t a l r e t u r n s , when the re turns from an a c t i v i t y depended upon both the magnitude of the resource a l l o c a t e d and the u t i l i t y f u n c t i o n of the p a r t i c u l a r a c t i v i t y . Necessary assumptions were: the re tu rns from d i f f e r e n t a c t i v i t i e s can be measured i n a common u n i t ; the re tu rn from any a c t i v i t y i s independent of the a l l o c a t i o n s to the o ther a c t i v i t i e s ; the t o t a l r e tu rn can be obta ined as the sum of the i n d i v i d u a l r e t u r n s . The assumptions concern ing independence of the a c t i v i t i e s and a d d i t i v i t y of the assoc i a ted re tu rns permit the t o t a l re tu rn from an a l l o c a t i o n p r o c e s s , R(x-j , x 2 , . . . ,x^) to be expressed as g-j (x-| ) +g 2 ( x 2 )+ , • • • >+9|<(xK) ' W N E R E 9 j ( x j ) 1 S t n e r e t u r n w n e n a quan t i t y of resource X j i s a l l o c a t e d to a c t i v i t y j . The requirement i s . t o . m a x i m i z e 20 R(x-| jXg,• •. jX^) where K. s x . = X , 3 = 1 J the t o t a l quan t i t y of resource a l l o c a t e d , and x-j . . . y x ^ are. the quan-t i t i e s a l l o c a t e d to a c t i v i t i e s 1 , 2 , . . . y K . Bel lman developed the b a s i c f u n c t i o n a l e q u a t i o n ; f k ( X ) = max [ g k ( x k ) + f k _ 1 ( X - x k ) 3 [1] 0 < x k < X f o r k = 1 , 2 , . . . , K , where f k ^ = maximum re tu rn when the q u a n t i t y X i s a l l o c a t e d to a c t i v i t i e s 1 through k and f Q ( X ) = 0 . Th is r e l a t i o n s h i p i s the mathematical f o rmu la t i on of Be l lman 's P r i n c i p l e of O p t i m a l i t y f o r m u l t i - s t a g e d e c i s i o n processes which he ' expressed a s : An opt imal p o l i c y has the proper ty tha t whatever the i n i t i a l s t a t e and i n i t i a l d e c i s i o n . a r e , the remaining d e c i s i o n s must con-s t i t u t e an opt imal p o l i c y w i th regard to the s t a t e r e s u l t i n g from the f i r s t d e c i s i o n . • By pe rm i t t i ng x k , the q u a n t i t y a l l o c a t e d to each a c t i v i t y to vary over a s p e c i f i e d range, a l t e r n a t i v e a l l o c a t i o n s of the q u a n t i t y X are eva lua ted . In the r e c u r s i v e r e l a t i o n s h i p each s e t of e v a l u a t i o n s , made when each a c t i v i t y i s i n t r o d u c e d , compares a l t e r n a t i v e a l l o c a t i o n s ' between the a c t i v i t y k and a c t i v i t i e s 1 through k - 1 . The opt imal se t of a l l o c a t i o n s made to a c t i v i t y k when a c t i v i t i e s 1 to k have been con-s i d e r e d , i s i d e n t i f i e d as x . ( X ) . When a l l a c t i v i t i e s have been inc luded 21 i n t h i s " forward pass" and f k ( X ) has been eva lua ted f o r 1 through K o a c t i v i t i e s the opt imal a l l o c a t i o n x k ' (X) to each a c t i v i t y i s i d e n t i f i e d from a "backward pass" a s : o XjJX) = Xj^X) f o r a c t i v i t y K, x. (X) = x. (X- E x . (X ) ) . f o r k = K - l , K - 2 , . . . , 1 , K K j=k+l J with the maximum re tu rn from the K-stage d e c i s i o n process eva lua ted as f K ( X ) . Th is o p t i m i z a t i o n procedure i s the bas i s f o r the fo rmu la t i on developed f o r the log a l l o c a t i o n problem. The m o d i f i c a t i o n s requ i red are d i scussed i n d e t a i l i n the f o l l o w i n g s e c t i o n s . The f i r s t s tep i s to rep lace the maximiz ing r e l a t i o n s h i p o f [1] by an analogous min im iz ing f u n c t i o n to correspond w i th the o b j e c t i v e of the problem. f k ( X ) = min [ c . ( x k ) + f k , ( X - x . ) ] [2] 0 <_ x k <_ X f o r k = 1 , 2 , . . . , K , where = minimum cos t of a l l o c a t i n g the quan t i t y X to a c t i v i t i e s 1 through k and f Q ( X ) = 0 , c k ( x k ) = c o s t of a l l o c a t i n g the quan t i t y x k to a c t i v i t y k. B. EXTENSION OF THE BASIC RECURSIVE EQUATION Equat ion [2] i s r e a d i l y adapted f o r the a l l o c a t i o n of two r e -sou rces . I f , i n a d d i t i o n to the quan t i t y X , a quan t i t y Y of a second resource i s a l l o c a t e d to K a c t i v i t i e s , f k ( X , Y ) = min [ c k ( x k , y k ) + f k _ 1 ( X - x k , Y - y | < ) ] [3] 0 < < X 0 < y k < Y f o r k = 1 , 2 , . , K , where f k ( X , Y ) = minimum re tu rn when the q u a n t i t i e s X,Y are a l l o c a t e d to a c t i v i t i e s 1 through k, and f o ( X , Y ) = 0 , c k ( x k , y k ) = cos t of a l l o c a t i n g the q u a n t i t i e s x k , y k to a c t i v i t y k. Equat ion [3] shows how two r e s o u r c e s , X and Y , may be a l l o c a t e d to K a c t i v i t i e s a t a minimum composite c o s t . I f , i ns tead of r e f e r r i n g to a l l o c a t i o n s , t o K activities we cons ide r a l l o c a t i o n s from K sources, equat ion [3] a l s o shows how the minimum cos t a l l o c a t i o n of two r e -sources from K sources may be d e r i v e d . The next m o d i f i c a t i o n i s to permi t a l t e r n a t i v e a l l o c a t i o n s from d i f f e r e n t t ime per iods and to r e l a t e the v a r i a b l e s more c l o s e l y to the s p e c i f i c a l l o c a t i o n problem. L e t , t = 1 , 2 , . . . , T be a sequence of t ime per iods cor responding to the p roduc t ion per iods i n a p lann ing c y c l e , 23 X and Y = t o t a l q u a n t i t i e s of pu lp logs and saw logs , r e s p e c t i v e l y , i n thousands of c u n i t s a l l o c a t e d dur ing the p lann ing c y c l e , X^ and Y.£ = q u a n t i t i e s of pu lp logs and s a w l o g s , - r e s p e c t i v e l y , i n thous-ands of c u n i t s a l l o c a t e d from a l l a v a i l a b l e sources i n pe r iod t , x t k a n c l y t k = Q u a n t i t i e s of pu lp logs arid saw logs , r e s p e c t i v e l y , i n thousands of c u n i t s a l l o c a t e d from source k i n pe r iod t , c t k ^ x t k ' y t k ^ = " n e t d e l i v e r e d log c o s t " ^ o f a l l o c a t i n g the q u a n t i t i e s x^^ and y ^ from source k i n pe r iod t , f t k ( X t , Y t ) = minimum "net d e l i v e r e d log c o s t " o f a l l o c a t i n g the q u a n t i -t i e s X^ and Y ^ from sources 1 through k i n pe r iod t , t h e n , = m i n w t c t k ( x t k ' y t k ) + f t , k - i ( x t - x t k » Y t - y t k ^ m o < x t k < x t f o r k = 1,2 K, t = 1 , 2 , . . . , T . I t i s impor tant to note the opt imal p o l i c i e s , the se t { x t k ^ t k } i d e n t i f i e d f o r each pe r i od t , are independent of a l l o ther p e r i o d s . Consequent ly , the minimum c o s t f u n c t i o n f ^ X ^ Y p , a s s o c i a t e d w i th each a l l o c a t i o n p o l i c y , i s . e v a l u a t e d on ly f o r the q u a n t i t i e s of l o g s * X^ and + The d e f i n i t i o n of t h i s composite c o s t i tem i s g iven i n the G lossa ry o f Terms, page x i x . Yj., which are a l l o c a t e d i n pe r iod t . A l i n k between per iods i s prov ided by the q u a n t i t i e s which may be s to red from one per iod to the n e x t . . I f l ess than the q u a n t i t i e s a l l o c a t e d i n pe r iod t are requ i red to s a t i s f y m i l l demands i n pe r iod t there w i l l n a t u r a l l y be a s u r p l u s . In the sequence of T time per iods these su rp luses are viewed as i n v e n t o r i e s of a l l o c a t e d resou rces . P r o v i d i n g an i n v e n t o r y , a l l o c a t e d i n per iod t , can be used to p a r t i a l l y or complete ly s a t i s f y the requirements of pe r iod t+1, the q u a n t i t i e s X^+-] and Y t + - | t ha t would o therwise be requ i red are co r respond ing l y reduced. With the p r o v i s i o n tha t the demands of any per iod must be s a t i s f i e d , the s e r i e s of per iods are cons idered as a T-s tage d e c i s i o n process i n which the r e c u r s i v e r e l a t i o n s h i p proceeds from t =1 to t = T. In t h i s a l l o c a t i o n process the magnitudes of the log volumes p a r t i t i o n e d between per iod t and pe r iod t-1 are r e s t r i c t e d to q u a n t i t i e s equal to the su rp lus developed i n t - 1 . For a g iven s e t of f i x e d demands f o r each per iod t '= 1 , 2 , . . . , T the q u a n t i t y of logs a l l o c a t e d i n per iods 1 through t determines the s i z e of the su rp lus i n pe r iod t . I f the t o t a l q u a n t i t i e s of pu lp logs and sawlogs demanded f o r T per iods are represented by X and Y then the f u n c t i o n a l equat ion which op t im izes resource a l l o c a t i o n between per iods may be expressed as F t ( X , Y ) min C f t K ( X t > Y t ) + F t . 1 ( X - X t . Y . Y t ) ] [5] f o r t = 1 , 2 , , . . J , where F t ( X , Y ) minimum cos t of a l l o c a t i n g the q u a n t i t i e s X.and 25 Y from a l l a v a i l a b l e sources i n per iods 1 through t , R.| , R 2 = f e a s i b l e reg ions of X^ and Y ^ , l i m i t e d by the resources a v a i l a b l e i n pe r iod t and the pragmat ic r e s t r i c t i o n imposed by the requirements of the p h y s i c a l problem tha t i f log p roduc t ion i n pe r iod t exceeds the m i l l demands of pe r iod t the su rp lus must not exceed the m i l l demands of pe r iod t+1. Log a l l o c a t i o n from each of the sources w i t h i n a p a r t i c u l a r time per iod i s independent of a l l o c a t i o n d e c i s i o n s made w i t h i n any o ther p e r i o d . - Consequent ly the s e t { f ^ X ^ Y ^ . ) } may be eva lua ted f o r T pe r -iods before the p a r t i t i o n i n g between per iods i s examined. However, i t i s compu ta t i ona l l y more conven ient to r e l a t e the minimum cos t f u n c t i o n , f t k ( X t , Y t ) , w i th F t ( X , Y ) immediately a f t e r the e v a l u a t i o n of f ^ U ^ t ) • Figure 4 shows the r e l a t i o n s h i p between ^ ( X ^ Y ^ . ) and F t ( X , Y ) when t h i s computat ional process i s adopted. Equat ions [4] and [ 5 ] , when the v a r i a b l e s are de f ined as i n t e g e r s , p rov ide the conceptual bas i s f o r the d e t a i l e d mathematical development o f the log a l l o c a t i o n model desc r ibed i n the f o l l o w i n g s e c t i o n s of t h i s chap te r . As a supplement to the d i s c u s s i o n a f l owcha r t showing the computat ional sequence of the formula ted model i s g iven i n F igure 3 , page 17; By r e l a t i n g each s tep desc r ibed w i th i t s coun te rpar t i n the f l owcha r t the manner i n which the two m u l t i - s t a g e d e c i s i o n processes are connected can be r e a d i l y i d e n t i f i e d . MINIMUM COST ALLOCATION FROM K SOURCES TIME PERIODS 1, 2 , • • • > k, ,K 1, • • ' f T k ( X l ' V ' • • • ' f l K ( X l ' V 2 , ^ 2 1 ^ X 2 ' Y 2 ^ ' ^ 2 2 ^ X 2 ' Y 2 ^ ' * * ' ^2k ^ X 2 ' Y 2 ^ ' " • • s ^2K^ X 2 ' Y 2 ^ • . t , f t " | ( X t ' V ' f t 2 ^ X t ' Y t ^ ' • " f t k ( X t ' V ' • '•' W W • J f j l ( X j , Y j ) , f y 2 ( X y , Y y ) , . " f T k ( X T ' Y T ) s ' f T K ^ X T ' Y T ) t F i g . 4 R e l a t i o n s h i p between f ^ C X ^ Y ^ and F t ( X , Y ) MINIMUM COST ALLOCATION BETWEEN PERIODS F ^ X . Y ) = f 1 K ( X 1 , Y ] ) , F 1 ( X - X 2 , Y - Y 2 ) ] , F t ( X , Y ) = m i n [ f t K ( X t , Y t ) + F t - l ( X - X t ' Y - Y t ^ ' F T ( X , Y ) = m i n [ f T K ( X T , Y T ) + FT - l ( X - X T ' Y - Y T ) ] av 27 C. APPLICATION OF THE EXTENDED RECURRENCE EQUATIONS TO LOG ALLOCATION The a l l o c a t i o n problem requ i res q u a n t i t i e s of pu lp logs and saw-logs to be d e l i v e r e d to m i l l yards i n s u f f i c i e n t q u a n t i t i e s to s a t i s f y m i l l demands over three time p e r i o d s . A maximum of f i v e log sources are a v a i l a b l e from which a l l o c a t i o n s must be made. The measure of . e f f e c t i v e n e s s has been de f ined as a combinat ion of d e l i v e r e d log c o s t s , the cos t of " c o l d - d e c k i n g " su rp lus logs and the re tu rn from log con-v e r s i o n . The cos ts of " c o l d - d e c k i n g " are i ncu r red when the quan t i t y of d e l i v e r e d logs i s su rp lus to cu r ren t m i l l demands. They are eva lua ted when a l t e r n a t i v e a l l o c a t i o n s of log q u a n t i t i e s between per iods are con-s i d e r e d . De l i ve red log cos ts and the re tu rns from log convers ion are each expressed as a f u n c t i o n of the q u a n t i t i e s cut or purchased from each source f o r each time p e r i o d . For d e s c r i p t i v e purposes , the exp ress ion "ne t d e l i v e r e d log c o s t " i s used to r e f e r to the va lue obta ined when the re tu rns from log convers ion are sub t rac ted from d e l i v -ered log c o s t s * The r e l a t i v e magnitudes of t h i s composite cos t index f o r d i f f e r e n t a l l o c a t i o n s from the a v a i l a b l e sources prov ide the bas i s f o r i d e n t i f y i n g opt imal log a l l o c a t i o n w i t h i n a time p e r i o d . In Chapter I I I the mathematical d e r i v a t i o n o f t h i s measure i s d i scussed i n d e t a i l f o r each sou rce . In t h i s chapter the method of comparison i s emphasized. 1. A l l o c a t i o n from Log Sources As p r e v i o u s l y i n d i c a t e d , due to permi t ted log i n v e n t o r i e s the t o t a l q u a n t i t y of pu lp logs or -sawlogs d e l i v e r e d i n a p a r t i c u l a r t ime 28 pe r iod may be l ess t h a n , equal t o , or more than the m i l l demands f o r tha t p e r i o d . The maximum quan t i t y of logs a l l o c a t e d may be l i m i t e d e i t h e r by the t o t a l log volume a v a i l a b l e or by the requirement tha t log su rp luses do not exceed m i l l demands of the per iod immediately f o l l o w i n g . The requ i red opt imal a l l o c a t i o n from log sources f o r q u a n t i t i e s ex tend-ing over the permi t ted range i s found from a ' r e c u r r e n c e r e l a t i o n s h i p based on equat ion [ 4 ] . L e t , v^^ = q u a n t i t y of logs i n thousands of cun i t s a v a i l a b l e from source k i n pe r iod t , v s t k = quan t i t y °t~ sawlogs i n thousands of c u n i t s a v a i l a b l e from source k i n pe r iod t , D x t and D y t = demand f o r pu lp logs and sawlogs , r e s p e c t i v e l y , i n thous-ands of c u n i t s i n pe r iod t . From equat ion [ 4 ] , f t k ( X t , Y t ) = min [ C t ^ X t ^ y t ^ + f ^ ^ ^ X ^ x ^ . Y ^ ) ] [6] X t k e R l y t k e R 2 X t k + y t k e R 3 f o r k = 1 , 2 , . . . ,-5, t = 1 , 2 , 3 , where- f t 0 ( X t , Y t ) = 0 , f o r the f o l l o w i n g r e s t r i c t i o n s on the magnitudes of X. and Y . ; -t+1 r j= t J t+1 0 < Y, < min[ E Dy. 1 j=t 3 k k 0 < X.+Y. <_ E v 1 z m=l where Dx 4 .= Dy 4 = 0 . The th ree reg ions R-j s R 2 and R^ r e f e r to the range of f e a s i b l e q u a n t i t i e s of pu lp logs or sawlogs tha t may be cut or purchased from each sou rce . The lower l i m i t s of these reg ions are de f ined by the r e -quirement tha t on ly non-negat ive a l l o c a t i o n s are p o s s i b l e and tha t the quan t i t y a l l o c a t e d from source k i s a t l e a s t s u f f i c i e n t to ensure the sum of the q u a n t i t i e s a l l o c a t e d from sources 1 through k-1 does not exceed the a v a i l a b l e log supp l y . The upper l i m i t i s de f ined f o r source k as the s m a l l e r of the magnitudes of X^. and Y^ or the a v a i l a b l e supply of logs from source k. Using the no ta t i on g iven above, the. reg ions are de f ined as {x tk f o r k = R-, = tk 0 < x t k < X t > f o r k + 1, {*tk ! y t k = V f o r k = . k-1 max[0 ; . Y t - ^ v s t m ] < y t k < m i n [ v s t k ;. Y ^ } f o r k f 1, R 3 = 30 { xtk^tk I 0 < x t k + y t k < v t k } f o r k = 1, k-1 ^ t k+y t k I maxfO ; V V V t J 1 x t k \ ^ vtk } ^ t f c " UK. " L U -i U l l ~ — LK. " L. m= I f o r k f 1, where t = 1 ,2 ,3 . The opt imal s e t o f a l l o c a t i o n s f o r each p a i r o f q u a n t i t i e s and Y t i s i d e n t i f i e d f o r the s e t {X^ ,Y t } i n a manner analogous to tha t des-c r i bed e a r l i e r . From the " forward pass" i n which the computat ion p ro-gresses from k = 1 to k = 5 the magnitudes of the q u a n t i t i e s x ^ k and y t k a s s o c i a t e d w i th f ^ ( X t , Y t ) are i d e n t i f i e d as x ^ k ^ X t * Y t ^ a n d y t k ^ X t ' Y t ^ ' L e t , . x t k ( X ^ . , Y ^ ) and y - t k ^ t ' ^ t ^ = Quant i ty of pu lp logs and saw logs , respec -t i v e l y , a l l o c a t e d i n pe r iod t from source k i n the " forward p a s s . " As each of these se ts of a l l o c a t e d q u a n t i t i e s are de r i ved they are r e t a i ned f o r the second computat ional p r o c e s s , the "backward p a s s . " Th is proceeds from k = 5 to k = 1 f o r each per iod t , and i s e s s e n t i a l l y a scanning process i n w h i c h , by app rop r ia te s u b t r a c t i o n , p a r t i c u l a r elements of the se ts { x t k ( X t , Y t ) } and ( y t k ( X t , Y t ) } are i d e n t i f i e d as , t he opt imal a l l o c a t i o n s x + t k ( X t > Y t ) and y t k ( X t . Y t ) : L e t , x t k ( X t , Y t ) and y ^ U ^ Y ^ ) = opt imal a l l o c a t i o n s of pu lp logs and sawlogs , .* supe rsc r i p t , r e f e r s to opt imal a l l o c a t i o n s f o r t ' f i x e d , 1 tha t i s , k i s a v a r i a b l e , t = parameter. 31 r e s p e c t i v e l y , from source k i n each per iod t , i d e n t i f i e d from a "backward p a s s . " The opt imal a l l o c a t i o n s , x t k ( X t , Y t ) and y t k ( X t , Y t . ) are i d e n t i f i e d as W w = W W ^ W = W W 5 * . . 5 • * W W = x t k ( V . 4 + / t j ^ t » ^ ) » V j = s k + / t j ^ x t » Y t » where k = 4 ,3 ,2 ,1 A s s o c i a t e d w i th each opt imal p o l i c y f o r a l l o c a t i n g the q u a n t i t i e s Xj. and Yj. i s the eva lua ted f u n c t i o n f ^ U ^ Y ^ ) . This, i s the minimum net d e l i v e r e d l og c o s t of a l l o c a t i n g the q u a n t i t i e s X^. and Y t from per iod t and i s cons idered as the u t i l i t y f u n c t i o n f o r pe r iod t f o r the s e t of q u a n t i t i e s {X^.,Y^.}. Th is f u n c t i o n i s i nco rpo ra ted i n the m u l t i - s t a g e d e c i s i o n p r o c e s s , developed i n the next s e c t i o n , from which opt imal a l l o c a t i o n s between each pe r iod are i d e n t i f i e d . 2. A l l o c a t i o n between Per iods The m i l l demands f o r the- p lann ing c y c l e of three time per iods must be s a t i s f i e d . With the assumption tha t e x i s t i n g log s tocks 32 d e l i v e r e d p r i o r to the p lann ing c y c l e are mainta ined a t a cons tant l e v e l , the t o t a l quan t i t y o f pu lp logs and sawlogs to be a l l o c a t e d , X and Y , equals the sum of the m i l l demands f o r the three p e r i o d s . When t h i s quan t i t y i s a l l o c a t e d among p e r i o d s , d i f f e r e n c e s i n the a v a i l a b i l i t y and d e l i v e r e d c o s t of logs may favour the accumulat ion of log i n v e n t o r i e s i n advance of planned m i l l demands. However, a l l o c a t i o n d e c i s i o n s must a l s o r e f l e c t the a d d i t i o n a l cos ts i ncu r red when log su rp luses are " c o l d -decked" f o r l a t e r u t i l i z a t i o n . The problem desc r i bed requ i res m i l l demands to .be s a t i s f i e d and , p rov id i ng the necessary logs are a v a i l a b l e , permits temporary log s u r -p luses to be accumulated i n per iods one and two. Because an excess of cu r ren t log d e l i v e r i e s over the m i l l demands of pe r iod t must not exceed the q u a n t i t i e s demanded i n pe r iod t+1 the magnitudes of the q u a n t i t i e s X and Y are a c c o r d i n g l y r e s t r i c t e d f o r each p e r i o d . The minimum q u a n t i t i e s a l l o c a t e d must permit the accumulated m i l l demands, t t E Dx . and E D y . , j= l 3 j= l J to be s a t i s f i e d . The maximum q u a n t i t i e s tha t may be a l l o c a t e d from 1 through t per iods are r e s t r i c t e d by the cumula t ive demands, t+1 t+1 E Dx . and E D y . , j=l J • j= l J the sum of a v a i l a b l e l o g s , t 5 t 5 E E v . and E E v s . , j= l m=l J m j=l m=l j m o r , i f a l l a v a i l a b l e logs from e a r l i e r t ime per iods have not been 33 u t i l i z e d , by the composite t o t a l o f t t E Dx . a n d E Dy. j= l J j= l J and the q u a n t i t i e s of logs a v a i l a b l e i n pe r iod t , 5 5 E v. and E vs . . m=l t m m=l t m These r e s t r i c t i o n s may be de f ined f o r each per iod t = 1,2,3 as t t+1 t 5 t 5 E Dx. < X < min[ E Dx. ; E E v . ; E Dx.+ E v ] , j = l . J j= l J j= l m=l j m j= l J m=l t m t t+1 t 5 t 5 E Dy. < Y < min[ E Dy. ; E E v s . ; E Dy.+ E v s . ] , j= l J ~ ~ j= l 3 j= l m=l J m j=l J m=l t m t t+1 t 5 t 5 E (Dx.+Dy.) < X+Y < min[ E (Dx.+Dy.) ; E E v . ; E (Dx.+Dy.)+ E v . J , j= l J J ~ j= l J J j= l m=l j m j= l J J m=l t m where Dx^ = Dy 4 =0, t t t t and E (Dx.+Dy.) <^  E v . ; E Dy. £ E vs . , j= l 3 3 j= l 3 j= l 3 j= l J 5 5 f ° r ^ - ^ / t m ' v " s " t l m E / s t m ' m= I m=1 where vs" t and v^ = volume of sawlogs and the volume of both sawlogs and p u l p l o g s , r e s p e c t i v e l y , tha t are cu t i n pe r iod t . The cos t of " c o l d - d e c k i n g " su rp lus d e l i v e r e d logs i n each per iod i s dependent upon the q u a n t i t i e s and Y^., the magnitudes of the m i l l demands, Dx^ and Dy^., and the u n i t v a r i a b l e cos t of " c o l d - d e c k i n g . " L e t , ch = u n i t v a r i a b l e log hand l ing cos t per thousand c u n i t s when su rp lus logs are p laced i n , and taken out o f , " c o l d - d e c k " s to rage areas i n m i l l y a r d s , g t U t , Y t ) = t o t a l v a r i a b l e log hand l ing cos ts i ncu r red i n pe r iod t by " c o l d - d e c k i n g " su rp lus logs from an a l l o c a t i o n o f the q u a n t i t i e s X^. and Y^., then g t ( X t , Y t ) = (max[0 ; X t -Dx t ]+max[0 ; Y t - D y t ] ) . c h f o r t = 1 ,2 ,3 . The recur rence r e l a t i o n s h i p by which the q u a n t i t i e s X and Y are . o p t i m a l l y a l l o c a t e d from the three time per iods may now be de r i ved from Equat ion [ 5 ] . . L e t , Fj.(X,Y) =minimum cos t of a l l o c a t i n g the q u a n t i t i e s X and Y from a l l a v a i l a b l e sources i n 1 through t p e r i o d s , then F t ( X , Y ) = min [ f t 5 ( X t , Y t ) + g t ( X t , Y t ) + F t _ 1 ( X - X t , Y - Y t ) ] , [7] X t e R 4 V R 5 X t + Y t e R 6 where F Q ( X , Y ) = 0 f o r t = 1 ,2 ,3 . The three reg ions R^, Rg and Rg de f i ne the f e a s i b l e magnitudes of X^ and Y^ and X^+Y^., the a l l o c a t i o n s made from the log sources a v a i l a b l e i n pe r i od t . For each pe r iod the d e r i v a t i o n of the minimum 35 magnitudes of and Y .^ r equ i res i d e n t i f i c a t i o n of the maximum volume of a d e l i v e r e d log su rp lus at the end of per iod t - 1 . In a d d i t i o n , these minima a l s o vary f o r each elemental p a i r of the se t {X ,Y} . Due to these requirements the d e f i n i t i o n s of the reg ions are based on the i d e n t i t i e s X t = X - ( X - X t ) , . Y t = Y - ( Y - Y t ) and X t +Y t -= X + Y - ( X + Y - X t - Y t ) . The t o t a l quan t i t y o f logs d e l i v e r e d by per iod t-1 cannot exceed t o t a l m i l l demands accumulated to per iod t , t t E Dx. and E D y . , j= l J j= l J and may be r e s t r i c t e d by the a v a i l a b l e supply of l o g s . The r e s t r i c t i o n may be e i t h e r the t o t a l quan t i t y a v a i l a b l e from per iods 1 through t - 1 , t-1 5 A o r , i f not a l l a v a i l a b l e logs were u t i l i z e d , by the sum of the m i l l de-mands accumulated to per iod t - 1 , . t-1 t-1 E Dx. and E Dy . , j= l 3 j= l J and the a v a i l a b l e log supply i n per iod t - 1 , 5 5 E v. , m and E vs . •, . m = 1 t - l , m m = 1 t - l , m By s u b t r a c t i n g the minimum of the three a l t e r n a t i v e s from the s e t {X,Y} the cor responding minimum magnitudes of the q u a n t i t i e s , X^.,Y^ and X^+Y^.are i d e n t i f i e d . Maximum q u a n t i t i e s of logs which may be a l l o c a t e d to per iod t are i d e n t i f i e d f o r {X,Y} by assuming tha t no log su rp luses have been 36 developed i n pe r iod t - 1 . A l l o c a t i o n s are then r e s t r i c t e d by e i t h e r the m i l l demands i n pe r iod t , Dx^ and Dy^, or by the a v a i l a b l e log s u p p l y , 5 : m= I The three reg ions are de f ined a s : t t-1 5 t-1 5 R„ = {X. |max[X- E Dx. ; X- E E v . ; X- E DX.- E v t , J 4 t j= l 3 j= l m=l j m j=l J m=l t - l ' m t-1 5 <_ X. <_min[X- E Dx. ; E v. ] } , z j=l J m=l z m t t-1 5 t-1 5 R, = {Y. |max[Y- E Dy, ; Y- E E v s . ; Y- E Dy . - E vs . 1 ] 5 z i=l 3 i=l m=l. j m i=l J m=l 1 - 1 ' m t-1 5 t t - i 5 t - i < Y < min[Y- E Dy. ; E vs . ] } , z i=l 3 m=l t m \ c = {X.+Y. |max[X+Y- E (Dx.+Dy.) ; X+Y- E E v . ; X+Y- E (Dx.+Dy.)- E v. b z z j= l 3 3 j= l m=l j m j= l J J m=l t _ t-1 £ X.+Y. < mi n [X+Y- E (Dx.+Dy.) ; E v . ] } z z i=l J 3 m=l z m J 5 5 m= 1 m= I and Dx Q = DyQ = Dx 4 = Dy 4 = 0 f o r t = 1 , 2 , 3 . The v a r i a b l e cos t component g t ( X ^ , Y ^ . ) , i nc luded i n Equat ion [ 7 ] , i s dependent upon the d i f f e r e n c e between cu r ren t log d e l i v e r i e s and cu r ren t m i l l demands. In a d d i t i o n to t h i s type of inven to ry cos t a second group of v a r i a b l e cos ts may a f f e c t the a l l o c a t i o n of logs between 37 p e r i o d s ; those tha t are dependent upon the t o t a l quan t i t y of logs remaining i n s torage a t the end of each p e r i o d . For pe r iod t t h i s q uan t i t y i s the d i f f e r e n c e between the accumulated volumes of logs p ro -duced i n per iods 1 through t and the accumulated m i l l demands. These cos ts r e f e r to the u n i t v a r i a b l e cos ts of ma in ta in ing log i n v e n t o r i e s and are t y p i c a l l y p re -de te rmined . For example, the "average cos t o f d e t e r i o r a t i o n o f log q u a l i t y " and the "average c a r r y i n g c o s t " are f r e -quent ly based on p r a c t i c a l s i m p l i f i c a t i o n s made p r i m a r i l y f o r account ing purposes. Even so they represen t r e a l cos ts w h i c h , a l though a r b i t r a r i l y exc luded from the hypo the t i ca l prob lem, may be r e a d i l y i nco rpora ted i n the a n a l y t i c framework developed f o r the model. L e t , G t ( X , Y ) = t o t a l v a r i a b l e s torage cos t i ncu r red i n pe r iod t f o r an a l l o c a t i o n of t h e : q u a n t i t i e s , X and Y , from 1 through t p e r i o d s , = u n i t v a r i a b l e s to rage cos t per thousand c u n i t s of logs remaining i n " c o l d - d e c k " i nven to ry a t the end of per iod t , t t then G . (X ,Y ) = C + . (X+Y- E Dx . - E Dy.) Z 1 "j=l 3 j= l 3 t t f o r E Dx. <_ X and E Dy. <_ Y. j= l J j= l J Because the c o s t te rm, G t ( X , Y ) , i s independent of the component q u a n t i t i e s t-1 t-1 X . , E X . , and z Y . . when i nc luded i n the f o rmu la t i on , Equat ion [7] i s rep laced by F t ( X , Y ) =• min [ f t 5 ( X t , Y t ) + g t ( X t > Y t ) + F t _ 1 ( X - X t , Y - Y - t ) ] + G t ( X , Y ) . X t E R 4 Y t £ R 5 W R 6 f o r t = 1 ,2 ,3 . The q u a n t i t i e s X and Y equal the accumulated m i l l demands 3 3 z Dx . and z Dy. j= l J j= l J when t = 3 . Consequent ly the minimum cos t of s a t i s f y i n g m i l l demands f o r the p lann ing c y c l e i s eva luated by F ^ X j Y ) . The opt imal p a r t i t i o n ing among the three t ime per iods i s i d e n t i f i e d from a "backward p a s s . " The q u a n t i t i e s s e l e c t e d are used to i d e n t i f y the minimum cos t s e t of a l l o c a t i o n s from each source f o r each p e r i o d . L e t , X^.(X,Y) and Y^.(X,Y) = quan t i t y of pu lp logs and sawlogs , r e s p e c t i v e l y , a l l o c a t e d from per iod t when the q u a n t i t i e s X and Y are o p t i m a l l y a l l o c a t e d from 1 through t per iods i n a " forward p a s s , " * * X t ( X , Y ) and Y t ( X , Y ) = opt imal a l l o c a t i o n of pu lp logs and :sawlogs , r e s -p e c t i v e l y , from per iod t , i d e n t i f i e d from a "backward p a s s , " t hen , X t ( X , Y ) = X t ( X , Y ) " Y t ( X , Y ) .= Y t ( X , Y ) _ f o r t '= 3 , X t ( X , Y ) Y t ( X , Y ) X . ( X - E X . ( X , Y ) , - Y- I Y . ( X , Y ) ) j=t+l J j=t+l J 3 3 Y . ( X - E X . ( X , Y ) , Y- Z Y . ( X , Y ) ) Z j=t+l J j=t+l J J f o r t = 2,1 L e t , O o x t k ( X , Y ) and y^.^(X,Y) = q u a n t i t y of pu lp logs arid saw logs , r e s p e c t i v e l y , a l l o c a t e d from source k i n each pe r iod t when the q u a n t i -t i e s X and Y are o p t i m a l l y a l l o c a t e d from a l l per iods i n the p lann ing c y c l e , then x \ . ( X , Y ) = x , . ( X , ( X , Y ) ,Y . (X ,Y ) ) " Z K Z K z z f o r t = 1 , 2 , 3 , ; t k ( X , Y ) = J t k ( X t ( X , Y ) , Y t ( X , Y ) ) J k = 1 . 2 . . . . . 5 . CHAPTER I I I DERIVATION OF NET DELIVERED LOG COSTS A. INTRODUCTION The p lann ing requirement i s to s a t i s f y the log demands of both m i l l s i n accordance w i th the economic o b j e c t i v e s of h igher management. These o b j e c t i v e s have been de f ined as the m in im iza t i on of the d i f f e r e n c e between two groups of log p roduc t ion cos ts and the va lue of d e l i v e r e d sawlogs. These three items a r e : (1) the c o s t of ha rves t i ng and t r a n s -po r t i ng pu lp logs and sawlogs to m i l l y a r d s , (2) the c o s t of " c o l d -deck ing " temporary log s u r p l u s e s , and (3) a measure of the re tu rn from the p rocess ing of d e l i v e r e d sawlogs . The d e r i v a t i o n of " c o l d - d e c k i n g " cos ts and t h e i r i n c l u s i o n i n the computat ional process f o r op t im i z i ng log a l l o c a t i o n s between time per iods has been desc r i bed i n Chapter I I . A l s o d i scussed was the method by which the composite c o s t f u n c t i o n s , c t i < ( x t | < > y . t | < ) » a r e c o m b i n e c l w n e n a l t e r n a t i v e a l l o c a t i o n s among,log sources are compared. In t h i s chapter a t t e n t i o n i s d i r e c t e d towards the d e r i v a t i o n of c ^ U ^ » v - t i < ) a n c ' a de-t a i l e d d i s c u s s i o n of items (1) and ( 3 ) . For the i d e n t i f i c a t i o n o f an opt imal a l l o c a t i o n p o l i c y those c o s t or revenue components tha t do not vary w i th a l t e r n a t i v e a l l o c a t i o n s need not be c o n s i d e r e d . They are i r r e l e v a n t f o r a l l o c a t i o n d e c i s i o n -making. The assumption tha t m i l l demands i n each per iod must be s a t i s -f i e d and the emphasis on sho r t - t e rm log a l l o c a t i o n f u r t h e r r e s t r i c t s the 41 s i g n i f i c a n t p roduc t ion elements to be e v a l u a t e d . In t h i s i l l u s t r a t i v e example no attempt i s made to de f ine a r b i t r a r i l y the s p e c i f i c composi-t i o n of each element in t roduced i n the f o r m u l a t i o n . S p e c i f y i n g the i n d i v i d u a l items tha t comprise each element i s not e s s e n t i a l to the d i s -cuss ion and would tend tp focus a t t e n t i o n u n n e c e s s a r i l y on the s p e c i f i c a d m i n i s t r a t i v e and f i n a n c i a l s t r u c t u r e , d e t a i l s o f ha rves t i ng methods, equipment and p o s s i b l e a l t e r n a t i v e u t i l i z a t i o n of the p h y s i c a l resources of a h y p o t h e t i c a l f o r e s t e n t e r p r i s e . Instead the emphasis i s d i r e c t e d towards i l l u s t r a t i n g how a v a r i e t y of l og -p roduc t i on sys tems, ranging from the s imple to the r e l a t i v e l y complex, can be r e a d i l y eva lua ted and i nc luded i n the formula ted model. In the f o l l o w i n g d i s c u s s i o n , each of the major elements de f ined i s assumed to con ta in on ly those cos t or revenue items i d e n t i f i e d as v a r i a b l e w i th respec t to a l t e r n a t i v e log a l l o c a t i o n s . Due to b a s i c d i f f e r e n c e s between the groups of cos t components requ i red to eva lua te each of the assumed ha rves t i ng systems the d e r i v a -t i o n of l og p roduc t ion and t r a n s p o r t a t i o n cos ts i s desc r ibed f o r each i n d i v i d u a l sou rce . The re tu rn per u n i t of d e l i v e r e d sawlogs i s assumed to be dependent upon average log d iameter and s p e c i e s , both of which are p r i m a r i l y r e l a t e d to the f o r e s t type of each source of l o g s . In a d d i t i o n , when a l l o c a t i o n s are made from sources two and t h r e e , the type of sawlog i s a l s o a f f e c t e d by the p ropo r t i on of sawlogs to pu lp logs and a log s o r t i n g p o l i c y . Because of t h i s i n t e r a c t i o n the measure developed to eva lua te the re tu rn from d e l i v e r e d sawlogs i s i nc luded i n a composite term which combines items (1) and ( 3 ) . These two items are desc r ibed 42 before the d e t a i l e d examinat ion of each log sou rce . 1. Return per Un i t of De l i ve red Sawlogs The term "sawlog net wor th" i s de f ined as the d i f f e r e n c e i n d o l l a r s per thousand c u n i t s between the f . o . b . m i l l p r i c e of manufactured lumber and v a r i a b l e m i l l convers ion c o s t s . To s i m p l i f y e v a l u a t i o n on ly two f . o . b . p r i c e s are used , the d i s t i n c t i o n being based on an assumed market p re fe rence f o r c e r t a i n t ree s p e c i e s . The group of spec ies des-ignated as those which command a premium lumber p r i c e w i l l g e n e r a l l y change w i th market f l u c t u a t i o n s . However, to permi t numerical i l l u s t r a -t i o n D o u g l a s - f i r i s used as an example of a p r e f e r r e d t ree spec ies and western hemlock as an example of a l e s s des i r ed spec ies f o r lumber manufacture. L e t , wp^ and wn^. = average f . o . b . m i l l p r i c e per thousand c u n i t s i n pe r iod t f o r a l l lumber.manufactured from D o u g l a s - f i r and western hemlock, r e s p e c t i v e l y , l e s s m i l l ya rd and a l l v a r i a b l e convers ion cos ts tha t are independent of log d iamete r , P t k = r a t i o of the volume of D o u g l a s - f i r logs to the q u a n t i t y y t k > D ^ = average log d iameter i n inches of the quan t i t y y ^ , mc = v a r i a b l e m i l l convers ion cos t per 100 minu tes , cm = time requ i red to produce one thousand cub ic f e e t ( ten c u n i t s ) of lumber, expressed as number of minutes X log d i a m e t e r - i n c h , 43 then tk" P r e l i m i n a r y sawmi l l schedules and market ing p o l i c i e s f o r the p lann ing c y c l e are assumed to be based on a supply of logs having a p a r t i c u l a r average log d iameter and a des ignated p ropor t i on o f Douglas-f i r . Q u a n t i t i e s of sawlogs meeting these s p e c i f i c a t i o n s may be a l l o -cated from source t h r e e , the major sawlog-produc ing a r e a . When sawlogs have average diameters l a r g e r than the s i z e s t i p u l a t e d v a r i a b l e log convers ion cos ts dec rease , and when the D o u g l a s - f i r content of processed sawlogs i s g rea te r than a s p e c i f i e d p ropo r t i on r e c e i p t s from lumber s a l e s i n c r e a s e . A c c o r d i n g l y , the e f f e c t o f the type of log from a l t e r -na t i ve log a l l o c a t i o n s on u n i t m i l l re turns i s measured by the d i f f e r -ence between two eva lua t i ons of sawlog net wor th . One of these va lues i s c a l c u l a t e d f o r the processed saw logs , the o ther i s based on the average log d iameter and D o u g l a s - f i r content i n i t i a l l y s p e c i f i e d f o r the p lann ing c y c l e . The d i f f e r e n c e , r e f e r r e d to as " d i f f e r e n t i a l sawlog net wor th" may be p o s i t i v e or nega t i ve . x ^ and y^^ = quan t i t y of pu lp logs and saw logs , r e s p e c t i v e l y , i n thous-ands of c u n i t s a l l o c a t e d from source k i n pe r iod t , L e t , Ds = average log d iameter i n inches of the quan t i t y y ^ s e l e c t e d from the t o t a l q u a n t i t y , x ^3 + y^3> a l l o c a t e d from source three when y . ^ <_ 2 . x + o , Ps = r a t i o of the volume of D o u g l a s - f i r logs to the quan t i t y 44 s e l e c t e d from the t o t a l q u a n t i t y , x ^ 3 + y ^ 3 » a l l o c a t e d from source three when y ^ ± ^ . x ^ , " d i f f e r e n t i a l sawlog net wor th" = d i f f e r e n c e i n d o l l a r s between the sawlog net worth of the quan t i t y y ^ conver ted i n t o lumber and the sawlog net worth of an equal quan t i t y of sawlogs having an average log diameter of Ds inches and con ta in i ng Ps.100 per cent by volume of D o u g l a s - f i r l o g s , c w t k ^ x t k ' y t k ^ = " d i f f e r e n t i a l sawlog net wor th" m u l t i p l i e d by the quan t i t y y t k , t hen , from the d e f i n i t i o n o f sawlog net worth g iven i n the prev ious s e c t i o n , and from the d e f i n i t i o n of " d i f f e r e n t i a l sawlog net wor th" g iven above, c w t k ^ x t k ' y t k ^ = { ( p t k " P s ^ w p t " w n t ^ " r n c • c m • ^ 1 / D t k " 1 / D s ^ } • y t k • For the lumber p r i c e s , sawmi l l convers ion cos ts and magnitudes f o r Ds and Ps assumed i n the hypo the t i ca l problem (Appendix C ) , the e f f e c t of log s i z e and spec ies content on the va lue of " d i f f e r e n t i a l sawlog net wor th" i s shown i n F igure 5 . 2. Net D e l i v e r e d Log Costs Log p roduc t ion and log t r a n s p o r t a t i o n cos ts are combined w i th " d i f f e r e n t i a l sawlog net wor th" to p rov ide the c o s t f u n c t i o n c t k ^ x t k ' y t k ^ " This f u n c t i o n was used i n the o p t i m i z a t i o n equat ion developed i n Chapter II f o r a l l o c a t i n g logs from a l t e r n a t i v e sou rces ; L e t , "ne t d e l i v e r e d log c o s t " = sum of the v a r i a b l e log p roduc t ion and log 46 t r a n s p o r t a t i o n cos ts combined w i th the " d i f f e r e n t i a l sawlog net wor th" of the quan t i t y y ^ , c t k ^ x t k ' ^ t k ^ = " n e t d e l i v e r e d log c o s t " of a l l o c a t i n g the q u a n t i t i e s x^^ and y ^ from source k i n pe r iod t , c c * t k ^ x t k s y t k ^ = s u m 0 T " the v a r i a b l e log p roduc t ion and log t r a n s p o r t a -t i o n cos ts i n c u r r e d when the q u a n t i t i e s and y^ are d e l i v e r e d to the p u l p m i l l and sawmi l l y a r d s , r e s p e c t i v e l y , then W t k ' W = ^ t k ^ t k ^ t k ^ ^ t k ^ t k ' ^ t k ) ' B. LOG COSTS FOR EACH SOURCE The purpose of the f o l l o w i n g s e c t i o n s i s to p resent a v a r i e t y of cos t f unc t i ons w h i c h , though hypo the t i ca l , • r e f l e c t log p roduc t ion p r o - , cesses t y p i c a l l y found i n i n d u s t r y . The d e r i v a t i o n of each f u n c t i o n requ i res i d e n t i f y i n g those cos t components which depend upon log a l l o -c a t i o n . For the s t a t e d prob lem, d e l i v e r e d l o g . c o s t s range from cons tan t u n i t d e l i v e r y charges to a combinat ion o f cos ts which requ i re the i d e n t i -f i c a t i o n of optimum log s o r t i n g and t r ad ing p o l i c i e s f o r each log a l l o c a t i o n . - The d i s c u s s i o n of each source i nc ludes a b r i e f p h y s i c a l d e s c r i p t i o n of log a v a i l a b i l i t y fo l l owed by the mathematical d e r i v a t i o n o f the requ i red cos t f u n c t i o n . A d e s c r i p t i v e f low char t of the e v a l u a -t i o n procedure i s p rov ided when cons idered to be warranted by the complex i ty of the mathematical development. As i n d i c a t e d i n F igure 1, page 9 , the log p roduc t ion p lanners are assumed to have no con t ro l over f o r e s t management p o l i c y . The maximum 47 q u a n t i t i e s tha t may be cut from each of the fou r log p roduc t ion areas i n each per iod are pre-determined f o r the du ra t i on of the p lann ing c y c l e . The quan t i t y of logs tha t may be purchased from the open log market i n each pe r iod i s l i m i t e d by market f o r c e s . Th is r e s t r i c t i o n does not prevent the exchange of a d d i t i o n a l logs when log t r ad i ng i s b e n e f i c i a l . The p r i c e of purchased l o g s , assumed to be constant w i t h i n each t ime p e r i o d , v a r i e s over the p lann ing c y c l e i n response to the log demands of competing o r g a n i z a t i o n s . 1. Log Costs f o r Source One Th is log p roduc t ion area s u p p l i e s on ly pu lp logs and i s a c c e s s i b l e i n each time p e r i o d . - An e x i s t i n g network of main haul roads prec ludes the n e c e s s i t y of major road c o n s t r u c t i o n w i t h i n the p lann ing c y c l e . However, i f the planned log p roduc t ion f o r any time per iod exceeds a s p e c i f i e d l e v e l , en larged land ings and a d d i t i o n a l spur roads are r e -q u i r e d . The p r o v i s i o n of these a d d i t i o n a l f a c i l i t i e s , wh i l e decreas ing v a r i a b l e log p roduc t ion c o s t s , requ i res c a p i t a l expend i tu re . L e t , vm-j = quan t i t y of l o g s , i n thousands of c u n i t s , which can be cut i n each per iod w i thou t changing u n i t v a r i a b l e c o s t s , c f = c o s t o f en l a rg i ng land ings and c o n s t r u c t i n g a d d i t i o n a l spur r o a d s . f o r source one, cxl^-j = v a r i a b l e c o s t per thousand c u n i t s of pu lp logs d e l i v e r e d from source one i n pe r iod t when the cut volume does not exceed the quan t i t y vm, c x 2 ^ = v a r i a b l e cos t per thousand c u n i t s of pu lp logs d e l i v e r e d from source one i n per iod t f o r the p o r t i o n of the cut vo l ume which exceeds the quan t i t y vm-|, then c d t l ( x t l - ^ t ^ = c x l ^ i .x .^-] f o r 0 <_ x ^ <_ vm^, c f+cx l^^ .vm^+cx2 t-j . ( x t i - vm- | ) f o r vm-| < x^^ <_ v^^ a . Net De l i ve red Log Costs No sawlogs are produced, consequent ly c w t i ( x t r y t i } = 0 a n d ^ t i ^ t i ^ t i ) = c d t i ( x t r y t i ? -Pu lp log p roduc t ion and t r a n s p o r t a t i o n cos ts are shown i n F igure 6 f o r the range of q u a n t i t i e s tha t may be a l l o c a t e d from source one. 2. Log Costs f o r Source Two The major c h a r a c t e r i s t i c s of t h i s p roduc t i ve area which a f f e c t log a l l o c a t i o n a r e ; i t s i n a c c e s s i b i l i t y i n the second t ime p e r i o d , a f o r e s t type from which both sawlogs and pu lp logs may be c u t , mandatory log s e l e c t i o n r u l es when both log products are r e q u i r e d , and the permi t -ted exchange of pee le r logs f o r sawlogs and/or p u l p l o g s . Due to log s o r t i n g and t r ad i ng a c t i v i t i e s the spec ies mix and average.d iameter of d e l i v e r e d logs are a f f e c t e d . A c c o r d i n g l y , the e v a l u a t i o n of " d i f f e r -e n t i a l sawlog net wor th" i s necessary f o r each t r ad ing p o l i c y examined. D e t a i l s of the e v a l u a t i o n of the f u n c t i o n c ^ 2 ^ x t 2 ' y t 2 ) a r e i n c ^ u d e d i n the f l o w c h a r t shown. in F igure 7. 220 r volume i n thousands of c u n i t s F i g . 6 ' De l i ve red Cost of Pulpwood from Source One c l t ( x t 2 , y t 2 ) = c f l + ( x t 2 + y t 2 ) . c l l t yes yes c M x t 2 , y t 2 ^ = c f l + c f 2 + v m 2 . c l l + ( x t 2 + y t 2 - v m 2 ) . c l 2 t yes Q l = ( x t 2 + y t 2 ) ( 1 _ p c ) ( 1 - r s ) - p l Q 2 = ( x t 2 + y t 2 ) ( 1 " p c ) ( 1 _ r s ) ( 1 - P 1 ) Q 3 = ( x t 2 + y t 2 ) ( l - p c ) . r s . p s Q 4 = ( x t 2 + y t 2 ) ( l - p c ) . r s . ( l - p s ) V i n = m a x [ ° i y t 2 - P c - ( x t 2 + y t 2 ) ] 4 ymax = m i ' n r y t 2 ; S Q j ] j=l J Z t = +00 t r = 0 y c = y mi n c st = E Q j - c n + p c . ( x t 2 + y t 2 ) . c p ( t r + p c ) ( x t 2 + y t 2 ) - y t 2 + y c + m T n [ E Q j - y c ; y t 2 - t r . ( x t 2 + y t 2 ) ] y .ess I c t t = ( E Q j - y c ) . c t x t + ( y t 2 - t r . ( x t 2 + y t 2 ) ) . c t y t + ( ( t r + p c ) ( x t 2 + y t 2 ) - y t 2 + y c ) ( c t t - t c ) yes dy = Q1.dpl+Q2.d1+ ( y t 2 - ( l - p c ) - Q 1 - Q 2 ) . d p s py = y t 2 - 0 - p c ) - Q 2 dy = Q 1 . d p l + ( y t 2 . ( l - p c ) - Q 1 ) . d l py = Qi dy = Q 1 .dpl+Q 2 .d l+Q 3 .dps+ ( y t 2 - ( i - p c ) - Q 1 - Q 2 - q 3 ) . d s py = D t 2 = ( d y + ( y t 2 - y c - t r - ( x t 2 + y t 2 ^ - d p + t r ^ x t 2 + y t 2 ^ d t ) / ' y t 2 p t 2 = ( p y + ( y t 2 " y c " t r - ( x t 2 + y t 2 ^ - p p + t r ^ x t 2 + y t 2 ^ p t ) / y t 2 C W t = ( ( P t 2 " P s ) ( w p t " w n t ) " m c - c m - ( 1 / D t 2 " 1 / D s ) ) - y t 2 I p - c s t + c t t - c w t yes yc = yc+.Ol yes V 0 <t> = yc y = t r t r = tr+.Ol p c . x t 2 > ( c p + c s s + c t t - t c ) c t 2 ( x t 2 , y t 2 ) = 0 F igure 7 F lowchar t o f the D e r i v a t i o n of Net De l i ve red Log Costs f o r Source Two 51 a . Log S o r t i n g and Trading An a n a l y s i s of the sawlog s e l e c t i o n r u l es and log t r ad ing a l t e r -na t i ves w i l l p rov ide a bas i s f o r examining the cos t components assumed f o r source two. A pr imary o b j e c t i v e of s o r t i n g i s to improve the net worth of sawlog a l l o c a t i o n s by i n c r e a s i n g average log s i z e and the p ropo r t i on of D o u g l a s - f i r logs i n the d e l i v e r e d quan t i t y of sawlogs. In t h i s i l l u s -t r a t i v e example one pee le r log c l a s s and fou r non-pee le r log c l a s s e s are recogn i zed . Each type i s cons idered s u f f i c i e n t l y w e l 1 - d i s t r i b u t e d to ensure the cut volume of each c l a s s i s p ropo r t i ona l to the t o t a l quan t i t y c u t , x t 2 + ^ t 2 * ^ f i x e d p ropo r t i on of pee le r logs i s i d e n t i f i e d by a se t of r u l es based on log d iamete r , l e n g t h , spec ies and freedom from any de fec ts p r o h i b i t i n g ro ta r y p e e l i n g . The fou r non-pee le r log c l a s s e s are grouped by s i z e and s p e c i e s . The s e l e c t i o n of sawlogs from these c l a s s e s f o l l o w s a r u l e which requ i res p r i o r i t y be g iven to the l a r g e r d iameter logs fo l l owed by p r i o r i t y to p re fe r red t ree s p e c i e s , tha t i s , D o u g l a s - f i r . A more comprehensive s e l -e c t i o n r u l e w i l l be requ i red i f a d d i t i o n a l log c l a s s e s are r e c o g n i z e d . P rov id i ng the harves ted volume of each c l a s s i s assumed to be p ropor -t i o n a l to the t o t a l cu t the methodology desc r ibed may be r e a d i l y adapted. Pee le r logs are e i t h e r t ranspor ted d i r e c t l y to the sawmi l l f o r lumber manufacture or t raded f o r an equal volume of sawlogs or p u l p l o g s . on the l o c a l log market . Due to the g r e a t e r u n i t va lue of pee le r logs an exchange reduces the t o t a l cos t of producing and d e l i v e r i n g l o g s . 52 However, when pee le r logs tha t would o therwise be used as sawlogs are t r a d e d , . t h e cos t reduc t i on may be more.than o f f s e t by a lower sawlog net wor th . The opt imal quan t i t y of pee le r logs t raded f o r sawlogs i s dependent not only upon the r e l a t i v e u n i t cos ts but a l s o upon the magni-tudes of the a l l o c a t e d volumes a n d y ^ - In a d d i t i o n to the quan t i t y t raded there i s a second.opt imal volume, the amount of sawlogs s e l e c t e d from the non-pee le r log c l a s s e s . These two opt imal volume components are i d e n t i f i e d f o r the q u a n t i t i e s , a n d y ^ , by an i t e r a t i v e procedure and represent the opt imal s o r t i n g and t r ad i ng p o l i c y f o r t h i s p a r t i c u l a r a l l o c a t i o n . When on ly pu lp logs are requ i red log s o r t i n g and t r ad ing i s r e s -t r i c t e d to the exchange of pee le r logs f o r an equal volume of pu lp logs w i th a consequent reduc t i on i n the d e l i v e r e d cos t of the quan t i t y x ^ . L e t , pc = r a t i o of the volume of pee le r logs to the a v a i l a b l e log supply f o r source two, t r = r a t i o of the volume of pee le r logs t raded f o r sawlogs to the quan t i t y of logs cut from source two, yc = p o r t i o n i n thousands of c u n i t s o f the quan t i t y y^ tha t 1 S s e l e c t e d from the non-pee le r log c l a s s e s of source two, l a rge log c l a s s = merchantable length non-pee le r logs having a minimum top d iameter of e i g h t i n c h e s , smal l l og c l a s s = merchantable lengths of logs having a minimum top d iameter of f ou r inches a f t e r pee le r logs and l a rge logs i n the non-pee ler log c l a s s have been manufactured, Q-j and Q 2 = volume of D o u g l a s - f i r and western hemlock, r e s p e c t i v e l y , conta ined i n the la rge log c l a s s , and Q 4 = volume of D o u g l a s - f i r and western hemlock l o g s , r e s p e c t i v e - ' l y , conta ined i n the smal l log c l a s s . For each log s o r t i n g and t rad ing p o l i c y the a l l o c a t e d q u a n t i t y , x t 2 + y t 2 ' 1 S P a r t i t i o n e d i n t o f i v e volume components. The method of p a r t i t i o n i n g and the i n t e r - r e l a t i o n s h i p s between these q u a n t i t i e s are i l l u s t r a t e d i n F igure 8 . Using the no ta t i on g iven above the f i v e com-ponents are i d e n t i f i e d a s : ( i ) volume of non-pee le r logs used as pu lp logs = 4 1 Q . - y c , j= l J ( i i ) volume of pee le r logs t raded f o r pu lp logs = p c . ( x t 2 + y t 2 ) - y t 2 + y c , ( i i i ) volume of sawlogs s e l e c t e d from non-pee le r logs = y c , ( i v ) volume of pee le r logs t raded f o r sawlogs.= t r . ( x t 2 + y t 2 ) , (v) volume of pee le rs used as sawlogs = y t 2 - y c - t r . ( x t 2 + y t 2 ) . As shown i n F igure 8 the volume of each of these f i v e components may be expressed as a f u n c t i o n of the three q u a n t i t i e s x ^ , y ^ 2 and y c , and of the r a t i o , t r . A c c o r d i n g l y , the magnitudes of yc and t r . deno te a l t e r n a t i v e log s o r t i n g and t rad ing p o l i c i e s f o r a .g i ven a l l o c a t i o n o f pu lp logs and sawlogs from source two. The d e r i v a t i o n of net d e l i v e r e d DELIVERED QUANTITIES t 2 y t 2 = QUANTITIES ALLOCATED x t 2 ' y r 2 PEELER SELECTION VOLUME OF NON-PEELER CLASSES ( l - p c ) ( x t 2 + y t 2 ) LOG SORT PEELER VOLUME p c ( * t 2 + y t 2 ) TRADING PULPLOG PULPLOG 2 Q j - y c j=i + p c . ( x t 2 + y f 2 ) - y t 2 + y c [ l ] [2] SAWLOG SAWLOG + t r . ( x t 2 + y t 2 ) + [ 3 ] F i g . 8 Volume Components of Quan t i t i es .A l l oca ted from Source Two 55 log cos ts requ i res i d e n t i f y i n g an opt imal log s o r t i n g and t r ad i ng p o l i c y f o r each elemental p a i r of the se t ^^2'^t2^' The i t e r a t i v e process used , i n which each a l t e r n a t i v e s o r t i n g and t r ad ing p o l i c y i s e v a l u a t e d , i nc ludes a l l u n i t v a r i a b l e cos ts i ncu r red i n the log p roduc t ion sequence commencing w i th log s o r t i n g . Due to p o s s i b l e changes i n log d iameter and the spec ies mix o f d e l i v e r e d sawlogs the r e t u r n , " d i f f e r e n t i a l sawlog net w o r t h , " ' i s a l s o i n c l u d e d . The com-p o s i t e f u n c t i o n used to eva lua te each p o l i c y i s d i scussed i n a l a t e r s e c t i o n . ' In the f o l l o w i n g s e c t i o n on ly those log p roduc t ion cos ts t ha t are not a f f e c t e d by log s o r t i n g o r ; t r a d i n g are d e r i v e d . b. Costs tha t are independent of Log S o r t i n g and Trad ing P o l i c i e s  i ) S t a r t - u p Costs Because of i t s i n a c c e s s i b i l i t y i n the second. t ime per iod log produc t ion i s r e s t r i c t e d to the f i r s t and t h i r d time p e r i o d s . In e i t h e r per iod the t o t a l se t -up cos t depends upon the ex ten t of main road con -s t r u c t i o n , the p r o v i s i o n of spur - roads and l a n d i n g s , and the c o s t o f assembl ing logg ing equipment. As an app rox ima t i on , the s i z e of the planned cut i s assumed to con t ro l the magnitude of a f i x e d c o s t component. L e t , -c f l = s t a r t - u p cos ts when logg ing i s i n i t i a t e d i n any per iod on source two, c f2 = an a d d i t i o n a l f i x e d cos t i ncu r red when the t o t a l volume 56 cut from source two i n any per iod i s g rea te r than the quan-t i t y vm 2 -i i ) V a r i a b l e Un i t Costs to Woods Landings This component i nc ludes the u n i t v a r i a b l e cos ts of f e l l i n g , buck-i n g , topping and forward ing the qu a n t i t y X t 2 + - Y t 2 z 0 w o o c ' s l a n d i n g s . I f t h i s quan t i t y exceeds a s p e c i f i e d volume, v m 2 , more p roduc t i ve equipment i s in t roduced and u n i t v a r i a b l e cos ts dec rease . L e t , c l l j . = v a r i a b l e cos t per thousand c u n i t s of logs yarded to woods land ings when the volume cut on source two i n pe r iod t does not exceed the q u a n t i t y v m 2 , c l 2 ^ = v a r i a b l e cos t per thousand c u n i t s of logs yarded to woods land ings f o r the p o r t i o n of the volume cut on source two i n pe r iod t .wh ich exceeds the q u a n t i t y v m 2 . These two c o s t i t e m s , s t a r t - u p cos ts and v a r i a b l e u n i t c o s t s , are combined i n the composite cos t f u n c t i o n , c l t ( x t 2 , y t 2 ) , shown i n F igure 9. L e t , c ^ t ^ x t 2 ' y t 2 ^ = P ° r t i ° n of the net d e l i v e r e d log cos t f o r source two i n . pe r iod t which i s independent o f l og s o r t i n g and t r ad ing p o l i c i e s expressed as a f u n c t i o n o f the cu t . vo lumes , x ^ 2 and y t 2 , then 57 F i g . 9 ' V a r i a b l e Log Costs to Woods-Landings f o r Source Two C l t ( x t 2 ' ^ t 2 ) = 58 0 f o r x ^ 2 = y t 2 ='0, c f l + c l l ^ . (x^ . 2 +y t 2 ) f o r 0 < * t 2 + y t 2 £ v m 2 , c f1+c f2+c l1^ . vm 2 +c l2^ . ( x^ 2 +y t 2 - vm 2 ) f o r vm 2 < x t 2 + y t 2 < v t 2 . c . Costs tha t are Dependent upon Log S o r t i n g and Trad ing P o l i c i e s The c o s t of s o r t i n g , load ing and t r a n s p o r t i n g a volume of logs equal to the q u a n t i t y , x f . 2 + y t 2 ' ^ s dependent upon the r e l a t i v e magnitudes of the f i v e volume components i d e n t i f i e d f o r each log s o r t i n g and t r a d -ing p o l i c y . The c o s t o f s o r t i n g i nvo l ves two a c t i v i t i e s , the s e l e c t i o n of the quan t i t y yc and the s e l e c t i o n of the quan t i t y o f . p e e l e r logs r e -qu i red f o r t r a d i n g . The cos t of l oad ing depends upon the r e l a t i v e volumes of pee le r logs and non-pee le r l o g s . T ranspo r t a t i on cos ts depend upon the magnitudes of the a l l o c a t e d q u a n t i t i e s , x t 2 and y t 2 , .and the quan t i t y of pee le r logs exchanged on the l o c a l log market. The cos ts o f s o r t i n g and load ing the quan t i t y  x^2+^tZ a r e de f ined f o r each log s o r t i n g and t rad ing p o l i c y as c s t ( x t 2 , y t 2 , y c , t r ) = sum of the u n i t v a r i a b l e cos ts of s o r t i n g and load ing when the q u a n t i t i e s , x ^ 2 and y ^ 2 , are a l l o c a t e d from source two i n pe r iod t f o r a log s o r t i n g and t r ad i ng p o l i c y de-noted by the quan t i t y y c and the r a t i o t r . Log t r a n s p o r t a t i o n cos ts of the quan t i t y ^^2+^tZ a r e de f ined as c t t ( x t 2 , y t 2 , y c - , t r ) = v a r i a b l e net t r a n s p o r t a t i o n cos t i ncu r red 59 when the q u a n t i t i e s and are d e l i v e r e d to m i l l yards from source two i n pe r iod t f o r a log s o r t i n g and t r ad ing p o l i c y de-noted by the quan t i t y yc and the r a t i o t r . The d e r i v a t i o n of these c o s t s , g iven i n Appendix A , examines the groups of v a r i a b l e cos ts tha t are i d e n t i f i e d f o r each a l l o c a t i o n of logs from source two. d . D i f f e r e n t i a l Sawlog -Net Worth There are three sources of logs from which the q u a n t i t y , y^' m a y be s u p p l i e d : pee le r logs used as saw logs ; logs from the l o c a l ; l o g . m a r k e t which rep lace t raded pee le r l o g s ; a s e l e c t e d p o r t i o n of the quan t i t y of non-pee le r logs which have been so r ted i n accordance w i th s p e c i f i e d log s e l e c t i o n r u l e s . The average log d iameter and .spec ies mix of d e l i v e r e d sawlogs de-pends upon the magnitude and log type of these three component q u a n t i -t i e s . The d e r i v a t i o n of the magnitudes of each component has been des -c r i b e d (page 5 3 ) , and the type of l og f o r each of the f i r s t two com-ponents i s known. The log diameter and p ropo r t i on o f D o u g l a s - f i r i n the quan t i t y y c , the t h i r d component, must be c a l c u l a t e d . Computat ional d e t a i l s are g iven i n Appendix A . The weighted averages of the log diameter and the p ropo r t i on of D o u g l a s - f i r logs i n the quan t i t y y t 2 > the d e l i v e r e d volume of saw logs , are used to c a l c u l a t e " d i f f e r e n t i a l sawlog net w o r t h . " Th is re tu rn i t e m , eva lua ted f o r each a l t e r n a t i v e log s o r t i n g and t r ad i ng p o l i c i e s , i s de f ined as i c w t ^ x t 2 ' y t 2 ' y C j t r ^ = " d i f f e r e n t i a l sawlog net wor th" of the d e l i v e r e d volume y^ when the q u a n t i t i e s , x ^ 2 and y ^ 2 , are a l l o -cated from source two i n pe r iod t f o r a log s o r t i n g and t r ad ing p o l i c y denoted by the quan t i t y yc and the r a t i o t r . D e t a i l s of the d e r i v a t i o n of t h i s term are g iven i n Appendix A. e . Optimal Log S o r t i n g and Trad ing P o l i c i e s An opt imal s o r t i n g and t rad ing p o l i c y i s i d e n t i f i e d f o r each p a i r of q u a n t i t i e s , x t 2 and y t 2 . Each p o l i c y i s eva lua ted by combining the " d i f f e r e n t i a l net wor th" of the sawlog a l l o c a t i o n w i th the two c o s t components a f f e c t e d by the q u a n t i t i e s so r ted and t r aded . By min im iz ing the r e s u l t i n g composite f u n c t i o n opt imal p o l i c i e s are i d e n t i f i e d f o r each pa i red element o f the s e t ^^'^tZ^' L e t , 7 - t ^ x t 2 ' y t 2 ^ = minimum va lue i n pe r iod t o f the sum of a l l v a r i a b l e log p roduc t ion and t r a n s p o r t a t i o n cos ts tha t are dependent upon the log s o r t i n g and t rad ing p o l i c i e s of source two l e s s the " d i f f e r e n t i a l sawlog net wor th" de r i ved f o r each p o l i c y , t hen , z t ( x t 2 » y t 2 ^ = m i n t c s t ^ x t 2 ' y t 2 ' y c ' t r ^ + c t t ^ x t 2 ? y t 2 ' y c ' t r ^ y c s R 1 t r e R 2 " G w t ^ x t ' 2 ' y t 2 ' y c ' t r ^ y c + t r . ( x t 2 + y t 2 ) e R 3 f o r y t 2 ^ 0 , where R 1 =;{yc|max[0 ; y t 2 - p c . ( x t 2 + y t 2 ) ] <_ yc < .m in [ y t 2 ; ( l - p c ) ( x t 2 + y t 2 ) ] } , R 2 = { t rl° £ t r 1 P c } > R 3 = { y c + t r . ( x t 2 + y t 2 ) |max[0 ; y t 2 - p c ; ( * t 2 + y t 2 ) ] <. y c + t r . ( x t 2 + y t 2 ) •<. y t 2 > . When pu lp logs on ly are requ i red a p o l i c y o f exchanging pee le r logs f o r an equal quan t i t y of pu lp logs i s mandatory and the magnitudes of yc and t r are z e r o . Then 4 Z t ( x t 2 > y t 2 ) = z Q j • ( c n + c t x t ) + p c . x t 2 . ( c p + c s s + c t t ^ - t c ) . The importance of i d e n t i f y i n g opt imal log s o r t i n g and t r ad ing p o l i c i e s v a r i e s f o r d i f f e r e n t a l l o c a t i o n s from source two. F igure 10 and F igure 11 show the response su r faces generated by e v a l u a t i n g a l l f e a s i b l e p o l i c i e s f o r x ^ 2 = 3 , y t 2 = 1 and x ^ 2 = 3 , y ^ 2 = 2. Inspec t ion of these su r faces and comparison of the r e l a t i v e ranges of y c , the volume of sawlogs s e l e c t e d from the non-pee le r log c l a s s e s , show the inc reased importance of sawlog net worth when- the upper l i m i t o f the range of a l t e r n a t i v e p o l i c i e s i s reduced. In each diagram the con-t inuous l i n e s between the su r face boundar ies represent the permi t ted range of the quan t i t y yc f o r each t r ad i ng p o l i c y denoted by the r a t i o t r . These l i n e s a l s o represen t the search procedure used to i d e n t i f y ^ t ^ x t 2 ' ^ t 2 ^ ^ o r e a c n P a i r ° f q u a n t i t i e s x ^ 2 and y ^ . D e t a i l s of the method used are g i v e n . i n the lower p o r t i o n of the f l owcha r t shown i n ^ D e f i n i t i o n s of i n d i v i d u a l components are g iven i n the G l o s s a r y ' of Terms, page x i i i . 10 Response Sur face f o r L o g . T r a d i n g and S o r t i n g P o l i c i e s f o r x - ^ .= 3 , = 1 F i g . 11 Response-Surface fo r . Log-Trad ing and S o r t i n g ' P o l i c i e s f o r x-|2 - 3 , y - ^ = 2 64 F igure 7. Th is pa r t of the computat ional sequence i nc ludes the e v a l u a -t i o n of s o r t i n g and load ing c o s t s , sawlog net w o r t h , and log t r a n s p o r t a -t i o n cos ts f o r each a l t e r n a t i v e p o l i c y examined. f . Net De l i ve red Log Costs When the opt imal log s o r t i n g and t r ad ing p o l i c y has been i d e n t i -f i e d the net d e l i v e r e d log cos ts f o r the q u a n t i t i e s x ^ 2 and y t 2 m a y D e der i ved as the sum of the v a r i a b l e cos ts tha t are independent of log s o r t i n g and t r ad ing p o l i c i e s p lus the sum of the v a r i a b l e cos ts tha t are de r i ved f o r an opt imal log s o r t i n g and t r ad ing p o l i c y minus the d i f f e r e n t i a l sawlog net worth of the q u a n t i t y y ^ , o r , C t 2 ( x t 2 ' y t 2 } = - c l t ( x t 2 * y t 2 ) + Z t ( x t 2 * y t 2 ) -3 . Log Costs f o r Source Three Th is p roduc t i ve area i s the main source of saw logs , a l though both pu lp logs and sawlogs are normal ly a l l o c a t e d i n each t ime p e r i o d . Log p roduc t ion cos ts d i f f e r f o r each log product .and change when the t o t a l quan t i t y cu t i n a s i n g l e per iod exceeds a s p e c i f i e d volume. Due to these cos t d i f f e r e n c e s , the combined c o s t of d e l i v e r i n g the q u a n t i -t i e s , x ^ and y ^ » may vary depending upon the log product ass igned l e a s t cos t p r i o r i t y . A c c o r d i n g l y each a l l o c a t i o n requ i res the 65 i d e n t i f i c a t i o n of an opt imal " c o s t i n g " p o l i c y . The average diameter and spec ies mix of a l l sawlogs from the area are assumed to remain unchanged, p rov i d i ng the volume of pu lp logs cut i s a t l e a s t equal to h a l f the sawlog a l l o c a t i o n . The net worth of sawlogs a l l o c a t e d from source three under these c o n d i t i o n s i s the s tandard f o r comparing the net worths of a l l logs d e l i v e r e d to the s a w m i l l . I f the p ropo r t i on of pu lp l ogs to sawlogs tha t i s a l l o c a t e d from t h i s source i n any t ime pe r iod i s l ess than the r a t i o 1 to 2 then both log s i z e and the content of D o u g l a s - f i r logs i s assumed to dec rease . L e t , vm-j = quan t i t y of logs i n thousands of c u n i t s which can be cut i n each pe r iod w i thou t changing u n i t v a r i a b l e c o s t s , x l and y l '= pu lp log and sawlog volume components, r e s p e c t i v e l y , i n thousands of c u n i t s , of the a l l o c a t i o n , - x ^ and y ^ , which together do not exceed the quan t i t y vrrig, x2 and y2 = pu lp log and sawlog volume components, r e s p e c t i v e l y , i n thousands of c u n i t s , of the p o r t i o n of the a l l o c a t i o n , x ^ and y ^ , which exceeds the q u a n t i t y vm^, c x l ^ g . ° y ^ 3 > c x ^ t 3 a n c ' c y ^ t 3 = v a n a D ^ e c o s t P e r thousand c u n i t s of the volume components x l , y l , x2 and y 2 , r e s p e c t i v e l y . - a . Log Produc t ion and T ranspo r ta t i on Costs When the volume x^g+y^g exceeds the quan t i t y virig the magnitude of each volume component x l , y l , x2 and y2 i s dependent upon the c o s t i n g p o l i c y . For example, g iven the cos t r e l a t i o n s h i p s shown i n F igure 12 66 where c x 2 t 2 < c x ^ 3 • a n c ' c y 2 t 3 < c y ^ t 3 ' s a w T ° 9 s a r e supp l i ed a t the lowest cos t when the quan t i t y y l i s a minimum. A c o s t i n g p o l i c y such as t h i s i s i l l u s t r a t e d i n F igure 12 f o r the log a l l o c a t i o n = 5 and y^2 = 3 . C o n v e r s e l y , when l e a s t cos t p r i o r i t y i s ass igned to pu lp logs the volume component x l i s a minimum. In the f o l l o w i n g d i s c u s s i o n a numerical procedure f o r i d e n t i f y i n g each opt imal se t of volume compon-ents i s de r i ved which i s a p p l i c a b l e when c x 2 ^ > c x l . ^ a n c l c y 2 t 3 * c y l t 3 . Log p roduc t ion and t r a n s p o r t a t i o n cos ts are c a l c u l a t e d as c d t 3 ^ x t 3 ' y t 3 ^ = x l • c x l t 3 + x 2 > c x 2 t 3 + y l ' c y 1 t 3 + y 2 c y 2 t 3 J x l = x g where and v t 3 y l = y t 3 x2 = 0 y2 = 0 . xl+x2 = x ^ y i+y2 = y t 3 x l+y l = vm, f o r x t 3 + y t 3 1 v m 3> f o r x t 3 + y t 3 * v m 3 ' b. Optimal Cos t i ng P o l i c i e s When x ^ 3 + y ^ 3 1. vm^, the magnitudes of the volume components are not a f f e c t e d by the c o s t i n g p o l i c y . However, when x+j+y+j > vm 3 the r e l a t i v e magnitudes of x l and y l , t h e h igh u n i t - c o s t volume components, must be i d e n t i f i e d . By s u b s t i t u t i o n , the exp ress ion g iven above f o r d e r i v i n g d e l i v e r e d log cos ts may be r e w r i t t e n , when x ^ + y . ^ > vm^, as Fig. - 12 Log .Cos t Components . fo r , Source Three x l . c x l t 2 + ( x t 3 - x l ) . cx2 t 2+(vm.^x ! ) . c y l ^ " K y ^ - v n i g + x l ) - c y 2 t 3 ? and the minimum c o s t f o r the q u a n t i t i e s x ^ and y ^ may be de r i ved as c d t 3 ^ x t 3 ' y t 3 ^ = m i n • c x ^ 3 + ( x t 3 " x ^ ' c x ^ t 3 + ^ v m 3 ~ x ^ " C y ^ t 3 +  x l e R l ( y t 3 - v m 3 + x 1 ) . c y 2 t 2 ] s where -R-| •= {x l |max[0 ; v m ^ - y ^ ] <_ x l <_ min [x^ 3 ; vm^]}. Al though the opt imal magnitude of the quan t i t y x l may be i d e n t i -f i e d i n the above equat ion by an i t e r a t i v e procedure a s imp le r method i s a v a i l a b l e . Th is a l t e r n a t i v e method a l s o f a c i l i t a t e s i d e n t i f y i n g the appropr ia te c o s t i n g p o l i c y . The exp ress ion f o r c o s t m i n i m i z a t i o n may b e - r e w r i t t e n as min [x l - c x l t g + U ^ - x l ) . c x 2 t 3 + ( v m 3 - x l ) . c y l t 3 + ( y t 3 ~ v m 3 + x ^ ^ , c y ^ t 3 xleR-j = min [ f u n c t i o n C x ^ g . y ^ g j V m g J + x l . ( c x l t 2 - c x 2 t 3 - c y l t 3 + c y 2 t 3 ) ] xleR-j For a g iven a l l o c a t i o n the f i r s t te rm, f u n c t i o n U ^ j y ^ j V i r i g ) , i s constant and minimum log p roduc t ion cos ts are acco rd i ng l y dependent upon the second term, x l . ( c x l ^ 2 - c x 2 ^ 3 - c y l ^ 2 + c y 2 ^ 3 ) . I f the magnitude of the exp ress ion ( c x l ^ 3 - c x 2 t 3 - c y l ^ 2 + c y 2 ^ 2 ) i s n e g a t i v e , then cos ts are minimized when the q u a n t i t y x l i s a maximum. I f the exp ress ion i s p o s i t i v e , then minimum cos ts r equ i r e a minimum magnitude f o r x l . Be -cause the component u n i t cos ts do not vary w i t h i n a p e r i o d , the l e a s t cos t p o l i c y need be i d e n t i f i e d on ly once f o r each p e r i o d . . I t does not d i r e c t l y depend on the magnitudes o f x ^ and y ^ . However, there are 69 p a r t i c u l a r a l l o c a t i o n s , i d e n t i f i e d be low, where s p e c i f y i n g a p r i o r i t y i s i n a p p r o p r i a t e . Acco rd i ng l y each a l l o c a t i o n from source three i n each per iod i s examined. L e t , I N D E X t ( x t 3 , y t 3 ) = numerical i n d e x , 0 , 1 , 2 , 3 , which r e f e r s to the opt imal c o s t i n g p o l i c y f o r source three where 0 = no a p p l i c a b l e p o l i c y , 1 = l e a s t cos t p r i o r i t y t o - p u l p l o g s , 2 = l e a s t c o s t p r i o r i t y to saw logs , and 3 = s i n g l e log product o n l y , then and where INDEX, 0 , x l y i x t 3 , x2 = 0 y t 3 , y2 = o f o r x t 3 + y t 3-1 v m 3 ' x2 = x t 3 - x l y l = vm 3~xl y 2 = y t 3 - v m 3 + x 1 f o r x t 3 + y t 3 * v m 3 : INDEX. = 1 t t x l max[0 ; v r r y y ^ ] f o r 0 <_ c x ^ 3 - c x 2 t 3 - c y l t 3 + c y 2 + , , t 3 : INDEX. = 2 x l = m in [x^ 3 ; vm 3] f o r 0 > c x l t 3 - c x 2 | . 3 - c y l t 3 + c y 2 t 3 ! f o r m i n [ x t 3 ; y t 3 ] ^ 0 ; Where c x ^ 3 _ c x 2 ^ 3 ~ c y ^ 3 + c y 2 ^ 3 = 0 the procedure i d e n t i f i e s a l e a s t cos t p r i o r i t y to pu lp log d e l i v e r i e s (INDEX. = 1 ) . Th is c o s t i n g 70 INDEX t = 3 x l = m i n [ x t 3 ; vm3D = max[0 ; v m 3 - y t 3 ] , f o r m i n [ x t 3 ; y t 3 ] = 0, D e l i v e r e d l og cos ts are c a l c u l a t e d as x t 3 ' C X l t 3 + y t 3 - e y 1 t 3 f 0 r X t 3 + y t 3 1 v m 3 ' x l . c x l t 3 + x 2 . e x 2 t 3 + y l . c y l t 3 + y 2 . c y 2 t 3 f o r ^ 3 ^ 3 " v m 3 ' c . D i f f e r e n t i a l Sawlog Net Worth The imp l i ed assumption i s made tha t a t l e a s t a l i m i t e d log s o r t i s requ i red when both sawlogs and pu lp logs are a l l o c a t e d from source th ree . An a d d i t i o n a l assumption i s made t h a t , p rov i d i ng a minimum p ro -p o r t i o n of pu lp logs i s cu t whenever sawlogs are a l l o c a t e d , the average s i z e and spec ies mix of d e l i v e r e d sawlogs are unchanged. As the sawlog net worth based on these logs i s used as a s tandard (page 4 3 ) , the va lue of " d i f f e r e n t i a l sawlog net w o r t h , " c a l c u l a t e d f o r sawlogs s e l e c t e d from p o l i c y i s not un ique. For t h i s s p e c i a l c o n d i t i o n a l l f e a s i b l e magni-tudes of the volume component x l and the r e l a t e d components x 2 , y l and y2 g ive an i d e n t i c a l c o s t . 71 t h i s area under these c o n d i t i o n s , i s z e r o . When l e s s than the minimum p ropo r t i on o f pu lp logs i s a l l o c a t e d the " s tanda rd " magnitudes of log s i z e and spec ies mix cannot be main-t a i n e d . For t h i s c o n d i t i o n " d i f f e r e n t i a l sawlog net wor th" has a nega-t i v e v a l u e . L e t , Ps = r a t i o o f the volume of D o u g l a s - f i r logs to the quan t i t y y ^ s e l e c t e d from the t o t a l q u a n t i t y , X ^ 3 + Y t 3 > a l l o c a t e d from source three when y t 3 <_ 2 . x t 3 , Ds = average log d iameter i n inches of the quan t i t y y^.g s e l e c t e d from the t o t a l q u a n t i t y , x ^ + y ^ j ? a l l o c a t e d from source th ree when y^g ..<_ 2.x^.g, = r a t i o of the volume of D o u g l a s - f i r logs to the quan t i t y t3 ' t 3 ' D t g = average log d iameter i n inches of the quan t i t y y ^ g . Assuming tha t the f o r e s t types on the area des ignated as source three are such tha t the r a t i o of D o u g l a s - f i r logs to the a v a i l a b l e t imber supply i s 0.6 and the average log d iameter i s e i g h t inches when no improvement by s o r t i n g i s p o s s i b l e , the magnitudes of and P ^ are c a l c u l a t e d as 8.0 D ' tk 8 . 0 + ( 2 x t 3 / y t 3 ) ( D s - 8 . 0 ) Ds f o r x t 3 = 0 , f o r x t 3 / y t 3 < 1 / 2 > f o r x t 3 / y t 3 . 1 1 / 2> and 72 tk Hence,. "0.6 0 . 6 + ( 2 x t 3 / y . t 3 ) ( P s - 0 ; 6 ) Ps ^ t S ^ t S ' W = ( P t k - P s ) ( w p t - w n t ) -mc.cm. ( 1 / D ^ - l / D s ) r t k f o r x ^ 3 = 0 , f o r x t 3 / y t 3 : < 1 / 2 ' f o r x t 3 / y t 3 1 / 2 ' f o r y t 3 =0 f o r x t 3 / y t 3 > 1 / 2 > f o r x t 3 / y t 3 * 1 / 2 ' The computat ional sequence fo l l owed to de r i ve log p roduc t ion cos ts and " d i f f e r e n t i a l sawlog net wor th" i s shown i n the accompanying f l owchar t (F igu re 13) . d . Net De l i ve red Log Costs The composite f u n c t i o n c t 3 ( x t 3 ' y t 3 ^ ' u s e c ' * ° e v a l u a t e log a l l o c a -t i ons from source three i n each time p e r i o d , i s de r i ved as 4 . Log Costs f o r Source Four In a d d i t i o n to log p roduc t ion a l t e r n a t i v e s the t o t a l demand f o r logs may be p a r t i a l l y s a t i s f i e d by purchase. The quan t i t y o f logs a v a i l a b l e f o r s a l e , t h e i r s i z e , spec ies content and p r i c e may a l l vary between t ime p e r i o d s . (Al though f o r log t r ad ing purposes the volumes a v a i l a b l e are assumed to be u n l i m i t e d , on ly a r e s t r i c t e d quan t i t y may INDEX - 0 INDEX - 2 xl - m1n[x^j;vmj] • INDEX • » xl • x t 3 x2 = 0 y2 • 0 xl - max x2 • " j 3 - x l y l - vnij-xl y2 » yt3-vmj+xl INDEX - 3 yes c d t 3 ( x t 3 > y t 3 > ' Xl.cxl t 3+yl.cyl t3+x2.cx2 t 3+y2.cy2 t3 yes yes D„- 8.0+2. (D»-8).xt3/yt3 P„= .6+2.(P«-.6).xt3/yt3 T c w t 3 ( x t 3 , y t 3 > " ( ( P t 3 _ P s ) ( w p t - w n t ) -mc.OT.(l/D t 3-l/Ds)) .y t 3 T -t3<xt3'»t3.) * 0 ^ t s ^ t s ' = c d t 3 ( x t 3 > y t 3 ) - c " t 3 ( x t 3 ' * t 3 ) F i g . 13 F lowcha r t - o f - t he - :De r i va t i on of .Net De l i ve red - Log Costs f o r Source Three 74 be purchased ir i each p e r i o d ) . Purchased logs-may • be .'used e i t h e r as sawlogs or pu lp logs bu t , due to the g rea te r t r uck i ng d i s t a n c e , a g rea te r u n i t v a r i a b l e t r a n s p o r t a t i o n cos t i s i ncu r red when logs are d e l i v e r e d to the p u l p m i l l . L e t , c X j . and cy^. = cos t of purchased logs per thousand c u n i t s d e l i v e r e d to the p u l p m i l l and s a w m i l l , r e s p e c t i v e l y , i n per iod t , pt^ = r a t i o i n pe r iod t o f the q u a n t i t y o f - D o u g l a s - f i r logs to the volume of logs a v a i l a b l e from the l o c a l log market , dtj . = average log diameter i n inches of the quan t i t y o f logs a v a i l a b l e from the l o c a l l og market i n pe r iod t . The combined v a r i a b l e d e l i v e r e d c o s t of the q u a n t i t i e s , and y ^ , i s de r i ved as c d t 4 ( x t 4 ' y t 4 } = x t 4 ' c x t + y t 4 - c y t ' f o r 0 " ± x t 4 + y t 4 ± v t 4 ? and " d i f f e r e n t i a l sawlog net wor th" as c w t 4 ( x t 4 ' y t 4 ) ( p t t - P s ) ( w p t - w n t ) -mc.cm. ( l / d t ^ . -1 /Ds ) •y t4 a . Net De l i ve red Log Costs Th is composite f u n c t i o n i s d e r i v e d . f o r purchased logs as c t 4 ( x t 4 ' y t 4 ) ' - ^ H ^ ^ - ^ h A ' ^ - -75 5. Log Costs f o r Source F i ve Th is p roduc t i ve area i s the second major source of pulpwood. A c c e s s i b l e i n each of the three p lann ing p e r i o d s , the f o r e s t type i s un-s u i t a b l e f o r sawlog p r o d u c t i o n . - The c o s t - o f d e l i v e r e d p u l p l o g s , nego t ia ted w i th a c o n t r a c t o r , i s a composite of two ra tes depending on the t o t a l cu t w i t h i n a p e r i o d . These log d e l i v e r y cos ts vary between-per iods and i n c l u d e a guarantee of a minimum sum f o r each per iod the c o n t r a c t o r i s r equ i red to ope ra te . However, as the company i s under no o b l i g a t i o n to schedule log d e l i v e r i e s from area f i v e i n any p e r i o d , when logs are a l l o c a t e d the quan t i t y i s u s u a l l y s u f f i c i e n t to cover the .agreed minimum payment: L e t , vrrig = quan t i t y of l o g s , i n thousands of c u n i t s , which can be cu t i n each per iod w i thout changing u n i t v a r i a b l e c o s t s , ms = minimum amount to be pa id to the c o n t r a c t o r f o r d e l i v e r e d l o g s , c x ^ t 5 = v a r i a D ^ e cos t per thousand c u n i t s of pu lp logs d e l i v e r e d from source f i v e i n pe r iod t when the cut volume does not exceed the quan t i t y vm^, cx2^.g = v a r i a b l e c o s t per thousand c u n i t s of pu lp logs d e l i v e r e d from source f i v e i n per iod t f o r the p o r t i o n of the cut volume which exceeds the quan t i t y vm^.: The d e l i v e r e d c o s t of pu lp logs a l l o c a t e d from source f i v e i s de r i ved as where max[X | .g .cx l^g ; ms] f o r 0 < x^g <_ vrrig, v m 5 . c x l t 5 + ( x t 5 - v m 5 ) . c x 2 t 5 f o r vm 5 < x t 5 < v i 5 1 ms < vmg .cx l ^g . a . Net D e l i v e r e d Log Costs No d e l i v e r i e s are made to the sawmi l l from t h i s s o u r c e ; conse-quent ly d i f f e r e n t i a l sawlog net worth i s zero and. c t 5 ( x t 5 ' y t 5 ) = c d t 5 < x t 5 » * t 5 ! " The d e l i v e r e d cos ts of p u l p l o g s , de r i ved as desc r ibed are shown i i i F igure 14. F i g . 14 D e l i v e r e d - C o s t of Pu-1 pwood' from Source F ive CHAPTER IV RESULTS A . ' DISCUSSION OF THE SOLUTION TO THE PROBLEM i Two quest ions were posed i n the log a l l o c a t i o n problem desc r i bed i n Chapter I. What q u a n t i t i e s of pu lp logs and sawlogs should be de-l i v e r e d i n each of three time per iods? How should these q u a n t i t i e s be a l l o c a t e d from the a v a i l a b l e sources? The recur rence r e l a t i o n s h i p s developed i n Chapter II showed how a l t e r n a t i v e s o l u t i o n s to these two quest ions cou ld be r a p i d l y compared. The economic -ob jec t i ve was the minimum sum of log p roduc t ion and t r a n s p o r t a t i o n c o s t s , the cos ts o f " c o l d - d e c k i n g " log su rp luses i n m i l l . ya rds , . and a measure of sawlog net wor th . In Chapter I I I the "net d e l i v e r e d log c o s t , " a composite term used to eva lua te a l t e r n a t i v e a l l o c a t i o n s w i t h i n each t ime p e r i o d , , o f logs supp l i ed from each source was examined i n d e t a i l . For two of the f i v e examples of t y p i c a l log produc t ion systems a d d i t i o n a l d e c i s i o n s were requ i red to de r i ve cos ts f o r a p a r t i c u l a r a l l o c a t i o n . In each c a s e , the necessary computat ions were inc luded i n the procedure f o r eva lua t i ng the c o s t f u n c t i o n , c t k ^ x t k t k ^ * T o demonstrate the computa-t i o n a l f e a s i b i l i t y of the mathematical f o rmu la t i on p resen ted . i n Chapters H a n d I I I a computer programme, DYALOG, was w r i t t e n . - A copy of t h i s programme, coded i n FORTRAN IV i s l i s t e d i n Appendix B. I t i nco rpo ra tes the phys i ca l c o n s t r a i n t s s p e c i f i e d f o r the hypo the t i ca l problem and f o l l o w s - t h e computat ional sequences of the f l owcha r t shown i n - F i g u r e ' 3 , • 79 • page 17. The i l l u s t r a t i v e inpu t data used f o r model development and t e s t i n g i s l i s t e d i n Appendix C , together w i th the complete s e t of ou t -put i n fo rma t ion generated by the cu r ren t v e r s i o n of DYALOG. Th is i n f o r -mation i s i n the form of conso l i da ted summaries of the r e s u l t of each major computat ional s tep i n the a l l o c a t i o n p rocess . These summaries p rov ide the manager w i th ope ra t i ona l i n fo rmat ion a t two l e v e l s . One, the c r i t i c a l examinat ion of c o s t r e l a t i o n s h i p s between log sources i s f a c i l i t a t e d . Such examinat ion i s necessary to i d e n t i f y where p o s s i b l e cos t economies are most e f f e c t i v e . ' Two, due to unforeseen changes, i t may not be p o s s i b l e to d e l i v e r the q u a n t i t i e s i n d i c a t e d as o p t i m a l . From the i n fo rma t ion a v a i l a b l e i n the summary t a b l e s , the manager can i d e n t i f y the l e a s t c o s t l y a l l o c a t i o n p o l i c y f o r an a l t e r n a t i v e d e l i v e r y schedu le . In t h i s s e c t i o n po r t i ons of these output t ab les are s e l e c t e d f o r examinat ion . The f i r s t t ab l e to be d i scussed g i ves the numerical s o l u -t i o n to the h y p o t h e t i c a l prob lem, the opt imal p o l i c y f o r a l l o c a t i n g logs f o r three time per iods from f i v e sou rces . The remaining t a b l e s , or po r t i ons of t a b l e s , are examined i n the order i n which they are d e r i v e d . By r e l a t i n g these summaries of the r e s u l t s obta ined by each s tep i n the computat ion to the computat ional sequence i t s e l f , shown i n F igu re 3 , page 17, the p r o c e s s . o f a l l o c a t i o n by dynamic programming i s numer i ca l l y e x e m p l i f i e d . 1. Optimal A l l o c a t i o n P o l i c y An opt imal a l l o c a t i o n of logs f o r the hypo the t i ca l problem i s g i v e n . i n Table I. The q u a n t i t i e s of pu lp logs and sawlogs shown i n TABLE I FIRST OPTIMAL ALLOCATION POLICY 80 J MINIMUM COST ALLOCATION POLICY FOR X=68,Y=28 OPTINAL ALLOCATICNS BETWEEN AND WITHIN PERIODS' TIME MILL ALL AREAS PRODUCTIVE AREAS PERIOD D.E.M.6M3. . I 2 - 3_ •__ 4 _ _ .5.. P/L S/L XT^ Yl0 XTK dYTK e XTK YT.K XTK YTK XTK YTK XTK YTK <1) 24 10 30 15 1 5 - 0 3 7 - 7 8 0 0 5 0 J 2 J _ 2D „ £ .-...I l,6u< 5. , 6 0 _jQL_. JQ 3_ . 5. 0 0 1 £L (31 24 9 22 8 6 0 0 0 7 . 8 4 0 5 0 MINIMUM COST = $1324765.8 (a) Each l og source i nc luded i n the same sequence as i n d i c a t e d , e . g . , "AREA V ' r e f e r s to source one , "AREA 2" r e f e r s to source two, e t c . J b L J ? I j = L l ( X 5 Y > (c) YT = Y t ( X , Y ) (d) X T K = x t k ( X , Y ) (e) YTK o y t k ( X , Y ) 81 co lumns f o u r and f i v e o f t h i s t a b l e have been d e f i n e d as X^ . (X ,Y ) and * Yt ( X , Y ) , t h e o p t i m a l a l l o c a t i o n s f r o m each t i m e p e r i o d . . S u c c e e d i n g co lumns, show how t h e s e q u a n t i t i e s a r e a l l o c a t e d f r o m t h e a v a i l a b l e l o g o o s o u r c e s as x t k ( X , Y ) and y + . j J X . Y ) . The o p t i m a l l o g s t o r a g e p o l i c y i s r e a d i l y i d e n t i f i e d by c o m p a r i n g t he i n d i c a t e d q u a n t i t i e s o f d e l i v e r e d l o g s w i t h t he m i l l demands o f each p e r i o d . F o r t h e a l l o c a t i o n p o l i c y shown i n T a b l e I, s i x t h o u s a n d c u n i t s o f p u l p l o g s and f i v e t h o u s a n d c u n i t s o f s a w l o g s a r e s t o r e d f r om p e r i o d one t o s u p p l e m e n t t h e l o g q u a n t i t i e s d e l i v e r e d i n the s e c o n d p e r i o d . Of t h e l o g s d e l i v e r e d i n t he s e c o n d p e r i o d , a s u r p l u s o f two t h o u s a n d c u n i t s o f p u l p l o g s and one t h o u s a n d c u n i t s o f s a w l o g s i s used i n the t h i r d t i m e p e r i o d . The o p t i m a l l o g a l l o c a t i o n p o l i c y g i v e n i n T a b l e I i s n o t u n i q u e . An a l t e r n a t i v e a l l o c a t i o n o f t h e t o t a l q u a n t i t y o f p u l p -l o g s d e l i v e r e d i n the t h i r d p e r i o d i s shown i n T a b l e I I . T h i s s o l u t i o n was i d e n t i f i e d f r o m a s e p a r a t e compu te r r un i n w h i c h t h e s e q u e n c e i n w h i c h each s o u r c e was i n c l u d e d i n t he c o m p u t a t i o n was r e v e r s e d . (The co lumn h e a d i n g s o f T a b l e I I must be s i m i l a r l y i n t e r p r e t e d ) . C o m p a r i s o n o f t he two s o l u t i o n s , i d e n t i c a l i n c o s t , shows t h e e q u i v a l e n c e o f s u p -p l y i n g t h e f i n a l f o u r t h o u s a n d c u n i t s o f t h e q u a n t i t y o f p u l p l o g s r e -q u i r e d i n t h e t h i r d p e r i o d f r o m e i t h e r t he l o g m a r k e t o r f r o m the c o n t r a c t o r - l o g g e d a r e a . R e f e r e n c e t o t h e c o s t s c h e d u l e s f o r t h e s e two s o u r c e s , g i v e n i n A p p e n d i x C , c o n f i r m s t h i s e q u i v a l e n c e a t t he n e c e s s a r y l o g p r o d u c t i o n l e v e l s . TABLE II SECOND OPTIMAL ALLOCATION POLICY 82 M I N I M U M COST ALLOCATION POLICY FOR X=68,Y=28 OPTIMAL ALLOCATIONS BETWEEN AND VvITHIN PERIODS TIME KILL ALL AREAS PRODUCTIVE AREAS PER IOC DEMAND . . . .JL 2. 3 „ 4_ .. ... . ...5 P/L S/L XT*5 YT0 XTK^YTK e XTK YTK XTK YTK XTK YTK XTK YTK ( 1) 24 IC 3G 15 5 0 0 0 7 8 3 7 15 0 ..(.2) 2.0. 9 _.. 1.6 5 7_ .0. D JD_ 3 5..... .. .0 0.: 6 .0_ (3) 24 <5 22 8 9 0 0 0 7 8 0 0 6 0 MINIMUM COST = $1324765.8 (a) Each log source included in reverse order to the sequence indicated, e . g . , "AREA 1" refers to source f i v e , "AREA 2" refers to source four, etc. * .(.b)_XT_-..X tU.-Y-). (c) YT = Y t(X,Y) (d) XTK = x t k (X ,Y) (e) YTK - y t k ( X , Y ) 83 2. Log A l l o c a t i o n from A l t e r n a t i v e Sources The a l l o c a t i o n s made a t each stage of the " forward p a s s , " the computat ional sequence i n which the f u n c t i o n ^ ( X ^ Y ^ ) i s s u c c e s s i v e l y eva lua ted f o r k = 1 , 2 , . . . , 5 , are l i s t e d f o r each pe r iod i n se ts of " p o l i c y t a b l e s " (Appendix C ) . The q u a n t i t i e s a l l o c a t e d i n the " forward pass" have been p r e v i o u s l y de f i ned as x + j< (X t ,Y t ) a n d ^ t k ^ t ^ t ^ " A ^ s o l i s t e d are three a d d i t i o n a l components, X^. and Y^., the magnitudes of the t o t a l q u a n t i t i e s a l l o c a t e d from each pe r iod and the eva lua ted f unc -t i o n , f t | < ( X t , Y t ) . The f i r s t s e c t i o n of each of the f i v e p o l i c y t ab les generated f o r the f i r s t pe r iod has been s e l e c t e d to i l l u s t r a t e how op-t imal a l l o c a t i o n s are de r i ved by the recur rence r e l a t i o n s h i p developed i n the model . These f i v e s e c t i o n s , g iven as Tables I I I to VII i n c l u s i v e , were de r i ved when the f i v e stages were s u c c e s s i v e l y i nc luded i n the com-p u t a t i o n a l sequence. Consequent ly , the process by which cos t m in im i z -a t i o n i s e f f e c t e d by dynamic programming can be numer i ca l l y t r a c e d . For example, when on ly one log source i s i nc luded i n the sequence, Table I I I , the t o t a l q u a n t i t i e s tha t may be a l l o c a t e d are r e s t r i c t e d by the a v a i l -ab le log s u p p l y , so X-| <_ 15 , Y-| = 0 and f ^ (X^ ,Y^ ) i s n e c e s s a r i l y equated w i th c ^ ( X - | , 0 ) , the net d e l i v e r e d l o g * c o s t . f u n c t i o n f o r : s o u r c e one i n the f i r s t t ime p e r i o d . With the a d d i t i o n o f a second source of logs (Table I V ) , the range of {X^,Y^} i s i nc reased and p a r t i t i o n i n g of. the q u a n t i t i e s X-| and Y ^ between the two sources i s p o s s i b l e . For each p a i r of q u a n t i t i e s X-j and-Yy a l l f e a s i b l e combinat ions of the q u a n t i t i e s x 1 2 ' y 1 2 ' X l ~ x T 2 a n c - Y l ~ y 1 2 a r e e v a l u a t e d . Only the p a r t i c u l a r magnitudes of x-|2 and y ^ which min imize the f u n c t i o n t " - ^ ^ » Y i ) a r e r e t a i ned as TABLE III ALLOCATIONS FROM SOURCE 1 FOR PERIOD 1 84 XT = X t , YT = Y t , X = x t l ( X t , Y t ) t Y » y t l ( X t , Y t ) , • RETURN = f t l ( X t , Y t ) . : J P_ULP_ . . . SAJlU2£_EBn0Jii3J-QJi_ _-__^__^___ _ PROD YT= 0 YT= 1 YT= 2 YT= 3 YT= 4 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 n 0. 1 1 0 1250C. 2 2 0 25000. 3 3 0 37500. 4 4 0 50000. 5 5 0 62500. 6 6 0 75000. 7 7 0 135000. 8 8 0 145000. 9 9 0 155000. 10 10 0 165000. I L . 11 . 0 1.7.5 Q.0.0 . 12 12 0 185000. 13 13 0 195000. 14 14 0 205000. 15 15 0 215000. TABLE IV ALLOCATIONS FROM SGURCE 2 FOR PERIOD 1 XT = X t , YT = Y t , X « x t 2 ( X t , Y t ) , Y = y t 2 ( V V s YC = y c , T = t r . 1 0 0 , RETURN = f t 2 ( X t , Y t ) : . J . PULP _S_AW.LC.G_ jE-RQDUCXlOfc PROD YT = 0 YT= -' 1 YT= = 2 YT= = 3 YT= = 4 -XT- X Y YC T RETURN X Y YC T RETURN X Y YC T RETURN X Y YC T RETURN X Y YC T RETURN 0 c 0 0.0 0 0 . 0 1 0.91 0 56619. 0 2 1.82 0 73237. 0 3 2 . 73 0 89856. 0 4 3.64 0 106475. i _ . c. ...0 _Q._Q.__ . Q „ _1.25Q0_. _ L J L _0_..8.2_ 9 64273. 1 _2_ 1.73 0 75928. 1 _3.. .2...6A. _Q_._ _8_3.CL0_8_. 1 A .3. .53 0__ 112390. 2 0 0 0 . 0 0 25000. 2 1 0.73 9 74739. 1 2 1.73 0 88428. 1 3 2 .64 0 100508. 2 4 3.46 0 117455. 3 0 c 0 . 0 0 37500. 3 1 0.87 3 84916. 1 2 1.73 0 100928. 1 3 2. 64 0 113008. 3 4 3.37 0 128028. 4 0 0 0 . 0 0 50000. 3 1 0.87 •3 97416. 1 2 1.73 0 113428. 1 3 2.64 0 125508. 4 4 3. 28 9 138292. 5 0 0 0.0 0 62500 . 3 1 0.87 3 109916. c 2 1.51 7 121907. 5 3 2.26 9 137524. 5 4 3.19 9 147685. 6__ c .0 J3 ._0„. _D_.. ._ .750CO.. _.__3_ 1 0.8 7.. _3._ 122416. 6 JL 3_._74_ 3 131031. 6 _3_ _2...19_ ^9 143218. 6 JL .3. 10 .9. . 157275. . 7 4 0 0.0 0 126380. 7 1 1.00 0 134463. 7 2 1.96 0 140196. 7 3 2.18/ 8 148815. 6 4 3.10 9 169775. .8 4 c 0 . 0 0 138 880. 8 1 1 . 0 0 0 144483. 8 2 2.00 0 150007. 7 3 2. 18 8 161315. 6 4 3. 10 9 182275. 9 9 0 0 . 0 0 148980. 9 1 1.00 C 154503. 8 2 2.00 0 162507. 7 3 2.18 8 173815. 6 4 3. 10 9 1947 75. 10 IC c 0. 0 0 1590C0. 9 1 1.00 0 167003 . 8 2 2.00 0 175007. 7 3 2.18 8 186315. 6 4 3.10 9 207275. 11.. 1.0. 0 0.0 0.__ _1.715CQ_. 9_ _1___L._30 0 1793.03. 8 _2_ 2..JHL _0__. L87507. 7 _3_ _2.».1.8_ _8_ _15.8_8_1.5.. 6 _ i». .3... 1.0. 9 .219775. 12 10 0 o.o 0 184000. 9 1 1.00 0 192003. 8 2 2.00 0 200007. 7 3 2.18 8 211315. 6 4 3. 10 9 232275. 1.3 c 0 0. 0 0 1950C0. 9 1 1 .00 0 204503. 8 2 2.00 0 212507. 7 3 2.18 8 223815. 6 4 3. 10 9 292275. 14 0 0 0 . 0 0 205000. 9 1 1.00 0 217003. 8 2 2.00 0 22 5007. 1 3 2 .64 0 283008. 6 4 3.10 9 302275. 15 0 0 0.0 0 215000. 9 1 1 .00 0 229503. 1 2 1.73 0 280928. 1 •3 2. 64 0 293008. 6 4 3. 10 9 312275. . 16.. 10.. JO. _Q_..0_._ -JDL _ .2 340 0.0. 1 1 . 0_._8.2_ _9___ _2Z92_73._ _. X _2_ _L. 73 0 230 9.2.8.. 1_ _3_ .2.. 6i+_ o._ 303008. 6 4 .3. 10 _9_ 322275. 17 2 0 0 . 0 0 279440. 2 1 0 .73 9 289739. 2 2 1.64 9 303546. 7 3 2.18 8 313815. 6 4 3.10 9 332275. 18 3 0 0.0 0 291660. 3 1 0.87 3 299916. 8 2 2.00 0 315007. 7 3 2.18 8 323815. 6 4 3. 10 9 342275. 19 4 0 0.0 0 30388G. 4 1 1.00 0 319 403. 8 2 2.00 0 32 5007. 7 3 2.18 8 333815. 6 4 3.10 9 352275. 20 5 0 C O 0 323900. 5 1 1.00 0 329423. 8 2 2.00 0 335007. 7 3 2.18 8 343815. 6 4 3. 10 9 362275. _ Z l _ . 6.. IL _Q._Q__ _ < _ . _ _ _33.39.2D_. 6_ _1_ _1..I3.Q_ J0._ _3_3_9_4_ij3.._ 8_ _2__2..O.0_ _0_ .343J3.0.7.. 7. _3_ _2...18_ .8 3 538.1 5. 6 . 4. _3.. 10 _9...„ 372275. 22 7 0 0.0 0 343940. 7 1 1.00 0 349463. 8 2 2.0G 0 35 5007. 7 3 2. 18 8 363815. 23 8 0 0 . 0 0 353960. 8 1 1 . 00 c 359483. 8 2 2.00 0 365007. 24 9 0 0 . 0 0 363980. 9 1 1.00 0 369503. 25 IC 0 0.0 0 374000. .... — . . _ — - . . . . . — . . TABLE V ALLOCATIONS FROM SOURCE 3 FOR PERIOD 1 XT = X t , YT = Y r X-« x t 3 ( X t , Y t ) , Y = y t 3 ( X t , Y t ) , P = INDEX, X l = x l , RETURN = f ^ x ^ y ^ . ) 86 PULP PROD -XT-YT= 0 X Y P Xl RETURN , .5JJ«ifiG_£fiG.D_UXI.LON YT= 1 YT= 2 X Y P XI RETURN X Y P XI RETURN YT= 3 X Y P XI RETURN YT= 4 X Y P XI RETURN 0 0 "0 0 0 0. 0 1 0 0 22238 . 0 2 0 0 44476. 0 3 0 0 66714. C 4 0 0 88952. 1 . . Q Q_ .___0._.0__ _. _ . i 2.5_0_Q.. __. 1 ..1 ..0 -27.00.0-._- 1_ __2._ 0. _1_ _4D_Q0O.._-_ ,3._ __0_ 1... .6092 8. ... . 1 _ 4__ 0. . 82381. 2 0 0 0 0 25000 . 1 1 0 1 39500. 1 2 0 1 52500. 2 3 0 2 67000. 2 4 0 2 80000. 3 0 0 0 0 37500. 1 1 0 1 52000. 1 2 0 1 65000. 2 3 0 2 79500. _j 4 2 3 9C0C0. 4 0 0 0 0 5000C. 1 1 0 1 64500. 1 2 0 1 77500. 4 3 2 4 91000. 4 4 2 4 10C0C0. 5 0 0 0 0 62500. 1 1 0 1 77000. 1 2 0 1 90000. 5 2 5 1C1GG0. 5 4 2 5 110000. 6 __o__. -P. .....0_.Q_ ___7-50_0.0.. 1 _ 1 .0 1 8950G.___ 6 2 2 J_ 102000. 6 3 2 6 111000. 6 . 4 2 6 120000. 7 1 0 0 1 89000 . 1 1 0 1 102000. 7 2 2 6 114C00. 7 3 2 6 123000. 7 4 2 6 132000. 8 2 0 0 2 1030CO. 2 1 0 2 116000. 8 2 2 6 126000. 8 3 2 6 135000. 8 4 2 6 144000. 9 3 0 0 3 117000 . 9 1 2 6 129000. 9 2 2 6 138000. 9 3 2 6 147000. 9 4 2 6 156000. 10 4 0 0 4 131000. 10 1 2 6 141000. 10 2 2 6 150000. 10 3 2 6 159000. 10 4 2 6 168000. 11 -P.. _..3....6__ 14.4000... _ i i 1_ _ 2 6 153000. Ll_ _.2_ _2_ -.6. 16,2(30X1.-.- 11 3 2 .6,._ 171000. l i . . -4__ 2 6_ .: 180-0.00. 12 12 0 3 6 156000. 12 1 2 6 165000. 12 2 2 6 174000. 12 3 2 6 183000. 11 4 2 6 192500. 13 13 0 3 6 16 8000. 13 1 2 6 177000. 13 2 2 6 186000. 12 3 2 6 195500. 11 4 2 6 205000. 14 14 Q 3 6 180000. 14 1 2 6 189000. 13 2 2 6 198500. 12 3 2 6 208000. 11 4 2 6 217500. 15 15 0 3 6 192GGO. 14 1 2 6 201500. 13 2 2 6 211000. 12 3 2 6 22050G. 11 4 2 6 230000. .16 15 0 _ 3 _ 6 _ 2045OA. ._ i.4_ u _2 6 214000.. 13 2 2 jfc 223.5.00. 1.2_ _3. 2 .6. _2.33.0_0.0-. l i . _.4 . .2 6 .242500. _ _ 17 15 0 3 6 217000. 14 1 2 6 226500. 13 2 2 6 236000. 12 3 . 2 6 245500. 11 4 2 6 255000. 18 15 0 3 6 229500. 14 1 2 6 239000. 13 2 2 6 248500. 12 3 2 6 258000. 10 2 2 6 300007, 19 15 0 3 6 242000. 14 1 2 6 251500. 13 2 2 6 261000. 11 1 2 6 303007. 11 2 2 6 312007. 20 15 0 3 6 254500. 14 1 2 6 264000. 12 0 3 6 306007. 12 1 2 6 315007. 12 2 2 6 324007. 2 i _ 15. ._>.._ 3 6_ 2.6Z0.0.0_._ __i> 1 _2_ 6 .3-08.000, 6. _2_ _.2_6 3.1700.0.. . 6. 3 _2 ..6. . 3.2.600.0. 6 .. 4_ _2. .6... .. 3.3 50.00.. 22 7 0 3 6 3110C0. 7 1 2 6 320000. 7 2 2 6 329000. 7 3 2 6 338000. 7 4 2 6 347000. 23 8 0 3 6 3 23000. 8 1 2 6 332000. 8 2 2 6 341000. 8 3 2 6 350000. 8 4 2 6 359000. 24 9 0 3 6 335000. 9 1 2 6 . 344000. 9 2 2 6 35300C. 9 3 2 6 362000. 9 4 2 6 371000. 25 10 0 3 6 347000. 10 1 2 6 356000. 10 2 2 6 365000. 10 3 2 6 374000. 10 4 2 6 383000. 26 __ l i _ 0 3 6_ 3_5_9J)J_XL. 11... ...1 .... _2_ 6 3 6.8.CL0 Q.. 1.1 _2_ .2 ,6 377I5J3JD-. ___U_ _3__ 2 6 386000. . .. .11. 4 _2 __.3_950.QJX. 21 12 0 3 6 371000. 12 1 2 6 380000. 12 2 2 6 389.000. 12 2 6 398000. 13 2 2 6 411007. 28 13 0 3 6 383000. 13 1 2 6 392000. 13 2 2 6 401000. 14 1 2 6 414007. 6 I 2 6 456815. 29 14 0 3 6 395000. 14 1 2 6 404000. 15 0 3 6 417007. 6 1 2 6 458007. 6 2 2 6 467007. 30 15 0 3 6 4C700C. 15 0 3 6 421503. 7 0 3 6 461007. 7 1 2 6 470007. 7 2 2 6 4790C7. 3.1 15 0 _3L 6 _ 4.2.6.O.O.O.. 7 _0_-.3. _6 4-6.5.50-3. .- 8_ 0 3-_6__ 47_3_Q.Q7. 8 _1 -_2_ .6-.. -__4.8.2.G071.- _8__ 2_ _ _ _ _ _ _6_ .49 10 07_. 32 7 0 3 6 470000. 8 0 3 6 4 77503. 9 0 3 6 48 5007. . 9 1 2 6 494007. 9 2 2 6 503007. 33 8 0 3 6 482000. 9 0 3 6 489503. 10 0 .3 6 497007. 10 1 2 6 506007. 10 2 2 6 515007. 34 9 0 3 6 494000. 10 0 3 6 501503. 11 0 3 6 509007. 11 1 2 6 518007. 11 2 2 6 527007. 35 10 0 3 6 506000 . 11 0 3 6 513.03. 12 0 3 6 521007. 12 1 2 6 530007. 12 2 2 6 539007. 36 11 0_. .3 _6_. 518000. 12 _XL_ 3 6 _5_23_50 3. 13 _ 0 . _3_ _6„_ .. 5330DJ. 1.3 _1 2-_6 _ ..5.42.0 0..7,__ 13_ _2_ _2__6. . -_5.5-l.00-7.. 37 12 0 3 6 5 30000. 13 C 3 6 537503. 14 0 3 6 545007. 14 1 2 6 554007. 38 13 0 3 6 542000 . 14 0 3 6 549503. 15 0 3 6 557007. 39 14 0 3 6 554000. 15 0 3 6 561503. TABLE VI A L L O C A T I O N S F R O M S O U R C E 4 F O R P E R I O D 1 87 XT = X t , YT = Y t , X = x t 4 ( X t , Y t ) 5 Y = y t 4 ( X t , Y t ) , RETURN = f t 4 ( X t . Y t ) , J P U L P - - S . A W . L . O G . . _ P . R , C C . U . C T . L C N : _ _ . P R O D Y T = 0 Y T = 1 Y T = 2 YT= 3 Y T = 4 - X T - X Y R E T U R N X Y R E T U R N X Y R E T U R N X Y R E T U R N X Y R E T U R N 0 G O 0 . 0 0 2 2 2 3 8 . 0 Q 4 4 4 7 6 . € 0 6 6 7 1 4 . 0 0 8 8 9 5 2 . 1 Q _ _ _ ^ Q 1 2 . 5 0 0 . . C 0...__-.-__X7_00.0. 0 0 4 0 0 0 0 . 0 0 6 0 9 2 8 . „ Q 0 _ . 8 . 2 3 8 1 . . . 2 0 0 2 5 0 0 0 . 0 0 3 9 5 0 0 . 0 0 5 2 5 0 0 . 0 0 6 7 0 0 0 . 0 0 8 0 0 0 0 . 3 0 0 3 7 5 0 0 . 0 0 5 2 0 0 0 . 0 0 6 5 0 0 0 . 0 0 7 9 5 0 0 . 0 0 9 0 0 0 0 . 4 Q Q 5 0 0 0 0 . 0 0 6 4 5 0 0 . 0 0 7 7 5 0 0 . 0 0 9 1 0 0 0 . 0 0 1 0 0 0 0 0 . 5 0 0 6 2 5 0 0 . 0 0 7 7 0 0 0 . 0 0 9 0 0 0 0 . C 0 1 C 1 0 C 0 . 0 0 1 1 0 0 0 0 . .6 . _jQ . _ J Q 7 3 J 0 O . a . _ _ 0 _ _ O :__8_9 .5 I31L . .0 0 L G . 2 0 J H L . : Q 0 L U O J O J O L . _ 0 _ Q 1 2 0 0 . 0 0 . . 7 . 0 0 8 9 0 0 0 . 0 0 1 0 2 0 0 0 . 0 0 1 1 4 0 0 0 . 0 0 1 2 3 0 0 0 . 0 0 1 3 2 0 0 0 . 8 0 0 1 0 3 0 0 0 . 0 0 1 1 6 0 0 0 . 0 0 1 2 6 0 0 0 . 0 0 1 3 5 0 0 0 . 0 0 1 4 4 0 0 0 . 9 0 0 1 1 7 0 0 0 . C 0 1 2 9 0 0 0 . 0 0 1 3 8 0 0 0 . 0 0 1 4 7 0 0 0 . 0 0 1 5 6 0 0 0 . 10 0 0 131000. . 0 0 141000. 0 0 150000. 0 0 159000. 0 0 168000. 11 0 0 1 4 4 0 0 0 . 0 0 15 3 0 0 0 . 0 0 1 6 2 0 0 0 . 0 0.. 17.lQ_Q.Q_. 0...0. 1.8.0.00.0... 12 0 0 156000. 0 0 165C00. 0 0 174000. C 0 183000. 0 0 192500. 13 0 0 168000. 0 0 177000. 0 0 186000. 0 0 195500. ,0 0 205000. 14 0 0 180000. C 0 189000. 0 0 198500. 0 0 208000. 0 0 217500. 15 0 0 192000. 0 0 201500. 0 0 211000. C 0 22G500. 0 0 230000. .1.6. 0 0 2 0 4 5 0 0 . 0 0 2 1 4 0 0 0 . 0 0 223 5 0 0 . 0 Q 233.000... 0 .2.42500. 17 0 0 217000. 0 0 226500 . 0 0 236000. 0 0 245500. 0 0 255000. 18 0 0 229500. 0 0 239000. 0 0 248500. 0 0 258000. 1 0 275000. 19 0 0 242000. 0 0 251500. 0 0 261000. 1 0 278000. 2 0 295000. 20 0 0 254500. 0 0 264000. 1 0 281000. 2 0 298000. 2 0 315000. 2.1 0 Q 26 ,7000 . 1 0 2 8 4 0 0 0 . ? 0 301000. 3 0 - .18000 _. 0 _ o ..3 3.50.00. 22 1 0 287000. 2 0 304000. ' 3 0 3 21000. G 0 338000. 0 0 347000. 23 2 0 .307000. 3 0 324000. 0 0 341000. 0 0 350000. c 0 35900C. 24 3 0 3 27000 . o- 0 344000. 0 0 353000. 0 0 362000. 0 0 371000. 25 0 0 347000. 0 0 356000. 0 0 365000 . 0 0 374000. C 0 3830GG. 2.6 JD 0 - ^ q o n r . _. 0 _Q ? 6 8 n n o . 0 0 3 7 7 0 0 0 . 0 ..Jj . 3 . a £ 0 0 J D . 0 Q „3_9.5.0O.£L. 27 0 0 371000. 0 0 380000. 0 0 389000. 0 0 398000. 0 0 411007. 28 0 0 383000. 0 0 392000. 0 0 401000. 0 0 414007. 1 0 431007. 29 0 0 395000. 0 0 4G400C. 0 G 417007. 1 0 434007. 2 0 451007. 30 0 0 407000. 0 0 421503. 1 0 437007. 2 G 454007. 3 . 0 471007. 3.1 0 0 426.0OO . 1 0 441 50^. . ? 0 457007. 3. 0 4J40.07.. 0 - ___.Q .4-9.10.0-7... 32 1 0 446000. 2 0 461503. 3 0 477007. G 0 494007. 0 0 503007. 33 2 0 466000. 3 0 481503. 0 0 497007. 0 0 506007. 0 0 515007. 34 3 0 486000. 0 0 501503. 0 0 509007. G 0 518 0 0 7 . 0 0 527007. 35 0 0 506000. V 0 513503. 0 0 521007. 0 0 530007. 0 0 539007. - . 36 G_ J X 51.800.0.. 0 n 5 ? 5 5 0 3 . 0 0 533(107. .0 .0 54-20.0-7. 0 0 5.5.1-00-7-. — 37 0 0 530000. 0 0 5 37 503. 0 0 545007. 0 0 554007. 1 0 571007. 38 0 0 542000. 0 0 549503. 0 0 557007. 1 0 574007. 2 0 591007. 3<5 0 n 5 5 4 0 0 0 . 0 0 5615Q3, 1 0 5 7 7 0 0 7 . 2 0 594007. 3 0 6 1 1 0 0 7 . ) TABLE VII ALLOCATIONS FROM SOURCE 5 FOR PERIOD 1 XT = X t , YT » Y t , X = x t 5 ( X t , Y t ) s Y » y t 5 ( X t , Y t ) , RETURN =. f t 5 ( X t , Y t ) .PULP . S_4W-Lj0_G___PRaD_U_CJJ0N . . PROD YT= 0 YT= 1 YT= 2 YT= 3 YT= 4 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 0 0. 0 Q 22238. 0 0 44476. C 0 66714. 0 0 88952. 1 G_ 0 1 2500. 0 __0 77000. 0 0 40000. 0 0 60928. o__ _0_ __82381... 2 0 0 25000. 0 0 39500. 0 0 52500. 0 0 67000. 0 0 80000. 3 3 0 36000. 0 0 52000. 0 0 65000. 0 0 79500. 0 0 90000. 4 4 c 48CC0. 3 0 63000. 3 0 76000. G 0 91000. G 0 100000. 5 5 0 60000. 4 0 75000. 4 0 88000. C 0 101000. 0 0 110000. 6 _5. 0 72500. 5 0 87000. 5 0 100000 . 0 0 111000. 0 _o_.__ 12Q0_0Q. 7 5 0 85000. 5 0 99500. 5 0 112500. 0 0 123000. 0 0. 13 2000. 8 5 0 97500. 5 0 112000. 5 0 125000. 0 0 135000. 0 0 144000. 9 5 0 110000. 5 0 124500. 5 0 137500. 0 0 147000. 0 0 15600C. 10 5 0 122500. 5 0 137000. 0 0 150000. 0 0 159000. 0 0 168000. 11 5 a l-3.5_0.CLa.__. ________5 a 14_9_.5_Q.Q-. 0 0 162000. ___ _..0 0. 171000. 0. __0. 180.00.0. 12 5 0 149000. 5 0 162000. 0 0 174000. G 0 183000. 3 0 192000. 13 5 0 163000. 5 0 176000. 0 0 186000. 3 0 195000. 3 0 204000. 14 5 0 177000. 0 0 189000. 3 0 198000. 3 0 207000. 3 0 216000. 15 5 0 191000. 3 0 201000. 3 0 210000. 3 0 219000. 4 0 22800C. .1.6 _3_..._0 2iIAQ.0_L.___ 3 0 21 3000. 3 0 722000. _4 .0 2310.00.. 5„JQL 2.40.000.. _ 17 3 0 216000. 3 0 225000. 4 0 234000. 5 0 243000. 5 0 252500. 18 3 0 228000. 4 0 237000. 5 0 246000. 5 0 255500. 5 0 265000. 19 4 0 240000. 5 0 24 9000. 5 0 258500. 5 0 268000. 5 0 277500. 20 5 0 252000. 5 0 261500. 5 0 271000. 5 0 280500. 5 0 29000C. 2.1 5 _0___2.6_4_50.0-. 5. 0 7 7 4 0 0 0 . 5 C_____2.8.33il0. = 5 0. 2S3.00O. 5__0. 30.230.0_. 22 5 0 277000. 5 0 286500. 5 0 296000. 5 0 305500. 5 0 315000. 23 5 0 289500. 5 0 299000. 5 0 308500. 5 0 318000. 6 0 334000. 24 5 0 302000. 5 0 311500. 5 0 321000. 6 0 337000. 7 0 353000. 25 5 0 314500. 5 0 324000. 6 0 34000G. 7 0 356000. 8 0 372000. 26 5__Q 3_27O_0_0_. 6 0 343000. 7___0 3_5_9J).0_Cl. 8 0 375000. 9_._0.____ .39.10.0.0.. 27 6 0 346000. 7 0 362000. 8 0 378000. 9 0 394000. 3 0 407000. 28 7 0' 365000. 8 0 381000. 9 0 397000. 3 0 410000. 3 0 419000. 29 8 0 3 84000. 9 0 400000. 3 0 413000* 3 0 422QQQ. 3 0 431000. 30 9 0 403000. 3 0 416000. 3 0 425000. 3 0 434000. 4 0 443000. 3_1__ 3 0 419000. 3 0 4 7 8 0 0 0 . 3 0 437000. 4 0_ 44.6.000.. _ - 5 0 455000 . 32 3 C 431000. 3 0 440000. 4 0 449000. 5 0 458000. c _rf 0 471007. 33 3 0 443000. 4 0 452000. 5 0 461000. 5 0 474007. 6 0 490007. 34 4 0 455000. 5 0 464000. 5 0 477007. 6 0 493007. 7 0 509007. 35 5 0 467000. 5 0 481503. 6 0 496007. 7 0 512007. 8 0 528007. 3_6__ . _ .3 0 486000. ft 0 500 503. 7 0 515007. 8 0 531007. 9 0 547007. 37 6 0 505000. 7 0 519503. 8 0 534007. 9 0 550007. 3 0 563007. 38 7 0 5 24000. 8 0 538503. 9 0 553007. 3 0 566007. 3 0 5 75007. 39 8 0 543000. 9 0 557503 . 3 0 569007. 3 0 578007. 3 0 587007. 89 X ^ U - J Y - J ) and y - ^ X - j ,Y- j ) . Th is s e l e c t i o n process i s repeated a t each stage i n the sequence and i s r e f e r r e d to as the " forward p a s s . " The i n -format ion l i s t e d i n the p o l i c y tab les enables the manager to examine the c o s t - e f f e c t i v e n e s s of i n c l u d i n g each s tage i n the sequence. The incrementa l e f f e c t s of s tage k on c o s t s , and the a l l o c a t i o n p o l i c y f o r 1 to k s t a g e s , are i d e n t i f i e d by comparing the magnitudes of x ^ ( X p Y y ) and y-jkCX-j »Y-j) w i th the cor responding magnitudes of X-j and Y-j , and by examining the v a r i a t i o n i n the c a l c u l a t e d f l k ( X - | , Y - ) - w i t h the p r e v i o u s l y de r i ved va lue of f- k - |(X-j,Y-|). Fur ther examinat ion o f the i n d i v i d u a l cos t f u n c t i o n s , c-| k (X- | ,Y-j) f o r a l t e r n a t i v e log sources w i l l i d e n t i f y the reason f o r c r i t i c a l p roduc t ion l e v e l s . Tables I I I , IV , V , V I , and VII each con ta in the three m a t r i c e s , x ] k ( X 1 , Y 1 ) , y l k ( X 1 , Y . ) and f ! k (X-, J - j ) . In a d d i t i o n Tables IV and V i n -c lude two mat r i ces which i d e n t i f y the p a r t i c u l a r log p roduc t ion p o l i c i e s adopted f o r s p e c i f i c a l l o c a t i o n s . In Table IV , which r e f e r s to the i n -c l u s i o n of the second source of logs i n the " forward p a s s , " these are i d e n t i f i e d by- the symbols "YC" and " T " . These symbols r e f e r to the elements i d e n t i f i e d as " y c " and " t r , " r e s p e c t i v e l y , i n Chapter I I I , and denote opt imal log s o r t i n g and t r ad ing p o l i c i e s . (For compactness the r a t i o , t r , i s expressed as a percentage i n Table I V ) . From the magni-tudes of "YC" and " T " , the manager can s p e c i f y the optimum q u a n t i t i e s of sawlogs to be s e l e c t e d from the non-pee le r l og c l a s s e s i n area two and determine the quan t i t y of p e e l e r logs to be exchanged f o r sawlogs . As shown i n Chapter I I I f o r each a l l o c a t i o n of the q u a n t i t i e s , x ^ - -and y ^ , the magnitude of yc and t r f u l l y s p e c i f y each s o r t i n g and t r ad i ng p o l i c y . 90 The a d d i t i o n a l columns i n Table V , which r e f e r s to the i n c l u s i o n of the t h i r d source of logs i n the " forward p a s s * " are denoted by the symbols " P " and " X T . " These symbols r e f e r to the term, INDEX, and the volume component, x l , r e s p e c t i v e l y , de f ined i n Chapter I I I , and s p e c i f y where, and i n what q u a n t i t i e s , h i g h - c o s t and low-cos t logs are d e l i v e r e d f o r each a l l o c a t i o n from the t h i r d p roduc t i ve a r e a . 3. Optimal A l l o c a t i o n among Sources When the " forward pass" i s comple ted, f o r example when t ~ t 5 ^ t ' Y t ^ •k -k has been c a l c u l a t e d , the opt imal a l l o c a t i o n s , ^ ( X ^ Y ^ ) and y ^ U ^ Y ^ . ) f o r each of the f i v e sources are i d e n t i f i e d by a "backward p a s s . " Th is scanning p rocedure , d i scussed i n Chapter I I , r equ i res the s e t of p o l i c y tab les to be entered i n reverse order and the computat ional path r e -t raced f o r each element of the s e t { X ^ Y ^ } / . A comprehensive summary of the opt imal p a r t i t i o n i n g o f a l l f e a s i b l e combinat ions of X^ and Y^ among a v a i l a b l e sources i s l i s t e d by DYALOG f o r each p e r i o d . Table VI I I shows the o p t i m a l , a l l o c a t i o n of a p o r t i o n of the s e t ' { X 1 S Y | } ass igned from the f i r s t p e r i o d . I t shou ld be noted t h a t , f o r t h i s p a r t i c u l a r p e r i o d , the magnitudes of X-| and Y-j can never be l e s s than m i l l demands. In t h i s r e s p e c t , a l though conven ien t , p r i n t i n g the opt imal a l l o c a t i o n s f o r X-| < 24 and Y-j < 10 i s perhaps m i s l e a d i n g . However, i n l a t e r pe r -i o d s , when i t i s p o s s i b l e to s a t i s f y m i l l demands comp le te l y , or ir i p a r t , from s to red l o g s , the more i n c l u s i v e l i s t i n g i s r e q u i r e d . The ;phys i ca l arrangement of Table VI I I f a c i l i t a t e s comparison of the a l l o c a t i o n s made when the q u a n t i t i e s X. and Y. v a r y . Th is c a p a b i l i t y TABLE V I I I OPTIMAL ALLOCATION WITHIN PERIOD 1 1ST OF PAIRED COLS . 2ND OF PAIRED COLS. 91 PULP ALLOC. S/L ALLOC. PULP PROD AREA SAWLOG PRODUCTION 7 8 9 1 0 11 12 13 14 15 16 17 18 19 0 (1) 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 o 0 0 c • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .12J___ 0 0 0 0 0 0 0 0 0 _0 0 0 0 0 0 n 0 0 0 c j GXC.... 0X0 0.1.0 0.1.0 0.10 0.1.0 . oxo 0X0. .. _0X0...__ 010 (3) 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 0 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 8 <4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (5) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 G 0 0 0 0 0 0 0 0 1 (1) 1 0 0 G c 0 0 o- 0 C 0 G 0 c 0 0 0 c 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (71 0 0 c _0_ 0_ .0 0 0 0 0 .0. _0 0 0 0 0 0 o • 0. 0 0 0 0 0 CLLO _ . ..0.1.0 _ 0.1.0 Q.LO __oxo_ 01.0.... 010 .010. . (3 ) 0 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 8 1 8 1 8 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 8 (4 } 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (5) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 2 (1 ) 2 0 1 0 1 0 0 0 0 0 0 0 0 0 0 C 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 i 2) 0 €L JQ_ _0 _.G_ jQ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.....0 0 0 1 9 CLIO CXO 0X0. 0X0_ 010 . .010 (3 ) 0 0 1 1 1 2 2 3 2 4 2 c 2 6 2 7 2 8 2 8 2 8 2 8 2 8 1 4 2 4 2 5 2 6 2 7 2 8 2 8 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 1 0 2 0 3 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (5) 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 c 0 G 0 0 0 0 0 0 3 (1 ) 0 0 2 0 2 0 1 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 C G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12) JOL JOL __-0_ 0 0 0 0 n 0 0 .0 0 0 0 0 0 0 0 0 0 0 0 _____ 0 _J3 0 1 9 1 9 0 ..9 OXO. oxo_ _.oxc.__- ..0.1.0 (3 ) 0 0 1 1 1 2 2 3 4 3 c 3 6 3 7 3 8 3 8 3 8 3 8 3 8 2 4 2 5 3 6 3 6 3 7 3 8 3 8 (4) 0 o 0 Q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (5 ) 3 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 4 <1) 0 0 o- 0 0 0 o- 0 0 0 0. 0 0 0 0 0 0 0 0 G 0 0 0 0 0 C 0 G 0 0 0 0 0 0 0 0 0 0 0 0 { 2) 0 0 0 n 0 0 0 _0__ 0 0 0 0 0 0 0 0 0 JOL_ 0 0 0 0 Q 0 0 0 2 8 1 8 1 9 1_9. 0 9 01.0. .. .0.10 (3) 0 0 1 l 1 2 4 3 4 4 4 «; 4 6 4 7 4 e 4 8 4 8 4 8 4 8 2 5 3 6 3 6 3 7 4 8 4 8 4 8 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 G 3 0 4 C 0 0 0 0 G .0 0 0 0 0 0 0 1 <5> 4 0 3 0 3 0 c 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 (1 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 G G 0 0 G G 0 0 0 0 0 0 0 0 c 0 c 0 0 0 0 0 0 ( 2 ) 0 0 0 n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0_0 _JQ 0 0 G 2 7 2 8 2 8 1 8. ___1_._9.__ . 1. 9 -1 9 (3) 0 0 1 l 1 2 5 _ 5 4 c c 5 6 c 7 5 8 5 8 5 8 5 8 5 8 3 6 3 6 3 7 4 8 4 8 4 8 4 8 (4) 0 0 0 0 0 0 0 0 0 0 6 6 0 0 0 0 0 0 0 1 0 2 0 3 0 4 0 0 0 0 0 0 0 0 0 0 0 1 0 2 (5 ) 5 0 4 0 4 0 0 0 0 Q 0 0 0 0 c C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Q o 6 ( 1) 1 0 0 0 0 0 0 0 0 0 0 0 -.12). _. O._..0__ _.JOL 0 Q-0__ _J3__JCL_ __C_ G. 0 0 (3) 0 0 1 1 1 2 6 3 6 4 6 5 (4) 0 o c C 0 0 0 0 0 C 0 0 (5) 5 0 5 0, 5 0 0 0 0 C c 0 0 0 JL-CL 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ? 5 2 6 7 7 _2__8. 2_8... _-_ 2 8 6 7 6 8 6 8 6 8 6 8 6 8 4 8 4 8 4 8 4 8 4 8 4 8 0 0 0 0 G 1 0 2 0 3 G 4 0 0 0 0 0 c 0 0 0 1 0 2 0 G 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 2 8 4 8 C 3 0 0 7 ( 1) 2 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 c 0 0 0 0 0 0 J2X-, 0—0 0 0 ___G 0 0 0 0 0 0 _CL_ ._JL 0 . 0 .0. 0 G 0. G_ 0 0 0 0 G._ G 2. 5. 3 6 3 7 ? 8 _2__8: -Z. .8- 2. 8 .. (3) 0 0 1 1 1 2 7 3 7 4 7 5 7 6 7 7 7 8 7 8 7 8 7 8 7 8 5 8 4 8 4 8 5 8 5 8 5 8 5 8 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 1 0 2 0 3 0 4 0 0 0 0 0 0 0 C 0 1 0 2 0 3 <5 ) 5 0 5 0 5 0 0 0 0 0 0 0 0 G 0 G 0 0 0 0 0 0. 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 ( 1 ) 3 0 2 Q 2 C 0 G 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 (Z).__ _.JLJ3._ _ 0 _ 0 G_D 0. 0 0 0 0_ 0 0 0 0 G _0 -0 0 0 0 0 0 0 2 _4-_. 2.5 3 6 3_ 7 2. 8 . —2-8.._. -2. -8. 2 .8 (3) 0 0 1 1 1 2 8 3 8 4 8 c 8 6 8 7 7 8 7 8 7 8 7 8 6 8 6 8 5 8 5 8 6 8 6 8 6 8 6 8 .4 3 0 0 0 0 0 0 0 0 0 C c 0 0 0 C 0 0 0 0 1 0 2 c 3 0 0 0 0 0 0 0 0 0 c 0 1 0 2 0 3 <5) 5 0 •5 0 5 0 0 0 0 G Q Q 0 0 0 C 0 0 •_' G P 0-.. Q 0 0 0 0 Q Q Q Q 0 0 0 0 0 Q 0 a 92 i s impor tant to the manager, not on ly as a convenient supplement to the p o l i c y t a b l e s , but a l s o as a rap id method of examining the e f f e c t on logg ing p lans of an enforced change i n the q u a n t i t i e s r e q u i r e d . For example, f o r the problem d e s c r i b e d , the opt imal p o l i c y s p e c i f i e d a t o t a l of t h i r t y thousand c u n i t s of pu lp logs and f i f t e e n thousand c u n i t s of sawlogs f o r d e l i v e r y i n the f i r s t pe r iod (Tables I and I I ) . I f , because of an emergency c l o s u r e o f the sawmi l l or because sawlog s torage became unexpectedly l i m i t e d , i t i s necessary to reduce the quan t i t y of sawlogs d e l i v e r e d i n the f i r s t pe r iod by one thousand cun i t s a r e v i s e d a l l o c a t i o n p o l i c y i s r e q u i r e d . Comparison of the opt imal a l l o c a t i o n s when X-| = 30 and Y-j = 15 w i th the a l l o c a t i o n s f o r X^ = 30 and Y^ = 14 shows not only a cut -back i n sawlog p roduc t ion from source two, but a l s o a changed a l l o c a t i o n f o r p u l p l o g s . (The two se ts of opt imal a l l o c a t i o n s are i n d i -cated i n the output t ab le shown i n Appendix C . p a g e 176). In the r e -v i sed a l l o c a t i o n , the p roduc t ion of one thousand c u n i t s of pu lp logs i s t r a n s f e r r e d from the c o n t r a c t o r - l o g g e d area to source two. In the absence of p o l i c y t ab les and the l i s t i n g o f the "backward p a s s , " the marginal cos t of the f i f t e e n t h one thousand c u n i t quan t i t y of sawlogs would have to be i d e n t i f i e d from the i n d i v i d u a l cos t f u n c t i o n s . Though the c a l c u l a t i o n s requ i red may be onerous , the requ i red decrease i n saw-log p roduc t ion would then be c o r r e c t l y ass igned to source two. However, i t i s unce r t a i n i f the d e s i r a b i l i t y of changing the pu lp log a l l o c a t i o n would s i m i l a r l y be r e c o g n i z e d . 93 4 . Log A l l o c a t i o n between Per iods The o p t i m i z a t i o n process f o r a l l o c a t i n g q u a n t i t i e s from each per iod i s analogous to the sequen t i a l method used f o r a l l o c a t i n g q u a n t i -t i e s from a l t e r n a t i v e log s o u r c e s . As f o rmu la ted , the f u n c t i o n F t ( X , Y ) i s eva lua ted by o p t i m a l l y a l l o c a t i n g each pa i red element of the range of q u a n t i t i e s , {X ,Y } , s p e c i f i e d f o r each s t a g e , between stage t and a l l p rev ious stages 1 through t - 1 . The r e s u l t i n g m u l t i - s t a g e process i s a " forward pass" i n which the q u a n t i t i e s o p t i m a l l y a l l o c a t e d from each pe r iod are i d e n t i f i e d as the se ts ( X t ( X , Y ) } and { Y t ( X , Y ) } . In only one respec t does t h i s " forward pass" d i f f e r i n p r i n c i p l e from the " forward pass" requ i red f o r a l l o c a t i n g logs from a l t e r n a t i v e sou rces . The order i n which each log source i s i nc luded i n each sequence i s u n c o n s t r a i n e d , but the " forward pass" f o r a l l o c a t i o n between per iods n e c e s s a r i l y p ro -ceeds i n the sequence s p e c i f i e d by the t ime p e r i o d s . Th is sequence t e r -minates when F~,(X,Y). has been c a l c u l a t e d f o r 3 3 X = E Dx. and Y = E Dy t=l 1 t=l z and i s f o l l owed by the i d e n t i f i c a t i o n o f * 3 3 * 3 3 X + ( E D x . , . E Dy.) and Y. ( E D x . , z Dy.) zm=-\ z m = i t - t m = 1 t f o r t = 3,2,1 by a "backward p a s s . " (The magnitudes of these q u a n t i t i e s *k ie k "k "ic o and the cor respond ing x t k ( X t , Y t ) and y t k ( X t , Y t ) , i d e n t i c a l to ^ ( X . Y ) o and y t k ( X , Y ) , have been g i v e n . i n Tables I and I I ) . 94 5. Optimal A l l o c a t i o n between Per iods Values of the minimum cos t f u n c t i o n F^ .X .Y ) and the magnitudes of the a l l o c a t e d q u a n t i t i e s X t ( X , Y ) and Y t ( X , Y ) f o r the f i r s t and second-time per iod are g iven i n Tables IX and X r e s p e c t i v e l y . The v a r i a t i o n i n the range of {X,Y} between per iods r e f l e c t s the i nc rease i n the accumu-l a ted q u a n t i t i e s of logs tha t may be a l l o c a t e d from 1 to t p e r i o d s . Be-cause both Tables IX and X are abbrev ia ted not a l l q u a n t i t i e s of sawlogs tha t are a l l o c a t e d between per iods are shown. Complete t ab les are g iven i n Appendix C. Except a t the lower l i m i t , when accumulated m i l l demands are e x a c t l y s a t i s f i e d , each pa i red element of {X,Y} i nc ludes a quan t i t y of su rp lus l o g s . The de fe r red u t i l i z a t i o n of s to red logs i s the a l t e r n a -t i v e to f u l l y s a t i s f y i n g m i l l demands from the logs c u r r e n t l y d e l i v e r e d i n each p e r i o d . For the f i r s t pe r iod i t i s assumed tha t any log su rp luses from an e a r l i e r p lann ing c y c l e do not a f f e c t the q u a n t i t i e s demanded. Consequent ly , f o r t h i s t ime pe r iod m i l l demands must be f u l l y s a t i s f i e d by cu r ren t log d e l i v e r i e s . Th is i s shown by the a l l o c a t e d q u a n t i t i e s i n Table IX. However, the va lue of the f u n c t i o n F^(X,Y) i nc ludes the c o s t of " c o l d - d e c k i n g " su rp lus logs f o r use i n the second p e r i o d . B , DISCUSSION OF THE FORMULATED MODEL The o p t i m i z a t i o n model desc r ibed i n t h i s t h e s i s was developed f o r a s p e c i f i c , though h y p o t h e t i c a l , log a l l o c a t i o n problem. As suggested 95 TABLE IX OP "FINAL A!_ L Q CA T I ON E 1 . T 1 . T E ~ N P E T R T O D~ 1 AND~AlL~PASf P E R T O D S ~ . LOG PRODUCTION ALLOCATED FROP. PERIOD 1  RANGE OF P/L DEMANDS RANGE OF SAWLOG DEMANDS FOR PERIODS 1 - 2 (10-19) FOR _P ER IPDS_.l - 2 .1.0 Y= 11 Y = 12 Y = 13 Y= 14 (24 TO 44) -X- XT YT RETURN d XT YT RETURN XT YT RETURN XT YT RETURN XT YT RETURN 24 24 10 4 2 8007. 24 11 442815. 24 12 467275. 24 13 490664. 24 14 512304. 25 25 10 444007. 25 11 458815. 25 12 485818 . 25 13 512 822. 25 14 534804. 26 26 10 4 60007. 26 11 481315. 26 12 508318. 26 13 535322. 2~6~ 14 557304. 27 27 10 482507. 27 11 503815. 27 12 530818. 27 13 557822. 27 14 578283. 28 28 10 505007. 28 11 526315. 28 12 553318. 28 13 579523. 28 14 592304. 29 29 10 527507. 29 11 548815. 29 12 575818. 29 13 593164. 29 14 607804. 30 30" 10 550007. 30 i l 570315. 30 ~12~ 592275. 30 13 608664. 30 14 623304. 31 31 10 571315. 31 11 583815. 31 12 607775. 31 13 624164. 31 14 638804. 32 32 10 585007. 32 11 599315. 3 2 12 6232 75. 32 13 639664. 32 14 661304. 33 33 10 600507. 33 11 614815. 33 12 638775. 33 13 662164. 33 14 683804. 34 34 10 616007. 34 11 630315. 34 12 657318. 34 13 684322. 34 706304." 35 35 10 631507. 35 11 652815. 35 12 679818. 35 13 706822. 35 14 728804. 36 3 6 10 654007. 36 11 675315. 36 12 702318. 36 13 729322. 36 14 752304. 37 37 10 676507. 37 11 697815. 37 12 724818. 37 13 751822. 37 14 775804. 38 38 10 699007. 38 11 720315. 3 8 12 747318. 38 13 774322. 38 14 799304. 39 39 10 721507. 39 11 743815. 39 12 770818. 39 13 797822. 39 14 822804. 40 4C 10 745007. 40 11 7673 15. 40 12 794318. 40 13 821322. 41 41 10 768507. 41 11 790815. 41 12 817818. ~~2 10 792007. 42 11 814315. 43 43 10 815507. (a) The symbols "XT", "YT", and "RETURN" in this table refer to the allocated quantities X J X . Y ) , Y, (X,Y) , - and-the-MtaJmu^ — — . — J - 1  96 TABLE X OPTIMAL ALLOCATION BETWEEN PERIOD 2 AND ALL PAST PERIODS LOG PRODUCTION ALLOCATED FROM PERIOD 2 RANGE OF P/L DEMANDS -RANGE OF SAWLOG DEMANDS FOR PERIODS 2 - 3 {"19- 28) FOR PERIODS 2 - 3 Yj_JL9 Y = 20 Y = 21 Y= 22 Y = 23 (44 TO 68) -X- XT YT RETURN a XT YT RETURN XT YT RETURN XT YT RETURN XT YT RETURN 44 14 5 887304. 14 5 901766. 14 5 928769. 14 5 955773. 14 5 982776. 45 14 5 902804. 15 5 921766. 15 5 948769. 15 5 975773. 15 .5 10G2776. 46 15 5 922804. 16 5 941766. 16 5 968769. 16 5 995773. 16 5 1022776. 47 16 5 942804. 17 5 963766. 17 5 990769. 17 5 1017773. 17 5 1044776. 48 17 5 964804. 18 5 985766. 18 5 1012769. 18 5 1039773. 18 5 1066776. 49 18 5 986804. 19 5 1007766. 19 5 1034 769. 19 5 1061773. 19 5 1088776. 50 19 5 1008804. 19 5 1030266. 19 5 1057269. 19 5 1084273. 19 5 11 11276. 51 19 5 1031304. 19 5 1052766. 19 5 1079769. 19 5 1106773. 19 5 1133776. 52 19 5 1053804. 19 5 1075266. 19 5 1102269. 19 5 1129273. 19 5 1156276. 53 19 5 1076304. 19 5 1097766. 19 5 1124769. 19 5 1151773. 19 5 1178776. 54 19 5 1098804. 19 5 1121266. 19 5 114826 19 5 1175273. 19 5 1202276. 55 19 5 1122304. 19 5 1144766. 19 5 1171769. 19 5 1198773. 20 5 12 52276. 56 19 5 1145804. 19 5 1168266. 19 5 1195269. 20 c 1248773. 20 6 1276273. 57 19 5 1169304. 19 5 1191766. 20 5 1245269. 20 6 1272769. 58 19 5 1192804. 20 5 1241766. 20 6 1269266. 59 20 5 1242804. 21 5 1267266. 60 21 5 1268304. (a) The symbols "XT", "YT", and "RETURN" in this table refer to the allocated quantities X (X S Y), Y (X,Y), and the minimum cost function F 2 (X,Y) . z 97 i n Chapter I and demonstrated by the d i s c u s s i o n s i n Chapters II and I I I , dec is ion -p rob lems must be s p e c i a l l y formulated be fore they can be so lved by dynamic programming. • The a n a l y t i c framework developed f o r the hypo-t h e t i c a l case prov ides a s u i t a b l e bas i s f o r f o rmu la t i ng comparable o p t i m i z a t i o n models f o r a v a r i e t y of analogous problems. In t h i s s e c t i o n some of the advantages and l i m i t a t i o n s of the dynamic programming method of p rob lem-so lv ing are i l l u s t r a t e d by examining s e l e c t e d f ace t s of the formulated model. Th is d i s c u s s i o n w i l l enable the g e n e r a l i t y of both the formulated model and i t s conceptual bas i s to be more r e a d i l y a s s e s s e d . 1. Mandatory Assumptions The recur rence r e l a t i o n s h i p between a v a i l a b l e log s o u r c e s , Equa-t i o n [ 6 ] , page 28, repeated here f o r conven ience , f t k ( X t , Y t ) = min C c t k ( x t k , y t k ) + f ^ x t k e R l ^ t k e R 2 x t k + * t k e R 3 f o r t = 1 , 2 , 3 , k = 1 2 5 and the analogous r e l a t i o n s h i p between t ime p e r i o d s , Equat ion [ 7 ] , page 34 , F t ( X , Y ) = min [ f t 5 ( X t , Y t ) + g t ( X t , Y t ) + F t _ 1 ( X - X t , Y - Y t ) ] X t _ R 4 Y t _ R 5 X.+Y.eR, t t 6 f o r t = 1 ,2 ,3 , 98 are each based on th ree necessary c o n d i t i o n s : ( i ) a l l o c a t i o n s from d i f f e r e n t log sources ( t ime per iods) are eva luated i n a common u n i t , ( i i ) the cos t of a l l o c a t i n g logs from any source (per iod) i s not dependent upon a l l o c a t i o n s made from other sources ( p e r i o d s ) , ( i i i ) the t o t a l c o s t o f a l l o c a t i n g a quan t i t y of logs from a group of sources (pe r iods ) i s de r i ved as the sum of the cos ts of the a l l o c a t i o n s from each i n d i v i d u a l source ( p e r i o d ) . The connec t ion between the two e q u a t i o n s , which represen t the formulated model when each reg ion R. i s s p e c i f i e d , may now be examined 3 more c l o s e l y . As d i scussed i n Chapter I I , the l i n k between time per iods i s prov ided by the permi t ted log s u r p l u s e s . A l t e r n a t i v e a l l o c a t i o n s • from per iods a f f e c t the s i z e of a log s u r p l u s . A c c o r d i n g l y , the c o s t of " c o l d - d e c k i n g " su rp lus logs i s dependent upon the magnitude of the a l l o c a t e d q u a n t i t i e s X^. and and i s one of the components of the t o t a l log a l l o c a t i o n c o s t . Th is a d d i t i o n a l c o s t , de f ined as g ^ X ^ . Y ^ ) , i s par t of the formula ted model and i s inc luded i n the second o p t i m i z a -t i o n process represented by Equat ion [ 7 ] . The connect ion between the two cos t m in im iza t i on equat ions i s emphasized i f the cos t f u n c t i o n , g t ( X ^ . , Y t ) , i s i nc luded w i th the minimum cos t f u n c t i o n , f ^ X * . , Y t ) , as h t ( X f V = V X t » ^ t ( V t > : and Equat ion [7] i s r es ta ted as F t ( X , Y ) = min [ h ^ X ^ Y ^ + F ^ - , ( X - X t , Y - Y t ) ] X^- e R 4 Y t £ R 5 X t + Y t e R 6 f o r t = 1 , 2 , 3 . Accep tab le v a r i a t i o n s i n the d e r i v a t i o n of the cos t components inc luded i n the model which do not contravene assumptions ( i ) , ( i i ) and ( i i i ) have been i l l u s t r a t e d i n Chapter I I I . A review of the net d e l i v -ered log c o s t f u n c t i o n s , c ^ ( < ( x t k ' - ^ t k ^ ' W 1 ' ^ c o n f i r m tha t the r e q u i r e -ments of dynamic programming are s a t i s f i e d i n a l l c a s e s . The e v a l u a t i o n , of log a l l o c a t i o n s made from source two i s of p a r t i c u l a r i n t e r e s t as an example of how a r e l a t i v e l y complex c o s t s t r u c t u r e may be accep tab ly fo rmu la ted . As d e s c r i b e d , ex tens i ve use i s made of r a t i o s based on ly on the a l l o c a t e d q u a n t i t i e s , x^ and y ^ , to de f i ne i n d i v i d u a l volume components. A major group o f these components i s eva luated as a com-pos i t e cos t f u n c t i o n , de f ined as z t ^ x t k ' y t k ^ ' T l i e e v a ^ u a t i o n of t h i s f u n c t i o n requ i res i d e n t i f y i n g the minimum cos t of a s e r i e s of a l t e r n a t i v e log s o r t i n g and t rad ing p o l i c i e s . Consequent ly , a l though the mathe-mat i ca l development i n v o l v e s the c o n s i d e r a t i o n of a range of a l t e r n a t i v e cos t e v a l u a t i o n s , the f u n c t i o n c t | < ( x t k ' y t k ^ 1 S always s p e c i f i c a l l y de f ined f o r each a l l o c a t i o n and i s not dependent on other a l l o c a t i o n s . A p o s s i b l e c o n f l i c t w i th t h i s requ i red independence may have developed wi th respec t to log t r a d i n g . An i d e n t i c a l source of logs was used to rep lace t raded logs as we l l as supp ly ing q u a n t i t i e s of purchased logs ("market" logs from source f o u r ) . However, the d e s c r i p t i o n of the 100 a l l o c a t i o n problem, i n which a maximum quan t i t y of logs f o r purchase i s s p e c i f i e d , e x p l i c i t l y excludes logs used f o r t r ad ing from t h i s l i m i t a t i o n . 2. M o d i f i c a t i o n of the Model In a d d i t i o n to d e v e l o p i n g , when r e q u i r e d , more d e t a i l e d d e r i v a -t i ons of i n d i v i d u a l cos t components there are three general ways i n which the formula ted model may be m o d i f i e d . . The f i r s t i s the expansion of the model by i n c r e a s i n g the number of s tages i n the a l l o c a t i o n p rocess . The fo rmu la t i on desc r ibed requ i res the examinat ion of f i v e sources f o r each t ime p e r i o d , e f f e c t i v e l y a t o t a l of f i f t e e n s o u r c e s , and the examinat ion of each per iod to g i ve a combined t o t a l of e igh teen s tages . A d d i t i o n a l stages of any des i r ed length may r e a d i l y be i nc luded i n the model ; They would be r e q u i r e d , f o r example, i f a more d e t a i l e d log a l l o c a t i o n p o l i c y i n which monthly or weekly log p roduc t ion p lann ing was necessary . Due to the r e c u r s i v e method of computa t ion , which requ i res the r e t e n t i o n on ly of opt imal a l l o c a t i o n s a t each s t a g e , the number of stages tha t may be added w i l l not normal ly be r e s t r i c t e d by computer s to rage c a p a c i t y . The i nc rease i n s torage requ i rements , approx imate ly p ropo r t i ona l to the i nc rease i n the number of s t a g e s , may be m in im ized , i f necessa ry , by c a l c u l a t i n g and p r i n t i n g out on ly the r e s u l t s . o f each " forward p a s s . " The requ i red "backward passes" can be made e i t h e r manual ly from the app rop r ia te se t of p o l i c y t ab les o r , w i th recourse to a v a i l a b l e "back-up" s t o r a g e , i n -c luded i n a segmented se t of programming i n s t r u c t i o n s . 101 A second way i n which the model may be mod i f ied i s i l l u s t r a t e d by c o n s i d e r i n g the opt imal a l l o c a t i o n not on ly of the demanded volumes of pu lp logs and sawlogs but a l s o of a quan t i t y of a t h i r d category of l o g s . The a d d i t i o n of a t h i r d " d e c i s i o n - v a r i a b l e , " Z , may be r e q u i r e d , f o r example, when the log demands of a veneer p l an t are o p t i m a l l y a l l o c a t e d together w i th the demands f o r pu lp logs and saw logs . -For t h i s requirement the cor responding ve rs ions of Equat ions [6] and [7] would be W W = m i n [ S k ( x t k ' ^ and F t ( X , Y , Z ) = m i n ( f t 5 ( X t , Y t , Z t ) + F t _ - ( X - X t , Y - Y t , Z - Z t ) ] w i th s u i t a b l e d e f i n i t i o n of a more complex s e t of f e a s i b l e r e g i o n s . T h e o r e t i c a l l y , the number of d e c i s i o n - v a r i a b l e s tha t may be i nc luded i n t h i s way i n the model i s u n l i m i t e d . However, because the a d d i t i o n of a d e c i s i o n - v a r i a b l e en la rges the s i z e o f each of the generated mat r i ces by one d imens ion , computer s to rage r e q u i r e m e n t s . r a p i d l y i n c r e a s e . Th is i nc rease i s an e f f e c t i v e de te r ren t to the i n c l u s i o n of more than three or fou r d e c i s i o n - v a r i a b l e s when fo rmu la t i ng a problem f o r s o l u t i o n by dynamic programming. Al though Bel lman (1967) has shown how a reduc t i on i n d i m e n s i o n a l i t y may be achieved by us ing s p e c i a l i z e d mathematical t echn iques , the "Curse of D i m e n s i o n a l i t y " i s one of the major l i m i t a t i o n s of dynamic programming. A t h i r d m o d i f i c a t i o n tha t may be d e s i r e d i s a reduc t i on i n the 102 s i z e of the volume un i t s a l l o c a t e d . In the formulated model the minimum permi t ted v a r i a t i o n i n an a l l o c a t e d quan t i t y of logs i s one thousand c u n i t s . The adequacy of opt imal log a l l o c a t i o n p o l i c i e s based on volume un i t s of t h i s magnitude i s dependent upon the d e t a i l r equ i red f o r log p roduc t ion p l a n n i n g . A s m a l l e r minimum s i z e i s almost mandatory i f a d d i t i o n a l t ime per iods are inc luded i n the annual p lann ing c y c l e r e p r e -sented by the model . - When more accura te p o l i c i e s are r e q u i r e d , two methods are p o s s i b l e . The f i r s t s imply i nc ludes a s m a l l e r volume u n i t i n an expanded " g r i d . " Th is method i s l i m i t e d due to the inc reased s torage requirements necessary f o r the en larged m a t r i c e s . The second method uses a "coarse g r i d " s o l u t i o n as the i n i t i a l a l l o c a t i o n f o r a more d e t a i l e d search procedure. Th is more genera l procedure depends upon the development of s u i t a b l e search procedures f o r i d e n t i f y i n g a more accura te opt imal s o l u t i o n based on the requ i red s i z e o f g r i d . 3 . S e n s i t i v i t y and E r ro r Analyses Bel lman (1967) i n h i s d i s c u s s i o n of the use fu lness of a computa-t i o n a l system commented, I t i s g e n e r a l l y t rue tha t i n an i n v e s t i g a t i o n o f a p h y s i c a l sys tem, we are not content w i t h . t h e de te rmina t ion of the opt imal behaviour of the system f o r any s i n g l e s e t of parameter v a l u e s . Ra the r , we wish to a l l ow the parameters to vary over a c r i t i c a l range of v a l u e s , and then to observe how the opt imal p o l i c y i s a f f e c t e d by these changes. I t i s by observ ing the change i n the s t r u c t u r e of these p o l i c i e s as the parameters change tha t we ga in the most v i t a l i n f o r m a t i o n . In the f i r s t s e c t i o n of t h i s chapter the d i s c u s s i o n of the s o l u -t i o n to the formula ted problem i l l u s t r a t e d how dynamic programming 103 a u t o m a t i c a l l y imbeds a s p e c i f i c problem w i t h i n a s e r i e s of analogous problems. As pa r t o f . t h e computat ional process the q u a n t i t i e s of logs a l l o c a t e d a t each s tage vary over a wide range. With opt imal p o l i c i e s i d e n t i f i e d and eva luated f o r each of these q u a n t i t i e s the manager has the " v i t a l i n f o rma t i on " r e f e r r e d to by Be l lman. Of perhaps g rea te r impor tance, however, i s an i n d i c a t i o n o f the s e n s i t i v i t y of the opt imal s o l u t i o n . What i s the necessary change i n the magnitude of a c o n s t r a i n t before the opt imal p o l i c y changes? What i s the e f f e c t o f a change i n a s p e c i f i c u n i t p r i c e or c o s t i tem on the opt imal a l l o c a t i o n p o l i c y ? The answer to t h i s type of ques t ion may be necessary before implementing an i n d i c a t e d a l l o c a t i o n p o l i c y i s even cons idered b y - r e s p o n s i b l e Management. To ob ta in the needed in fo rma t ion a s p e c i f i c s e n s i t i v i t y a n a l y s i s i s requ i red i n which parameters of i n t e r e s t are v a r i e d i n t u r n , and i n comb ina t ion , and i n d i c a t e d changes i n opt imal p o l i c i e s no ted . The number and type of parameters s e l e c t e d depend upon t h e i r s i g n i f i c a n c e and the degree of conf idence p laced on t h e i r assumed magnitude or v a l u e . In an ope ra t i ona l a p p l i c a t i o n , i t i s most l i k e l y tha t the ex ten t of a s e n s i t i v i t y a n a l y s i s would a l s o depend upon the a c c e p t a b i l i t y of an i n d i c a t e d opt imal p o l i c y to>the respon-s i b l e dec i s i on -make r . A log p roduc t ion schedule de r i ved from a "magic box" which r a d i c a l l y d i f f e r s from c o n v e n t i o n a l l y - d e r i v e d schedules w i l l c e r t a i n l y be s c r u t i n i z e d m o s t . c l o s e l y . U n f o r t u n a t e l y , f o r the hypo the t i ca l a l l o c a t i o n problem of t h i s t h e s i s , a t h e o r e t i c a l s e n s i t i v i t y a n a l y s i s of the i n d i c a t e d opt imal p o l i c i e s would be of l i m i t e d i n t e r e s t . Based on i l l u s t r a t i v e data the 104 a d d i t i o n a l s p e c i f i c a t i o n of the degree of r e l i a b i l i t y of each parameter would be spur ious and the development o f a bas i s f o r s e l e c t i n g s i g n i f i -cant parameters f o r a n a l y s i s i r r e l e v a n t to the main d i s c u s s i o n . Reference has been made i n t h i s s e c t i o n to computat ional l i m i -t a t i o n s w i th respec t to the s torage c a p a c i t y of computer memory. Due to d i f f e r e n c e s between va r ious e l e c t r o n i c computer i n s t a l l a t i o n s no attempt has been made to quan t i f y these r e s t r i c t i o n s . They are depen-den t , not only on the s p e c i f i c computing dev ice and the p a r t i c u l a r f o r -mu la t ion o f the a l l o c a t i o n mode l , but a l s o upon the computer code used f o r programming i n s t r u c t i o n s and the i ngenu i t y o f the programmer.' A s i m i l a r d i f f i c u l t y i n s p e c i f i c a t i o n i s encountered when d i s c u s s i n g the computat ional e r r o r s assoc i a t ed w i th the formula ted model . The problem of " s t a b i l i t y " which may a r i s e when d i s c r e t e se ts of va lues are used to approximate a cont inuous f u n c t i o n has been by-passed i n the f o r m u l a t i o n . The p h y s i c a l problem was desc r ibed o r i g i n a l l y i n d i s c r e t e terms and a l l a l l o c a t i o n s are i n t e g r a l m u l t i p l e s of a s p e c i f i e d volume u n i t of one thousand c u n i t s . The inpu t data was assumed to be p r e c i s e l y known and a l l f o r e c a s t s were assumed c o r r e c t f o r p lann ing purposes. Numerical e r r o r s , t h e r e f o r e , are e s s e n t i a l l y those due to t r u n c a t i o n and round ing-o f f . The magnitude of both of these types of e r r o r are dependent upon the computing system a v a i l a b l e and the sequence i n which mathematical man ipu la t ions are made. As pa r t of t h i s t h e s i s , DYALOG, a computer coded l i s t o f programming i n s t r u c t i o n s f o r an IBM 360, Model 67 , was prepared to v e r i f y the computat ional f e a s i b i l i t y of the formula ted model . (A copy of t h i s programme i s g iven i n Appendix B ) . 105 A necessary pa r t of the development o f DYALOG inc luded the v e r i -f i c a t i o n of each of the computat ions necessary to eva lua te each log a l l o c a t i o n p o l i c y . A review of each t e s t would be of e x c e p t i o n a l l y l i m i t e d va lue and would i n d i c a t e only the w e l l - r e c o g n i z e d f a c t tha t both t r u n c a t i o n and round-o f f e r r o r do e x i s t . However, as an example of the computat ional p r e c i s i o n a t t a i n e d by a p a r t i c u l a r s e t of programming i n s t r u c t i o n s , coded i n FORTRAN IV f o r an IBM 360, Model 67 computer i n -s t a l l e d i n the Computing Centre a t the U n i v e r s i t y of B r i t i s h Co lumb ia , the f o l l o w i n g b r i e f a n a l y s i s i s g i ven i n Table X I . The computat ion s e l -ected i s shown i n F igure 7 , page 50 , and i s requ i red f o r the d e r i v a t i o n o f the re tu rn component " d i f f e r e n t i a l sawlog net wor th" f o r source two. As d i scussed i n Chapter I I I , page 6 0 , and i n g rea te r d e t a i l i n Appendix A , pages 116 to 127, the e v a l u a t i o n of the term c w j . ( x t - , y t 2 j y c , t r ) r e -qu i res a s e r i e s of de r i ved volume components to be combined before the magnitudes of the average d iamete r , D t 2 . and average content of Douglas-f i r l o g s , P t 2 . can be d e r i v e d . For i l l u s t r a t i v e purposes o n l y , a spe-c i a l computer run was made i n which a l l c a l c u l a t i o n s were made i n " s i n g l e -p r e c i s i o n " a r i t h m e t i c . (For c a l c u l a t i o n s of t h i s type the use of " d o u b l e - p r e c i s i o n " a r i t h m e t i c i s a s tandard recommendation by Centre s p e c i a l i s t s ) . The i n fo rma t ion g iven i n Table XI permits comparison be-tween three se ts o f f i g u r e s . The f i r s t se t l i s t e d con ta ins s i x t e e n s i g n i f i c a n t f i g u r e s . These were c a l c u l a t e d manual ly w i th a desk c a l -c u l a t o r . The two se ts of computer -der ived f i g u r e s r e f e r to c a l c u l a t i o n s , made w i th " s i n g l e - p r e c i s i o n " a r i t h m e t i c and " d o u b l e - p r e c i s i o n " a r i t h -m e t i c , r e s p e c t i v e l y . D e t a i l s of the manner i n which the computat ions TABLE XI PRECISION OF THE MAGNITUDE OR VALUE COMPUTED BY "DYALOG" FOR SELECTED VARIABLES AFFECTED BY LOG SORTING AND TRADING POLICIES FOR SOURCE TWO • Log q u a n t i t i e s a l l o c a t e d ; x ^ 2 = 3 M c u n i t s , y t 2 = 2 M c u n i t s y c (M c u n i t s ) 3 t r ( r a t i o ) ' 3 S e l e c t e d V a r i a b l e s D e r i v e d Magn i tude or V a l u e T rue C a l c u l a t i o n Computed by "DYALOG" S i n g l e P r e c i s i o n Doub le P r e c i s i o n 1.95 .00 D^2 ( i n c h e s ) P t 2 ( r a t i o ) d c w | ( d o l l a r s ) e 1 2 . 0 0 2 -50 .857 250 2 2 9 4 . 3 6 4 090 814 413 (6 ) 12 .002 62 .857 245 9 2294,419 1 2 . 0 0 2 500 000 000 03 .857 249 999 999 998 8 2 2 9 4 . 3 6 4 090 814 428 1.55 .09 D t 2  P t 2 1 3 . 3 5 2 50 .807 256 2412 .267 740 123 572 (3) ( S i n g l e p r e c i s i o n run made o n l y f o r t r = 0) 1 3 . 3 5 2 499 999 999 99 .807 249 999 999 999 7 2412 .267 740 123 557 1.55 .00 D T -P t 2 1 3 . 8 0 2 50 . . . .807 250 3026 .169 534 504 618 (7) 13 .802 49 . .807 249 5 3026 .147 1 3 . 8 0 2 499 999 999 99 .807 249 999 999 999 9 3026 .169 534 504 610 a Q u a n t i t y o f s a w l o g s s e l e c t e d f o r n o n - p e e l e r c l a s s e s . -^ P o r t i o n o f x ^ + y t 2 t r a d e d f o r s a w l o g s . c D t 2 ~ a v e r a g e d i a m e t e r o f the d e l i v e r e d q u a n t i t y y t 2 > d e r i v e d as d e s c r i b e d i n Append i x A . C l p t 2 = a v e r a 9 e D o u g l a s - f i r c o n t e n t o f t he d e l i v e r e d q u a n t i t y , y t 2 , d e r i v e d as d e s c r i b e d i n Append i x A . e c w ! = c w ! ( x . - , y . y c . t r ) = d i f f e r e n t i a l s a w l o g ne t w o r t h f o r l o g s o r t i n g & t r a d i n g p o l i c y denoted by q u a n t i t y y c & r a t i o t r , 1 * t _ - t . = ( ( P t - - . 8 ) ( w p t - w n t ) - 2 8 8 0 ( 4 3 . 6 5 ) ( l / D t 2 - l / 1 2 . 0 ) ) . y t 2 . o G. 107 were made are g iven i n Chapter I I I , i n F igu re 7 , and a l s o i n DYALOG (Appendix B ) . By i n s p e c t i o n i t may be deduced tha t the magnitude of the -15 numerical e r r o r in t roduced a t t h i s s tage i s of the order o f 0 .5 ± 10 when d o u b l e - p r e c i s i o n a r i t h m e t i c i s used. When the magnitude of the t o t a l cos t of a l l o c a t i n g the demanded q u a n t i t i e s X and Y i s • r e q u i r e d to the neares t d o l l a r the necessary p r e c i s i o n i s of the order of 1 0 " ^ . Consequent ly there i s cons ide rab le l a t i t u d e f o r a more d e t a i l e d fo rmu la -t i o n and , f o r a l l p r a c t i c a l purposes , the numerical e v a l u a t i o n may be cons idered e r r o r - f r e e . CHAPTER V CONCLUSIONS Th is t h e s i s has desc r ibed an op t im i z i ng model f o r the a l l o c a t i o n of logs f o r a hypo the t i ca l s i t u a t i o n . The problem requ i red the a l l o c a -t i o n of q u a n t i t i e s of pulp logs and saw logs , demanded f o r three time p e r i o d s , from f i v e a v a i l a b l e sou rces . An a n a l y t i c framework was d e v e l -oped by fo rmu la t i ng t h i s a l l o c a t i o n problem as two connected m u l t i - s t a g e d e c i s i o n processes i n which a stage represented e i t h e r a t i m e . p e r i o d or a source of l o g s . Each of these a l l o c a t i o n processes was expressed as a recur rence e q u a t i o n , so l ved by dynamic programming. Both the f e a s i b l e reg ions and the economic o b j e c t i v e i nco rpo ra ted i n each of these equat ions were n e c e s s a r i l y d i c t a t e d by the cond i t i ons of the s p e c i f i e d problem. D i f f e r e n t ways i n which these r e l a t i o n s h i p s cou ld be mod i f ied to accommodate a wide v a r i e t y of log a l l o c a t i o n problems were d i scussed and examined. The major f ea tu re o f t h i s method of f o rmu la t i on i s the a c c e p t a -b i l i t y of n o n - l i n e a r r e l a t i o n s h i p s when a model i s designed to be so lved by dynamic programming. Th is c h a r a c t e r i s t i c i s p a r t i c u l a r l y impor tant when log p roduc t ion cos ts f o r a l t e r n a t i v e sources are compared; Sub-s t a n t i a l c a p i t a l expend i tures g e n e r a l l y accompany the i n i t i a l d e v e l o p -ment o f a wood-harves t ing programme on a log p roduc t ion a r e a . F r e q u e n t l y , i n a d d i t i o n to these " s t a r t - u p " c o s t s , f u r t h e r c a p i t a l ou t lays are i n t e r m i t t e n t l y needed as p roduc t ion i n c r e a s e s . I f , i n order to ob ta i n 109 "cons tan t re tu rns to s c a l e , " a " s t e p - f u n c t i o n " i s approximated by a l i n e a r r e l a t i o n s h i p , the ensuing d i s t o r t i o n may c rea te d i f f i c u l t i e s i n cos t compar isons . • To emphasize the f a c i l i t y w i th which n o n - l i n e a r f unc t i ons may be i n c o r p o r a t e d , a wide v a r i e t y of cos t components were in t roduced f o r the d e r i v a t i o n of the "net d e l i v e r e d log c o s t " f unc t i ons used i n the t h e s i s . They i nc luded cons tan t u n i t log c o s t s , assumed i n each t ime per iod f o r source f o u r , and ranged to the complex cos t s t r u c t u r e developed f o r source two. For t h i s l a t t e r s o u r c e , the a v a i l a b l e wood supply was assumed to be a composite of va r ious log c l a s s e s . The "net d e l i v e r e d log c o s t " of these l o g s . i n c l u d e d groups of cos ts a f f e c t e d by log s o r t i n g and . t r ad ing p o l i c i e s . For each p a i r of q u a n t i t i e s a l l o c a t e d , an opt imal log s o r t i n g and t r a d i n g p o l i c y was i d e n t i f i e d , the dependent cos t components e v a l u a t e d , and a composite cos t d e r i v e d . Th is p a r t i c u l a r d e r i v a t i o n we l l i l l u s t r a t e d the manner i n which a d e t a i l e d c o s t s t r u c t u r e can be formula ted i n accordance w i th the r e -quirements of dynamic programming. One of the necessary cond i t i ons f o r s o l v i n g a l l o c a t i o n problems by t h i s method i s tha t the e v a l u a t i o n of an a l l o c a t i o n from a p a r t i c u l a r source i s dependent on ly upon the q u a n t i -t i e s a l l o c a t e d and i s not a f f e c t e d by a l l o c a t i o n s from other s o u r c e s . Because of t h i s c o n d i t i o n , on ly those c o s t , o r r e t u r n , components i n a p roduc t ion cha in t ha t are a f f e c t e d by a l l o c a t i o n d e c i s i o n s need be i d e n t i f i e d f o r i n c l u s i o n i n a " s t r i n g " o f dependent cos ts or b e n e f i t s . The amalgamation of i n d i v i d u a l components can b e , u t i l i z e d to d e r i v e " second-o rde r " o p t i m i z a t i o n s , as i l l u s t r a t e d by the procedure 110 used to d e r i v e log cos ts f o r source two. An opt imal s o r t i n g . a n d t r ad i ng p o l i c y f o r each log a l l o c a t i o n from t h i s source was de r i ved by m in im i z -ing the composite sum of on ly r e l e v a n t cos ts and r e t u r n s . Having de-r i v e d an opt imal log m ix , the remaining elements i n the p roduc t ion cha in were then i n c l u d e d . In t h i s way, a complex a n a l y t i c a l problem was so lved d i r e c t l y . A f u r t h e r s i m p l i f i c a t i o n i s p o s s i b l e i f the s ta ted o b j e c t i v e i s p r i m a r i l y to i d e n t i f y , r a t h e r than to e v a l u a t e , an opt imal p o l i c y . When t h i s i s the c a s e , r e l a t i v e magnitudes only of a l l oca t i on -dependen t c o s t s , or r e t u r n s , w i l l s u f f i c e . Th is concept was i l l u s t r a t e d by the re tu rn component, " d i f f e r e n t i a l sawlog net w o r t h , " which e s s e n t i a l l y i s a measure of the d i f f e r e n c e between the r e s i d u a l va lues of d e l i v e r e d and " s tanda rd " sawlogs . In a d d i t i o n to these t e c h n i c a l advantages, the computat ional method by which the formulated model i s so l ved generates s u b s i d i a r y i n -fo rmat ion of cons i de rab le ope ra t i ona l va lue to the p lanne r . Sets of opt imal s u b - p o l i c i e s are generated by the " forward pass" as each stage i s i nc luded i n the d e c i s i o n - p r o c e s s . By r e t a i n i n g these s u b - p o l i c i e s , the necessary i n fo rma t ion i s a v a i l a b l e to d e r i v e opt imal p o l i c i e s i f demand schedules are changed. In a d d i t i o n , as demonstrated w i th the output t ab les generated by the computer-coded programme, DYALOG, i f unforeseen changes i n log p roduc t ion c a p a b i l i t i e s o c c u r , the l e a s t c o s t l y depar ture from a planned opt imal p o l i c y can be r e a d i l y i d e n t i f i e d . The mathematical model represented by DYALOG may be used i n a number of ways. I t can be adapted f o r problems analogous to the in hypo the t i ca l case and used f o r sho r t - t e rm log a l l o c a t i o n . - DYALOG has been designed to permi t m o d i f i c a t i o n s to be r e a d i l y i nco rpo ra ted and i s reasonably e f f i c i e n t ( the present v e r s i o n of DYALOG r e q u i r e s l e s s than one hundred seconds of machine computat ion time when processed by an IBM 360, Model 67 sys tem) . I f p lann ing requirements extend over a longer term the model cou ld be used r e p e t i t i v e l y , w i th the app rop r ia te se ts of p lann ing i n f o r m a t i o n , to d e r i v e a s e r i e s of p e r i o d i c minimum c o s t a l l o c a t i o n s f o r each year i n a p lann ing c y c l e . Each s e t of " s h o r t -term" minimum cos t a l l o c a t i o n s may then be cons idered as a s i n g l e " s t a g e " i n a long- te rm m u l t i - s t a g e d e c i s i o n process f o r o v e r a l l op t im-i z a t i o n . A t h i r d a p p l i c a t i o n would use the developed model as a sub-model i n a more comprehensive system. In t h i s case the prime purpose of o p t i m i z a t i o n would be to de r i ve pa r t of the inpu t i n fo rma t ion requ i red by the major system. Conve rse l y , the developed model i t s e l f may be used as the framework f o r connect ing a s e r i e s of d e t a i l e d sub-sys tems. For each of these p o s s i b i l i t i e s , the a n a l y t i c framework developed i n t h i s t h e s i s g i ves a f l e x i b l e computat ional system t o i d e n t i f y opt imal d e c i s i o n - r u l e s f o r complex log a l l o c a t i o n p rocesses . However, the u l t ima te worth of t h i s concep t , i n any a p p l i c a t i o n , depends upon the awareness, r e s o u r c e f u l n e s s , and i ngenu i t y of the opera t ions p lanne r . LITERATURE CITED LITERATURE CITED Ar imuzu , T. "Regu la t i on of the cut by dynamic programming," Journa l of  Japanese Operat ions Res. 1, 1958. Be l lman , R. Dynamic Programming. P r i n c e t o n : P r i nce ton U n i v e r s i t y P r e s s , 1957. C a r l s s o n , B. Rout ines f o r sho r t - range p lann ing of l ogg ing o p e r a t i o n s . B u l l e t i n No. 5. Skogsarbe ten , S tockho lm, 1968. C u r t i s , F. H. " L i n e a r programming the management of f o r e s t p r o p e r t y , " Journa l of F o r e s t r y , 6 0 ( 9 ) , 1962. D o n n e l l y , R. H. L i nea r programming i n plywood manufac tu r ing . Proceed-ings of seminar on opera t ions research i n the f o r e s t i n d u s t r i e s . San F r a n c i s c o , 1965. Dowdle, B. Log a l l o c a t i o n d e c i s i o n making. Proceedings of seminar on opera t ions research i n the f o r e s t i n d u s t r i e s . San F r a n c i s c o , 1965. H o o l , J . N. "A dynamic programming - Markov cha in approach to f o r e s t p roduc t ion c o n t r o l , " Fo res t Sc ience Monograph 12, 1966. K i d d , W. E . , E. F. Thompson, and P. H. Loepner. " F o r e s t r e g u l a t i o n by l i n e a r programming - a case s t u d y , " Journa l of F o r e s t r y , 6 4 ( 9 ) , 1966. : Lonner, G. A system f o r sho r t - t e rm p lann ing of l o g g i n g , s t o r i n g , and  t r a n s p o r t a t i o n of wood. B u l l e t i n No. 6. Skogsarbe ten , S tockho lm, 1968: Loucks , D. F. "The development of an opt imal programme f o r sus ta ined y i e l d management," Journa l o f F o r e s t r y , 6 2 ( 7 ) , 1965. N a u t i y a l , J . C. and P. H. Pearse . "Op t im iz ing the convers ion to s u s -ta ined y i e l d - a programming s o l u t i o n , " Fo res t S c i e n c e , 13 (2 ) , 1967. P e a r s e , P: H. and S . Sydneysmith. "Method f o r a l l o c a t i n g logs among severa l u t i l i z a t i o n p r o c e s s e s , " Fores t Products J o u r n a l , 1 6 ( 9 ) , 1966. : 114 Schreuder , G. F. Optimal f o r e s t investment d e c i s i o n s through dynamic programming. B u l l e t i n No. 72. School of F o r e s t r y , Ya le U n i v e r s i t y , 1968. Thompson, E. F. and D. P. R i c h a r d s . "Us ing l i n e a r programming t o .deve lop l ong - t e rm , l e a s t cos t wood procurement s c h e d u l e s , " Pulp and Paper  Magazine of Canada. Convent ion i s s u e , 1969. Wo l f e , R. K. and D . M . Ba tes . "Resource a l l o c a t i o n model f o r a hypo-t h e t i c a l paper company," T a p p i , 51 (11 ) , 1968. APPENDICES APPENDIX A 117 APPENDIX A COSTS OF LOG SORTING, LOADING AND TRANSPORTING AND DIFFERENTIAL SAWLOG NET WORTH FOR SOURCE TWO Each of these components i s dependent upon the log s o r t i n g and t r ad ing p o l i c y denoted by the quan t i t y yc and the r a t i o t r . As shown i n Chapter I I I , pages 60 to 6 4 , the e v a l u a t i o n of c ^ 2 ^ x t 2 ' ^ t 2 ^ r e ~ qu i res m in im iz ing a composite f u n c t i o n , ^ ( x ^ - . y ^ ) -Th is Appendix g i ves the d e r i v a t i o n of each component of t h i s f u n c t i o n . S o r t i n g and Loading Costs S o r t i n g i nvo l ves the s e l e c t i o n of a quan t i t y of pee le r l o g s . f o r t r ad ing and the s e l e c t i o n o f a sawlog component from four non-pee le r log c l a s s e s i n accordance w i th pre-determined p r i o r i t i e s . L e t , cp and cn = v a r i a b l e c o s t of l oad ing per thousand c u n i t s of pee le r logs and non-pee le r l o g s , r e s p e c t i v e l y , ess = v a r i a b l e cos t per thousand c u n i t s f o r log s e l e c t i o n and temporary s t o c k - p i l i n g a t woods l a n d i n g s . The v a r i a b l e cos ts o f load ing the q u a n t i t i e s , x^ and y ^ . i s assumed to depend.upon the r e l a t i v e volumes of pee le r and non-pee le r l o g s . The cos ts shown i n F igure 15 were de r i ved^ f o r each of the i n d i c a t e d magnitudes f o r the content of pee le r l o g s , from the exp ress ion \ 118 volume loaded i n thousands of c u n i t s F i g . 15 Combined Cost of Loading Pee le r and Non-Peeler Logs 119 p c . ( x t 2 + y t - ) . c p + ^ Q j . c n . The combined c o s t of s e l e c t i n g the pee le r log components ( i tems i i and i v on page 53) f o r each s o r t i n g and t rad ing p o l i c y , when the magnitude of the r a t i o pc = . 0 9 , i s shown i n F igure 16. V a r i a b l e cos ts were de r i ved from the exp ress ion { ( t r + p c ) ( x t 2 + y t 2 ) - y t 2 + y c } . c s s . Because sawlog s o r t i n g i s assumed to r e q u i r e the s e l e c t i o n of the s m a l l e r of the two q u a n t i t i e s i n v o l v e d , the t o t a l cos t o f s o r t i n g the quan t i t y y c , shown i n F igure 17, i s de f ined as 4 min[ E Q j -yc ; y t 2 - t r . ( x t 2 + y t 2 ) ] . e s s . j= l The combined sum of these c o s t s , expressed as a f u n c t i o n of x t 2 ' y t 2 ' y c a n c '  z r ' 1 S de f ined as c s t ( x t 2 , y t 2 , y c , t r ) E Q j . c n + p c . ( x t 2 + y t 2 ) . c p + ( t r + p c ) ( x t 2 + y t 2 ) - y t 2 + y c 4 +min[_E^Q J . -yc ; y t 2 + t r . ( x + 9 + y + 9 ) ] • e s s , t 2 , J , t 2 ' where y t 2 f 0. z Qj . c n + p c . x ^ . (cp+css) , where y t 2 = 0 . F i g . 16 Cost of S e l e c t i n g the Quan t i t y of P e e l e r Logs Required f o r Trad ing 121 F i g . 17 ' C o s t per.Thousand Cuni ts . of S o r t i n g Non-Traded Logs 122 Log T ranspo r ta t i on Costs The t o t a l u n i t cos t of t r a n s p o r t i n g the q u a n t i t i e s x t 2 and y t 2 to the s e p a r a t e l y l oca ted m i l l yards i s a f f e c t e d by t h e - r e l a t i v e magnitudes of pu lp logs and sawlogs . The inc reased hand l ing cos ts i ncu r red by a l l t raded q u a n t i t i e s i s reduced by the pre-determined p r i c e d i f f e r e n c e between p e e l e r logs and replacement l o g s . L e t , c t x ^ and c t y t - v a r i a b l e cos t per thousand cun i t s of t r a n s p o r t i n g logs d i r e c t l y from source two to the p u l p m i l l and s a w m i l l , r e s p e c t i v e l y , i n pe r iod t , c t t ^ = v a r i a b l e cos t per thousand c u n i t s i ncu r red i n pe r iod t when logs are t ranspor ted from source two to the log mar-ket and , a f t e r exchange, an equal log volume i s d e l i v e r e d to e i t h e r p u l p m i l l or s a w m i l l , t c = f i x e d p r i c e d i f f e r e n c e per thousand c u n i t s between t raded p e e l e r logs and an equal volume of replacement l o g s . The v a r i a b l e c o s t o f t r a n s p o r t i n g the q u a n t i t y x ^ ( i tems i and i i on page 5 3 ) , f o r a p a r t i c u l a r log t r ad ing and s o r t i n g p o l i c y , i s de r i ved from 4 ( _ E ^ Q j - y c ) . c t x t + { p c . ( x t 2 + y t 2 ) - y t - + y c } . c t t t and the cor responding c o s t f o r the q u a n t i t y y t 2 , ( the sum of the combined components i i i , i v , and v on page 5 3 ) , i s found from { y t 2 - t r . ( x t 2 + y t 2 ) > - c t y t + t r . ( x t 2 + y t 2 ) . c t t t . 123 The p a r t i a l l y o f f s e t t i n g re tu rn from the q u a n t i t i e s t raded (volume components i i and i v on page 53) reduces the t r a n s p o r t a t i o n cos t by { ( t r + p c ) ( x t 2 + y t - ) - y t - + y c } . t c . When these items are combined the v a r i a b l e t r a n s p o r t a t i o n cos ts f o r the a l l o c a t i o n of the quan t i t y x ^ 2 and y ^ 2 , f o r a t r ad i ng and s o r t -ing p o l i c y denoted by the quan t i t y yc and the r a t i o t r , are de f ined as c t t ( x t 2 , y t 2 , y c , t r ) (_z Q j - y c ) . c t x t + { y t 2 - t r . ( x t 2 + y t 2 ) } . c t y t + { ( t r + p c ) ( x t 2 + y t 2 ) - y t 2 + y c } ( c t t t - t c ) , where y t 2 f 0 , I Q . . c t x . + p c . x . - . . ( c t t . - t c ) .1=1 where Hi D i f f e r e n t i a l Sawlog Net Worth Sawlog net worth depends upon average log d iameter and spec ies content o f the quan t i t y o f d e l i v e r e d sawlogs . As shown i n F igure 8 , page 54, a quan t i t y of sawlogs d e l i v e r e d from source two con ta ins three component volumes. The r e l a t i v e p ropor t ions - of these q u a n t i t i e s , which depend upon the log s o r t i n g and t r ad ing p o l i c y , determine the average log d iameter and spec ies mix o f the d e l i v e r e d volume y t 2 . Hence d i f f e r -e n t i a l sawlog net worth i s dependent upon the s o r t i n g and. t r ad ing p o l i c y . 124 L e t , dp = average log d iamete r , i n i n c h e s , of pee le r l o g s , pp = r a t i o o f the volume of D o u g l a s - f i r logs i n the pee le r log c l a s s to the t o t a l volume of pee le r l o g s , d t t = average log d iamete r , i n i n c h e s , of the quan t i t y of logs a v a i l a b l e from the l o c a l log market i n per iod t , ptj. = r a t i o i n pe r i od t o f the quan t i t y o f D o u g l a s - f i r logs to the volume of logs a v a i l a b l e from the l o c a l log market . The average top d iameter and spec ies mix of the volume component, y c , so r ted from the non-pee le r log vo lumes, must be d e r i v e d . Both de-pend upon the type of log i n each log c l a s s i nc luded i n the quan t i t y yc and the s e l e c t i o n r u l es i n e f f e c t . L e t , rs = r a t i o o f the volume of the smal l log c l a s s to the t o t a l volume of non-pee le r c l a s s e s o f l o g s , pi = r a t i o o f the volume of D o u g l a s - f i r logs i n the l a rge log c l a s s to the t o t a l volume of the l a rge log c l a s s , ps = r a t i o o f the volume of D o u g l a s - f i r logs i n the smal l log c l a s s to the t o t a l volume of the smal l log c l a s s , d ip = average log d iameter o f D o u g l a s - f i r logs i n the l a rge log c l a s s , dsp = average log diameter of D o u g l a s - f i r logs i n the smal l log c l a s s . d l = average log d iameter of western hemlock logs i n the l a rge log c l a s s , 125 ds = average log d iameter of western hemlock logs i n the smal l l og c l a s s . For the q u a n t i t i e s x ^ 2 and y ^ - the vo lume, -Q^ , of each non-pee le r log c l a s s i s de r i ved as Q-, = ( l - p c 2 ) ( x t 2 + y t 2 ) ( l - r s ) . p l , . Q 2 = ( l - p c 2 ) ( x t 2 + y t 2 ) ( l - r s ) ( l - p l ) , Q- = ( l - p c 2 ) ( x t 2 + y t 2 ) . p s . r s , Q 4 = ( l - p c 2 ) ( x t 2 + y t 2 ) ( l - p s ) . r s . L e t , d y t ( x t 2 , y t 2 , y c , t r ) = average log d iamete r , i n i n c h e s , o f the quan t i t y yc when the q u a n t i t i e s x t 2 and y t 2 are a l l o c a t e d from source two i n pe r iod t f o r a log s o r t i n g and t r ad ing p o l i c y de-noted by the quan t i t y yc and t h e . r a t i o t r , P y t ^ x t 2 9 y t 2 ' y c ' t r ) = r a t i ° 0 T * the volume of D o u g l a s - f i r logs to the quan t i t y yc when the q u a n t i t i e s x ^ 2 and y t 2 are a l l o c a t e d from source two i n pe r iod t f o r a log s o r t i n g and t r ad ing p o l i c y denoted by the quan t i t y yc -and the r a t i o t r . When p r i o r i t y o f l a rge log s i z e , f o l l owed by p r i o r i t y of p r e -f e r r e d t ree s p e c i e s , i s ass igned to saw logs , the average log diameter and the D o u g l a s - f i r content of the quan t i t y yc are de r i ved from 126 d y t ( x t - , y t 2 , y c s t r ) = 0 d ip f o r yc = 0 , f o r 0 < yc <^  Q-j 2 { Q 1 . d l p + ( y G - Q 1 ) . d l } / y c f o r Q, < yc <_ z Q. j=l J {Q , . d lp+Q ? . d l+ (yc - z Q . ) . dsp } / yc .i=l J 2 3 f o r z Q. < yc < z Q . , j - 1 J j= l J {Q, .d lp+Q-.d l+Q- .dsp+(yc- z Q . ) . d s } / y c and P y t ( x t 2 ' y t 2 ' y c > t r ) 0 1 Q i / y c ( y c - Q 2 ) / y c (Qn+QgJ/yc where 3 f o r z Q. < j= l J yc <_ 4 £ 0 . , j= l J f o r yc = 0 , f o r 0 < yc i Q] > f o r Q- < yc 2 < z j= l 2 f o r I Q. < j= l J yc <_ 3 j= l J 3 f o r z Q . < j= l \ yc <_ 4 s 0 . , j= l J 4 m a x [ 0 ; ; y t 2 - p c . ( x t 2 + y t 2 ) ] < yc < m i n [ y , 2 ; z Q . ] . j= l J 127 The average log diameter and spec ies mix of the d e l i v e r e d quan-t i t y , y t 2> i s de r i ved as the weighted averages of the log diameter and spec ies mix of the three component volumes, y c , t r . ( x t 2 +y . j . 2 ) a n d y t 2 ~ y c ~ ^ r ' ^ x t 2 + y t 2 ^ ' ^ e c a u s e the r e l a t i v e magnitudes of these compon-. e n t s , as we l l as the de r i ved d y t and py^ , are a f f e c t e d by the log s o r t -ing and t r ad ing p o l i c y , weighted averages are c a l c u l a t e d f o r a l l permi t ted magnitudes of yc and t r . These averages are used to eva lua te the cor responding d i f f e r e n t i a l sawlog net wor ths , C W U x t 2 ' y t 2 ' y c » t r ^ = { ^ P t 2 " P s ^ ( w P t " w n t ^ - m c • c m • ( 1 / D t 2 - l / D s ) } . y t 2 , where P t 2 . = { y c . p y t + t r . ( x t 2 + y t 2 ) . p t t + ( y t 2 - y c - t r . ( x t 2 + y t 2 ) ) . p p } / y t 2 D t 2 = { y c . d y t + t r . ( x t 2 + y t 2 ) . d t t + ( y t 2 - y c - t r . ( x t 2 + y t 2 ) ) . d p } / y t 2 . APPENDIX B APPENDIX B LISTING OF THE COMPUTER-CODED PROGRAMME "DYALOG" A. CODED TERMS USED IN "DYALOG" This s e c t i o n l i s t s and de f ines the code names of the more impor-tan t v a r i a b l e s used i n "DYALOG." Code Name D e f i n i t i o n BETAN = c s t ( x t 2 ? y t 2 , y c , t r ) + c t t ( x t 2 , y t 2 . y c . t r ) - c w j . ( x t 2 , y t 2 , y c . t r ) B E T A 1 " c t k ( x t k ' y t k ) + f t . k - l ( X t - x t k ' Y f y t k ) BETA2 - V X t > V + M X t > V + F t - l ( X - X t , Y - Y t ) C(M,1) = c x l t l C(M,2) = c x 2 t 2 C(M,3) = c t t t C(M,4) = c l l t C(M,5) = c l 2 t C(M,6) = c t x t C(M,7) = c t y . Code Name C(M,8) C(M,9) C(M,10) . C(M,11) C(M,12) C(M,13) C(M,14) C(M,15) CD(NXTK,NYTK) CH CM CMS CN CP CSLW CSS D e f i n i t i o n c x l t 3 c x 2 t 3 c y 1 t 3 C y 2 t 3 c x t c y t c x l t 5 c x 2 t 5 C t k ( x t k ' y t k > ch cm ms cn cp c w j . ( x t - . y t 2 , y c , t r ) ess Code Name D DEL DELTA DELTB DIA DIAM DL DP DPL DPS DS DT(L) FNEW(NXT,NYT) FOLD(NXT.NYT) FSNEW(KX,KY) D e f i n i t i o n d y t ( x t 2 , y t 2 , y c s t r ) 4 E Q . - y c j= l J c s t ( x t 2 , y t 2 , y c , t r ) c t t ( x t 2 . y t 2 . y c , t r ) D t 3 D t 2 d l dp dpi dps ds d t t f tk<XfV ft.k-^Xt»Yt)-F t ( X , Y ) , Code Name FSOLD(KX,KY) G(NXT.NYT) GRADE IN(NXTK,NYTK) KX KY L LMV(K) M MX0P(M,K) MY0P(M,K) N NDXT(M) NDYT(M) NG(NXTK,NYTK) NNSV D e f i n i t i o n F^T.CX.Y).. g t ( x t , Y t ) c x l t 3 - c x 2 t 3 - c y l t 3 + c y 2 t 3 -I N D E X t ( x t 3 J y t 3 ) X Y t-1 v r y t x t k ( X , Y ) y t k (x ,Y ) , t+1 D x t Dh x l t 5 133 Code Name NNV NSDX(M) NSDY(M) NSET2(1) NSET2(2) NSV(M,K) NV(M.K) NVSTK(M,K) NVTK(M,K) NXO(K,NXT,NYT) NXOP(M,K,KX,KY) NXT D e f i n i t i o n t 5 E E v . k j= l k=l J K t E Dx. . j= l J t s Dy. j= l J c f l c f2 k E VS j = l t j E V t i j = l t J VS tk r t k xtk<XfV •k x t k ( X t , Y t ) , expressed i n terms of X and Y NXTK t^k 134 Code Name D e f i n i t i o n NXTO(M,KX,KY) = X t ( X , Y ) NYO(K,NXT,NYT) = NYOP(M,K sKX,KY) =• y t k ( X t , Y t ) , expressed i n terms of X and Y NYT = Y t NYTK = y t k NYTO(M,KX,KY) = Y t ( X , Y ) P = p y t ( x t 2 , y t 2 . y c , t r ) PC PP PS pc PL = pi PLY = t r PP PREF - P t 3 PREM = P t _ ps PT(M) = p t t Q1.Q2.Q3-and Q4 = Q-, , Q 2 , Q 3 and ; Q 4 Code Name RS SET1 SIZE SMC STAND TC VA VAL WN(M) XTK XI YC YTK rs c f Ds mc Ps t c c w t 3 ( x t 3 , y t 3 ) c w t 4 ( x t 4 , y t 4 ) w n t x t k x l yc APPENDIX B ( C o n t ' d . ) B. LISTING OF "DYALOG" FORTRAN IV G COMPILER MAIN 09-14-70 22:11:47 PAGE 0C01 C C ' • DYALOG *• C C DYNAMIC PROGRAMME FOR LOG ALLOCATION FOR GIVEN DEMANDS  C 0001 IMPLICIT REAL*8 <A-H.O-Z) 0002 DIMENSION L M V ( 5 )» P T ( 3 ) »~DT ( 3 ) , ND XT ( 5 ) . NO YT ( 5) , N SE T2 < 2 ) , WP ( 3 ) , M N ( 3 ) , 1NSDX (5) ,NSDY (5. ,NXT 1 ( 3 )', NYT1 ( 3 } , NO ( 70 ), MSY( 70 ) ,N ST( 20) 0003 DIMENSION NVTK (3i5) tNV( 3,51 ,MX0P(3,5 ) ,MYGP(3,5 ) , NVSTK (3 , 5 ) , NS V ( 3 , 6 1) ,C(3,15),TEMP( 16 111 ) ,NPL(16-11),CD( 16, 11),IN< 16,11) ,NG(16,11)»NCX 2(45.20) ,NP(45,2G),G (45t20 ) ,IND<45,20),YO(45,20),FNEw(45,20) , FOLD ( 4 _ _ _ _ _ 3 3 _ , J ^ 445,20),NY0(5,45,20 ),NXOP<3, 5,45,20),NYOP( 3, 5,4 5,20) C CLEAR OUT STATIC-0004 000 5 . C. DATA NSDX/5*0/fNSDY/5*0/tFSNEW/2100*0.0/,NXTO/6300*0/,NYTO/6300*0/ DATA NXOP/13500*0/-NYOP/13500*0/ 0006 0007 CRANK 57 IN NECESSARY INPUT DO 57 1=1,3 READ (5,203 )(NVTK{I , J ) , J= 1,5),.C<I,J),J=l,15) 0008 0009 READ ,5 ,204)< ( NVSTK { I , J ) , J= 1, 5 ) , 1= 1, 3 ) , ( PT < J 1, J = 1, 3 ), ( DT ( J ) , J -1, 3 ) READ (5,2G4)(LMV{J) ,J = 1,5) ,(NDXT(J) ,J = 1,5) , (NCYT(J),J = 1,5), {W P(J ) , 1J=1 ,3) . (WN( J ) ..1=1 .3) o o i o ~ 0011 0012 200 READ (5,205) CN,CP,CSS,TC,SMC,CM,CMS,SET1,(NSET2(J),J=1,2),CH,MN READ ( 5, 206) R S , PL , P S , PP , DPL , DPS , DL ,D S , DP , S I 7.E , ST AND, PC FORMAT(1H1) 0013 0014 0015 201 202 203 FORMAT(IH ) FORMAT(///////) FORMAT(512,2F5.0,5F4 .0,8F5.0) 0016 0017 0018 204 2G5 206 FORMAT! 1512 ,6F5.2I F0RMAT(4F4.G,F4.1,F5.l,2F5.0t215,F4.0,I2) F0RMAT(4F3.2,7F4.1,F3.2) -0019 0020 2C8 209 FORMAT,53X,'SAMPLE DATA FCR DYALOG'/52X,24<•-')/) FOR MAT( '0 *,18X,*WOOC SUPPLY ,M CUNITS*,16X,'MILL DEMANDS ,M CUNITS 1» .16X,«LUMBER PRICES /CUNIT*/18X » 23('-»),14X.24(•-•). 14X.22('-•)/2 0021 210 20X,'CUTTING LIMITS (NVTK) • ,25X, • P/L S/L* ,20X,'FIR HEMLOCK' 3/33X,'AREAS',18X,'PERIOD (NDXT) (NCYT)•,19X,MWP)',7X,'(WN)•) FOPMAT(14X,'PERIOD • , 5 I 5 , 12X,12,18,17) 0022 0023 0024 211 212 213 FORMAT ( 16X, I2,3>,5I5,12)<,I2,I8,I7,14X,2(5X.»F6.1)) FORMAT(58X,12,18,17) FORMAT( 17X.*MAX. CUT OF SAWLOGS (NVSTK)'/33X,* AREAS'-18X.'SAWMILL 0025 214 1C0NVERSI0N COEFF.',13X,'COLD-DECKING COST /M CUNITS'/14X,'PER I CD ' 2,5I5,9X,27('-') ,11X,29(«-')) FORMAT(16X,I 2,3>,515,16X,'CM =',F7.2,30X, • CH-' ,F6.0/T6X,12,3X,5 15, 0026 216 116X,'SMC=',F7.2/16X, I2,3X,5I5) FORMAT('0',13X,'PRODUCTION LEVELS FCR COST CHANGES'/16X,•AREAS -', 1I3.4T5/16X, »l MV - • , I 3.41 5/16X,•MN ', ? ( • 1 .. (4X,»-•).T5/1 0027 217 FORMAT('0 *,54X,'LOG CHARACTERIST ICS'/54X,21( '-» )/4X,'AREA•,I 2,26X, l'AREA',12,11X,2(19X,'AREA* ,12),13X,'AREA« ,I 2/15X, •VOL. RATIOS•,12X 2,'DIAMETERS',21X,MSTANDARD S/L )',12X ,'(MARKET LOGS)*) 0C28 218 F0RMAT(3X,'PULPLOGS PEELER < PC ) 1 ., F5..2, • PEELER (DP 1)«,F5 .0,4X,'LCG 01 AM. (SIZE)' »F5.1,4X,'PERIOD FIR/HEM. DIAM PU 2LP LCGS * ) 137 FORTRAN IV G COMPILER MAIN 09-14-70 22:11:47 PAGE 0002 0029 0030 2 1 9 F O R M A T . 5 X , ' O N L Y « , 8 X , ' S M A L L L O G S ( R S ) ' , F 5 . 2 , » S M A L L F I R - ( D P S ) ' , 1 F 4 . 0 , 4 X , ' F I R / H E M . ( S T A N D ) * , F 4 . 1 , 1 4 X , * (PT ) (DT) O N L Y ' ) 2 2 0 F O R M A T ( 1 5 X , ' F I R / H E M R A T I O S * , 1 3 X , * — H E M . - ( D S ) ' , F 5 , 0 , 3 I X . I 2 , 5 X , F 5 1 . 2 . F 7 . 1 / 1 7 X , ' S M A L L L O G S ( P S ) ' , F 5 . 2 , ' L A R G E F I R - ( D P L ) ' , F 4 . C , 3 1 X 138 .00 31 . 0032 2,I2,5X,F5.2,F7.1/17X,'LARGE LOGS <PL>• , F 5 . 2 , 5 X , H E M . - (DL)*,F5 3.0,-IX, 12,5X,F5.2,F7.1/17X.'PEELER (PP)',F5.2) .2.2.1 ECJiM.AJl(_Jli-._.aX.,l^ 1EA',12,2(21X,'AREA* ,12 ,7X),2(12X,'AREA' ,12)) 222 FORMAT( IX,'PERIOD C(L,1) C(L,2) ',5(« C(L , ' , I 1 , ' )•) ,3X,'C(L ,8) 1 C(L .9) ' .3(2('C(L. ' , 12, ' ) ' ),2X) )  003 3 0034 223 FORMAT!3 X,I2 » 5 X»2F7.0,2X,5F7.0,2X,2F7.0,2F8.Q,2(2X,2F8.0>) 224 FORMAT('0*/27X,'SETUP COSTS - FIXED',23X,'ADDITIONAL VARIABLE COST IS ./M CUNITS'/2fSX,21 ('-' ),21X.37('-« )/) _ _ . 0035 2 25 FORMAT(29X,•SET1•,4X,F8.0 ,23X,* SORTING COSTS',13X,« (CSS)* ,F6 .0/29X 1 N S E T 2 ( 1 )', 17,24X,'LOADING COSTS - PEELER (CP)',F7.0/29X,'NSET 22(2)',I7,38X,'- NON-PEEL. (CN)' .F7.0/29X,'CMS',6X,F7.0,23Xt 'PEELER 0036 ..-0.0.3-7.-003 8 0C39 0040 3 PREMIUM PR ICE',6X, • (TC)',F7.0) D O 36 1=1 ,70 -.3 6 N.0.( J J_= Ir.l PRINT 200 PRINT 208 PRINT 2C9  0041 0042 -0.0.43 CC44 0045 PR INT 210,(NO(I) , 1=2,6),N0(1),NDXT(1),NDYT(1) DO 26 1=1,3 -2.6 PJ.iN_T_2J-L,-N^ 1(1) ,WN(I) PRINT 212,NC(5) ,NDXT(5),NCYT(5) PRINT 213,(NO(I),1 = 2,6) 0046 .0.0-4-7-004 8 0049 0050 P R I N T 2 1 4 , N O ( 2 ) * ( N V S T K ( 1 , J ) , J = 1 , 5 ) , C M , C H , N O ( 3 ) , ( N V S T K ( 2 , J ) , J = L , 5 ) , 1 S M C , N G ( 4 ) , ( N V S T K ( 3 , J ) , J = 1,5) _ P _ P J . N : L _ 2 J _ 6 . . , X ^ . . P R I N T 2 1 7 , ( N O ( J ) , J = 2 , 6 ) P R I N T 2 1 8 , P C , D P , S I Z E P R I N T 2 1 9 , R S , C P S , S T A N D  0051 -00.52-00 5 3 0054 0C55 PRINT 2 20,OS,NO(2) , PT {1 ) , DT (1 ), PS .DPL,NO(3 ),PT(2),DT(2 ),PL,DL,NO(4 1),PT( 3) ,DT(3) ,PP 38 PRINT 222 ,(NC(I),1=4,8),(N0(J) , J = l l ,16) DO 38 1=1,3 PRINT 2 23 ,K0(141 ) , (C( I,J) ,J=1,15)  0056 0057 _0-C3..8_ 0C59 0060 0061 PRINT PRINT _NH-=4.Q AX = 1 . AY = 8. AZ=6. 224 22 5,SET1,CSS,NSET2( 1),CP,NSET2(2),CN,CKS,TC OD + 0 OD+0 OD-01 0062 0063 J3-Q.6A. 0065 0066 0067 TEST=5.OD-01 DO 6 1=1,3. JW.„(JJ_W.i__(XL^^ WP<I) = WP(I)*1C00.CD+O NV(I,D=NVTK( 1,1) N S V ( I , 1) = 0 006 8 006 9 JX0J..O-NSV(1 ,2)=NVSTK( 1,1) 00 6 J = 2 , 5 -N.S-VjJ-,J±lJ._^ FORTRAN IV G COMPILER MAIN 09-14-70 22: 11:47 PAGE 0003 0071 0072 0073 0074 6 NV(I , J )=NVU,J -1)+NVTK<I , J ) DO 9 1=2,5 NSDX ( .1 ) =NSDX < 1-1 )+NDXT { I ) 9 NSD Y( I)=NSDY( I—1)+NDYT( I) 139 / 0075 0076 NNV=2 NNSV=1 r.nNf.FNTRATF ON FAf.H PFRinO TN SUCCESSION \ 0077 0C78 0079 N = 3 2 3 M'-N-l L = N-2 0080 0081 0082 N NV = NNV +NV( L » 5 ) NNSV=NNSV+NSV(L,6) DO 13 11=1,70 0083 00 84 0085 00 13 JJ=1,30 FSOLD(I I,JJ)=FSNEW(II , J J ) 13 FSNEWlI I , J J ) = Q.OD+0 0086 00 87 0088 DO 16 J=l ,45 DO 16 IJ=1,2C NP(J .TJ ) = C 0089 0090 0091 YO(J,1J)=0.OD+O NCX(J,IJ)=0 IND < J , IJ ) = 0 0092 0093 0094 FNEW(J,IJ)=0.00+0 DO 16 1=1,5 NXC <I,J ,IJ1=0 0095 0096 16 NYO ( I , J , I J ) = 0 VAL=(PT(L)-STAND)*.WP(L)-WN(L))-SMC*CM*(SIZE-CCNSIDER LOGS FRCM SUCCESSIVE AREAS D H L ) )/(SIZE*DT(L) ) 0097 0098 0099 K=l 40 DO 24 1=1,45 DO 24 J=1.20 0100 0101 v 0102 FOLDU , J)=FNEW( I , J) 24 FNEWC-I, J) = 0.0D+G DO 22 1=1,16 0103 0104 -V: 0105 DO 22 J = l ,11 NPL ( I , J ) = 0 TFMP. I,J)=0.0D+0 0106 0107 0108 22 CD(I , J)=0.OD+G VLM=LMV(K) VTK=NVTK(L,K) 0109 O H O 0111 MA X X=ND XT ( M) + ND XT(N) +1 IF.MAXX.GT.(NV{L,K)+1)) MA X X=N V ( L , K ) + 1 MAXY=NOYT(M)+ NDYT(N ) +1 0112 0113 0114 IF.MAXY.GT. <NSV(L ,K+1) + 1) ) MAXY=NS V ( L , K+l ) +1 NYT= 1 45 NXT=1 0115 0116 0117 11 IF<<NXT+NYT-2).GT.NV.L,K)) GO TO 50 ALPHAl=9.00+07 NYTK=1 0118 0119 0120 IF C (NYT-1) .GT.NSV(L,K)) NYTK = NYT-NSV{L,K) 8 NXTK=NXT I F ( K . E Q . l ) GO TO 41 0121 0122 0123 NXTK=NXT+NYT-NYTK-NV(L,K-1) I F ( N X T K . L T . l ) NXTK=1 -1 IF(( NXTK+NYTK) . R T . 1 NVTKH .K )+?) ) GO TT 39 FORTRAN IV G COMPILER MA IN 09-14-70 22: 11:47 PAGE 0C04 0124 0125 0126 0127 XTK=NXTK-1 YTK=NYTK-1 XY=XTK+YTK IF(CO(NXT K - NVT K).NE « 0.) GO TO 101 140 0128 0129 0130 013 1 CALCULATE TOTAL VARIABLE LOGGING COSTS FOR AREAS 1 GO TO (1,2,3*4.5) , K X 0 . _ P A . r 4 Y _ S _ M A L R _ A . R . E . A _ ^ . 1 IF(NXTK.EQ.l) GO TO 101 CDCNXTK ,NYTK) = XTK*C(L ,1) IF(NXTK.LE.(LMV(K )+l) ) GO TO 101  0132 0133 0134 0135 0136 CD(NXTK,NYTK)=SET1+VLM*C(L,1)+(XTK-VLM)*C(L,2) GO TO 101 _C OilEULTJ.UJQ_^ 2 IF(XY.EQ.O.) GO TO 101 CF=NSET2(1) CL=XY»C(L,4) 0137 0 1 38 .0139. 0140 0141 68 IF(XY.LE.VLM) GO TO 68 CF=NSET2( 1 )+NSET2( 2 ) X.L=V1M_.CXL.,A.)JL( XYnV_LJDJLCJ.L.,_5J_ XYU=XY*(AX-PC) IF(YTK.NE.O. ) GO TO 69 0143 GO TO 3 2 CAST ABOUT FOR TRADING POLICIES CLASSIFY NGN-PFF! ER V (11 UM F COMPONENTS 0 144 69 Q1=XYU*(AX-RS)*PL 0145 Q2=XYU#(AX-RS)#<AX-PL ) 0146 Q3_XYU*RS*PS 0147 0148 .01.4.9_ 0150 0151 0152 Q4=XYU* (AX-PS )*RS YNIN=YTK-PC*XY YHAX=YTK IF(YMAX.GT.XYU) YMAX=XYU BETAN_9.Q0.G7  0153 0154 J3.1.5.5. 0156 0157 PLY=O.CD+0 31 Y C = Y MIN _2.8 IJEJJJ_C±JlL..Y.^ DEL=XYU-YC IF(CEL.GT.(YTK-PLY*XY)) DEL=YTK-PLY*XY COST OF SORTING/LOADING  0158 ... 0.1.59 0160 0161 DELTA=PC*XY*CP+XYU*CN+(XY*(PC+PLY)-YTK+YC+DEL)*CSS COMBINED TRANSPORT COSTS AND BONUS _D.ELJJirJ_XJ_U_lY^ 1L,3 )-TC) D=DPL*YC P=YC 0162 0163 J0A.6JL 0165 0166 0167 IF(YC.LE.Ql) GO TO 52 D=Ql*DPL+(YC-Q1 )*DL J E _ 0 1 IF{YC.LE.(Q1+C2)) GO TO 52 D=Q1#DPL+Q2*DL+(YC-Q1-Q2)*DPS P=YC-C2  0168 0169 JOL1J_0_ IF(YC.LE.(Q1+C2+G3)) GO TO 52 D=Ql*DPL+Q2*DL + Q3*DPS+( YC.-Q 1-Q2-Q3 5 *D S .P_-_Gl_t.C3 in o o o uu U> < o_ r-CM (\l o r-i sr r-4 I CT* O Z < ct LU O (_> o _ < Ct t ~ ct o 00 00 _- _ * — . — (_! _c —I -J OO ~—* ^ —« I I— (— — * * >- >- ~-_ l _ l 2 o- a. 3: > > i + + — tt. o. -J o a — # -a- u. _c > >- ~ _ J _ i # a — > a I _ U > t— I oo _ I I— __ >- U J w ct + + Uv -J a a. — II H >- — _i <. a. k-t > Q Q.> I (_> >• I X >-II II LU Ct 00 (M in r—i <N ro sr fs. t — i«~ r~ I r-4 r-4 i - 4 o o o o r-CM O l>0 (_> I C O a a 3 . z in o l c_ o o o r-4 l_> + <l O, CD 00 LU O LU O II z <i x a. 1 H z oo( < LU X cd a. I -J LU < x! — H LL 1—4 LU 00 li i t Z h-< > X 2 I! X Z Z < «-I — I U J CL co z o r- co r- r- c-r-4 r-4 l—f O O O o > II it r - (M >- o Z I • a it O • x _ 4 Z + — O o_ >-E: II LU U I - >-o j (\) - o *! i < i o 2 3 O LU, + LU _ l •-• a. a o x > i it — SS • «a. LU I— • CG >- + CL U • + O LL z o <1- II (\i O it a (— • >-UJ z _ J •> • it >- H _ J x Q- 2 u- o CT- o >~i <\f r i <r r~- co ro oo co oo r~t »H r-4 [ i—I 1-4 r-4 O O C i O O O LU LU CC «a a: l _ J x ct || O II o u o iC oo t — o >• O -J *; n s sc O oo X X I - Z j 2 o </> e>! z o o 3L LU CL . I — C > II >-< LL. in vo, r- cc a> co co oo oo oo o o o o o o o o a. _i D Q. O —< LL X l 00 z I—] LU 00, t -LU # (~! ^ • H-sd >-l - w H —! ^ l -L U 00 L i I LU t—i oo # O «• o 2: oo I z at i 00 • LU LU H- > <_) < U  <t OC <t X u o o — <l X - J <t — 3 l!| < 00 u"> z <t X o LU <l t-j 00 M <l II U, LU CC a. •» o o z —• < s: i - _J oo > I • LL LU LU _ l CC • a > w x II ~-< LL o i — i o i r - 4 ( \ J m <r a\ O' fw-^  ^ - J r-^ r—4 o o o o o II a. x ^ o • 2-a J , X 3> ~ 111 LL O ^ »- X o + o U) h- I x — il m oo • r x _j -~ O r ->c i -I CM X CO | • u o 2. II <I — I LU Ct > O O w < ~ LL Ct U-<-• O i-< Lt^ vO CO o (JN <jit CT> O i - 4 i - 4 1-4 i—I r-4 ( M o o o ! o o o IN) II z i t t — X O z q IM o * - 4 X II H K- t -> > z z ^: i t »— i — CL X X X z 2 II H Z Q X M CJ m • - I (NJ m >r m N O o o a o o o CM (M f\j CM (M t\J o o o o o o >-l Z < > I o * LL) iti H CQ o > it x ct) >- II q l ^ •! —. |— oj <-<>-•. r-4 2 Oj r-4 2 • r U J O • ~- t — i t v~ O X r - O X * 2 X j r - Z H O J o D X 2 f-| O O i t K x II it r -h TO (? o O O; O r - i CM (M CM CM o o o o > o it t ->• + CM LU r - t O0 » < _ J X — <_J o ct 3 it a I— x O II o •— _ l i t K LU >~ O 2 *—i ^ oo it K- f— 3 X O 2 UJ S o r - i oo 2 O o sr ct LU O O o ct a. o O o 3 a cj —J r - Ct O O r -r-4 o r - i o (_ oo O 2. O O II • i t O r -• > O 2 LU • • i t it r -t - X X 2 LL O >— O o o in 1— CM rn ,4- i n r - t r - t r - 4 1—4 r-4 CM CM CM* CM CM O O O O O a r- i o * 3 X j +1 I—I # SL —J > I! — > vr as r - i » • ^ . _ J r-«- X o z # — it o H U X II — — 3L it J r- >j > Z I— <- CD it t- it X K 2 X *— *ir O LL O vO r~ co CM CM CM o o o FORTRAN IV G COMPILER MAIN 09-14-70 22: 11:47 PAGE 0CG6 0219 0220 0221 101 BETA1=C0(NXTK,NYTK)+FCLD(NXT-NXTK+1,NYT-NYTK+1) COMMENCE IDENTIFYING LOCAL OPTIMA IFIBETA1.GE.ALPHA1 ) GO TO 43 A L PHA1= BE TA 1  142 0222 0223 -022-4. 0225 0226 0 227 LACCA1=XTK MU1=YTK NPLY=NPL(NXTK,NYTK) NN = NG(N'XTK ,NYTK) IP=IN(NXTK,NYTK) 0228 0_22S_ 0230 0231 0 232 CAUTIOUSLY INCREMENT CURRENT DELIVERIES 43 NXTK=NXTK+1 I.E(j^jaK-X.E_..llXTl-J3JDJCL^J 39 NY7K=NYTK+l IF((NYTK-1).GT.NVSTK(L fK)) GO TC 46 IF(NYTK .L E »NYT) GO TO 8  0233 ..-0.234. 023 5 0236 0237 CONSIDER AREAS LOGGED SO FAR FOR RELATIVE OPTIMA 46 FNEW(NXT.NYT)=ALPHA1 NML{_tL,J^ .XI-,-NXTJL=J-A&3XA_l NY0(K,NXT,NYT)=MU1 IF.K.NE.2) GO TO 29 YC(NXT, NYT) = Y0O  0238 0239 .02.4.0-0241 0242 29 NP(NXT,NYT)=NPLY IFCK.NE.3) GO TO 51 -fiCJL{J^LiLNYjJ^.NM_ IND(NXT,NYT)=IP CRANK IN MORE LOG DELIVERIES FOR PERIOD 51 NXT = NXT+1 0 243 0244 .0.2.4.5-0246 0247 0248 IF( (NXT-1) .LE.<NDXT(|«) + ISDXT(N) ) ) GO TO 11 50 NYT=NYT+1 IFJJ-NXInlJ_.J3X IF((NYT-1).LE 44 DO 14 JJ=1,4 IF(J J . E Q . l ) PRINT J J^S^.LJ-K±l_LL-JlD-jrji_4A_ (NDYT(M)+NDYT(N))) GO TO 45 200 0249 0250 JX2.5.1-0252 0253 0254 NSST=(JJ-1)*5+1 NSSP=JJ*5 JM.X.PH.NAL(.L-,.K.)J=.N.S.SX±2_ IFIMAXP.GT.MAXX ) MAXP=MAXX (MSPP = MAXY IF(NSPP.GT.NSSP) NSPP=NSSP 0255 0256 .0257 0258 0259 0260 KKP=(XAXF + NH-1)/NH+ 1 NST(1)=0 -D0__lA^K.L = 2.dK.K.(V_ NST(KL)=MAXP IF ( KL.LT. KK«V) N: S T (K L ) =N h MA XQ=NST(KL )  0261 0262 -0.263. 0264 0265 NNN=NST(KL-1)+l PRINT 226,K,L -2.2.6.„_E0.R.MATX/-4^ 1*PULP',50X,'SAWLOG PRODUCT ION') GO TO (12,19, 17,12,12),K 12 PRINT 227.(N0(J).J=NSST,NSSP) ...I.2.,JL.-F_C.P.....P-E.R..I.C.DX,.I2 /-///.// 2X  (    . J = NSST,N SP) FORMAT (2X ,' PR CD • , 5 ( 11 X , • YT= •, 12, 9 X ) /2 X, •-XT-' , 5 ( 6 X, ' X 1',5X)/5X,5(6X,16(•-'),3X5) _ D . Q _ A 8 _ J ^ j m , . l ! l A J L Q . 0266 .0267 227 RETURN FORTRAN IV G COMPILER M A I N 09-14-70 22:11:47 PAGE 0007 0268 0269 0270 0271 48 228 IF{(I+NSPP).GT.(NV(L,K)+2)) NSPP=NV(L,K>-1 + 2 I F ( ( I - l ) . E Q . ( ( ( I - l ) / 5 ) * 5 ) ) PRINT 201 PRINT 228,NO<I) , (NX0(K, I ,J),NYC(K,I ,J),FNEW{I,J),J=NSST,NSPP) FORMAT(3X,I 2,5(5X,2 I 3,F11.0,3X) )  143 -< 0272 0 273 _0274_ 0 27 5 0276 GO TO 2 5 19 PRINT 229,(NO(J), J=NSST,NSSP ) .2.2.9 FJlfL_AIJj2_l_-^ 1 Y YC T RETURN • )/5X ,5(5X,20(•-*) )) DO 49 I=NNN,MAXQ N  IF ( (I+NSPP) .GT. (N V ( L , K ) + 2 ) ) NSPP=NV (L , K ) - 1 + 2 _,J5J_5X, 0277 0278 I F ( ( 1 - 1 ) . E Q . ( ( ( 1 - 1 ) / 5 ) # 5 ) ) PRINT 201 49 PR INT 231,NO( I ) , ( N X O ( K , I , J ) , N Y O ( K , I , J ) , Y O ( I , J ) , N P ( I , J ) , F N E W ( I , J ) , J 1 = NSS.T.,.N.S.PP.) . . 0279 0280 0281 231 FORMAT.3X,I2»5(5X,21 2,F5.2,12,F9 .C) ) GO TO 25 17 PRINT 2 32 , (NC(J),J=NSST,NSS P) 23 2 FORMAT? 2X,•PROD',5(11X,•YT=»,12,9X>/2X,•-XT-«,5(5X, • X 1TUPN • >/5X,5(5X,20('-*)) ) DO 20 I-NNN.MAXC 0282 .13.2 8JL 0284 0285 028 6 P XI RE 20 IF((l+NSPP).GT.(NV(L,K)+2)) NSPP=NV(L,K) IF((1-1) .EQ.((( 1-1)/5 )*5)) PRINT 2G1 •1 + 2 1, J=NSST,NSPP) 0287 233 FOPMAT(3X,I2,5(4X»3I3,I2,F10.0)) 0.28.8 25 IF(MAXY.LE.NSSP) GO TO 18 0289 IF(MAXQ.N E.NH) PRINT 200 0290 IF(MAXQ.EG.NH) PRINT 201 G291 14 CONTINUE 0292 18 K= K +1 0293 IF(K.LE.5) GO TO 40 CONCLUDE THE FORWARD PASS AND START THE BACKWARD PASS FOR PERIOD 0294 MX = NDXT < N) + NDXT(M ) +1 0295 5 5 MY=NDYT(N)+NDYT(M)+1 0296 60 NXT=MX 0297 NYT=MY 0298 K = 5 - 0.2.9.9 6 5 NXOP(L.K.MX.MY)=NXO(K.NXT.NYT) 0300 NYOP(L,K,MX,MY)=NYO(K,N XT,NYT) 0 301 NXT=NXT-NXOP(L,K,MX,MY) 0302 NYT=NYT-NYOP(L,H,MX,MY) 0303 K=K-1 0304 IF(K.NE.O) GO TO 65 0 30 5 MY-MY-1 0306 IF(MY.NE.O) GO TO 60 0307 MX=MX-1 0308 IF(MX.NE.O) GO TO 55 CITE RESULTS FOR LATEST PERIOD 0309 PRINT 200 0 3.1.0 nn 58 .1 = 1 ,MAXX 0311 NSP-MAXY f 0312 IF((J+NSP).GT.(NV(L,5)+2)) NSP=NV(L,5)-J+2 5-' 0313 IF((J-1 ) .NE.( ( J - l l / 9 * 9 ) ) GO TO 56 0314 PRINT 234,L 0315 234 FORMAT(48X,'OPTIMAL ALLOCATION WITHIN PERIOD*,12) . _ 03.1.6 PRINT 2 36 FORTRAN IV G COMPILER MAIN 09-14-70 22:11-47 PAGE 0008 0317 0318 0319 236 FORMAT(99X,* 1ST OF PAIRED COLS. - PULP ALLOC.'/99X,'2ND OF PAIRED ICOLS. - S/L A L L O C ) PRINT 237 237 FORMAT{IX -'PULP *,51X, 'SAWLOG PRODUCTION') 144 0320 0321 0 5.2 2 PRINT 238,(N0(JX) , JX=1,20) 238 FORMAT.IX,'PROD AREA',3X,I2,19(4X,I2)»/,lX,129(lH-)) 56 DO 58 LL=. ,5 0323 0324 0325 IF(LL.NE.l) GO TO 61 PRINT 239,NO(J),LL,(NXOP.L,LL,J,JJ) ,NYOPIL,LL,J,JJ) ,JJ=1,NSP) 23 9 F0RMAT(2X,I2,3X,' (' ,11,* )•,20<2X,212) ) 0326 0327 .... 0 32 8 GO TO 30 61 PRINT 241,LL,{NXOP(L,LL,J,JJ),NYOP.L,LL,J,JJ),JJ=1,NSP) 241 FORMAT.7X.'{« .11.«)•,?0.2X,2T2)) 0329 0330 03 31 30 IF(LL.EQ.5) PRINT 201 58 CONTINUE IF.MAXX.NE. (NDXT.m+NDXT.NJ+l ) ) PRINT 200 0332 03^3 CURRENTLY REQUIRED PRODUCTION KY=NSDY(M)+1 67 KX=NSDX(M)+1 0334 70 IF((KX+KY).GT.NNV.OR.(KX+KY).GT.(NSDX(M)+NSDY(M)+NV(L,5)+2)) GO TO 0335 1 - 3 MSY(KX)=K Y 0336 0337 033.8 ALP HA 2= +9.E0 8 NYT=KY IF.M.FQ.2) GO TO 62 0 339 0340 0 341 NYT=KY-NN SV+NS V(L,6 ) + 1 IF(NYT.LT.(KY-NSDY(N))) NYT=KY-NSDY(M) IF(NYT.LT.{KY-NSDY(L)-NSV(L-1,6))) NYT=KY-NSCY.L)-NSV(L-l,6) 0342 0343 0344 IF(NYT.LT.l) NYT=1 62 NXT=KX IF.M.F0.2) GO TO 34 0345 0346 0347 NXT = KX-NS DX(M) IF{NXT.LT.(KX+KY-NYT-NNV+NV(L,5 ) + 2) ) NXT=KX+KY-NYT-NNV+NV(L,5)+2 IF.NXT.LT.{KX+KY-NYT-N$DX.L)-NSDY<L>-NV(L-1,5))) NXT=KX+KY-NYT-NSC 0348 - - 034-9 1X(L)-NSCY(L)-NV(L-1,5) IF.NXT.LT.l) N'XT-1 34 IF((N XT +N YT).GT.(NV(1 . 5) + 2).OR.(NXT + NYT).GT.(KX + KY-NSDX(L)-NSCY<L ) 0350 0351 1)) GO TO 54 XX=NXT-NDXT( M)-l IF(XX.LT.G.) XX=0. 0352 0353 YY=NYT-NDYT{M)-1 IF.YY.LT.O.) YY = 0. CALCULAT F I OG COSTS OF COLD-DECKING LOG SURPLUS FOR NXT.NYT 0354 0355 G(NXT,NYT)=(XX+YY)*CH v 10 2 BETA2=FNEWCNXT,NYT)+G.NXT ,NYT)+FSOLD(KX-NXT+1,KY-NYT+1) C-LECT THE BEST ALLOCATION BETWEEN THIS AND ALL PAST PERIODS 0356 0357 03 5 8 IF(BETA2.GE.ALPHA2) GO TO 47 ALPHA2= BETA2 I ArT.A? = NXT-1 0359 0360 0361 MU2=NYT-1 47 NXT=NXT+1 IF(NXT.LE.(KX-NSDX(L>)) GO TO 34 -0362 0363 - 0-3.6 4_. 54 NYT-NYT+1 IF{NYT.GT.NSV(L,6)+1) GO TO 83 TF( NYT. f F . ( KY-N.<.nY I M )) GO TO 62 FORTRAN IV G COMPILER MAIN 09-14-7C 22:11:47 PAGE 0009 036 5 0366 036 7 COUPLING FUNCTION EVALUATION 83 FSNEW{KX,KY)=ALPHA2 NXTO <L ,K.X ,KY ) = LAM0A2 NYTO.L ,KX ,KY) = MU2 145 ) 0368 0369 0370 KX-KX+1 IFUKX-1) .LE.NSCX(N) ) GO TO 7G 53 KY=K Y+1 0371 0372 IF{KY.LE.NNSV.AND.KY.LE.< NSDY(M)+NSV(L,6)+l).AND.KY.LE.(NSDY(N)+l) 1) GO TO 67 NSTX=NS0X(M)+1 0373 0374 037 5 DO 21 IX=1,2 NS PY=NSDY(M)+5* IX NSTY=NSDY(M) + 5*{ IX-D+l 0376 0377 0378 NSX=NNV-NSTY NSY=NSPY IF.N.E0.5) GO TC 59 0379 0380 0381 PRINT 202 PRINT 242 ,L ,L 242 FORMAT.3 8X»'OPTIMAL ALLOCATION BETWEEN PERIOD',12.' AND ALL PAST P 0382 1ERI0DSV/47X,'LOG PRODUCTION ALLOCATED FROM PERIOD',12) PRINT 243 ,L,M,NSDY(M),NSDY(N),L,M,(NO(J),J=NSTY,NSPY) ,NSOX{M),NSDX 1(N ) 0383 243 FOPMAT<1H0,4X,»RANGE OF P/L DEMANDS',32X, 'RANGE OF SAWLOG DEMANDS IFOR PERIODS' ,12 ,« -',12,' . • , 12 , '-* , 12,• ) •/5X,•FOR PERIODS«,I 2, • -2«,I2.3X,5(16X,'Y= », 12 )/9X , •( »,I 2, ' TO',13,»)•) 0384 0 38 5 03 86 PRINT 244 244 FORMAK 13X,'-X-t ,14X,5(6X,'XT Y T RETURN' ) /5X ,126 ( • - ' ) ) DO 15 IT=NSTX,NSX 0387 0388 0389 11 = 1T-1 IF(MSYCIT).LT.NSY) NSY=MSY(IT) PRINT 245, I I , {N XTO(L, IT , J ), N YTO (L , I T , J ) , F SNE W< I T , J ) ,J=NSTY,NSY) 0390 0391 0392 245 FORMAT. 13X, 12,16X ,5 (4X ,2 13 ,F1C0) ) IF( IT • EQ•({ IT/2 )*2) ) PRINT 201 15 CONTINUE 0393 0394 0395 IF{ IX.EQ.1) PRINT 200 21 CONTINUE N=N+1 0396 CEASE NUMBER CRUNCHING FOR AWHILE AND START AGAIN IF.N.LE.5) GO TO 23 CHOOSE GLOBAL OPTIMAL FCR ALL PERIODS ANC AREAS 0397 0398 0399 59 NY=NSDY(4)+1 NX=NSDX.4)+1 M=3 0400 0401 0402 7 I=NXTO(M,NX,NY)+1 J=NYT0(M,NX,NY)+1 DO 10 K=l,5 0 403 0404 0405 MXOP(M,K)=NXOP(M,K,I,J) 10 MY OP { M, K)=NYOP(M,K, I,J) LX=NX-NXTO(M,NX-NY) 0406 0407 040 8 L Y=NY-NYTO(M,N X,N Y) NX T1 ( M) = NXT 0 {M, N X , NY ) NYTKM)=NYTO(M,NX,NY) 0409 0410 0411 NX=LX NY = LY M = M-1 J FORTRAN IV G COMPILER MAIN 09-14-70 22:11:47 PAGE 0010 0412 IF(M.NE.O) GO TO 7 CAN STOP DECISION-MAKING AND PUBLISH RESULTS 146 0413 PRINT 200 0414 PRINT 202  0415 PRINT 230,NSDX(5) ,NSDY(5) ,(N0(J) ,J = 2,6) 0416 230 FORMAT(* 0 *, 4 3 X ,'MINIMUM COST ALLOCATION POLICY FOR X=',I 2,•,Y =',I 2 L/_/5J-X.,X0£JjmL.^ 2 MILL',1IX,'ALL AREA S' ,2 3X,'PRODUCT IVE AREAS'/24X,'PERIOD CE 3MAND' ,23X,5(19 ) /24X ,6('-'),3X,8(«-'),9X,61{'-')/33X,'P/L S/LSlOX 4, ' XT VT' ,8X ,51 » XTK YTK' ) ) "  0417 0418 3 5 DO 35 1=1,3 PRINT 235,1,NDXT(1+1),N DY T{1 + 1),NXT 1(I),NYT1( I ) *(MXOP{I,J),MYOP(I, 1J),J=1,5) 0419 0420 0421 235 FORMAT!1H0,24X, •(',II,«)',2X,2(215,8X),5(214,IX)) PRINT 201 PRINT 240,FSNEW(NSDX( 5) + l ,NSDY( 5J + 1) 0422 240 FORMAT('0',53X,'MINIMUM COST = $',F9.1) 0 423 PRINT 200 _ -0424 STOP 0425 END TOTAL MEMORY REQUIREMENTS 0470*6 BYTES APPENDIX C APPENDIX C ( C o n t ' d . ) B. LISTING OF OUTPUT TABLES GENERATED BY "DYALOG" 148 APPENDIX C OUTPUT TABLES GENERATED BY "DYALOG" A. SYMBOL GUIDE FOR TABLE SERIES The f o l l o w i n g tab les i l l u s t r a t e the computat ional sequence fo l l owed i n the model. A l l output t ab les generated by .DYALOG (Appen-d i x B) are i nc luded to f a c i l i t a t e the d e t a i l e d examinat ion of the opt imal log a l l o c a t i o n p o l i c y f o r the hypo the t i ca l problem. The d e r i -va t i on of these tab les has been d i scussed i n Chapter IV together w i th i l l u s t r a t i o n s of t h e i r use fu lness to the woods manager. Due to the need f o r compactness the columns i n these tab les are des ignated by h i g h l y abbrev ia ted symbols. In c e r t a i n cases the mean-ing of a symbol v a r i e s f o r d i f f e r e n t types of t a b l e s . To f a c i l i t a t e i n t e r p r e t a t i o n the groups of symbols used f o r each s e r i e s of t ab l es are de f ined i n the f o l l o w i n g gu ide . Type of Table Symbols D e f i n i t i o n Sample data f o r DYALOG, "Code-names" Def ined f o r DYALOG page 151 i n Appendix B. A l l o c a t i o n from sources f o r XT X^ each p e r i o d , pages 152-172 YT Y t 180-187 X W t ) 193-200 y »tk«t 'V Type of Table Symbols D e f i n i t i o n RETURN f t k ( V Y t } YC yc T t r . 1 0 0 P I N D E X t ( x t 3 , y t 3 ) XI x l A l l o c a t i o n between XT X t ( X , Y ) p e r i o d s , pages 178-179 YT Y t ( X , Y ) 191-192 RETURN F t ( X , Y ) Optimal a l l o c a t i o n XT X t ( X , Y ) p o l i c y , page 204 YT Y t ( X , Y ) XTK x t k ( X , Y ) YTK' y + J X , Y ) . S A M P L E D A T A F O R D Y A L O G 151 W O O D S U P P L Y , M C U N I T S . M I L L D E M A N D S , M C U N I T S L U M B E R P R I C E S / C U M T C U T T I N G L I M I T S ( N V T K ) P /L S / L ARFAS P F R T O D ( N D X T ) ( N D Y T ) F I R H E M L O C K (WP) ( W N ) P E R I O D 1 1 1 5 2 8 2 1 0 0 3 4 5 G O O 1 5 4 9 1 2 4 1 0 8 3 7 2 2 0 9 1 1 0 . 0 9 0 . 0 8 0 . 0 7 0 . 0 3 1 0 2 1 5 4 9 3 2 4 9 4 0 0 M A X . C U T O F S A W L O G S ( N V S T K ) 1 0 0 . 0 8 0 . 0 P E R I O D 1 1 0 2 1 0 A R E A S S A W M I L L C O N V E R S I O N CCEFF. C O L D - D E C K I N G C O S T /M C U N I T S 8 4 G CM = 4 3 6 . 5 0 CH= 3 5 0 0 . 2 0 3 0 0 2 6 3 G S M C = 2 8 8 . C O 8 4 0 P R O D U C T I O N L E V E L S F O R C O S T C H A N G E ' S A R E A S - 1 2 3 4 5 LMV - 6 4 6 0 5 M N - - - - 2 L O G C H A R A C T E R I S T I C S A R E A 1 AREA 2 AREA 3 A R E A 4 A R E A 5 P U L P L O G S O N L Y V O L . R A T I O S P E E L E R S P A L L L O G S ( PC ) ( R S ) G I A P E T E R S ( S T A N D A R D S / L ) 0 . C 9 P E E L E R ( D P ) 1 6 . LOG D I A M . ( S I Z E ) 0 . 7 0 S M A L L F I R - ( D P S ) 7 . F I R / H E M . ( S T A N D ) 1 2 . 0 0 . 8 ( M A R K E T L O G S ) P E R I O D F I R / H E M . D I A M (PT) ( D T ) P U L P L O G S O N L Y F I R / H E M R A T I O S S M A L L L O G S ( P S ) L A R G E L O G S ( P L ) HEM.- ( D S ) 8 . 0 . 6 0 L A R G E F I R - ( D P L ) 1 4 . 0 . 8 0 - HEM.- ( D L ) 1 4 . • 1 G . 7 5 1 4 . 0 2 0 . 6 5 1 2 . 0 3 0 . 7 5 1 4 . 0 P E E L E R ( PP) 0 . 7 5 R A S T C I O G G T N G OflSTS / M CUNITS P E R I O D A R E A 1 C C L . l l C ( L , 2 ) C(L»3 ) A R E A 2 AREA 3 C ( L , 4 ) C ( L , 5 ) C ( L , 6 ) C ( L , 7 ) C ( L , 8 ) C ( L , 9 ) C ( L , 1 0 ) C ( L t l l ) A R E A 4 C ( L , 1 2 ) C ( L , 1 3 ) A R E A 5 C ( L , 1 4 ) C I L . 1 5 ) 1 2 . . . 3 1 2 5 0 0 . 1 0 C 0 0 . 1 3 5 0 0 . 1 1 0 0 0 . . 1 2 5.0.0.. LOO-OIL. 7 GOO. 0 . 7 0 G 0 . 4 4 0 0 . 2 2 0 0 . 5 0 0 0 . 4 5 0 0 . 1 4 0 0 0 . 1 2 0 0 0 . 1 3 0 0 0 . 9 0 0 G . 0 . C . 0 . 0 . 1 7 0 0 0 . 1 3 0 0 0 . 1 4 0 0 0 . 1 3 0 0 0 . 4 4 n n . ? ? n n . 5 0 0 0 . 5 C O 0 . 1 4 0 0 0 . 1 2 0 0 0 . 1 3 0 0 0 . 9 0 0 0 . 2 0 0 0 0 . 2 4 0 0 0 . 2 2 0 0 0 . 2 6 0 0 0 . 2 0 0 0 0 . 2 4 0 0 0 . 1 2 0 0 0 . 1 9 0 0 0 . 1 4 0 0 0 . 2 0 0 0 0 . 12.0.0-0-. 2 0 - 0 . 0 - 0 . S E T U P C O S T S - F I X E D A D D I T I O N A L V A R I A B L E C O S T S /M C U N I T S S E T ! snonr. S O R T I N G C O S T S LOSS.) 1-5-O.C-N S E T 2 U ) 4 0 0 0 0 L O A D I N G C O S T S N S E T 2 I 2 I 1 0 0 0 0 C M S 3 0 C G C . P E E L E R P R E M I U M - P E E L E R - N O N - P E E L . P R I C E (CP) 2 5 0 0 . ( C N ) 3 0 0 0 . ( T C ) 5 0 0 0 . ALLOCATIONS FROM SOURCE 1 FOR PERIOD 1 1S2 = . : : J PULP_ SLAWL.aQ_JlBjamC_TJ.CjJV| _ PROD YT= C YT= 1 YT= 2 YT= 3 YT= 4 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN ' 0 0 0 0 . __1. _ l___L__^Z5_a_,_ 2 2 0 25000. 3 3 0 37500. 4 4 0 50000. 5 5 0 6 2 500. 6 . 6 0 7_5j0J3_O__ 7 7 0 135000. 8 8 0 145000. 9 9 Q 155000. 10 10 0 16 5000. 11 11_._0. 17.5J10JX. 12 12 0 185000. 13 13 0 195000. 14 14 0 205000. 15 15 0 215000. ALLOCATIONS FROM SOURCE 2 FOR PERIOD 1 153 PULP. PROD -XT- X Y YT= C YC T RETURN X Y Y 7= 1 YC T RETURN _5„mj0iLJ-RJ3.CLLCXIIlN Y 7= 2 X Y YC T RETURN X Y YT= 3 YC T RETURN X Y YT= 4 YC T RETURN 0 „_L 2 3 4 0 C 0.0 0 X. „C C O Q_ 0 0 0.0 0 C 0 0.0 0 0 0 0.0 0 0. J2.5_G.0__. 25000 . 37500. 50000. 0 J _ 2 3 3 1 0.91 0 J_J1...8.2__9.. 1 0.73 9 1 0.87 3 1 0.87 3 56619. J>42_73_._ 74739. 84916. 97416. 0 2 1.82 0 _1_2_1 ._7_3__Q 1 2 1.73 0 1 2 1.73 0 1 2 1.73 0 73237. .__75__L2I3_._ 88428. 100928. 113428. C 3 2.73 0 J__3__2_._64__.0_ 1 3 2.64 0 1 3 2.64 0 1 3 2.64 C 89856. _ja.aoaa._ 100508. 113008. 125508. 0 4 3.64 0 J.__i__3._5.5_.Q 2 4 3.46 0 3 4 3.37 0 4 4 3.28 9 106475. .1123 90. 117455. 128028. 138292. 5 _6._ 7 8 9 0 0 0.0 0 ..a jo. c o x>_ 4 0 0.0 0 4 0 C O 0 9 0 0.0 0 62500. ._7__aao_._ 126380 . 138880. 148980. 1 0.87 3 J_a.J37._3__ 1 1.00 0 1 1.00 0 1 1.00 0 109916. .1.2 24.1_6_,_ 134463. 144483. 154503. 5 2 1.51 7 _6__2_ 1_.J_4__.3_. 7 2 1.96 0 8 2 2.00 0 8 2 2.00 0 121907. _1_3J„0_31_._ 140196. 150007. 162507. 5 3 2.28 9 _6___3_2.._19__9_ 7 3 2.18 8 7 3 2. 18 8 7 3 2.18 8 137524. JA321_8.._ 148815. 161315. 173815. 5 4 3.19 9 6 4 3.10 9 6 4 3.10 9 6 4 3.10 9 6 4 3. 10 9 147685. 157275. 169775. 182275. 194775. 10 LI 12 13 14 10 0 0.0 0 Ji3__0_..a._0 0_ 10 0 0.0 0 0 C 0.0 0 . 0 0 0.0 0 159000. JJJ.5_Q0_._ 184000. 19 5000. 205000. 9 1 1 .00 0 _2__1__L._0_Q._0_ 9 1 1.00 0 9 1 1.00 0 9 1 1.00 Q 167003. 179 503. 192003. 204503. 217003. 8 2 2.00 0 175007. 8 2 2.00 0 187507. 8 2 2.00 0 200C07. 8 2 2.00 0 212507. 8 2 2.00 0 225007. 7 3 2.18 8 _7__3__2_. 1.8__8_ 7 3 2.18 8 7 3 2.18 8 1 3 2 .64 0 186315. _1_9_8_8J.5.._ 211315. 223815. 283008. 6 4 3.10 9 6__4_3..J.0_ 9-6 4 3.10 9 6 4 3.10 9 6 4 3. 10 9 20 7275. _2.1S.7 75. 232275. 292275. 302275. 15 0 0 0.0 0 215000. _16. 1.0_ _Q_D___0___0 2.3A0_XC_._ 17 2 0 0.0 0 2 79440. 18 3 0 0.0 0 291660. 19 4 0 0.0 0 303880. 9 1 1.00 0 1 1 0.82 9 2 1 0.73 9 3 1 0.87 3 4 1 1.00 0 229503. _2J_9_2_73_._ 289739. 299916. 319403. 1 2 1.73 0 280928. J_2_l.._7.3__0_2_9___9.2_8_._ 2 2 1.64 9 303546. 8 2 2.00 0 315007. 8 2 2.00 0 325007. 1 3 2.64 0 J__3_2...6j4__0_ 7 3 2. 18 8 7 3 2.18 8 7 3 2.18 8 293008. 3_03.0_18_t_ 313815. 323815. 333815. 6 4 3. 10 9 _6__4_3_.J.0_9_ 6 4 3.10 9 6 4 3.10 9 6 4 3.10 9 312275. .32.2275. 332275. 342275. 352275. 20 _2.1_ 22 23 24 5 0 0.0 0 _6__0__0....0 G_ 7 0 C O 0 8 0 0.0 0 S C 0.0 0 3 23900. 333_9_2.Q_._ 343940. 353960. 363980. 5 1 1.00 0 _6_.l_l__0i)__f3_ 7 1 1.00 G 8 1 1.00 0 9 1 1.00 o 329423. 33_9_4A3_._ 349463. 359483. 369503. 8 2 2.0G 0 335007. 8 2 2.00 0 345007. 8 2 2.00 TK 355007. 8 2 2.00 0 365007. 7 3 2.18 8 JL_3_2..J_3_8_ 7 3 2.18 8 343815. .3.53_8J.5.._ 363815. 6 4 3.10 9 _6__4_3_. JO.. 9_ 362275. 3122.75. 25 IC 0 0.0 0 374000. ALLOCATIONS FROM SOURCE 2 FOR PERIOD 1 154 PULP. . . SAWLOG PRODUCTION _ „ „ _ PROD YT-= 5 VT= 6 Y T= 7 YT= 8 YT= 9 -XT- X Y YC T RETURN X Y YC T RETURN X Y YC T RETURN X Y YC T RETURN X Y YC T RETURN 0 C 5 4.55 0 13C893. C 6 5.46 0 145312. 0 7 6.37 0 159731. 0 6 7.28 C 174150. C 9 8.19 0 188568. . 1 1 .5 4.46..0... 126786 1 6 5.37 C 141189. 1 7 6.28 0 155597. 1 8 7.19 0 170007. 1 _9_ 8 . 10. 0 184419. 2 2 5 4.37 0 126948. 2 6 5.28 0 137216. 2 7 6.19 0 151592. 2 8 7.10 0 1&5979. 1 9 8.10 0 196919. 3 3 5 4.28 0 137358. 3 6 5.19 0 146783. 3 7 6.10 0 156266. 2 8 7.10 0 178479. 1 9 8.10 0 209419. 4 4 5 4.19 0 148023. 4 6 5. 10 0 157304. 3 7 6. 10 0 168766. 2 8 7.10 0 190979. 1 9 8. 10 0 221919. 5 5 5 4.10 9 158164. 4 6 5.10 0 169804. 3 7 6.10 0 181266. 2 8 7.10 G 203479. 1 9 8.10 C 234419. ..6 5 5 4. 10 9 170664. 4 6 5. 10 0 182304. 3 7 6.10 0 193766. 2 8 7.10 0 215979.. ...1. _9 .8 .10. „0 24 6919. 7 5 5 4.10 9 183164. 4 6 5.10 0 194804. 3 7 6.10 0 206266. 2 8 7.10 0 228479. 1 9 8.10 0 259419. 8 5 5 4.1G 9 195664. 4 6 5.10 0 207304. 3 7 6.10 0 218766. 2 8 7.10 0 240979. 1 9 8.10 0 319419. 9 5 5 4.10 9 208164. 4 6 5.10 0 219804. 3 7 6.1G 0 231266. 2 8 7.10 0 300979. 1 9 8.10 0 329419. 10 5 5 4.1C 9 220664. 4 6 5.10 0 232304. 3 7 6.10 0 291266. 2 8 7.10 G 310979. 1 9 8.10 0 339419. 11 _ _ 5 .5. -4..-10_9-__Z3.3_1.6-4.. 3_6„5_.X9_Q_.2_9JX8_3_. 3JL6._LGL^___30X2.6J... 2_8_7_..1..0_0 3.209_7J_. l__a_8_,_10_..Q._. 349419... . 12 2 5 4.37 0 291948. 3 6 5.19 0 301783. 3 7 6.10 0 311266. 2 8 7.1G 0 330979. 1 9 8.10 0 359419. 13 2 5 4.37 0 301948. 3 6 5.19 G 311783. 3 7 6.10 0 321266. 2 8 7.10 0 340979. 1 9 8.10 0 369419. 14 2 5 4.37 0 311948 . 3 6 5.19 0 321783. 3 7 6. 10 0 331266. 2 8 7.10 0 350979. 1 9 8.10 0 379419. •15 2 5 4.37 0 321948. 3 6 5.19 C 331783. 3 7 6.10 0 341266. 2 8 7.10 0 360979. 1 9 8.10 0 389419. 1A 2_ 5_4_.3_7._0 3_3J__a_._ 3 6 5.19 0 3A1JJ33_. 3_7_6-._lJ3_0___35_1.2.6_6_. 2_3_7_U.0_0 3JJ3 £L7_9.. l_9_8..U.0_.a. _39__41.9. 17 2 5 4.37 0 341948. 3 6 5.19 0 351783. 3 7 6.10 0 361266. 2 8 7.10 0 380979. 18 3 5 4.28 0 352358. 3 6 5.19 0 361783. 3 7 6.10 0 371266. 19 4 5 4. 19 0 363023. 4 6 5 .10 Q 372304. :  20 5 5 4.10 9 373164. ALLOCATIONS FROM SOURCE 2 FOR PERIOD 1 155 — ' < PULP . „ SAliLjOj__JLm^ _ PRCD YT=10 YT=11 YT=12 YT=13 YT = 14 -XT- . X Y YC T RETURN X Y YC T RETURN X Y YC T RETURN X Y YC T RETURN X Y YC T RETURN 0 CIO 9.10 0 202S87. __ _____ 0.1 .<L_9_._1.0__Q___2_1.5A8_7_.. 2 010 9.10 0 227987. 3 010 9.10 0 240487. 4 010 9. 10 0 252987. 265487. 2J_I.9L8.7_.. 337987. 347987. 3579 67. 367987. 3_7_7_9_eJl.. 387987. 397987. 4C7987. 15 C1C 9.10 0 417987. 5 _6_ 7 8 9 010 9.10 0 _0JCLJL.J0_.O_ 010 9.10 0 010 9.10 0 01C 9.10 0 10 CIO 9.10 0 ... II O.0_.9...10._C. 12 CIO 9.10 0 13 010 9.10 0 14 CIO 9.10 0 ALLOCATIONS FROM SOURCE 3 FOR PERIOD 1 i § 6 PULP S.AMLJJG PRODUCTION PROD YT= 0 Y7= 1 YT= 2 YT= 3 YT= 4 -XT- X Y p XI RETURN X Y P XI RETURN X Y P XI RETURN X Y P X l RETURN X Y P XI RETURN 0 0 0 0 0 0. 0 ! 0 C 22238. 0 2 0 0 44476. 0 3 0 0 66714. 0 4 0 0 88952. _1 - _ JL. c 0 0 0 12500. 1 1 - _ L J 2 J 0.0.0. 1 2 0 1 40000. 1 3 0 1 . 60928. 1 4 JO J _..8 2381. 2 0 0 C 25000. 1 1 0 1 39500. 1 2 0 1 52500. 2 3 0 2 67000. 2 4 0 2 800 00. 3 0 0 0 0 37500 . 1 1 0 1 52000. 1 2 0 1 6 5000. 2 3 0 2 79500. 3 4 2 3 90000. 4 0 0 0 0 50000. 1 1 0 1 64500. 1 2 0 1 77500. 4 3 2 4 91000. 4 4 2 4 100000. 5 0 0 0 0 62500 . 1 ! 0 1 77000. 1 2 0 1 90000. 5 3 2 5 101000. 5 4 2 5 110000. _6_. __ 0 0 0 _0_ 75000. 1 1 0 1 __89_500. 6 2 2 6 102000. 6 _3. __2 __>_.._ .111000. 6 4 2 6 120000. 7 1 G 0 1 89000. 1 1 0 1 102000. 7 2 2 6 114000. 7 3 2 6 123000. 7 4 2 6 132000. 8 2 0 0 2 1030C0. 2 1 0 2 116000. 8 2 2 6 126000. 8 3 2 6 135000. 8 4 2 6 144000. 9 3 0 0 3 117000. 9 1 2 6 129000. 9 2 2 6 138000. 9 3 2 6 147000. 9 4 2 6 156000. 10 L 0 0 4 13 10C0 . 10 ! 2 6 141000. 10 2 2 6 150000. 10 3 2 6 159000. 10 4 2 6 168000. 11 — 1.1. -0 6 144000. 11. 1 2 6 153000. IJ __2_ 2 6 162000. I J 3 2. 6 1.71.0.00.. _ 1.1. _ _ i L _ _2_ _6 J8J30.0Q. 12 12 0 3 6 156000. 12 1 2 6 165000. 12 2 2 6 174000. 12 3 2 6 183000. 11 4 2 6 192500. 13 13 0 3 6 168000. 13 1 2 6 177000. 13 2 2 6 186000. 12 3 2 6 195500. 11 4 2 6 205000. 14 14 0 3 6 180000. 14 1 2 6 189000. 13 2 2 6 198500 . 12 3 2 6 208000. 11 4 2 6 217500. 15 15 0 3 6 192000 . 14 l 2 6 201500. 13 2 2 6 211000. 12 3 2 6 220500. 11 4 2 6 230000. ..... 1.6 15 0 3 6 204500. 14 I 2 6 214Q.0O. 13 2 2 6 223500. 12 _3_ 2 6 2.33.00.0.. - J i - ._A_ _2_ _6 242500. 17 15 0 3 6 217000 . 14 1 2 6 226500. 13 2 2 6 236000. 12 3 2 6 245500. l l 4 2 6 255000. 18 15 0 3 6 229500. 14 1 2 6 239000. 13 2 2 6 248500. 12 3 2 6 25800C. 10 2 2 6 300007. 19 15 0 •x 6 242000. 14 1 2 6 251500. 13 2 2 6 261000. 11 1 2 6 303007. 11 2 2 6 312007. 20 15 0 3 6 2 54500. 14 ! 2 6 264000. 12 0 3 6 306007. 12 1 2 6 315007. 12 2 2 6 324007. .21 15 0 3 .6 6 1 2 6 3 08J1Q.O. h 2 2 6 3J7000 . 6 _3_ __2._6 3.2.6.0.00.. ..... 6 _4 _2_ _6_ 3.3.5000. 22 7 0 3 6 311000 . 7 1 2 6 320000. 7 2 2 6 329000. 7 3 2 6 338000. 7 4 2 6 347000. 23 8 0 3 6 323000. 8 1 2 6 332000. 8 2 2 6 341000. 8 3 2 6 350000. 8 4 2 6 359000. 24 9 0 3 6 3 35000. 9 1 2 6 344000. 9 2 2 6 353000. 9 3 2 6 362000. 9 4 2 6 371000. 25 IC 0 3 6 34 7000. 10 I 2 6 356000. 10 2 2 6 365000. 10 •3 2 6 374000. 10 4 2 6 38 3000. 26 JL 1 _ 0 3 6 359000. U I 2 J>_ 368000. 11 _2_ 2 6 377000. 11 3 2 6 386000. 1.1. ._4 . _2_ 6 3J9.5X3JD.O. 27 12 0 3 6 371000. 12 1 2 6 380000. 12 2 2 6 3 8 9000. 12 3 2 6 398000. 13 2 2 6 411007. 28 13 0 "3 6 383000. 13 1 2 6 392000. 13 2 2 6 401000. 14 1 2 6 414007. 6 1 2 6 456815. 29 14 0 3 6 39 5000. 14 1 2 6 404000. 15 0 3 6 417007. 6 1 2 6 458007. 6 2 2 6 467007. 30 15 . 0 3 6 407000. 15 0 3 6 421503. 7 0 3 6 461007. 7 1 2 6 470007. 7 2 2 6 479007. 31 1.5 ..0... _3... 6 4 26000. 7 0 3 463.5.0.3.. 8 0 3 <S 473007. 8 J L . _ ._2__6 4.8.2.0.0 7-. ft 2 2 6 -A9JOG7. ..... 32 7 0 3 6 470000 . 8 0 3 6 . 477503. 9 0 3 6 485007. 9 1 2 6 494007. 9 2 2 6 503007. 33 8 0 3 6 482000. 9 0 3 6 489503. 10 0 3 6 497007. 10 1 2 6 506007. 10 2 2 6 515007. 34 9 0 3 6 494000. 10 0 3 6 501503. 11 0 3 6 509007. 11 1 2 6 518007. 11 2 2 6 527007. 35 10 0 3 £ 506000. 11 0 3 6 513 503. 12 o 3 6 521007 . 12 1 2 6 5 30007. 12 2 2 6 5390C7. .36 IJ 0 3 6 .5.1800.0. 12 0 3 6 52.5.5.C.3... I 3 0 3 6 333007. 1.3 _ i _ 2 6 54.2.0.07. . 13 2 2 6 551007. 37 12 0 3 6 530000. 13 0 3 6 537 503. 14 0 3 6 545007. 14 1 2 6 554007. 38 13 0 3 6 542000. 14 0 3 6 549503. 15 0 3 6 557007. 39 14 0 3 6 5 54000. 15 0 3 6 561503. ALLOCATIONS FROM SOURCE 3 FOR PERIOD 1 W7 P.ULR. PROD -XT-YT- 0 P XI RETURN YT= 1 P X l RETURN _SJmX£_PJ10mCX.I.0jNi Y 7= 2 X Y P XI RETURN YT= 3 P XI RETURN YT= 4 P XI RETURN 40 15 0 3 6 5660C0. ' ALLOCATIONS FROM SOURCE 3 FOR PERIOD 1 m < PULP _S_AfcLL.C-G_ PRODUCT TON PROD YT= 5 YT= 6 YT_ 7 . YT= 8 YT= 9 - XT- X Y P XI RETURN X Y P XI RETURN X Y P XI RETURN X Y P XI RETURN X Y P XI RETURN C 0 5 0 0 111190. 0 6 0 0 133428. 0 7 3 0 151666. 0 8 3 C 169904. 0 0 0 0 188568. 1 __1^ _5_ a..i 1.04-0-9.5-. . 1 6 _2„1_ . 1.2.19,55.. 1 7 2 1 139916. 1 -.8— _2_1_. 15.7.936.. • 0 ..OL JCL_0__. _1.844_19. ... 2 2 5 2 2 96741. 2 6 2 2 113857. 2 7 2 2 131222. 2 8 2 2 148762. 0 1 0 0 188217. 3 5 2 3 99C00. 3 6 2 3 108000. 3 7 2 3 124667. 3 8 2 3 141619. 1 2 G 1 191592. 4 4 5 2 4 109000. 4 6 2 4 118000. 4 7 2 4 127000. 4 8 2 4 136000. 4 8 2 4 192619. 5 5 5 2 5 1190CC. 5 6 2 5 128000. 5 7 2 5 137000. 5 8 2 5 146000. 4 8 2 4 20C273. 6 6 _5. 2 6 129000. 6 6 2 6 138000. 6 7 2 6 147000. 6 _8_„ _2__6__. J.5 6.0-0-0. 5 :_a. 2 5 210273. 7 7 5 2 6 141000. 7 6 2 6 150000. 7 7 2 6 159000. 7 8 2 6 168000. 6 8 . 2 6 220273. 8 8 5 2 6 15300G. 8 6 2 6 162000. 8 7 2 6 171000. 7 8 2 6 180500. 6 8 2 6 230739. 9 9 5 2 6 165000 . 9 6 2 6 174CGC 8 7 2 6 183500. 7 8 2 6 193000. 6 8 2 6 240916. 10 10 5 2 6 17 7OC0. 9 6 2 6 186500. 8 7 2 6 196000 . 7 8 2 6 205500. 7 8 2 6 252916. 1.1 10 5 2 6 189500 • 9 6 2 6 19.9-0 OH. 8 7 2 6 -2-0-85-0.D-. -7 _ 8 _ 2 6 2.18000.. 7 _8_ _2_.6__. .2.65416.. 12 10 5 2 6 202000. 9 6 2 6 211500. 8 7 2 6 221000. 7 8 2 6 230500. 5 6 2 5 276815. 13 10 5 2 6 214500. 9 6 2 6 224000. 8- 7 2 6 233500 . 7 8 2 6 243000. 6 6 2 6 286815. 14 10 5 2 6 227000 . 9 6 2 6 236500. 8 7 2 6 246000. 6 6 2 6 288007. 6 7 2 6 297007. 15 1C 5 2 6 239500. 9 6 2 6 249000. 7 5 2 6 291007 . 7 6 2 6 300C07. 7 7 2 6 3090G7. 16. _1.Q_ _5_ __2_6. 23___0_. 8 4 2 6 _ 2-9 4-0-0 7_. 8 5 2 6 303007. 8 _ 6 _ _2_-6 _31.2.007_._ _8__ _7_. _2._6 ..32 10.0.7. 17 c 3 2 6 297007. 9 4 2 6 306007. 9 5 2 6 315007. 9 6 2 6 324007. 8 7 2 6 3335C7. 18 10 3 2 6 309007. 10 4 2 6 31800 7. 10 5 2 6 32 700 7. 9 6 2 6 336507. 8 7 2 6 346007. 19 11 3 2 6 321007. 11 4 2 6 330007. 10 5 2 6 3 39 507. 9 6 2 6 349007. 8 7 2 6 358507. 20 12 3 2 6 333007. 11 4 2 6 342507. 5 7 2 5 352000. 5 8 2 5 361000. 8 7 2 6 37 1007. 21 6 —5_ __2__6_ __34400G . 6 __6_ 2 6 3 53000. 6 7 2 6 362000. 6 _8_ _2_6_._ 3.71000. ... -8_ __7_. . 2 6 -.3.8350.7.. . 22 7 5 2 6 3560CC. 7 6 2 6 365000. 7 7 2 6 374000. 7 8 2 6 383000. 8 7 2 6 396GC7. 23 8 5 2 6 368000. 8 6 2 6 377000. 8 7 2 6 386000. 9 6 2 6 399007. 6 8 2 6 445739. 24 9 5 2 6 380000. 9 6 2 6 389000. 10 5 2 6 402C07. 6 7 2 6 446916. 6 8 2 6 455916. 25 10 5 2 6 392000. 11 4 2 6 405007. 7 6 2 6 449916. 7 7 2 6 458916. 7 8 2 6 467916. 26 1.2_ _3_ 2 6 408007. 8 _5_ 2 6 452916. 8 6 2 6 461916. 8 7 2 6 4.70.9.1.6.. ._ 4 6 2 4- 4818.15-. . 27 2 2 5 455815. 5 3 . 2 5 464815. 5 4 2 5 473815. 5 5 2 5 482815. c 6 2 5 491815. 28 6 2 "2 6 465815. 6 o 2 6 474815. 6 4 2 6 483815. 6 5 2 6 492815. 6 6 2 6 501815. 29 6 3 2 6 476007. 6 4 2 6 485007. 6 5 2 6 494007. 6 6 2 6 503007. 6 7 2 6 512007. 30 7 3 2 6 488007 . 7 4 2 6 497C07. 7 5 2 6 506007. 7 6 2 6 5150G7. 7 7 2 6 5240G7. 31 __8_ __3_ 2 6 500007. 8 4 -2 6 ..509.Q07.. 8 5 2 6 518007. 8 _6_ 2 6 _».27OJ0 7.._ 8. _-7— _2_6 .53.6007. 32 9 3 2 6 512CC7. 9 4 2 6 521007. 9 5 2 6 530007. 9 6 2 6 5 39007. 33 10 3 2 6 5240C7. 10 4 2 6 533C07. 10 5 2 6 542007. 34 11 3 2 6 5360C7. 11 4 2 6 545007. r 35 12 3 2 6 5480C7. -ALLOCATIONS FROM SOURCE 3 FOR PERIOD 1 < .. PULP SAKLJCG.. PRODUCTION _ .... ... PROD YT=10 YT=11 YT=12 YT=13 YT = 14 -XT- X Y P XI RETURN X Y P XI RETURN X. Y P XI RETURN X Y P XI RETURN X Y P XI RETURN 0 0 0 0 0 2029 67. 0 1 0 0 225225. 0 2 o o' 247463. 0 3 0 0 269701. 0 4 0 0 291939. 1 0 1 0 0 206657. _ 1 2 0 1 228568. 1 2 0 1_ -242.98.7. 1 _ 3 _ 0 1 263915. 1 4 0 1 2853.68.. 2 1 2 0 1 210007 . 1 2 0 1 224419. 1 3 0 1 245348. 1 4 0 1 266800. 2 4 0 2 282987. 3 1 2 0 1 205979. 1 3 0 1 226908. 3 8 2 3 248094. 2 4 0 2 264419. 2 5 2 2 281161. 4 4 8 2 4 - 209237. 4 8 2 4 225856. 4 8 2 4 242475. 2 5 2 2 262721. 3 6 2 3 278GG7. 5 4 8 2 4 211928. 4 8 2 4 224008. • 3 6 2 3 245216. 3 6 2 3 259592. • 3 6 2 3 273979. 6 5 8 .2. 5 £21928. 5 __.8_ ._2_ ..5... 2.3 4 m s . . .. 4 _8_ .2 4 . 253455.. _ 4 8 2 4 .262948. 4. _ 8 _ _2_ .4 273216. 7 6 8 2 6 231928. 6 8 2 6 244008. 5 8 2 5 263455. 5 8 2 5 272948. 4 8 2 4 282783. 8 7 8 2 6 243928. 7 8 2 6 256008. 6 8 2 6 273455. 6 8 2 6 282948. 5 8 2 292783. 9 7 8 2 6 256428. 7 8 2 6 268508. 4 8 2 4 283685. 6 8 2 6 293358. 6 8 2 6 302783. 10 4 8 2 4 267031. 4 8 2 4 279218. 4 8 2 4 293275. 6 8 2 6 304023. 6 8 2 6 313304. 11 4 7 2 4 275 815 . 4 8 2 4 284815. 5 8 2 5 30 3.275. 6 _ 8 _ 2. 6 3.14164. 7 _2_ .6 ..32 5.304. 12 c 7 2 5 28 5815. 5 8 2 5 294815. 6 8 2 6 313275. 7 8 2 6 326164. 7 8 2 6 337804. 13 6 7 2 6 295815. 6 8 2 6 3C4815. 7 8 2 6 325275. 7 8 2 6 338664. 7 8 2 6 350304. 14 6 8 2 6 306007. 7 8 2 6 316815. 7 8 2 6 337775. 7 8 2 6 351164. 7 8 2 6 362804. 15 7 8 2 6 318007. 7 8 2 6 229315. 7 8 2 6 350275. 7 8 2 6 363664. 7 8 2 6 375304. 1 6 7 8 2—6 330507.' 7 8 2 6 341815. 7 8 2 6 36.2775. ... _ 7 8. _2 6 .3.76.1.64. 7 8 2 6 38.7-80.4. 17 7 8 2 6 343007. 7 8 2 6 354215. 7 8 2 6 375275. 7 8 2 6 388664. 7 8 2 6 40C304. 18 7 8 2 6 355507. 7 8 2 6 366815. 7 8 2 6 387775. 7 8 2 6 401164. 4 8 2 4 457783. 19 7 8 2 6 368CC7. 7 8 2 6 379315 . 7 8 2 6 40C275. 4 8 2 4 457948. 4 8 2 4 467783. 20 7 8 2 6 38C507 . 7 8 2 6 3 91815. 4 8 2 4 458275. 4 8 2 4 467948. 4 8 2 4 477783. 21 7 __8_ 2 6 393007. ..5 8 2 5 449.008 . _ _ 4 8 ^2^4. 46 8275. 4. 3 _2_4. 4.77948. 4. .8.. 2. .4 487783. 22 6 8 2 6 446928. 6 8 2 6 459008. 4 8 2 4 478275. 5 8 2 5 487948. 4 8 2 4 497783. 23 7 8 2 6 458928. 4 8 2 4 469815. 4 8 2 4 488275. 6 8 2 6 497948. 5 8 2 5 507783. 24 4 7 2 4 47C815. 4 8 2 4 479815. 4 8 2 4 498275 . 6 8 2 6 508358. . 6 8 2 6 517783. 25 4 7 2 4 480815. 4 8 2 4 489815. 4 8 2 4 506275. 6 8 2 6 519023. 6 8 2 6 528304. 26 4 7 2 4 490815. 4 8 ? ' 4 493815.. 5 8 2.3 3JjB__7_5 . 6 8 2 6. _5.2S1.64. 7 8___ _2_ .6 .54.03.04. 27 5 7 2 5 5Q0815. 5 8 2 5 509815. 6 8 2 6 528275. 7 8 2 6 541164. 28 6 7 2 6 510815. 6 8 2 6 519815. 7 8 2 6 54C275. 29 6 8 2 6 5210C7. 7 8 2 6 531815. 30 7 8 2 6 5330C7 . ALLOCATIONS FROM SOURCE 3 FOR PERIOD 1 160 P.ULP . __ SAMI PC- PRODUCTION . . PROD YT-15 YT=16 YT=17 YT=18 YT=19 -XT- X Y p XI RETURN X Y P XI RETURN X Y P XI RETURN X Y P XI RETURN X Y P XI RETURN 0 0 5 0 0 314177. 0 6 0 0 336415. 0 7 3 0 354653. 0 8 3 0 372891. J L . _ _ 1_ 5 0 1 3.07.0-8.2.. 1 6 .2.. 1 3 24946. 1 7 •p 1 34-2-903.. 1 __8_ 2 J 36092.3. 2 2 5 2 2 299728. 2 6 2 2 316844. 2 7 2 2 334209. 2 8 2 2 351749. 3 3 6 2 3 296568. 3 6 2 3 310987. 3 7 2 3 327654. 3 8 2 3 344606. 4 3 6 2 3 292419. 3 7 2 3 309086. 4 8 2 4 324568. 4 8 2 4 338987. 5 -a 7 2 •a 290646. 4 8 2 4 306007. 4 8 2 4 320419. 5 8 2 5 348987. 6 __8_ ...2 4 28759?. 4 .8 -2_ 4 301979. 5 8 2 5 330419. 6 __8_ _2_.6 358987.. 7 _____ 4 8 2 4 292266. 5 8 2 5 311979. 6 8 2 6 340419. 7 8 2 6 370987. 8 5 8 2 5 302266. 6 8 2 6 321979. 7 8 2 6 352419. 7 8 2 6 383487. 9 6 8 2 6 312266. 7 8 2 6 333979. 7 8 2 6 364919. 7 8 2 6 395987. 10 7 8 2 6 324266. 7 8 2 6 346479. 7, 8 2 6 377419. 7 8 2 6 408487. 11 7_ __a. _2_6 33.6J66 . 7 8 _2_ 6 358.979. 7 8 _2_ 6 389919. 7 _8_ 2 6 4209.8 7. 12 7 8 2 6 349266. 7 8 2 6 371479. 7 8 2 6 402419 . 7 8 2 6 433487. 13 7 8 2 6 361766. 7 8 2 6 383979. 7 8 2 6 414919. 7 8 2 6 445987. 14 7 8 2 6 3742 66. 7 8 2 6 396479. 7 8 2 6 427419. 4 8 2 4 503987. 15 7 8 2 6 386766. 7 8 2 6 408979. 4 8 2 4 485419. 4 8 2 4 513987. J 6 7 8 _2_Ja_ 39 92.6.6 . 4 8 2 4 4-6.6.9.79. 4 8 ? 4 495419. 4 8 2 4 5.2.35.87.. 17 4 8 2 4 457266. 4 8 2 4 476979. 4 8 2 4 505419. 4 8 2 4 533987. 18 4 8 2 4 467266. 4 8 2 4 486979. 4 8 2 4 515419. 4 8 2 4 543987. 19 4 8 2 4 477266. 4 8 2 4 496979. 4 8 2 4 525419. 4 8 2 4 553987. 20 4 8 2 4 487266. 4 8 2 4 506979. 4 8 2 4 535419. 5 8 2 5 563987. .21 4 8 ...2—4.. . 4-9 7.2.6.6.. 4 8 ? 4 516,979. 5 8 2 5 545419. 6 _8_ _2_ 6-. 573J3.8-7.. 22 4 8 2 4 507266. 5 8 2 5 526979. 6 8 2 6 555419. 7 8 2 6 585987. 23 5 8 2 5 517266. 6 8 2 6 536979. 7 8 2 6 567419. 24 6 8 2 6 527266. 7 8 2 6 548979. 25 7 8 2 6 539266. ALLOCATIONS FROM SOURCE 4 FOR PERIOD 1 161 " : < _e.u._e _ „ ; SAWi,(iiLj-.emuj:TiQN • PROD VT= 0 YT= 1 YT= 2 YT= 3 YT- 4 -XT- X Y RETURN X Y RETURN " X Y RETURN X Y RETURN X Y RETURN 0 0 0 C. C O 22238. 0 0 44476. 0 0 66714. C 0 88952. 1 0 Oj 1_5J_0_._ 0 0 27000. 0 0 _AOjDM. _ 0 _ 0 60928. i l _ f l L _JL23.8..1... 2 0 0 25000. 0 , 0 39500. 0 0 52500. 0 0 67C00. 0 0 80000. 3 C O 37500. 0 0 52000. 0 0 65000- 0 0 79500. 0 0 90000. 4 0 0 50000. 0 0 64500. 0 0 77500. 0 0 91000. 0 0 100000. 5 G O 62500. 0 0 77000. 0 0 90000. 0 0 101000. C 0 110000. _ _ 6 . . 0 Q ZSOflft* 0 0 8 9 5 0 0 . _ J L 0 1 0 2 0 0 0 . _J5_ _ Q XlXSfflL* 0 0 1 2 0 0 0 0 . 7 0 0 89000. 0 0 102000. 0 0 .114000. 0 0 123000. 0 0 132000. 8 . 0 0 103000. 0 0 116000. 0 0 126000. 0 0 135000. 0 0 144000. 9 0 0 117000. 0 0 129000. 0 0 . 138000. 0 0 147000. 0 0 156000. 10 0 0 131000. 0 0 141000. 0 0 150000. 0 0 159000. 0 0 168000. 1.1 J P - _ . J P , 1 4j4J3JQj_* . . 0 Q___ ......1.5.300J1.. 0___0 I J A Z Q O J X . 0__JOL_ XTJLJl&O.., 0 0 1 .8J0.0.0.0... 12 0 0 156000. 0 0 165000. 0 0 174000. 0 0 183000. G 0 192500. 13 0 0 168000. 0 0 177000. 0 0 186000. 0 0 195500. 0 0' 205000. 14 0 0 180000. 0 0 189000. 0 0 198500. 0 0 . 208000. 0 0 217500. 15 0 0 192000. 0 0 2C15CC. 0 C 211000. 0 0 220500. 0 0 230000. 1.6 0 0__ 2JQA5JOO......._. Q 0 _2.1AQ____,____ _ Q 0_ 2235iOg____._. Q^^^J2333m^ G__J3. 242500. 17 0 0 2 17000. 0 0 226500. 0 0 236000. 0 0 245500. 0 0 255000. 18 0 0 229500. 0 0 239000. 0 0 248500. 0 G 258000. 1 0 275000. 19 0 0 242000. 0 0 251500. 0 0 261000.. 1 0 278000. 2 0 295000. 20 0 0 254500. 0 0 264000. 1 0 281000. 2 0 298000. 3 0 315000. . 2 1 _J_ Q _L6J_0.0JDL. 1 £ _ 2.8A0QLCL.__ 2 C 1 0 1 , 0 0 0 . 3. . 0 __3X8JDUQ_. _ 0 _ _ 0 3 3 5 . 0 0 . . 0 . . . . 22 1 0 287000. 2 0 304000. 3 0 321000. 0 0 338000. 0 0 347CC0. 23 2 0 307000. 3 0 324000. 0 0 341000. 0 0 350000. 0 0 359000. 24 3 0 327000. 0 0 344000 . 0 0 353000. 0 0 362000. 0 0 371000. 25 0 0 347000. 0 0 356000. 0 0 365000. G O 374000. 0 0 383000. 2b JO 0- MADC-CU-, 0 0 3i_a0_3_L. 0 0 ..OQJMt. • " _0—JO. ; 3J.6JB110.___. 0 0 __3_9.5J3.0.0_.. 27 0 0 3710.00. 0 0 380000. 0 0 389000. 0 0 398000, 0 0 411007. 28 0 0 383000. 0 0 392000. 0 0 4 0 1 0 0 0 . C 0 " 414007. 1 0 431007. 29 0 0 " 395000. 0 0 404000. ' 0 0 41700 7. 1 0 434007. 2 0 451007. 30 0 0 407000. 0 0 421503. 1 0 437007. 2 . 0 454007. 3 0 471007. .3.1 __J0- 0 • • 4_26.0.0.g. • ,1 ff 4JJ3fl3L. 2 O___^3J__0J_._ 3L_j6______3__3U_X_ 0 0 • 3-SQ.0.0J... 32 1 0 446000. 2 C 461503. 3 0 477007. 0 0 494007. G O 503007. 33 2 0 466000. 3 0 481503. 0 0 497007. 0 0 506007. 0 0 515007. 34 3 0 486000. 0 0 501503. 0 0 509007. 0 0 518007. 0 0 527007. 35 0 0 506000. 0 0 513503. 0 0 521007. C 0 530007. 0 0 539007. 3J6_ _ Q 0 5JL8.QJ..0. : 0 0 , 5.2J.5_B-t 0 _ C J52amJL. _ 0 fi 5___.0XL7_. 0. 0 — 5.510.07... 37 . Q ^ - - 0 - 530000. C 0 5.37503. 0 0 545007. 0 0 554007. 1 0 571007. 38 0 0 542000.. 0 0 549503. 0 0 557007. 1 0 574007. 2 0 591007. 33. Q Q 554000. 0 0 561503. 1 0 5 7700 7. 2 0 594007. 3 0 6 1 1 0 0 7 . J ALLOCATIONS FROM SOURCE 4 FOR PERIOD 1 1 6 2 -< P . U L . P _ PROD -XT-Y T= 0 Y RETURN YT= 1 RETURN . JSu&i_LOX__J_BLaauXXLQ_J YT= 2 X Y REtURN YT- 3 RETURN YT= 4 X Y RETURN 40 0 0 566000. 1 0 5815C3. 2 C 597007. 3 0 614007. 41 1 0 586000. 0 601503. 3 0 617007. 4 0 6.3.4 0.0 7_, 42 2 0 606000. 3 0 621503. 4 0 637007. 43 3 0 • 626000 . 4 0 641503. 44 4 0 646000. 4 0 631007. ALLOCATIONS FROM SOURCE 4 FOP PERIOD 1 163 . . < .P.UL.P S.A.W.L.C.G_P_RC.D.U.CJJ.C.N : PROD YT= 5 YT= 6 Y 7= 7 YT= 8 YT= 9 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 0 111190. , 0 0 133428. 0 G 151666. C 0 169904. 0 0 188568. 1 G_ 0 104095. 0 0 121959. 0 0 139916. 0 0 157936. 0 1 181.43.9. . ... 2 0 0 96741. G 0 113857. 0 0 131222. 0 0 148762. 0 1 172265. 3 0 0 99000. 0 0 108000. 0 G 124667. c G 141619. 0 1 165122. 4 0 0 109000. C 0 118000. 0 0 127000. 0 0 136000 . 0 1 159503. 5 0 0 119000. 0 0 128000 . 0 0 137000. 0 0 1460GO. 0 1 169503. 6 0 0 1290D0. 0 0 138000. 0 0 1470DG. 0 0 156000 . 0 1 1X9.50 3. -7 0 0 141000. 0 0 150000. 0 0 15900G. 0 0 168000. 0 1 191503. 8 0 0 153000. 0 0 162000. 0 0 171000. 0 0 180500. C 1 204003. 9 0 0 16 5000. 0 0 174000. 0 0 183500. 0 0 193000 . 0 1 216 503. 10 0 0 177C00. c 0 186500. 0 0 196000. c G 2055CG. C 1 229003. 1 1 0 0 189 500. 0 0 199000. 0 0 208500. 0 0 218000. 0 1 241.5.0.3.. 12 0 0 202000. 0 0 211500. G 0 221000. 0 G 230500. 0 1 254003. 13 0 0 214500. 0 0 224000. 0 0 233500. 0 0 243000. 0 1 266503. 14 0 0 2 27000. 0 0 236500. G 0 246000. 1 0 263000. 1 1 286503. 15 0 0 23950C. C 0 249000. 1 0 266000 . 2 0 283000. 2 1 306503. _16_ 0 0 252000. 1 0 269000. 2 0 2 86000. 3 0 303000 . 0 0 32100 7.. 17 1 0 2 7200 0. 2 0 289000. 3 0 3G6000. 4 G 323000. 0 0 333507. 18 2 0 292000. 2 0 3 09000. 4 0 326000. G 0 336507. G 0 346007. 19 3 0 312000. 4 0 329000. 0 G 339507. . C G 349007. 0 0 358 50 7. 20 4 G .33 2000. 0 0 342507. 0 C 352000. 0 0 361000. c 0 371007. Z l 0 _0 344000. n 0 3.5.3G.0.G. 0 0 362000. 0 0 371000. 0 n .38350.7. 22 0 0 356G00. 0 0 365000 . 0 0 374000. G 0 383G00. 0 0 396007. 23 0 0 368000. G 0 377000. G 0 386000. 0 0 399007. 1 0 416007. 24 0 0 380000. C 0 389000. 0 0 402007. 1 0 419007. 2 0 436007. 25 0 0 392000. c 0 40 5007. 1 0 422007. 2 G 439GC7. 3 0 456007. Z 6 „ 0 0 40800 7. 1 0 425007. 2 0 4420G7. 3 _ 0 _ ___45.9_0.CL7_. 4 0 476007. 27 1 0 4280G7. 2 0 445007. 3 0 462007. 4 0 479CC7. 0 0 491815. 28 2 0 448CG7. 3 0 465GC7. 4 0 482007. 0 G 492815. 0 0 501815. 29 3 0 468007. 0 0 485007 . 0 0 494007. C 0 503007. 0 0 512007. 30 0 0 4880C7. G G 497G07. 0 0 506007. 0 0 515007. 0 0 524007. .3.1 0 0 5000G7. 0 0 50 900 7. 0 0 518007. n 0 527007. 0 0 536007. 3 2 0 0 512GC7. G 0 521007. 0 0 530007. 0 0 539007. 1 0 556007. 33 0 0 5 2400 7. 0 0 533G07. 0 0 542007. 1 0 5590G7. 2 0 576007. 34 0 0 5360G7. G 0 545007. 1 0 562007. 2 0 579007. 3 0 596007. 35 0 0 5 4 80C7. 1 0 565007. 2 0 582007. 3 0 599007 . 4 0 6160C7. .36 1 _ J Q . 568007. 2 0 5R5007. 3 G 6G2007. 4 G 619GG7. 37 2 0 588CC7. _ 0 605007. 4 0 622007 . 38 3 0 608007. 4 0 625007. 39 4 0 628007. ALLOCATIONS FROM SOURCE 4 FOR PERIOD 1 T64 PJJ.L.P SAW.L.Q.G PRODUCTION PROD Y T = 1 0 YT=11 YT-12 YT=13 YT=14 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN . 0 0 0 202987. 0 0 225225. 0 0 24746 3. 0 0 269701. G 0 291939. _ 1 . O L 2 204943. 0 __3_ . 2.2.84 4.6.. 0 C 242987. G 0 263915. 0 0 235368. 2 G 2 195768. 0 3 219272. 0 4 242775. G 0 266800. 0 0 282987. 3 0 2 .188626. 0 3 212129. 0 4 235633. G 0 264419. 0 0 281161. 4 • 0 2 183007. 0 3 206510. 0 4 230014. G 0 262721. 0 0 278007. 5 G 2 192007. 0 3 216510. 0 4 240014. 0 0 259592. 0 G 273979. 6 0 2 203007. 0 3 226510. 0 4 250014. 0 0 262948. 0 0 273216. 7 0 2 2150C7. 0 3 238510. 0 4 262014. 0 0 272948. c G 282783. 8 0 2 227507. 0 3 251010. 0 0 273455. 0 0 282948. 0 0 292783. 9 0 2 24000 7. 0 3 263510. 0 0 283685. C 0 293358. 0 0 302783. 10 0 2 252507. c 3 276010. 0 0 293275. 0 0 304023. 0 0 313304. J L 1 . - 0 2 265007. 0 0 284R15. 0 0 303275. G _ 0 . 314164. 0 _ 0 . _ 325.304. 12 C 2 27 7 5C7. c 0 294815. 0 0 313275. 0 0 326164. c 0 337 804. 13 0 2 290007. 0 0 304815. 0 0 325275. 0 0 338664. 0 0 350304. 14 0 0 306007. 0 0 316815. 0 0 337 77 5. 0 0 351164. 0 0 362804. 15 0 0 3 18007 . 0 G 329315. 0 0 35C275. 0 0 363664. 0 0 375304. 16 0 0 33X1507. Q 0 34.L8 1.5. QL 0 362775. G _0_ 376164. 0 0 387804. 17 0 0 343007. 0 0 354315. 0 0 375275. 0 G 388664. 0 0 400304. 18 0 0 355507. 0 0 366815. 0 0 387775. 0 G 401164. 1 G 42030 4. 19 0 0 368007. 0 0 379315. 0 0 400275. 1 0 421164. 2 0 440304. 20 0 0 380507. 0 0 391815. 0 1 415318. 0 2 438822. 3 0 460 304. 2.1 ..JO 0 393007. 1 0 411815 . 1 I 435318. 1 2 458822. 4 _0_ 4.8.0.30-4.. 22 1 0 413007. 2 0 431815. 2 1 455318. 2 2 478822. 0 0 497783. 23 2 0 43300 7. 3 0 451815. 3 1 475318. c 0 497948. 0 0 507783. 24 3 0 4530C7. 4 0 471815. 0 0 498275. G 0 508358. 0 0 517783. 25 4 0 473007. 0 0 489 815. 0 0 5G8275. 0 0 519023. 0 0 5 2830 4. 26 0 0 49081 5. ._0 0 499815. 0 0 518275. 0 0 529164. 0 0 540304. 27 G 0 5G0815. G 0 509815. 0 0 528275. 0 0 541164. 1 0 560304. 28 0 0 510815. 0 0 519815. 0 0 540275. 1 0 561164. 2 0 580304. 29 0 0 5 21GC7. G 0 531815. 0 1 555318. 0 2 578822. 3 0 600304. 30 0 0 533007. 1 0 551815. 1 1 5 7 5 318. 1 2 598822. 4 0 620304. . 3 1 . 0 5530C7. ? 0 571815. 2 1 595318 . 2 2 618822. 32 2 G 573007. 3 0 591815. 3 1 615318. 33 3 G 593GG7. 4 0 611815. 34 4 0 6130C7. ALLOCATIONS FROM SOURCE 4 FOR PERIOD 1 165 < ... P.U.LP SAWLCG PRODUCT T ON PROD YT = 15 YT=16 YT=17 YT-18 YT=19 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 0 314177. 0 0 336415. 0 0 354653. 0 0 372891. 0 1 396394. L 0 0 3070 82. 0 _ .a 324946. 0 0 342903. 0 0 360923. 0 _1_ ___3_84_»26.. 2 0 0 299728. 0 0 316844. 0 0 334209. 0 0 351749. ' 0 1 375252. 3 0 0 29656 8. 0 0 310987. 0 0 327654. 0 0 344606. 0 1 368109. 4 0 0 292419. 0 0 309086. 0 0 324568. 0 0 33 8987. 0 1 362490. 5 0 0 290646. 0 0 306007. 0 0 320419. 0 1 343923. c 2 367426. .6 0 .0 287592. 0 0 301979. 0 1 325483. G 2 348986. 0 3 372490. 7 0 0 292266. 0 0 311979. 0 1 335483. c 2 358986. 0 3 382490. 8 0 0 302266. G 0 321979. 0 1 345483. 0 2 368986. 0 3 392490. 9 0 0 312266. 0 0 333979. 0 1 357483. 0 2 380986. 0 3 4-04490. 10 0 0 324266. 0 0 346479. 0 1 369983. 0 2 393486. 0 3 416990. 1.1 _ o 0 336766. 0 0 3 5 8979. 0 1 382483. 0 2 405986. 0 3 429490. 12 0 0 349266. 0 0 371479. 0 1 394983. 0 2 418486. 0 3 441990. 13 0 0 361766. 0 0 38 3979. 0 1 407483. 0 2 430986. 0 3 454490. 14 0 0 374266. 0 0 396479. 0 1 419983. 0 2 443486. 0 3 466990. 15 0 0 386766. 0 0 408979. 0 1 432483. 0 2 455986. 0 3 479490. 16 0 0 395.2.6.6. 0 1 422769. 0 2 446273. 0 3 469776. 0 4 493280. 17 1 0 419266. 1 1 442769. 1 2 ' 466273. 1 3 489776. 0 3 -. 547490. 18 2 0 439266. 2 1 462769. 2 2 486273. 0 2 533986. 0 3 557490. 19 3 0 459266. 3 1 482769. 0 1 520483. 0 2 543986. 0 3 567490. 20 4 0 4 7926 6. 0 0 506979. 0 1 53G483. 0 2 553986. 0 3 577490. 21 0 0 4J9_I2_6.6_. 0 0 51 6-979. 0 1 540483. 0 2 563986. 0 3 58749.0. 22 0 0 507266. 0 0 526979. 0 1 550483. 0 2 573986. 0 3 597490. 23 0 0 517266. 0 0 536979. 0 1 560483. 0 2 583986. 0 3 607490. 24 0 0 527266. 0 0 548979. 0 1 572483. 0 2 595986. 0 3 619490. 25 0 0 539266. 0 1 562769. 0 2 586273. 0 3 609776. 0 4 6 33 280. 26 .1 „0_ 559266. 1 1 582769. 1 2 606273. 1 3 629776. 27 2 0 579266. 2 1 602769. 2 2 626273. 28 3 0 599266. 3 1 622769. 29 4 0 619266. ALLOCATIONS FROM SOURCE 5 FOR PERIOD 1 166 < PULP . : SAWLOG PRODUCTION ______ .. PROD YT= 0 YT= 1 YT= 2 YT= 3 YT= 4 -XT- X Y RETURN > Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 0 0. 0 0 22238. G 0 44476. 0 G 66714. C 0 88952. 1-: 0 0 125GO. 0 0 27000. 0 0 40000. 0 0 60928. 0 0 .82 381. 2 0 0 25000. 0 0 39500. 0 0 52500. 0 0 670GG. 0 0 80000.' 3 3 0 36000. G 0 52000. 0 0 65000. 0 0 ' 79500. 0 0 9000C. 4 4 0 48000. 3 0 63000. 3 0 76000. G 0 91000. 0 G 100000. 5 . 5 0 60G00. 4 c 75000. 4 0 88000. G 0 101000. G 0 110000. 6 5 0 72500. 5 0 87000. 5 0 100000. C _£» 111000. 0 0 120000. 7 5 0 85000. 5 0 99500. 5 0 112500 . 0 0 123000. G 0 13200C. 8 5 0 97500. 5 0 112000. 5 0 125000. G 0 135000. 0 0 144000. 9 5 0 110000. 5 0 124500. 5 0 137500. 0 G 147000. 0 0 156000. 10 5 0 122500. 0 137000. 0 0 15C000. 0 0 159000. C 0 168000. 11 5 135000. 5 0 149500. 0 0 16 2000. 0 _0 0 1 71000. 0 _ 0 _ 1.80000. 12 5 0 149000. 5 0 162000. 0 0 174000. 0 18 3000. 3 0 19200C. 13 5 0 163000. 5 0 17 6000. 0 0 186000. 3 0 195000. 3 0 204000. 14 5 0 177000. 0 0 189000. 3 0 198000. ra 0 2G7000. 3 0 216000. 15 5 0 191000. 3 0 2C10C0. 3 0 21GG0G. 3 0 219000. 4 0 2 28000. 16 3 0 2 04000. 3 0 213000 . 3 0 222000. 4 0 231000. 5 0. 240000. 17 3 0 216000. 3 0 225000. 4 0 234000. 5 0 243000. 5 0 252500. 18 3 G 228000. 4 0 237000. 5 0 24600G. 5 0 255500. 5 0 265000. 19 4 0 240000. 5 0 249000. 5 0 258500. 5 0 268000. 5 0 2775GC. 20 5 0 25200G. 5 0 261500. 5 0 271000. 5 0 280500. 5 0 290000. 2.1._ 5_ 0 2645GG. 5 0 274000 . 5 0 283500. 5 ..G___ 293000. 5 0 ..3.02 500. 22 5 0 27700G. c 0 286500. 5 c 296000. 5 0 305500. e . 0 31500G. 23 5 0 289500. 5 0 299000. 5 G 308500* 5 0 318G00. 6 0 334000. 24 5 0 3C200G. 5 0 311500. 5 0 321000. 6 0 337000. 7 0 353C0G. 25 5 0 314500. 5 0 324000. 6 C 340000. 7 0 356000. 8 0 372000. 26 5 0 327000. 6 0 343000. 7 0 359000. 8 0 37.5J3.Oa. 9 _jO__ 3_9_i o.o.a. 27 6 0 346000. 7 0 362000. . 8 0 378000. 9 0 394000. 3 0 407000. 28 7 0 " 365000. 8 0 381000. 9 0 39700G. 3 0 410000. 3 G 419000. 29 8 0 384000. 9 0 400000. 3 0 413000. 3 0 422000. 3 0 431C0G. 30 9 0 403000. 3 0 416000. 3 0 425000. 3 0 434000. 4 0 443000. 31 .3 0 419000. 3 0 428000. 3 0 437000. 4 0 446000. 5 0 4.5.5.0.0 0. 32 3 0 4 31000. 3 0 440000. 4 G 449000. 5 0 458000. 5 G 471007. 33 3 0 443000. 4 0 452000. 5 0 461000. -> 0 474007. 6 0 490007. 34 4 0 '455000. . 5 0 464000. 5 0 477007. 6 0 4930G7. 7 0 509007. 3 5 5 0 467000. 5 0 481503 . 6 0 496007. 7 0 512007. 8 0 528007. 36 5 0 4860G0. 6 0 500503. 7 0 5 1 500 7 . 8 .0 531007. 9 .0 54-7.0G7. 37 6 0 505000. 7 0 519503. 8 0 5 3400 7. 9 0 550007. 3 0 5 6300 7. 38 7 0 5 24000. 8 0 538503 . 9 0 553007. 3 0 5660G7. 3 0 575C07. 39 8 0 543000. 9 0 5575G3. ? 0 569007. 3 0 578007. 0 587G07. J ALLOCATIONS FROM SOURCE 5 FOR PERIOD 1 167 . — , . — : : . . < - -PULP- —-'_- . S.4ViLjDjG_PJ?aDJU-CXL0j_ - . PRO D YT = C YT= 1 YT= 2 YT= 3 YT= 4 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 40 .9 0 562000. 3 0 573503. 3 0 5 81007. 3 0 590007. 4 0 599007. 4.1 . - 3 n 578000. 3 D 585503 . 3 "0 59300 7. 4 0 6020O7. 5 0 6.L1.0O7-.— 42 3 0 590000. 3 0 597503. 4 0 605007. 5 0 614007. 6 0 63000 7. 43 3 0 602000. 4 0 609503. 5 0 617007. 6 0 6 3 3007. 7 0 649007. 44 4 0 614000. 5 0 621503. 6 0 6 36007. 7 0 652007. 8 0 668007. / A L L O C A T I O N S FROM S O U R C E 5 F O R P E R I O D 1 m : : : < P U L . P L : S A W . U Q j L _ e . _ D m c i ^ PROD Y T = 5 YT= 6 Y T = 7 YT= 8 Y T = 9 - X T - X Y R E T U R N X Y R E T U R N X Y R E T U R N X Y R E T U R N X Y R E T U R N 0 0 0 1 1 1 1 9 0 . 0 0 1 3 3 4 2 8 . 0 0 1 5 1 6 6 6 . G 0 1 6 9 9 G 4 . G 0 1 8 8 5 6 8 . 1 . 0. 0 1 0 4 0 9 5 . 0 0 1.2.1-9-59-.. 0 0 1 3 9 9 1 6 . 0 0 1 5 7 9 3 6 . 0 0 1 8 1 4 3 9 . 2 0 0 9 6 7 4 1 . 0 o. 1 1 3 8 5 7 . 0 Q 1 3 1 2 2 2 . 0 0 1 4 8 7 6 2 . 0 0 1 7 2 2 6 5 . 3 0 0 9 9 0 0 0 . G 0 1 0 8 G 0 0 . G 0 1 2 4 6 6 7 . 0 0 1 4 1 6 1 9 . C 0 1 6 5 1 2 2 . 4 0 0 1 C 9 0 0 0 . 0 G 1 1 8 0 0 0 . 0 0 1 2 7 0 0 0 . 0 0 1 3 6 0 0 0 . 0 0 1 5 9 5 0 3 . 5 0 0 1 1 9 0 0 0 . G 0 1 2 8 0 0 0 . 0 0 1 3 7 0 0 0 . G 0 1 4 6 0 0 0 . G 0 1 6 9 5 0 3 . ft 0 0 - 1 2 9 0 0 0 . 0 0 1 3 8 0 0 0 . 0 0 1 4 7 0 0 0 . 0 _ .0_ 1 5 6 0 0 0 . - 0 0 17-9 503 . . - ... 7 0 0 1 4 1 0 0 0 . 0 •o 1 5 0 0 0 0 . 0 0 1 5 9 0 0 0 . c G 1 6 8 G 0 O . 0 0 1 9 1 5 0 3 . 8 0 0 1 5 3 0 0 0 . 0 0 1 6 2 0 0 0 . 0 0 1 7 1 0 0 0 . 0 0 1 8 0 5 0 0 . 0 0 2 G 4 G 0 3 . 9 G 0 1 6 5 0 0 0 . 0 0 1 7 4 0 0 0 . 3 0 1 8 3 0 0 0 . 3 0 1 9 2 0 0 0 . 3 0 2 1 5 5 0 3 . 10 0 0 1 7 7 0 0 0 . 3 0 1 8 6 0 0 0 . 3 0 1 9 5 0 0 0 . 3 0 2 0 4 0 0 0 . 3 0 2 2 7 5 0 3 . . . 1.1 3 0 1 8 9 0 0 0 . 3 0 1 9 8 0 0 0 . 3 fl 2 0 7 0 0 0 . 4 -0 -_.. 2.1.60.0.0. 4 _____ 2.3.9.503-...- : ... 12 3 0 2 0 1 0 0 0 . 3 0 2 1 0 0 0 0 . 4 0 2 1 9 0 0 0 . 5 0 2 2 8 0 C G . 5 0 2 5 1 5 0 3 . 13 3 0 2 1 3 0 0 0 . 4 0 2 2 2 G G 0 . 5 0 2 3 1 0 0 0 . 5 G 2 4 0 5 0 0 . 5 0 2 6 4 0 0 3 . 14 4 0 2 2 5 0 0 0 . 0 2 3 4 0 0 0 . 5 0 2 4 3 5 0 0 . 5 0 2 5 3 0 0 0 . 5 0 2 7 6 5 0 3 . 15 5 0 23 7 0 0 0 . c 0 2 4 6 5 0 0 . 5 0 2 5 6 0 0 0 . 5 0 2 6 5 5 0 0 . 5 0 2 8 9 0 0 3 . _L6 5 0 2 4 9 5 0 0 . 5 0 2 5 9 0 0 0 . 5 0 2 6 8 5 0 0 . 5 G 2 7 8 0 0 0 . 5 __0_. ___3.0_150.3_. 17 5 0 2 6 2 0 0 0 . 5 0 2 7 1 5 0 0 . 5 0 2 8 1 0 0 0 . c G 2 9 0 5 0 0 . 5 0 3 1 4 0 0 3 . 18 5 0 2 7 4 5 0 0 . 5 0 2 8 4 0 0 0 . 5 0 2 9 3 5 0 0 . 5 0 3 0 3 0 0 0 . c 0 3 2 6 5 0 3 . 19 5 0 2 8 7 0 0 0 . 5 0 2 9 6 5 0 0 . 5 0 3 0 6 0 0 0 . 6 0 3 2 2 0 0 0 . 6 0 3 4 5 5 0 3 . 20 5 0 2 9 9 5 0 0 . c 0 3 0 9 0 0 0 . 6 0 3 2 5 0 0 0 . 7 0 3 4 1 0 0 0 . 7 0 3 6 4 5 0 3 . ___21. 5 . 0 3 1 2 0 0 0 . 6 0 3 2 8 0 0 0 . 7 0 3 4 4 0 0 0 . 8 _c_ 3 6 0 0 0 0 . 5 0 3-81-00-7. 22 6 0 3 3 1 0 0 0 . 7 0 3 4 7 0 0 0 . 8 0 3 6 3 0 0 0 . 9 0 3 7 9 0 0 0 . 5 0 3 9 3 5 0 7 . 23 7 0 3 5 0 0 0 0 . 8 0 3 6 6 0 0 0 . 9 0 3 8 2 0 0 0 . 5 0 3 9 6 5 0 7 . 5 0 4 C 6 0 0 7 . 24 8 0 3 6 9 0 0 0 . 9 0 38 5 0 0 0 . 3 0 3 9 8 0 0 0 . 3 0 4 0 7 G 0 G . 5 0 4 1 8 5 0 7 . 25 9 0 3 8 8 0 G 0 . 3 G 4 0 1 0 0 0 . 3 r '-r 4 1 0 0 0 0 . 3 0 4 1 9 0 0 0 . 5 0 4 3 1 0 0 7 . _ . 26 3 . 0 4 0 4 0 0 0 . 3 0 4 1 3 0 0 0 . 3 0 4 2 2 0 0 0 . 4 0 4 3 1 0 0 0 . 5 0 4 4 3 5 0 7 . 27 3 0 4 1 6 0 0 0 . 3 0 4 2 5 0 0 0 . 4 0 4 3 4 0 0 0 . 5 0 4 4 3 0 0 0 . c 0 4 5 6 0 0 7 . 28 3 0 4 2 8 0 0 0 . 4 0 4 3 7 0 0 0 . 5 0 4 4 6 0 0 0 . 5 0 4 5 9 0 0 7 . 6 0 4 7 5 0 0 7 . 29 4 0 4 4 0 0 0 0 . 5 0 4 4 9 0 0 0 . 5 0 4 6 2 0 0 7 . 6 0 4 7 8 C 0 7 . 7 0 4 9 4 0 0 7 . 30 5 0 4 5 2 0 0 0 . 0 4 6 5 0 0 7 . 6 0 4 8 1 0 0 7 . 7 0 4 9 7 0 0 7 . 8 0 5 1 3 0 0 7 . 3.1 5 . 0 4 6 8 0 0 7 . 6 0 4 8 4 0 0 7 . 7 0 5 0 0 0 0 7 . 8 __0__ 5 1 6 0 G 7 . 9 __0_ ._ 5 3 2 0 0 7 . -3 2 6 0 4 8 7 0 C 7 . 7 0 50 3 0 0 7 . 8 0 5 1 9 0 0 7 . 9 0 5 3 5 0 0 7 . 3 0 5 4 8 0 0 7 . 33 7 0 5 0 6 0 0 7 . 8 0 5 2 2 0 0 7 . 9 0 5 3 8 0 0 7 . 3 0 5 5 1 0 0 7 . 3 0 5 6 0 0 0 7 . 34 8 0 5 2 5 0 G 7 . 9 0 5 4 1 0 0 7 . 3 0 5 5 4 0 G 7 . 3 0 5 6 3 0 0 7 . 3 0 5 7 2 0 0 7 . 3 5 9 0 5 4 4 0 0 7 . 3 0 5 5 7 0 0 7 . 3 G 5 6 6 0 0 7 . 3 0 5 7 5 0 0 7 . 4 0 5 8 4 0 0 7 . 3 6 - 3 0 5 6 0 0 0 7 . 3 0 5 6 9 0 0 7 . 3 0 5 7 8 0 0 7 . 4 0 S R 7 G 0 7 . __5 0 59.600 7-. _ 3 7 3 0 5 7 2 0 0 7 . 3 0 5 8 1 0 0 7 . 4 0 5 9 0 0 0 7 . 5 0 5 9 9 0 0 7 . 6 0 6 1 5 0 0 7 . 38 3 0 5 8 4 0 0 7 . 4 0 5 9 3 0 0 7 . 5 0 6 0 2 G 0 7 . 6 0 6 1 8 0 0 7 . 7 0 6 3 4 0 0 7 . 3 9 4 0 5 9 6 0 0 7 . 5 0 6 0 5 0 0 7 . 6 p 6 2 1 0 0 7 . 7 0 6 3 7 0 0 7 . .... 8 ..ft. 6 5 3 0 0 7 . ._ J ALLOCATIONS FROM SOURCE 5 FOR PERIOD 1 169 P-ULP S.A-W.L.O.G_BR.Q.D.U.C.T.LON :  PROD YT= 5 YT= 6 Y T= 7 YT= 8 YT= 9 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 40 5 0 . 608Q07. 6 0 624007. 7 0 640007. 8 0 656007. 9 0 672007. 41 6 0 627007. 7 0 643007. 8 0 659007. 9 0 675007. 9 0 69200.7.. 42 7 0 646007. 8 0 662007. 9 0 678007 . 9 0 695007. 9 0 712007. 43 8 0 6650G7. 9 0 681007. 9 .0 698007. 9 0 715007. 9 0 732007. 44 9 0 684007. 9 0 701007. 9 0 718007. 9 0 735007. 9 0 752007. ALLOCATIONS FROM SOURCE 5 FOR PERIOD 1 170 • : < P U L . P _ _ , SmLjO£_(_aG.DUXJJj3.N. , PROD YT-10 YT=11 YT = 12 YT=13 YT=14 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 0 202987. 0 0 225225. 0 0 247463. C 0 269701. 0 0 291939. .0 0 204943. 0 _0 228446,. 0 0 242987. 0 0 263915. o 0 2.85368. 2 0 0 195768. 0 0 219272. 0 0 242775. 0 0 266800 . 0 0 282987. 3 0 0 188626. 0 0 212129. 0 0 235633. 0 0 264419. 0 0 281161. 4 0 0 183007. 0 0 206510. 0 0 230014. 0 0 262721. 0 0 278007. 5 0 0 193007. 0 0 216510. 0 0 240014. C 0 259592. 0 0 273979. 6 0 0 20300 7. 0 0 2 26510. 0 0 250014. 0 ._0.._ 262948. 0. ...0 _ 2 7.3 216... 7 0 0 215007. 0 0 . 238510. 0 0 262014. 0 0 272948. 0 0. 282783. 8 0 0 227507. 0 0 251010. 0 0 273455. 0 0 282948. 0 0 292783. 9 3 0 239007. 3 0 262510. 0 0 283685. 0 0 293 3 58. 0 0 302783. 10 3 0 251007. 3 0 274510. 0 0 293275. 0 0 304023. 0 0 313304. - _ 11 ._ 4 0 263007. 0 0 284815. 0 0 30327 5. 0 _ 0 _ _ 314164. 0 0 32530.4. 12 5 0 275007. 0 0 294815. . 0 0 313275. 0 0 326164. 0 0 337804. 13 5 0 287507. 0 0 304815. 0 0 325275. 0 0 338664. 0 349304. 14 5 0 300C07. 0 0 316815. 0 0 337775. 3 0 350164. 3 0 361304. 15 5 0 312507. 0 0 329315. 3 0 349275. 1 0 362164. 4 0 373304. . .1.6 5 0 3 2 5007. 3 0 340 815. 3 0 361275. 4 0 374164. 5 0. .3.8.5.3.0.4.. 17 5 0 337507. 3 0 352815. 4 0 373275. 5 0 386164. 5 G 397804. 18 5 0 350CC7. 4 0 364815. 5 0 3 85275. 5 0 398664. 5 0 410304. 19 4 0 3660C7. 5 0 376815. 5 0 397775. 5 0 411164. 5 0 422804. 20 5 0 378007. • 5 . 0 389315. 5 0 410275. 5 0 423664. 5 0 435304. 21 5. 0 390507. 5 0 401815. 5 0 422775. 5 0 436.164.. 5 0 447804. .. 22 5 0 403007. 5 0 414315. 5 0 435275. 5 0 448664. 5 0 460304. 23 5 0 415507. 5 0 426815. 5 0 447775. 5 0 461164. 6 0 479304. 24 5 0 428007. 0 439315. 5 C 460275. 6 0 480164. 7 0 498304. 25 5 0 440507. 5 0 451815. 5 0 475318 . 5 0 498822. 8 0 517304. 2J? 5 0 453007. 6 0 470815 . 6 0 494318. 6 0 5178??. 9 _0. __.53630_... 27 . 6 0 472007. 7 0 489815. 7 0 513318. 7 0 536822. 3 0 553783. 28 7 o '• 491007. 8 0 508815. 8 0 532318. 3 0 555023. 3 0 564304. 29 8 0 510007. 9 0 527815. 9 0 551318. 3 0 565164. 3 0 576304. 30 9 0 529007. 3 0 545815. 3 0 564275. 3 0 577164. 4 0 588304. .3.1 3 n 546815 . 3 0 5 5 5 8 1 5 . 3 0 576275. 4 0 -5.8.9.1.64. 5 n 60.0304.. 32 3 0 557007. 3 0 567815 . 4 0 588275. 5 0 601164. 6 0 619304. 33 3 0 569007. 4 0 579815. 5 0 600275. 6 0 620164. 7 0 638304. 34 4 0 581007. 5 0 591815. 5 0 615318. 5 0 638822. 8 0 657304. 35 5 0 593007. 6 0 61C815. 6 0 634318. : 6 0 6578 22. 9 0 676304. 3-6 . 6 0 612007. 7 0 629815. 7 __X3 6 53318. 7 0 6 7 6 8 2 2 . 9 0 6.9.6304... 37 7 0 631007. 8 0 648815. 8 0 672318. 8 0 695822. 9 0 716304. 38 8 0 6 5OC0 7. 9 0 667815. 9 0 691318. 9 0 714822. 9 0 736304. 39 9 n 669007. 9 0 687815. 9 0 711318. 9 0 734822. 9 o 756304. ALLOCATIONS FROM SOURCE 5 FOR PERIOD 1 171 < PULP , SA.W1£'G__P_R£'.0.U.CI.I.C.N , _ PROD YT-10 YT=11 YT=12 YT=13 YT=14 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 40 9 0 689C07. 9 0 707815. 9 0 731318. 9 0 754822. A l 9 0 709007. 9 0 72781 5. ..._9__.0 75.131.8_. . . 42 9 0 729007. 9 0 747815. 43 9 0 7490C7. ( ALLOCATIONS FROM SOURCE 5 FOR PERIOD 1 PULP _ . SAW.LCG PRODUCTT GN PROD YT = 15 YT= 16 YT=17 YT = 18 YT-19 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 0 314177. C 0 336415. 0 0 354653. G 0 372891. 0 0 396394. _ ~ . 1 . .0 0 307082. 0 0 324946. 0 0 342903. . 0 0 360923. 0 _0_ 3.8A426.. 2 0 0 299728 . 0 0 316844. 0 0 334209. G 0 351749. 0 0 375252. 3 0 0 296568. 0 0 310987. 0 0 327654. 0 0 344606. 0 0 368109. 4 0 0 292419. 0 0 30 90 86. 0 0 324568. 0 0 338987. 0 0 362490. 5 0 0 290646. 0 0 306007. 0 0 320419. 0 0 343923. 0 0 367426. 6 0 0 _ 2.87 59.2.. 0 0 301979. 0 0 325483. 0 0 348986. 0 0 372490. 7 0 0 292266. 0 0 311979. 0 0 335483. 0 0 358986. 0 0 382490. 8 0 0 302266. G 0 321979. 0 0 345483. 0 0 368986. 0 0 392490. 9 0 0 312266. 0 0 333979. 0 0 357483. 0 0 380986. 0 0 404490. 10 c 0 324266. 0 0 346479. 0 0 369983. 0 0 393486. 0 0 416990. 1.1 _ 0 0 3 3 6.766. 0 357979. 3 0 3 8 1 4 8 3 . 3 o 404986. 3 .0.. 428490. 12 3 0 348266. 3 0 369979. 3 0 393483. 3 0 416986. 3 0 440490. 13 3 0 360266. 4 0 381979. 4 0 405483. 4 0 428986. 4 0 45249C. 14 4 0 372266. 5 0 393979. 5 0 417483. 1 5 0 440986. 5 0 464490. 15 .5 0 384266. c 0 406479. 5 0 4 2998 3. 5 0 453486. c 0 476990. 16 5 0 396766. 5 0 418979. 5 n 442483. 5 0 465986. 5 0 489490. 17 5 0 409266. 5 0 431479. 5 0 454983. 5 0 478486. 5 0 5C1990. 18 5 0 421766. 5 0 443979. 5 0 467483. 5 0 490986. 5 0 514490. 19 5 0 434266. 5 0 456479 . 5 0 479983. 5 0 503486. 5 0 526990. 20 5 0 446766. 5 0 468979. 5 0 492483. 5 0 515986. 0 539490. 21 5 . _0_ 4 592 6 6 . 5 0 482769. 5 0 506273. 5 0 529776. 5 __0_ 5.5.3 2 80.. 22 6 0 478266. 6 0 501769. 6 0 525273. 6 0 548776. 6 0 572280. 23 7 0 497266. 7 0 520769. 7 0 544273. 7 G 567776. 7 0 591280. 24 8 0 516266. 8 0 539769. 8 0 563273. 8 0 586776. 8 0 610280. 25 9 0 535266. 9 0 558769. 9 0 582273. 9 0 605776. 9 0 629280. _.2.6 3 0 553266. 3 0 572979. 3 0 596483. 0 61 9986. 3 _ Q _ 6A34-.9.0.. 27 3 0 563266. 3 0 584979. 3 0 608483. 3 0 631986. 3 0 655490. 28 3 0 575266. 4 0 596979. 4 0 6 20 48 3. 4 0 643986. 4 0 667490. 29 4 0 587266. 5 ' 0 608979. 5 0 632483. c _• 0 655986. 5 0 679490. 30 5 0 599266. 5 0 622769. K _v 0 646273. 5 0 669776. 5 0 693280. 3.1... -6_ JQ . 6 0 6 4 1 7 6 9 . 6 0 665273. 6 0 688776. 6 0 71 2280. 32 7 0 637266. 7 0 660769. 7 0 684273. 7 0 707776. 7 0 731280. 33 8 0 656266. 8 0 679769. 8 0 703273. 8 0 726776. 8 0 750280. 34 9 0 675266. 9 0 6 .8769. 9 0 722273. 9 0 745776. 9 0 769280. 35 9 0 695266. 9 0 718769. 9 0 742273. 9 0 765776. . 3.6 9 0 715266. 9 0 738769. 9 0 762273. 3 7 9 0 735266. 9 0 758769. 38 9 0 755266. OPTIMAL ALLOCATION WITHIN PERIOD 1 7.73 1ST OF PAIRED COLS. - PULP ALLOC. 2ND OF PAIRED COLS. - S/L ALLOC. PULP SAWLOG PRODUCTION PROD AREA 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0 ( 1 ) G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 G 0 0 0 0 0 G 0 G 0 G 0 0 0 0 0 0 0 0 0 0 0 0 . ( 2J 0 0 c 0 0 n 0 fl n f c 0 0 0 0 c 0 0 0 9 0 1 0 G I G 0 10 G I G G I G G I G OlC OlC _0..1.0_._ G I G (3 ) 0 0 0 1 0 2 0 3 0 4 c 5 0 6 0 1 c 8 0 G 0 G 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 8 (4 ) G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 G 0 0 0 0 0 G 1 <5 . 0 0 0 G 0 0 0 G 0 C G G 0 0 0 G 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (1) 1 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 G 0 0 0 0 L2J 0 0 0 n 0 0 0 0 0. G 0 G 0 0 0 0 0 0 0 0 0 0 0 0 G I G 010 010 010 01 0 010 0.10— 010 (3) 0 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 8 1 8 1 8 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 8 (4 ) 0 0 G G 0 0 G G 0 0 0 0 0 0 G 0 0 0 0 1 0 2 0 3 0 G 0 0 0 G 0 C 0 0 0 G 0 0 G 1 (5) 0 G 0 G 0 G 0 0 C 0 0 0 G 0 0 c 0 0 0 c 0 G 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 C 0 2 ( 1 ) 2 0 1 C 1 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 c 0 G C G 0 0 C C 0 0 0 G G C 0 0 _ C2J 0_ G 0 0 0 0 0 n 0 (1 0 G 0 0 G G 0 0 0 0 0 0 0 0 0 _0 1 9 010 G I G 010 010 G L G 010 (3) G 0 1 1 1 2 2 3 2 4 2 5 2 6 2 7 2 8 2 8 2 8 2 8 2 8 1 4 2 4 2 5 2 6 2 7 2 8 2 8 (4 ) 0 0 G 0 0 c 0 0 0 G 0 G 0 0 0 0 0 0 0 1 0 2 0 3 0 4 0 G G 0 G c 0 C G 0 0 0 0 1 (5) G 0 0 G 0 0 0 0 0 0 0 0 0 c C C 0 0 0 G G 0 0 0 0 0 G G 0 0 0 0 0 0 0 0 G 0 0 0 3 ( 1) C 0 2 0 2 0 1 r V- G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 G G c 0 G 0 0 C G 0 0 (.2J 0 0 G 0 0 0 0 0 G 0 0 0 0 0 G G 0 0 0 0 0 n 0 0 0 J O 1 9 1 9 0 9 010. G I G 0.1.0 0.10 (3) 0 0 1 1 1 2 2 3 3 4 3 5 3 6 3 7 3 8 3 8 3 8 3 8 3 8 2 4 2 c -> 3 6 3 6 3 7 3 8 3 8 (4 ) 0 0 0 0 0 0 0 G 0 0 G 0 G 0 0 C 0 0 0 1 0 2 0 3 0 4 G 0 0 0 0 0 0 0 0 0 0 0 c 1 (5 ) 3 0 0 0 0 0 0 C 0 0 0 0 0 0 G G G 0 0 G 0 G G 0 0 G 0 G 0 0 0 0 O 0 0 0 0 0 0 0 4 ( 1 ) 0 0 0 c 0 0 G c 0 G c G 0 0 0 C 0 0 0 0 0 0 0 0 G 0 r 0 G 0 0 0 c 0 c 0 c 0 0 0 0 f.2J _ 0 _ 0 n 0 0 0 0 0 0 n 0 a 0 0 0 0 0 fi 0 G 0 0 0 0 0 0 2 8 1 8 1 9 1 9 0 9 _0.1.0_ 010 (3 ) 0 0 1 1 1 2 4 3 4 4 4 5 4 6 4 7 4 8 4 8 4 8 4 8 4 8 2 5 3 6 3 6 3 7 4 8 4 8 4 8 (4 . 0 0 G G 0 C 0 0 0 G G 0 G 0 0 0 0 0 0 1 0 2 0 3 0 4 0 0 0 0 0 0 .0 G 0 0 0 0 C 1 (5) 4 0 3 G 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 G 0 G 0 G 0 0 0 0 0 0 0 0 G 0 5 < 1 ) 0 0 0 C 0 c 0 G c G u 0 0 c 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 n V. 0 0 0 G 0 0 G 0 (2J 0 0 0 0 0 _0__ __0_ 0 0 0 0 n 0 0 0 0 0 0 0 0 0 0 G 0 0 0 2 7 2 8 2 -8 1 8 1. _9 _1__9...._ - J L 9. (3) 0 0 1 1 1 2 5 3 5 4 5 5 5 6 5 7 5 8 5 8 5 8 5 8 5 8 3 6 3 6 3 7 4 8 4 8 4 8 4 8 (4 ) 0 0 0 0 0 G 0 G 0 G 0 c 0 0 C G 0 0 0 1 0 2 0 3 0 4 0 0 0 0 G 0 0 0 0 C 0 1 G 2 (5) 5 0 4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 G G 0 0 G C G 0 0 0 0 0 0 0 0 6 (1 ) 1 0 c G 0 0 G G • 0 0 C G 0 0 C C 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 G 0 (.2.) 0_ 0 0_ 0 0_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f! 0 0 G 0 0 _o .2. 5 2_ 6 2 J L _ _2_ _8__ 2_ 8 2. 8 _2.. _8.__ (3) 0 0 1 1 1 2 6 3 6 4 6 5 6 6 6 7 6 8 6 8 6 8 6 8 6 8 4 8 4 8 4 8 4 8 4 8 4 8 4 8 (4 ) 0 0 0 0 0 G 0 0 0 0 0 G 0 C 0 0 0 0 0 1 0 2 0 3 0 4 G 0 0 0 0 0 0 0 G 1 0 2 0 3 (5) 5 0 5 0 5 0 0 0 •0 0 0 0 . 0 0 0 c 0 0 0 0 0 0 0 0 0 G 0 G 0 0 0 0 0 0 0 0 0 0 0 0 7 (1 J 2 0 1 0 1 0 0 0 0 0 0 0 0 0 G c c C G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . G 0 0 0 0 G (.2.) 0 0 n 0 n 0 0 0 0 0 0 G 0 n ft 0 0 _0 0 0 0 n n 0 0 -0 2 5 3 6 3 7 2-_ 8 ~ 2--8 2 8. . _2-.8-. (3 ) 0 0 1 1 1 2 1 3 7 4 7 c 7 6 7 7 7 8 7 8 7 8 7 8 7 8 5 8 4 8 4 8 5 8 5 8 5 8 c 8 (4) 0 0 G 0 0 0 0 G 0 0 0 0 0 G G C 0 0 0 1 0 2 0 3 0 4 G 0 0 0 G 0 0 0 0 1 0 2 0 3 (5) 5 0 5 0 5 0 0 G 0 G G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G c G G 0 0 0 0 0 0 0 G 0 0 8 (1 ) 3 0 2 0 2 C 0 0 0 C 0 0 0 0 0 c 1 0 1 0 1 0 1 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (.2.) 0 0 0 0 0 n o_ 0 G 0 0 0 0 0 0 0 0 0 0 n 0 n 0 _o 2 4 2 5 3 6 3 7 2 8 2. .8 _2_8 2 .8 . <3 ) 0 0 1 1 1 2 8 3 8 4 8 c 8 6 8 7 7 8 7 8 • 7 8 7 8 6 8 6 8 5 8 5 8 6 8 6 8 6 8 6 8 (4) 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 G 0 1 0 2 G 3 0 0 G 0 0 G 0 0 0 G 0 1 0 2 0 3 ( 5 ) •5 0 c 0 5 0 G c 0 G 0 0 0 0 Q Q 0 0 0 0 p 0 G G 0 Q 0 o 0 0 Q 0 __. c. 0 .a. a a .._ XL-. O P T I M A L A L L O C A T I O N W I T H I N P E R I O D 1 1 S T O F P A I R E D C O L S . 2 N D O F P A I R E D C O L S . 174 P U L P A L L O C . S / L A L L O C . P U L P P R O D A R E A S A W L O G P R O D U C T I O N 7 8 9 10 11 1 2 1 3 14 1 5 1 6 1 7 1 8 19 9 ( 1 ) 4 0 3 0 3 0 0 C 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 G 0 0 ( 2 , _Q_ 0 0 G Q_ JL_ o.. CL J L . 0 0 0 C . __G_ 0 0 0 0 0 0 0 0 5 4 3 5 3 _6__ __3__7_ 2. _8„ __2 . 8 _ .2_8__ ._.2 .8 . . ( 3 ) 0 0 1 .1 1 2 9 3 9 4 9 5 9 6 6 7 6 8 6 8 6 8 6 ___ 8 4 8 6 8 6 8 6 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 c 0 0 G 0 0 0 G 0 0 0 0 0 0 0 0 0 1 0 2 0 3 0 0 0 0 0 0 0 C 0 0 0 1 0 2 0 3 ( 5 ) 5 0 5 0 5 0 n V 0 0 0 0 0 0 0 3 c 3 0 3 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ( 1 ) 5 0 4 G 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 G 0 0 G 0 1 0 1 0 1 0 1 0 _ J . 2 J . _ . -_Q_ 0 0 0 0 0 0 c 0 c 0 0 0 0 0 G ' 0 0 0 0 0 0 0 0 6 4 4 5 4 6 3 7 2 8 2 _8__ _2._ 8_.. ._2 8...... ( 3 ) 0 0 1 1 1 0 2 1 0 3 1 0 4 10 5 7 6 7 7 7 8 7 8 7 8 7 8 4 8 6 8 6 8 7 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 0 0 0 •o 0 G 0 0 0 0 0 0 0 0 0 G 0 1 0 2 0 3 0 0 0 0 • 0 0 0 0 0 C 0 1 0 2 C 3 <5 ) 5 0 5 0 0 0 0 0 0 0 0 0 3 G 3 c 3 0 3 0 3 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 ( 1 ) 6 0 5 C 0 G c 0 G G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 C 0 G 0 0 0 0 G 0 _ (2J _ 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 7 3 6 4 5. _ 5 _ „_A_ 6. 3 7 _2_ 8_ 2 _ 8 _ _ __2__8__ _ 2 _ . 8 . ( 3 ) 0 0 1 1 1 1 2 11 3 11 4 8 5 8 6 8 7 7 8 7 8 7 8 4 8 5 8 6 8 7 8 7 8 6 8 6 8 6 8 6 8 ( 4 ) 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 C 0 0 0 1 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 C 3 <5) 5 0 5 0 0 0 0 0 0 0 3 0 3 0 3 0 4 0 4 C 4 0 0 c 0 G 0 0 0 0 0 G 3 0 3 0 3 0 3 0 ( 1 ) 6 0 6 0 0 G 0 G C G • c 0 0 0 0 G 0 C 0 G 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 .(2.L_ CL.O__ 0 0 __Q. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 3 6 _A__ __5_5.. 4 6 3 7 2___8 _ _ 2 _ J B _ __2„ _8_ 2_8 ( 3 ) 1 0 1 1 12 2 1 2 3 9 4 9 5 9 6 8 7 , 7 8 7 8 7 8 5 8 6 8 7 8 7 8 6 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 1 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 ( 5 ) 5 0 5 0 0 0 0 0 3 0 3 0 3 0 4 G 5 0 5 0 5 0 0 0 0 0 G 0 0 0 3 0 3 0 3 0 3 0 3 0 1 3 ( 1 ) 6 0 6 C 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 c 0 0 0 0 0 0 ( 2 ) ._ a. Jp__ _ _ J L 0 0 0 0 0 0 0' 0 0 0 0 0 o. 0 0 0 0 0 0 7 3 6 4 5 _ 5 . _ _ _ » _ 6 .3 7 2 8 __2. 8 2. _8_. __2_ -8 ( 3 ) 2 0 2 1 1 3 2 1G _* 1 0 4 IC 5 9 6 8 7 7 8 7 8 7 8 6 8 7 8 7 8 6 8 7 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 0 0 0 0 0 0 " 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0 0 G 0 0 0 0 0 0 0 •0 1 0 2 0 3 ( 5 ) 5 0 5 0 0 0 3 0 3 0 3 0 4 0 5 0 5 0 5 0 5 0 0 0 0 0 0 G 3 0 3 0 4 0 4 0 4 0 4 0 1 4 (1 ) 6 0 0 0 0 0 0 0 0 c 0 0 0 0 1 0 2 0 2 G 2 G 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 _ L 2 J _ _ ._.Q. _Q_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 i 6 4 5 _ 5 . _ __A ,6 3 7 . _ _ 2 . 8 2 8 2 8_. ..JL. .8 ( 3 ) -a. 0 14 1 1 1 2 1 1 3 1 1 4 10 c 9 6 8 7 7 8 7 8 7 8 7 8 7 8 6 8 7 8 7 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 G 0 0 0 0 0 0 0 1 0 2 0 3 ( 5 ) 5 0 0 0 3 0 3 0 3 0 4 0 5 0 5 0 5 0 5 0 5 ' 0 0 0 0 0 3 0 3 0 4 0 5 G 5 0 5 0 5 0 1 5 ( 1 ) 6 0 0 0 0 0 0 0 0 c 0 0 1 0 2 C 3 0 3 0 3 0 1 0 G 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 -t.2J.__. __CL _0_ G 0 0 0 0 0. __Q_ _ 0 _ 0_ 0 0 0 0 0 0 0 0. _0__ .0 0 7 3 6 4 5 _5__ -_4_ .6 _ 3 _7_ __2. .8. _ _ 2 _8 . 2. -8__ ___2. 8.... ( 3 j 4 0 12 1 12 2 12 3 11 4 I C 5 9 6 e 7 7 8 7 8 1 8 7 8 6 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 ( 4 ) 0 G " 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 C 0 0 0 0 0 0 0 0 0 0 C 0 1 0 2 0 3 ( 5 ) 5 0 3 0 3 0 G 4 0 5 0 5 0 5 0 5 0 5 0 5 0 0 0 3 0 3 0 4 0 5 0 5 0 5 0 5 0 5 0 1 6 ( 1 ) 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 G 4 0 4 0 4 0 c C 0 0 0 0 0 0 1 0 2 0 2 0 2 0 2 0 J.2.L- 0. 0 0 0 0 0 0 0 OL 0 0 0 0 0 0 0 0 0 0 0 0 0 7 3 6 4 5 5 ... .4-..6__ __3L _ 7 _ _8_ 2 8 2. JL - _ 2 . 8 .3) 1 3 0 1 3 1 1 3 2 12 3 11 4 10 c 9 6 8 7 7 8 7 8 7 8 6 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0 G 0 0 0 0 0 G 0 0 0 1 0 2 0 3 ( 5 ) 3 0 3 c 3 0 4 0 0 5 0 5 0 5 0 5 0 5 0 5 0 ^ 0 3 0 4 G 5 0 0 5 G 5 0 5 0 5 0 17 ( 1 ) 0 0 0 0 0 0 0 0 _ 1 0 2 0 3 0 4 G 5 0 5 0 5 0 0 0 0 0 0 \i 1 0 2 0 3 0 3 0 3 0 3 0 ( 2 ) _0._ JQ 0_ 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 3 6 4 5 _ 5 _ __A. 6 •2. 7 2. 8 2 8 ... „ 2 _ „ 8 _ __2. .8 . ( 3 ) 1 4 0 1 4 1 1 3 2 1 2 3 11 4 10 c 9 6 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 0 0 0 0 0 0 0 0 0 0 • 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 ( 5 ) 3 0 3 0 4 0 5 o 5 0 c 0 5 0 5 0 5 0 5 0 5 0 3 0 4 Q 5 0 5 0 5 0 __?.. P 5 ,0 „5 0 0 J OPTIMAL ALLOCATION WITHIN PERIOD 1 ^ 7 5 1ST OF PAIRECCOLS. - PULP ALLOC. 2ND CF PAIRED COLS. - S/L ALLOC. PULP SAWLOG PRODUCTION PROD AREA Q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 18 ( 1 ) 0 G G G 0 0 1 G 2 C 3 0 A 0 5 0 6 0 6 0 6 0 0 G G 0 1 0 2 0 .3 0 A 0 A 0 A 0 A G (2.) G 0 0 0 0 0 0. 0 0 0 0 Ct 0 f. 0_ c '0 0 0 0 0 0 7 6 A 5 5 A 6 3 7 2 8 2 8 2. 8_ 2 8 (3) 15 0 1A 1 13 2 12 3 11 A 10 5 9 6 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (4 ) 0 0 0 0 0 0 0 G c C 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 0 0 0 G 0 0 G 0 1 0 2 C 3 (5) 3 0 A 0 5 0 5 0 5 0 5 0 5 0 5 G 5 0 5 0 5 0 A G 5 0 5 G 5 G 5 0 5 0 5 0 5 0 5 0 19 (1 ) 0 0 0 0 1 0 2 0 0 A 0 5 0 6 0 6 0 6 0 0 0 0 0 1 0 2 0 3 0 A 0 5 0 5 0 5 0 5 0 (.2.L .....XL 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 G 0 8 2 7 3 6 A 5 5 A 6 3 7 2 8 2 8 2.. Ji_ 2 .8 ... (3) 15 0 1A 1 13 2 12 3 11 A 10 5 9 6 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (4) 0 0 G 0 0 G 0 G 0 c 0 0 0 G 0 G 0 0 0 1 0 0 0 0 G 0 0 0 0 0 0 0 0 0 ' 0 1 0 2 0 3 (5) A 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 6 0 6 0 A 0 5 0 5 G 5 G 5 0 5 0 5 G 5 0 5 0 5 0 20 (1 ) 0 0 1 C 2 0 3 0 A 0 .5 c 6 G 6 C 6 0 6 G 0 0 1 0 2 0 3 0 A 0 5 0 6 G 6 c 6 0 6 0 L2J_ 0„ JDL Q_ JL_ -.0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 3 2 7 3 6 A 5 5 A 6 3 7 2_ 8 2 _8_ 2_ _8_. _ 2 .8 _ . (3) 15 0 1A 1 13 2 12 3 11 A 10 c 9 6 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (4 > 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 C 0 0 0 1 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 ( 5) 5 0 5 C 5 0 5 0 5 0 5 G 5 0 6 0 7 0 7 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 21 ( 1 ) 1 0 2 0 3 0 A 0 5 0 6 0 6 0 6 0 6 0 0 0 1 0 2 0 3 0 A 0 5 0 6 C 6 c. 6 0 6 0 6 0 .(.2.) 0 0 0 C 0 0 0 0 0 0 G 0 0 0 0 G 0 0 8 2 8 2 7 3 6 A 5 5 A 6 3 7 3 7 3 7 3 7 3 7 . ... (3) 1 5 0 14 1 1 3 2 12 3 11 A 10 5 9 6 8 1 7 8 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 1 0 2 0 3 G A (5 ) 5 0 5 0 5 0 «. 0 5 0 c 0 6 0 7 0 8 G 5 G 5 0 5 0 5 0 5 0 5 0 5 0 5 G 5 0 5 0 5 0 22 ( 1 ) 2 0 3 0 A C 5 C 6 0 6 0 6 0 6 c 6 Q 1 0 2 0 3 0 A G 5 0 6 C 6 c 6 C 6 0 6 G 6 0 (2J 0 0 0 G 0 0 - 0 0 0 0 0 0 0 0 0 0 ' 0 0 £ 2 8 2 7 3 6 A 5 5 4 6 3 7 3 7 3 _7_ 3. _7._ 3 7 <3) 1 5 0 1A 1 13 2 12 3 11 A 10 5 9 6 8 7 7 8 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (A ) 0 0 0 G 0 G 0 0 0 0 G 0 G -0 G G 0 G 0 0 0 G 0 0 0 0 G 0 0 0 0 0 .0 1 0 2 0 3 C A (5 ) 5 0 5 0 5 0 5 G 5 G 6 0 7 0 8 0 9 0 5 0 5 0 c 0 5 0 5 G 5 0 6 0 6 0 6 0 6 0 6 0 23 ( 1 ) 3 0 4 0 5 0 6 0 6 0 6 c 6 G 6 C 1 0 2 0 3 0 A 0 5 G 6 0 6 0 6 0 6 0 6 0 6 G 6 0 (.2_L 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 8 2 8 2 8 2 7 3 6 4 5 5 4 6 3 _3„ 7 3 7 3_ „7_ 3 7 (3) 15 0 14 1 12 2 12 3 11 A 10 5 9 6 8 7 9 6 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (A ) 0 0 0 G 0 0 G 0 f\ 0 0 G 0 G 0 C 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 1 0 2 0 3 G 4 (5) 5 0 5 0 5 0 5 0 6 0 7 0 8 0 9 G 5 0 5 0 5 0 5 0 5 0 5 G 6 0 7 0 7 0 7 0 7 0 7 G 24 ( 1 > A 0 5 0 6 0 6 0 6 C 6 0 6 G 15 C 15 0 3 C A 0 5 0 6 0 6 0 6 0 6 0 6 G 6 0 6 0 6 0 (.2.) 0 0 G 0 0 0 0 0 0 Q 0 0 0 0 0 0 0 0 8 2 8 2 7 3 6 _4 __5_ 5 A 6 3 7 3 7 3 7 3 _7._ 3 7 (3 1 15 0 1A 1 13 2 12 3 11 A 10 5 9 6 6 7 6 8 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (A ) 0 0 0 0 0 0 0 C 0 C 0 c 0 C 0 C 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 0 4 (5) 5 G 5 0 5 0 6 0 7 0 8 0 9 0 3 0 3 0 5 G 5 0 5 0 5 0 6 G 7 0 8 0 8 G 8 0 8 0 8 0 25 11) 5 0 6 0 6 c 6 0 6 G .6 0 15 G 15 0 15 0 A 0 5 0 6 0 6 0 6 0 6 0 6 0 6 G 6 0 6 0 6 0 (2J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 8 2 8 ? 7 3 7 •a. 7 3 4 6 3 7 3 7 3 7 3 7 3. 7 _ (3) 15 0 1A 1 13 2 12 3 11 A 10 C. _• 7 6 7 7 7 8 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (A ) 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 G 0 G 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 1 0 2 G 3 0 4 (5) 5 G 5 0 6 0 7 0 8 0 9 0 3 0 3 0 3 0 5 0 5 0 c; 0 5 0 5 G 8 0 9 0 Q 0 9 0 9 0 9 0 26 (1 ) 6 0 6 0 6 G 6 0 6 0 15 G 15 0 15 C 15 G 5 0 6 0 6 0 6 0 6 0 6 0 15 0 15 G 15 0 15 0 15 0 . (.2.) n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 2 8 2 7 3 7 3 7 3 A 6 3 7 2 8 2 8 2__ _8_ 2 8 (3 ) 15 0 1A 1 13 2 12 3 11 A 8 8 6 8 7 7 8 8 7 7 8 7 8 7 8 7 8 7 8 5 8 6 8 6 8 6 8 6 8 (4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 G 0 0 1 0 2 0 0 0 0 0 0 0 1 0 2 0 3 (5) 5 0 6 c 7 0 8 0 9 0 .., 3, Q 3 0 3 0 A Q 5 0 5 0 6 0 6 P 6 0 9 0 •a 0 a 3 JL ., ,_L P 3 0 J OPTIMAL ALLOCATION WITHIN PERIOD 1 176 1ST OF P A I R E D C O L S . - PULP A L L O C . 2ND OF P A I R E D C O L S . - S / L A L L O C . PULP SAWLOG PRODUCTION PROD AREA 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 27 < 1) 6 0 6 0 6 0 6 0 15 0 15 0 15 0 15 0 15 0 6 0 6 0 6 G 6 0 6 0 15 0 15 0 15 G 15 0 15 0 15 0 (.2J 0_ 0 0 c 0 0 0 0 0 0 G C 0 C 0 c CL 0 8 2 8 2 7 3 7 3 7 3 3 6 3 7 2 8_._ 2 8 2. 8 _.._2 .8. (3 ) 15 0 14 1 13 2 12 3 9 4 9 5 9 6 8 7 7 8 8 7 7 8 7 8 7 8 7 8 6 8 6 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 '0 0 0 G 0 1 0 2 0 0 G 0 0 C 0 1 C 2 0 3 (5 ) 6 0 7 G 8 0 9 0 3 0 3 0 3 0 4 C 5 0 5 0 6 0 7 0 7 0 7 0 3 0 3 0 3 C 3 0 3 0 3 0 28 ( 1 ) 6 0 6 0 6 0 15 0 15 0 15 0 15 0 15 0 6 0 6 0 6 0 6 0 6 0 15 0 15 0 15 G 15 0 15 0 15 0 15 0 (.2J_ _ G _ G •o 0 0 0 0 G 0 0 0 0 0 0 0 0 8 2. 8 2 8 2 7 3 7 3 4 5 4 6 3. 7 2 8. 2 _8_ _2_ _8_ __2.. _8. (3) 15 0 14 1 13 2 10 3 10 4 10 5 9 6 8 7 9 6 8 7 7 8 7 8 7 8 6 8 6 8 7 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 0 0 0 G 0 G 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 G G 1 0 0 0 0 0 C 0 C 0 1 0 2 0 3 (5 ) 7 G 8 0 9 0 3 0 3 0 3 0 4 0 5 G 5 0 6 G 7 0 8 0 8 0 3 G 3 0 3 0 4 G 4 0 4 0 4 0 29 ( 11 6 C 6 G 15 0 15 0 15 0 15 0 15 0 6 0 6 0 6 0 6 0 6 0 6 0 15 G 15 0 15 0 15 C 15 0 15 0 15 0 ( 2 J _ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 2 8 2 8 2 8 2 7 3 7 3 5 5 4 6 3 7 2 8 2 8 2 _8._ _2_ 8_ <3> 15 0 14 1 11 2. 11 3 11 4 10 5 9 6 10 5 9 6 8 7 7 8 7 8 7 8 6 8 7 8 7 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 1 0 0 G 0 - 0 C 0 C 0 1 0 2 G 3 (5 ) 8 0 9 0 3 0 2 0 3 0 4 0 c 0 .> c 6 c 7 0 8 0 9 0 9 G 3 0 3 0 4 0 5 0 5 0 5 0 5 0 30 ( 1 ) 6 0 15 0 15 0 15 0 15 0 15 G 6 G 6 G 6 0 6 G 6 0 15 0 15 G 15 0 15 0 15 G 15 0 15 0 15 0 15 G 12.) 0 ..0 0 0 • 0 0 0 0 C 0 0 0 8 2 8 2 8 2 8 2 8 2 7 3 6 4 5 _5.__. _4_ 6 3 7 3 _7__ __3_ _7 „_3. _7_ 3_ _7_ ( 3 ) 15 0 12 1 12 2 12 3 11 4 10 5 11 4 10 5 9 6 8 7 7 8 5 8 6 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 ( 4 ) G 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 G 0 G 0 0 0 0 0 1 0 2 0 3 G 4 <5) 9 0 3 0 3 c 3 0 4 0 5 0 5 0 6 0 7 0 8 0 9 G 3 G 3 0 3 0 4 0 5 0 5 0 5 0 5 0 5 0 31 ( 1 ) 15 0 15 0 1 5 C 15 G 15 0 6 0 6 0 6 G 6 0 6 G 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 (.2.) G _o_ G_ 0 0 0 0 0 0 0 8 2 8 2 8 2 8 2 8 2 7 3 7 3 6 4 5 5 4 _ 6 _ 3 7 3 7 3 7 3 7 3 _7_ ( 3 ) 13 0 13 1 13 2 12 3 11 4 12 3 11 4 10 5 9 6 8 7 6 7 6 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 ( 4 ) 0 0 0 G 0 0 G 0 0 0 0 G 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 G 0 0 0 0 0 1 0 2 0 3 0 4 (5 ) 3 0 3 0 3 0 4 0 5 0 5 0 6 0 7 c 8 0 9 0 3 0 0 3 G 4 0 5 0 6 0 6 0 6 0 6 0 6 0 32 (1) 15 0 15 0 15 0 15 G 6 C 6 C 6 0 6 G 6 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 G 15 0 15 0 15 G 12.) 0. _0 0 0 0 0 0 0 R 2 8 2 8 2 8 _ 2 _ 8_ _ 2 _ 8 2 8 2 7 3 6 4 5 _5 __A. 6 3 7 3 7 3 _ 7 _ _3_ _7__ __.3_ 7 ( 3 ) 14 0 14 1 13 2 12 3 13 2 12 3 11 4 10 5 9 6 6 7 6 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (4 ) 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 G 0 0 0 0 G 0 0 G 0 G 0 1 0 2 0 3 0 4 (5 ) 3 0 3 0 4 0 5 0 5 0 6 0 7 0 8 C 9 0 3 0 3 0 3 0 4 0 5 G 6 0 7 0 7 0 7 0 7 0 7 G 33 (1 ) 15 0 15 0 15 0 6 0 6 G 6 0 6 0 6 G 15 c 15 c 15 G 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 <.2J __0_ 0 0 0 0 0 8 2 8_ _2 8 2 8 2 8 2 8 2 8 2 8 2 7 6 4 5 5 4 6 3 _ J _ . _ 3 _ J L _ _ 3 _ 7 3 _7_ 3. .7. ( 3 ) 15 0 14 1 13 2 14 1 12 2 12 3 11 4 10 5 7 6 7 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (4 ) 0 o • 0 0 0 0 0 0 0 0 G 0 0 0 0 G 0 0 0 0 0 0 G G 0 C 0 G 0 0 0 0 0 1 0 2 0 3 0 4 ( 5) 3 0 4 G 5 0 5 0 6 C 7 0 8 0 9 0 3 0 3 0 3 G 4 C 5 C 6 C 7 0 8 0 8 0 8 0 8 0 8 0 34 (1 ) 15 0 15 0 6 0 6 0 6 0 6 0 6 c 15 c 15 G 15 G 15 G 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 _ < 2 J _ _ _0_ .0 0 0 8 2 8 2 8 _2 8 2 8 2 8 _ 2 _ 8 2 R 2 R 2 7 3 7 3 7 3 4_ 6 3_ 7 3 7 3 __7 3 _7_ __.3 7 (3 ) 15 0 14 1 15 G 14 1 12 2 12 3 11 4 8 c 8 6 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (4 ) G 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 G 2 0 0 0 0 0 1 0 2 0 3 0 4 ( 5 ) 4 0 0 5 G 6 C 7 0 8 0 9 0 3 0 3 0 3 0 4 0 5 0 5 G 5 0 8 0 9 0 9 0 9 0 9 0 9 0 35 (1 ) 15 0 6 0 6 0 6 0 6 o 6 0 15 0 15 0 15 G 15 G 15 0 15 0 15 G 15 G 15 0 15 0 15 0 15 0 15 0 <_2J 0 Xl_ 9_ 1 8 2 8 2 8 2 8 _2 8 2 R 2 8 2 8 2 3 _2_ _7_ 3 7 3 7 3 4 .•6 3 7 3 7 3 7 3_ _7_ (3 ) 15 0 15 0 15 0 14 1 13 2 12 3 9 4 9 c 9 6 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (4 ) 0 0 0 G G G 0 0 0 0 0 0 . 0 0 G G 0 0 0 0 0 0 G G 0 1 0 2 0 0 1 0 1 1 1 2 1 3 <5) 5 0 5 q 6 G 7 G 8 0 0 3 c q 3 0 4, 0 5 0 6 0 6 0 6 0 0 9 0 0 9 Q -_L Q OPTIMAL ALLOCATION WITHIN PERIOD 1 1ST OF PAIRED COLS. 2ND OF PAIRED COLS. 177 PULP ALLOC. S/L ALLOC. PULP PROD AREA SAWLOG PRODUCTION 7 8 9 10 11 12 13 14 15 16 17 18 19 36 (1) 6 0 6 c 6 0 6 0 6 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 C 15 0 15 0 15 0 . .2LL LO. 0 9 1 8 ? 8 _2_ -.8 2 8 ? 8 2 8 ? 8 2 8 2 8 2 7 3 7 3 7- 3 „4-_6__ _3_ JJ— 3 7 3—7... ... (3) 15 0 15 0 15 0 14 1 13 2 10 3 10 4 10 5 9 6 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 (4) 0 0 0 C 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 1 0 2 0 2 1 2 2 (5) 5 0 6 0 7 0 8 0 9 0 •a 0 3 0 3 0 4 0 5 0 6 0 7 0 7 0 7 0 9 0 9 0 9 0 9 0 37 { 1 > 6 G 6 0 6 0 6 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 G 15 0 15 0 15 0 —J2_L 10 -0 9 1 8 2 8 2 8 2 8_ _2_ 8 2 8 2 8 2 8 2 7 ..3 7 3 7 3 4 6 3 7 -__3_ J L . (3) 15 0 15 G 15 ___ 0 14 1 11 2 11 3 11 4 10 5 9 6 8 7 7 8 7 8 7 8 7 8 7 8 7 8 7 8 > (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 2 0 3 0 3 1 (5) 6 0 7 0 8 0 9 0 3 0 3 0 3 0 4 0 5 0 6 0 7 0 8 0 8 0 8 G 9 0 9 0 9 0 38 ( 1 ) 6 0 6 0 6 0 15 0 15- C 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 G 15 0 15 0 12 L ._-l.0_.-0- 9L 1 8 _2_ 8_ _2__ 8_2__ a. 2 8 2 8 _2_ 8 2 8 ? 8 2 7 3 7 3 7. _3__ 4_ 6 3 7 (3) 15 0 15 0 15 0 12 1 12 2 12 3 11 4 10 5 9 6 8 7 7 8 7 8 7 8 7 8 7 g 7 8 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .0 0 0 G 0 0 0 0 0 1 0 2 3 0 4 0 (5) 7 0 8 0 9 0 3 0 3 0 3 0 4 0 5 0 6 0 7 G 8 0 9 0 9 C 9 0 9 0 9 0 39 (1 ) 6 0 6 G ' 15 0 15 0 15 C 15 0 15 0 15 G 15 0 15 0 15 0 15 0 15 0 15 0 15 0 - (.2.L 1 0 0- 9 1 8 ? 8 ? 8 2 8 2 8 ? 8 ? 8 _2_ 8 2 8 2 7 3 7 3 7 3 4 6 (3) 15 0 15 0 13 0 13 1 13 2 12 3 11 4 10 5 9 6 8 7 7 8 7 8 7 8 7 8 7 8 (4) 0 0 0 0 0 0 0 0 0 G 0 G 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 2 4 0 (5) 8 0 9 0 3 0 3 0 3 0 4 0 5 0 6 c 7 0 8 0 9 0 9 0 9 C 9 G 9 0 40 (1 ) . 6 0 15 0 15 0 15 0 15-0 15 0 15 c 15 0 15 0 15 0 15 0 15 0 15 0 15 0 (.2.) LO 0 9 1 8 ? 8 2L 8 2 8 2 8 2 8 2 .8 2 8 2 8 2 7 3 7 •a 7 3 (3) 15 0 13 0 14 0 14 1 13 2 12 3 11 4 10 9 6 8 7 7 8 7 8 7 8 7 8 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0-. 0 0 0 0 0 0 1 0 2 0 2 1 2 2 (5) 9 0 3 0 3 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 9 0 9 0 9 C 9 0 41 (1) 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 C 15 0 15 C 15 0 15 0 15 0 12.) i o n 9 1 8 2 . 8 2 8 2 8 2 8 ? 8 2 8 2 R 2 8 2 7 3 7 3 (3 ) 13 0 14 0 15 0 14 1 13 2 12 3 11 4 10 5 9 6 8 7 7 8 7 8 7 8 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 0 3 1 (5) 3 0 3 G 3 0 4 0 5 0 6 0 7 0 8 0 9 0 9 0 9 0 9 0 9 0 42 U ) 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 -JI.2J. -_1JO_0_ 9- 1 8 2 8 2 8 ? 8 2 8 2 8 2 8 2 8 7 8 2 7 3 <3 ) 14 0 15 0 15 0 14 1 13 2 12 3 11 4 10 5 9 6 8 7 7 8 7 8 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 1 0 2 0 3 0 4 0 (5) 3 0 3 0 4 0 c 0 6 0 7 0 8 0 9 0 9 0 9 0 9 0 9 0 43 < 1) 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 J2.L - ia_o 9 1 ,8 _2__. 8 ? 8 2 8 2 8 2 8 2 8 2 8_2 8 2 (3) 15 0 15 0 15 0 14 1 13 2 12 3 11 4 1G c 9 6 8 7 7 8 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 G 2 0 3 0 4 0 (5) 3 0 4 0 5 0 6 0 7 0 8 0 9 0 9 0 9 G 9 0 9 0 44 (1) 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 15 0 _._2.)_ 1.0 -0 9 _1_ 8 ? 8 2 8 2 8 2 8 2 .8 .2 a 2 8 2 (3) 15 0 15 0 15 0 14 1 13 2 12 3 11 4 IC 9 6 8 7 (4) 0 0 0 0 0 0 n 0 0 0 0 0 1 0 2 0 3 0 4 0 (5) 4 0 q 0 6 0 7 o 8 0 9 Q 9 0 9 Q 9 0 9 Q 178 < OPTIMAL ALLOCATION BETWEEN PERIOD 1 AND ALL PAST PERIODS LOG PRODUCT ION ALLOCATED FROM PERIOD 1 RANGE OF P/L DEMANDS FOR PERIODS 1 - 2 Y= 10 RANGE Y OF SAWLOG = 11 DEMANDS FOR PERIODS 1 Y=12 - 2 (10-Y = 19) 13 _Y .= 14 . (24 TO 44) -X- XT YT RETURN XT YT RETURN XT YT RETURN XT YT RETURN XT YT RETURN 24 25 24 25 10 10 428007. 444GG7. 24 25 11 1 1 442815. 458815. 24 25 12 12 467275. 485818. 24 25 13 13 490664. 512822. 24 25 14 14 512304. 534804. 26 27 26 27 10 10 460007. 482507. 26 27 11 11 481315. 503815. 26 27 12 12 508318. 530818. 26 27 13 13 535322. 557822. 26 27 14 14 5 57304. 578283. 28 29 2 8 29 10 10 505007. 527507. 28 29 11 11 526315. 548815. 28 29 12 12 553318. . 575818. 28 29 13 13 579523. 593164. 28 29 14 14 5 92304. 6C7804. 30 31 30 31 10 10 5500G7. 571315. 30 31 11 11 570315. 583815. 30 31 12 12 592275. 607775. 30 31 13 13 6C8664. 624164. 3C 31 14 14 623304. 638804. 32 33 32 33 10 10 585007. 6005G7. 32 33 11 11 599315. 6148 15. 32 33 12 12 623275. 638775. 3 2 33 12 13 639664. 662164. 32 33 14 14 661304. 683804. 34 35 34 35 10 10 616007 . 631507. 34 35 11 11 630315. 652815. 34 35 12 12 657318. 679818,. 34 35 13 13 684322. 706822. 34 35 14 14 706304. 728804. 36 37 36 37 10 10 654007 . 6765G7. 36 37 11 11 675315. 697815. 36 37 12 12 702318. 724818. 36 37 13 13 729322. 751822. 36 37 14 14 752304. 775804. 38 39 38 39 10 10 699007 . 721507. 38 39 11 11 7203 15. 743815. 38 39 12 12 747318. 77G818. 3 8 39 13 13 774322. 797822. 38 39 14 14 799304. 822804. 40 41 40 41 10 10 745007. 7685G7. 40 41 11 11 767315. 790815 . 40 41 12 12 794318. 817818. 40 13 821322. 42 4 3 -42 43 10 10 792007. 8155G7. 42 11 814315. -OPTIMAL ALLOCATION EETWEEN PERIOD 1 AND ALL. PAST PERIODS LOG PRODUCTION ALLOCATED FROM PERIOD 1 RANGE OF P/L DEMANDS FOR PERIODS 1 - 2 Y= 15 RANGE Y OF1 SAWLOG = 16 DEMANDS FOR PERIODS 1 Y=17 - 2 (10-Y= 19) 18 Y= 1.9. -. (24 TO 44 ) -X- XT YT RETURN XT YT RETURN XT YT RETURN XT YT RETURN XT YT RETURN 24 25 24 2 5 15 15 533766. 556266. 24 25 16 16 560769. 583269. 24 25 17 17 587773. 61G273. 24 25 18 18 614776. 627276. 24 25 19 19 641780. 664280. 26 27 26 27 15 1 5 577766. 591266. 26 27 16 16 600979. 616479. 26 27 17 17 627983. 643483. 26 27 18 18 654986. 67G486. 26 27 19 19 681990. 697490. 28 29 28 29 15 15 606766 . 622266. 28 29 16 16 631979. 647479. 28 29 17 17 658983. 674483. 28 29 18 18 685986. 701486. 28 29 19 19 712990. 728490. 30 31 30 31 15 15 637766. 660266. 30 31 16 16 664769. 687269. 30 31 17 17 691773. 714273. 30 31 18 18 718776. 741276. 30 31 19 19 745780. 768280. 32 33 32 33 15 15 662766. 705266. 32 33 16 16 7G9769. 732269. 32 33 17 17 736 7 73. 759273. 32 33 18 18 763776. 786276. 3 2 33 19 19 790780. 813280. 34 35 34 35 15 15 727766. 751266. 34 3 5 16 16 754769. 778269. 34 35 17 17 781773. 805273. 34 35 18 18 808776. 832276. 34 19 835780. 36 37 36 37 15 15 774766. 798266. 3 6 37 16 16 801769. 825269. 36 17 828773. 38 38 15 821766. i ALLOCATIONS FROM SOURCE 1 FOR PERIOD 2 1 8 0 P U . L P _____ . SAWLOG PRODUCTION . ; _ PROD YT= 0 Y T= 1 YT= 2 YT= 3 YT= 4 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 G O G. _1 1 Q 13500. 2 2 0 27000. 3 3 0 40500. 4 4 0 54000. 5 5 0 67500. 6 6 0 81000 . 7 7 0 142000. 8 8 0 153000. ALLOCATIONS FROM SOURCE 2 FOR PERIOD 2 181 .PJJ.LP_ PROD -XT- X Y YT = 0 YC T RETURN SAWLOG PRGDUCTICN X Y YT- 1 YC T RETURN X Y YT= 2 YC T RETURN X Y YT= 3 YC T RETURN X Y YT= 4 YC T RETURN 0 JL. 2 3 4 0 0 0.0 JOL_0_J1.J__ 0 0 0.0 G 0 0.0 0 0 0.0 G JL C, L35.C.0 0 27000, 0 40500< 0 54G00, 5 _6_ 7 8 C 0 0.0 0 675GG. ,.0.._0...0...0 0._ 8.1.0.00.. 0 0 0.0 0 142000. C 0 0.0 0 153000. ALLOCATIONS FROM SOURCE 3 FOR PERIOD 2 182 PULP PROD -XT-YT= 0 P XI RETURN _S AW LOG PRODUCT I ON YT= 1 P XI RETURN YT= 2 P XI RETURN YT= 3 P XI RETURN YT= 4 X Y F XI RETURN c 0 0 0 0 0 . 0 1 0 0 21238. 0 2 0 c 42476. 0 3 0 0 63714. 0 4 0 0 84952. 1 0 0 0 0 13 500. 1 1 -0 1 31000. 1 2 G 1 45000. 1 3 0 1 64928. 1 4 0 1 85381. 2 0 0 0 0 27000 . 1 1 0 1 44500. 1 2 0 1 5850G. 2 3 0 2 76000. 2 4 0 2 90000. 3 0 0 0 0 40500 • 1 1 0 1 58000. 1 2 0 1 72000. 2 3 0 2 89500. 3 4 1 2 103000. 4 C 0 0 0 54000. 1 1 G 1 715GG. 1 2 0 1 8550G. 2 3 0 2 103000. 4 4 1 2 116000. 5 c G 0 0 67500 . 1 1 0 1 85G0G. 1 2 G 1 99000. 2 3 0 2 116500. 4 4 1 2 129500. 6 0 0 0 0 81000. 1 1 G 1 98500. 1 2 0 1 112 500 . 2 _3_ _ ? _ 2_ 130000. 4 4 1 2 1430G0. 7 1 0 0 1 98000 . 1 1 0 1 112000. 1 2 0 1 126GG0. 2 3 0 2 143500. 4 4 1 2 156500. 8 2 0 0 2 115000. 2 1 0 2 129000. 2 2 0 2 143000. 2 3 0 2 157000. 4 4 1 2 170000. 9 •a 0 0 3 12 2000. 3 1 0 3 146000. 3 2 0 3 160000. 5 3 1 3 173000. 4 4 1 2 183500. 10 4 0 G 4 149000. 4 1 0 4 163000. 6 2 1 4 176G0G. 5 3 1 3 186500. 4 4 1 2 197000. 11 5 0 __0_ 5 166000. '7 1 _1_ 5 179000. 6 . 2 1 4 18950G. 5 _3_ __1_ 3 200000. 3 4 1 2 ... .2 5.600.0. 12 8 0 3 6 182000. 7 1 1 5 192500. 6 2 1 4 203000. 4 3 1 3 259000. 4 4 1 2 2690 00. 13 8 0 3 6 195500. 7 1 1 5 206000. 5 2 1 4 262000. 5 3 1 3 272000. 14 8 0 i 6 209000. 6 1 1 265000. 6 2 1 4 275000. 15 7 0 - 6 2680CC. 7 1 1 5 278G00. 16 8 G 3 6 281000. ALLOCATIONS FROM SOURCE 3 FOR PERIOD 2 183 -< -P-ULE-PROD -XT-YT= 5 P XI RETURN YT- 6 P XI RETURN SM.IS&-R&DSLWSJJS1M YT- 7 X Y P XI RETURN YT= 8 X Y P XI RETURN YT= 9 P XI RETURN 0 0 ^ C C 106190. 0 6 0 0 127428. . L 1 5 0 1 106095. 1 6 1 0 122959. 2 2 5 1 1 105741 . 2 6 1 0 121857. 3 ' 3 5 1 1 113000. 2 6 1 0 135357. 4 3 5 1 1 126500. 2 6 1 0 148657. 5 3 5 1 1 140000. 2 6 1 0 162357. 6 3 _J5_ 1 .1 153500. 2 6 1 G 175857. 7 3 5 1 1 167000. 2 6 1 0 189357. 8 3 "5 1 1 180500. 2 6 1 0 202 857. 9 3 5 1 1 19 4000 . 2 6 1 0 263857. 10 3 5 1 1 255000. 2 6 1 0 274857. -JU 3 5 1 1 . 2J5AOJ3D. ALLOCATIONS FROM SOURCE 4 FOR PERIOD 2 184 P_.UL.P_ SAWLOG PRODUCTION ._ PROD YT= 0 YT= 1 YT= 2 YT= 3 YT= 4 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 c 0. ' G 0 21238. 0 0 42476. 0 0 63714. 0 0 84952. L. . 0 0 13500. 0 0 31000. 0 0 45000. 0 0 64928. 0_ _J0 8 5.3.8 1. 2 0 0 27000. 0 0 44500. 0 0 58 500. 0 0 76000. G 0 90000. 3 o 0 4050G. 0 0 58000. 0 0 72000. 0 0 89500. 0 0 1G3000. 4 0 0 54000. 0 0 7150G. 0 0 85500. G 0 103000. 0 0 116000. 5 0 0 67500. 0 0 E5000. 0 0 99000. G 0 116500. C 0 129500. 6 _0__ _ 0 81000. 0 0 9 8 500. 0 0 112500. 0 _0 130000. 0 0 143000. 7 0 0 98000. 0 0 112000. 0 0 126000. 0 0 143500. 0 .0. 156500. 8 0 0 115000. 0 0 129000. 0 0 143000. 0 0 157000. 0 0 170000. 9 0 0 132000. 0 0 146000. 0 0 160000. 0 0 173000. 0 0 183500. 10 0 0 149000. c 0 163000. 0 0 176000. 0 0 186500. 0 0 197C00. '.1.1 .._G... 0 . 166000. 0 0 179000. 0 0 189500. 0 _0 200000. 1 _0 219000. 12 c c 182000. 0 0 192500. 0 0 203000. 1 0 222000. 2 0 241000. 13 0 0 195500. 0 0 20 6000. 1 0 225000. 2 0 244000. 3 0 263000. 14 0 0 209000. 1 0 228000. 2 0 247000. 3 0 266000. 2 0 313000. 15 1 0 231000. 2 0 250000. 3 0 269000. 2 0 316000. 3 0 335000. . ...__16 __2_ _D 253000. 3 0 272000. 2 319000. 3 JO .. 338000. 17 3 0 275000. 2 0 322000. 3 0 341000 . 18 2 0 325000. 3 0 344000. 19 0 347000. J ALLOCATIONS FROM SOURCE 4 FOR PERIOD 2 185 .PULP-PROD -XT-YT= 5 Y RETURN YI= 6 Y RETURN 3AW.L.0i3_E8-0mCJJi]Jvi YT= 7 X Y RETURN YT= 8 X Y RETURN YT= 9 Y RETURN 0 0 0 106190. C 0 127428. 0 1 154928. 0 2 182428. 0 3 2C9928. 1 0 0 106095. 0 0 122959. 0 1 150459. 0 2 177959. 0 _3 .205459. 2 0 0 105741. 0 0 121857. 0 1 149357. 0 2 176857. 0 3 204357. 3 0 0 113000. 0 0 135357. 0 1 162857. 0 2 190357. 0 3 217857. 4 0 0 126500. 0 0 148857. 0 1 176357. 0 2 203857. 0 3 231357. 5 0 0 140000. 0 0 162357. 0 1 189857. 0 2 217357. 0 3 244857. ._6 o__ J>_ 1533QQ. . 0 0 175857. 0 1 203357. 0 2 230857. 0 _3___ _._2 5.83 5.7.. . . 7 0 0 167000. 0 0 189357. 0 1 216857. 0 2 244357. G 3 271857. 8 0 0 180500. 0 0 202857. 0 1 230357. 0 2 257857. 0 3 285357. 9 0 0 194000. 0 1 221500. 0 2 249000. 0 3 276500. 0 3 346357. 10 1 0 216000 . 1 1 243500. 1 2 271000. 0 2 329857, 0 3 357357. 1.1 2_ 2 3 8000. 2 1 265500. 0 2 321000. 0 _3_ 348500. 12 3 0 260000. 1 1 315500. 1 2 343000. 13 2 0 310000. 2 1 337500. 14 3 0 332000. J A L L O C A T I O N S F R O M S O U R C E 5 FOR P E R I O D 2 1 8 6 < . P U L P S.AWLOLG P R O D U C T I O N PROD Y T = 0 YT= 1 Y T = 2 Y T = 3 Y T = 4 - X T - X Y R E T U R N X Y R E T U R N X Y R E T U R N X Y R E T U R N X Y R E T U R N 0 0 0 0 . 0 0 2 1 2 3 8 . 0 G 4 2 4 7 6 . G 0 6 3 7 1 4 . 0 0 8 4 9 5 2 . . 1 0 0 1 3 5 0 0 . J ) . 0 3 1 0 0 0 . 0 0 4 5 0 0 0 . 0 0 6 4 9 2 8 . 0 _o_ 8 5 3 8 1 . . 2 0 0 2 7 0 0 0 . 0 0' 4 4 5 0 0 . 0 0 5 8 5 0 0 . 0 0 7 6 0 0 0 . 0 0 9 0 0 0 0 . 3 0 0 40 5 0 0 . 0 0 5 8 0 0 0 . 0 0 7 2 0 0 0 . 0 0 8 9 5 0 0 . 0 0 1 0 3 0 0 0 . 4 0 0 5 4 0 0 0 . 0 0 7 1 5 0 0 . 0 0 8 5 5 0 0 . 0 0 1 0 3 0 0 0 . 0 0 1 1 6 0 0 0 . 5 0 0 67 500 . 0 0 8 5 0 0 0 . 0 0 9 9 0 0 0 . c 0 1 1 6 5 0 0 . 0 0 1 2 9 5 0 0 . 6 0 0 8 1 0 0 0 . 0 0 9 8 5 0 0 . 0 0 1 1 2 5 0 0 . 0 0 1 3 0 0 0 0 . 0 0 1 4 3 0 0 G . . 7 3 0 9 6 0 0 0 . 0 0 1 1 2 0 0 0 . 0 0 1 2 6 0 0 0 . 0 0 14 3 5 0 0 . 0 0. 1 5 6 5 0 0 . 8 3 0 1 G 9 5 0 0 . 3 0 1 2 7 0 0 0 . 3 0 1 4 1 0 0 0 . 0 0 1 5 7 0 0 0 . 0 0 1 7 0 0 0 0 . 9 3 0 1 2 3 0 0 0 . 3 0 1 4 0 5 0 0 . 3 0 1 5 4 5 0 0 . 3 0 1 7 2 0 0 0 . G 0 1 8 3 5 0 0 . . 10 4 0 1 3 7 0 0 0 . 3 0 1 5 4 0 0 0 . 3 0 1 6 8 0 0 0 . 3 0 1 8 5 5 0 0 . 0 0 1 9 7 0 0 0 . 11 .0 1 5 1 0 0 0 . 4 0 1 6 8 0 0 0 . 4 0 1 R 2 0 0 0 . 3 0 _.L9_9.0.Q£L. 3 0 2 1 2 0 0 0 . 12 5~ 0 1 6 3 0 0 0 . 5 0 1 8 2 0 0 0 . 5 G 1 9 6 0 0 0 . 4 0 2 1 3 0 0 0 . 3 0 2 2 5 5 0 0 . 13 5 0 1 8 5 0 0 0 . 5 0 1 9 9 0 0 0 . 5 0 2 1 3 0 0 0 . c 0 2 2 7 0 0 0 . 3 0 2 3 9 0 0 0 . 14 5 0 2 0 2 0 0 0 . 5 0 2 1 6 0 0 0 . 5 0 2 3 0 0 0 0 . 3 0 2 4 2 0 0 0 . 4 0 2 5 3 0 0 0 . 15 5 0 2 1 9 0 0 0 . 5 0 23 3 0 0 0 . 3 0 2 4 5 0 0 0 . 4 0 2 5 6 0 0 0 . 5 0 2 6 7 0 0 0 . 5 0 2 3 6 0 0 0 . 3 0 2 4 8 0 0 0 . 4 0 2 5 9 0 0 0 . 5 0 27.00.0.0.. 6 0 2 8 7 0 0 0 . 1 7 3 0 2 5 1 0 0 0 . 4 0 2 6 2 0 0 0 . 5 0 2 7 3 0 0 0 . 6 0 2 9 0 0 0 0 . 7 0 3 0 7 0 0 0 . 18 4 0 2 6 5 0 0 0 . 5 0 2 7 6 0 0 0 . 6 0 2 9 3 0 0 0 . 7 0 3 1 0 0 0 0 . 7 0 3 2 9 0 0 0 . 19 5 0 2 7 9 0 0 0 . 6 G 2 9 6 0 0 0 . 7 0 3 1 3 0 0 0 . 7 0 3 3 2 0 0 0 . 7 0 3 5 1 0 0 G . 20 6 0 2 9 9 0 0 0 . 7 0 3 1 6 0 0 0 . 7 0 3 3 5 0 0 0 . 7 0 3 5 4 C 0 0 . 7 0 3 7 3 0 0 0 . _ 21 _0_ 3 1 9 0 0 0 . 7 0 3 3 8 0 0 0 . 7 0 3 5 7 0 0 0 . 7 0 3 7 6 0 0 0 . 7 „0_ 4.2300.0.. 22 7 0 3 4 1 0 0 0 . 7 0 3 6 C 0 0 0 . 7 0 3 7 9 0 0 0 . 7 0 4 2 6 0 0 0 . 7 0 4 4 5 0 0 0 . 23 7 0 3 6 3 0 0 0 . 7 0 3 3 2 0 0 0 . 7 r\ 4 2 9 0 0 0 . 7 0 4 4 8 0 0 0 . 24 7 0 3 8 5 0 0 0 . 7 0 4 3 2 0 0 0 . 7 0 4 5 1 0 0 0 . 25 7 0 4 3 5 0 0 0 . 7 0 4 5 4 0 0 0 . 26 7 0 4 5 7 0 0 0 . •• ALLOCATIONS FROM SOURCE 5 FOR PERIOD 2 167 ..P.U.L.P S__VJXJC.6 PRODUCTION PROD -XT- X Y YT= 5 RETURN X Y YT- 6 RETURN X Y YT= 7 RETURN X Y YT= 8 RETURN X Y YT= 9 RETURN 0 1 0 CL 0 0 106190. 106095. 0 0 0 0 127428. 122959. 0 0 0 0 154928. 150459. 0 0 0 0 182428. 177959. 0 0 0 0 209928. 205459. 2 3 4 0 0 0 0 0 0 105741. 113000. 126500. 0 •0 C 0 0 0 121857. 135357. 148857. C 0 0 0 0 .0 149357. 162857. 176357. 0 0 0 0 0 0 176857. 190357. 203857. 0 0 0 0 0 0 204357. 217857. 231357. 5 6 0 0 0 0 140000. 153 500. 0 0 0 0 162357. 175857. 0 0 0 0 189857. 203357. 0 0 0 0 2173 57. 230 857. 0 0 0 0 244857. 258357. 7 8 9 0 0 0 0 0 0 167000. 180500. 194000. 0 0 3 0 0 0 189357. 202857. 217857. 0 0. 3 0 0 0 216857. 230357. 245357. 0 0 3 0 0 0 244357. 257857. 272857. 0 0 3 0 0 0 271857. 285357. 300357. 1.0 -. U -3 3 0 0 209000. 222500. _ —* 0 0 2313 57. 244857. 3 3 0 0 258857. 272357. 3 3 0 0 286357. 299857. 3 3 0 0 313857. 3.2.735 71. 12 13 14 3 A 5 0 0 0 236000. 250000. 264000. 4 5 5 0 0 0 258857. 272857. 291500. 4 5 5 0 0 0 286357. 300357. 319000. 4 5 5 0 0 0 313 8 57. 327857. 346500. 4 5 6 0 0 0 34135 7. 355357. 375357. 15 16 6 7 0 0 284000. 304000 . 6 7 0 0 311500. 331500. 6 7 0 0 3 39000. 359000. 6 7 0 0 366 500 . 386500. 7 6 0 _0._ \ 395357. f_7_3_5_7_. 17 18 19 7 7 7 0 0 0 326000. 348G00. 370000. 7 7 7 0 0 0 353500. 375500. 425500 . 7 7 7 0 0 0 381000. 431000. 453000. 6 7 0 0 438500. 458500. 7 0 467357. 20 _ 2.1 7 7 0 0 4 20000. 442000. 7 0 447 500. OPTIMAL ALLOCATION WITHIN PERIOD 2 188 1ST OF PAIRED COLS. - PULP ALLOC. 2ND OF PAIRED COLS. - S/L ALLOC. PULP SAWLOG PRODUCTION PROD AREA Q 1 2 3 4 5 6 7 3 9 1G 11 12 13 14 15 16 17 18 19 < 0 C 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (2 ) c 0 0 G 0 c 0 c c c G 0 0 0 0 c 0 0 0 0 13 ) 0 0 0 1 0 2 0 •a 0 4 0 e 0 6 0 6 0 6 0 6 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 (5 ) 0 0 0 0 0 0 0 0 o 0 c G 0 0 0 c 0 0 .0 0 1 (1) 1 0 0 0 0 0 0 0 0 '0 0 0 0 0 0 0 0 0 0 0 ( 2 ) 0 0 0 0 0 G 0 0 0 c 0 0 0 0 0 0 0 0 0 0 (3 ) 0 0 1 1 1 2 1 3 1 4 1 5 1 6 1 6 1 6 1 6 (4) G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 (5 ) 0 0 0 0 0 0 0 Q 0 0 0 c 0 0 0 0 0 0 0 0 2 11) 2 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . ( 2 ) 0 0 0 0 0 0 0 c G 0 0 G 0 0 0 0 0 0 0 0 <3 > 0 0 1 1 1 2 2 3 2 4 2 c 2 6 2 6 2 6 2 6 (4) 0 0 0 0 0 0 0 c 0 C 0 0 0 0 0 1 0 2 0 3 (5 ) 0 0 0 0 0 c G 0 0 0 0 0 0 0 0 0 0 0 0 0 3 (1) 3 0 2 0 2 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1.2J. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 •0 0 0 (3 ) 0 0 1 1 1 2 2 3 3 4 -3 5 2 6 2 6 2 6 2 6 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 (5 ) 0 0 0 0 0 0 0 0 c c 0 0 0 0 0 0 0 0 0 0 (1) 4 0 3 0 3 0 2 0 0 0 1 0 2 0 2 0 2 0 2 0 12 1 0 0 0 0 _ Q _ 0 0 0 0 0 0 0 0 0 0 0 0__0 0 0 (3 ) 0 0 1 1 1 2 2 3 4 4 c 2 6 2 6 2 6 2 6 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 (5 ) 0 0 0 0 0 0 0 0 G 0 0 c 0 G 0 0 0 0 C 0 (1) 5 0 4 0 4 0 3 0 1 0 2 0 3 0 3 c 3 0 3 0 12 )___ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 .0 0 0 (3 ) 0 0 1 1 1 2 2 •x 4 4 •a 5 2 6 2 6 2 6 2 6 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 1 0 2 0 3 (5 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 ( 1 ) 6 0 5 0 5 0 4 0 2 0 3 0 4 0 4 0 4 0 4 0 (2 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (3 ) 0 0 1 1 1 2 2 3 . 4 4 •a c 2 6 2 t 2 6 2 6 (4) 0 o - 0 0 0 0 0 C 0 0 0 0 0 0 0 1 0 2 0 3 (5 ) 0 0 0 0 0 0 0 0 0 0 G G 0 0 0 G 0 0 0 6 7 (1) 4 0 6 0 6 0 5 0 3 0 4 0 5 0 5 0 5 0 5 0 (2 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (3 ) 0 0 1 1 1 2 2 3 4 4 •3 C 2 6 2 6 2 6 2 6 (4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 (5 ) 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 (1) 5 0 4 0 4 0 6 0 4 0 5 0 6 0 6 c 6 0 6 0 (2 ) 0 0 0 0 0 0 0 <1 G 0 0 0 0 0 G 0 0 0 0 0 (3 ) 0 0 1 1 1 2 2 3 4 4 3 c 2 6 2 6 2 6 2 6 (4 1 0 0 0 0 0 0 c 0 0 0 0 0 0 0 0 1 0 2 0 3 (5 1 •3. 0 i 0 3 0 0 0 o c 0 0 0 0 0 C 0 0 0 0 OPTIMAL ALLOCATION WITHIN PERIOD 2 1ST OF PAIRED COLS. - PULP ALLOC. 2ND OF PAIRED COLS. - S/L ALLOC. PULP SAWLOG PRODUCTION PRCO AREA 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 9 ( 1 ) 6 G 5 Q c 0 4 0 0 5 0 4 0 4 0 4 0 4 0 < 2J 0 0 0 Q , 0 0 0 n 0 0 • G_ 0 G 0 C G o G n G 13) 0 0 1 1 1 2 2 3 4 4 3 5 2 6 2 6 2 6 2 6 (4 ) 0 0 0 0 0 G 0 C C 0 0 0 0 0 G 1 0 2 0 3 (5) 3 0 3 0 3 0 3 0 0 0 0 0 3 0 3 0 3 0 3 0 10 (1 ) 6 0 6 C 6 0 c c 6 G 4 0 5 G 5 0 5 0 5 0 <2J CL 0 0 0 0 0 0 0 0 G 0 0 0 0 0 c 0 0 0 0 (3) 0 0 1 1 1 2 2 3 4 4 3 5 2 6 2 6 2 6 2 6 <4 ) 0 0 0 0 0 G 0 G 0 G 0 0 0 0 0 1 0 2 0 3 (5) 4 0 3 0 3 0 3 0 0 0 3 0 3 0 3 G 3 0 3 G 11 (1 ) 6 0 6 € 6 0 6 0 4 C 5 0 6 G 6 0 6 0 6 0 <2_) 0 0 0 0 0 0 0 0 0 G 0 0 0 0 G c 0 0 G 0 (3 1 G G 1 1 1 2 2 3 4 4 3 5 2 6 2 6 2 6 2 6 (4 ) 0 0 0 0 0 0 0 0 0 G G 0 0 0 0 1 0 2 0 3 (5) 5 n 0 4 0 3 n 3 0 3 0 3 0 3 0 3 G 3 G 12 (1 ) 6 o - 6 0 6 G 6 0 5 G 6 c 6 G 6 c 6 0 6 0 (2J 0 0 0 0 0 0 0 0 C G G 0 0 G 0 0 0 0 0 0 (3) 1 0 1 1 1 2 2 3 4 4 3 5 2 6 2 6 2 6 2 6 (4 ) 0 0 0 0 0 0 0 0 0. G 0 G 0 G G 1 0 2 0 3 (5) 5 0 5 0 5 0 4 G 3 0 3 0 4 0 4 0 4 0 4 0 13 (1 ) 6 0 6 0 6 0 6 G 6 G 6 0 6 0 6 G 6 0 6 0 (2J 0 0 0 0 0 0 0 0 0 G 0 0 0 G C G 0 0 0 0 (3) 2 0 2 1 2 2 2 3 4 4 3 5 2 6 2 6 2 6 2 6 (4 ) 0 0 0 G 0 0 G 0 G 0 0 0 0 G G 1 0 2 G 3 (5 ) 5 0 5 0 5 0 5 0 3 0 4 0 5 0 5 0 5 0 5 G 14 ( 1 ) 6 0 6 0 6 0 6 0 6 0 6 G 6 0 6 G 6 0 6 0 (2J 0 0 0 0 0 0 . 0 0 0 0 0 0 G c G 0 0 G 0 0 ( 3) 3 0 3 1 3 2 5 3 4 4 3 5 3 5 3 5 3 5 2 6 (4) 0 0 G 0 0 G 0 0 0 c 0 0 0 1 0 2 0 3 0 3 (5) 5 0 5 0 5 0 3 0 4 0 5 0 5 0 5 0 5 0 6 G 15 ( 1 ) 6 0 6 G 6 0 6 G 6 G 6 0 6 G 6 0 6 0 6 C 12.)_. . ._0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 ( 3 ) 4 0 4 1 6 2 5 3 4 4 3 5 3 5 3 5 3 5 2 6 (4) 0 0 0 0 0 C 0 G 0 0 0 0 0 1 0 2 0 3 0 3 (5) 5 0 5 0 3 0 4 0 ' 5 0 6 0 6 0 6 0 6 0 7 0 16 ( 1 ) 6 0 6 0 6 0 6 C 6 C 6 0 6 0 6 G 6 0 • 8 0 (.2.1 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 Q n (3) 5 0 7 1 6 2 5 3 4 4 3 5 3 5 3 5 3 5 ' 2 6 (4 ) 0 0 G 0 0 0 0 0 0 0 0 0 0 1 0 2 0 3 0 3 (5) 5 0 3 0 4 0 5 0 6 0 7 0 7 0 7 0 7 G 6 0 17 ( 1 ) 6 0 6 c 6 0 6 G 6 0 6 0 6 0 6 c 8 G 8 0 (.2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G (3 ) 8 0 7 1 6 2 5 3 4 4 3 5 3 5 3 5 3 5 2 6 (4) 0 0 G 0 0 G G 0 0 0 1 0 1 1 1 2 0 3 0 3 <5> 3 0 4 0 5 0 6 0 7 0 7 0 7 0 7 0 6 G 7 G , . _ _ ^ OPTIMAL A L L O C A T I O N WITHIN PERIOD 2 190 1ST OF P A I R E D C O L S . - PULP A L L O C . 2ND OF P A I R E D C O L S . - S / L A L L O C . PULP SAWLOG PRODUCTION PROD AREA 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 < 18 (1 ) 6 0 6 0 6 0 6 0 6 0 6 0 6 0 8 0 8 0 J..2J 0 0 0 0 0 0 G O O O O O O O O O 0 0 ( 3 ) 8 0 7 1 6 2 5 3 4 4 3 5 3 5 3 5 3 5 (4 ) 0 0 0 0 0 0 0 0 1 0 2 0 2 1 0 2 0 3 15) 4 0 5 0 6 0 7 0 7 0 7 0 7 0 7 0 T O 19 (1 ) 6 0 6 0 6 0 6 0 6 0 6 0 8 0 8 0 .L2.L o o o o o o o o J I _ O 0—0 o_a-___0-_0-( 3 ) 8 0 7 1 6 2 5 3 4 4 3 5 3 5 3 5 (4 ) 0 0 0 0 0 0 1 0 2 0 3 0 1 1 1 2 ( 5) 5 0 6 0 7 0 7 0 7 .0 7 0 7 0 7 0 20 ( 1 ) 6 0 6 0 6 0 6 0 6 G 8 0 8 0 i2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( 3 ) 8 0 7 1 6 2 5 3 4 4 3 5 3 5 (4 ) 0 0 0 0 1 0 2 0 3 0 2 0 2 1 ( 5) 6 0 7 0 7 0 7 0 7 0 7 0 7 0 21 ( 1 ) 6 0 6 0 6 0 6 0 8 0 8 0 (2 ) Q Q 0 0 0 0 0 0 0 0 0 0 ( 3 ) 8 0 7 1 6 2 5 3 4 4 3 5 ( 4 ) 0 0 1 0 2 0 3 0 2 0 3 C (5 ) 7 0 7 0 7 0 7 0 7 0 7 0 2 2 ( 1 ) 6 0 6 0 6 0 8 0 8 0 (2 ) 0 0 0 0 0 0 0 0 0 0 < 3 ) 8 0 7 1 6 2 5 3 4 4 (4 ) 1 0 2 0 3 0 2 0 3 0 ( 5 ) 7 0 7 0 7 0 7 0 7 G  2 3 (1 ) 6 0 6 0 8 0 8 0 _JL2J- 0_0_ 0 0 0 0 0 0 13) 8 0 7 1 6 2 5 3 ( 4 ) 2 0 3 0 2 0 3 0 (5 ) 7 0 7 0 7 0 7 0  24 ( 1 ) 6 0 8 0 8 0 (_2J 0 0 0 . 0 0 0 •  (3 ) 8 0 7 1 6 2 .4) 3 0 " 2 0 3 0 ( 5 ) 7 0 7 0 7 0  2 5 {1 ) 8 0 8 0 ( 2.)_„ 0 0 0 0 (3 ) 8 0 7 1 (4 ) 2 0 3 0 (5 ) 7 0 7 0 26 (1 ) 8 0 L2J D__CL ( 3 ) 8 0 <4 ) 3 0 (5 ) 7 0 191 i : J OPTIMAL ALLOCATION BETWEEN PERIOD 2 AND ALL PAST PERIODS LOG PRODUCTION ALLOCATED FROM PERIOD 2 RANGE OF P / L DEMANDS RANGE OF SAWLOG DEMANDS FOR PERIODS 2 - 3 (19 - 2 8 ) ... - _ F O J _ EE.B10JD.S 2 - 3 ____ = 19 Y= 20 Y=21 Y = 22 ... __Y. =23 (44 TO 6 8 ) - X - XT YT RETURN XT YT RETURN XT YT RETURN XT y T RETURN XT YT RETURN 44 14 5 6 8 7 3 0 4 . 14 5 9 0 1 7 6 6 . 14 5 9 2 8 7 6 9 . 14 5 9 5 5 7 7 3 . 14 5 9 8 2 7 7 6 . 45 14 5 9 0 2 8 0 4 . 15 c 9 2 1 7 6 6 . 15 5 9 4 8 7 6 9 . 15 5 9 7 5 7 7 3 . 15 5 1 0 0 2 7 7 6 . 46 15 5 9 2 2 8 0 4 . 16 5 9 4 1 7 6 6 . 16 5 9 6 8 7 6 9 . ~5 _ 9 9 5 7 7 3 . 16 5 1 0 2 2 7 7 6 . 47 16 5 9 4 2 8 0 4 . 17 5 9 6 3 7 6 6 . 17 5 9 9 0 7 6 9 . 17 5 1 0 1 7 7 7 3 . 17 5 1 0 4 4 7 7 6 . 43 17 5 9 6 4 8 0 4 . 18 5 9 8 5 7 6 6 . 18 5 1 0 1 2 7 6 9 . 18 5 1 0 3 9 7 7 3 . 18 5 1 0 6 6 7 7 6 . 49 18 5 9 8 6 8 0 4 . 19 5 1 0 0 7 7 6 6 . 19 5 1 0 3 4 7 6 9 . 19 5 1 0 6 1 7 7 3 . 19 5 1 0 8 8 7 7 6 . 50 19 5 ICC 8 8 0 4 . 19 5 1 0 3 0 2 6 6 . 19 5 1 0 5 7 2 6 9 . 19 5 1 0 8 4 2 7 3 . 19 5 1 1 1 1 2 7 6 . 51 19 5 1 0 3 1 3 0 4 . 19 5 1 0 5 2 7 6 6 . 19 5 1 0 7 9 7 6 9 . 19 5 1 1 0 6 7 7 3 . 19 5 1 1 3 3 7 7 6 . 52 19 5 1 0 5 3 8 0 4 . 19 5 1 0 7 5 2 6 6 . 19 5 1 1 0 2 2 6 9 . 19 5 1 1 2 9 2 7 3 . 19 5 1 1 5 6 2 7 6 . 53 19 5 1 0 7 6 3 0 4 . 19 5 1 0 9 7 7 6 6 . 19 5 1 1 2 4 7 6 9 . 19 5 1 1 5 1 7 7 3 . 19 5 1 1 7 8 7 7 6 . 54 19 5 1 0 9 8 8 0 4 . 19 5 1 1 2 1 2 6 6 . 19 5 1 1 4 8 2 6 9 . 19 5 1 1 7 5 2 7 3 . 19 5 1 2 0 2 2 7 6 . 55 19 5 1 1 2 2 3 0 4 . 19 tx 1 1 4 4 7 6 6 . 19 5 1 1 7 1 7 6 9 . 19 5 1 1 9 8 7 7 3 . 20 5 1 2 5 2 2 7 6 . 56 19 5 1 1 4 5 8 0 4 . 19 5 1 1 6 8 2 6 6 . 19 5 1 1 9 5 2 6 9 . 20 5 1 2 4 8 7 7 3 . 2G 6 1 2 7 6 2 7 3 . 57 19 5 1169 3 0 4 . 19 5 1 1 9 1 7 6 6 . 20 5 1 2 4 5 2 6 9 . 20 6 1 2 7 2 7 6 9 . 58 19 5 1 1 9 2 8 0 4 . 2G 5 1 2 4 1 7 6 6 . 20 6 1 2 6 9 2 6 6 . 59 20 5 1 2 4 2 8 0 4 . 21 5 1 2 6 7 2 6 6 . 60 21 5 1 2 6 8 3 0 4 . 192 < OPTIMAL ALLOCATION EETWEEN PERIOD 2 AND ALL PAST PERIODS LOG PRODUCTION ALLOCATED FROM PERIOD 2 RANGE OF P/L DEMANDS FOR PERIODS ? - 3 Y = 24 RANGE Y OF SAWLOG = 25 DEMANDS FOR PERIODS 2 Y=?6 - 3 ( 19-Y = 28) 27 y= .28 . (44 TO 68) -X- XT YT RETURN XT YT RETURN XT YT RETURN XT YT RETURN XT YT RETURN 44 45 14 5 15 5 1009780. 1C29780. 14 15 6 6 1037280. 1057280. 14 7 15 7 106478G. 1084780. 14 15 8 8 1092280. 1112280. 14 15 9 9 1121137. 1141137. 46 47 16 5 17 5 1049780. 1071780. 16 17 6 6 1077280. 1099280. 16 7 17 7 1104780. 1126780. 16 16 8 8 1132280. 1154780. 15 15 9 9 1163637. 1186137. 48 49 18 5 19 5 1093780. 1115780. 18 18 6 6 1121280. 11437 80. 17 7 17 7 1149280. 1171780. 16 16 8 8 1177280. 1199780. 15 15 9 9 1208637. 1231137. 50 51 19 5 19 5 1138280. 1160780. 18 18 6 6 1166280. 11887 80. 17 7 17 7 1194280. 1216780. 16 18 8 8 1222280. 1271780. 17 17 9 9 1280637. 1303137. 52 53 19 5 19 5 1183280. 1205780. 18 20 6 6 1211280. 1260780 . 19 7 19 7 1266280. 1288780. 18 8 1294280. 54 55 20 5 20 6 1255780. 1279776. 20 6 1283280. i ALLOCATIONS FROM SOURCE 1 FOR PERIOD 3 133 < JPJJ.LjP. . __ ____S_K.UQ^^ PROD YT = 0 YT= 1 YT= 2 YT= 3 YT= 4 - XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 0' ' 0. .1.. 1 . 0 12500. 2 2 0 25000. 3 3 0 37500. 4 4 0 5000G. 5 5 0 62 500. . 6... . 6 _ .JJ 15.0M.. 7 7 0 135000. 8 8 0 145000. 9 9 0 155000. 10 0 165000. ALLOCATIONS FROM SOURCE 2 FOR PERIOD 3 m -P.ULP-PROD -XT- X Y YT= 0 YC T RETURN X Y Y7= 1 YC T RETURN _S.AML.O.G_P_B-aD.UCJ„.OJ.) . YT=. 2 X Y YC T RETURN YT= 3 X Y YC T RETURN YT= 4 X Y YC T RETURN 0 _ 1 _ 2 3 4 0 0 0.0 j Q _ £ _ & . _ Q _ 0 0 0.0 0 0 0.0 0 0 0.0 0 _0_ 0 •0 JL 0 . J2.5.Q.C.._. 2 5000. 37500 . 50000. 0 1 0.91 0 J_1_JL..B.2_9_ 1 1 0.82 9 1 1 0.82 9 1 1 0.82 9 57119. _5A6JB3_._ 77183. 89683. 102183. 0. 2 1.8 2 0 _0_2_1...8.2_JX. 0 2 1. 82 0 0 2 1. 82 0 0 2 1.8 2 0 74237. __6_3_7_,_ 99237. 111737. 124237. 5 _6. 7 8 9 G 0 0.0 _C._0_0..-0_ 2 0 0.0 2 0 0.0 C 0 0.0 0 _0_ 0 0 0 62 500. __75.0CtC, 126940, 139440, .1 5 5000, 1 J_ 1 1 1 10.82 9 J_1.JJ.2_J_. 1 0.8 2 9 1 0.82 9 1 0.82 9 114683. A221S3..-139683. 199683. 209683. 0 2 1.82 0 _0_2_L,_2_0_ 0 2 1.82 C 0 2 1.82 0 0 2 1 .8 2 0 136737. JA9.2.3i._ 209237. 219237. 229237 . 10 J.l_ 12 0 0 0.0 0 j . _ _ _ a . _ o_ 2 0 0.0 0 165000 .2.1.722.0. 229440 1 1 0 . 8 2 9 j_i_a._e.2_9_ 219683. _2.2_9.6_8.3_._ C 2 1.82 0 239237. ALLOCATIONS FROM SOURCE 3 FOR PERIOD 2 1 9 5 P.UL.P S.A.W.L.O.G PRODUCTION PROD -XT- X Y Y T= 0 P XI RETURN X YT= 1 Y P XI RETURN X Y YT= 2 P XI RETURN X Y YT= 3 P XI RETURN X Y YT= 4 P XI RETURN 0 1 C 0 0 0 0 0 0 0 0. • 12 500 . 0 1 1 0 0 1 0 1 22238. 270.0.0.. G 1 2 _2_ 0 0 0 1 44476. 40000. 0 1 3 3 0 0 0 1 66714. 60928. 0 1 4 __4_ 0 C _0_1 8 89 52. .3.23.8.1. 2 3 4 0 0 0 0 G 0 0 G 0 0 0 0 2 5000. 3 7 500. 50000. 1 1 .1 1 0 1 1 0 1 .1 0 1 39500. 52000. 64500. 1 1 1 2 2 2 0 1 0 1 0 1 52500. 65000. 77500. 2 2 4 3 3 3 G 2 0 2 2 4 6700G. 79500. 91000. 2 •x 4 4 4 4 C 2 2 3 2 4 80000. 90000. 100000. 5 6 0 0 0 0 0 c 0 0 62500. 75000 . 1 1 1 0 1 1 0 1 77C00. 89500. 1 6 2 __2 •0 1 2 6 90000. 102000. 5 6 3 _3_ 2 5 2 6 -101000. 111000. G 6 4 4 2 5 2 6 11CC00. 1200G0. 7 8 9 1 2 3 0 0 0 G 1 .0 2 0 3 89GCG. 103000 . 117000 . 1 2 9 1 0 1 1 0 2 1 2 6 102000. 116000. 129000. 7 8 9 2 2 2 2 6 2 6 2 6 114C0G . 126000. 138000. 7 8 9 3 3 3 2 6 2 6 2 6 12300C. 135000. 147CG0. 7 8 9 4 4 4 2 6 2 6 2 6 132000. 144000. 156000. 10 . - .1.1 4 1.1. 0 _0.._ G 4 3 6 1310GC . 144000. 10 1.1 1 2 6 1 2 6 141GGG. 15 3 C 0.0.. 1C 1 1 2 __2. 2 6 2 6 15C000. J62000 . 10 1 1 3 2 6 2 6 159000. 1 7.L0.G.0. 10 1 1 4 _4_ 2 6 __2_6 168000. .18.000.0.... _ 12 13 14 12 13 14 0 0 0 3 6 3 6 3 6 156000. 168000 . 180000. 12 13 14 1 2 6 1 2 6 1 2 6 165000. 177000. 1890GG. 12 13 13 2 2 2 2 6 2 6 2 6 17400 0. 186000. 198 500 . 12 12 12 3 3 3 2 6 2 6 2 6 18300 0 . 195500. 2G8C0G. 11 11 11 4 4 4 2 6 2 6 2 6 192500. 205000. 217500. 15 1.6 15 _1.5_ G _0__ 3 6 _3_6. 19 2000 . 2045CG. 14 14 1 2 6 1 2 6 2C150C. 214.00.0.. 12 13_ 2 2 2 6 2 6 211C00. 223500. 12 12 3 _3 _ 2 6 2 6 220500. 2330.00.. 11 U 4 4 2 6 2 6 230000. 242500. _ . 17 18 19 15 15 15 0 0 0 3 6 3 6 3 6 217C00. 2295C0 . 242000 . 14 14 14 1 2 6 1 2 6 1 2 6 2265C0. 239000. 251500. 13 13 13 2 2 2 2 6 2 6 2 6 236000. 248500. 261G00. 12 12 12 3 3 3 2 6 2 6 2 6 24550C. 25800G. 309940. 11 11 11 4 4 4 2 6 2 6 2 6 255000. 306940. 319440. 20 21 15 1.5 0 .0 3 6 3 6 _ 254500. 267000 . 14 14 1 2 6 1 2 6 264C0C. 315.94.G... 13 1 3 2 2 2 6 2 6 312940 . 325440. 12 11 3 _3 _ 2 6 2 6 222440. 336.0XLO. 1C 1.1 4 4 2 6 2 6 333000. _34.5J10.0..__,. 22 23 24 15 15 14 0 0 0 3 6 3 6 3 6 318940. 331440. 34 5000. 14 13 14 1 2 6 1 2 6 1 2 6 328440. 342G00. 354000. 12 13 13 2 2 2 2 6 2 6 2 6 339000. 351000. 4G322G. 12 12 12 3 3 3 2 6 2 6 2 6 348000. 400220. 412440. 11 11 4 4 2 6 2 6 397220. 409440. ALLOCATIONS FROM SOURCE 3 FOR PERIOD 3 1.96 -< PULP-PROD -XT-YT= 5 P XI RETURN YT- 6 P XI RETURN SAUI nfi PRODUCTION YT= 7 X Y P XI RETURN YT= 8 P XI RETURN YT= 9 Y P XI RETURN 0 Q 5 0 0 111190. 0 6 0 0 13342 8. 0 7 3 0 151666. 0 8 3 0 169904. 0 7 3 0 225903. _1 1 5 0 1 104095. 1. 6 2 1 1.21959. 1 _7_ _2_ 1 139916. 1 . _8_ _2_ 1 1.57936. 1 7 2. 1. —21.4154. 2 2 5 2 2 96741. 2 6 2 2 113857. 2 7 2 2 131222. 2 8 2 2 148762. 2 7 2 2 205459. 3 5 2 3 9 90C0. 3 6 2 3 108000. 3 7 2 3 124667. 3 8 2 3 141619. 3 8 2 3 198738. 4 4 5 2 4 109000. 4 6 2 4 118000. 4 7 2 4 127000. 4 8 2 4 136000. 4 8 2 4 193119. 5 5 5 2 5 119000. 5 6 2 5 128000. 5 . 7 2 5 137000. 5 8 2 5 146000. 4 8 2 4 2006 83. 6 ._5.._ _2_.6_ 1290.00.. 6 6 __2_ _6 13-80.00. 6 _7_ 2 6 1470-00.. 6 _8_ _2_ 6 13J_CL0J1. 5 8 2 .5 __ 21.0683. 7 7 5 2 6 141000. 7 6 2 6 150000. 7 7 2 6 159000. 7 8 2 6 168000. 6 8 2 6 220683. 8 8 5 2 6 153000. 8 6 2 6 162000. 8 7 2 6 171000. 7 8 2 6 180500. 7 8 2 6 232683. 9 9 5 2 6 165000. 9 6 2 6 174000. 8 7 2 6 183 500. 7 8 2 6 193000. 7 8 2 6 245183. 10 10 5 2 6 17 7000. 9 6 2 6 186500. 8 7 2 6 196000. 7 8 2 6 205500. 7 8 2 6 257683. .1.1.-_ 1.0_ _5._ 2 6 18930.0. 9 6 _2_ 6 199000. 8 7 2_ 6 258J5.0.Q.. 7 8 _2_ _6__ .2.1.8.0.0-0.. 7 _.8_ _2_6_ _ 2.70183. 12 10 5 2 6 202000. 9 6 2 6 211500. 8 7 2 6 221000. 7 8 2 6 230500. 7 8 2 6 282683. 13 10 5 2 6 214500 . 9 6 2 6 224000. 8 7 2 6 233500. 7 8 2 6 243000. 7 8 2 6 295183. 14 10 5 2 6 227000. 9 6 2 6 236500. 8 7 2 6 246000. 7 8 2 6 294940. 7 8 2 6 307683. 15 10 5 2 6 239500 . 9 6 2 6 249000. 8 7 2 6 297940. 7 8 2 6 307440. 4 8 2 4 365683. 16 10 5 2 6 252000. 9 6 2 6 300940. 8 7 2 6 310440. 6 _8_ _2_ _6_-___.21.0Jia. 5__J3 2. ..5_ —3736 83. 17 10 5 2 6 303940. 9 6 2 6 313440. 7 7 2 6 324000. 7 8 2 6 333000. 6 8 2 6 385683. 18 10 5 2 6 316440. 8 6 2 6 327000. 8 7 2 6 336000. 7 8 2 6 385220. . 7 8 2 6 39 76 83. 19 9 5 2 6 3 300CC. 9 6 2 6 339000. 8 7 2 6 388220. 7 8 2 6 397440. 20 10 5 2 6 342000 . 9 6 2 6 391220. 8 7 2 6 400440. 2.1 1.0 5 2 _6_ 334.22.GL. 9 6 2 6 403440. 22 10 5 2 6 4C644G. J ALLOCATIONS FROM SOURCE 4 FOR PERIOD 3 197 < PUL P. S_AWJ-CG PRODUCT ION PROD YT= o YT= 1 YT= 2 YT= 3 " Y T = V -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 G 0. 0 0 22238. 0 0 44476. c C 66714. ' 0 0 88952. .0 0 12500. 0 . 0 2700G. 0 0 40000. 0 0 60928. 0 0 .8.23.8.1.. 2 0 0 25000. 0 0 39500. 0 0 52 500. c 0 67C00. G 0 80000. 3 0 G 37500. 0 0 52000. 0 0 65000. 0 G 795GC. C G 9 0 0 0 0 . 4 0 0 500GG. G G 64500. 0 0 77500. G 0 91000* 0 0 10GOOG. 5 C 0 62500. 0 0 77000. 0 0 90000. c c 1C1GQ0. G 0 11G000. 6 0 0 7 5000. 0 0 89500. 0 0 102000. 0 0 111000. G 0 12000C. 7 0 0 89000. 0 C 1020G0. 0 G 1 1 4 0 0 0 . G G 123000. C 0 132000. 8 0 0 1030G0. 0 0 116000. 0 0 126000. 0 0 135000. n. 0 14400C. 9 0 0 117000 . 0 0 129000. 0 0 138000. 0 G 147000. 0 0 156000. 10 0 0 131000. G 0 141000. 0 0 150000. 0 0 15900G. 0 0 168000. .0 _ Q . 144000. 0 0 153000. 0 0 162000. 0 _ 0 _ 17100.0... 0 0 1.8.0.00-0.. 12 0 0 1 5 6 0 0 0 . 0 0 16 5000. 0 0 174000. C G 183000. c 0 192 500. 13 0 0 168000. 0 G 177000. 0 0 186000. 0 0 195500. 0 0 2050GG. 14 0 0 180000. 0 0 189000. 0 0 198500. G 0 2G800G. 0 0 217 500. 15 0 G 192000. 0 0 20150G. G 0 211000. c 0 22G50G. G 0 23GG0G. . .__1_6 0_ _0_. 204500. 0 G 214000. 0 G 223500. 0 0 233000. 0 0 24250Q. 17 0 0 217000. 0 0 226500. 0 0 236000. 0 0 245500. 0 0 255000. 18 G 0 229500. 0 G 239000. 0 0 248500. 0 0 258000. 1 0 275G0G. 19 0 0 24 2000. 0 G 251500. 0 0 26100G. 1 G 278000. 2 0 295000. 20 0 0 254500. 0 0 264000. 1 0 281000. 2 G 29800C. 3 0 315000. . 21. 0 _Q._ 267000. 1 0 2.84.00.0-. 2 0 301000. 3 _.G. 318000. 4 0 ' . 3 . 3 . 5 . 0 . 0 0 . 22 1 0 287000. 2 0 304000. 3 0 321000. 4 0 3 3 8 0 0 0 . 1 0 365000. 23 2 0 307000. 3 0 324000. 4 0 341000. 1 G 368000. 2 0 38500G. 24 3 0 3 2 7000. 4 0 344000. 1 0 371000. 2 0 388000. 3 0 405000. A J ALLOCATIONS FROM SOURCE 4 FOR PERIOD 3 PULP- SAW! 00, PRODUCTION PROD YT= 5 YT= 6 Y T- 7 YT= 8 YT= 9 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN C 0 0 111190. C 0 133428. 0 ' 0 151666. 0 0 169904. G 1 193407. 1 0 0 10409 5. 0 0 12 1959. 0 0 139916. G G 1.579.36. 0 __1_ 18.1439... 2 0 0 96741. 0 0 113857. 0 0 1312 2 2. 0 0 148762. c 1 172265. 3 0 0 9 9 0 0 0 . 0 0 108000. 0 0 124667. 0 0 141619. 0 1 165122. 4 0 0 109000. 0 0 118000. 0 0 127000. 0 G 136000. 0 1 159503. 5 0 0 119000. 0 0 128000. 0 0 137000. 0 0 146000. 0 1 169503. 6 .0_ _0-_ . 1.2.9J3-0-0-. 0 0 138-000. 0 0 147000. G 0 156000. 0 _1_ 1 79 503 . 7 0 C 141000. 0 0 150000. 0 0 159000. C 0 1 6 8 0 0 0 . c 1 191503. 8 0 0 153000. 0 0 162000. 0 0 171000. 0 G 180500. 0 1 204C03. 9 0 0 165000. 0 0 174000. 0 0 18 3 500. G 0 19 3G00. 0 1 216503. 10 0 0 < 177000. 0 0 186500. 0 0 196000. 0 0 205500. 0 1 229003. 1.1 Q_ 0L_ 189500. 0 0 199000. 0 0 208500. G _ 0 _ 21 RGr.n. - 0 -1_ 241503. 12 0 0 2G2000. 0 0 211500. 0 0 221000. 0 0 2 30 500. c 1 254003. 13 0 0 214500. 0 0 224000. 0 0 233500. 0 0 243000. 0 1 266503. 14 0 0 2 27000. 0 0 236500. 0 0 246000. 1 0 263000. 1 1 286503. 15 0 0 239500. 0 0 249000. 1 0 266000. 2 0 283000. 2 1 306503. . . ..1.6 0 0 252000. 1 0 269000 . 2 0 286000. 3 0 303000. 3 _1_ 3 265.03. .. 17 1 0 272000. 2 0 289000. 3 0 306000. 4 0 323000. 0 1 356503. 18 2 0 292000. 3 0 309000. 4 0 326000. 1 0 353000. 1 1 376503. 19 3 0 312000. 4 0 329000. 1 0 356000. 2 0 373G0C. 2 1 396503. 20 4 0 332000. 1 0 359000. 2 0 376000. 3 0 393000. 3 1 416503. 21 1 0 362000. 2 0 379000. 3 0 396000. 4 413G00. 3 . X»_ 45_768 3. 22 2 0 382000. 3 0 399000. 4 0 416000. 3 _____ 0 457440. 4 0 477683. 23 3 0 40 2 0 0 0 . 4 0 419000. 3 0 460440. 4 0 477440. 24 4 0 422000. 3 0 463440. 4 0 480440. ALLOCATIONS FROM SOURCE 5 FOR PER IOC 3 199 P_UL.P___.__ , , SJWLaG_e.RD.DU.C_TJ.Q_. : PROD YT= 0 YT= 1 YT = 2 YT= 3 YT= 4 -XT- X Y RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 0 0. 0 0 22238. 0 0 44476. 0 0 66714. 0 0 88952. _ l - _ ___.0_ 0 12500. 0 .0 27000. n 0 40000. 0 0 60928. 0 _D_._ - 823.81..__ 2 0 0 25000. 0 C 39500. 0 0 52500. 0 0 67000 . 0 0 80000. 3 3 0 36000. 0 0 52000. 0 o • 65000. c 0 79 500. 0 0 90000. 4 4 G 480G0. 3 0 63000. 3 0 76000 . 0 0 91000. G 0 10000C. 5 5 0 60000. 4 0 75000. 4 0 88000. 0 0 101000. 0 0 110000. 6 . 5 0 72500. 5 0 87000. 5 0 100000 . 0 _0_ 111000. 0 0 12_0_0jD0_. 7 5 0 8 5000 . 5 0 99500. 5 0 112500. 0 0 123000. 0 0. 132000. 8 5 0 97500. 5 0 112000 . 5 0 125000. 0 0 135000. 0 0 144000. 9 5 0 110000. 5 0 124500. 5 0 137500. 0 0 147000. 0 0 15600G. 10 5 0 122500. 5 0 137000. 0 0 150G0C. 0 G 159G00. 0 0 168000. _ . J.1 5 0 135000. c 0 1.49500. 0 0 162000. 0 _ 0 L _ 171000. 0 0 1 80000. 12 5 0 149000. . 5 0 162000. 0 0 174000. 0 0 183000. 3 0 192000. 13 5 0 163000. 5 0 1.76000. 0 0 186000. 3 0 195000. 3 0 204000. 14 5 0 177000. 0 0 189000. 3 0 198000. 3 0 • 207000 . 3 0 216000. 15 5 0 191000. 3 0 201000. 3 0 210000. 3 0 219000. 4 0 228000. 16 3 _0 2-0 4.0_0.0.. 3 _ 0 213000. 3 0 222000. 4 0 231000. 5 0 24000C. 17 3 0 216000. 3 0 225000. 4 0 234000. 5 0 243000. 5 0 25250G. 18 3 0 2 2 8000. 4 0 237000. 5 0 246000. 5 0 2 5 5500. c 0 265000. 19 4 0 240000. 5 0 24900G. 5 0 258500. 5 0 268000 . 5 0 277500. 20 5 0 252000. 5 0 261500. 5 0 271000. E G 2 80500. c 0 290G0G. 21 5 0 2.6 4.5JQ.CL. 5 0 2_7 4.0.0.0, 5 0 2 83 500. 5 0 293000 . 5 0 3G2500. 22 5 0 277000. 5 0 286500. 5 0 296000. 5 0 305500. 5 0 315000. 23 5 0 289500. 5 0 299000. 5 0 308500. 5 0 318000. 5 0 335000. 24 5 0 302000. 5 0 311500. 5 0 321000. 5 0 338000. 5 0 355000. ALLOCATIONS FROM SOURCE 5 FOR PER IOO 3 2 0 0 . PULP SAWLOG PRESDUCT I ON PROD YT= 5 YT- 6 YT= 7 Y T= 8 YT= 9 -XT- X Y . RETURN X Y RETURN X Y RETURN X Y RETURN X Y RETURN 0 0 C 11119C. 0 0 133428. 0 0 151666. G 0 169904. . G 0 193407. 1 CL_ 0 1 0 4 0 9 5 . 0 0 121959. 0 0 139916. 0 0 157936. 0 0 13.14 3.9. 2 0 0 96741 . 0 0 113857. 0 0 131222. G 0 148762. 0 0 172265. 3 0 0 99000. 0 0 108000. 0 0 124667. 0 0 141619. 0 0 165122. 4 0 •o 109000. 0 0 118000. 0 0 127000. 0 0 136000. 0 0 159503. 5 0 0 119000. 0 0 128000. 0 0 137000 . 0 0 146000. c 0 169503. 6 0 0 129000. 0 0 138000. 0 0 147000. 0 0 156000. 0 0 179503. 7 0 0 141000. 0 0 150000. 0 0 159000. G 0 168000. 0 0 191503. 8 0 0 153000. 0 0 162000. 0 0 171000 . 0 0 180500. 0 0 204003. 9 0 0 165000 . 0 0 174000. 3 0 183000. 3 0 192000. 3 0 215 50 3 . 10 0 0 177000. 3 0 186000. 3 0 195000 . 3 0 204000. 3 0 227503. 1.1 3 0 189000. 3 0 19 8000. 3 0 207000. 4 0 216000. 4 0 23.9.5.03.. 12 3 0 201000. 3 0 210000. 4 0 219000. 5 C 228000. 5 0 251503. 13 3 0 213000. 4 0 222000. 5 0 231000. 5 0 240500. 5 0 264003. 14 4 0 225000. 5 0 234000. 5 0 243500. 5 0 253000. 5 0 276503. 15 5 0 237000. c 0 246500. 5 0 256000. 5 0 265500. c 0 289003. . 1.6 5 0 2 4 9 5 0 0 . 5 0 259000. 5 0 2 68 500. 5 0 278000. 5 0 301 50 3. 17 5 0 262000. 5 0 271500. 5 0 281000. 5 0 2905G0. 5 0 314003. 18 5 0 274500. 5 0 284000. 5 0 293500. 5 0 3G3000. 5 0 326503. 19 5 0 287000. 5 0 296500. 5 0 306000. 5 0 323000. 5 0 346503. 20 5 0 299500. c 0 309000. 5 0 326000. 5 0 343000 . 5 0 366503. .2.1 5 0 312000. 5 0 329000. 5 0 346000. 5 0 363000. 5 _0_ 3.8.6 5.0 3 . 22 5 0 332000. 5 0 349000. 5 0 366000. c _> 0 383000. 6 0 406503. 23 5 0 3 5 2000. t. -> 0 369000. 5 0 .386000. 6 0 40 3000. 7 0 426503. 24 5 0 372000. 5 0 389000. 6 0 406000. 7 0 4230GO. 8 0 4 4 6 5 0 3 . O P T I M A L A L L O C A T I O N W I T H I N P E R I O D 3 1 S T O F P A I R E D C O L S . 2 N D O F P A I R E D C O L S . _§1 P U L P A L L O C . S / L A L L O C . P U L P P R O D A R E A S A W L O G P R O D U C T I O N 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 16 17 1 8 1 9 0 ( 1 ) 0 0 0 0 0 o- • 0 0 0 0 0 0 0 0 0 0 0 G 0 0 ( 2 > G O 0. 0 G 0 0 0 0 0 c G _ _ G G C 0 G 0 0 < 3 ) 0 0 0 1 0 2 0 3 0 4 0 G 6 0 7 0 8 0 8 < 4 ) 0 0 .0 0 0 Q 0 0 0 0 0 0 G 0 . 0 0 0 0 0 1 ( 5 ) 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 G 0 0 0 0 1 ( 1 ) 1 0 o' •o 0 G 0 0 0 0 0 0 0 0 0 G 0 0 0 0 12.). 0 0 0 0. 0 0 0 G G C G 0 0 0 0 G 0 0 0 0 ( 3 ) 0 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 8 ( 4 ) G 0 0 0 0 G 0 0 0 0 0 0 0 0 b 0 0 0 0 1 ( 5 ) 0 0 0 0 0 G 0 0 G 0 0 0 0 0 c 0 0 C 0 D ( 1 ) 2 0 1 G 1 0 0 C 0 0 0 0 0 0 0 0 0 0 0 0 < . 2 J 0 0 D Q 0 0 0 0 G C 0 G 0 C 0 G 0 0 0 0 ( 3 ) 0 0 1 1 1 2 2 3 2 4 2 c 2 6 2 7 2 8 2 8 ( 4 ) 0 G 0 0 0 0 0 G 0 0 0 0 0 0 0 G 0 0 0 1 ( 5 ) 0 0 0 G G C G 0 0 0 0 0 0 0 G G c G 0 G ( 1 ) 0 0 2 0- 2 0 1 G c 0 G 0 0 0 o 0 0 0 0 0 1 2 J 0 0 0 G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( 3 ) 0 0 1 1 1 2 2 3 3 4 3 5 3 6 3 7 3 8 3 8 ( 4 ) C 0 0 0 0 0 G C 0 0 0 0 0 0 0 0 0 0 0 1 ( 5 ) 3 0 0 0 0 0 0 0 C 0 0 0 0 C 0 G G G 0 G ( 1 ) G 0 G 0 0 0 G C 0 C 0 0 0 0 0 G 0 0 0 0 ( 2 ) 0 0 0 0 0 c 0 G 0 C 0 G 0 c 0 G 0 0 0 0 ( 3 ) 0 0 1 1 I 2 4 3 4 4 4 5 4 6 4 7 4 8 4 8 ( 4 ) 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 1 ( 5 ) 4 0 3 0 3 G 0 0 G 0 0 0 0 0 0 C 0 c 0 c ( 1 ) G 0 0 G •o 0 c c 0 G 0 0 0 G 0 0 0 G 0 0 J . 2 _ ) - _ . 0 0 G _ J 3 ' 0 0 0 0 0 0 0 0 0 G 0 G 0 G 0 0 ( 3 ) 0 G 1 1 1 2 5 3 5 4 5 5 6 5 7 c -* 8 5 8 ( 4 ) 0 0 C 0 0 G G G 0 0 0 0 0 0 0 G 0 0 0 1 ( 5 ) 5 0 4 0 4 0 0 0 0 0 0 G 0 G C C G G 0 0 6 ( 1 ) 1 0 C .0 O C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( 2 ) G O 0 0 0 0 0 0 O G 0 0 0 0 O G 0 0 G O ( 3 ) 0 0 1 1 1 2 6 3 6 4 6 5 6 6 6 7 6 8 6 8 ( 4 ) G O ' 0 G 0 0 0 0 G O 0 0 0 0 0 0 0 0 0 1 ( 5 ) 5 0 5 0 5 0 O G O G O G 0 0 0 C C C 0 0 7 ( 1 ) 2 0 1 0 1 0 0 0 G O 0 0 G O 0 0 0 0 0 0 ( 2 ) G O 0 0 0 0 G O 0 0 0 0 G O G G 0 0 0 0 ( 3 ) 0 0 1 1 1 2 7 3 7 4 7 5 7 6 7 7 7 8 7 8 ( 4 ) 0 0 0 0 0 0 O G 0 0 0 0 0 0 0 0 0 0 0 1 ( 5 ) 5 0 5 0 5 0 0 0 0 0 0 0 G O O C G G 0 0 8 ( 1 ) 3 0 2 0 2 0 G G 0 0 0 0 0 0 0 0 1 0 1 0 ( 2 ) 0 0 O G 0 0 0 0 G G G Q 0 0 G O G O 0 0 . ____ . .. ( 3 ) 0 0 1 1 1 2 8 3 8 4 8 5 8 6 8 7 7 8 7 8 < 4 ) G O 0 0 G O 0 0 G O 0 0 . 0 0 0 0 0 0 0 1 ( 5 ) 5 0 5 0 5 0 O G O C 0 0 0 0 C O O C 0 0 J OPTIMAL ALLOCATION WITHIN PERIOD 3 202 1ST OF PAIRED COLS. - PULP ALLOC. 2ND OF PAIRED COLS. - S/L ALLOC. PULP SAWLOG PRODUCTION PROD AREA 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 < 9 (1 ) 4 0 3 0 3 0 0 0 0 G G 0 0 0 0 G 0 0 0 0 ( _ _ _ G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0_ G 0 G (3) 0 0 1 1 1 2 9 3 9 4 9 5 ' 9 6 6 7 6 8 6 8 (4 J 0 0 0 0 0 0 0 0 0 C 0 G 0 0 0 C 0 0 0 1 (5) 5 0 5 0 5 0 0 0 0 0 • 0 0 0 0 3 0 3 0 3 0 10 (1 ) 5 0 4 •o 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 _ _ ( . _ . ) _ . 0_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o • 0 0 (3) G 0 1 1 10 2 10 3 10 4 IC 5 7 6 7 7 7 8 7 8 • (4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (5) 5 0 5 0 0 0 0 0 0 G 0 0 3 0 3 0 3 0 3 0 11 (1 ) 6 0 5 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 (2J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (3) 0 0 1 1 11 2 11 3 11 4 8 c 8 6 8 7 7 8 7 8 (4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 0 0 1 ( 5 > 5 0 5 0 0 0 0 0 0 0 3 0 3 0 3 0 4 0 4 0 12 (1 ) 6 0 6 0 0 0 0 0 0 0 0 0 0 0 0 C 0 C 0 0 _ 12) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (3) 1 0 1 1 12 2 12 •3 c 4 c c 9 6 8 7 7 8 7 8 (4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 C 0 C 0 0 0 1 (5) 5 0 5 0 0 0 0 0 3 0 3 0 3 0 4 0 5 0 5 0 13 (1 ) 6 0 6 0 0 0 0 0 0 0 0 0 0 G 0 0 1 0 1 0 121 _0 0 0 0 _ 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (3 ) 2 0 2 1 13 2 10 _ IC 4 IC c 9 6 8 7 7 8 7 8 (4) 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 1 (5) 5 0 5 0 0 0 3 0 3 G 3 0 4 0 5 0 5 0 5 0 14 (1 ) 6 0 0 0 0 0 0 C 0 G 0 0 0 0 1 0 2 0 2 0 (?) 0 ..0 0 0 0 0 . 0 0 0 0 0 0 _D_ 0 0 0 0 0 (3 ) "2 0 14 1 11 2 11 3 11 4 IC c _ _ _ _ _ 9 6 8 7 7 8 7 8 (4 ) 0 0 0 0. 0 0 0 0 0 0 ' 0 0 0 0 0 C 0 C 0 1 (5) 5 0 0 0 3 0 3 0 3 G 4 0 5 0 5 0 5 0 5 G 15 (1 ) 6 0 0 0 0 0 0 0 0 0 G 0 1 0 2 C 3 0 3 0 (2J D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (3 ) 4 0 12 1 12 2 12 _ 11 4 10 c 9 6 8 7 7 8 7 8 <4) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (5) 5 0 3 0 3 0 3 0 4 C 5 0 5 0 5 0 5 0 5 0 16 (1 ) 0 0 0 0 0 0 0 0 0 G 1 0 2 0 3 G 4 0 4 0 (.2.) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (3 ) 13 0 13 1 13 2 12 3 11 4 10 c 9 6 8 7 7 8 7 8 (4) 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 C 0 0 0 1 ( 5 ) 3 0 -> 0 3 0 4 0 5 0 5 0 5 0 5 0 5 0 5 0 17 (1) 0 0 0 0 0 0 0 0 1 0 2 0 3 0 4 C 5 0 5 0 - (.2J 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (3 ) 14 0 14 1 1 2 2 12 •2 11 4 IC 5 9 6 8 7 7 8 7 8 (4 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 1 (51 0 0 4 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 OPTIMAL ALLOCATION WITHIN PERIOD 3 1ST OF PAIRED COLS. - PULP ALLOC. 2ND OF PAIRED COLS. - S/L ALLOC. PULP SAWLOG PRODUCTION PROD AREA 0 1 2 3 4 5 6 7 ' 8 9 10 11 12 13 14 15 16 17 18 19 18 (1) 0 0 G 0 0 0 1 0 2 0 3 G 4 0 5 0 6 G 6 0 (.2 ) 0 0 0 G C 0 G_ Q C 0 c G 0_ C G G 0 G 0 0 (3 ) 15 0 14 1 13 2 12 3 11 4 10 c 9 6 8 7 7 8 7 8 < M 0 0 0 0 • 0 G 0 G 0 0 0 0 0 0 0 0 0 0 0 1 (5 ) 3 0 4 •0 5 0 5 0 5 0 5 C 5 0 5 0 5 0 ' 5 0 (1) 0 0 0 0 1 G 2 0 3 0 0 5 0 6 0 6 0 6 0 <_2J 0 0 J3 0 0 0 0 G G c CL- G 0 0 0 0 0 0 0 0 (3) 15 0 14 1 13 2 12 3 11 4 IO 5 9 6 8 7 7 8 7 8 (4) 0 G 0 0 c u G 0 0 G 0 0 0 0 0 G 1 0 1 1 (5 ) 4 0 5 0 5 0 c 0 5 0 5 0 5 0 5 C 5 0 5 G 19 20 (1) 0 0 1 0 2 0 3 0 4 0 5 0 6 0 6 0 6 0 6 0 L2 ) 0_J3 • .0 0 0 G 0 0 0 G 0 0 0 0 Cl_C 0 0 0 0 <3) 15 0 14 1 13 2 12 3 11 4 10 9 6 8 7 7 8 7 8 ' < M 0 0 G c G 0 0 G G 0 0 0 0 G 1 0 2 0 2 1 (5 ) 5 0 5 0 5 0 5 0 5 0 5 0 «; ._< 0 5 0 5 0 5 0 21 (1) 1 0 2 0 3 0 4 0 5 G 6 0 6 0 6 0 6 D 6 0 (2J .0 .0__ 0 0 0 0 0_D G 0 G 0 0 0 G 0 0 0 G 0 (3) 15 0 14 1 13 2 12 3 11 4 1C 5 9 6 8 7 7 8 7 8 < M 0 0 G 0 0 0 0 C G 0 0 0 1 0 2 0 3 0 3 1 (5) 5 0 5 0 5 0 5 G 5 C 5 0 5 0 5 0 5 G 5 0 22 < 1) 2 0 3 0 • 4 G 5 0 6 0 6 0 6 0 6 G 6 0 6 0 _ _ < . 2 J _0 0 0 G _0_ 0 0 0 0 0 0 G _ 0 C 0 _0 0 0 0 0 (3) 15 0 14 1 13 2 12 3 11 4 10 5 9 6 8 7 7 8 7 8 < « ) 0 0 C 0 0 C 0 G 0 0 1 0 2 0 3 0 4 0 3 1 (5 ) 5 0 5 G 0 c C 5 G 5 0 5 0 5 0 5 0 6 G 23 (1) 3 0 4 C 5 0 6 G 6 0 6 0 6 0 6 0 6 0 6 0 (_2J . a_o_ .0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G 0 (3) 15 0 14 1 13 2 12 3 11 4 1G 5 9 6 8 7 7 8 7 8 ( 4 ) G 0 0 G 0 0 0 0 1 0 2 0 3 0 4 0 4 G 3 1 (5 ) 5 0 5 0 5 G c G 5 0 5 0 5 G 5 0 6 0 7 C 2 4 (1) 4 0 5 0 6 0 6 G 6 G 6 0 6 0 6 0 6 0 6 0 - (.2.) 0 0 0 0 0 0 0 0 C 0 0 0 0 0 0 0 0 0 0 0 (3) 15 0 14 1 13 2 12 3 11 4 10 5 9 6 8 7 7 8 7 8 < M 0 0 0 0 0 0 1 0 2 0 3 0 4 0 4 0 4 0 3 1 (5 ) 5 0 5 0 5 0 c 0 5 0 5 G 5 C 6 G 7 0 8 G 204 < MINIMUM COST ALLOCATION POLICY FOR X=68 ,Y=28 OPTIMAL ALLOCATIONS BETWEEN AND WITHIN PERIODS TIME MILL ALL AREAS PRODUCTIVE AREAS JLEJiLG.C C.E-MAN.O : 1 2 3 4 5 P/L S/L XT YT XTK YTK XTK YTK XTK YTK XTK YTK XTK YTK ( 1) 24 10 30 15 15 0 3 7 7 8 0 ( F 5 C~ -(.2) _2C S 1.6 5 6_ 0 _0 0 3 5 .0 0 7 G__ (3) 24 9 22 8 6 0 0 0 7 8 4 0 5 0 MINIMUM COST = $1324765.8 INDEX 206 INDEX Page A b s t r a c t i i i Acknowledgment . . . . • . . • . • . . x i i Appendices 115 Conc lus ions 108 Costs of Log S o r t i n g , Loading and T ranspor t i ng and D i f f e r e n t i a l Sawlog Net Worth f o r Source Two 116 D e r i v a t i o n of Net De l i ve red Log Costs 40 The Formulat ion and Development of the Model . 16 G lossary of Terms . . . . . . . • . x i i i I n t roduc t i on . 1 L i s t i n g of the Computer-Coded Programme DYALOG 128 L i s t of F igures x i L i s t of Tables x L i t e r a t u r e C i t e d . . . . . . . . . . . 112 Output Tables Generated by DYALOG 137 Resu l t s 78 Table of Contents . v i / 

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