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Some mathematical programming models in the design and manufacture of plywood Raghavendra, Bangalore Gururajachar 1982

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SOME MATHEMATICAL PROGRAMMING MODELS IN THE DESIGN AND MANUFACTURE OF PLYWOOD  by  BANGALORE GURURAJACHAR RAGHAVENDRA B . S c , Bangalore U n i v e r s i t y , 1967 M.Sc,  U n i v e r s i t y o f Mysore, 1969  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in  THE FACULTY OF GRADUATE STUDIES ( F a c u l t y o f Commerce and B u s i n e s s A d m i n i s t r a t i o n ) We accept t h i s t h e s i s as conforming to t h e r e q u i r e d s t a n d a r d .  THE UNIVERSITY OF BRITISH COLUMBIA May, 1982  Bangalore G u r u r a j a c h a r Raghavendra  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree at the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may department or by h i s or her  be granted by  the head of  representatives.  my  It is  understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be allowed without my  permission.  Department of  Commerce & B u s i n e s s  Administration  The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6  (3/81)  written  (ii) ABSTRACT  One f a c t o r o f wood l o s s i n the manufacture o f plywood i s i m p l i c i t i n t h e form o f e x c e s s t h i c k n e s s i n plywood due t o t h e c h o i c e o f veneer t h i c k n e s s e s and plywood d e s i g n s used i n assembly.  The t h i c k n e s s and  d e s i g n s c u r r e n t l y i n use appear t o have come l a r g e l y from t r a d i t i o n and t h e r e i s no e v i d e n c e i n t h e l i t e r a t u r e t o show what c o n s t i t u t e s t h e most economical veneer t h i c k n e s s e s and plywood d e s i g n s f o r a m i l l .  The  problem o f d e t e r m i n i n g them i s v e r y complex s i n c e many t y p e s o f plywood are assembled i n each m i l l  as some i n t e g r a l m u l t i p l e c o m b i n a t i o n o f a  few veneers s a t i s f y i n g the 'balanced d e s i g n ' and o t h e r s t r u c t u r a l specifications.  The consumption o f l o g s i s dependent on the excess  t h i c k n e s s i n plywood and t h e economics o f t h e m i l l  f u r t h e r depend on  how e f f i c i e n t l y a g i v e n s e t o f veneers and d e s i g n s are used t o s a t i s f y the o r d e r f i l e requirements.  In t h i s d i s s e r t a t i o n , t h e s e a s p e c t s o f  t h e Plywood Design and M a n u f a c t u r i n g (PDM)  problem a r e addressed u s i n g  a mathematical programming approach. The problem o f f i n d i n g t h e o p t i m a l veneer t h i c k n e s s e s , a s s o c i a t e d plywood d e s i g n s and product mix i s f o r m u l a t e d as a n o n - l i n e a r mixed i n t e g e r mathematical programming model.  U t i l i z i n g t h e s t r u c t u r e o f the  c o n s t r a i n t s and by s e l e c t i n g a p p r o p r i a t e v a r i a b l e s t o branch on, i t i s demonstrated t h a t the PDM  problem can be s o l v e d e f f i c i e n t l y t h r o u g h  an i m p l i c i t enumeration a l g o r i t h m i n v o l v i n g a t r e e s e a r c h procedure. The subproblem t o be s o l v e d a t each f e a s i b l e node o f t h e t r e e i s a L i n e a r M u l t i p l e Choice Knapsack  (LMCK) problem whose s o l u t i o n can be  obtained e x p l i c i t l y f o l l o w i n g i t s c o e f f i c i e n t structure.  A computer  code i s w r i t t e n i n FORTRAN f o r t h e i m p l i c i t enumeration a l g o r i t h m .  (iii)  Data o b t a i n e d  from a plywood m i l l  model and t h i s code. of the mill  i n B.C. i s a n a l y s e d  u s i n g t h e PDM  I t i s demonstrated t h a t theaannual n e t revenue  can be s u b s t a n t i a l l y i n c r e a s e d through t h e use o f t h e PDM  model. The PDM model i s f u r t h e r extended t o m i l l  situations involving  more than one s p e c i e s and v a r y i n g o r d e r f i l e r e q u i r e m e n t s . is reformulated  The model  i n each case and i t i s demonstrated t h a t e s s e n t i a l l y  the same t r e e s e a r c h procedure can be used t o s o l v e a l l t h e s e models. When t h e o r d e r f i l e i s independent o f s p e c i e s , t h e subproblem t o be s o l v e d a t each node o f t h e t r e e i s a G e n e r a l i z e d Network i s shown t h a t t h i s G e n e r a l i z e d Network  problem.  It  problem can be reduced t o a  G e n e r a l i z e d T r a n s p o r t a t i o n problem u t i l i z i n g t h e s t r u c t u r e o f t h e c o e f f i c i e n t s and s o l v e d as an o r d i n a r y T r a n s p o r t a t i o n problem.  When  the o r d e r f i l e i s dependent on s p e c i e s , t h e subproblem decomposes i n t o several  L i n e a r M u l t i p l e Choice Knapsack problems.  I f more than one  s p e c i e s o f veneer can be mixed w i t h i n a plywood p a n e l , t h e subproblem i s a l i n e a r programming  problem.  The PDM model i s f u r t h e r shown t o be a s p e c i a l case o f a d i s j u n c t i v e programming  problem.  F o l l o w i n g t h e development o f t h e PDM model,  methods t o determine t h e e f f i c i e n c y o f plywood d e s i g n s and t h e optimum number o f veneer t h i c k n e s s e s f o r a plywood m i l l  are developed..  (iv) TABLE OF CONTENTS  ABSTRACT  ( i i )  ACKNOWLEDGEMENTS CHAPTER 1  •  (vi i i )  INTRODUCTION  1  1.1  Introduction  1  1.2  O p e r a t i o n s Research  1.3  Wood Losses i n Plywood. Manufacture  11  FORMULATION OF THE PROBLEM  14  2.1  The Veneer T h i c k n e s s Problem  14  2.2  Terminology  16  2.3  The Design and M a n u f a c t u r i n g Problem  19  2.4  F o r m u l a t i n g t h e PDM Model  22  2.5  D i s c u s s i o n and E x t e n s i o n s  34  SOLUTION TO THE PROBLEM  38  3.1  S o l u t i o n t o t h e PDM Model  38  3.2  An I m p l i c i t Enumeration  40  3.3  B r a n c h i n g Tests and Bounds  48  3.4  Computational  60  CHAPTER 2  CHAPTER 3  CHAPTER 4  i n Plywood I n d u s t r y ....  R e l a t e d t o Plywood Designs  Algorithm  Results  6  EXTENSIONS TO MORE SPECIES  63  4.1  E x t e n s i o n s t o t h e PDM Model  63  4.2  O r d e r f i l e Independent o f S p e c i e s  64  4.3  O r d e r f i l e Dependent on S p e c i e s  81  4.4  Mix o f S p e c i e s w i t h i n a Panel  85  DISJUNCTIVE PROGRAM  91  5.1  D i s j u n c t i v e Programming  91  5.2  The PDM as a D i s j u n c t i v e Program  93  CHAPTER 5  (v)  CHAPTER 6  NUMBER OF VENEER THICKNESS  101  6.1  The Optimum Number o f Veneer T h i c k n e s s e s  101  6.2  A l t e r n a t e Face Veneer T h i c k n e s s  110  CONCLUSIONS  113  CHAPTER 7 BIBLIOGRAPHY APPENDIX I :  H L i s t i n g o f FORTRAN Program  APPENDIX I I : Plywood M i l l  Data  5  120 133  APPENDIX I I I : R e s u l t s o f PDM Model: Four Veneers  136  APPENDIX IV: R e s u l t s o f PDM Model: Three Veneers  138  APPENDIX V:  140  R e s u l t s o f PDM Model: A l t e r n a t e Face Veneer ...  (vi) LIST OF TABLES  1.  Number o f Veneers f o r Face, Core and Centre  17  2.  Example o f Plywood Designs w i t h Four Veneers  19  3.  P e r m i s s i b l e C o n s t r u c t i o n A l t e r n a t i v e s f o r Three P l y Plywood  4.  27  Permissible Construction A l t e r n a t i v e s f o r Five Ply Plywood  5.  28  P e r m i s s i b l e C o n s t r u c t i o n A l t e r n a t i v e s f o r Seven P l y Plywood  6.  28  P e r m i s s i b l e C o n s t r u c t i o n A l t e r n a t i v e s f o r Nine P l y Plywood  29  7.  Symmetry i n Core/Centre Veneers  54  8.  Bounds on x^ and x^  56  9.  PDM Subproblem Example  76  10.  S p e c i e s Lay-up A l t e r n a t i v e s  86  11.  Computation o f t h e Upperbound  12.  Optimum Number o f Veneer T h i c k n e s s and Design E f f i c i e n c y . .  ZQ  107 108  (vii)  LIST OF FIGURES  1.  A Flowchart o f the Algorithm to Solve the PDM  Problem  59  2.  The PDM Subproblem as a Network Flow Problem  74  3.  F e a s i b l e Region o f Plywood Design C o n s t r a i n t  92  (vii i)  ACKNOWLEDGEMENTS  I am g r e a t l y i n d e b t e d t o P r o f . Derek A t k i n s and P r o f . Daniel Granot, co-chairmen o f my t h e s i s committee, f o r t h e i r v a l u a b l e g u i d a n c e , encouragement and a s s i s t a n c e d u r i n g t h e development o f t h i s t h e s i s .  I  am g r a t e f u l t o P r o f . G.G. Young and Dr. D.H. W i l l i a m s o f t h e F a c u l t y o f F o r e s t r y and Dr. I s r a e l  Zang, V i s i t i n g P r o f e s s o r i n Management  S c i e n c e , f o r many h e l p f u l comments and s u g g e s t i o n s . Many thanks t o Barbara o f an o t h e r w i s e  complicated  S t r o u t s f o r her prompt and e f f i c i e n t t y p i n g manuscript.  I am t h a n k f u l t o my w i f e , L e e l a and son, P r a s a d , f o r t h e i r understanding  and encouragement throughout t h e w r i t i n g o f t h i s  patience thesis.  I owe a deep sense o f g r a t i t u d e t o my b r o t h e r Vasu, mother Kamalabai and my l a t e f a t h e r G u r u r a j a c h a r support  f o r a l l t h e i r h e l p , encouragement and  leading to thew r i t i n g o f this  Financial  support  thesis.  r e c e i v e d from t h e Canadian Commonwealth S c h o l a r s h i p  Committee as w e l l as some support  r e c e i v e d from NSERC g r a n t s A-4181  and A-4743 a r e g r a t e f u l l y acknowledged.  1 CHAPTER 1  1.1  Introduction  In t h e c o n v e r s i o n o f l o g s t o veneer and plywood, wood l o s s e s o c c u r at several  stages due t o f a c t o r s such as t h e s i z e and shape o f l o g s and  processing l i m i t a t i o n s .  The y i e l d o f plywood i s g e n e r a l l y 45-55 per  cent o f t h e l o g i n p u t by volume, t h e r e m a i n i n g b e i n g c o n v e r s i o n l o s s e s i n t h e form o f r e s i d u a l s and l o s s e s due t o s h r i n k a g e and c o m p r e s s i o n . Though much o f t h e r e s i d u a l s i s c o n v e r t e d i n t o byproducts such as wood c h i p s and hog f u e l and used e l s e w h e r e , t h e i r economic v a l u e i s c o n s i d e r a b l y reduced.  In t h e c o n t e x t o f d w i n d l i n g f o r e s t r e s o u r c e s , i n c r e a s i n g  m a n u f a c t u r i n g c o s t s and f l u c t u a t i n g and c o m p e t i t i v e market t h e importance o f e f f i c i e n t u t i l i z a t i o n o f t i m b e r i s a l l  structure,  t h e more  g r e a t e r now than i t was e v e r b e f o r e . Though many o f t h e f a c t o r s o f wood l o s s a r e e i t h e r b i o l o g i c a l o r t e c h n o l o g i c a l , t h e r e a r e some a r e a s i n plywood manufacture where decision-making or p o l i c y factors contribute to the reduction i n the y i e l d o f plywood.  One such f a c t o r i s i m p l i c i t i n t h e form o f excess  t h i c k n e s s i n plywood due t o t h e c h o i c e o f veneer t h i c k n e s s e s and plywood d e s i g n s used i n assembly.  W h i l e t h e e x t e n t o f wood l o s s i n t h i s form  may appear t o be s m a l l , t h e s a v i n g s any improvement i n t h i s a r e a may b r i n g about can prove t o be s u b s t a n t i a l .  The importance stems from  t h e f a c t t h a t p r o c e s s e d wood w i t h an added v a l u e from manufacture is lost indirectly.  The veneer peel t h i c k n e s s and plywood d e s i g n s  c u r r e n t l y i n use i n most m i l l s have come l a r g e l y from age o l d t r a d i t i o n with i n t u i t i v e  "improvements',  i f any, over t i m e .  There i s no e v i d e n c e  2 i n the p u b l i s h e d l i t e r a t u r e to show what c o n s t i t u t e s the most peel t h i c k n e s s e s and plywood designs f o r a m i l l  and how  t o f i n d them.  r e c e n t y e a r s , t e c h n o l o g i c a l i n n o v a t i o n s have l e a d t o improved processes  i n the plywood i n d u s t r y .  economical In  manufacturing  Adoption o f s o p h i s t i c a t e d q u a n t i t a t i v e  t e c h n i q u e s and computers as a i d s i n d e c i s i o n - m a k i n g  and process  control  have brought c o n s i d e r a b l e s a v i n g s t o t h e i n d u s t r y .  Y e t , t h i s one  aspect  o f the wood l o s s and plywood d e s i g n problem has not been g i v e n t h e a t t e n t i o n it  deserves. The  problem o f d e t e r m i n i n g the optimum veneer t h i c k n e s s e s and  a s s o c i a t e d plywood d e s i g n s  i s v e r y complex due to t h e f o l l o w i n g  Many t y p e s o f plywood are assembled by each plywood m i l l few ( t h r e e o r f o u r ) veneer t h i c k n e s s e s .  reasons.  using only a  G e n e r a l l y , a l l the plywood so  assembled s h o u l d conform t o t h e 'balanced d e s i g n ' , a requirement r e g u l a t e s t h e d i r e c t i o n o f g r a i n and the o r d e r i n which veneers  which of  d i f f e r e n t s p e c i e s and t h i c k n e s s e s can be assembled i n t o plywood. S p e c i f i c a l l y , t h e assembly o f veneers symmetrical  s h o u l d be such t h a t t h e y are  about t h e c e n t r a l p l y ( i e s ) .  There s h o u l d be a t l e a s t  one  balanced d e s i g n f o r each type o f plywood w h i c h , i n a d d i t i o n , s h o u l d s a t i s f y other s t r u c t u r a l  requirements  and s p e c i f i c a t i o n s .  The  production  o f plywood u s i n g any d e s i g n a l t e r n a t i v e i s c o n d i t i o n a l upon t h a t d e s i g n a l t e r n a t i v e b e i n g f e a s i b l e f o r t h e balanced d e s i g n requirement specifications.  When veneer t h i c k n e s s e s are themselves  other  decision variables,  t h e r e l a t i o n between consumption o f l o g s and the p r o d u c t i o n o f is non-linear.  and  veneers  The v o l u m e t r i c wood l o s s i n t h e form o f excess t h i c k n e s s  i n plywood, o r e q u i v a l e n t l y , the consumption o f l o g s , a l s o dependsoon the d i s t r i b u t i o n of the o r d e r f i l e requirements. mill  f u r t h e r depend on how  The economics o f the  e f f i c i e n t l y a s e t o f veneer t h i c k n e s s e s and  3 plywood d e s i g n s are u t i l i s e d t o form t h e b e s t product mix to c o n s t r a i n t s on r e s o u r c e a v a i l a b i l i t y  subject  and product demand.  A s s o c i a t e d w i t h the veneer t h i c k n e s s and plywood d e s i g n problem d e s c r i b e d above i s t h e problem o f d e t e r m i n i n g the optimum number o f veneer t h i c k n e s s e s f o r a m i l l .  G e n e r a l l y , the h i g h e r t h e number o f  veneer t h i c k n e s s e s used by a m i l l , t h e lower i s the wood l o s s and l o g costs.  However, a h i g h e r number o f veneer t h i c k n e s s e s i s a s s o c i a t e d  w i t h h i g h e r setup c o s t s i n p e e l i n g , d r y i n g , s t o r a g e , l a y - u p , h a n d l i n g and r e c o r d - k e e p i n g .  On t h e o t h e r hand, the lower t h e number o f veneer  t h i c k n e s s e s , t h e h i g h e r i s the wood l o s s and l o g c o s t s , but t h e c o s t s are reduced  considerably.  setup  The problem o f d e t e r m i n i n g t h e optimum  number o f veneer t h i c k n e s s e s which balances the t r a d e - o f f between the two i s t h e r e f o r e a d i r e c t sequel o f t h e veneer t h i c k n e s s , plywood d e s i g n and product mix  problem.  In t h i s d i s s e r t a t i o n , we address  t h e above a s p e c t s o f t h e s e  problems u s i n g a q u a n t i t a t i v e approach.  We  f o r m u l a t e t h e problem o f  d e t e r m i n i n g the veneer t h i c k n e s s , a s s o c i a t e d plywood d e s i g n s and mix as a mathematical  two  product  programming model i n which the o b j e c t i v e f u n c t i o n  and some o f t h e c o n s t r a i n t s are n o n - l i n e a r and, i n a d d i t i o n , some o f t h e d e c i s i o n v a r i a b l e s are r e s t r i c t e d t o 0-1  values.  The model t a k e s on  d i f f e r e n t forms depending on the f a c t o r s c o n s i d e r e d i n t h e problem and mill  practices.  Some o f t h e general problems are c o n s i d e r e d and, i n  each c a s e , s o l u t i o n procedures are developed. plywood m i l l  which e x p l o i t the s t r u c t u r e i n t h e model  A computer code i s w r i t t e n and data o b t a i n e d from a  i s a n a l y s e d to demonstrate the s u i t a b i l i t y o f the model.  To our knowledge, t h e r e are no l i t e r a t u r e r e f e r e n c e s which d i r e c t l y c o n s i d e r the e v a l u a t i o n o f optimum veneer t h i c k n e s s e s , plywood  4 d e s i g n s and product mix.  Whatever l i t t l e work done i n t h i s f i e l d  t o the study o f s t r e n g t h or s t r u c t u r a l  relates  p r o p e r t i e s o f known t h i c k n e s s e s  and d e s i g n s ( C o l e b e c k and N o r t h c o t t ( 1 9 5 8 ) , N o r r i s , Werren and McKinnon (1961)).  However, t h e r e i s c o n s i d e r a b l e work done i n t h e area o f optimum  product mix f o r plywood u s i n g , m o s t l y , l i n e a r programming t e c h n i q u e s .  We  c o n s i d e r t h i s and o t h e r O p e r a t i o n s Research a p p l i c a t i o n s i n t h e plywood i n d u s t r y i n s e c t i o n 1.2  of t h i s chapter.  In a d d i t i o n , some work i s done  i n the general a r e a o f wood l o s s e s i n plywood p r o d u c t i o n .  We  briefly  d e s c r i b e t h i s work as w e l l , a n a l y s e the f a c t o r s o f wood l o s s and z a t i o n t e c h n i q u e s a s s o c i a t e d w i t h them i n s e c t i o n  optimi-  1.3.  The o r g a n i s a t i o n o f t h e r e s t o f t h i s d i s s e r t a t i o n i s as  follows.  In c h a p t e r two we p r e s e n t t h e t e r m i n o l o g y and f a c t o r s a s s o c i a t e d w i t h the plywood m a n u f a c t u r i n g process and t h e d e s i g n problem.  We d e f i n e the  d e c i s i o n v a r i a b l e s and develop the c o n s t r a i n t s and the o b j e c t i v e o f the problem as f u n c t i o n s o f t h e s e v a r i a b l e s .  We  f u r t h e r demonstrate  how  the  balanced d e s i g n requirement can be a c h i e v e d through the development o f a s e t o f d e s i g n c o e f f i c i e n t s and a s s o c i a t e d i n d i c a t o r v a r i a b l e s we f o r m u l a t e t h e Plywood Design and M a n u f a c t u r i n g (PDM) n o n - l i n e a r mixed i n t e g e r mathematical We problem PDM  problem, we demonstrate  by t h e i m p l i c i t enumeration We  problem as a  programming model.  p r e s e n t an i m p l i c i t enumeration i n chapter three.  and  a l g o r i t h m f o r s o l v i n g the  PDM  U t i l i s i n g the c o n s t r a i n t s t r u c t u r e i n the how t h e s i z e o f t h e s e a r c h t r e e , employed a l g o r i t h m , can be s u b s t a n t i a l l y  f u r t h e r show t h a t t h e subproblem  reduced.  t o be s o l v e d a t each node o f t h i s  s e a r c h t r e e . i s a L i n e a r M u l t i p l e Choice Knapsack (LMCK) problem whose s o l u t i o n can be o b t a i n e d e x p l i c i t l y f o l l o w i n g i t s c o e f f i c i e n t We d e s c r i b e a computer code w r i t t e n i n FORTRAN f o r t h e  structure.  implicit  5 enumeration a l g o r i t h m t o s o l v e t h e PDM problem. a n a l y s e data from a plywood m i l l  Using t h i s code, we  i n B r i t i s h Columbia and show t h a t f o r  t h e p a r t i c u l a r c o n f i g u r a t i o n o f t h e problem parameters, the s o l u t i o n o b t a i n e d from our model i s f a r s u p e r i o r than t h o s e c u r r e n t l y used. E x t e n s i o n o f the PDM model t o m i l l  s i t u a t i o n s i n v o l v i n g more than  one s p e c i e s and v a r y i n g o r d e r f i l e r e s t r i c t i o n s a r e c o n s i d e r e d i n chapter four.  The PDM model i s r e f o r m u l a t e d i n each case and i t i s  demonstrated t h a t e s s e n t i a l l y t h e same t r e e s e a r c h procedure can be used to solve a l l  t h e s e models.  When the o r d e r f i l e i s independent o f s p e c i e s ,  the l i n e a r subproblem t o be s o l v e d a t each node o f t h e t r e e i s a G e n e r a l i z e d Network problem.  I t i s shown t h a t t h i s G e n e r a l i z e d Network problem can  be reduced t o a G e n e r a l i z e d T r a n s p o r t a t i o n problem f o l l o w i n g some p r o p e r t i e s o f t h e PDM model.  When the o r d e r f i l e i s dependent on s p e c i e s ,  t h e subproblem decomposes i n t o s e v e r a l l i n e a r m u l t i p l e c h o i c e knapsack problems whose s o l u t i o n can be g i v e n e x p l i c i t l y .  I f more than one  s p e c i e s o f veneer can be mixed w i t h i n a p a n e l , t h e subproblem i s a l i n e a r program. In c h a p t e r f i v e , we show how the PDM model can be n a t u r a l l y c a s t as a s p e c i a l case o f a D i s j u n c t i v e Programming problem.  We a n a l y s e  the c h a r a c t e r i s t i c s o f t h i s D i s j u n c t i v e Programming problem and show how the PDM problem can be viewed as a l a r g e s c a l e l i n e a r programming problem. In c h a p t e r s i x , we c o n s i d e r the problem o f d e t e r m i n i n g t h e optimum number o f veneer t h i c k n e s s e s f o r a plywood m i l l .  Using computer  o f t h e i m p l i c i t enumeration a l g o r i t h m and d a t a from t h e plywood  codes mill  we d e r i v e t h e maximum revenue a s s o c i a t e d w i t h v a r i o u s number o f veneer thicknesses.  Together w i t h h y p o t h e t i c a l data on s e t u p c o s t s we  demonstrate  6 how t h e optimum number o f veneer t h i c k n e s s e s can be  1.2  O p e r a t i o n s Research  determined.  In The Plywood I n d u s t r y  E x t e n s i v e i n f o r m a t i o n on O p e r a t i o n s Research  (OR) t e c h n i q u e s i n  f o r e s t r y and t h e f o r e s t products i n d u s t r y (plywood, lumber, pulp and paper m i l l s and o t h e r p r o c e s s i n g i n d u s t r i e s u s i n g wood as t h e  raw  m a t e r i a l ) are a v a i l a b l e from t h e b i b l i o g r a p h i e s o f Bare ( 1 9 7 1 ) , M a r t i n and Sendak (1973) and F i e l d (1976, 1977).  However, much o f the work  r e p o r t e d i n t h e s e s o u r c e s r e f e r s t o a p p l i c a t i o n o f OR t e c h n i q u e s i n f o r e s t r y o p e r a t i o n s such as h a r v e s t i n g , l o g g i n g , f o r e s t management and o t h e r s i m i l a r a r e a s .  A general a p p r a i s a l o f p o s s i b l e OR  applications  i n t h e f o r e s t products i n d u s t r y m e n t i o n i n g the areas where t h e s e can be e f f e c t i v e l y used i s g i v e n by Holmes (1976).  Diaz (1974) has  a ' b i b l i o g r a p h y ' o f OR i n lumber p r o d u c t i o n and  'other f o r e s t r y  industries'.  prepared  However, h i s b i b l i o g r a p h y s u f f e r s from the drawback t h a t  i t n e i t h e r e x c l u s i v e l y contains l i t e r a t u r e r e l a t e d to the f o r e s t products i n d u s t r y nor i s e x h a u s t i v e o f a l l industry.  OR a p p l i c a t i o n s i n the  In o r d e r t o complement t h e s e s o u r c e s w i t h r e c e n t l i t e r a t u r e  and u n r e p o r t e d use o f OR i n the i n d u s t r y , w i t h p a r t i c u l a r r e f e r e n c e t o plywood and s a w m i l l i n g , a s u r v e y was (Raghavendra ( 1 9 7 9 ) ) .  undertaken  Columbia  OR work r e l a t e d t o t h e plywood i n d u s t r y as  found from p u b l i s h e d l i t e r a t u r e and i n d u s t r i a l Columbia  in British  are b r i e f l y described i n t h i s s e c t i o n .  practices in B r i t i s h The d e s c r i p t i o n i s  by t h e t e c h n i q u e employed f o r s o l u t i o n such as L i n e a r Programming, Dynamic Programming and S i m u l a t i o n .  c  7 1.2.1  L i n e a r Programming  A s . i s the case i n many o t h e r i n d u s t r i e s , the most w i d e l y used OR methodology i n plywood i n d u s t r y i s L i n e a r Programming ( L P ) .  From  s i m p l e t r a n s p o r t a t i o n problems to complex o p e r a t i o n a l p l a n n i n g models i n v o l v i n g l o g a l l o c a t i o n , p r o d u c t i o n s c h e d u l i n g and  distribution,  t h i s technique  i s e x t e n s i v e l y used.  Numerous i n t r o d u c t o r y a r t i c l e s  d e s c r i b i n g how  LP can be e f f e c t i v e l y used i n d i f f e r e n t c o n t e x t s  of  the plywood i n d u s t r y can be found i n the l i t e r a t u r e [ r e f e r Ijlare.(1971), Diaz ( 1 9 7 4 ) , M a r t i n and Sendak (1 973)  and  F i e l d (1976, 1 9 7 7 ) ] .  A p p l i c a t i o n o f L i n e a r Programming t o plywood manufacture has been c o n s i d e r e d by B e t h e l and H a r r e l ( 1 9 5 7 ) , Koenigsberg ( 1 9 6 6 ) , Ramsing (1965, 1968), E v e r e t t (1967) and  (1960),  Donnelly  Lee (1968).  An  e x t e n s i v e account o f an a c t u a l a p p l i c a t i o n o f LP i n plywood p r o d u c t i o n i s d e s c r i b e d by Kotak (1976).  The model, developed  f o r the plywood  and  hardboard d i v i s i o n o f Canadian F o r e s t P r o d u c t s , New  W e s t m i n s t e r , BC  and  i n use s i n c e 1969,  b a s i c a l l y determines an optimum balance  the a v a i l a b l e wood mix and o r d e r f i l e requirements c o n t r i b u t i o n margin o f t h e d i v i s i o n . an annual  between  so as t o maximize the  The model s e r v e s as a b a s i s f o r  o p e r a t i n g plan g i v i n g d e t a i l s o f p r o d u c t i o n s c h e d u l e s ,  t r a c k o f r a w - m a t e r i a l i n v e n t o r y , o r d e r f i l e requirements :  on a b i w e e k l y b a s i s .  keeping  and t a r g e t s  The s t r a t e g i e s a r e f u r t h e r r e v i s e d on a day  day b a s i s w i t h s c h e d u l e s  f o r l o g p e e l i n g , veneer d r y i n g , press  and s h i p p i n g based on a v a i l a b i l i t y o f raw m a t e r i a l s and performance up to the day.  A financial  a n a l y z e t h e e f f e c t o f v a r i a t i o n s due on t h e o p e r a t i n g income.  The  to  production  cumulative  v a r i a n c e a n a l y s i s i s prepared  t o p r i c e and mix o f  raw-materials  net c o n t r i b u t i o n margin o f the d i v i s i o n  to  8  i s r e p o r t e d t o have i n c r e a s e d by an average o f one m i l l i o n per y e a r d u r i n g the p e r i o d 1969-1975 due Wellwood (1970) d e s c r i b e s how  dollars  t o the use o f t h i s model.  the o r d e r f i l e requirements  of different  types o f plywood can be l i n k e d t o p r e s s , d r y i n g and l a t h e s c h e d u l e s LP.  using  H i s a r t i c l e i s s i m i l a r t o t h a t o f Kotak but g i v e s g r e a t e r d e t a i l  on s c h e d u l i n g m a t e r i a l f l o w a t d i f f e r e n t p r o c e s s i n g c e n t r e s and  has  the o b j e c t i v e o f m i n i m i z i n g the p e n a l t y a s s o c i a t e d w i t h unused veneer and downgrading o f veneer. hypothetical  Yaptenco and W y l i e (1970) c o n s i d e r a  i l l u s t r a t i o n i n which a c h a r a c t e r i s t i c p r o d u c t i o n  problem o f a plywood m i l l  i n v o l v i n g l a t h e s , d r y e r s , edge g l u e r s ,  p a t c h e r s and glue spreaders f o r m u l a t i o n through  scheduling  i s brought down to l i n e a r programming  ' a l g e b r a i c and d i f f e r e n c e e q u a t i o n s ' .  Dobson  (1971) d e s c r i b e s the use o f LP f o r the a l l o c a t i o n o f l o g s t o plywood, sawmill and open market s a l e s . There a r e s e v e r a l o t h e r LP models t h a t are a c t u a l l y i n use i n m i l l s but are not r e p o r t e d i n t h e l i t e r a t u r e (Raghavendra  (1979)).  Many o f t h e s e models are meant f o r i n t e r m e d i a r y s t a g e s o f manufacture and t h e i r r e s u l t s may i s considered.  hot prove t o be o p t i m a l when the o v e r a l l  N e v e r t h e l e s s , t h e s e serve..theypurpose  problem  of analysing  e f f i c i e n c y o r p r o d u c t i v i t y f a c t o r s i n a d e c e n t r a l i z e d framework. such LP i s used t o g i v e l a t h e schedules  for peeling different  One  species  and veneer t h i c k n e s s e s f o r each one o f the l a t h e s w i t h t h e o b j e c t i v e of minimizing p e e l i n g time.  Another LP i s meant f o r improving  i n d r y i n g s i n c e the d r y i n g process  efficiency  i s a b o t t l e n e c k i n plywood manufacture.  The d r y i n g time o f veneer depends on the s i z e and type o f d r y e r , the number o f d e c k s , s p e c i e s and t h i c k n e s s o f veneer and the o b j e c t i v e o f t h e LP i s t o m i n i m i z e  total  drying time.  A t h i r d LP model i s  9 d e s i g n e d t o g i v e optimum p r e s s i n g s c h e d u l e s on a weekly b a s i s w i t h c o n s t r a i n t s on p r e s s i n g t i m e , g l u e - s p r e a d e r o r d e r f i l e r e q u i r e m e n t s and  press c o n f i g u r a t i o n s ,  setup time between b a t c h e s .  There are s e v e r a l o t h e r areas i n t h e i n d u s t r y where LP models have been used or recommended f o r use. general  how  S i t t e r (1969) d e s c r i b e s  in  LP can be e f f e c t i v e l y used i n an i n t e g r a t e d woodworking company  c o n s i s t i n g o f p u l p m i l l , s a w m i l l , and  veneer m i l l s .  Klamecki (1978)  u t i l i s e s LP t o d e t e r m i n e the l e a s t c o s t energy mix complex c o n s i d e r i n g a l t e r n a t i v e s o u r c e s from m i l l gas and s o l a r energy.  Holecek (1975) and  f o r a f o r e s t products residues, o i l , natural  M c K i l l o p (1974) d e s c r i b e  the  use o f LP as a systems model f o r an i n t e g r a t e d f o r e s t products f i r m in  California.  1.2.2,  Simulation  Simulation  s t u d i e s i n the plywood i n d u s t r y have been made i n  the area o f veneer p e e l i n g and (1977) s i m u l a t e d  drying only.  Resch and  the softwood veneer d r y i n g o p e r a t i o n  t h e optimum f l o w o f veneer through two  j e t dryers.  Scheurman so as t o determine  Simulated  computer  runs demonstrated the e f f e c t o f veneer t h i c k n e s s , d r y i n g t e m p e r a t u r e , and  p r e - s o r t i n g o f veneer on d r y i n g time and  (1969) d e s c r i b e an a n a l y t i c a l  Tobin and  Bethel  procedure t o e v a l u a t e the q u a l i t y and  q u a n t i t y o f veneers r e c o v e r a b l e simulation.  costs.  i n the r o t a r y c u t t i n g process u s i n g  10 1.2.3  Dynamic Programming  B a i l e y (1970, 1972,  1973)  uses Dynamic Programming f o r l o g  a l l o c a t i o n d e c i s i o n s by a n a l y s i n g log  supply  and  Mann (1972) and  hypothetical  problems r e l a t e d to  from s e v e r a l a r e a s w i t h n o n - l i n e a r c o s t s . Briggs  Pnevmaticos ,  (1978) demonstrate t h a t Dynamic Programming  can be used f o r bucking o f t r e e - l e n g t h or long l o g s i n t o small  logs.  When the v a l u e s o f small l o g s are known, the c u t t i n g process i s viewed as a s e q u e n t i a l  d e c i s i o n process i n which the optimal  bucking a t s e v e r a l  'stages'  values  from  o f the l o g l e n g t h are determined t h r o u g h  Dynamic Programming r e c u r s i o n .  1.2.4  Non-Li near Programming  Klamecki (1978) has  used Non-Linear Programming t o d e t e r m i n e the  optimum l a t h e s e t t i n g s f o r p r o d u c i n g the b e s t q u a l i t y o f veneer.  Three  b a s i c v a r i a b l e s , namely, k n i f e rake a n g l e , r o l l e r bar compression  and  k n i f e r o l l e r l e a d i n the veneer p e e l i n g process are r e s t r i c t e d physical considerations  as l i n e a r c o n s t r a i n t s .  i s based on the f o r m a t i o n and  and  The  objective function  s e v e r i t y of lathe checks.in  i s e x p r e s s e d as a n o n - l i n e a r  the veneer  f u n c t i o n o f the t h r e e d e c i s i o n v a r i a b l e s .  For v a r i o u s process parameters such as s t r e s s , shear and i n the c u t t i n g p r o c e s s ,  strength  the optimum l a t h e s e t t i n g s are d e r i v e d  a Non-Linear Programming model.  by  using  11 1.2.5  Other OR Techniques  Tyre and S c r e p e t i s (1978) propose an i n v e n t o r y system based on weight s c a l i n g o f roundwood l o g s and d e s c r i b e a method f o r c o n t r o l o f v e n e e r , sawntimber and pulpwood volumes.  Traditional  inventory control  t e c h n i q u e s such as EOQ a r e r e p o r t e d t o be used (Raghavendra  (1979))  i n t h e a r e a o f l o g boom c o n t r o l , i n - p r o c e s s veneer o r lumber and f i n i s h e d p r o d u c t s i n v e n t o r y .  inventory  Ramalingam (1976) i s r e p o r t e d ( i n  B r i g g s (1978)) t o have used a Branch and Bound approach t o t h e t r e e b u c k i n g problem.  S u c c e s s i v e r e d u c t i o n s o f t h e t r e e stems o r l o n g l o g s  at the merchandiser i s postulated to f o l l o w a f i n i t e - h o r i z o n s t a t e Markov (1980)).  discrete  Process by Luken (1978) (as r e p o r t e d i n Luken e t a l .  Economic  impact o f f o r e s t based i n d u s t r i e s i s a n a l y s e d u s i n g  L e o n t i e f i n p u t - o u t p u t models by Reimer (1969) and Raizada and N a u t i y a l (1974).  1.3  Mood Losses In Plywood Manufacture  The f o r e g o i n g a n a l y s i s o f l i t e r a t u r e s o u r c e s and a c t u a l  practice  o f OR i n t h e plywood i n d u s t r y i n d i c a t e t h e wide v a r i e t y o f problems t h a t can be t a c k l e d t h r o u g h some f a c e t o f t h e OR methodology.  Examined  c l o s e l y , t h e s u c c e s s o f many OR t e c h n i q u e s l a y i n t h e i r a b i l i t y t o g i v e e f f i c i e n t operational  s t r a t e g i e s which, e i t h e r d i r e c t l y or i n d i r e c t l y ,  reduces wood l o s s i n t h e c o n v e r s i o n o f t r e e s t o l o g s , veneer and plywood. Even when t h e o b j e c t i v e o f an OR model i s t h e m a x i m i z a t i o n o f v a l u e , i t i m p l i c i t l y r e s u l t s i n e f f i c i e n t u t i l i s a t i o n o f t h e p r i n c i p a l rawm a t e r i a l namely, wood.  This l e a d s us t o t h e b a s i c q u e s t i o n s : why  12 wood l o s s e s o c c u r i n t h e c o n v e r s i o n o f l o g s t o veneer and plywood and how can i t be prevented o r reduced. E x t e n s i v e i n f o r m a t i o n on t h e areas o f wood l o s s i n plywood manufacture,  f a c t o r s r e s p o n s i b l e for,them and t h e e x t e n t o f t h e s e  l o s s e s are a v a i l a b l e i n t h e l i t e r a t u r e [ Dobie and Hancock ( 1 9 7 2 ) , Woodfin ( 1 9 7 3 ) , N a g a r a j u , Raghavendra and Venkataraman ( 1 9 7 4 ) , M e r i l u o t o ( 1 9 6 5 ) , Heiskanen (1962)].  ( 1 9 6 6 ) , B r a c k l e y ( 1 9 6 8 ) , B a l d w i n (1975) and Wood  Many o f t h e f a c t o r s o f wood l o s s i n plywood a r e i n t e r a c t i v e .  However, t h e y can be b r o a d l y c l a s s i f i e d i)  Biological  Factors:  into three categories:  Due t o t h e i n h e r e n t n a t u r e and  v a r i a b i l i t y o f t h e raw m a t e r i a l .  Losses due t o r o u n d i n g o f l o g s ,  d e f e c t s i n wood, s h r i n k a g e i n d r y i n g and compression can be c l a s s i f i e d t o f a l l ii)  i n pressing  into t h i s category;  Technological Factors:  Due t o p r o c e s s i n g l i m i t a t i o n s o r  n o n - a v a i l a b i l i t y o f better technology.  Core l o s s e s which  occur  due t o t h e l i m i t a t i o n o f t h e s p i n d l e i n veneer l a t h e , r o u n d i n g l o s s e s due t o c e n t e r i n g e r r o r s , l o s s due t o spur t r i m and h a n d l i n g l o s s e s a r e some o f t h e t e c h n o l o g i c a l l o s s e s i n plywood manufacture iii)  Decision-Making/Design  f a c t o r s r e s p o n s i b l e f o r wood  and  Factors:  Due t o m a n u f a c t u r i n g  or decision-making p r a c t i c e s w i t h i n a given t e c h n o l o g i c a l  designs setup.  At l e a s t p a r t o f t h e bucking l o s s e s i n t h e l o g y a r d , trimming l o s s e s caused by o v e r s i z e a l l o w a n c e f o r veneer and l o s s i n t h e form o f excess t h i c k n e s s i n plywood a r e some o f t h e d e c i s i o n making f a c t o r s r e s p o n s i b l e f o r wood l o s s .  Investigation of the b i o l o g i c a l or technological  f a c t o r s o f wood  l o s s e s i n plywood manufacture .is beyond t h e scope o f t h i s  dissertation.  Among t h e d e c i s i o n - m a k i n g o r d e s i g n f a c t o r s r e s p o n s i b l e f o r wood l o s s e s , e v a l u a t i o n o f some o f t h e s e f a c t o r s w i t h an economic o b j e c t i v e has r e c e i v e d some a t t e n t i o n i n r e c e n t y e a r s .  The problem o f l o g bucking  f o r m a x i m i s i n g v a l u e , though not s p e c i f i c a l l y i n t h e c o n t e x t o f plywood manufacture,  has been s t u d i e d by Conway ( 1 9 7 8 ) , Pnevamaticos  and Mann  ( 1 9 7 2 ) , B r i g g s ( 1 9 7 8 ) , Ramalingam ( 1 9 7 6 ) , L e f e b v r e (1978) and Western F o r e s t Products L a b o r a t o r y (1978).  E v a l u a t i o n o f the t r i m allowance  i n veneer f o r plywood has been s t u d i e d t o some e x t e n t by Hawkins and C l a r k e (1970) and Raghavendra and Nagaraju  (1975).  The concept o f  e v a l u a t i n g veneer peel t h i c k n e s s e s and plywood d e s i g n s f o r economic o p t i m a l i t y i s , t o o u r knowledge, not c o n s i d e r e d i n t h e l i t e r a t u r e . A c o n t r i b u t i o n o f t h i s t h e s i s i s t h e development o f a q u a n t i t a t i v e t e c h n i q u e t o d e r i v e t h e optimum veneer t h i c k n e s s e s , plywood d e s i g n s and product mix.  The c h a r a c t e r i s t i c s o f t h i s problem a r e d e s c r i b e d  i n t h e next c h a p t e r .  14 CHAPTER 2  2.1  The  Veneer Thickness  Problem  Most plywood m i l l s manufacture a v a r i e t y o f plywood t y p e s , each v a r y i n g , a p a r t from s p e c i e s and  surface q u a l i t y c h a r a c t e r i s t i c s , in  the number, t h i c k n e s s and o r d e r o f veneers or p l i e s and the t h i c k n e s s o f plywood.  In any one m i l l  total  l o g s a r e peeled t o one o f t h r e e ,  f o u r or f i v e b a s i c veneer t h i c k n e s s e s and a l l the plywood types assembled as some i n t e g r a l m u l t i p l e combination  are  o f t h e s e veneers.  The  veneers as w e l l as t h e plywood assembled from them should meet some s p e c i f i c a t i o n s with regard to t h i c k n e s s , s t r e n g t h , s t i f f n e s s , q u a l i t y and o t h e r f a c t o r s .  Most o f the plywood so produced  a l s o conform t o the "Balanced Design",  surface  should  a r e q u i r e m e n t which r e g u l a t e s  the o r d e r i n which veneers o f d i f f e r e n t s p e c i e s and t h i c k n e s s e s  can  be assembled i n t o plywood. The  use o f a l a r g e number o f veneer peel t h i c k n e s s e s tends to  i n c r e a s e the c o s t o f plywood because o f the added c o s t s o f p e e l i n g , d r y i n g , s t o r a g e , h a n d l i n g , l a y - u p and r e c o r d - k e e p i n g and N o r t h c o t t ( 1 9 5 8 ) ) .  involved  (Colebeck  A l t e r n a t i v e l y , i n a b i l i t y t o manufacture c e r t a i n  types w i t h i n plywood t h i c k n e s s s p e c i f i c a t i o n , the p o s s i b i l i t y  of  most plywood types ending up i n h i g h e r than r e q u i r e d t h i c k n e s s  and  requirement, o f s p e c i f i c customer o r d e r s d a c t a t e t h e use o f two or more peel t h i c k n e s s e s . having  For example, the two  5 veneers adding up t o 20.5  mm  plywood types 5 p l y 20.5  t h i c k n e s s and  7 p l y 20.5  mm  mm  7 veneers a d d i n g up to the same t h i c k n e s s cannot both be assembled u s i n g a s i n g l e veneer t h i c k n e s s w i t h i n a t o l e r a n c e o f - 0.5  mm.  having  15 S i m i l a r l y , a 7 p l y 20.5 mm  plywood and a 7 p l y 23.5 mm  plywood cannot  both be assembled t o w i t h i n + 0.5 mm u s i n g a s i n g l e veneer t h i c k n e s s . The economics o f plywood manufacture depend of  not o n l y on the c o s t s  l a b o r , raw m a t e r i a l , equipment and s e r v i c e s but a l s o on the d e s i g n s  used i n the assembly o f plywood.  T r a d i t i o n a l l i n e a r programming  models  which have been h i t h e r t o used and a r e b e i n g e x t e n s i v e l y used [ Dobson ( 1 9 7 1 ) , Kotak ( 1 9 7 6 ) , Lee (1968) and Ramsing  (1965, 1968)] m a i n l y  c e n t r e around o p t i m a l product mix o f l o g g r a d e s , s p e c i e s and veneer lay-up a l t e r n a t i v e s .  However, one b a s i c i n f o r m a t i o n which goes  t h e s e LP models as i n p u t i s the e x i s t i n g s e t o f veneer peel  into  thicknesses,  a s s o c i a t e d y i e l d f a c t o r s and d e s i g n or c o n s t r u c t i o n a l t e r n a t i v e s .  There  i s no e v i d e n c e i n p u b l i s h e d l i t e r a t u r e t o c l a i m t h a t t h e veneer t h i c k n e s s e s and t h e a s s o c i a t e d d e s i g n s so used a r e t r u l y the best f o r any mill.  particular  Most o f the plywood d e s i g n s c u r r e n t l y used appear t o have been  developed " p a r t l y from t h e o r e t i c a l  c o n s i d e r a t i o n s , p a r t l y from t r a d i t i o n  and p a r t l y from m a n u f a c t u r i n g r e q u i r e m e n t s " ( C o l e b e c k and N o r t h c o t t The f o r e g o i n g a n a l y s i s n a t u r a l l y r a i s e s the f o l l o w i n g two  (1958)).  questions:  (1)  what i s the optimum number o f veneer t h i c k n e s s e s f o r a m i l l , and  (2)  g i v e n t h e number o f veneer t h i c k n e s s e s , what s h o u l d be t h e s e  t h i c k n e s s e s and what plywood d e s i g n s o r c o n s t r u c t i o n a l t e r n a t i v e s a r e the  best f o r a m i l l .  The answer t o t h e f i r s t q u e s t i o n depends on  how  best the " b e s t " veneer t h i c k n e s s e s a r e f o r each number i n the second problem and how r e s p o n s i v e the changes i n c o s t s a r e t o t h e a d d i t i o n o f each veneer t h i c k n e s s .  Both t h e s e , i n t u r n , a r e dependent on the  d i s t r i b u t i o n of the m i l l ' s o r d e r f i l e requirements, l o g a v a i l a b i l i t y and o t h e r r e s o u r c e r e s t r i c t i o n s .  16 I t i s demonstrated i n t h i s c h a p t e r t h a t t h e problem o f f i n d i n g t h e optimum veneer t h i c k n e s s e s can be f o r m u l a t e d as an o p t i m i z a t i o n model which i s a mixed 0-1 n o n - l i n e a r programming problem.  Some o f  t h e t e r m i n o l o g y r e l a t e d t o plywood d e s i g n s and t h e m a n u f a c t u r i n g p r o c e s s a r e d e s c r i b e d i n s e c t i o n 2.2.  The veneer t h i c k n e s s , plywood d e s i g n  and product mix problem i s d e s c r i b e d i n terms o f t h i s t e r m i n o l o g y i n s e c t i o n 2.3.  The model f o r m u l a t i o n w i t h d e t a i l s o f t h e d e c i s i o n  v a r i a b l e s , t h e c o n s t r a i n t s and t h e o b j e c t i v e f u n c t i o n a r e p r e s e n t e d i n s e c t i o n 2.4.  In s e c t i o n 2.5 a d i s c u s s i o n o f t h e c h a r a c t e r i s t i c s  o f t h e o p t i m i z a t i o n model, i t s v a r i a t i o n s and e x t e n s i o n s a r e p r e s e n t e d .  2.2  Some Terminology R e l a t e d t o Plywood Design and Manufacture  B e f o r e p r e s e n t i n g t h e mathematical f o r m u l a t i o n o f t h e plywood d e s i g n problem  some of- the t e r m i n o l o g y a s s o c i a t e d w i t h t h e manufacture  o f plywood a r e c o n s i d e r e d .  G e n e r a l l y , most plywood s h e e t s c o n s i s t  o f an odd number o f l a y e r s o r p l i e s o f veneer bonded t o g e t h e r by an a d h e s i v e i n such a way t h a t t h e g r a i n d i r e c t i o n o f a d j a c e n t p l i e s i s a t r i g h t a n g l e s t o each o t h e r . from an even number o f p l i e s  In r e c e n t y e a r s , however, plywood  assembled  i s a l s o being made ( P a r a s i n ( 1 9 7 6 ) , COFI  (1978)). In a plywood sheet w i t h odd number o f p l i e s , c o u n t i n g from t h e t o p o r bottom v e n e e r , a)  The f i r s t and t h e l a s t veneers a r e c a l l e d f a c e v e n e e r s ,  b)  A l l even numbered veneers w i t h g r a i n d i r e c t i o n  perpendicular  t o t h a t o f t h e f a c e a r e c a l 1 e d c o r e veneers o r c r o s s - b a n d and  17 c)  A l l o t h e r odd numbered v e n e e r s , i f any, w i t h g r a i n  direction  p a r a l l e l to that o f the face are c a l l e d centre veneers. The  number o f veneers f o r f a c e , c o r e and c e n t r e i n plywood s h e e t s  odd  number o f p l i e s and t h e i r r e s p e c t i v e p o s i t i o n s would t h e r e f o r e be as  l i s t e d i n Table 1.  In an e v e n - p l y plywood, t h e f a c e , core and c e n t r e  veneers a r e s i m i l a r l y d e f i n e d r e l a t i v e t o t h e i r p o s i t i o n of grain  having  and a l i g n m e n t  direction.  # o f veneers rfor  # of plies i n plywood  Core  Face  Centre  -  3  2(1,3)  1(2)  5  2(1,5)  2(2,4)  K3)  7  2(1,7)  3(2,4,6)  2(3,5)  9  2(1,9)  4(2,4,6,8)  3(3,5,7)  Table!:  Number o f Veneers f o r Face, Core and Centre  A s p e c i f i c a t i o n which d e s c r i b e s t h e number, t h i c k n e s s , s p e c i e s o f veneer and t h e o r d e r  i n which t h e y a r e assembled i n t o a plywood sheet  i s c a l l e d t h e d e s i g n o r c o n s t r u c t i o n o f plywood.  S i n c e t h e number o f  veneers f o r f a c e , c o r e and c e n t r e p l i e s can be determined once t h e number o f p l i e s i n plywood i s known ( T a b l e 1 ) , a d e s i g n  can be s p e c i f i e d  by  d e s c r i b i n g t h e t h i c k n e s s and s p e c i e s f o r each one o f f a c e , core and centre  veneers.  The  balanced d e s i g n / c o n s t r u c t i o n  r e q u i r e s t h a t w i t h i n a plywood  panel, t h e s p e c i e s , t h i c k n e s s and d i r e c t i o n o f g r a i n o f veneers  should  be symmetrical  from  about t h e c e n t r a l  ply(ies).  This i s s t i p u l a t e d  18 c o n s i d e r a t i o n s r e l a t e d t o t h e s t r e n g t h p r o p e r t i e s and warping o f t h e panel.  Together w i t h general  t h a t w i t h i n a plywood  mill  p r a c t i c e s , a balanced design  implies  panel  a)  a l l f a c e veneers should  be o f t h e same t h i c k n e s s and s p e c i e s ,  b)  a l l core veneers s h o u l d  be o f t h e same t h i c k n e s s and s p e c i e s ,  c)  a l l c e n t r e veneers should  be o f t h e same t h i c k n e s s and s p e c i e s  and d)  t h e s p e c i e s and/or t h i c k n e s s i n any one group a ) , b) o r c ) above might be t h e same as t h o s e o f t h e o t h e r ( s ) .  An unbalanced d e s i g n o r m o d i f i e d  c o n s t r u c t i o n r e f e r s t o plywood  panels which v a r y from t h e r e q u i r e m e n t s f o r a balanced d e s i g n  i n that  the g r a i n d i r e c t i o n , s p e c i e s and/or t h i c k n e s s o f i n n e r p l i e s may be unbalanced about t h e c e n t r a l p l y ( i e s ) . B r i e f l y , t h e process o f plywood manufacture i n v o l v e s t h e c o n v e r s i o n o f l o g s t o v e n e e r , d r y i n g t h e veneer t o remove e x c e s s i v e m o i s t u r e and g l u i n g , a s s e m b l i n g and p r e s s i n g t h e veneers t o form a plywood The  panel.  t h i c k n e s s o f veneer before t h e d r y i n g process i s c a l l e d t h e green  thickness while that a f t e r drying i s c a l l e d the dry thickness. thicknesses  Plywood  g e n e r a l l y r e f e r t o t h e t h i c k n e s s o f t h e f i n i s h e d panel  f o r market, a f t e r a c c o u n t i n g  f o r sanding l o s s e s , i f n e c e s s a r y .  ready  Throughout  our a n a l y s i s veneer t h i c k n e s s e s r e f e r t o t h e green t h i c k n e s s and plywood thicknesses  r e f e r t o t h e t h i c k n e s s o f unsanded  panels.  F i n a l l y , s i n c e most m i l l s manufacture a v a r i e t y o f plywood, a plywood t y p e i s n o r m a l l y and  designated  t h e t h i c k n e s s o f plywood.  by t h e number o f p l i e s i t c o n t a i n s  Thus,a 7 p l y 20.5 mm plywood i m p l i e s  t h a t t h i s plywood has seven veneers i n i t and t h a t i t s t h i c k n e s s i s  19 20.5  millimetres. Using t h e n o t a t i o n s and t e r m i n o l o g y i n t r o d u c e d above, we can now  d e s c r i b e t h e plywood d e s i g n and m a n u f a c t u r i n g problem.  2.3  D e s c r i p t i o n o f t h e Plywood Design and M a n u f a c t u r i n g Problem  C o n s i d e r an example o f f o u r veneer peel t h i c k n e s s e s and a sample of  plywood t y p e s and a s s o c i a t e d d e s i g n s as p r e s e n t e d i n Table 2.  are  t a k e n from t h e a c t u a l p r a c t i c e s o f a plywood m i l l  Plywood P l i e s Type  Design  Columbia  Plywood  Veneers f o r  Thickness(mm) A l t e r n a t i v e Face  in British  These  Excess(+)  Core  Centre Thickness(mm)  inches  Green  Dry  or Shortage(-)  1  3 p l y 7.5  (.1)  1/10  1/10  -  8.07  7.60  +1.33%  2  3 p l y 9.5  :(i)  1/10  3/16  -  10.36  9.74  +2.53%  (ii)  1/8  1/8  -  10.06  9.46  -0.42%  3  5 p l y 12.5  CD  1/10  1/10  1/10  13.46  12.65  +1.20%  4  5 p l y 15.5  (i)  1/10  1/8  3/16  17.06  16.04  +3.48%  (ii)  1/10  1/7  1/7  17.27  16.24  +4.77%  (iii)  1/8  1/8  1/8  16.76  15.76  +1 .68%  (iv)  1/8  1/7  1/10  17.32  16.28  +5.03%  (D  1/10  1/10  1/8  20.16  18.96  +2.49%  (ii)  1/10  1/8  1/10  20.83  19.58  +5.84%  1/8  1/10  1/10  20.16  18.96  +2.49%  5  7 p l y 18.5  (iii)  Table 2:  Plywood Designs w i t h Four Veneer T h i c k n e s s e s ; Veneer T h i c k n e s s i n mm ( i n c h e s ) a r e : 2.69 ( 1 / 1 0 ) , 3.35 ( 1 / 8 ) , 3.96 (1/7) and 4.98 (3/16).  .  20 whose d e s i g n and m a n u f a c t u r i n g c h a p t e r 3.  problem  will  be f u l l y c o n s i d e r e d i n  The veneer t h i c k n e s s e s used were 2.69, 3.35, 3.96 and 4.98 mm  corresponding  approximately  respectively.  t o 1/10, 1/8, 1/7 and 3/16 o f an i n c h  For some plywood types t h e r e  i s more than one d e s i g n  a l t e r n a t i v e , which i s d e s c r i b e d i n t h e form o f veneers f o r f a c e , core and c e n t r e p l i e s .  The 'green'  t h i c k n e s s o f plywood r e p r e s e n t s t h e sum  t o t a l o f t h e t h i c k n e s s o f veneer i n them w h i l e t h e 'dry' t h i c k n e s s r e f e r s t o t h e a c t u a l f i n a l t h i c k n e s s o f plywood a f t e r a c c o u n t i n g f o r l o s s e s due t o s h r i n k a g e line additions.  i n d r y i n g , compression i n p r e s s i n g and g l u e -  The l a s t column, excess o r s h o r t a g e , r e p r e s e n t s t h e  percentage d e v i a t i o n from t h e intended t h i c k n e s s o f t h e plywood induced  by t h e c h o i c e o f d e s i g n .  shortages excesses  type  Within permissible tolerances,  preceeded by a n e g a t i v e s i g n i n d i c a t e s a v i n g s i n wood w h i l e preceeded by a p o s i t i v e s i g n i n d i c a t e l o s s o f wood.  The a c t u a l  e x t e n t o f wood l o s s o r g a i n can be computed by m u l t i p l y i n g t h e a b s o l u t e d e v i a t i o n s w i t h t h e r e s p e c t i v e q u a n t i t i e s o f plywood produced. We emphasise t h a t t h e volume o f wood l o s s o r g a i n depends not o n l y on t h e veneer t h i c k n e s s and plywood d e s i g n but a l s o on how e f f i c i e n t l y t h e d e s i g n s a r e used t o meet t h e o r d e r f i l e under c o n s t r a i n t s o f l o g a v a i l a b i l i t y , machine c a p a c i t i e s and o t h e r m i l l  restrictions.  T h i s l e a d s t o two a s p e c t s o f t h e problem, namely, ( 1 ) t h e veneer t h i c k n e s s and d e s i g n problem  and (2) t h e product- mix and  manufacturing  problem. In t h e veneer t h i c k n e s s and d e s i g n problem, t h e l i s t o f plywood t y p e s , t h e i r s p e c i f i c a t i o n s and t h e number o f veneer peel are  known. The o b j e c t i v e i s :  thicknesses  21 (i) (ii)  To determine t h e t h i c k n e s s and To s p e c i f y how t h e s e veneers s h o u l d be assembled so as t o produce a l l  t y p e s o f plywood w i t h i n  specification.  On the o t h e r hand, i n t h e product mix and m a n u f a c t u r i n g problem t h e veneer t h i c k n e s s e s , t h e plywood d e s i g n s , t h e a v a i l a b i l i t y o f l o g s , t h e demand f o r end p r o d u c t s and o t h e r m i l l  r e s t r i c t i o n s a r e known.  The  objective i s : (i)  To f i n d t h e optimum q u a n t i t y o f veneers o f each s p e c i e s and t h i c k n e s s t o be produced and  (ii)  To f i n d the optimum q u a n t i t y o f plywood t o be assembled under each d e s i g n a l t e r n a t i v e .  The two a s p e c t s o f t h e problem a r e i n t e r r e l a t e d as veneer t h i c k n e s s and plywood d e s i g n s a r e i n p u t t o the product mix p a r t o f t h e problem. The o b j e c t i v e f o r both problems combined  would be ( i ) t o m i n i m i z e  i m p l i c i t wood l o s s i n t h e form o f excess t h i c k n e s s i n plywood w h i c h , as a r e s u l t , would m i n i m i z e t o t a l revenue.  l o g consumption, o r ( i i ) t o maximize  net  When more than one s p e c i e s w i t h v a r y i n g l o g c o s t s a r e used,  t h e two o b j e c t i v e f u n c t i o n s need not n e c e s s a r i l y g i v e the same r e s u l t s . T h i s can happen, f o r example, when an e x p e n s i v e s p e c i e s has a lower y i e l d compared t o an i n e x p e n s i v e s p e c i e s .  Recognizing that the ultimate  t r a d e - o f f can be measured i n terms o f v a l u e , we use m a x i m i z i n g  of  net revenue as t h e o b j e c t i v e . The product mix p a r t o f t h e problem can be s o l v e d u s i n g  linear  programming (Kotak ( 1 9 7 6 ) , Dobson ( 1 9 7 1 ) , Lee (1968) and Ramsing (1965, 1968)) which i s now an a c c e p t e d m i l l  practice.  However, the veneer  t h i c k n e s s and plywood d e s i g n problem as p r e s e n t e d above has not been c o n s i d e r e d t o date. Whatever l i t t l e r e s e a r c h has been undertaken i n  22 t h i s regard r e l a t e s to strength or s t r u c t u r a l  p r o p e r t i e s o f known  t h i c k n e s s e s and d e s i g n s ( C o l e b e c k and N o r t h c o t t ( 1 9 5 8 ) , N o r r i s , Werren and McKinnon (1961) and B i b i i s , Hsu and Chiu ( 1 9 7 2 ) ) . In t h i s t h e s i s , we c o n s i d e r s i m u l t a n e o u s l y both a s p e c t s o f t h i s problem, r e f e r r e d t o as t h e Plywood Design and M a n u f a c t u r i n g (PDM) problem.  In t h e PDM problem we seek t h e veneer t h i c k n e s s e s , a s s o c i a t e d  plywood d e s i g n s and q u a n t i t i e s o f veneers and plywoods t o be produced which w i l l maximize t h e n e t revenue f o r a m i l l .  2.4  F o r m u l a t i n g t h e Model  To s i m p l i f y t h e p r e s e n t a t i o n , we do not c o n s i d e r e x p l i c i t l y f a c t o r s such as s p e c i e s , l o g g r a d e s , s u r f a c e q u a l i t y o f v e n e e r s , plywood grades and machine c a p a c i t i e s i n t h e f o r m u l a t i o n o f t h e PDM model i n t h i s c h a p t e r .  These f a c t o r s can be e a s i l y i n c o r p o r a t e d i n  t h e model as demonstrated by t h e e x i s t i n g plywood L.P. models (Lee ( 1 9 6 8 ) , Ramsing (1965, 1968), Kotak ( 1 9 7 6 ) ) .  F u r t h e r , though we  c o n s i d e r i n our f o r m u l a t i o n plywood d e s i g n s w i t h an odd number o f veneers h a v i n g balanced d e s i g n s , o u r model can be extended t o e v e n - p l y c o n s t r u c t i o n and/or unbalanced d e s i g n s . o f some o f t h e s e i n s e c t i o n 2.5.  We d i s c u s s t h e i m p l i c a t i o n s  Constraint c o e f f i c i e n t s are i l l u s t r a t e d  f o r plywood t y p e s h a v i n g up t o n i n e p l i e s and t h r e e veneer t h i c k n e s s e s but can be extended t o any number o f p l i e s and any number o f veneer thicknesses.  S i z e s o f veneer and plywood sheets a r e e x p r e s s e d i n  e q u i v a l e n t s o f t h e s t a n d a r d s i z e o f plywood  (8  1  x 4' o r 2.44 m  x 1.22 m).  Decision Variables  Let K be t h e number o f veneer t h i c k n e s s e s i n general the number o f plywood t y p e s .  and N be  Define t h e d e c i s i o n v a r i a b l e s as f o l l o w s :  th x^ =  k  veneer t h i c k n e s s ( i n mm); x-| i s c o n v e n t i o n a l l y t r e a t e d  as t h e t h i c k n e s s o f t h e face L^ =  Quantity o f logs peeled  veneers,  i n t o veneer t h i c k n e s s x^ ( i n c u b i c  metres), =  Q u a n t i t y o f veneer sheets  produced o f t h i c k n e s s x^ ( i n  number o f s h e e t s o f s i z e e q u i v a l e n t t o 2.44 m x 1.22m  size  of plywood), P. .=  Q u a n t i t y o f plywood o f type i produced u s i n g c o n s t r u c t i o n a l t e r n a t i v e j ( i n number o f sheets o f s t a n d a r d  s i z e 2.44 m  x 1.22 m o r e q u i v a l e n t ) , 6..=  An i n d i c a t o r (0-1) v a r i a b l e f o r plywood type i made u s i n g construction alternative j  k  =  l,2...k;  j = l , 2 . . . n . and i = 1,2...N.  G e n e r a l l y , t h e number o f veneer t h i c k n e s s e s , K, used i n most m i l l s i s t h r e e o r f o u r and seldom more than f o u r peel t h i c k n e s s e s a r e used.  The  number o f d e s i g n a l t e r n a t i v e s , n., depends on t h e number o f p l i e s i n plywood. The C o n s t r a i n t s  In t h i s s e c t i o n we w i l l i n t h e PDM problem.  describe the various c o n s t r a i n t s a r i s i n g  a)  Veneer_Jhickness  Tolerances standards  Jolerance  and_Constraints  f o r veneer t h i c k n e s s a r e n o r m a l l y l a i d down i n company  or s p e c i f i c a t i o n s r e l a t e d t o the product.  Council o f  I n d u s t r i e s o f B r i t i s h Columbia (COFI (1978)) standards plywoods, f o r example,  veneer t h i c k n e s s e s , we  To e s t a b l i s h t o l e r a n c e s f o r  should f i r s t c o n s i d e r the c a p a b i l i t i e s o f t h e  veneer p e e l i n g l a t h e o f a m i l l .  Most p e e l i n g l a t h e s can produce o n l y  a d i s c r e t e l y f i n i t e s e t o f veneer t h i c k n e s s e s . l a t h e s might be capable  a c c u r a c y may  for exterior  s p e c i f y , t o l e r a n c e s f o r face veneers s e p a r a t e l y  from those f o r c o r e o r c e n t r e veneers.  range o f v a l u e s .  Forest  of producing  However, some p e e l i n g  veneer t h i c k n e s s e s i n a  continuous  B u t , veneer t h i c k n e s s e s beyond a c e r t a i n degree o f  not be p o s s i b l e from p r a c t i c a l  considerations.  Thus, we  l e t the veneer t h i c k n e s s t a k e o n l y d i s c r e t e v a l u e s , i f n e c e s s a r y , by t r a n s f o r m i n g the range o f peel t h i c k n e s s e s i n t o a d i s c r e t e s e t i n s t e p s o f , say,1/10  of a m i l l i m e t r e .  Let T = {T-j, Tgj-.-T^} be the s e t c o n t a i n i n g a l l p o s s i b l e veneer t h i c k n e s s e s the p e e l i n g l a t h e ( s ) o f a m i l l T^ and  T  £  be the subset  k  where m(k)  T  k  = U \  c o r e / c e n t r e veneers r e s p e c t i v e l y .  T , 2  k  Of t h i s , l e t  representing a l l thicknesses within tolerances  s p e c i f i e d f o r f a c e and  x e  can produce.  k  T  m ( k ) k  } ,  Then  k = 1,2,...K  i s the number o f elements i n T.. i f k = 1 and t h a t i n T f c  otherwise. be e x p r e s s e d  Since x  k  can t a k e on o n l y one v a l u e i n T ,  as f o l l o w s :  k  i t can  (2.  25 (2.1 A)  m(k) km  m=l  = 1  (2.IB) if x  and  m (2.1 C)  km  0  Clearly, in all  otherwise  t h e c o n s t r a i n t s o f t h e PDM model i n which x  k  appears i t  can be r e p l a c e d by t h e r i g h t hand s i d e o f (2.1A) and i n t h a t case t h e o t h e r two c o n s t r a i n t s ( 2 . I B ) and (2.1C) a r e t o be i n c l u d e d  explicitly.  However, f o r s i m p l i c i t y i n p r e s e n t a t i o n , we r e t a i n x^ as a v a r i a b l e t h a t can assume one v a l u e from t h e s e t  b)  as e x p r e s s e d by ( 2 . 1 ) .  P1_ywood_Jhickness_ Con s t r a i n ^  These a r e perhaps t h e most c o m p l i c a t e d c o n s t r a i n t s i n t h e PDM problem due t o t h e f o l l o w i n g r e a s o n s : i)  A l l t h e N plywood t y p e s a r e t o be assembled u s i n g some p e r m i s s i b l e c o m b i n a t i o n o f one o r more of. t h e x ^ ' s , w i t h t h e number o f p l i e s adding t o 3, 5, 7 o r 9 ( o r h i g h e r , i f t h e case d i c t a t e s ) ;  ii)  There s h o u l d be a t l e a s t one c o n s t r u c t i o n a l t e r n a t i v e s a t i s f y i n g t h e b a l a n c i n g r e q u i r e m e n t and t h i c k n e s s t o l e r a n c e f o r each plywood t y p e ;  iii)  There might be more than one plywood t y p e having t h e same number o f veneers i n i t but d i f f e r i n g i n t h i c k n e s s , and  iv)  There might be more than one plywood t y p e h a v i n g t h e same t h i c k n e s s but d i f f e r i n g i n t h e number o f p l i e s .  ,  All  o f t h e s e problems were overcome i n our f o r m u l a t i o n  e v a l u a t i o n o f t h e balanced c o n s t r u c t i o n r e q u i r e m e n t . plywood types w i l l  by a c a r e f u l Since a l l the  have f a c e v e n e e r s , t h e c o n v e n t i o n t h a t  i s the face  veneer l e a d s t o t h e f a c t t h a t a l l c o n s t r u c t i o n a l t e r n a t i v e s w i l l a t l e a s t two veneers o f x^.  Analysis o f the balancing  have  requirement w i t h  the number o f veneers r e q u i r e d f o r f a c e , c o r e and c e n t r e veneers i n d i c a t e t h a t t h e r e a r e o n l y a few p e r m i s s i b l e c o m b i n a t i o n s o f veneers i n which a plywood o f a g i v e n number o f p l i e s can be assembled.  I f K=3,  a 3 p l y plywood can be assembled i n one o f 3 a l t e r n a t e ways and  t h e r e a r e 9 p o s s i b l e ways f o r each one o f h i g h e r  having odd number o f p l i e s ( 5 , 7, 9 o r h i g h e r odd).  ply construction These p e r m i s s i b l e  c o n s t r u c t i o n a l t e r n a t i v e s s p e c i f y i n g t h e veneers f o r f a c e , c o r e and c e n t r e , u s i n g some o r a l l o f t h e t h r e e veneer t h i c k n e s s e s  are l i s t e d  i n Tables  3, 4, 5 and 6 f o r plywood made o f 3, 5, 7 and 9 p l i e s r e s p e c t i v e l y . S i m i l a r p e r m i s s i b l e c o n s t r u c t i o n a l t e r n a t i v e s can be l i s t e d number o f veneer t h i c k n e s s e s plies.  f o r any  and/or plywood made w i t h any number o f  Using Tables 3-6, t h e balanced c o n s t r u c t i o n and plywood  thickness  t o l e r a n c e can be s p e c i f i e d by t h e f o l l o w i n g s e t o f c o n s t r a i n t s :  M  (  1  ^ i j  )  a  i j k  X  k  <  b  i  +  M(l-6  (2.2)  )  (2.3)  I f plywood type i i s assembled construction alternative j  0  Otherwise  f o r a l l j = 1,2 , . . . n.  and  i = 1, 2, ... N,  using (2.4)  27 where, a  .. =  iNumber o f veneers o f t h i c k n e s s x, used i n c o n s t r u c t i o n a l t e r n a t i v e j f o r plywood t y p e i ;  The a ^ ^ a r e t a k e n  from Tables 3 - 6, b -'"(b.^)  -  Lower (Upper) t o l e r a n c e f o r t h i c k n e s s o f plywood t y p e i ,  M  =  A l a r g e p o s i t i v e number,  n..  =  Number o f p e r m i s s i b l e c o n s t r u c t i o n a l t e r n a t i v e s f o r t h e  1  i**  Construction Alternative ( j )  1  t y p e o f plywood.  Number o f Veneers o f x  l  X  2  X  3  Face  Veneers f o r Core Centre  1.  3  0  0  x  l  X  l  2  2  1  0  x  l  X  2  3  2  0  1  x  l  X  3  Table 3:  P e r m i s s i b l e C o n s t r u c t i o n A l t e r n a t i v e s f o r Three P l y Plywood  28 Number o f Veneers o f  Construction Alternative ( j )  x  2  X  3  Face  Veneers f o r Centre Core  5  0  0  x  l  x  l  2  4  1  0  x  l  X  l  3  4  0  1  x  l  X  l  X  3  4  .3  2  0  x  l  X  2  X  l  5  3  0  2  X  3  X  l  6  2  3  0  X  2  X  2  7  2  0  3  X  3  X  3  8  2  2  1  X  2  X  3  9  2  1  2  X  3  X  2  1  Table 4:  l  x  X-i  l  x  X,  l x x  2  P e r m i s s i b l e C o n s t r u c t i o n A l t e r n a t i v e s f o r F i v e p l y Plywood.  Number o f Veneers o f  Construction Al t e r n a t i v e ( j )  x  2  X  3  Face  1  7  0  0  x  2  5  2  0  X,  3  5  0  2  x  4  4  3  0  5  4  0  6  2  7  x  l  x  l  x  l  X  2  l  x  l  X  3  x  l  X  2  X  l  3  x  l  X  3  X  l  5  0  x  l  X  2  X  2  2  0  5  x  l  X  3  X  3  8  2  3  2  x  l  X  2  X  3  9  2  2  3  x  l  X  3  X  2  Table 5:  l  Veneers f o r Centre Core  P e r m i s s i b l e C o n s t r u c t i o n A l t e r n a t i v e s f o r Seven P l y Plywood  29 Construction Alternative (j)  V  :  X x  l  9  1  Number o f Veneers o f 2  X  Face  3  0  0  X, x  l  Veneers f o r Core Centre  x  l  x  X  l  l  X  2  2  6  3  0  3  6  0'  3  x  l  x  l  X  3  4  5  4  0  x  l  X  2  X  l  5  5  0  4  X,  X  3  X  6  2  7  0  x  X  2  7  2  0  7  X,  X  3  X  3  8  2  4  3  x  X  2  X  3  9  2  3  4  X,  X  3  X  2  Table 6:  l  l  1 x  2  P e r m i s s i b l e C o n s t r u c t i o n A l t e r n a t i v e s f o r Nine P l y Plywood  Observe t h a t i n c o n s t r a i n t ( 2 . 2 ) , |-| k  a  -j| | x  1  <  <  represents the t o t a l  t h i c k n e s s o f veneer which i s t r e a t e d as t h e t h i c k n e s s o f plywood.  In  a c t u a l p r a c t i c e , t h i s would not be t h e same as s h r i n k a g e i n d r y i n g , compression i n p r e s s i n g and s p r e a d i n g o f g l u e between veneer l a y e r s a f f e c t t h e f i n a l t h i c k n e s s o f plywood.  Appropriate correction factors  can be used i n t h e a c t u a l a p p l i c a t i o n o f t h e model. From Tables 3 - 6 i t can be observed t h a t when t h e r e a r e t h r e e veneer t h i c k n e s s e s , n. = 3 f o r t h r e e p l y plywood and n. = 9 f o r 5, 7 o r 9 p l y plywood.  In g e n e r a l , s i n c e x-j i s t r e a t e d as f a c e veneer by  c o n v e n t i o n , t h e number o f p o s s i b l e c o m b i n a t i o n s i n which K veneer t h i c k n e s s e s 2 can be used f o r c o r e and c e n t r e veneers would be K x K = K .  As t h r e e  p l y plywood won't have c e n t r e p l i e s i n i t , t h e c o r r e s p o n d i n g number o f c o m b i n a t i o n s f o r them would o n l y be K. 3 p l y plywood and n. = K  Thus, i n g e n e r a l , n.. = K f o r  f o r any h i g h e r p l y (odd) plywood.  I t should  be noted a g a i n here t h a t most plywood m i l l s use 3, 4 o r a t most 5 veneer  t h i c k n e s s e s and we need not t h e r e f o r e be concerned about l a r g e number o f c o n s t r u c t i o n a l t e r n a t i v e s a s s o c i a t e d w i t h h i g h e r v a l u e s o f K.  c)  C o n s t r a i n t _ L i n k i n g _ L o g Consumption_to  Veneer  Production  To r e l a t e t h e q u a n t i t y o f veneer produced t o t h e consumption o f l o g s , t h e f o l l o w i n g assumption i s made  (we d i s c u s s t h e i m p l i c a t i o n s  o f t h i s assumption i n s e c t i o n 2 . 5 ) . The volume o f veneer  obtainable  from a l o g remains t h e same i r r e s p e c t i v e o f t h e t h i c k n e s s o f veneer which i t i s c o n v e r t e d .  into  This i s e q u i v a l e n t t o s a y i n g t h a t t h e number  o f veneer s h e e t s o f a f i x e d s i z e o b t a i n a b l e from a l o g i s i n v e r s e l y p r o p o r t i o n a l , t o t h e veneer t h i c k n e s s i n t o which i t i s p e e l e d .  With  t h i s a s s u m p t i o n , t h e s e v a r i a b l e s s a t i s f y t h e e q u a t i o n y x ^ v ^ = L^, o r  yx v k  k  - L  k  = 0  fora l l k  (2.5)  Where y i s a c o r r e c t i o n f a c t o r f o r y i e l d o f veneer from l o g s and f o r u n i t s and dimensions o f t h e t h r e e v a r i a b l e s x , v k  if x  k  i s in millimetres, v  k  k  and L  k >  For example,  i s t h e number o f veneers o f s i z e 2.6 m x  1.4 m, L- i s i n c u b i c metres and t h e y i e l d o f veneers i s 60% o f l o g 7  k  Volume, then y i s g i v e n by  y = (0.001 x 2.6 x 1.4)/0.6 = 0.006067  d)  Log A v a i l a b i l i t y _ C o n s t r a i n t  These a r e t y p i c a l q u a n t i t y o f l o g s peeled l e s s than o r equal  c o n s t r a i n t s o f resource  availability.  The  i n t o d i f f e r e n t veneer t h i c k n e s s e s should be  to the quantity o f logs a v a i l a b l e .  T h i s i s given by  31 K E  L  k=l  < W  (2.6)  K  where W is the quantity of logs a v a i l a b l e .  Using (2.5) as a d e f i n i t i o n  for L^, (2.6) can be replaced by  K y  Z :x: v  k=l  K  «W  •  (2.7)  K  A plywood construction alternative can be used for production of plywood only i f that alternative is feasible for plywood assembly and thickness tolerance.  This is equivalent to saying that i f 6.. = 0  for any particular ( i , j ) , then the corresponding P . . must be zero. With non-negativity constraints on P . . this can be expressed by ' J  P.. < M 6  or,  1 j f  P . . - M6.. < 0 for a l l i , j 1J "I J where M is a large positive number as in (2.2).  f)  (2.8)  Cgnstraints_Linking_Ven^  The quantity of veneers of each thickness used by various construction alternatives for different types of plywood should be within the total quantity of veneers produced of that thickness.  This is expressed by  the constraint E z a. . . P . . < v. , or i j 1  J  K  1  J  N  n.  z  z . a. .. P. . - v. < 0  i=l  1  j=l  1  J  K  1  J  K  K  for a l l k  (2.9)  32 The slack in this constraint represents the excess quantity of veneer produced, but not used, in any of the construction a l t e r n a t i v e s .  P§[T§D^ZPr^§rril§_  g)  Cons t r a i nts  The quantity of each type of plywood produced should meet the demand or o r d e r f i l e requirements.  These are specified by typical  constraints  of the form  n. E P.. > d. j=l  for a l l i  1  1  J  (2.10)  1  where d. is the demand for product i F i n a l l y , a l l the decision variables used in the formulation are required to be non-negative,  V  L  k- i j p  >  i.e.,  —  0  for a l l k = l,2...Kj, j = l , 2 . . . n . and i  The Objective  =1,2.  Function  The objective i s to maximize net revenue for the m i l l .  If we  assume that revenues and costs are l i n e a r functions, the objective function can be expressed as  Max  where  N  n.  E  E  i=l  j=l  1  K r.P.. 1  1  J  -  C  E  k=l  L  . K  < -"> 2  33 r . = Revenue per plywood sheet o f type i C  = Cost o f l o g per  A g a i n , u t i l i s i n g (2.5)  - Cy.  1  1  The  as a d e f i n i t i o n f o r L^, t h i s can  N n. E E r.P.. i=l j=l  Max  unit  1 J  OverallFormulation  K E k=l  x.v K  o f the PDM  problem can  N  n.  (2.12)  Problem  E  E  1=1  j=l  Max  be w r i t t e n  constraints  as:  K r.P..  1  1  - Cy E • k=l  1 J  x.v. K  K  S u b j e c t to  x x  G  k  b,  T  -  L  =  { T  k  1 *'k  M(l-  = (0,  6 j j  2 T  ) <  1}  n. E  1  6. .  > 1  K y  £ x. v. < k=l P. . - Ms. . < U iJ K  w  K  0  "i( ) k k  T  ' 'k 1^ *  as  K  Using the n o t a t i o n , d e c i s i o n v a r i a b l e s and above, the PDM  be w r i t t e n  }  '  • « ! - « , . )  introduced  34 ..N  n.  n  i  :-E  J-l P  i j '  P. . > d. 1 J  V  1  k  >  0  f o r a l l i = 1.2...N; j = l , 2 , . . . n  1  and k = 1.2...K.  The above is:'a' foritiuliation o f t h e PDM problem as a mathematical  programming  problem i n which t h e o b j e c t i v e f u n c t i o n and some o f t h e c o n s t r a i n t s are  n o n - l i n e a r and, i n a d d i t i o n , some o f t h e d e c i s i o n v a r i a b l e s a r e  r e s t r i c t e d t o 0-1 v a l u e s .  The s i z e o f t h e problem depends on f a c t o r s  such as t h e number o f veneer t h i c k n e s s e s , t h e number o f plywood t y p e s and t h e number o f p l i e s i n each o f them.  2.5  D i s c u s s i o n and E x t e n s i o n s  In t h e f o r m u l a t i o n o f t h e PDM model i n a r e a s r e l a t e d t o t h e product mix p a r t , we have d e v i a t e d from t h e t r a d i t i o n a l ( 1 9 7 6 ) , Ramsing o f veneer  ( 1 9 6 5 ) , Lee ( 1 9 6 8 ) ) .  L.P. models (Kotak  In p a r t i c u l a r , t h e q u a n t i t y  and plywood a r e d e f i n e d i n terms o f t h e number o f s h e e t s  r a t h e r than volume,and  y i e l d f a c t o r s a r e used as d i r e c t p e r c e n t a g e s  r a t h e r than i n v e r s e m u l t i p l i e r s o r r e c o v e r y r a t i o s .  This o f f e r s  s e v e r a l advantages i n t h e d e s i g n problem s i n c e t h e c o n s t r u c t i o n a l t e r n a t i v e s can be s p e c i f i e d i n terms o f number o f v e n e e r s , t h e f e a s i b l e d e s i g n s can be i d e n t i f i e d through 8.. and veneer consumption can be d i r e c t l y  35 l i n k e d t o p r o d u c t i o n o f plywood t h r o u g h s.. and a... . The model assumes t h a t the volume o f veneer from a l o g remains the same i r r e s p e c t i v e o f t h e t h i c k n e s s o f veneer i n t o which i t i s p e e l e d . T h i s assumption i s made i n t h e absence o f r e l e v a n t i n f o r m a t i o n and i s v a l i d t h e o r e t i c a l l y , s i n c e , the volume o f wood p e e l e d from a l o g remains the same.  However, i n p r a c t i c e , t h i c k e r veneers may r e s u l t i n l o w e r  v o l u m e t r i c y i e l d due t o f a c t o r s such as s p l i t s o r l a t h e checks i n veneer. In such c a s e s , i f i n f o r m a t i o n i s a v a i l a b l e on t h e r e l a t i o n between veneer t h i c k n e s s and y i e l d , i t can be i n c l u d e d i n t h e model i n t h e form o f y^, a f a c t o r dependent on k, r e p l a c i n g y i n c o n s t r a i n t ( 2 . 5 ) . r e s t o f the model f o r m u l a t i o n o r s o l u t i o n procedure w i l l a f f e c t e d by t h i s change. b^  The  not be  The plywood t h i c k n e s s t o l e r a n c e b.'" and  i n c o n s t r a i n t (2.2) r e f e r t o d e s i g n t o l e r a n c e s and not the t h i c k n e s s  t o l e r a n c e o f an i n d i v i d u a l individual  plywood sheet a t a random p o i n t .  panel t o l e r a n c e s are a v a i l a b l e t h e y can be a d j u s t e d i n d e s i g n  f o r chance v a r i a t i o n s t h r o u g h s t a t i s t i c a l limits.  I f only  c o n c e p t s such as 3 - sigma  The o r d e r f i l e r e q u i r e m e n t s s p e c i f i e d i n c o n s t r a i n t (2.10) can  be o f t h e ' l e s s than o r equal t o ' form o r , may  combine both t y p e o f  inequalities. In t h e f o r m u l a t i o n o f t h e PDM model we c o n s i d e r e d o n l y t h o s e plywood t y p e s w i t h odd number o f p l i e s h a v i n g balanced d e s i g n s as t h e y c o n s t i t u t e t h e m a j o r i t y o f t h e p a n e l s c u r r e n t l y manufactured.  The model  can be e a s i l y extended t o even p l y and/or unbalanced d e s i g n s o f plywood. For e v e n - p l y plywood w i t h balanced d e s i g n s , d e s i g n c o e f f i c i e n t s t o t h e a... o f t a b l e s 3 - 6  similar  can be c o n s t r u c t e d f o r any number o f p l i e s .  For example, w i t h K = 3, f o r a f o u r - p l y plywood, t h e r e would be t h r e e d e s i g n a l t e r n a t i v e s r e p r e s e n t e d by t h e v e c t o r s ( 4 , 0, 0 ) , ( 2 , 2, 0) ( 2 , 0, 2) f o r ( a  n  .  a  - H ^ -  ^  and  ply °°d can be made w i t h unbalanced w  36 d e s i g n s , t h e number o f d e s i g n a l t e r n a t i v e s n. w i l l  increase considerably,  e s p e c i a l l y f o r those plywood types having l a r g e number o f p l i e s .  However,  i f t h e r e a r e r e s t r i c t i o n s on t h e f a c e veneers t o be o f t h e same t h i c k n e s s , the i n c r e a s e i n t h e number o f d e s i g n a l t e r n a t i v e s w i l l and they can be handled  n o t be s u b s t a n t i a l  w i t h i n t h e framework o f o u r PDM model.  We have t r e a t e d  as t h e f a c e veneer by c o n v e n t i o n  more r e s t r i c t i o n s on t h e t h i c k n e s s and s u r f a c e q u a l i t y  since there are  characteristics  o f f a c e veneer than any o t h e r veneer i n a plywood s h e e t . in the manufacturing  Additionally,  p r o c e s s , f a c e veneers must be peeled t o t h e f u l l  l e n g t h o f plywood sheets w h i l e core veneers which go a c r o s s t h e f a c e can be peeled  i n l e n g t h s r e l a t i v e t o t h e w i d t h o f plywood.  Having  more than one f a c e veneer t h i c k n e s s would t h e r e f o r e r e s u l t i n more s c h e d u l i n g , h a n d l i n g , s o r t i n g and s u r f a c e p r e p a r a t i o n c o s t s .  However,  having a second f a c e veneer t h i c k n e s s might r e s u l t i n b e t t e r d e s i g n s for  some plywood t y p e s .  Such a s i t u a t i o n , i . e . having more than one  f a c e veneer t h i c k n e s s , can a l s o be i n c l u d e d w i t h i n t h e framework o f our PDM model.  We i l l u s t r a t e t h i s b r i e f l y through our plywood m i l l  example i n c h a p t e r s i x . E x i s t i n g a l g o r i t h m s f o r l a r g e s c a l e problems can s o l v e e i t h e r i n t e g e r l i n e a r programs o r n o n - l i n e a r programs i n c o n t i n u o u s v a r i a b l e s . The  unique f e a t u r e s o f t h e PDM model i s t h a t i t c o n t a i n s both d i s c r e t e  and c o n t i n u o u s  v a r i a b l e s as w e l l as n o n - l i n e a r i t i e s i n t h e c o n s t r a i n t s  and o b j e c t i v e f u n c t i o n . problem t o c o n t i n u o u s  R e l a x i n g t h e i n t e g e r v a r i a b l e s o f t h e PDM  values w i l l  produce a n o n - l i n e a r non-convex  o p t i m i z a t i o n problem i n which a l o c a l optimum i s n o t n e c e s s a r i l y a g l o b a l one.  Thus, t o our knowledge, no a l g o r i t h m o r s o l u t i o n  t h a t can produce a g l o b a l  procedure  s o l u t i o n t o t h e PDM model i s a v a i l a b l e .  37 One o f t h e i m p o r t a n t c o n t r i b u t i o n s o f t h i s t h e s i s i s t h e development o f an e f f i c i e n t a l g o r i t h m f o r s o l v i n g t h e PDM problem.  By  u t i l i s i n g t h e c o n s t r a i n t s t r u c t u r e i n t h e model and by s e l e c t i n g a p p r o p r i a t e v a r i a b l e s t o branch o n , we d e v e l o p an e f f i c i e n t  implicit  enumeration a l g o r i t h m t o d e r i v e a g l o b a l s o l u t i o n t o t h e PDM problem. T h i s a l g o r i t h m i s d e s c r i b e d i n t h e next c h a p t e r . When f a c t o r s such as s p e c i e s , l o g g r a d e s , veneer types and plywood grades a r e i n c l u d e d , t h e s i z e o f t h e problem i n c r e a s e s manyfold. each s i t u a t i o n , however, t h e essence o f t h e problem f o r m u l a t i o n  In remains  t h e same s i n c e t h e d e s i g n a s p e c t o f t h e problem ( c o n s t r a i n t s (2.1) through (2.4)) i s unaffected. changes.  Only t h e p r o d u c t mix p a r t o f t h e problem  A s e t o f o p t i m a l veneer t h i c k n e s s e s and o p t i m a l plywood  d e s i g n s f o r a PDM problem w i t h one s p e c i e s need not remain t h e o p t i m a l s o l u t i o n when more than one s p e c i e s a r e i n c l u d e d i n t h e PDM model. c o n s i d e r some o f t h e s e e x t e n s i o n s t o t h e PDM problem, a n a l y s e t h e c o r r e s p o n d i n g s t r u c t u r e s and s o l u t i o n procedures i n c h a p t e r f o u r .  We  38 CHAPTER 3  3.1  S o l u t i o n t o t h e PDM Model  From c h a p t e r 2 we r e c a l l  t h a t t h e plywood d e s i g n and m a n u f a c t u r i n g  (PDM) problem can be f o r m u l a t e d as t h e f o l l o w i n g n o n - l i n e a r mixed 0-1 programming  Z  problem. N n. K z E r . P . . - C y z x.v. i-1 j = l k=l  = Max  Q  1  u  1  1 J  K  K  s.t. b  L 1  -M(l-6  1 j  )<  L a ^ x ^ b ^ + M C l - ^ j )  n. E  6. . > 1  1  K y  x. v. < W  E  k-1  K  K  P. . - Ms:. < 0 N  n.  E  E  a. .. P. . - v. < 0  1  1-1 j - l  1 j k  1 J  k  n. E  1  j-l  P. . > d. 1 J  1  6. . G {0,1}  x e T = {T k 'k k x  u  1  T ' 'k ' 2  T  m ( k )  'k  }  i  39  f o r a l l k = 1,2,...K; j = 1,2,...n^ and i = 1,2,...N. T  k  i s t h e s e t o f veneer t h i c k n e s s e s f o r x , k = 1,2,...K. k  r e c a l l t h a t by c o n v e n t i o n  Further,  i s t h e f a c e veneer t h i c k n e s s and i n most  m i l l s K, t h e number o f veneer t h i c k n e s s e s i s g e n e r a l l y t h r e e o r f o u r . We d e v e l o p an i m p l i c i t enumeration a l g o r i t h m f o r s o l v i n g t h e PDM problem.  The e f f i c i e n c y o f o u r i m p l i c i t enumeration a l g o r i t h m stems  from t h e f a c t t h a t f o r a g i v e n s e t o f veneer t h i c k n e s s e s t h e PDM problem reduces t o a s p e c i a l s t r u c t u r e L i n e a r M u l t i p l e Choice Knapsack (LMCK) problem [ Zemel (1980), G l o v e r and Klingman ( 1 9 7 9 ) ] . each one o f t h e s e LMCK problems can be s o l v e d e x p l i c i t l y .  Moreover, Further,  we show t h a t due t o t h e s p e c i a l s t r u c t u r e o f t h e PDM problem, we can reduce s u b s t a n t i a l l y , from t h e o u t s e t , t h e number o f veneer t h i c k n e s s e s t h a t need t o be c o n s i d e r e d  i n t h e i m p l i c i t enumeration a l g o r i t h m .  Our i m p l i c i t enumeration a l g o r i t h m was coded i n FORTRAN and was used t o s o l v e some r e a l w o r l d problems. mill  Data o b t a i n e d  from a plywood  i n B r i t i s h Columbia was used t o e v a l u a t e t h e PDM model and t h e  algorithm.For  t h e p a r t i c u l a r c o n f i g u r a t i o n o f t h e problem parameters  used i n t h e study o u r model gave a s e t o f veneer t h i c k n e s s e s and plywood d e s i g n s which can i n c r e a s e t h e annual n e t revenue by more than 6.8 p e r c e n t . The plan o f t h i s c h a p t e r  i s as f o l l o w s .  In s e c t i o n 3.2 we d e s c r i b e  our i m p l i c i t enumeration a l g o r i t h m and a n a l y z e t h e LMCK problems.  In  s e c t i o n 3.3 we show how we can use t h e s t r u c t u r e o f t h e PDM problem t o a c c e l e r a t e t h e performance o f t h e i m p l i c i t enumeration a l g o r i t h m , w h i l e i n s e c t i o n 3.4 we p r e s e n t o u r computational  results.  40 Though much o f the c o n t e n t s o f t h i s c h a p t e r i s m a t h e m a t i c a l , the  i m p l i c a t i o n s o f t h e end r e s u l t a r e s t r a i g h t f o r w a r d .  What i s  shown here i s t h a t a s e e m i n g l y c o m p l i c a t e d n o n - l i n e a r mixed  0-1  mathematical programming problem can be s o l v e d e f f i c i e n t l y f o r a g l o b a l s o l u t i o n by e x p l o i t i n g t h e s t r u c t u r e s i n t h e model.  In f a c t ,  the  main j o b o f t h e computer code w r i t t e n f o r our a l g o r i t h m t o s o l v e  the  PDM problem i s more o f a house-keeping n a t u r e than one i n which  complicated o p t i m i z a t i o n routines are involved.  3.2  An I m p l i c i t Enumeration A l g o r i t h m t o S o l v e the PDM  Problem  Our i m p l i c i t enumeration a l g o r i t h m can be, i n a v e r y r u d i m e n t a r y manner, d e s c r i b e d as f o l l o w s : A l g o r i t h m A:  (Rudimentary a l g o r i t h m f o r s o l v i n g t h e PDM  * Step 1 :  Let x^. = x^ , x  * e t  T^, t = 1, 2,  K-1 denote a p r e v i o u s l y  u n s e l e c t e d s e t o f v a l u e s f o r the t h i c k n e s s e s x^, x^, PDM problem. (x-|  problem)  I f none e x i s t s , t e r m i n a t e .  ...»  x^_^ i n t h e  O t h e r w i s e , denote by  , ..., x ^ _ i ) t h e PDM problem i n which x  t  = x  t  PDM  , t = 1 , ...,  K-1,  and go t o s t e p 2. Step 2:  Attempt t o fathom PDM  go t o s t e p 1. Step 3:  (x^ ,  x^_-| ).  I f successful,  Otherwise,  S o l v e PDM  (x^  x^_^  ).  Store the optimal s o l u t i o n i f  b e t t e r than t h e incumbent, and go t o s t e p 1. Clearly, algorithm A w i l l  t e r m i n a t e a f t e r a f i n i t e number o f  i t e r a t i o n s w i t h an o p t i m a l s o l u t i o n t o t h e PDM  problem.  In t h e  d i s c u s s i o n which f o l l o w s we show how t o execute e f f i c i e n t l y s t e p 3 o f a l g o r i t h m A.  In s e c t i o n 3.3.we d e v e l o p t e s t s which a s s i s t i n  41 fathoming t h e PDM A.  (x  ,  1  X K  _-|  ) problem i n s t e p 2 o f .  algorithm  F u r t h e r , we show i n t h a t s e c t i o n how we can e l i m i n a t e , from t h e  * outset, values o f  = x  , t = 1, 2, .... K-1 which are not c o n s i s t e n t  t  w i t h an o p t i m a l s o l u t i o n o f our PDM 3.2.1  The L i n e a r M u l t i p l e Choice Knapsack Problem  We w i l l  c o n s i d e r now t h e PDM problem i n which a l l  *  T^ f o r k = 1, 2, ... K.  G  j = 1, 2, ..., n.. and i = 1, 2, I •  i f  (  <5 . =  i  b  i 1  )  t h i c k n e s s e s have  *  been d e t e r m i n e d , i . e . x^ = x^ , x£ all  problem.  L  <  For  N, l e t  * iik k k=l a  x  l j k  k  <  b 1  i  U  (3.1)  Otherwise  fi Note t h a t i f E" s*.= 0 f o r any i , then plywood t y p e i cannot be J ' * assembled w i t h the s e t o f t h i c k n e s s e s x^ , k = 1, 2, K, and 1  t h e r e f o r e t h e r e m a i n i n g PDM problem i s i n f e a s i b l e . n. hand, E 6*. 1 f o r al 1 i = 1 , 2 , ... N, 1 e t j=l  I f , on t h e o t h e r  1  1 J  I. = l j | s * . = 1}  (3.2)  I . r e p r e s e n t s the i n d e x s e t o f f e a s i b l e d e s i g n a l t e r n a t i v e s f o r plywood t y p e i w i t h x P.. - Ms.. < 0  k  = x^ , k = 1 , 2, ..., K.  Now,  the c o n s t r a i n t s  and P.. > 0 - V - ' i , j i n t h e PDM problem can e q u i v a l e n t l y  be r e p l a c e d by  P. . > 0  Ar  j e l . ; i = l , 2 , i : . J  (3.3)  Thus, when the thicknesses  are assigned the values x^  k = 1, 2, . . . , K, the PDM problem reduces to N Max  K *  E  E  j G I.  i=l  E  r . P . . - Cy I  1  T  1  T  k=l  J  J  K  k  s.t. y  s x. v. < k=l K  w  K  N E  a. .. P. . - v. < 0 ^ !J  E  1=1 j e I.  k  P. . > d.  E  p  k  1J-  k  V  >  0  for a l l k = 1, 2, . . .K ;  *  denote by PDM (x-j * * PDM (x-j  , x  2  j e I.  *  , x *  2  and i = 1, 2, . . . N.  Let  * , . . . x^ ),  problem (3.4) - (3.8).  , . . . x^ ) is a linear program.  Notice further  Clearly, that  at an optimal solution to this problem,(3.6) would be s a t i s f i e d as an equality. v, = k  N E i = 1  Substituting v^,  .  E e  a...P.. ijk  in (3.5) produces N E b. .P. . 1=1 J e I. ^ ^ E  u  -V-  given by k  —  43 where  b  i j  j/«ijk k x  =  —  Note t h a t b. . > 0 V i ,  j e I . s i n c e y > 0, x *  e x i s t s a t l e a s t one k f o r each j such t h a t a..  ,  (  3  :  n  )  > 0, a. . > 0, and t h e r e t  > 0.  Substituting  (3.9) i n t h e o b j e c t i v e f u n c t i o n (3.4) we get t h a t  N K E E r . P . . - Cy E i=l j e I k=l 1  1 J  k  i  * N x, v. = 2 E f..P.. 1=1 j e i . U U  (3".12)  k  where K r  ij  =  r  i "  C y  E k  a = 1  iJk k X  = r . - Cb..  Thus, t h e PDM  (3.13)  ( x * , x ^ » ..., x£ ) has been reduced t o an  optimization  problem o f t h e form: N  E  Z = Max  E  1=1: J  G  r..P. . I•  s .t. N E  .-" E  -b,,P,, < W (3.14)  E  P.j > 0  P. • > d.  -V- i = 1, 2  N and  j e  I.  44 ic -  ic  The PDM (x^ , x  2  ic  , ... x ^ ) problem given  as a l i n e a r m u l t i p l e c h o i c e  by (3.14) i s e a s i l y  knapsack (LMCK) problem.  s l i g h t l y from t h e t r a d i t i o n a l  recognized  (It differs  LMCK [Zemel ( 1 9 8 0 ) , G l o v e r and Klingman  (1 9 7 9 ) ] , i n t h a t , t h e l a t t e r problem has ( i ) e q u a l i t y s i g n i n t h e m u l t i p l e choice  c o n s t r a i n t s , ( i i ) d. = 1 f o r a l l i , and ( i i i )  as t h e o b j e c t i v e ) .  minimization  Using p r o p o s i t i o n 3.1 which f o l l o w s , we show t h a t  due t o t h e s p e c i a l s t r u c t u r e o f t h e c o e f f i c i e n t s i n t h e PDM (x-| , * * x  , . . . , - x ^ ) problem, (3.14) can be s o l v e d e x p l i c i t l y , w i t h o u t  2  h a v i n g t o go t h r o u g h any o f t h e LMCK a l g o r i t h m s 3.2.2  An E x p l i c i t S o l u t i o n f o r t h e PDM ( x  1  ,  [ s u c h as Zemel  (1980)].  x^ ) Problem  Let b, , i  =  1 J  Min j G I  {b  }  ^ i  (3.15)  1 J i  and o b s e r v e from (3.13) i t f o l l o w s  that  r . , = r . - Cb. . i i l  J  1  =  l  J  Max {r..} j e I.  ¥ i  (3.16)  1 J  P r o p o s i t i o n 3.1:  P - = 0 ij  Proof:  There e x i s t s an optimum s o l u t i o n t o (3.14) such t h a t  V j f J . and ¥ i = 1 ,2,...N  Suppose (3.17) does not h o l d .  That i s , t h e r e e x i s t s ,a s o l u t i o n  (P'...) which i s optimum f o r (3.14) w i t h 1=1' IJ  i = p, t G r  (3.17)  but P' > 0 ,p"t  \ {Jp}- C o n s i d e r t h e new s o l u t i o n given by  f o r some  45 p'  +  pt  P'.  pj.  o •  P  IJ  for  f i r s t that  (3.14)  = p.  J=J  i  = p,  J'=t  (3.18)  Otherwise  PI.  Observe  i  (P..) l J  i s feasible for  (3.14).  Indeed,-.(P'...) lJ  feasible  implies:  *  CD  >  *  z  (ii) j  * U  0  I.  e  =  P .1 .J  z J e  ;  I  I.  j e i .  P'.,, > d. 1 ^  N  N  z  z  and ( i i i )  -V- i  i=l  U  Z  = z  b. . P i . U  i  fj  =  N Z i=l  <  b . .P'. . + b , P ! .  j  (=.  U  u  pJp  Pt  b . . P1. . s i n c e , 1 J iJ  z j e i  - b  .P' pt pt  b . < b . ¥ j J PJp PJ from  Next,  suppose b . pt  one J  e  (3.16)  Ip m i n i m i z i n g b ^ that  r  On t h e o t h e r then  it  i s such t h a t  b . = b . ; i . e . t h e r e e x i s t s more pt pJp  for i = p in (3.15).  , = r . and t h e r e f o r e PJp Pt hand,  follows  if  P  tPpt  +  r  from ( 3 . 1 6 )  pJ  p  P  pJ  j  z e  Jp i s u n i q u e f o r i that  r  ,  p  =  r  pJ  p  [ P  pJ  p  +  r. .P.. ij  it =  ij  pt]  r . P . ' + r . P' + PJp P J p Pt Pt  follows z  i  i . e . bpj  and o b s e r v e p  P  j_  = p;  > r p  p  r  z i  Then  z j  G  T  .  < b^., that  6  I  P  (3.15)  than  from r..P'. . = V  ij  j e  u  Ip,  ,  46  * Therefore, z z r . . P . . >s z i j G I. i j G I. 1 J  1 J  (P . .) and t h e p r o o f o f p r o p o s i t i o n 1J 1  Following  Max  proposition N Y, r. 1=1  r..P'.., c o n t r a d i c t i n g t h e o p t i m a l i t y o f J  3.1  J  follows.  3.1, t h e LMCK reduces t o s o l v i n g  P.  K  K  N  Z b. P.  s.t.  i=l * n  n  P. > d.  Where b. l.  (3.19)  -V- i = 1 , 2, ... N.  = b. , and r . i J.  <W  '  = r . , = r . - Cb.  l .  i J.j  i .  I  This i s a Linear  Knapsack problem i n bounded v a r i a b l e s . N (3.19) i s f e a s i b l e i f and o n l y i f z b. d. < W. Let 1=1 K  Observe  that  1  Max  If r  P"  (3.20)  > 0, t h a t i s , i f t h e r e e x i s t s a t l e a s t one plywood t y p e w i t h  p o s i t i v e n e t p r o f i t , t h e s o l u t i o n o f (3.19) i s g i v e n by d. W -  i  Z b.d.  i  p (3.21)  47 If r  P•  ^ Q , t h a t i s , i f t h e net p r o f i t o f a l l t h e plywood  types  i s n o n - p o s i t i v e , t h e s o l u t i o n o f (3.19) i s g i v e n by  * P  i .  =  d  ¥  i  Transforming  i = 1 , 2 , ... N  (3.22)  these r e s u l t s t o our LMCK problem, we have t h a t t h e  *  *  s o l u t i o n t o t h e PDM (x-|  ) i s g i v e n by  i P,  i 1  <W  =  P  '  j  J = J =  J  p  i f  i  r  pJ ^° p  E b. . d.  -  1*>  l J  i  1  (3.23) 1  =  P-  j  =  J  p  i f  r P  J  >  0  P 0  Otherwi se  From (2.9) i t f o l l o w s t h a t t h e q u a n t i t y o f veneers t o be produced i s g i v e n by N  = i =zl  a i  J  •V- k l  k  1  J  (3.24)  i 7^  Thus, when t h e x^'s a r e a s s i g n e d v a l u e s x^. , k = 1 , 2, ... K, we have shown t h a t t h e PDM problem can be e x p l i c i t l y s o l v e d . to s....  The s o l u t i o n  P.. and v. would be as g i v e n by ( 3 . 1 ) , (3.23) and (3.24)  respectively.  In o r d e r t o produce (P..-..) as g i v e n by ( 3 . 2 3 ) , we need  t o f i n d t h e i n d i c i e s J . , i = 1 , 2 , ... N and t h e index p.  The c o m p u t a t i o n a l  d i f f i c u l t y i n g e n e r a t i n g (P*.) i s t h e r e f o r e 0 C [ I -j I + |I I +•••+ |I I + N). 2  N  48 3.3  Branching Tests  In t h e f i r s t  a n d Bounds  part  o f t h i s s e c t i o n we d e v e l o p bounds on t h e v a l u e  o f x k when v a l u e s o f x bounds,  for t = 1,  2,  . . .  how a n u p p e r  bound on t h e o b j e c t i v e  second p a r t ,  from t h e o u t s e t ,  of  emanating  * problem.  v)  we u s e t h e s t r u c t u r e o f t h e PDM p r o b l e m t o  branching tests which,  these  s e c t i o n , we  l i n e a r s u b p r o b l e m c a n be e s t a b l i s h e d f o r a l l b r a n c h e s  f r o m t h e node a s s o c i a t e d w i t h t h e PDM (  In  develop  can s u b s t a n t i a l l y reduce t h e  *  *  number o f s e t s o f t h i c k n e s s e s x-j , in  Using  function value  *:  the  K-1.  and making use o f t h e r e s u l t s from t h e p r e v i o u s  demonstrate the  are fixed  x ^  that  n e e d t o be c o n s i d e r e d  s t e p 1 o f o u r a l g o r i t h m A.  3.3.1  B o u n d s on x ^ a n d an U p p e r b o u n d o f t h e PDM (x-)  ,  . . . x ^ -|)  * T. ...Assume t h a t  x  t  =  x  on t h e O b j e c t i v e  Function  Value  Problem.  *  ^  > t x  T^, t = 1 ,  e  2,  . . .  K-1 a n d l e t  K-1  rU. • ^=- | Ei j at • t  (3.25)  •V- i , j  -4.x.  t  Further, thickness  for  For a l l i , j  l e t T^min X | (  max a n d T K " ' a A d e n o t e t h e minimum a n d maximum  , i . e . , TKmin  -  M  f o r which a i - j K > 0 ,  b  L  /  b < K T  m a X K  >  {  TKm}]and  TKmax  =  M  *  {T^ 1 1 1 }.  x  let  - T . .  - T — ~ a ijK  n  possible  J  U  - T - .  ^ a ijK  L  L  >V" K  n  and a „ „ > 0 l  j  K  (3.26)  49 D.j r e p r e s e n t s t h e s e t ;of a l l d e s i g n a l t e r n a t i v e s w i t h a . . j » 0 which K  * might be f e a s i b l e f o r plywood t y p e i w i t h x^ = x^ , t = 1, 2, ... K-1 and X|<  T|<.  e  max  Define Max , I bY  Min  hi,  i 1 a. U  J  max  (3.27)  1  and ..Min ,  mm  X.  K  min  = Max  (3.28)  ijK I f f o r some i and j , ai . .  v  = 0 b u t b. < T.. < b., t h e n plywood  t y p e i can be assembled w i t h o u t x T™ .  min  K  i n which case s e t X..  K  max v" '  _min  =  lcl  and i « x  Let  x  K  D. l  <T.. < b ?  'ijK  = 0}  (3.29)  and D! U D.  (3.30)  Now, d e f i n e  "; C>  min  X  Xt  -  K  —0.31)  and max  X,  =  M  j {X-  Observe t h a t i f % set o f x x  K-i)  ^  s  t  n  m u  n  >X  X  }  m a x I >  —(3.32)  , t h e PDM problem i s i n f e a s i b l e f o r t h i s  , t = 1, 2, ... K - 1 , and t h e node a s s o c i a t e d w i t h t h e PDM (x^ , fathomed.  Further, f o r values o f x  K  such t h a t x  K  >  X^  ax  50 or  x  <  x  m  K  i  n  K  '  ( * » ••• K - 1 ^  P D M  x  X  i  1 S  n f e  asible.  An upperbound on t h e v a l u e o f t h e o b j e c t i v e f u n c t i o n o f t h e PDM problem when  =  , t = 1, 2, ... K-1 can now be computed by a r e l a x a t i o n * * o f t h e PDM (. x-| , . . . x ^ _ i ) problem. R e c a l l from p r o p o s i t i o n 3.1 t h a t i f the LMCK i s f e a s i b l e , then i n an o p t i m a l i,  P  s o l u t i o n t o ( 3 . 1 4 ) , f o r each K o n l y f o r t h a t index j f o r which b. . = z y a...x. i s minimum.  >0 IJ  lj  Thus, s i n c e y i s a c o n s t a n t  IJK  K  and when x ~ i s known, P.. would be p o s i t i v e fc  i j  o n l y f o r t h a t j f o r w h i c h (T. . + a . . x ) i s a minimum, j e D.. K  K  Using  t h e s e r e s u l t s , a r e l a x a t i o n o f t h e c o n s t r a i n t s r e l a t e d t o plywood t h i c k n e s s can be s p e c i f i e d as f o l l o w s . Determine, f o r each i ,  i f t h e r e e x i s t s x^ such t h a t  K  —  (3.33)  b[ < T.j + a ^ x j . < b^ f o r a t l e a s t one j e D.  ?—  (3.34)  4 (C 'V VT 6  n  A  and  The r e l a x a t i o n i s t h a t t h e v a l u e ( s ) o f x may d i f f e r f o r each i . I f such an x  *  n K  K  s a t i s f y i n g (3.33) and (3.34)  does not e x i s t f o r some i ,  PDM (x.j ,...x _.|) problem i s i n f e a s i b l e and t h e node a s s o c i a t e d K  x  t  =  x  * t » t  the  * =  ^' > 2  ••• -K-1 i s fathomed.  Assuming t h a t an x.  with  i satisfying K  (3.33) and (3.34) e x i s t s f o r each i d e f i n e  b^ = j ^ n . ^ T . j + 3 . ^ x ^ ( 3 . 3 3 ) and (.3.34) a r e f e a s i b l e }  b. r e p r e s e n t s  t h e minimum p o s s i b l e t h i c k n e s s f o r plywood type i ,  (3.  given  51 t h a t x. = x. , t = 1, 2, ... K-1 t t  and x„ i s an element o f (X ™ , l\ K  X  l n  An upperbound on t h e o b j e c t i v e f u n c t i o n v a l u e o f t h e PDM  (x-j ,  m a x I /  ) n  K X K  _-|>)  problem i s g i v e n then by the o p t i m a l v a l u e , Z, o f the o p t i m i z a t i o n problem g i v e n below N  1 = Max  f  Z r'. p. 1-1 1  s .t.  N  ,.  z b.P. < 1=1  1  1  P. > l  1  W/y ^  —  d. i  where r . = r . -  Cyb.  ( 3 . 3 6 ) i s a L i n e a r Knapsack Problem (LKP) N  ,  s b.d. >W/y, i=l 1  1  i  (3.36) i s infeasible.  i n bounded v a r i a b l e s .  Otherwise,  i  Max{r./b.}, the s o l u t i o n o f ( 3 . 3 6 ) i s g i v e n 1 1  i f (r  P  If  /b ) = P  by  1  i f  P i  P  i = p if r < 0 P  P  i  { w/y  v  z iri>  b.d. ' 1  i = p i f r^  > 0  and  _  Z =  N , * z r.P. 1=1  I f Z, as g i v e n by ( 3 . 3 8 ) , i s l e s s than the c o r r e s p o n d i n g  v a l u e o f the  incumbent s o l u t i o n i n s t e p 3 o f a l g o r i t h m A, the node a s s o c i a t e d w i t h  52 x  t  =  * t > t  x  incumbent  =  1'  ... K-1  s o l u t i o n then  should  be made.  large,  this  computed  2,  c a n be f a t h o m e d .  b r a n c h i n g on t h e v a l u e  I f t h e number o f e l e m e n t s  interval  can f u r t h e r  branching t e s t s elements, not  that  further  x e  of  in  v  n  (xJ' , nln  (XKmin,  Xl/max) n j  XKmax)  is  be s u b d i v i d e d and bounds on Z c a n be  i n t h e same way a s a b o v e  s i n c e t h e number o f e l e m e n t s  _ Z i s higher than the  If  f o r each such s u b d i v i s i o n .  in  i s g e n e r a l l y small  follow eliminate  However,  and s i n c e  a s u b s t a n t i a l number o f  divisions of the interval  n (X^min,  other these  X ^ m a x ) may  be n e c e s s a r y .  3.3.2  Branching Tests  from t h e S t r u c t u r e  We u s e t h e s p e c i a l reduce,  at the outset,  by a l g o r i t h m  Reduction  1:  s t r u c t u r e o f t h e PDM p r o b l e m t o s u b s t a n t i a l l y the size of the tree that  h a s t o be s e a r c h e d  A.  (Distinct  Since the a . . . ' s  K  o f t h e PDM Model  1  J  Veneer  Thicknesses)  a r e c o n s t r u c t e d i n s u c h a way t h a t  a l l feasible  K  values of  E a . . . x . with x = x . , f o r some s f t , c a n a l s o be o b t a i n e d _1 1 J K S C f x t , we c a n assume t h a t i n t h e PDM p r o b l e m K  k  with x  xs  For  example,  f x  c o n s i d e r x^ = 2 . 5 , x 2  the design a l t e r n a t i v e for  a 5-ply  1 < s, t < K  t  plywood.  nine, That  (a^-j ,  i s ,  (3.  = 2 . 5 , Xg = 4 . 0 a n d K = 3 . a  i92'  a  i93^  =  ^ '  ^»  Suppose is feasible  E a ^ x ^ = 2x-j + x 2 + 2 x 3 •= 1 5 . 5 mm i s k—1  53 within  the  permissible lower  same o r d e r i n g  of  the  plies  and u p p e r  for  face,  tolerances.  Then,  c o r e and c e n t r e  veneers  produced (3,  0,  by u s i n g t h e d e s i g n a l t e r n a t i v e f i v e i n w h i c h 3 2). Indeed, z a i 5 ( < x k = 3x.j + 2 x 3 = 1 5 . 5 mm.  Reduction  2:  Due t o a  ijk  ^  o r  '  t  (Symmetry i n (Core/Centre the  i e  D a  symmetry i n t h e ^  using d i f f e r e n t x  =T  a n c i n  but  , x  , 2 < s ,  , s  S£ set of  t h i c k n e s s e s x, , I  by u s i n g t h e For veneer  I  set of  example,  x^  ...  second s e t of designated  thicknesses  set  II.  designs j  plywoods  the  same  Then t h e  x  ,  ...x  ,  = 2.4,  S  x, , x2  i52  can  coefficients be  produced  plywood  suppose  produced  by  c a n be p r o d u c e d  as  I  three  thicknesses.  or  ...x  and x^ in Table  by x-j  = 2.4,  for five  well  K.  ,  = 2.7  as s e t  be g i v e n  having  a  Explicitly,  ...x.  2  ...x  T a b l e 7 shows t h a t  alternative of  < K .  design  designs  thicknesses.  S-j  thicknesses designated  i5T  be  ^  thicknesses  let  9  of the  identical  set of  (a  can  the  Veneers)  construction  requirement,  symmetric  = T  S-j  the  9  precisely  all  ,  ...X,,.  = 3.1  be one s e t  7.  Let  x2  = 3.1,  the x^  =  possible values  p l i e s these  of  two  sets  2.7, of produce  54 Number o f PI i e s  Design Alternative,  3  5  Table 7:  Symmetry  Set  Plywood T h i c k n e s s I Set  1  7.2  7.2  2  7.5  7.9  3  7.9  1  12.0  12.0  2  12.3  >12.7  3  12.7  —>12.3  4  12.6  5  13.4  6  12.9  ^14.1  7  14.1  ~~^>12.9  8  13.; 3  ^ _ ^ » 1 3 . 7  9  13.7  >13.3  j  ^ " - ^  II  7.5  ^ ^ ^ 1 3 . 4 > 12.6  i n Core/Centre Veneers.  The a b o v e o b s e r v a t i o n i s v a l i d K > 3 and f o r a l l x  , x 1  such t h a t  f o r a n y number o f p l i e s , 2 < s , , s , < K.  T h e r e f o r e , we  2  can assume t h a t  our s e t o f thicknesses  problem i s such  that  xc > xc s^ s -j  f o r any  x ^ , x^, . . .  for a l l 2 < s, < s x  0  x^, i n t h e PDM  < K  (3.40)  c  R e d u c t i o n 3 : ( U p p e r Bound on Face V e n e e r T h i c k n e s s ) Let  T,  be t h e minimum p o s s i b l e  veneer t h i c k n e s s ,  i . e . T,  < T.m  -V-m, k .  55 In a n y p l y w o o d t y p e ,  a t l e a s t two v e n e e r ' s t h i c k n e s s s h o u l d be x 1 ,  the face veneer t h i c k n e s s . type  i , t h e minimum t o t a l  c a n have of type  i s (L.  I f L.  i s t h e number o f p l i e s  thickness the remaining (L.  - 2)T^.  i n plywood  - 2) veneers  I f bY i s t h e u p p e r b o u n d on p l y w o o d t h i c k n e s s  i , t h e maximum t h i c k n e s s t h e two f a c e v e n e e r s c a n h a v e i s  U [ b.. X.j  (Li-2)T^].  for that  Thus,  an u p p e r bound X-j on t h e f a c e v e n e e r t h i c k n e s s i s [ b^ - ( - 2 ) T - j ] / 2 . When a l l p l y w o o d  plywood type  a r e c o n s i d e r e d t h i s u p p e r bound i s g i v e n by ^| [  (bV  n  Suppose T 1 m a x veneer. x]  Max  {T.|m}  < X1 = Mi n  mm i  b. i  r  U  f o r example,  - (L.-2)T, , i ' l I J 2 f '  with ^  specified  plywood t h i c k n e s s f o r a l l i .  l  (3.41)  l  t h e 12 p l y w o o d t y p e s l i s t e d  by a m i l l  x  that  max T  These a r e t a k e n from t h e a c t u a l l i s t  follows  - (L.-2)T^)/2] .  i s t h e maximum p o s s i b l e t h i c k n e s s f o r f a c e  Then i n o u r PDM p r o b l e m we must have  Consider, below.  =  types  = 2 . 4 mm, T 1 m a x  i n Table 8  o f plywood t y p e s  produced  = 3 . 2 mm a n d bY i s + 0 . 5 mm o f t h e From Column 3 o f t h e t a b l e  it  that  < X  1  Min  Notice that specification  m m j  b  i  "  (Li-2>T1  3.2  = 2.80  a 3 p l y 7 . 5 mm p l y w o o d c a n n o t be p r o d u c e d w i t h i n  i f X^ e x c e e d s 2 . 8 mm.  o n l y make t h e p r o b l e m i n f e a s i b l e .  V a l u e s o f x.| h i g h e r t h a n X 1  would  56  Plywood i  Table  Type  -  Ply L. 1  b  Thickness mm  1  ? - <V » i 2  T  b  2  i  -  2X  L.-2 i  1  3  -  7.5  2.80  1.40  2  3  -  9.5  3.80  3.40  3  5  -  12.5  2.90  2.13  4  5  -  15.5  4.40  3.13  5  7  -  18.5  3.50  2.48  6  7  -  20.5  4.50  2.88  7  7  -  22.5  5.50  3.28  8  9  -  23.5  3.60  2.49  9  9  -  25.5  4.60  2.77  10  9  -  27.5  5.60  3.20  11  9 ..-  28.5  6.10  3.34  12  9  30.5  7.10  3.49  8:  Bounds  -  1  on x-| a n d x ^ .  t h Reduction  4:  ( L o w e r Bound on K  From r e d u c t i o n 3 , face veneer all of  t  = 1,  as given  t h i c k n e s s and f r o m 2,  ... K - 2 .  a plywood w i t h  given  L.  by ( 3 . 4 1 )  (3.40)  i s t h e maximum  i t follows  S i n c e 2X^ i s t h e maximum  t h e two f a c e v e n e e r s ,  for  Thickness)  a lower  plies  that  possible  x^ > x^_t f o r  possible thickness  bound Ij^, on t h e v e n e e r  thickness  XK,  and l o w e r t o l e r a n c e f o r t h i c k n e s s b ^ , i s  L by ( b . - 2X^ ) / ( L i - - 2 ) .  S i n c e t h i s i s t r u e f o r a l l plywood t y p e s Max L " b o u n d o n x ^ i s o b t a i n e d by . [ (b.. - 2 X ^ • ) ' / ( L . - 2 ) ] . If ( T m } i s t h e minimum p o s s i b l e t h i c k n e s s o f x ^ , we have  minl o w_.errimnj  the  ~l m  v  Max  i  I L.  - 2  f ' 'K  (3.42)  57 The r e d u c t i o n example  in the l a s t  i m p l i e d by ( 3 . 4 2 ) column o f Table  i s i l l u s t r a t e d using the previous 8 with  TKm1n  = 2.40, ^  and b^ i s - 0 . 5 mm o f t h e s p e c i f i e d t h i c k n e s s f o r a l l i .  2.40  Observe t h a t fication  a 9 p l y 3 0 . 5 mm p l y w o o d  i f x „ i s less  than  3 . 4 9 mm.  w o u l d o n l y make t h e PDM p r o b l e m To a p p r e c i a t e consider a typical table 8 with  problem w i t h  cannot  be a s s e m b l e d w i t h i n  Values  of the reductions  12 t y p e s  o f plywood  number o f s e t s o f v e n e e r  the  PDM p r o b l e m r e d u c e s f r o m a maximum o f 1 7 7 , 1 4 7  If  plywood  branching tests the  reduction  above  -  t h e two f a c e v e n e e r s  by ( 3 . 3 9 )  -  (3.42),  as l i s t e d i n  some o f t h e In p a r t i c u l a r ,  would depend on t h e d e s i g n  by ( 3 . 4 1 )  i n a plywood  - (3.42),  an e f f i c i e n t a l g o r i t h m  t o 1 2 , 4 0 0 by t h e  may h a v e t o be m o d i f i e d .  given  Then  need t o be c o n s i d e r e d i n  unbalanced d e s i g n s ,  Using t h e a n a l y s i s developed specified  (3.39)  (3.42).  s p e c i f i e d by ( 3 . 4 0 )  c o n s i d e r e d a n d t h e bounds if  thicknesses that  c a n be made w i t h given  speci-  v o f x „ lower than X „  :m(.l) = 9 , a n d m ( k ) = 27 f o r k = 2 , 3 , 4 .  s p e c i f i e d by ( 3 . 3 9 )  example  = 3.49  the  reductions  In t h a t  infeasible.  t h e importance  K = 4,  = 2.80  sheet  and ( 3 . 4 2 )  alternatives  w o u l d n o t be v a l i d  c a n be o f d i f f e r e n t  thicknesses.  i n s e c t i o n 3 . 3 . 1 and t h e r e d u c t i o n s  we c a n now p r e s e n t ,  i n greater  detail,  f o r s o l v i n g t h e PDM p r o b l e m a s f o l l o w s .  Al g o r i t h m B : Step 0:  (An  algorithm for  Initialize  (x-j,  x^,  * Step 1 :  Let  x^  , x^.  values  Step  2:  go t o  K  ,  M A X  K  Otherwise,  Zg  t  = 1,  2,  ...  K-1  (3.39)  -  (3.41).  If  be a  x^, . . .  previously  x ^  none e x i s t s ,  of  the  terminate.  step 2 .  Evaluate  > X  M I N  and  problem)  the thicknesses x.j,  X  K  M L N  and  X  K  Je  X  x^,)  e T^.,  for  PDM p r o b l e m w h i c h s a t i s f y Otherwise,  ...  PDM  *  = xt  unselected set of  s o l v i n g the  PDM  (x|  compute  from  M A X  (.3.31)  and  (3.32).  If  Je  ,  . . .  x ^ )  is  infeasible  Z.'frPm ( 3 . 3 8 ) .  If  Z is less  and  go  than  to  step  ZQ go t o  1.  step  1.  Otherwise, *  Step 3 :  Let  xK e T  K  n (X  K  M I N  Je  ,  X  K  M A X  )  along with  (  Jc  , ...  X ]  x. _ ) K  1  be a p r e v i o u s l y u n s e l e c t e d s e t o f t h i c k n e s s e s s a t i s f y i n g ( 3 . 3 9 ) (3.42).  If  Step 4 : *  none e x i s t s ,  Solve the  * ,  x2  if  better  ,  In  ...  x^  using ( 3 . 2 3 ) .  than the  the  solutions  incumbent.  x^,  ZQ = -°° as t h e are generated.  As  the  s o l u t i o n and u p d a t e  terms of the of  solve the number o f  practical  problems,  . and t h e c o r r e s p o n d i n g v a l u e o f initial  initial  value  Clearly,  any o t h e r  node.  Otherwise,  and u p d a t e  algorithm B will  it  whenever  terminate  s o l u t i o n to  algorithm,  the  v a r i a b l e s and c o n s t r a i n t s presented  in  Fig.  1.  ZQ  we c a n  the  better in a  PDM  computational  p r o b l e m w o u l d m a i n l y d e p e n d on t h e  Algorithm B i s  Zg  step 3 .  i t e r a t i o n s w i t h an o p t i m a l  is true with  time r e q u i r e d to  A flowchart  Go t o  c a n be u s e d a t t h e  number o f  problem.  Store  implementation of t h i s algorithm for  the m i l l  start with  in  Otherwise,  LMCK p r o b l e m a s s o c i a t e d w i t h t h e t h i c k n e s s e s  the e x i s t i n g values of  finite  step 1 .  *  x-j  for  go t o  -  i n the  size  problem.  59  Figure 1 :  A Flowchart  of  the Algorithm  to  Solve the  PDM  Problem.  60 3.4  Computational  Our  Results  i m p l i c i t enumeration  c o d e d i n FORTRAN.  As t h e d e s i g n c o e f f i c i e n t s a n d t h e number  design alternatives thicknesses, Input data the y i e l d  a l g o r i t h m t o s o l v e t h e PDM p r o b l e m was  f o r each product  depended  s e p a r a t e codes were w r i t t e n  correction factor  set o f a l l p o s s i b l e veneer  for different  f o r face veneer  revenues,  f i l e requirements  order  thickness, l i s t  o f l o g s t o be c o n v e r t e d  the optimal  s o l u t i o n , marginal  design alternatives to  listing  that  t h e PDM model  enumeration in  (K = 4 ) .  o f plywood  the optimal  value  o f wood,  the  types,  veneer  t o e a c h t h i c k n e s s , maximum  their  tolerances. thicknesses,  net  revenue  a l l the feasible  and t h e q u a n t i t y  each d e s i g n a l t e r n a t i v e .  o f t h e FORTRAN p r o g r a m  thicknesses  log cost,  and t h e r e s p e c t i v e t h i c k n e s s  f o r each t y p e o f plywood  be p r o d u c e d u n d e r  v a l u e s o f K.  f o r c o m p r e s s i o n and s h r i n k a g e ,  The c o d e s a r e c a p a b l e o f g i v i n g a s o u t p u t ,  at  veneer  thicknesses of the peeling l a t h e ( s ) ,  upper t o l e r a n c e  quantity  on t h e number o f  f o r t h e code were t h e a v a i l a b i l i t y o f l o g s , factor,  of  Appendix  f o r t h e PDM p r o b l e m w i t h  The p u r p o s e o f t h e c o d e i s o n l y t o  of  plywood  I gives a four  veneer  demonstrate  c a n be s o l v e d e f f i c i e n t l y u s i n g o u r i m p l i c i t  a l g o r i t h m a n d no e x p e r t i s e  i s c l a i m e d on t h e e f f i c i e n c y  coding. For  t e s t i n g t h e s u i t a b i l i t y o f o u r PDM model  enumeration obtained  algorithm  for real-world  from a plywood m i l l  was m o s t l y m a n u f a c t u r i n g laid  (1978)).  Currently  the m i l l  representative  i n B r i t i s h C o l u m b i a was u s e d .  exterior  by t h e C o u n c i l o f F o r e s t  situations,  and t h e i m p l i c i t  plywood  Industries  data  The m i l l  adhering to the s p e c i f i c a t i o n s d f B r i t i s h Columbia  was u s i n g f o u r  veneer  (COFI  t h i c k n e s s e s (K = 4)  61 as l i s t e d i n T a b l e  2.  The number o f p o s s i b l e v e n e e r  c o u l d be c o n s i d e r e d w i t h i n amounted order  t h e framework  o f veneer  t o m ( l ) = 9 a n d m ( k ) = 27 f o r k = 2 ,  f i l e consisted o f twelve  types,  their  yield  factor,  thickness  types  thicknesses  thickness  3 and 4 .  o f plywood.  The  Details  e x i s t i n g designs, the a v a i l a b i l i t y of l o g s , revenue  and o r d e r  f i l e requirements  s p e c i f i c a t i o n s are given  When t h e m i l l ' s  i n appendix  of plywood,  net  Starting with  initial  s o l u t i o n , our i m p l i c i t enumeration  p r o b l e m w e r e u s e d on t h e U n i v e r s i t y computer  with  K , was f o u r , solution Details  a WATFIV c o m p i l e r . t h e code t o o k  design a l t e r n a t i v e For  than  t h e annual  that  loss  quantities  o f plywood  and s u c h o t h e r  net revenue  net revenue  mill's  products  f r o m o u r model  i n appendix  from t h e f a c t  III.  used i n t h e  was 6 . 8 6 % h i g h e r  that  designs  t h e wood  reduced from t h e  metres.  p o s s i b i l i t i e s of manufacturing veneer  each  t h i c k n e s s e s and plywood  to 1647.4 c u b i c  three  an o p t i m a l  $14,337,370.  o f t h e problem parameters  obtained  to the above, from o n l y  thicknesses,  t o be p r o d u c e d u n d e r  information are given  The i n c r e a s e r e s u l t e d  7944.7 c u b i c metres  In a d d i t i o n  AMDAHL 4 7 0 V8  t h i c k n e s s e s , the f e a s i b l e  i n t h e form o f e x c e s s t h i c k n e s s i n plywood  current  of  as an  f o r t h e PDM  2 9 . 3 s e c o n d s o f CPU t i m e and g a v e  f o r the e x i s t i n g s e t o f veneer  u s e d by t h e m i l l .  this  When t h e number o f v e n e e r  the p a r t i c u l a r configuration  study,  p o s s i b l e annual  o f B r i t i s h Columbia's  o f t h e c o r r e s p o n d i n g optimum v e n e e r  design a l t e r n a t i v e s ,  their  thicknesses  a l g o r i t h m codes  t o t h e PDM p r o b l e m w i t h a maximum  plywood  II.  t h e maximum  was $ 1 3 , 4 1 6 , 6 9 4 .  mill's  of the  data with the e x i s t i n g s e t o f veneer  for the mill  tolerance  log cost,  and d e s i g n s were used i n a l i n e a r p r o g r a m , revenue  that  thicknesses instead of  a l l the four  62 was t e s t e d products  u s i n g our  of the mill  specifications not  IV.  plywood  The maximum  thicknesses  with  three  Details  of  veneer the  p o s s i b l e net  revenue, set  there  in addition  to  was s u b s t a n t i a l  analysis of and t h e i r  the  r e s u l t s of the  comparison  i s given  plywood  that  all  the  thickness  t h i c k n e s s e s , a f a c t w h i c h was veneer  data are given  in  associated with three  c o r r e s p o n d i n g wood l o s s was 3 6 1 2 . 1  an i n c r e a s e o f savings  up c o s t s a s s o c i a t e d w i t h t h e  was f o u n d  c o r r e s p o n d i n g optimum  revenue  was $ 1 3 , 9 3 0 , 6 7 0 and t h e  that  It  d e s i g n s and s u c h o t h e r  form o f e x c e s s t h i c k n e s s i n plywood indicates  K = 3.  c o u l d be a s s e m b l e d w i t h i n  using only  known e a r l i e r .  thicknesses,  PDM model  PDM model in chapter  peel for six.  veneer in  cubic metres.  3.83% i n annual  p o s s i b l e in the  fourth  appendix  thickness. different  the This  net  form o f  reduced  A more number o f  detailed thicknesses  63 CHAPTER 4  4.1  Extensions  In  the  problem, in the  to the  PDM  formulation  Problem  of  the  Plywood  assembly of  a d d i t i o n to  species or plywood all  the  end p r o d u c t s .  However, most  specified specieswise.  The  factors  and s o l v e d  As we w i l l  show i n t h i s  variations  the  Changes  i n the d e n s i t y or  thickness for  making a p p r o p r i a t e veneer  west  incorporate  i n the  does not  veneer. to fir  In  affect  thickness  of a species  are  and c o m p r e s s i o n  significant deviations  such as Douglas  i n c l u s i o n o f more t h a n  p r o d u c t mix  solved at tree  if  any p a r t i c u l a r s p e c i e s , i t  veneer  to  the  occur  c a n be a d j u s t e d such a c a s e ,  the thickness of i n the  during  Pacific  in  by  the a North-  region. The  the  However,  corrections  species of  chapter,  i n green veneer  specific gravity  t h i c k n e s s c a n be e x p r e s s e d r e l a t i v e  standard  of  efficiently.  PDM p r o b l e m .  pressing.  factors.  p r o b l e m c a n be r e f o r m u l a t e d  n o r m a l l y n u l l i f i e d by t h e c o r r e s p o n d i n g s h r i n k a g e d r y i n g and hot  used  might  may be i n d e p e n d e n t  i n c l u s i o n o f more t h a n o n e s p e c i e s i n t h e model  the design part of  plywood  plywood m i l l s  l o g c o s t s and y i e l d  t h i s , the o r d e r f i l e requirements  d e s i g n and m a n u f a c t u r i n g  these  due t o  (PDM)  we had e a r l i e r c o n s i d e r e d t h e c a s e w h e r e o n l y one s p e c i e s i s  u s e more t h a n o n e s p e c i e s w i t h v a r y i n g In  D e s i g n and M a n u f a c t u r i n g  part  of  problem.  e a c h f e a s i b l e node o f  assumes d i f f e r e n t  model.  the  one s p e c i e s i n t h e m o d e l ,  the  The  however,  l i n e a r subproblem to  i m p l i c i t enumeration.'s  f o r m s d e p e n d i n g on t h e  factors  be  search  included  H o w e v e r , e s s e n t i a l l y t h e same i m p l i c i t e n u m e r a t i o n  affects  in  the  algorithm  64 described  i n the  extensions  of  previous  the  PDM  chapter  orderfile  In  the  model,  this  problem w h i c h ,  portation  problem.  required  the  i n the  that,  search tree  problems.  when v e n e e r s  different  4.2  Orderfile  4.2.1  involving  various  l i n e a r subproblem  some s p e c i a l  Independent  In  is a  into  section 4.4,  a Generalized a standard  Transpresent  algorithm.  In  i s dependent  on  separable  Linear  we c o n s i d e r t h e  PDM  Trans-  section  associated Multiple  situation  s p e c i e s c a n be m i x e d w i t h i n a p l y w o o d  of  the Generalized  c a s e , a l i n e a r subproblem  decomposes  these  in the  s u b p r o b l e m and  i m p l i c i t enumeration  in t h i s  when  structure  c a n be r e d u c e d t o of t h i s  some o f  we show t h a t  s i t u a t i o n when t h e o r d e r f i l e  Choice Knapsack of  due t o  in turn,  We d e m o n s t r a t e  w i t h a node o f  s p e c i e s , the  We s o l v e an e x a m p l e  we c o n s i d e r t h e  species.  PDM model  Network problem reduces to  portation  4.3,  of  the  in section 4 . 2 ,  We show t h a t ,  Generalized  modifications  to  f i r s t case,  i s independent  Network problem.  solve these  problem.  We now c o n s i d e r e x t e n s i o n s situations.  c a n be u s e d t o  panel.  Species  Formulation  Define  new d e c i s i o n v a r i a b l e s a n d c o e f f i c i e n t s o f t h e model  as  follows: v^  =  Quantity of (in  P..  veneer  number o f  = Quantity of  sheets  sheets of  Plywood o f  equivalent).  thickness  standard type  construction alternative s i z e or  of  j  k from s p e c i e s s  size or  equivalent).  i , s p e c i e s s , made (in  number o f  using  sheets of  standard  r.  =  Revenue  w"s  =  Quantity of  Cs  =  Cost  per u n i t  y  =  Yield  factor  Plywood t y p e ' i ,  species s.  of  l o g , species s  for  veneer  N;  j  All  v a r i a b l e s and p a r a m e t e r s  of  other  Formulating  PDM model  ZQ  = Max  u  = 1, 2 ,  i  the  problem i n the  i n t h i s case would  N  S  E  E  i=l  s=l  n. E1 j=l  1 S  1^ k=l n.  >1  6..  1  j=l  1 J  P.. ijs  -  Mfi. . < 0 °1J  K  y  s  E k  = 1  S iEj s=l  5  < W  Vks  s  n. =E { j=l 1  0  ' 1 }. P. ^ s  >  d. 1  P 1 J S  ($/cu.mtr.)  = 1, 2,  ...  -  E  s=l  s  E s  and k = 1 , 2 remain  same way a s i n C h a p t e r  K Cy  n.  t h e e a r l i e r model  be  S r.  s .t.  E  ...  mtrs.)  from s p e c i e s s .  ...  S;  ($)  logs a v a i l a b l e , species s (cu.  s = 1, 2 ,  unchanged. the  for  k=l  x.v. k  k s  2,  66 k  x  G  k  T  P. .  Note t h a t  T  k  —  m ( k ) }  { 4  > 0  -  9 )  (4.10)  ks  a l l s = 1, Again,  J  1  , v.  i js for  " ' ^ k * k>  2,  . . .  t h e above  S;  i = 1,  2,  ...N; j = 1 , 2 , . . . n . a n d k = 1 , 2 ,  i s a n o n - l i n e a r mixed  i n t h e a b o v e model  integer  represents  programming  the quantity  ..K.  problem.  of  logs  p e e l e d t o t h i c k n e s s x^ from s p e c i e s s and z y x ^ v ^ r e p r e s e n t s t h e t o t a l n. k s quantity of logs of species s . Also, z P.. i s the quantity of 1 J S j=l p l y w o o d t y p e i made w i t h s p e c i e s s a n d S n. Z z1 P . . i s t h e t o t a l q u a n t i t y o f plywood type i . 1 J S s=l j = l 4.2.2  The G e n e r a l i z e d  The to  evaluating  PDM p r o b l e m g i v e n  the i n d i c a t o r variables  be a s d e f i n e d  i n (3.1)  ( 4 . 5 ) reduces  i  j G i . k  s  by ( 4 . 1 ) - ( 4 . 1 0 )  , x^  G T^,  k = 1,  c a n be v e r i f i e d  6.. u s i n g ( 3 . 1 ) .  and ( 3 . 2 ) r e s p e c t i v e l y ,  f o l l o w i n g arguments  show t h a t  Subproblem  f e a s i b i l i t y o f a s e t o f t h i c k n e s s e s , x^  t h e extended  Then,  Network  L e t s. . . a r i d  of the previous  s i m i l a r t o those used i n t h a t  chapter,  2,  ,  by I.  chapter. we c a n  to  1  J  K  k  1 J S  -  or z z b. . 1 j G I. ^ S  P.,  < W  1 J S  S  where b. . ijs  = y ^s  k  K z a . ., x,' = 1 ljk k  (4.11)  67 The o b j e c t i v e  function  reduces  to  K . i  l  . s T ] e I.  i lsb P i 1j JsS  s r  s  T,  T  E  s r . . P. . s ^ s U  T,  II  1 j  z  , i j e l  i  z C  s  i  s s( y s s  z  a  k=l  * i 1i Jk Kx kK > p i 1 iJ sS  s  where K r  ijs  =  r  i s "  r.  C  -  sys  k  z  = 1  * ijkxk  a  (4 1 2 )  C b.. s IJS  is  V  ic  The  ic  l i n e a r subproblem a s s o c i a t e d w i t h a s e t o f t h i c k n e s s e s  , x2 , . . .  H  '  X  C  '  it  xk  ,  w o u l d t h e n be  E T . . P. . s ^  Z = Max E E i j e I.  s  s.t. E  i  b..  E  j e I.  1  J  P.. S  E E P s j e I. 1  P  (4.13)  >  >d J  0  =1  for a l l i , (4.13)  (GT)  problem,  -V-  s  A/-  i  s  b  f  i , sand  j e  Network  I.  (GN)  p r o b l e m a n d when  i s s p e c i a l i z e d to a Generalized  b.. c a n be i n t e r p r e t e d ij s  p r o d u c e one u n i t  (4.13)  1  i s known a s a G e n e r a l i z e d  |I.|  to  ijs  <W  1 J S  o f plywood  type  a s t h e amount  o f wood  Transportation required  i , species s , using design  alternative  68 j  and r „  of  i s the corresponding unit  s  (4.13)  i s that  constraints. that  each v a r i a b l e  This  special  we c a n a s s o c i a t e  represent  net revenue.  i n " i t a p p e a r s a t most t w i c e  structure  is further  feature  i n the  r e f l e c t e d by t h e  fact  a g r a p h w i t h a GN p r o b l e m i n w h i c h t h e n o d e s  t h e c o n s t r a i n t s and t h e u n d i r e c t e d a r c s , t h e v a r i a b l e s .  S i n c e each v a r i a b l e constraints,  h a s a t m o s t two n o n - z e r o c o e f f i c i e n t s  t h e b a s i s o f a GN p r o b l e m has some s p e c i a l  w h i c h f a c i l i t a t e s i t s s o l u t i o n i n much f a s t e r program  (Kennington  (1979),  P h i l l i p s and G a r c i a - D i a z  as g i v e n  The u n i q u e  and Helgason  by ( 4 . 1 3 )  (1980), (1981)).  Elam,  i n the  structure  time than a l i n e a r Glover  and Klingman  H o w e v e r , o u r GN p r o b l e m  r e d u c e s t o a GT p r o b l e m a s shown  in the following  section.  4.2.3  The G e n e r a l i z e d  Recall  that  by r . . = r. 1JS 1i s J  -  S.ubproblem  t h e c o e f f i c i e n t s i n o u r GN p r o b l e m C b. . s i j  = b. i J.s  Then,  Transportation  ii  e  i t follows  r. is  r.  =  are related  Suppose  Min { b . . . } j  (4.13)  I,  (4.14)  J  that  -  C b. s iJ,s  Max t r . . } j 6 I, 1 J S  (4.15)  Proposition if  4.1 :  more t h a n  For any i  one e x i s t s )  = 1,  2,  . . . N , the index  minimizing  b. . I  for  a l l s = 1,  Proof:  2,  =  y s  p  4.2:  Min { z a , . . x j e I. k 1 J K  •  there Z = V r  pJ  Suppose exists  a n c  '  } K  an o p t i m a l  solution to ( 4 . 1 3 ) ,  does  not hold  (P^-s)  which  f o r some i a n d s . i s optimum  That i s ,  for (4.13)  with  > 0 f o r some i = rp , s = f a n d t e I \ { J } . P P  c o n s 1 c  *  e r  P-  n  + P p J  P*.  t  f  e  n e w  P b  P  t f  Jpf  solution  pi P  given  i  by  = P» J = J p » s = f  t f  i = p , j = t , s = f  P..  i n which  -V- i , s  a solution  b u t P' ptf  f ^ ^  (4.16)  of b..js since  if,.  and  Proof:  i s ( a r e ) t h e same  of s.  There e x i s t s  = 0  i j s  indicies,  J  from t h e d e f i n i t i o n  i s independent  Proposition  in (4.14)  (or  ...S.  The p r o o f f o l l o w s  and s a . . . x .  J  J.  Otherwi se  Assume  70 Observe t h a t  (4.17)  i s feasible  (?'. . ) i s a f e a s i b l e  (1)  Pj  j s  >0  solution,  ¥  f o r o u r GN s u b p r o b l e m ( 4 . 1 3 ) . i t follows that  since b . f > b . > 0, ptf pJpf  i , j , s  ptf .  (ii)  z  E  Since  P i , . + ! hh7 T  P*is  "Ui  i/'J*  .-,  _1  IP'p t f  " » « *" p y  s i n c e  > d  ie  ic  P'  and ( i i i ) b . _P . . + b . -P-"-. - = 0 + b . ptf ptf p j p f pJpf pJpf  b  so  . PJpf  P b  pi  t f  pJ f P  P  t f  P1 + b P1 PJpf PJpf ptf ptf  that  z  z  1 j €= I .  b.. P.. ^ S ^  =  Further,  observe that  that  E  j G I. W s  b . . P'. 1JS 1JS  -V-  s.  i f b . . . i s such t h a t ptt  more t h a n o n e j e I p  from ( 4 . 1 5 )  E i  S  <  exists  +  = r  p  b . = b . r , ptf P^pf  minimizing bp^s i n (4.14).  j ^ and t h e r e f o r e  i.e. Then,  There i t follows  71 On t h e o t h e r j  J  f  it  P  hand,  , j e i . i  follows  from  s  if  (i)  Jp  i s unique  or ( i i ) the index t (4.15)  that  r  ,  i s such t h a t  > r  f  for i = p,  . f .  i.e. b  p J  f  <  b . < b ,,., PJpT P^T  Now, o b s e r v e  bp..f, then  that  P  r' prP*4.r + r p, J -f P p* J f f = r p. j -f | PI ' p. J -f + - rb^ f P' p t f tf ptf +  p  p  p  p  > V i / p j f p p > r  .  f  p  p  +  p r  P' p J  r  +  p J  J  f p  p  f  tf  P  f r o m  r . . P '  p  <4-14>  r  p  from  p  t  (4.15)  f  * Therefore  E E 1 j e I.  Er.. P . . s 1 J S ^ s  optimality of (P1.. rpj  <0  Er.. P 1 . . s 1 J S ^  this  r e s u l t h o l d s even  by d r o p p i n g t h e f a c t o r  arguments  as above.  , c o n t r a d i c t i n g the s  b  p t f  /b  p J  ^ in  when (4.17)  The p r o o f o f P r o p o s i t i o n 4 . 2  follows. Propositions  solution is  The f a c t t h a t  c a n be shown  and u s i n g s i m i l a r then  ).  > E E 1 j e I.  4.1  and 4 . 2 would t o g e t h e r  t o t h e g e n e r a l i z e d network  made w i t h more t h a n o n e s p e c i e s ,  imply that  problem ( 4 . 1 3 ) ,  i n an o p t i m a l  i f a plywood  the design a l t e r n a t i v e  type  f o r them  w o u l d be t h e s a m e . From p r o p o s i t i o n s 4 . 1 to  and 4 . 2 i t f o l l o w s t h a t  solving Z-  Max  ? ! V  S  P . .  S  s.t.  Eb.  P.  . V S V s  <W  s  -\f s  t h e GN p r o b l e m r e d u c e s  72  z P . >d 1  s  ¥1  1  (4.18)  s P  where,  b.  . -  unit  o f plywood  ¥  i , s  and r .  I  Again,  and r .  0  b.  15^  problem.  >  1's  •>  „ -  r.  l'S  .  b ^ > s i s t h e amount type  i , species  species  P I y w o o d Type  and plywood  b  revenue.  types  Observe t h a t  •  s  as  'destinations'.  ....  2  Vl  M  2-l  r  . . . .  "i-s  . . . .  r  l-s  b  •Vs  r  2-S  ..... .  N '  Vl  d  . . . .  r  . . . .  2  b  N-S  h a v e more t h a n  problem i n t h a t  one maximal  N-l  N-1  r  P  •  N-S  N-S  . . . .  -  s o l u t i o n p r o c e d u r e o f a GT p r o b l e m d e v i a t e s Transportation  Supply  ....  2-S  >  demand  standard  2-S P  >  The  ( 4 . 1 8 ) can  f o r m o f a GT p r o b l e m w i t h t h e  P  (s)  J^,  (i) 1  1  Transportation  o f wood r e q u i r e d t o p r o d u c e o n e  i n the following tabular  as ' s o u r c e s '  i s a Generalized  s , using the design a l t e r n a t i v e  i s the corresponding unit  be r e p r e s e n t e d  This  IJ^S  the basis  from t h a t  of a  g r a p h o f a GT c a n  c o n n e c t e d subgraph o r component.  The d u a l  variables feasible  a s s o c i a t e d w i t h a GT a r e u n i q u e s i n c e a n y solution will  constraints.  have a s many b a s i c v a r i a b l e s as t h e number  Algorithms  t o s o l v e a GT p r o b l e m have  the l i t e r a t u r e [Balas  and I v a n e s c u  (1964),  (1964),  Taha ( . 1 9 7 1 ) ,  Glover  Balas  The  (1966),  and K l i n g m a n sign  (1973))  B a l a s and I v a n e s c u in that  deviates (1964),  the standard  i n t h e demand c o n s t r a i n t s .  Eisemann  (1964),  Lourie  (1 9 7 3 ) ] .  from s t a n d a r d  Taha ( 1 9 7 1 ) ,  GT p r o b l e m s w i l l  H o w e v e r , we c a n s o l v e  the  present  for  a l l columns h a v i n g a l l o c a t i o n ( E P -  the dual  s (ii)  f o r a l l rows  supply  4.2.4  for  variable(s)  in  be z e r o  ) more t h a n demand  h a v i n g w e i g h e d a l l o c a t i o n (Eb.  (i)  ( d . ) and  g  P^.s)  less  than  the  s  S c a l i n g t h e GT p r o b l e m t o a T r a n s p o r t a t i o n  to solving  (4.18)  i t as a T r a n s p o r t a t i o n  network  problems w i t h  having gain/loss f a c t o r s .  a s a GT p r o b l e m ,  For example,  a n d two p r o d u c t s Fig. 2.  Problem  i t i s possible to  problem f o l l o w i n g a s c a l i n g  gains  (Truemper  (1976)). flow  procedure  Observe  t o t h e amount  t h e GT p r o b l e m ( 4 . 1 8 )  (N = 2 ) c a n be t r a n s f o r m e d  that  the  problem w i t h the a r c s  S p e c i f i c a l l y , i n such networks  f l o w e n t e r i n g an a r c n e e d n o t be e q u a l  the a r c .  in  (4.18)  s  GT p r o b l e m c a n be r e p r e s e n t e d a s a n e t w o r k  of  would  columns,  (W ).  Alternate solve  equality  (4".18) c a n be p u t i n t h e s t a n d a r d  and a s l a c k .  form n o t i n g t h a t  GT  Glover  have  f o r m o f a GT p r o b l e m by t h e a d d i t i o n o f a dummy row a n d N + l one f o r e a c h p r o d u c t  of  been d e s c r i b e d i n  and K l i n g m a n  probdem a s r e p r e s e n t e d by ( 4 . 1 8 )  problems (for e x a m p l e ,  non-degenerate  t h e amount  of flow  leaving  w i t h two s p e c i e s (S = 2 ) i n t o a network  a s shown  74  Fig.2:  In  The PDM S u b p r o b l e m a s a N e t w o r k F l o w  Figure  Problem  2 , S-| a n d S 2 a r e n o d e s a s s o c i a t e d w i t h t h e two s p e c i e s  and P.j a n d P^ a r e n o d e s a s s o c i a t e d w i t h t h e two p r o d u c t s .  S1  is a  'super s o u r c e ' , a c o n s o l i d a t i o n of a l l the supply (or species) and T 1 nodes.  i s a 'super s i n k ' , The t h r e e  arc capacity, The number a unit factors  of  numbers  a c o n s o l i d a t i o n o f a l l t h e demand  In a p u r e n e t w o r k  w o u l d a l l be e q u a l  quantity  the  q3 a i n f a c t o r ,  to unity.  o f plywood o f type r.' ls  = (r.  i  i n w h i c h b.1  p.  associated with  /b. ) i s the revenue i •s  c a n be v i e w e d # s  of species s , i s  generated i .  ),  by t h e  The s u p p l y  ( d . . ) r e s t r i c t i o n s o f t h e GT p r o b l e m c a n be i n c l u d e d a s in the f i g u r e .  node S g t o p r o d u c t  the v a r i a b l e  respectively.  = (1/b..  produced per one u n i t  I•s  a r c c a p a c i t i e s a s shown species  'flow'  Our p r o b l e m ( 4 . 1 8 )  o f one u n i t o f s p e c i e s s t o plywood t y p e  ( W g ) a n d demand  product)  flow problem these l o s s / g a i n  as o n e o f a l l o c a t i n g s p e c i e s t o p r o d u c t s  allocation  per u n i t  b e l o w an a r c r e p r e s e n t s t h e l o s s / g a i n f a c t o r  the  (or  i n b r a c k e t on t o p o f t h e a r c r e p r e s e n t t h e l o w e r  u p p e r a r c c a p a c i t y and r e v e n u e  'flow'.  nodes  of  (4.18)  node  If  f..  i s the 'flow'  P.. i n t h e n e t w o r k ,  by f .  = P.  > s  b.  # s  .  from  i t is related  Our GT p r o b l e m  to  (4.18)  75 would then revenue)  be e q u i v a l e n t  network  flow  to the f o l l o w i n g minimal  problem with  cost  ( o r maximal  gains.  Max v y r ! f. i s vs.; i s  s .t. fs'j  " f j s - ' V  3  T1)  t  T  3  f r j  where,  <  " f j r  f  j r  >'  F  i s the index o f a l l intermediary  i n the network,  nodes  (i.e.,  e x c l u d i n g S1 and  F i s t h e flow a v a i l a b l e a t the super  source  S1  ( = zW ) a n d F i s t h e minimum f l o w r e q u i r e d a t t h e s u p e r s s  V  ( - zd..).  network Since  Truemper  problem w i t h  (4.18)  gives  Transportation  a scaling  procedure  by w h i c h a  g a i n s c a n be r e d u c e d t o a p u r e n e t w o r k  i s a Generalized  problem obtained  example  (1976)  Transportation  by T r u e m p e r ' s  problem.  This  scaling  problem,  problem.  t h e pure  procedure y i e l d s a  i s i l l u s t r a t e d through  sink  network  standard  the following  o f a PDM s u b p r o b d e m .  Example 4 . 1 :  C o n s i d e r t h e f o l l o w i n g example . ( h y p o t h e t i c a l )  s u b p r o b l e m w i t h two s p e c i e s , veneer  F i r (F)  (N=4)  and f o u r  3.35,  3 . 9 6 a n d 4 . 9 8 mm f o r k = 1 ,  and Hemlock  (H),  four  o f a PDM products  t h i c k n e s s e s (K=4) w h i c h a r e s e t t o x ^ = 2 . 6 9 , 2,  3 and 4 r e s p e c t i v e l y ;  C- = 3 5 . 0 0 ,  76 CH  = 30.00,  YF = 0.006067,  Other data  f o r t h e PDM p r o b l e m a r e g i v e n  Plywood Type, i  Ply -  L. l  Thick (mm)  r  ($) r  9.  Demand d. i  iH  Design  a... Uk  j  -  7.5  4.3  4.1  171107  1  3  0  0  0  2  5  -  12.5  6.1  5.8  502289  1  5  0  0  0  2  4  1  0  0  1  2." 2  0  1  2  2  1  2  0  1  5  2  0  0  2  4  3  0  0  Table  5  7  9:  -  -  15.5  We w i l l the b. . gives  7.6  18.5  Input data Orderfile  For  iF  i n Table  3  4  (4.14)  Revenue  W p = 7 0 , 0 0 0 a n d WH = 8 0 , 0 0 0 .  1  3  of  YR = 0 . 0 0 6 2 7 6 ,  8.9  7.4  350192  8.6  423394  f o r E x a m p l e 4 . 1 ; A PDM s u b p r o b l e m w i t h S = 2 , Independent  of Species.  p r o c e e d now w i t h s o l v i n g t h i s  PDM s u b p r o b l e m .  Computation  -V- i , j , s u s i n g ( 4 . 1 1 ) a n d e v a l u a t i o n o f t h e minimum J.  = 1,  1,  2 and 1 f o r i = 1 ,  these design a l t e r n a t i v e s ,  b. i •s  f o r m o f a GT p r o b l e m  below.  from  2 , 3 and 4 r e s p e c t i v e l y .  and r . l •s  are presented  i n the tabular  77 Products 2  1  . 3  Supply  4  Sp^ecie>s\v .08160  .04896 F  .10101 4.0645  3.2440  2.5864  .10449  .08441  .05065  .12225  algorithms  and Ivanescu  (1973)) o r transformed the  502289  into  (1964),  a standard  s c a l i n g p r o c e d u r e o f Truemper  method given  for illustration. above  350192  Taha ( 1 9 7 1 ) ,  Observe t h a t  Glover  Transportation  and t h e n  solved.  -  423394  p r o b l e m c a n be s o l v e d a s a GT p r o b l e m u s i n g  (Balas  80,000  >  171107  above  < 4.8062  >  Demand  The  4.2651  3.2677  70,000  .12646  H 2.5806  < 4.6213  known and Klingman  problem  using  We u s e t h e l a t t e r  t h e c o n s t r a i n t s o f t h e GT p r o b l e m  c a n be w r i t t e n a s  . 0 4 8 9 6 P . c + . 0 8 1 6 0 P 0 .- + . 1 0 1 0 1 1 •F 2•F  P~ ,= + . 1 2 2 2 5 3-F  P. _ < 7 0 , 0 0 0 4-F (4.20)  .05065  P, n + . 0 8 4 4 1 1 'H  P0 „ + .10449 P , . u + .12646 P . „ < 80,000 2*H 3 H 4-H (4.21)  P  1-F  +  P  1.H  >  — ( 4 . 2 2 )  1 7 1 1 0 7  P  2-F  + P  2-H  >  5 0 2 2 8 9  — ( 4 . 2 3 )  P  3 . F  + P  3 - H  >  3 5 0 1 9 2  — ( 4 . 2 4 )  P  4.F  + P  4-H  >  4 2 3 3 9 4  — ( 4 . 2 5 )  There e x i s t s (4.20)  -  a s e t o f m u l t i p l i e r s , one each f o r each o f c o n s t r a i n t s  (4.25),  such t h a t  the gain factors  ( c o e f f i c i e n t s on t h e LHS  78 of  (4.20)  Phillips as  and ( 4 . 2 1 ) )  and G a r c i a - D i a z  the i n i t i a l  .08160,  c a n be t r a n s f o r m e d (1981)).  constraint,  .10101  and .12225  to unity  In p a r t i c u l a r ,  (Truemper ( 1 9 7 6 ) , starting with  i f we u s e t h e m u l t i p l i e r s 1 , for the constraints  (4.20)  (4.20)  0.9667,  to (4.25)  0.04896, respectively,  we g e t  .04896  F  P, + I • P, 1 •H  .04896  P, + I •  .04896  P  H  +  .08160  P  H  +  .10101  P  H  +  .12225  P.  H  .10101 .12225  F P F P. F P F  setting  of  P'  0  3  4>  P  1  Is  = .12225  4 s  (4.26)  problem,  .08160  +  =  P H 2>  1M  9  3>  = .04896  P4>s  to (4.31)  .10101  +  .04896  .08160  and  F  .08160 P„  P  1  +  •s,  .10101  P'2 s= Y  P  +  3-H±  .08160  712225  P  .12225  P  2-s'  <  7o,ooo-  -.(4.26)  4>P  P. < 7 7 3 3 6 . 0  —-  (4.27)  8377.40  —-  (4.28)  >  40986.78  —-  (4.29)  >  35372.89  —-  (4.30)  >  51759.92  —-  (4.31)  4  p ,  H  3 s=  = .10101  P, 3-s  f o r s = F, H , a l l t h e c o e f f i c i e n t s o n t h e LHS  are transformed  in a tabular  3-F  P  form,  would  to unity.  The r e s u l t i n g  transportation  be a s f o l l o w s  ""Products 2  3  4  52.8268  39.7549  40.2386  37.8020  ST70.000  52.7083  40.0453  42.2245  39.3145  <  1  Supply  S p e c i e s >s  F H  Demand  >  >  8377.4  40986.8  77336.0  > 35372.9  51759.9  _  The numbers revenues and  i  i n each c e l l  of  =1,  ...  4.  The a b o v e  signs  total  and  (ii)  this  problem to  the  P'  = 19216.4,  P'2p  and  variables P4.p  P'-^  we g e t  to  objective  4.2.5  standard  P].  = 343256.5  Generalized  function value  Implicit  chapter  thicknesses of  on x K  given  by  relaxation  (3.31)  to  a Generalized  branches  by  Then,  enumeration  Branching i s 2,  ...  and  K-1.  ('3.38),  by  the  (3.33)  = 9796.8,  P3>H  P'4H  the  =  changes  and  original  the  corresponding  (4.1)  -  (4.10),  done on t h e  -  node c a n be o b t a i n e d  for  (3.34),  problem,  =  350192.0,  = 0 as  The  reductions  l i n e a r subproblem  Transportation  Converting  veneer  i n the  (3.42)  in  number  and t h e  are e q u a l l y a p p l i c a b l e here.  however,  a  solution:  algorithm described  initially All  form o f  Problem  s p e c i f i e d by ( 3 . 3 9 )  (3.32)  s i n c e the  from t h i s  Overall  H  balanced.  these to  problem.  F>  the  inequality  = P3.p  PDM p r o b l e m a s g i v e n  s p e c i f i e d by  (3.35).  = P2-H  1 # R  the  search tree  in  and P  4 p  = 502289.0,  for  bound on Z g i v e n the  P2-p  Enumeration  = 1,  P'  Transforming  $6,158,270.  t  nodes o f t h e  -  Transportation  same i m p l i c i t  x^,  u  = 35372.9,  Z =  3 c a n be u s e d .  have  demand a r e n o t  is  To s o l v e t h e o v e r a l l e s s e n t i a l l y the  =  = 3924^2.0,  p  P4-H  the  = P'^p  P'3H  =  standard  f o r m and s o l v i n g y i e l d s t h e  = 40986.8,  = P ^  = 80137.5,  solution  s u p p l y and t o t a l  '  s  s  in the  the c o n s t r a i n t s  problem are  P'.j >  variables  problem i s not  problem s i n c e ( i )  41963.1  transportation  associated with the transformed  Transportation  1 F  this  this  let  b^.  The  problem. be a s  bounds upper  Following defined  in t h i s case reduces  an u p p e r  bound on Z f o r  as a s o l u t i o n o f  (4.32)  all below:  80 Z = Max z  r1.  z  ,  P.  S .t.  P. r s  Sbl  ^  i  <W  s  // yJ  s  (4.32) s  1 , s  1  P. ^ 0 i •s  ¥  i , s  where r.' i -s  Again,  = r. is  C  y s ^s  b\  i  i s a GT p r o b l e m and c a n be s o l v e d u s i n g m e t h o d s  (4.32)  outlined  * earlier. t  The  node a s s o c i a t e d w i t h  = 1 , 2 , ... K-1 c a n be f a t h o m e d  the xK  incumbent i n the  bounds our  Zg ..  set  If,  (XKmin,  however, X  M A X K  PDM p r o b l e m .  Zn = Min 0 s.t.  Consider  EW I s  5  +  1 obtained  if  is  found  i n which x t  in  (4.32) i s l e s s l a s t veneer  necessary,  GT p r o b l e m u s i n g t h e  the dual  of  = xt  (4.18) given  Recall  S  0,  from d u a l i t y  obtained can  s  u s i n g any  be o b t a i n e d  t.  than  thickness  additional structure  in  by  1  1  b. n + t. > r. 1•S S 1 1•s  n  ,  Ed.t. i  s  the  subproblem  b r a n c h i n g on t h e  ) n Tk  c a n be c a l c u l a t e d f o r  the  < 0  theory  (4.33) 1 '  -V-  that  i , s  an u p p e r  f e a s i b l e s o l u t i o n to  using the  structure  of  bound on (4.33).  the  (4.18) can An e f f i c i e n t  constraints  in  be upperbound  (4.33).  81 Observe t h a t  >r.  n S  in (4.33),  the constraints  /b.  s  T S  -V-  lead  to  T S  and t. > r. 1 T  n  - b . S  -V-  i , s  S  T S  Let  n  = Max { r . .  .  S  /b..  T S  }  T S  and t.  Then, in  = Max  Z  {r..  -  s  b.. ii } s  s  = EW n + E d . t . w i l l S s s . 1 1 S 1  (4.18).  Intuitively,  the o b j e c t i v e  function  LMCK p r o b l e m . 50.9497,  t-,  g i v e an u p p e r  J  this value  = 0, t  i s s h a r p s i n c e when S = 1 ,  a t an o p t i m a l  = -1.0330,  2  Z  _i  bound  For t h e example  bound on t h e v a l u e o f  solution of the  considered e a r l i e r , t  3  = - 1 . 0 5 8 6 and t  4  Z  =  Z,  corresponding  = 52.8286, = -1.6369.  = This  i gives  Z:  percent 4.3  = $6,191 , 2 2 0 , which d e v i a t e s  Orderfile  dependent  that  Dependent  s e c t i o n we f o r m u l a t e  t h e PDM p r o b l e m when t h e  on s p e c i e s a n d a n a l y z e  M u l t i p l e Choice  implicit  a  on S p e c i e s  the subproblem obtained  results  Z by h a l f  only.  In t h i s is  from the a c t u a l  Knapsack  of chapter  3.  enumeration  in this  its structure. c a s e decomposes  orderfile  We d e m o n s t r a t e into  Linear  p r o b l e m s w h i c h c a n be s o l v e d e x p l i c i t l y  For s o l v i n g t h e o v e r a l l  PDM p r o b l e m ,  algorithm described i n the previous  using  t h e same  c h a p t e r can  82 be u s e d , w i t h objective  few changes  i s dependent  on s p e c i e s ,  constraint  (4.7)  to  z  P,,c > d.  1  where d .  -V-  i s the quantity  g  other  i , s  (4.34)  lb  1JS  All  o f t h e bound on t h e  function.  When t h e o r d e r f i l e changes  i n the computation  o f plywood  required o f type i , species s .  c o n s t r a i n t s and t h e o b j e c t i v e  function  of section 4.2  remain  unchanged.  4.3.1  The S e p a r a b l e  LMCK p r o b l e m  *  *'.  When v e n e e r k = 1,  2,  . . .  thicknesses are assigned values  K,  5. .  we c a n show t h a t  Max  zz  s i j  and I.  a r e as g i v e n  by ( 3 . 1 )  the l i n e a r subproblem reduces  z  e  ,  e T^ f o r  and ( 3 . 2 )  respectively,  to  r. . P . . ^ ^  I.  s .t.  Z .  Z .  b. . P. . < W ijs ijs  T  t A  s  i  Z  P. . > d.  J G I .  P  Where b . .  and r . .  1  ijs  J  S  >  1 5  0  ¥  U  ,rx  (4.35)  j  a r e as d e f i n e d  ^  T  i  a n d  by ( 4 . 1 1 )  s  -  and ( 4 . 1 2 )  respectively.  83 (4.35)  i s separable into  e a c h s,  S distinct linear  t h e problem i s o f t h e  programming  problems.  For  form:  u e i V u s  Max  s.t.  z i  z  Observe t h a t , problem.  b . . P. .  j e I,  1 J S  1  J  < W  S  5  (4.36)  j  zz P . . e I.  P  i j s  > d.  1 J S  1 5  > 0  ¥  f o r each s ,  Also,  defining  i j e i .  (4.36)  b . , „ and r . . S  1  r e s p e c t i v e l y we c a n show t h a t applicable is  given  here.  i s a Linear c  1 u.j  Multiple  as i n ( 4 . 1 4 )  Knapsack  and ( 4 . 1 5 )  S  p r o p o s i t i o n s 4.1  From c h a p t e r  Choice  and 4 . 2 a r e e q u a l l y  3 , we know t h a t  the solution of  (4.36)  by  t P , J = J,-  i  is  1 = P, j  * P.. =  z d. t P  1  us  = Jp  if r  p J p S  <0  . b. .  1 5 1  J  i  s  1  pJ s  =  p  '  j  =  J  p  i  f  r  pj s > ° P  p  (4.37)  Otherwise  where r  /b  P  The  J S V  index  J S p  Pp J  S  = Max { r .  . i  /b.  c  U . S ' U.S i i  1  J  S1  }  = Max , ,c i .j e  J  p maximising ^ r - j j . s / b n - j . s ^  m  y  j  differ  I.  US'  1JS  J  from s p e c i e s t o  species:  84 4.3.2  The  Implicit  Enumeration  To s o l v e t h e o v e r a l l enumeration  algorithm of  PDM p r o b l e m  t h i c k n e s s e s x^.,  bounds  s p e c i f i e d by ( 3 . 3 1 ) ,  slightly In t h i s  The  upper  = 1,  v  c a s e , the  K-1.  and  All  B r a n c h i n g i s done the  by 1 i n  as a s o l u t i o n  on  branching t e s t s  (3.39)-(3.42)  given  same i m p l i c i t  are  (3.38)  and  applicable  i s to  i n c l u s i o n o f more s p e c i e s i n t h e  be  modified  PDM m o d e l .  of  P.  z Ezr. 1 s  ...  (3.32)  case, Z i s obtained  Max  2,  bound on Z ,  f o l l o w i n g the  in this  c h a p t e r 3 c a n be u s e d .  veneer  here a l s o .  t  Algorithm  s  v  s  s.t.  Eb. P.  < W J /yJ  1  v s  ? 1  s  >d.  P. TS  where  b|. and r i  LMCK i n o u r  s  -V-  a r e as g i v e n  PDM p r o b l e m ,  i , s  IS  the  in  (4.32).  solution of  Following (4.38):is  the  properties  given,  for  each  s,  by d  i r/'P  is  i -s  Ws/y  r 0  and  <  i = P if rp.s  'PS s  b.d. i  t  p  1  1 S 1  =  P  i  f  Otherwise  r  p-s  >  0  of  0  85 * = xt  The node w i t h x t from  (4.40)  is less  , t  = 1,  than the  2,  ...  algorithm.  4.4  S p e c i e s w i t h i n a Plywood  In some p l y w o o d m i l l s , a plywood  sheet.  This  i s fathomed  incumbent v a l u e o f  enumeration  Mix o f  K-1  Zg  _ Z obtained  if  i n the  implicit  Sheet  more t h a n  one s p e c i e s m i g h t  i s n o r m a l l y done when t h e r e  be u s e d  are t r a d e o f f  a s s o c i a t e d w i t h t h e c o s t and y i e l d f a c t o r s o f d i f f e r e n t  within benefits  species of  veneer.  However, t h e s e mixed s p e c i e s panel  a r e a s s e m b l e d i n s u c h a way t h a t  balanced design requirement  affected  The v e n e e r s  for  a n y one o f  i s not  f a c e , core or centre  same s p e c i e s o r b e l o n g t o a g r o u p o f properties. of  In  such c a s e s ,  v e n e e r s . ; s u c h as  'fir  i m p o s e d by s p e c i f i c the  formulation  of  only  there might for  plies  would  species. be o f  the  species having s i m i l a r physical be a d d e d r e s t r i c t i o n s on  face veneers'  customer o r d e r s . the  by t h e l a y - u p o f  the  In t h i s  or other  lay-up  restrictions  s e c t i o n , we i l l u s t r a t e  PDM p r o b l e m i n s u c h a s i t u a t i o n and a n a l y z e  its  structure. We d e f i n e a l a y - u p a l t e r n a t i v e or For  group o f example,  groups o f of  species for if  fir  species f i r only,  the  be a p l a n w h i c h s p e c i f i e s t h e s p e c i e s  f a c e , c o r e and c e n t r e v e n e e r s hemlock  s p e c i e s and t h e r e  d e s i g n would  (H)  and o t h e r  possible lay-up  be u s e d i n t h e  (j)  panel.  s p e c i f y the  i n a plywood  s p e c i e s (0)  a r e r e s t r i c t i o n s on t h e  be a s s p e c i f i e d i n T a b l e 1 0 .  design alternatives to  (F),  to  form  face veneer  alternatives  (1)  for  As e a r l i e r , we l e t  number o f v e n e e r s o f  panel.  three to  be  a balanced thee  each t h i c k n e s s  86 Lay-up  Species  Alternative,  Table 10:  4.4.1  1  Face  Core  Centre  1  F  F  F  2  F  F  H  3  F  F  0  4  F  H  F  5  F  H  H  6  F  H  0  7  F  0  F  8  F  0  H  9  F  0  0  Species Lay-up  Formulation  of  the  D e f i n e new d e c i s i o n as  for  alternatives.  PDM Model  v a r i a b l e s and c o e f f i c i e n t s o f t h e  PDM  problem  follows. P..j1  =  Quantity of j  and l a y - u p  s i z e or r.ji  =  =  type i , construction  alternative  1  ( i n number o f  alternative  sheets of  standard  equivalent).  Revenue f o r revenue  d..-|  plywood o f  product  i , lay-up alternative  i s independent  Demand f o r sheets of  product  of  1,  i , lay-up  r^  = r-  for  alternative  standard s i z e or e q u i v a l e n t ) ;  independent  of 1,  d  n  = dn- f o r  all  1.  1  ($);  all 1  If  (in  If  1. number  demand  is  of  87 a  ijkls  =  M u m  '  of veneers of species s , thickness k, used in  : , e r  lay-up alternative 1 and construction alternative j product 1  =  for  i.  1, 2 . . . L; L i s the number of lay-up a l t e r n a t i v e s .  All other variables and parameters of the problem remain unchanged. The a - i - j ^ i s - '  known numbers similar to a ^ ^ and are related to them by  a r e  i. . i i . " a . . i hjkls  "V"l— 1 , 2 ,  "  ...L  ljk and  -V- i ,  j, k  The model would be  Max  z z z r^P^  -  zC^  z x ^  s.t. b  Z  1  -M(l-6 .)<za.. x <bU  L  i j  6. .  j  Z  1  k  k  >1  /s k ks < s x  V  W  ?  Z  . ^  n  a - .,  P. -n -  Tjkls l j l z P . > j l  (4.41)  IJ  ijl  Z  M(l-6..)  J  P. . , - M 6 . . < 0  Z  +  j  l  d., 1 1  V.  ks  <  0  88  6. . = ( 0 ,  e x  P  This  k  T  =  1}  i j l '  v  1  { T  'k  l l  ks  >  2 'k '  T  k  '  *  0  'k  m(k)} *  i - J > k.  i s a g a i n a n o n - l i n e a r mixed  4.4.2  The  Linear  2,  Max  ... K t h e  z  z  i  j e I.  integer  programming  problem.  Subproblem  *  When t h e v e n e e r k = 1,  1 and s  t h i c k n e s s e s a r e a s s i g n e d v a l u e s x^ = x ^ , r e s u l t i n g problem reduces to  sr... 1  *  x^ e  T^,  solving  P..,  s.t.  ? j l l . ? Usl 1Jl b  E P j G Ii 1  P  >d J I  P. Ul  where,  by  b..  (3.2).  dependent  = jy a K  s l  s  < H  1 j k l s  s  *  5  +1,1  (4.42)  1 1  >  x*,  0  r.^  =  -  C b S  S  s  1 j s l  and I .  i s as  In t h e a b o v e model we have assumed t h e o r d e r f i l e t o on t h e l a y - u p a l t e r n a t i v e .  alternative,  If  it  t h e demand c o n s t r a i n t i n ( 4 . 4 2 )  i s independent is replaced  by  defined  be  of the  lay-up  89 z z P.., 1 j e I. 1 J I  fed.,  where,  1  is  the  o f whether  or n o t ,  quantity  plywood t y p e  i  i s t h e demand  for  of  the o r d e r f i l e  i s dependent  the  r e s u l t i n g subproblem  wood o f  j  do n o t  thicknesses are  on b o t h  reduce to  optimal  s o l u t i o n to the  The o p t i m a l  each s p e c i e s ,  illustrated  of  by t h e  Example 4 . 2 : P l y 22mm,  logs of  w i t h two  3 . 9 0 and 4 . 8 1  CH  = 30.00,  and r .  2, YF  0).  of 1  the  any  subproblems i n an have  one d e s i g n  alternative.  particular  plywood  t h e amount  c o s t and y i e l d  of  wood  factors, This  the  is  example.  respectively.  (F,  The  F,  H)  veneer  and ( F ,  Other c o e f f i c i e n t s o f Y^  = 0.0072,  type,  7  H,  for  F)  t h i c k n e s s e s are  mm and t h e d e s i g n a l t e r n a t i v e s  = 0.006067,  b.  r..-,  t y p e may  e a c h s p e c i e s and t h e o r d e r f i l e .  t h e model  are  (5,  are:  Wp = 8 7 . 7 2 9 , W  0,  0,  2.50, 2)  Cp = 3 5 . 0 0 , = 55.080  R  =8.0.  Suppose t h e o r d e r f i l e d.j = 1 0 0 0 . P112  relative  and  Consequently,  associated with  lay-up alternatives  3.10,  3,  alternative,  a plywood  for  ,  C o n s i d e r a PDM s u b p r o b l e m w i t h one p l y w o o d  c o r e and c e n t r e  (2,  alternative  Since b . .  more t h a n  alternative  their  following  face,  and  lay-up  and/or  t y p e w o u l d d e p e n d on t h e t r a d e - o f f  availability  x„).  l i n e a r subproblem,  d e s i g n and l a y - u p  required of  ...  simpler structure.  alternative  lay-up  *  s p e c i e s and t h e  any o t h e r  more t h a n one l a y - u p  (x,,  on t h e  produce one u n i t  and l a y - u p  *  are dependent  i .  i s a l i n e a r program,  s p e c i e s s required to  using design alternative  when t h e v e n e e r  product  1  Irrespective alternative  d.  Then t h e  = 5 0 0 . 0 and  P^21  i s independent  solution of = 500.0.  this  of  lay-up  subproblem  Suppose  is  alternative  and  P^^  = 0,  the o r d e r f i l e  is  = P-|22  dependent  on t h e ' 1 ayrup . w i t h i - c L y ; = ' 5 0 0 , to  60.000,  all  other  d.^ =  500 and  i s changed from  c o e f f i c i e n t s remaining unchanged.  s o l u t i o n o f t h e r e s u l t i n g s u b p r o b l e m w o u l d be P ^ ^ P  121  =  ^1.14  a n c  *  P-j22  =  i n d i c a t i n g that  a n d more t h a n o n e l a y - u p a l t e r n a t i v e  = 0,  Then P^^  55.080  the =  255.32,  more t h a n one d e s i g n  c a n be i n t h e  final  solution.  91 CHAPTER 5  5.1  Disjunctive  The integer formed of  Programming  PDM model (0-1)  formulated  mathematical  i n chapter  programming  integer  variables  6.. a n d x.  i n the constraints  enumeration  algorithm  resulted  that  and a h o s t  programs  with  attention  of other  logical  d i s j u n c t i v e programming,  be t r a n s f o r m e d constraints introduced  into  (Balas  (2.2)  -  variables  has been f o c u s s e d on t r e a t i n g programming  For e x a m p l e ,  t h e 0-1  o f t h e PDM model  of this  programming  l i n e a r program w i t h  integer  problems as  An o u t g r o w t h  approach  problem can  disjunctive  variables  f o r plywood  f e a s i b i l i t y c a n be o v e r c o m e  < bV  implicit  s o l u t i o n t o such a problem  i n w h i c h an i n t e g e r  in the formulation  b^" < z a . . . x .  and n o n -  linearities.  conditions.  (1979)).  ( 2 . 4 ) by t h e l o g i c a l  (2.1C))  The e f f i c i e n c y o f t h e  non-convex  an e q u i v a l e n t  t o l e r a n c e and d e s i g n  -  and t h e n o n - l i n e a r i t i e s i n t h e c o n s t r a i n t s  f u n c t i o n were r e d u c e d t o  In r e c e n t y e a r s , m u c h  linear  (2.5).  (2.1A)  region  due t o t h e p r e s e n c e  by b r a n c h i n g on x ^ , t h e i n t e g e r  were e x p l i c i t l y e v a l u a t e d  programming  (through  mixed  The f e a s i b l e  i s non-convex  i n seeking a global  from t h e f a c t  and o b j e c t i v e  is  problem.  by t h e c o n s t r a i n t s o f t h i s model  linearities  <$..  two i s a n o n . - r l i n e a r  5.. thickness  by r e p l a c i n g  constraints  condition  f o r a t l e a s t one j ,  for a l l i .  As a s e t o f c o n s t r a i n t s , t h i s c a n be e x p r e s s e d by  (5.1)  92  b  i *  "njk*k * » i  V  j Where t h e  symbol  these constraints w i t h two  plies  and t h e  veneer  V stands at  veneer  disjunction  l e a s t one j .  thicknesses.  design a l t e r n a t i v e s  t h i c k n e s s e s x-j and x , , . variables,  the  satisfying  constraint  (5.1)  figure  3.  r e g i o n would plywood  for  type.  implying that The  the  non-convexity  c a n be i l l u s t r a t e d by a s i m p l e e x a m p l e  continuous  in  —  1  <-' 5  k  s h o u l d be s a t i s f i e d f o r  type  ¥  are  Then,  (3,0)  or  (5.2)  and  (2,1)  is  o f one  type for  -  x2  be s t a r - s h a p e d  type  implied  plywood  two are  plane as  illustrated  i n c l u d e d , the  such s t a r - s h a p e d  by  has 3  the  x-j and x 2  i n t h e x-|  would  one p l y w o o d  i n t e r s e c t i o n of  plywood  assuming t h a t  feasible region  When more t h a n be t h e  Suppose the  constraint  region of  feasible each  2  93 The not  alternate  overcome  approach of  problems  associated with  by d i s j u n c t i o n .  However,  other  programming  non-convex  linear  programming  interesting the  problem.  the  problems  of  We s t u d y  (Balas  the  of  logical  the  variables  as i t  some i n t e g e r  as d i s j u n c t i v e  (1979)).  treating  integer  non-convexity  treatment  problems w i t h  properties  possibilities  e l i m i n a t i n g the  conditions In  this  is  implied  programs  some o f  or  have l e a d t o we  some  explore  PDM p r o b l e m as a d i s j u n c t i v e  implications of  and  programming  chapter,  does  i t s properties  programming on t h e  PDM  model.  5.2  The  PDM as a D i s j u n c t i v e  Expressing constraints  by  the  plywood  (5.2)  Programming  t h i c k n e s s t o l e r a n c e and d e s i g n  instead of  (2.2)  the  PDM p r o b l e m t o a d i s j u n c t i v e  the  f e a s i b l e design constraint  and t h e  non-linearities in  alternative  approach,  programming  problem.  Recall values  the  -  (2.4)  does not  programming  (2.8)  (2.5)  feasibility  itself  problem s i n c e  without  continue  the o v e r a l l  from c h a p t e r  Problem  8.. becomes  to  exist.  identifying  complicated  However,  PDM p r o b l e m c a n be c a s t as a  3 t h a t when v e n e e r  lead  thicknesses are  r e s u l t i n g subproblem i s a l i n e a r program a n d ,  in  from  an  disjunctive  assigned particular,  * an LMCK g i v e n set of the  (3.14).  ~k  = (x,, I  "k  x0,  c.  S i n c e <$.. and h e n c e 1J  thicknesses through  LMCK a r e d e p e n d e n t  -k  X  veneer  by  on t h e  "k  ...  x„)  "k  and P  (3.1)  set of veneer  are dependent  (3.2),  the c o n s t r a i n t s  thicknesses.  = ('P.,,.) i s an o p t i m a l  that  for  the of  Suppose  s o l u t i o n to  the  1J  from t h e  implicit  enumeration  algorithm. I  then  on  1  "k  I\  PDM p r o b l e m o b t a i n e d  and  I.  any o t h e r  set of  veneer  thicknesses X  I  I  = (x-p  Observe  i Xg. ... x ^ ) ,  '  X  *  f  for  '  X , the  I  *  and a s s o c i a t e d 6.. and corresponding  I.,  LMCK g i v e n  P  *  = (P..)  need not  be  feasible  by  i Max  E  E  ,  r..P.  .  ,  b. .P. . < W  s .t. E  E  i  j e  In.  1 J  1 J  ,  E  P. . > "  j e l ,  P  where,  b..  dependent  1  KO  V  = y E a . . . x. on t h e  d.  set of  i,  je  lj  -  Cb. . .  and r . . = r . veneer  Thus,  the  ( P . .)  t h i c k n e s s e s and i n an o p t i m u m  are  also  solution  * to  the  PDM p r o b l e m , w h a t m a t t e r s  LMCK a s s o c i a t e d w i t h the as  X  •k  "k  = (x-| ,  is that ie  ...  PDM p r o b l e m c a n be f o r m u l a t e d  x^).  (P..)  must  be f e a s i b l e f o r  From t h e s e  as a d i s j u n c t i v e  the  observations, programming  problem  follows. Suppose  distinct (2.1)  -  X  sets of (2.4).  can e q u i v a l e n t l y problem.  = (x-| , veneer Let  ...  xK),  for  various  values of  thicknesses feasible  Q be t h e  index  set of a l l  for  h,  represent  the design  such h.  The  be s p e c i f i e d by t h e : f o l l o w i n g d i s j u n c t i v e  constraints PDM model programming  95 ZQ  = Max  s.t. r  ~ ? • h iV?j < °  1  Z  i  E i  Z  E  j e  J e  I. T  lj  I J  I J  P <U  b  h  r  1 J  1 J  (5.3)  V < he Q  where. if  = (j|bf  9  disjunction for  distinct  at  feasible  -  example, on h .  by d e f i n i n g  h and a d j u s t i n g  the  that the  The  of  constraint  program  However, a variable  set of  in that  (5.3) set  coefficients  brackets  should  i n Q i s the  number  feasible  for  K f r o m a b o v e by n k=l  (.5.3) d e v i a t e s  (1 9 7 9 ) )  in  elements  thicknesses,  and i s bound  i n the  Balas  constraints  number o f  veneer  (2.4)  disjunctive  are dependent  h.  sets  results  format  (5.6)  l e a s t one  disjunction  for  (5.5)  V indicates  (2.1)  (see,  (5.4)  feasible}  X  constraints  Our  and X h  r. - Cb'..  IJ  hold  < bU  I ijk k  y  The  < za. . . x j  in  (5.3)  being  the  c a n be b r o u g h t i n t o  consisting of appropriately.  Z and  design  m(k).  The  non-convex.  from s t a n d a r d (5.3)  the  problems  variables the  (P^.)  standard  (P\ .) f o r  A unique  of  all  feature  of  96 (5.3)  i s that  disjunctive  5.2.1  Dual  Balas  the  some o f  i s the  problems  they would  Disjunctive  all  h while  be d e p e n d e n t  a 'dual'  for  i s not  to  be c o n f u s e d w i t h t h e  Interestingly,  though  the o r i g i n a l  'dual'  of  (5.3)  LL  = Min  U  -  the o r i g i n a l  l i n e a r programming  i s given  standard  h.  t h e d i s j u n c t i v e programming  (This  i s a convex  on  in  Program  between  it  the  same f o r  relations  'dual'''of the  of  so d e f i n e d .  LP).  (5.3)  (1979) d e f i n e s  and s t u d i e s  of  RH!S o f  programming  The  dual  the  p r o b l e m and classical  Following  the  dual  problem i s non-convex, problem.  problem  the  Balas,  by  U  s .t.  Wn  • r . .e  A  +  h  +  b  2 d.t^ >  •v- i , j G i :  t}>0  i j n  <  h  .h  n , t , i  where t h e  (5.7)  >I  he q  in  0  h ^  „  e > 0  •V- i  symbol A s t a n d s  brackets  for  conjunction  s h o u l d be s a t i s f i e d f o r  programming  problem.  by s o l v i n g ,  for  Observe t h a t  each h ,  the  problem  all  implying that  h G Q.  a s o l u t i o n to  (5.7) (5.7)  the  constraints  i s a convex c a n be  linear  obtained  97 Uh  = Min  -  Wn  h  zd.tj i  s .t. b. U  .n - t . ^  r.. ij  i  -V-  follows  LL u  (5.8)  Ar i  that  = Max [ U h ] h  (5.9)  i s p r e c i s e l y the dual  w i t h x^  e l ! (5.8)  n\ t}>0 and i t  i , j  = x^,  k = 1,  2,  ...  of  an LMCK s u b p r o b l e m o f t h e  K.  From t h e  in  c h a p t e r 3 and u s i n g c o m p l e m e n t a r y  of  (5.8)  i s given  the  LMCK d e r i v e d  slackness c o n d i t i o n s , the  solution  by  p.  i .  . h , h h t , = b. n -  h r.  l.  l  solution of  PDM p r o b l e m  / r  n \  (5.11)  where, b ^ / ^ M b ^ }  — ( 5 . 1 2 )  and h _ r  5.2.2  i . -  Max . j e*lf  Relation  J\ ,  r  C  < ij> r  between  the  =  r  i  "  , h C b  ,  i .  Disjunctive  and  h  =  {  ^Pij)  I  '(- lV P  s a t 1 s f 1 e s  ,~v  — ( 5 . 1 3 )  P r o g r a m and i t s  Dual  Suppose P  c  constraints of  (5.3)}  98 h  h , t p  = {(n Assume  that the  Regularity  Condition  then  there  exists  the  relation  Then,  characterized proof,  see  Theorem 5 . 1 dual  (5.7)  following  h  (5.3)  Q such t h a t the  condition  is  theorem  satisfied  for  h>.  holds.  feasible  and  f  disjunctive  following  (.5.7) are  and  (5.7)  is  infeasible,  = <j>.  program  and i t s  w h i c h we s t a t e  dual  without  can  proof  be (for  (1979)).  Assume  the  problems  b)  One o f  the  i n f e a s i b l e or  its  If  of  that the  regularity  disjunctive  condition.  program  (5.3)  Then e x a c t l y  and  its  one o f  the  solution  and  situations.hold:  Both  solution  (Balas):  between  satisfy  a)  We assume  | constraints regularity  (Balas):  two  e )  following  e  by t h e  Balas  h  i n the exists  traditional  are  feasible;  problems  is  infeasible;  has no f i n i t e  sequel  that  and t h a t  Zg  dual  both = Ug.  i s given  by  e a c h has an o p t i m a l  the  other  one  is  either  optimum. problems Now,  are  since  feasible, (5.7)  t h a t an  is a linear  optimum program  99  n  En  = Max  n  h  u  s .t. " *  n  •  E  A  h e Q  and  E x h G Q  The d u a l  h  T  h U i j  hrij  +  f l  d.X  ij  <  0  <0  (5.14)  i  J  h  Z  h  n , X  j»  h  >  >  n  i , j G  V  0  = 1.  of the 'dual'  block-angular  as g i v e n  structure,  by ( 5 . 1 4 )  w i t h one b l o c k o f c o n s t r a i n t s  l i n k e d through  t h e common c o n s t r a i n t  duality  of linear  theory  f e a s i b l e , U Q = r\ . Q  i s a l i n e a r program w i t h  x  =1.  h G Q t h a t when ( 5 . 7 ) u  programs  Observe t h a t  E  if  It  is  ( 5 . 3 ) then feasible  the corresponding  for  follows  and ( 5 . 1 4 )  P.., f o r some H e Q i s  H for  f o r each  h, from  are  feasible  H  W  s o l u t i o n w i t h y^. = P.. a n d x  = 1  (5.14).  In t h e b l o c k - a n g u l a r  structure  of the constraints  of (5.14), W  and d . o c c u r i n a l l t h e b l o c k s once e a c h f o r each h .  x^ i n  acts  thicknesses  as a s c a l i n g f a c t o r  Since the factors  a  for different  s e t s o f veneer  t h e y s c a l e a r e W and d . , b o t h  independent  (5.14)  of h,  X* 1 . it  h' 1 f o l l o w s t h a t ( 5 . 1 4 ) w i l l have an o p t i m a l s o l u t i o n i n w h i c h x = 1 for 1 ' h ; f some h G Q a n d \ = 0 f o r a l l h e Q \ " { h } . Exception to t h i s s o l u t i o n i s t h e p o s s i b i l i t y o f m u l t i p l e s o l u t i o n t o ( 5 . 1 4 ) i n w h i c h c a s e more  1 00 than  one s e t o f veneer  value. that to  Barring  situations  an o p t i m a l  of multiple  solution of (5.14)  our o r i g i n a l  this  t h i c k n e s s e s c a n y i e l d t h e same o b j e c t i v e  disjunctive  situation  RHS o f many d i s j u n c t i v e  the  'dual'  and t r a n s f o r m e d offer  reasons.  First,  be c o m p u t e d  Third,  t h e number  be s u b s t a n t i a l l y 3 , with  linear  for practical  t h e number  should  each b l o c k  (5.13).  a r e dependent may o n l y  c a n be f o r m u l a t e d  easier solutions  Secondly,  h,  be an o p t i m a l  problem  programs  t o an e q u i v a l e n t  large.  chapter  follows  solution  programs  since  The d u a l  of  be a r e l a x a t i o n  as a d i s j u n c t i v e  program  of  situations  o f p o s s i b l e elements  o f rows large.  K = 4,  having  f o r j e I.  there  anywhere  f o r t h e plywood  14 t o 170  case of a d i s j u n c t i v e properties.  enumeration  very  h b ^ . and  h r.^  by ( 5 . 1 4 )  would  mill  of  variables  the i m p l i c i t  i t shows  several  data  c a n be a maximum o f 1 7 7 1 4 7 b l o c k s between  does  s h o u l d be i d e n t i f i e d .  and c o l u m n s o f t h e LP g i v e n For e x a m p l e ,  it  in Q i s generally  advantages  over  due.to  f o r each such f e a s i b l e s e t o f t h i c k n e s s , h h e x p l i c i t l y and P . .  program  as i n ( 5 . 1 4 ) ,  program a p p r o a c h does n o t t h e r e f o r e  its  it  (Incidentally,  on h .  The d i s j u n c t i v e  Nevertheless,  (5.14),  problem).  Though t h e PDM model  not  also  to  for a l l disjunctive  programs  f o r such d i s j u n c t i v e  the o r i g i n a l  will  programming  n e e d n o t h o l d good  the  solutions  function  offer  of  and 14 any  algorithm of chapter  constraints.  computational 3.  t h a t t h e PDM p r o b l e m c a n be c a s t a s a s p e c i a l programming  problem and g i v e s  an i n s i g h t  into  101 CHAPTER 6  6.1  The  Optimum Number o f  The  PDM model  Veneer  presented  Thicknesses  i n the  e a r l i e r chapters determines  the  v e n e e r t h i c k n e s s e s , a s s o c i a t e d ' p l y w o o d d e s i g n s and t h e  corresponding  p r o d u c t mix  The maximum  revenue  for  a g i v e n number o f v e n e e r t h i c k n e s s e s .  so o b t a i n e d  and d o e s n o t Generally, operational  i s dependent  include the  the lower the  on t h e  setup or overhead number o f  costs associated with  record-keeping  but  number o f  costs associated with  peeling, drying,  result  i n h i g h e r ••revenue. For  benefits  6.1.1  are determined  would  to the at  any p a r t i c u l a r m i l l  by t h e t r a d e - o f f  gap,  appropriate  vertical thickness.  This  t h e c l i p p e r where t h e v e n e e r s  sorted peel  t h i c k n e s s e s , the  h i g h e r would  the  form  of  number  of  c o s t s but t h e y the  also overall  two.  the  peeling lathe,  there  l a t h e s e t t i n g s s u c h as  i s f o l l o w e d by c h a n g e s  the veneer. be t h e  is  gap a n d p r e s s u r e b a r c o m p r e s s i o n  are c l i p p e d to  by t h e t h i c k n e s s and s i z e o f  it.  Thicknesses  change t h e  gap o r e x i t  in the  therefore,  K,  a s s e m b l y and  higher the  between t h e  t h i c k n e s s i s changed a t  be a s e t t i n g t i m e r e q u i r e d t o  horizontal  The  i s the operational  S e t u p C o s t s A s s o c i a t e d w i t h More P e e l  Everytime a veneer  the  higher  lower  storage,  i n h i g h e r wood l o s s e s  e x c e s s t h i c k n e s s i n plywood and l o w e r r e v e n u e . t h i c k n e s s e s , the  thicknesses,  veneer t h i c k n e s s e s , the  they also r e s u l t  veneer  peel  net  different The  setup costs  in  sorting  sizes  and  h i g h e r t h e number i n p e e l i n g and  of  102 c l i p p i n g of  veneers.  b) PryiD9_2f_yD?§!T§ e  When v e n e e r s are to time  are d r i e d to  be a d j u s t e d  to  in conventional  and t h e  initial  reduce the moisture  proper s e t t i n g s of dryers.  moisture  thickness  content  setup costs  c)  of  species, sizes  quality  In  are  as w e l l  (full,  factors).  stored  both  as the dry  the  The  in the  in  thicknesses imply  times s i n c e veneers  half,  s t r i p s and  storage  higher  are  classified  fishtails)  surface  smoothness,  and h a n d l i n g c o s t s  g r e e n end  end ( a f t e r d r y i n g  g l u i n g and a s s e m b l y o f handling f a c i l i t i e s .  more d e s i g n a l t e r n a t i v e s .  e)  loss  (after  but  double  p e e l i n g but  before  before  assembly).  Assembling_Veneers  and m a t e r i a l  higher  be a t i m e  each time a change  c h a r a c t e r i s t i c s ( b a s e d on k n o t s ,  l a t h e c h e c k s and o t h e r  d)  speed/  ing_and_Storage_of_Veneers  basis of  since veneers  and d r y i n g  There would  more v e n e e r  problem i s m u l t i p l i e d several  and s u r f a c e  drying)  Again,  driers  d e p e n d on s p e c i e s , t h i c k n e s s  veneers. setup,  the  drying.  Hand!  This on t h e  in  i s made.  temperature  These s e t t i n g s  associated with the changes.in dryer veneer  the  content,  Changes  s c h e d u l i n g c o s t s and h i g h e r  veneers,  higher  More v e n e e r  K i m p l i e s more  thicknesses lead  in design alternatives setup time  between  result  space  to in  batches.  Recordkeeping  I n c r e a s e d number o f  peel  thicknesses result  in  increased costs  of  103 record-keeping. different  Routine  s p e c i e s and t y p e s  a s s i g n i n g veneers different types  of  s c h e d u l i n g problems  dryers  of  of  r e l a t e d to  logs to d i f f e r e n t  different  species, size,  veneer  of  thicknesses,  g r a d e and t h i c k n e s s  and a s s e m b l y o f a c o m b i n a t i o n o f  plywood  conversion  these  into  to  several  become more c o m p l i c a t e d w i t h t h e a d d i t i o n o f  each peel  thickness. All perhaps  6.1.2  the  above f a c t o r s  exponentially,  Benefits  plywood  form o f is  more p e e l  type.  i n c r e a s e i n the  p o s s i b l e net  number o f  they r e s u l t  i n plywood  veneer  revenue  setup costs number o f  increase,  peel  thicknesses.  Thicknesses  thicknesses result  Consequently,  excess thickness  t h e maximum  when t h e  with  A s s o c i a t e d w i t h More P e e l  Generally, per  i n d i c a t e that the  i n more d e s i g n  i n l o w e r wood l o s s  and h i g h e r r e v e n u e s . obtainable  t h i c k n e s s e s i s K.  alternatives  from t h e  That  in  Suppose  the Z^.  PDM model  is,  K Z „ = Max s E r . P . . 1 1 J i j  Cy  E  k  to  the  applies  to  single species model;  is  c o n s t r a i n t s as d e s c r i b e d i n c h a p t e r t w o .  objective  the o p t i m i z a t i o n f e a s i b l e and  > 1  K  model  lv  (6.1)  X.V  k=l  subject  appropriate  usual  -  function is  If  value  more t h a n one s p e c i e s s h o u l d be u s e d ] .  infeasible for  is f i n i t e ,  for  it  all  some K.  s a t i s f i e s the  K  Let  When t h e  [(6.1)  is"used,.the Z^  =  if  problem  relation  (6.2)  104 (6.2)  follows  from t h e f a c t t h a t  a t l e a s t as many d e s i g n o p t i o n s extra  peel t h i c k n e s s .  6.1.3  U p p e r Bound on  Though t h e maximum n e t number o f be a n y  peel  with  K + 1 t h i c k n e s s e s , one w i l l  as w i t h  revenue  K p l u s t h e added b e n e f i t  increases with  in the value of  c a n be p o s t u l a t e d t h a t  rate of  as K i n c r e a s e s , u l t i m a t e l y v a n i s h i n g a t difficult  to determine  such a value o f  -Z-Q, f o r t h e o b j e c t i v e f u n c t i o n v a l u e f o r t h e s i n g l e s p e c i e s model o f Recall  from chapter 3 t h a t  *  *  v a l u e s x-|, Further,  x2,  K,  the  i n c r e a s e i n Z^  some v a l u e o f K,  K.  In  K  K,  b. < 1  •••  xK,  for  i n an o p t i m a l  c a n be c a l c u l a t e d  when v e n e e r t h i c k n e s s e s a r e a s s i g n e d  s o l u t i o n to t h i s only for  that  . . . K, i t m u s t s a t i s f y K * U z a . . 1.x. < b for IJK  each i ,  independent b^  of  upperbound,  t h e r e s u l t i n g s u b p r o b l e m i s a LMCK p r o b l e m .  K  j  LMCK p r o b l e m , f o r  each i ,  f o r w h i c h b. . i s minimum.  K.  * = y J a i j . k x k >y  at  l e a s t one j  = x^  P.. Observe  * , x^  and f o r a l l  e  T^  ,  i  1  K  Thus,  the  is  PDM p r o b l e m .  t h a t when v e n e e r t h i c k n e s s e s a r e a s s i g n e d v a l u e s x k 2,  it  Though i t  *  I  fact,  *  w o u l d be n o n - n e g a t i v e  k = 1,  not  decreases  the exact value of  Z , for a l l  the  t h e r e would  the o b j e c t i v e f u n c t i o n .  the marginal  of  i n c r e a s e ! i n the  t h i c k n e s s e s , beyond a c e r t a i n v a l u e o f  improvement  have  z a...x. Then, L b,  *  it  I  i s bound f r o m b e l o w by b . , follows that,  and r . j  = r.  -  which  is  f o r each i ,  Cb.... < r..  -  L Cyb...  Consequently,  105 for  any  i  and j ,  {r../b..}is  "10  r„ -  From ( 6 . 3 ) it  Cyb*"  M a v  and t h e  follows that  bound f r o m a b o v e  . ^ F . d .  -  the  LMCK p r o b l e m a s d e r i v e d i n c h a p t e r  u p p e r b o u n d on  W-  V  fr.  s o l u t i o n to  the  +  E  correction the  2^, f o r  i&LzLAj r  |  Cybj:),  factor,  CybL  b.  C^, i s u s e d f o r  bound  and ( 6 . 4 )  since  implies that  permrssible thickness 6.1.4  in  Z^ f o r  En  k=  1  IJK  i j K K  {FT / b . } .  x  plywood types  i  Intuitively, for  all  i  the  upper-  i s meaningful  are assembled to  i s no l o s s  i n the  form o f  the  minimum  excess  Efficiency  K.  Then t h e  x 100 0  a  L b.  case  K  revenue  for  a set of  and p l y w o o d d e s i g n s and ZQ i s t h e u p p e r b o u n d  =  If  1  ZJJ r e p r e s e n t s t h e maximum n e t  all  A  a  plywood.  Design  thicknesses  M  K * z a . - . x . and i n t h a t  by C_  t h i c k n e s s and t h e r e  Suppose  of  all  K  =  s h r i n k a g e and c o m p r e s s i o n , t h e n  * L b e i n g o b t a i n e d when C , Ea..|X, = b . it  by  — ( 6 . 4 )  i s r e p l a c e d by (bir / C f ) .  T  i s given  p  = yb\ and (F^ /b^)  plywood t h i c k n e s s i s g i v e n  (6.3)  K,  /  t in  any  3,  b.d.X  J . w h e r e , 77 = ( r . -  by  "I J  on t h e  veneer value  ratio  (6.5)  106 can of  be t e r m e d peel  the design e f f i c i e n c y ,  t h i c k n e s s e s and p l y w o o d  indicate  better  of  ZQ i s d e p e n d e n t  K but  the  peel  designs.  the  how good a s e t o f  extent  of  Values  t h i c k n e s s e s and d e s i g n s . on i t ,  further  veneer  K.  rating of  it  any,  this  set  independent ignoring  gives  t h i c k n e s s e s and plywood if  is  efficiency  Nevertheless,  improvements,  for  Eg c l o s e r t o 1 0 0 %  S i n c e Zg  Eg m e a s u r e s t h e  setup costs associated with  as to  a percentage  an  indication  designs are  from h i g h e r  number o f  and peel  thicknesses.  Determining  6.1.5  with  any  particular m i l l ,  all  the  factors is  obtainable  K.  given  by t h a t  Z^  bound The  K.  K for  total  set  up c o s t  v  be t h e  c o r r e s p o n d i n g maximum Then  = Z^  which the  from above of  by the  net  of  benefit  Zg w h i l e 0^, i s upper  veneer  bound  -  associated  number o f  net  veneer  revenue  0^ g i v e s t h e  net  benefit  thicknesses for  the m i l l  is  R^ i s maximum.  Observe  that  unrestricted.  Zg,  the  thicknesses is  design e f f i c i e n c y  i l l u s t r a t e d through  and the  example.  Example 6 . 1 :  plywood  l  PDM m o d e l .  t h e o p t i m u m number o f  this  0„ be t h e  The o p t i m u m number  derivation  following  let  Thicknesses  d e s c r i b e d i n s e c t i o n 6 . 1 . 1 , when t h e  Let  from the  associated with  in  Optimum Number o f  For  thicknesses  is  the  The  example.  plywood The  plywood  thicknesses (b!r),  as l i s t e d i n a p p e n d i x  mill  II.  the For  data types  used i n c h a p t e r (N  revenue these  = 1 2 ) , the (r..)  considered  lower t o l e r a n c e  and t h e  plywood  3 is again  types,  orderfile b^,  r.  (d.)  and  on are ( / b . )  107 r e q u i r e d t o d e t e r m i n e ZQ w e r e c o m p u t e d u s i n g ( 6 . 4 ) a n d a r e p r e s e n t e d in  Table  11.  Plywood Type p l i e s - Thick  7.5  .04518  2.7188  60.1791  3  -  9.5  .05809  2.7670  47.6361  5  -  12.5  .07745  3.3893  43.7625  5  -  15.5  .09681  4.2117  43.5043  7  -  18.5  .11617  4.8340  41 . 6 1 0 5  7  -  20.5  .12908  5.2822  40.9219  7  22.5  .14199  5.5304  38.9499  9  -  23.5  .14844  6.1045  41 . 1 2 4 0  9  -  25.5  .16135  6.4528  39.9923  9  -  27.5  .17426  7.0010  40.1759  9  -  28.5  .18071  7.2751  40.2579  9  -  30.5  .19362  7.8233  40.4054  = p = 1.  1  Computations  =  a  =  x  Then,  f o r t h e U p p e r b o u n d Z.  that  {FT/b.}  Er.d. + l f l  Thus,  M  _  N  (r./b.)  _  From t a b l e 11 , we h a v e  Z  r. l  3  Tab! e l l :  so t h a t  b. ui  = 60.1791  f r o m ( 6 . 4 ) we h a v e  / 300000 -  E b.d.  ifl— "* \  2.7188  ~04518  $14,679,760  f o r t h e plywood m i l l  data o f chapter 3 (appendix  net revenue can never exceed $ 1 4 , 6 7 9 , 7 6 0 , veneer t h i c k n e s s e s used.  I I ) , t h e maximum  i r r e s p e c t i v e o f t h e number o f  108 Computer model  codes o f t h e i m p l i c i t  were u s e d w i t h  from 1 t o 5 .  the m i l l ' s  enumeration  data to determine  The maximum n e t r e v e n u e  Z^,  i n t h e form o f excess t h i c k n e s s d e r i v e d in  Table  the  12.  For comparison p u r p o s e s ,  e x i s t i n g s e t o f veneer  presented. computed  for values of K  and t h e q u a n t i t y  f r o m t h e PDM model  o f wood  Maximum Revenue ( $ ) ,  i x i s t i n g 13,416,694 (4)  loss  are given  the corresponding values  for  t h i c k n e s s e s and a s s o c i a t e d d e s i g n s a r e a l s o  u s i n g t h e v a l u e o f ZQ = 1 4 , 6 7 9 , 7 6 0 o b t a i n e d  h  *  lv  The d e s i g n e f f i c i e n c i e s f o r e a c h o n e o f t h e s e s e t s  i n column 4 o f t a b l e  K  a l g o r i t h m o f t h e PDM  above  were  and a r e p r e s e n t e d  12.  Wood L o s s i n Excess Thickness (cu. mtr.)  7,944.7  Design Efficiency (%), ED  Estimated Setup C o s t *  Net Benefit  ($),6 K  91 . 4 0  ($)  3,714,770  RK  9,701 , 9 2 4  1  Infeasible  1 ,118,870  2  Infeasible  1 , 6 6 9 , 1 60  3  13,930,670  3,612.1  94.90  2,490,090  11 , 4 4 0 , 5 8 0  4  14,337,370  1 ,647.4  97.67  3,714,770  10,622,600  5  14,562,530  558.3  99.20  5,541 , 7 9 0  9,020,740  Estimated Table  12:  For products  from 0 K = 750,000exV Optimum Number o f V e n e e r  Thicknesses  and Design  K = 1 a n d 2 , t h e p r o b l e m was i n f e a s i b l e i m p l y i n g t h a t o f the m i l l  cannot  o n e o r two p e e l , t h i c k n e s s e s .  be a s s e m b l e d w i t h i n  IV).  This  For K = 3 , t h e o p t i m a l  i s h i g h e r than  the  specifications using  w e r e 2 . 6 , 3 . 2 a n d 4.4mm a n d t h e maximum n e t r e v e n u e (appendix  Efficiency  veneer  thicknesses  was $ 1 3 , 9 3 0 , 6 7 0  the corresponding value  for the  109 existing only  set of  better  in the  four  t h i c k n e s s e s by 3 . 8 3 % .  designs existed  for  the m i l l  setup costs associated with the  possible.  When K = 4 ,  t h e optimum v e n e e r corresponding net  (appendix  if  thicknesses designs higher  Thus,  for  any r e a s o n ,  that  for  When K = 5 ,  were  the m i l l ,  wood l o s s  revenue  of  set of  $225,160 over  PDM m o d e l .  with  additional  was  $920,676  revenue,  However,  this  available  the m i l l .  d a t a on s e t u p c o s t s were It the  was p o s t u l a t e d t h a t number o f  peel  Table  12.  The  under  l a s t column o f  plywood  presented. further  table  types  The  was 5 5 8 . 3 c u b i c in  additional  thicknesses  obtained  s h o u l d be  fifth  peel  number o f  given R  design  The  K  =  thickness.  some  K  hypothetical  related  The  " \>  thicknesses  revenues. with  estimated  i n column 5 Z  compared  peel  e x i s t i n g d a t a on  gives  of  be s i g n i f i c a n t .  comparison purposes,  are  Since  from  veneer  the  3.7  benefits  i n plywood  assumption  this  3.2,  12  s e t u p c o s t s were e x p o n e n t i a l l y  this  2.7,  the  t h i c k n e s s e s by 0 K = 7 5 0 , 0 0 0 e x p ° ' ^ K . .  setup costs obtained  6.86%  for  are not  four  used along w i t h  the  to  $14,562,530.  i n c r e a s e i n revenue  For  plywood  was  setup costs a s s o c i a t e d with the  for  peel  Z^,  D a t a on s e t u p c o s t s a s s o c i a t e d w i t h d i f f e r e n t were n o t  (equal  thicknesses results  t h e one f o r  3.1,  veneer  t h i c k n e s s e s were 2 . 4 ,  t h e model  veneer  savings  $14,337,370  t h i c k n e s s e s and  t h i c k n e s s e s would not  five  further  exists.  form o f e x c e s s t h i c k n e s s  from t h e the  peel  not  t h i c k n e s s were a l s o  keeping four  was 9 9 . 2 0 % i n d i c a t i n g t h a t  number o f  This  peel  revenue  34 f e a s i b l e d e s i g n a l t e r n a t i v e s  i n the  of  set)  veneer  d e t a i l e d r e s u l t s of  higher  metres.  current  a s s o c i a t e d net  efficiency attained any  the  substantial  set of  revenue  indicates that  t h i c k n e s s e s were 2 . 5 ,  intends  a better  the optimal  and 4 . 8 mm and t h e there  the m i l l  resulting in additional than  but  fourth  3 . 9 and 4 . 8 mm and t h e III).  This  t n e  of estimated  110 net at  benefits.  For  this set of  K = 3 implying that  data,  t h e maximum n e t  t h e o p t i m u m number o f  benefit  veneer  occurs  thicknesses  is  two,  treated  three.  6.2  Alternate  In t h e x-j t o  quality  Veneer  formulation  be t h e  from t h e  Face  of the  face veneer  fact  that  Thickness  there  sheet with  sheet.  The  in chapter  t h i c k n e s s by c o n v e n t i o n .  a face veneer  face veneer  than  full  thin  veneer  rework  size  having t h i n veneers veneer  rather  are e a s i e r with this,  for  have t h e for  the width  serious  of  holes,  than w i t h  construction  (COFI  the  length of the  l o g s m u s t be p e e l e d t o  for  core veneers  they  manufacturing  the  face veneers  full  implying  length of  c a n be i n l e n g t h s  or  splits  In a d d i t i o n  panel  since a  repairs  p i t c h pockets or  (1978)),  strips  log for  Further,  surface  size  i s advantageous  thick veneers.  the  to  must that the  relative  to  panel.  convention  that  l i m i t a t i o n of  enumeration used f o r  the  follows  plies  from  From  from a g i v e n  a t h i c k veneer.  grain d i r e c t i o n along the  while that  The  for  be a f u l l  only.  as f a c e v e n e e r  such as knot  t h i n veneers  standard  face veneers  panel  than that  inner  s i z e d veneers  plies  s h e e t s c a n be o b t a i n e d  c a u s e d by f a c t o r s  convention  t h o s e on t h e  small  a n d f i s h t a i l s c a n be u s e d as c o r e o r c e n t r e  more  This  should normally  b l e m i s h - f r e e s u r f a c e whereas  considerations,  we had  a r e more r e s t r i c t i o n s on t h e t h i c k n e s s and  c h a r a c t e r i s t i c s of  i n a plywood  PDM model  the  algorithm to  face-veneer,  x-j  i s the  face veneer  c a p a b i l i t i e s of solve i t .  s u b j e c t to  If  a l l of  the  thickness PDM model  i s not or the  a implicit  more t h a n o n e t h i c k n e s s c a n them m e e t i n g t h e  relevant  be  Ill specifications, framework  of  such a s i t u a t i o n  the  PDM m o d e l .  a few m o d i f i c a t i o n s be t h e  same.  Recall Suppose  in that  t h a t most  c a n be two  associated with i n some o f  being d u p l i c a t e d for  a five-ply  are  generated  represented (a..ji, it  a  ij2'  by  a  chapter  x^. t o in  3,  be t h e  (3.41)  x-j  and a.. ^  a.^  with  that would  K = 3,  (2,  3,  for  (3,  be 6 ,  15,  3 - 6 .  15  2,  0)  resultant and 18  leads to  for  For  design  and  4,  (2, such  number plywood  few  the  in tables  in table  2  thicknesses.  However,  already  Eliminating  K = 3 the  it  may be e x c l u d e d .  and a ^  virtually  by i n t e r c h a n g i n g  in tables  0),  veneer  then  when a d d i t i o n a l  a_..p  requires  paragraphs.  four  and x 2 ,  would  the  this 3 - 6 .  example,  alternatives the  2,  1)  designs for  duplications, of  design  made  with  respectively. PDM model  some o f  the  face veneer  and ( 3 . 4 2 )  or  w h i c h c a n be o b t a i n e d  would be r e p e a t e d .  7 and 9 p l i e s  following  use t h r e e  face veneers,  vectors  In s o l v i n g t h e of  mills  interchanging  ij3^  (n^)  i n the  within  algorithm  its effectiveness  the design a l t e r n a t i v e s  plywood  c a n be v e r i f i e d  5,  enumeration  and s u c h a l t e r n a t i v e s  by t h e  alternatives 3,  implicit  case but  plywood  more d e s i g n a l t e r n a t i v e s  can r e s u l t  be i n c o r p o r a t e d  We i l l u s t r a t e t h e s e  there  columns  The  can a l s o  using the  branching thickness  would  change,  implicit  tests for  enumeration  algorithm  need m o d i f i c a t i o n .  Treating  f = 1,  2,  respectively  the to  bounds  derived  Where,  Tp  i s t h e maximum  Following the  arguments  branching test  x,  <  x  p e r m i s s i b l e face veneer  s i m i l a r to (3.40)  that  would  in chapter  3,  thickness.  we c a n show  be r e p l a c e d by t h e  that  f o l l o w i n g two  tests:  (6.8)  2  and for  3 "< s ,  With the m o d i f i c a t i o n s of  chapter  of  the  3 (appendix  PDM model  results  obtained  with  both 2 . 5  be o b s e r v e d  alternate  as t h e 12.5 other  types  of  it  should  results  are  the  implicit  are  the  plywood  k e e p i n g two  types  circumstances.  designs  3.1  obtained this  veneer  It  can mm,  mm o n l y  mm and 5  face veneers.  ply  For  all  h a v i n g e i t h e r 2 . 5 mm o r  3,  the  for  figure  solution of i s used f o r  t h i s model does not  3.1  was  the  a l g o r i t h m to  solve  it  that  each  for  plywood  $14,387,280.  include the  the  mm  LMCK  t h i c k n e s s e s as f a c e v e n e e r s . demonstrate  V.  and 3 . 8 mm  3 ply 9.5  3 ply 7.5  The  these designs are v a r i a b l e s  in chapter  here o n l y to  enumeration  K = 3.  i n appendix  be a s s e m b l e d u s i n g 3 . 1  plywood  be n o t e d t h a t  types  data  effectiveness  presented  c a s e were 2 . 5 ,  However, a l l  revenue  the  mill  thicknesses for  o n l y one d e s i g n , a l t e r n a t i v e  presented  plywood  b e i n g u s e d as f a c e v e n e e r s .  alternate  existed.  The maximum n e t  associated with  the  and t h a t  plywood,  w o u l d be s u c h t h a t  However,  in t h i s  3 0 . 5 mm must  LMCK p r o b l e m and as shown  type.  the  be a s s e m b l e d u s i n g 2 . 5 mm as t h e  as f a c e v e n e e r s the  m o d i f i e d model  from t h e s e r e s u l t s t h a t  face veneer  mm must  face veneer  mm v e n e e r s  9 p l y 2 8 . 5 mm and 9 p l y  (6.9)  described above,  thicknesses  and 3 . 1  K  was u s e d t o d e m o n s t r a t e  from t h i s  The o p t i m u m v e n e e r with  II)  < s„ <  PDM model  c a n be u s e d u n d e r  costs These and varying  113 CHAPTER 7  7.1  Conclusions  In t h i s w h i c h was  dissertation, a real-world  hitherto  not  c o n s i d e r e d due t o  and s o l v e d u s i n g a q u a n t i t a t i v e stems  from the  manufacture methods  of  resources for  is lost  The plywood  In  and f l u c t u a t i n g utilization  i t was e v e r problem o f  has been  of  i s developed.  problem so f o r m u l a t e d  written  exploit  and d a t a  p r a c t i c a l i t y of Variations  the  all  i s shown t h a t  market  raw m a t e r i a l  formulated  the  added  problem  from better  supply of  timber  structure  the  is all  The the  veneer  t h e more  need greater  thicknesses, associated the  net  as a m a t h e m a t i c a l  for  a global  efficiency, of structures  integer  for  this  (0-1)  A computer to  veneer  programming  algorithm arises  i s analyzed  model.  a set of  s o l u t i o n u s i n g an  in the model.  from a plywood m i l l  revenue  programming  des'igns f o r  The n o n - l i n e a r m i x e d  implicit from code  demonstrate  the  model.  and e x t e n s i o n s  the  value  of  n o n - a v a i l a b i l i t y of  f e a s i b l e plywood  have been c o n s i d e r e d and t h e i r It  with  mix which maximizes  i s solved  algorithm.  a b i l i t y to  the  formulated  thicknesses  is  and c o m p e t i t i v e  has been  importance  dwindling  determining a set of  evaluating  its  The  significance  before.  A method o f  enumeration  the  t h e wake o f  d e s i g n s and p r o d u c t  a plywood m i l l  approach.  i m p l i c i t l y due t o  practical  i t s complexity  that a processed material  evaluation.  efficient  now t h a n  fact  problem o f  of  under d i f f e r e n t  s o l u t i o n procedures  PDM p r o b l e m  w h i c h c a n be c a s t as a s p e c i a l  t h e model  i s a non-convex  case of  circumstances  have been  programming  a disjunctive  program.  analysed. problem Following  114 the development procedure,  of  ethods  the mathematical  factor  list  PDM model designs veneer  veneer  implication of  plywood m i l l  of  plywood  derive  and t h e sizes  provides  alternatives  LP m o d e l s ,  the  choice of  PDM model  potential  to  may be u s e d .  e x i s t i n g ones. existed  veneers  abridged  about.  the  plywood  model  to  the  model of  decisions product  c a n , at  u s e d by a  in generating  can  if  absence  d e s i g n s and t h e PDM model  thicknesses  verify  way,  In t h e  grades,  PDM model  mix.  worst, mill.  a solution  i n s i g n i f i c a n t when c o m p a r e d w i t h  can b r i n g  in  expensive,  The v e n e e r  and d e s i g n s c u r r e n t l y  yield  s u c h as l o g  c a n be u s e d as a b a s i s f o r  t i m e and money i n v o l v e d  it  factors  In t h i s  before.  are c o n s i d e r e d , the  is relatively  benefits  the  A  logs,  be c o m p u t a t i o n a l l y  t h i c k n e s s e s , plywood  factors  and  developed.  cost of  factors  as a s o l u t i o n o f t h i s  PDM model  set of  The c o m p u t a t i o n a l the  prove  which never  veneer  relevant the  i n c l u s i o n of  PDM model  than  are  thicknesses, associated  i n a d e t a i l e d l i n e a r programming  indeed better  end up w i t h  grades  the  veneer  If  designs obtained  they are  When a l l  a mill  o r d e r f i l e and s u c h o t h e r  the optimal  and p l y w o o d  solution  d i s s e r t a t i o n are s t r a i g h t - f o r w a r d .  produce mix.  be u s e d as i n p u t  on t h e  this  types,  an a b r i d g e d v e r s i o n o f  mill  thicknesses for  and i t s  plywood d e s i g n s  c a n u s e i t s d a t a on l o g a v a i l a b i l i t y ,  to  and p l y w o o d  model  to determine the e f f i c i e n c y of  t h e o p t i m u m number o f The  programming  the  to  115 BIBLIOGRAPHY B a i l e y , G.R. 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A L i n e a r P r o g r a m m i n g Model f o r t h e A l l o c a t i o n o f Logs f o r t h e M a n u f a c t u r e o f an Optimum P l y w o o d M i x . D.B.A. T h e s i s , U n i v e r s i t y o f Oregon, Eugene, Oregon. Ramsing, K.D. 1968. Forest Products  L i n e a r Programming f o r J o u r n a l , ! 8 (.4), 9 8 - 1 0 1 .  Plywood Mix  Problem.  R a i z a d a , H . C . and J . C . N a u t i y a l . 1974. An I n p u t - O u t p u t Model o f O n t a r i o Forest Based I n d u s t r i e s . Canadian Journal of Forest Research 4 ( 3 ) , 372-380.  119 Reimer , D.R. 1969. D e v e l o p m e n t o f a L e o n t i e f I n p u t - O u t p u t Model a n d i t s A p p l i c a t i o n i n t h e A n a l y s i s o f t h e Economic Impact o f a New F o r e s t I n d u s t r y . Ph.D D i s s e r t a t i o n , Purdue U n i v e r s i t y , Lafayette, Indiana. R e s c h , H. a n d H . L . S c h e u r m a n . 1977. Computer S i m u l a t i o n o f t h e Softwood Veneer Drying O p e r a t i o n . Forest Products J o u r n a l , 27 ( 2 ) , 3 2 - 3 5 . S i t t e r , R.M. 53 ( 5 ) , Taha,  H. New  1969. 31-35.  1971. York.  LP f o r B e t t e r  Operations  Profit  Research.  Levels.  BC  Lumberman,  MacMillan Publishing  Company,  T o b i n , L.R. and J . S . B e t h e l . 1969. Veneer R e c o v e r y P r e d i c t i o n and A n a l y s i s Through Computer S i m u l a t i o n . Wood a n d F i b e r , 1 ( 2 ) , 97-107. T r u e m p e r , K. 1976. An E f f i c i e n t Networks, 6 , 151-160. Tyre,  Scaling  Procedure f o r Gains  Networks.  G . L . and G.D. S c r e p e t i s . 1978. S y s t e m f o r Roundwood I n v e n t o r y and C o n t r o l o f V e n e e r , Sawn t i m b e r a n d P u l p w o o d V o l u m e s . Forest P r o d u c t s J o u r n a l , 28 ( 1 ) , 4 0 - 4 1 .  Wagner, H.M. 1975. Prentice-Hall,  P r i n c i p l e s o f O p e r a t i o n s R e s e a r c h ; (2nd E n g l e w o o d C l i f f s , New J e r s e y .  Edition).  W e l l w o o d , E.W. 1971. S c h e d u l i n g and C o n t r o l o f a Douglas F i r Plywood P l a n t A i d e d by L i n e a r P r o g r a m m i n g . Forest Products J o u r n a l , 21 ( 1 1 ) , 42-50. Western Forests Products L a b o r a t o r y . 1978. Program Review 1 9 7 7 - 7 8 . W e s t e r n F o r e s t P r o d u c t s L a b o r a t o r y ( N o w , F o r i n t e k Canada C o r p o r a t i o n ) , Vancouver, B r i t i s h Columbia. Wood, A . D . 1963. Plywoods o f t h e W o r l d ; T h e i r Development, Manufacture and A p p l i c a t i o n . J o h n s t o n and Bacon L i m i t e d , L o n d o n , E n g l a n d . W o o d f i n , R.O. Species.  Jr. 1973. Wood L o s s e s i n P l y w o o d P r o d u c t i o n F o r e s t P r o d u c t s J o u r n a l , 23 ( 9 ) , 9 8 - 1 0 6 .  Four  Y a p t e n c o , R.Q. and A . E . W y l i e . 1970. A Q u a n t i t a t i v e Approach to Plywood P r o d u c t i o n S c h e d u l i n g . F o r e s t P r o d u c t s J o u r n a l , 20 ( 3 ) , 54-59. Z e m e l , E. 1980. The L i n e a r M u l t i p l e C h o i c e K n a p s a c k P r o b l e m . R e s e a r c h , 28 ( 6 ) , 1 4 1 2 - 1 4 2 3 .  Operations  120 APPENDIX Listing of  the  Enumeration  c c  I  FORTRAN P r o g r a m f o r  Algorithm;  Four Veneer  the  Implicit  Thickness  C IMPLICIT ENUMERATION ALGORITHM FOR THE PLYWOOD C DESIGN AND MANUFACTURING (PDM) MODEL C C B.G.RAGHAVENDRA C FACULTY OF COMMERCE AND BUS. ADMN. C UNIVERSITY OF BRITISH COLUMEIA C VANCOUVER, BC C C NUMBER OF VENEER THICKNESSES (K) = 4 C THJ.S CODE CAN CURRENTLY HANDLE UPTO 25 TYPES C OF PLYWOOD MADE WITH 3,5.7 OR 9 PLIES. IT CAN C BE EXTENDED TO ANY NUMBER OF PLIES AND PLYWOOD C TYPES. C $COMPILE REAL X(4) ,XSTAR(4),BL(25) ,BU(25),T(50) ,TPLY(25) . THICK ( 25 , 1 6 ) REAL TW(25,16).TP(25).R(25),B(4).DEM<25),PLY3(4.4),P(25,16) REAL PLY5(16.4),PLY7(16.4).PLYS(16.4).BSUM(25, 16 ) .RSUM(25.16) REAL WOOD(25) ,REV(25),PSTAR(25.16),V(4) REAL TSTAR(25.16),RSTAR(25,16),TWSTAR(25,16),BSTAR(4),LSTAR(4) REAL XMIN(25),XMAX(25) ,TH(25. 16) ,BD(25 ) .PD(25).WD(25),TS(25) INTEGER ID(2 5, 16).IDENT(25) INTEGER I DELTA(25. 16).1PLY(25),I STAR(25. 16 ) ,ITEST(25),JBEST(25) C C PLY3(J,K),...PLY9(J,K) ARE THE DESIGN COEFFICIENTS C DO 10 d= 1 ,4 10 READ,(PLY3(J,K),K=1,4) DO 11 J= 1 . 16 11 READ, (PLY5(J.K ) ,K=1,4) DO 12 J= 1 . 16 12 READ,(PLY7(J,K),K=1,4) DO 13 J= 1 . 16 13 READ,(PLY9(J,K),K=1,4) C C INPUT DATA FOR THE PROBLEM; 'NUMBER' IS THE NUMBER C OF PLYWOOD TYPES FOLLOWED BY DETAILS OF ITS NUMBER C OF PLIES.THICKNESS,LOWER AND UPPER TOLERANCES,ORDERFILE C AND REVENUE C READ,NUMBER DO 14 1=1.NUMBER 14 READ,IPLY(I).TPLY(I),BL(I).BU(I).DEM(I),R(I) C C 'N2' IS THE NUMBER OF VENEER THICKNESSES AVAILABLE C AND 'TFU' IS THE UPPER TOLERANCE ON FACE VENEER C THICKNESS. 'XSTAR(K)' AND ' ZSTAR' ARE THE EXISTING C THICKNESSES AND THE CORRESPONDING NET REVENUE. C READ. N2  121  READ.(T!I),1=1,N2) READ.W,C,Y,CF,TFU READ.(XSTAR(K),K=1,4),ZSTAR C C C C  THE FOLLOWING SECTION PRINTS THE INPUT DATA AS READ BY THE COMPILER FOR VERIFICATION PURPOSES.  PRINT220 FORMAT!'1',5X,'INPUT DATA FOR THE PLYWOOD DESIGN PROBLEM') PRINT221 22 1 FORMAT ( ' ' . 5X , ' ' ) PPI NT 22 2.NUMB E R 222 FORMAT!'-'.10X.'NUMBER OF PLYWOOD TYPES= ' . 17X,I 3 ) PRINT223.W 223 FORMATf '-' , 10X, 'WOOD AVAILABILITY (CU.MTRS.) = ' . 11X,F10.2 ) PRINT224 . C 224 FORMAT( '-' , 10X, 'LOG COST (S/CU.MTRS. )-' , 19X.F6.2) PRINT225.Y 225 F O R M A T ! 1 0 X . 'CORRECTION FACTOR FOR Y I ELD= ' , 12X,F10.6) PR I NT 2 26.C F 226 FORMAT! '-', 10X , 'CORRECTION FACTOR FOR SHRINKAGE^' .8X.F6.2) PRINT227.TFU 227 FORMAT! '-', 10X, 'UPPER LIMIT FOR FACE VENEER (MM) =' .7X.F6.2) PRINT228 228 FORMAT! '-', 10X, 'OTHER DETAILS OF PLYWOOD TYPES:') PRINT229 229 FORMAT( '-' ,2X. 'SL.NO. ' ,2X. 'PLY' ,2X. 'THICKNESS' .2X. 'UPPER LIMIT' ,2X C, 'LOWER LIMIT',2X. 'REVENUE '. 2X, 'ORDER FILE' > PRINT230 230 FORMAT( ' ' , 18X. ' (MM ) ' ,8X. ' (MM) ' ,9X, ' (MM)',7X,'($)') DO 23 1 1=1.NUMBER 231 PRINT232, I , I PLY( I ) ,TPLY( I ) , BL( I ) .BU1 I ) ,R( I ) .DEM( I ) 2 32 FORMAT! '-'.4X,I2.5X,I2.5X,F4. 1.8X.F4. 1.9X.F4. 1.7X.F4. 1.5X.F8.0) KSET = 0 TKMAX=TFU NODE=0 IMPR=0 IBND=0 C C COMPUTATION OF THE BOUNDS IMPLIED BY THE BRANCHING C TESTS (3.41 )-(3.42 ) C DO 180 M=1,N2 IF(TfM).LT.TFU)GO TO 180 MAXIMUM GO TO 18 1 180 CONTINUE 181 MXN0DE=MAXIM1*N2**3 PRINT 182 182 FORMAT!'1',5X.'RESULTS OF THE PLYWOOD DESIGN PROBLEM WITH 4 VENEER CS ' ) PRINT 185 185 FORMAT(' ',5X.' C-' ) PRINT183 183 F O R M A T ! 1 0 X , ' R E S U L T S OF THE BRANCHING TESTS') PRINT184 184 FORMAT! ' ' , 10X ,' . ') DO 20 1=1,NUMBER IF(IPLY(I).E0.5)G0 TO 21 220  122  IF(IPLY(I).EO.7)G0 TO 22 IF(IPLY(I ) .EO.9 )GO TO 23 TMAX = (BU(I)-T( 1 ))/2. GO TO 20 21 TMAX=(ESU( I ) - 3 . *T( 1 ) )/2 . GO TO 20 22 TMAX=(BU(I )-5 . *T( 1 ) )/2. GO TO 20 23 TMAX = (BU(I ) -7 . * T( 1 ) )/2. 20 IF(TMAX.LT.TKMAX)TKMAX=TMAX PRINT24,TKMAX 24 FORMATf'-'.5X,'MAXIMUM PERMISSIBLE FACE VENEER THICKNESS FROM (3.4 C1) = ' , F8 . 2 . ' MM') TKMIN=0. DO 25 1 = 1 ,NUMBER IF( IP L Y(I ) .EO.5)G0 TO 26 I F(I PLY (I ) .EO.7)G0 TO 27 IF(I PLY(I ) .EQ.9)G0 TO 28 TMIN =(BL(I ) -2. •TKMAX ) GO TO 25 26 TMIN=(BL(I)-2. 'TKMAX )/3. GO TO 25 'TKMAX)/5. 27 TMIN=(BL(I ) -2. GO TO 25 23 TMIN=(BL( I )-2 .' TKMAX)/7. 25 IFfTMIN.GT.TKMIN)TKMIN=TMIN PR I NT 29.TKMIN '.5X.'MINIMUM PERMISSIBl E THICKNESS FOR X4 FROM (3.42)= 29 FORMAT( MM ' ) C6X.F8.2 DO 17 M=1,N2 IF(T(M ) .LT.TKMAX)G0 TO 17 MAX 1 =M GO TO 18 17 CONTINUE 18 DO 19 M=1,N2 IF(T(M).LT.TKMIN)G0 TO 19 MIN4=M GO TO 199 19 CONTINUE THIS SECTION IS INTENDED TO PRINT AN ITERATIVE SUMMARY AS AND WHEN IMPROVED VENEER THICKNESSES ARE FOUND. 199 PRINT195 195 FORMAT('- ,5X,'RUN STATISTICS: PRINT 194 194 FORMAT(' . 5X . PRINT196 196 FORMAT('- . 5X.'IMPROVED VENEER SETS FOUND SO FAR AND CORRESPONDING C THICKNESS AND OBJECTIVE' ) PRINT 1960 1960 FORMAT(' ' . 5X FUNCTION VALUE ARE AS FOLLOWS:') PRINT 197 197 FORMAT('- , 5X, 'NUMBER' , 10X, 'VENEER THICKNESS' ,5X, 'OBJ. FN. VALUE ( C$ ) ' )  PRINT74.(XSTAR(K).K=1.4),ZSTAR 74 F O R M A T ( 2 X . ' E X I S T I N G SET'.3X,4F6.2,2X.F15.2) DO 30 L1=1,MAX1 X(1)=T(L1) DO 31 L2=1,N2  C C C C  IF(L2.GT.N2-2)G0 TO 31 IF(L2.EO.L1)GO TO 31 X(2 )=T(L2 ) DO 311 L3= 1,N2 IF(L3.GT.N2-1)GO TO 311 I F(L3 . LE.L2)GO TO 311 IF(L3 . EO.L1)GO TO 31 1 X(3)=T(L3 ) COMPUTATION OF THE BOUNDS ON X4 FROM ( 3.31)-(3.32 ) AND THE INFEASIBILITY TESTS ASSOCIATED WITH THEM.  402  403 404  401 4 10  412  413 4 14  DO 400 1=1.NUMBER I DENT(I )=0 XMIN(I ) = T(N2) XMAX(I)=T(MIN4) IF(IPLY(I ) .E0.5)G0 TO 410 IF( IPLY( I ) .EO.7)G0 TO 420 IF(IPLYI I ) .EO.9 )G0 TO 430 DO 401 J=1.4 ID(I.J)=0 SUM=0. DO 402 K=1.3 SUM = SUM + X(K ) + PLY3(J.K ) TH(I .J ) = SUM * C F IF(PLY3(d.4 ) .EO.0)G0 TO 403 AMIN=(BL(I)-TH(I.J))/(PLY3(J.4CCF) AMAX=(BU( I ) - T H ( I , \J ) (/IPLY3IJ.4 ) *' C F ) IF(AMIN.GT.T(N2).OR.AMAX.LT.T(MIN4))GO TO 401 IF(AMIN.LT.T(MIN4))AMIN=T(MIN4) IF(AMAX.GT.T(N2 ) )AMAX = TIN2) GO TO 404 IF(TH( I ,J ) . LT . EL ( I ) .OR.THf I , J .) .GT . BU( I ) )G0 TO 401 AMIN = T(MIN4 ) AMAX = T(N2 ) ID(I .J ) = 1 I DENT(I ) = IDENT(I ) + 1 IF(AMIN.LT.XMIN(I))XMIN(I)=AMIN IF(AMAX.GT.XMAX(I ))XMAX( I ) =AMAX CONTINUE IF(I DENT(I) .EO.0)GO TO 311 GO TO 400 DO 4 11 J= 1 , 16 ID(I,d)=0 SUM=0. DO 4 12 K=1,3 SUM=SUM+PLY5(d.K)»X(K) TH(I.J)=SUM*CF IF(PLY5(J,4).EO.OJGO TO 413 AMIN=(BL(I)-TH(I,d))/(CF*PLY5(d.4)) AMAX=(BU(I)-TH(I.d))/(CF*PLY5(J.4)) IFrAMIN.GT.T(N2).OR.AMAX.LT.T(MIN4))G0 TO 411 IF(AMIN.LT.T(MIN4))AMIN=T(MIN4) IF(AMAX.GT.T(N2))AMAX=T(N2) GO TO 4 14 I F ( T H ( I , 0 ) . L T . B L ( I ) . O R . T H ( I , J ) .GT.BU(I ) )G0 TO 411 AMIN = T(MIN4 ) AMAX = T(N2) ID(I,J)=1 IDENT(I) = IDENT(I )+1  124  411 420  422  423 424  421 430  432  433 434  431 400  440  IF(AMIN.LT.XMIN(I))XMIN(I)=AMIN IF(AMAX.GT.XMAX(I))XMAX(I ) = AMAX CONTINUE IFfIDENT(I).EO.O)GO TO 311 GO TO 400 DO 421 J=1,16 ID(I,J)=0 SUM=0. DO 422 K=1,3 SUM=SUM+X(K)-PLY7(J,K) TH(I,d)=SUM'CF IF(PLY7(J,4).EO.O)GO TO 423 AMIN=(BL(I)-TH(I.d))/(CF»PLY7(d.4)) A M A X = (BU(I)-TH(I,d))/(CF*PLY7(d,4)) IF(AMIN.GT.T(N2 ) .OR.AMAX.LT.T(MIN4 ) )G0 TO 421 IF(AMIN.LT.TfMINJ))AMIN=T(MIN4) IF(AMAX.GT.T(N2) ) AMAX = T(N2) GO TO 424 IF(TH( I .d) .LT.BL(I ) .OR.THI I .d> .GT .BUI I) )G0 TO 421 AMIN=T(MIN4) AMAX = T(N2) ID( I .J ) =1 IDENT(I ) = IDENT(I )+ 1 IFfAMIN.LT.XMINI I ) )XMIN( I )=AMIN IF(AMAX.GT.XMAX(I))XMAX(I)=AMAX CONTINUE IF(IDENT(I).EO.0)G0 TO 311 GO TO 400 DO 43 1 d=1.16 ID( I .d ) =0 5UM=0. DO 432 K=1,3 SUM=SUM+PLY9(d.K)»X(K) TH(I,d)=SUM*CF IF(PLY9(d.4 ) .EO.0 )G0 TO 433 AMIN=(BL(I)-TH(I.d))/(CF"PLY9(d.4)) AMAX=(BU(I)-TH(l,d))/(CF*PLY9(d.4)) IF(AMIN.GT.T(N2).OR.AMAX.LT.T(MIN4))GO TO 431 IF(AMIN.LT.T(MIN4))AMIN=T(MIN4) IF(AMAX.GT.T(N2) )AMAX = T(N2 ) GO TO 434 IF(TH(I.d ) .LT .BL( I ) OR.TH(I,d ) .GT.BU(I ) )G0 TO 431 AMIN=T(MIN4) AMAX = T(N2) ID(I,J)=1 I DENT(I ) = IDENT(I)+1 IF(AMIN.LT.XMINII))XMIN(I)=AMIN IF(AMAX.GT.XMAX(I ))XMAX( I ) = AMA X CONTINUE IF(IDENTfI ) .EO.O)GO TO 311 CONTINUE X4MIN=TKMIN X4MAX=T(N2) DO 440 1 = 1 ,NUMBER IF(XMIN(I).GT.X4MIN)X4MIN=XMIN(I) IF(XMAX(I).LT.X4MAX)X4MAX=XMAX(I) IF(X4MIN.GT.X4MAX)G0 TO 311 DO 441 M=MIN4,N2 IF(T(M).LT.X4MIN)G0 TO 441 MINX4=M  GO TO 442 441 CONTINUE 442 DO 443 M=MIN4,N2 IF(T(M) .LT.X4MAX)GO TO 443 MAXX4=M GO TO 444 443 CONTINUE 444 IF(MINX4.GT.MAXX4)G0 TO 311 C C C  COMPUTATION OF THE UPPER BOUND ON Z FROM (3.38): 'ZBOUND' IS THE UPPER BOUND. DO 450 1=1.NUMBER BD(I)=BU(I) I DENT(I )=0 IF(IPLY( I ) .EO.5)G0 TO 45 1 IF( I PLY(I ) .EO.7)G0 TO 452 IF(IPLY(I).EQ.91G0 TO 453 DO 454 d= 1 . 4 IF(ID(I,d).EQ.O)GO TO 454 DO 4540 M = MINX4,MAXX4 5UMA=TH(I,d)+T(M)*CF*PLY3(d,4) IF(SUMA.LT.BL( I ) .OR.SUMA.GT.BU(I ))G0 TO 454 I DENT( I ) = IDENT( I )+1 IF(SUMA.LT.BD( I ) )BD(I )=SUMA 4540 CONTINUE 454 CONTINUE IF(I DENT(I ) .E0.O)G0 TO 311 GO TO 450 45 1 DO 455 d= 1 , 16 I F(ID(I .J) .E0.O)G0 TO 455 DO 4550 M = MI NX4,MA XX4 SUMA=TH(I.d)+T(M)*CF*PLY5(d.4) IF(SUMA.LT.BL(I).OR.SUMA.GT.BU(I))G0 TO 455 I DENT(I ) = IDENT( I )-M I F ( SUMA.LT.BD(I ) )BD(I)=SUMA 4550 CONTINUE 455 CONTINUE I F(I DENT(I ) .EO.0)GO TO 311 GO TO 450 452 DO 456 J=1.16 IF(ID(I,J).EO.O)G0 TO 456 DO 4560 M=MINX4.MAXX4 SUMA=TH(I,d)+T(M)»CF*PLY7(d,4) IF(SUMA.LT.BL(I).OR.SUMA.GT.BU(I))G0 TO 456 I DENT( I) = IDENT( I )+1 IF(SUMA.LT.BD(I ) )BD(I) = SUMA 4560 CONTINUE 456 CONTINUE IF(IDENTtI).E0.O)G0 TO 311 GO TO 450 453 DO 457 d=1.16 IF(ID(I.d).EO.0)G0 TO 457 DO 4570 M = MINX4.MAXX4 SUMA = TH( I , d) + T(M)-XF*PLY9(d.4) I F( SUMA.LT.BL(I).OR.SUMA.GT.BU(I ) )G0 TO 457 I DENT(I) = IDENT(I)+1 IF(SUMA.LT.BD(I))BD(I)=SUMA 4570 CONTINUE 457 CONTINUE IF(I DENT(I) .EO.O)GO TO 311  450  CONTINUE TSMAX=0. DO 4 6 0 1=1,NUMBER TS(I)=(R(I)-(C*Y*BD(I))/CF)/((BD(I) Y)/CF) I F ( T S ( I ) . L E . T S M A X )G0 TO 460 TSMAX = T S ( I ) IBE5T=I CONTINUE SW00D=O. DO 4 6 1 1 = 1 , N U M B E R >  460  I F ( I . E O . I B E S T l G O TO 461 PD(I)=DEM( I ) WD(I ) = ( B D ( I ) " P D ( I ) * Y ) / C F SWOOD=SWOOD+WD(I ) 46 1 C O N T I N U E P D ( I B E S T ) = ( W - S W O O D ) / ( ( B D ( I E E S T ) • i )/CF  462  )  I F ( P D ( I B E S T ) . L T , D E M ( I B E S T ) ) G 0 T O 311 ZB0UND=O. DO 4 6 2 1=1,NUMBER ZBOUND = Z B O U N D + ( R ( I ) - ( C » Y * B D ( I ) ) / C F ) * P D ( I ) I F ( Z B O U N D . G T . Z S T A R ) G 0 TO 198 I END =IBND+1 GO T O 3 11  S O L V I N G T H E LMCK P R O B L E M A F T E R B R A N C H I N G T E S T S AND B O U N D S : 'ZNODE' IS THE O B J E C T I V E FUNCTION V A L U E FROM (3.14). 198  DO 32 L4=MINX4,MAXX4 X(4)=T(L4 ) I F ( L 4 . L E . L 3 ) G 0 TO 32 I F ( X ( 4 ) . E 0 . X ( 2 ) ) G 0 TO 32 I F ( X ( 4 ) . E Q . X ( 1 ) ) G 0 TO 32 N O D E =NODE+ 1 DO 4 0 1=1.NUMBER ITEST(I )=0 I F ( I P L Y ( I ) . E O . 5 I G 0 TO 4 1 I F ( I P L Y ( I ) . E O . 7 ) G 0 T O 42 I F ( I P L Y ( I ) . E O . 9 ) G 0 TO 43 DO 44 J=1.4 I D E L T A ( I , J ) =0  THICK(I,J)=TH(I,J)+X(4)*PLY3(J,4)»CF I F ( T H I C K ! I .J) . L T . B L ( I ) . O R . T H I C K ! I ,J) . G T , B U ( I ))G0 IDELTA(I.J) =1 I TEST( I ) = ITEST( I ) + IDELTA( I.J) 44 CONTINUE I F ( I T E S T ( I ) . G E . 1 )G0 TO 40 GO T O 32 4 1 DO 4 5 J = 1 , 16 IDELTA(I.J)=0  TO  44  TO  45  THICK(I,J)=TH(I.J)+X(4)*PLY5(J,4)*CF  45  42  I F ( T H I C K ( I , J ) . L T , B L ( I ) .OR.THICK* I ,J) IDE L T A ( I , J ) = 1 ITEST(I) = ITEST(I) +IDELTA(I,J ) CONTINUE I F ( I T E S T ( I ) . G E . 1)G0 GO T O 3 2 DO 4 6 J = 1 . 16 IDE L T A ( I , J ) = 0  TO  40  THICK(I,J)=TH(I,d)+X(4)*PLY7(J,4)*CF  .GT.BU(I))G0  127  46  43  IF(THICK(I,J).LT.BL(I).OR.THICK! I D E L T A ( I , J)=1 I TEST(I ) = ITEST( I )+IDELTA(I,J) CONTINUE I F ( I T E S T f I ) . G E . 1)GO TO 40 GO T O 3 2 DO 4 7 J = 1 , 1 6 I D E L T A ( I , J)=0 THICK!I.J)=TH(I.J)  I,J  ) . G T . B U ( I ) ) G O TO  I F ( T H I C K ( I . J ) . L T . B L ( I ) . OR . T H I C K ! I , J ) . GT . B U I . I ) ) G 0 IDELTA(I.J)=1 47  40 C C C C  ITEST(I)=ITEST(I)+IDELTA(I.J) CONTINUE I F ( I T E S T l I ) . G E . 1 )G0 TO 40 GO T O 32 CONTINUE  SETTING MAXIMUM  50  LIMITS NUMBER  ON C O M P U T A T I O N S ; ' KSET' IS THE O F L M C K P R O B L E M S TO B E S O L V E D .  KSET=KSET+1 IF(KSET.GT.5000)GO DO 5 0 K=1.4 B(K)=Y*X(K) TWMAX=0. DO 51 1=1.NUMBER T P( I ) = 0 . IF( I P L Y l I ) . E O . 5 ) G 0 IFI I P L Y l I ) . E Q . 7 ) G 0 IF(IPLYl I ) .EO.9)G0 DO 5 5 J= 1 . 4 TWII.J)=0. RSUM(I,J)=0. BSUMII,J)=0.  611  55 52  612  TO  200  TO TO TO  52 53 54  IFIIDELTA(I.J).EQ 0 ) G 0 TO 55 DO 6 11 K=1 , 4 BSUMII,J )=BSUM(I,J)+B(K)*PLY3(J,K) RSUMI I , J ) = R ( I ) - B S U M I I , J ) * C TWII.J)=RSUM(I.J)/BSUM(I.J) I F ITWI I . J ) . L T . T P ! I ) ) G 0 T O 5 5 T P ( I )= TW( I . J ) JBESTI I ) = J IFITPIIl.LT.TWMAX)GO TWMAX = T P ( I ) IBEST=I CONTINUE GO T O 5 1 DO 5 6 J = 1 , 16  TO  55  TWII,0)=0. RSUMtl,J)=0. BSUMtI.J)=0. I F ( I D E L T A ( I , J ) . E O . O ) G O T O 56 DO 6 1 2 K=1,4 BSUM(I,J)=BSUM(I,J)+B(K)*PLY5(J,K) R S U M ( I , J ) = R(I ) - C * B S U M ( I . d ) TW(I,J)=RSUM(I,J)/BSUM(I.J) I F ( T W ( I , J ) . L T . T P ( I ) )G0 TO 56 TP(I)=TW(I,J) JBEST(I ) = J IF(TP(I).LT.TWMAX)G0  TO  56  TO  46  47  128  5G 53  TWMAX=TP(I) IBEST=I CONTINUE GO T O 51 DO 5 7 J = 1 , 1 6 TW(I.J)=0. RSUM(I , J ) = 0 . BSUM(I.J)=0.  I F ( I D E L T A ( I , J ) . E Q . O I G O TO 57 DO 6 13 K = 1 . 4 6 13 B S U M ( I . J ) = B S U M ( I , J ) + B ( K ) * P L Y 7 ( J . K ) RSUM(I,J)=R(I)-C*BSUM( I.J) TW(I.J)=RSUM(I,J)/BSUM( I.J) I F ( T W ( I . J ) . L T . T P ( I ) )GO T O 5 7 TP(I )=TW(I.J) JBEST(I )= J  57 54  I F ( T P < I ) . L T . T W M A X )GO T O 5 7 TWMAX=TP(I) IBEST=I CONTINUE GO T O 51 DO 5 8 J = 1 . 16 TW(I,J>=0. RSUM(I,J)=0. B SUM(I.J)=0.  I F ( I D E L T A f I . J ) . E O . O I G O TO 58 DO 6 14 K= 1 . 4 6 14 B S U M f I . J ) = B S U M ( I , J ) + B ( K ) * P L Y 9 ( J . K ) RSUMI I , J ) = R ( I ) - C * B S U M ( I . J ) TW(I.J)=RSUMII.J)/BSUM(I.J) I F ( T W ( I . J ) . L T . T P ( I ) ) G 0 TO 58 TP(I )=TW(I.J) JBEST(I)=J  58 51  61  69 60  I F f T P ( I ) . L T .TWMAX)G0 TO 58 TWMAX=TP(I) IBEST=I CONTINUE CONTINUE SUMB=0. DO 6 0 1 = 1 . N U M B E R I F ( I . E O . I B E S T ) G 0 TO 6 9 N I =4 I F ( I P L Y ( I ) . G T . 3 ) N I = 16 DO 6 1 J = 1 . N I P(I.J)=0. I F ( J . N E . J B E S T ( I ) ) G 0 T O 61 P(I.J)=DEM(I) . WOOD(I ) = B S U M ( I , J ) * P ( I . J ) REV(I )= R S U M ( I . J ) * P ( I . J ) CONTINUE SUMB=SUMB+WOOD(I) GO T O 6 0 IP=I CONTINUE NI=4 I F ( I P L Y ( I P ) . G T . 3 ) N I = 16 DO 7 0 J = 1 , N I P(IP.J)=0. I F ( J . N E . J B E S T ( I P ) )G0 TO 70 P(IP,d)=(W-SUMB)/BSUM(IP.J)  129  I F ( P (I P . J ) . L T . D E M ( I P ) ) G O REV(IP 70  TO 32  )=RSUM(IP,J)*P(IP.J)  CONTINUE ZNODE=0. 00  7 1  7 1  1=1,NUMBER  ZNODE=ZNODE+REV(I)  C C  UPDATING  C  IT  THE INCUMBENT  FOR I T E R A T I V E  SOLUTION  AND  PRINTING  SUMMARY.  C IF(ZNODE.LE.ZSTAR)GO  TO 32  IMPR=IMPR+1 Z S T AR = Z N O D E WOODVL=TWMAX DO 72  72  K= 1 . 4  XSTAR(K)=X(K) PRINT75.IMPR.(XSTAR(K),K=1.4).Z5TAR  75  FORMAT('O'.5X.I3.9X.4F6.2.2X.F15.2) DO  73  1=1.NUMBER  N I =4 IF(  I P L Y ( I ) . G T . 3 ) N I = 16  DO  73  J=1.NI  PSTAR!I.J)=P(I.J) R ST A R ( I  .J)=RSUM( I , J )  TSTAR(I,J)=THICK(I.J) T W S T A R f . I . J I = TW ( I . J ) 73  ISTARfI.J)=IDELTA!I.J)  32 31  CONTINUE 1 CONTINUE  31  CONTINUE  30  CONTINUE 1 F(KSET  . G T . 0 )G0  TO 100  C C  MESSAGE  IF  THE PROBLEM  IS  INFEASIBLE.  C P R I N T 101 101  FORMAT!'1',5X.'THE CISFYING GO  PROBLEM  CONSTRAINTS  IS  INFEASIBLE:  (2.2)-(2.4)  DOES  VENEER  THICKNESS  SET  SAT  NOT E X I S T ' )  TO 999  C C  MESSAGE  C  SOLVED  IF  T H E NUMBER  EXCEEDS  OF LMCK  A PRESET  PROBLEMS  TO BE  LIMIT.  C 200  PRINT201  201  FORMAT('1',5X,'NUMBER COO:  PROGRAM  TERMINATED  OF F E A S I B L E  VENEER  PREMATURELY.  THICKNESS  CURRENT  SETS  RESULTS  EXCEEDS  50  ARE PRESENTED'  C) C C  THIS  C  OF T H E I M P L I C I T  SECTION  GIVES  A SUMMARY  OF T H E PERFORMANCE  ENUMERATION ALGORITHM.  C 100  PR I NT  193  FORMAT! ' - ' , 5 X , 'MAXIMUM P O S S I B L E C  193,MXNODE  PROBLEM  SETS  OF V E N E E R  THICKNESS.ORIGINAL  = ' ,5X,16 )  PRINT470.IBND 470  FORMAT! ' - ' , 5 X . 'NUMBER CVE  OF T I M E S  UPPER  BOUND  ON Z  (3.38)  WAS E F F E C T I  =',8X,16)  PRINT 1 9 2 , N O D E 192  FORMAT('-',5X,'NUMBER CTS=',9X,16)  OF V E N E E R  SETS  EVALUATED AFTER  BRANCHING T E S  130  PRINT191,KSET 191  F O R M A T ( 5 X . ' N U M B E R C12X.16) DO 7 7 K = 1 . 4 V(K)=0. DO 7 8 1=1,NUMBER I F ( I P L Y l I ) . E 0 . 5 )GO IF(IPLYl I ) .EO.7)G0 IF(I P L Y ( I ) . E Q . 9 ) G 0 DO 1 7 3 J = 1 . 4  173 170  OF V E N E E R  SETS  EVALUATED  FOR LMCK  PROBLEM='  TO 170 T O 171 T O 172  I F ( I S T A R U , J ) . E O . O J G O TO 173 V(K)=V(K)+PLY3(J,K)*PSTAR(I.J) CONTINUE GO T O 7 8 DO 1 7 4  J = 1 , 16  I F ( I S T A R ( I , J ) . E O . 0 ) G O T O 174 V ( K ) = V ( K ) + P L Y 5 ( J . K ) *PSTAR( I.J) 174 C O N T I N U E GO T O 7 8 17 1 DO 1 7 5 J = 1 . 1 6  175  I F I I S T A R f I . J ) . E O . O I G O T O 175 V(K)=V(K)+PLY7(J,K)*PSTAR(I.J) CONTINUE  172  GO T O 7 8 DO 1 7 6 J = 1 . 16  176 78  77  N  I F I I S T A R I I . J ) . E O . 0 ) G O T O 176 V(K>=V(K)+PLY9(J.K)*PSTAR(I.J> CONTINUE CONTINUE BSTARIK )=Y*XSTAR(K ) LSTARIK)=V(K ) *BSTARIK ) CONTINUE  C C C  THIS  SECTION  GIVES  THE SOLUTION  TO T H E PDM P R O B L E M .  P R I N T 102 102 103  FORMAT I ' 1 ' , 5 X . ' R E S U L T S : ' ) P R I N T 103 FORMAT ( ' ' . 5 X . ' ') PRINT104,(XSTAR(K).K=1,4)  104  FORMAT!'0'.5X,'OPTIMAL PRINT177  VENEER  THICKNESSES  177  FORMAT!'O',5X,'QUANTITY OF L O G S P R I N T 1 7 7 0 . ( L S T A R I K ) ,K=1 , 4 )  ARE ( M M ) : ' , 4 F 8 . 2 )  FOR C O R R E S P O N D I N G  ')  1770  FORMAT(' ',5X.'THICKNESS (CU. MTRS.):'.15X.4F8.0) PRINT 105.ZSTAR 105 F O R M A T ! ' 0 ' , 5 X , ' O B J . F N . V A L U E AT O P T I M A L S O L U T I O N ($):',F20.2) PRINT 190,WOODVL 1 9 0 F O R M A T ! ' O ' . 5 X , ' M A R G I N A L V A L U E OF WOOD I $ / C U . M T R . ) : ' , F 1 9 . 4 ) P R I N T 106 106 F O R M A T ! ' O ' . 5 X , ' C O R R E S P O N D I N G D E S I G N A L T E R N A T I V E S AND P R O D U C T M I X CARE AS FOLLOWS: ' ) P R I N T 107  107  108  111  FORMAT!'-',5X.'PLYWOOD TYPE'.8X.'DESIGN C S ' . 5 X . 'QUANTITY ' , 10X, 'NET REVENUE' ) PRINT108  ALTERNATIVE',5X,'THICKNES  F O R M A T ( ' ' , 4 9 X , ' ! DRY - M M ) ' , 8 X , ' (r> ) ' . 1 3 X , ' $ / S H E E T ' DO 1 1 0 1 = 1 , N U M B E R PRINT 1 1 1 , I P L Y ( I ) , T P L Y ( I ) F O R M A T ( ' - ' , 5 X , 1 2 , ' P L Y ' , 2 X , F 4 . 1, ' M M ' )  )  131  IF(IPLY( I ) .EO.5)G0 IF(IPLY(I).EO.7)G0 IF(I PLY(I).E0.9)G0 DO 1 1 5 d=1 , 4  GO T O 1 10 DO 1 1 7 d= 1 , 16 I F ( I S T A R ( I , d ) . E Q . O ) G O TO 117 PRINT118.(PLY5(d,K),K=1,4).TSTAR(I ,d) .PSTAR(I ,d ) ,RSTAR(I,d ) FORMAT(' '.22X.4F5.0.7X,F5.2.5X.F12.1,8X,F8.4) CONTINUE GO T O 1 10  112  118 117 113  DO 1 1 9  120  I F ( I S T A R ( I , d ) . E O . O ) G O TO 119 P R I N T 1 2 0 . ( P L Y 7 ( d . K ) . K = 1 , 4 ) .TSTAR( I , d ) . P S T A R ( I , d ) , R S T A R ( I , d ) FORMAT( ' ' ,22X,4F5.0,7X,F5.2,5X.F12. 1.8X.F8.4)  119  121 122 110 999 C C C C C  5 5 5  . 16 , d ) . E O . O ) G O T O 122 PLY9(d.K),K=1.4) ,TSTAR(I.d).PSTAR(I.d),RSTAR(I,d) .22X.4F5.0.7X.F5.2.5X.F12. 1.8X.F8.4)  T H I S L A S T S E C T I O N IS THE INPUT D A T A . THE FIRST 52 S T A T E M E N T S ARE T H E D E S I G N C O E F F I C I E N T S . THESE A R E F O L L O W E D BY S P E C I F I C M I L L DATA U S E D FOR ILLUSTRATION IN CHAPTER THREE.  $DATA 3 0 0 0  2 2 2 7  d= 1 , 16  CONTINUE GO T O 1 1 0 DO 1 2 2 d= 1 I F ( I STAR( I PRINT121.( FORMAT( ' ' CONTINUE CONTINUE STOP END  114  3 3 3 2 2 2 2 2 2  112 113 114  I F ( I S T A R ( I , d ) . E O . O ) G O TO 115 P R I N T 1 1 6 . ( P L Y 3 ( d , K ) , K = 1 . 4 ) , T S T A R( I , d ) . P S T A R ( I , d ) , R S T A R ( I , d ) FORMAT( ' ' . 2 2 X . 4 F 5 . 0 . 7 X . F 5 . 2 . 5 X . F 1 2 . 1 . 8 X . F 8 . 4 ) CONTINUE  116 115  2 2 2 5 4 4 4  TO TO TO  1 0  0  o  0 1 0 0 1  0 0 0 1 0 0 0 1 0 0 0 1 2 0 0 3  0 2 0 0 3  0 0 2 0 0 3  0 0 2  0 0  0 0 0 2 1 0 2 0 1 1 2 0 1 0 2 0 2 1 0 1 2 0 2 0 0  0  0 2  132  4 4 4 2 2 2 2 2 2 2 2 2 9 6 6  3 0 0 5 0 0 3 3 2 2 0 0 0 3 0  0 3 0 0 5 0 2 0 3 0 3 2 0 0 3  0 0 3 0 0 5 0 2 0 3 2 3 0 0 0  6 0 0 3 5 4 0 O 5 0 4 0 5 0 0 4 2 7 0 0 2 0 7 0 2 0 0 7 2 4 3 0 2 4 0 3 2 3 4 0 2 3 0 4 2 0 4 3 2 0 3 4 12 3 7 . 5 7 . 0 8 . 0 1 7 1 1 0 7 . 4.3 3 9.5 9.0 10.0 106378.0 4.8 5 12.5 12.0 13.0 5 0 2 2 8 9 . 0 6.1 5 15.5 15.0 16.0 3 5 0 1 9 2 . 0 7.6 7 18.5 18.0 19.0 4 2 3 3 9 4 . 0 8.9 7 20.5 20.0 21.0 4 4 3 4 4 2 . 0 9.8 7 22.5 22.0 23.O 2 9 5 2 . 0 10.5 9 23.5 23.0 24.0 6 1 3 5 . 0 11.3 9 25.5 2 5 . 0 26.0 11172.0 12.1 9 27.5 27.0 28.0 2738.0 13.1 9 28.5 28.O 29.0 1278.O 13.6 9 30.5 30.0 31.0 38.0 14.6 27 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.8 4.9 5.0 3 0 0 0 0 0 . 0 35.0 0.006067 0.94 3.2 2.64 3.35 3.96 4.98 13416690.  3.5 4.7  133 APPENDIX  II  Data from a Plywood M i l l  1.  Number o f v e n e e r t h i c k n e s s e s ( K ) :  2.  Veneer t h i c k n e s s e s ( x k ) :  3.  Log  4.  Log a v a i l a b i l i t y ( W ) :  5.  Veneer y i e l d f a c t o r  Cost  (C):  35.00  2.69,  in  B.C.  4  3.35,  3 . 9 6 and 4 . 9 8 mm  $/cu.mtr. 300000 c u . m t r .  (y):  0.006067  ( c o r r e s p o n d s t o a 60% y i e l d  volume) 6.  S h r i n k a g e and c o m p r e s s i o n f a c t o r  7.  Upper  8.  Number o f  limit  (Cf):  0.94  on f a c e v e n e e r t h i c k n e s s ( T ^ ) :  plywood types  (N):  12  3 . 2 0 mm  by  134 9.  Plywood t y p e s  P l y w o o d Type Plies-Thickness  and d e s i g n s :  (mm)  Design Alternative  2.69  Number o f v e n e e r s o f 3.35 3.96 4.98  3  -  7.5  1)  3  0  0  0  3  -  9.5  i)  2  0  0  1  5  -  12.5  i)  5 4  0 1  0 0  0 0  2 2  2 1  0 2  1 0  5 4  2 3  0 0  0 0  2 4  5 0  0 3  0 0  2 2  0 2  5 3  0 0  6  3  0  0  6 5  0 4  3 0  0 0  5 2  0 7  4 0  0 0  6 2  0 4  0 3  3 0  2 5  0 0  7 0  0 4  ii) 5 -  15.5  1) ii)  7  -  18.5  i) i i)  7  -  20.5  i) i i)  7  -  22.5  i) ii)  9 -  23.5  9 -  25.5  D i) i i)  9 -  27.5  i) i i)  9 -  28.5  i) i i)  9 -  30.5  i) ii)  135 10.  P l y w o o d t h i c k n e s s t o l e r a n c e , r e v e n u e and o r d e r f i l e :  Number o f PI i e s  T h i c k n e s s (mm ) Specified Lower Upper Limit Limit  Revenue ($/Panel)  Orderfile ('# o f P a n e l s 2.44mrnx 1 . 2 2 m )  3  7.5  7.0  8.0  4.3  171107  3  9.5  9.0  10.0  4.8  106378  5  12.5  12.0  13.0  6.1  502289  5  15.5  15.0  16.0  7.6  3501 92  7  18.5  18.0  19.0  8.9  423394  7  20.5  20.0  21.0  9.8  443442  7  22.5  22.0  23.0  10.5  2952  9  23.5  23.0  24.0  11.3  6135  9  25.5  25.0  26.0  12.1  :111 72  9  27.5  27.0  28.0  13.1  2738  9  28.5  28.0  29.0  13.6  1278  9  30.5  30.0  31.0  14.6  38  11.  Maximum p o s s i b l e  12.  Wood l o s s thickness  Net Revenue  ($):  i n t h e form o f e x c e s s i n plywood ( c u . m t r . ) :  13416694  7944.7  136 APPENDIX Results  from the  III PDM Model  with  Four Veneer T h i c k n e s s e s  1.  Optimal  2.  Quantity of thicknesses  3.  4.  5.  veneer t h i c k n e s s e s  logs for corresponding (cubic metres):  Objective function value optimal s o l u t i o n ($): Marginal value of ($/cubic metre): Wood l o s s thickness  (mm):  2.5  3.1  3.9  4.8  205848  48886  31799  13468  at 14337370.00  wood  i n the form o f e x c e s s i n plywood ( c u b i c m e t r e s )  59.5003  1647.4  137 6.  Plywood  design a l t e r n a t i v e s  P l y w o o d Type ply-Thickness (mm) 3 -  7.5  Design Alternative  and p r o d u c t  mix:  Number o f V e n e e r s o f 2.5 3.1 3.9 4.8  Quantity of Plywood, # of Panels,2.44mx 1.22 m  i) ii)  3 2  0 1  0 0  0 0  2371653  -  3 -  9.5  i)  2  0  0  1  106378  5 -  12.5  1)  ii)  4 3  1 2  0 0  0 0  502289  i) ii)  2 2  0 2  3 0  0 1  3501 92  5 -  15.5  -  _  7 -  18.5  1)  4  3  0  0  423394  7 -  20.5  i) ii) iiiT  5 4 2  0 0 3  0 3 2  2 0 0  443442  1) ii)  4 2  0 3  0 0  3 2  2952  1)  5  4  0  0  6135  i) ii)  6 2  0 7  3 0  0 0  D  2 6  4 0  3 0  0 3  2738  1278  7 -  22.5  9 -  23.5  9 -  25.5  9 -  27.5  ii) 9 -  28.5  1)  2  3  4  0  9 -  30.5  i)  2  0  7  0  -  _  ' i l l 172  -  38  138  APPENDIX Results  f r o m t h e PDM Model w i t h Veneer  1.  Optimal  2.  3.  4.  IV Three  Thicknesses  v e n e e r t h i c k n e s s e s (mm):  2.6  3.2  4.4  Quantity of logs for corresponding thickness (cubic metres):  231533  32087  36380  Objective function value optimal s o l u t i o n ( $ ) :  13930670  at  M a r g i n a l v a l u e o f wood ($/cubic metre):  55.8657  Wood l o s s thickness  3612.1  i n t h e form o f e x c e s s i n plywood ( c u b i c m e t r e s ) :  139 6";  P l y w o o d d e s i g n a l t e r n a t i v e s and p r o d u c t P l y w o o d Type Ply - Thickness (mm) 3  7.5  3  -  9.5  5  -  12.5  5  7  -  -  15.5  18.5  Design Al t e r n a t i v e  mix:  Number o f v e n e e r s 2.6 3.2 4.4  of  Quantity of Plywood, # o f p a n e l s , 2.44m x 1.22m  i) ii)  3 2  0 1  0 0  2277072  1)  2  0  1  106378  1) ii)  5 4  0 1  0 0  502289  i) ii)  3 2  0 2  2 1  1) ii)  5 4  2 3  0 0  423394  -  -  _  350192  -  7  -  20.5  1)  5  0  2  443442  7  -  22.5  i) ii)  4 2  0 3  3 2  _  2952 6135  9  -  23.5  i)  6  3  0  9  -  25.5  1)  2  7  0  9  -  27.5  i)  6  0  3  2738  9  -  28.5  1)  5  0  4  1278  9  -  30.5  1)  2  3  4  11172  38  140 APPENDIX V  Results  1.  Optimal  2.  Quantity  of  logs for  Face v e n e e r s  4.  Objective  (mm):  Marginal  6.  Wood l o s s  function value  (cubic metres):  3.8  185670  82869  31461  14,387,280  form  thickness in  3.1  at  v a l u e o f wood ( $ ) : i n the  Veneers  2.5  2.5  solution ($):  5.  Face  corresponding  (cubic metres):  3.  optimal  /  PDM M o d e l ; A l t e r n a t e  v e n e e r t h i c k n e s s e s (mm):  thickness  excess  from the  59.5003  of  plywood 1403.0  3.1  141  7.  Plywood d e s i g n a l t e r n a t i v e s  Plywood Plies -  3  -  Type Thick  (mm)  Design Al t e r n a t i v e *  7.5  -  9.5  5  -  12.5  -  15.5  -  18.5  -  20.5  0  2  1  106378  4 4 3  1 0 2  0 1 0  502289  2 0 0 1  0 4 3 2  3 1 2 2  5 4 3  0 3 4  2 0 0  4 2 0 3  0 3 7 2  3 2 0 2  2 0  0 4  5 3  5 4  4 •5  0 0  2 4  7 2  0 3  4 3 2  3 4 4  2738  A  2 2 3  A i i) A  0 0  6 5  3 4  1278  A  0  2  7  38  D  A  i)  i)  i)  i) i i) i i i) A iv) A  7  -  22.5  i) i i) A  9  -  23.5  i) i i •) A  9  -  25.5  i) i i) A  9  -  27.5  i) i i) iii)  9  9  '*  -  -  'A'  28.5  30.5  Quantity of Plywood, # o f Panels  2383259  i i) iii) A 7  of  0 0  ii) A i i i) A iv) A 7  Number o f V e n e e r s 2.5 3.8 3.1  0 1  ii) i i i) 5  mix:  3 2  i i) 3  and p r o d u c t  i)  i)  indicates alternative  face veneer  t h i c k n e s s o f 3 . 1 mm  -  _  350192  _  423394  -  _  -  —  443442 _  2952 6135  11172  -  

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