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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 i n THE FACULTY OF GRADUATE STUDIES (F a c u l t y o f Commerce and Business Ad 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 the required standard. THE UNIVERSITY OF BRITISH COLUMBIA May, 1982 Bangalore Gururajachar Raghavendra In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Commerce & Business Administration The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6 (3/81) ( i i ) ABSTRACT One f a c t o r o f wood l o s s i n the manufacture of plywood i s i m p l i c i t i n the form o f excess thickness i n plywood due to the choice o f veneer thicknesses and plywood designs used i n assembly. The thickness and designs c u r r e n t l y i n use appear to have come l a r g e l y from t r a d i t i o n and there i s no evidence i n the 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 economical veneer thicknesses and plywood designs f o r a m i l l . The problem o f determining them i s very complex since many types of 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 combination of a few veneers s a t i s f y i n g the 'balanced design' and other s t r u c t u r a l s p e c i f i c a t i o n s . The consumption of logs i s dependent on the excess thickness i n plywood and the economics of the 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 given set of veneers and designs are used to 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 , these aspects o f the Plywood Design and Manufacturing (PDM) problem are addressed using a mathematical programming approach. The problem of f i n d i n g the optimal veneer t h i c k n e s s e s , a s s o c i a t e d plywood designs and product mix i s formulated as a no n - l i n e a r mixed int e g e r mathematical programming model. U t i l i z i n g the 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 appropriate v a r i a b l e s to branch on, i t i s demonstrated that the PDM problem can be solved e f f i c i e n t l y through an i m p l i c i t enumeration algorithm i n v o l v i n g a t r e e search procedure. The subproblem to be solved at each f e a s i b l e node of the t r e e i s a Linear 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 s t r u c t u r e . A computer code i s w r i t t e n i n FORTRAN f o r the i m p l i c i t enumeration algorithm. ( i i i ) Data obtained from a plywood m i l l i n B.C. i s analysed using the PDM model and t h i s code. I t i s demonstrated that theaannual net revenue of the m i l l can be s u b s t a n t i a l l y increased through the use of the PDM model. The PDM model i s f u r t h e r extended to 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 species and varying o r d e r f i l e requirements. The model i s reformulated i n each case and i t i s demonstrated that e s s e n t i a l l y the same t r e e search procedure can be used to solve a l l these 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 subproblem to be solved at each node o f the t r e e i s a Generalized Network problem. I t i s shown t h a t t h i s Generalized Network problem can be reduced to a Generalized Transportation problem u t i l i z i n g the s t r u c t u r e o f the c o e f f i c i e n t s and solved as an o r d i n a r y Transportation problem. When the o r d e r f i l e i s dependent on s p e c i e s , the subproblem decomposes i n t o several Linear M u l t i p l e Choice Knapsack problems. I f more than one species o f veneer can be mixed w i t h i n a plywood panel, the 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 to 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. Following the development of the PDM model, methods to determine the e f f i c i e n c y o f plywood designs and the optimum number o f veneer thicknesses f o r a plywood m i l l are developed.. ( i v ) TABLE OF CONTENTS ABSTRACT ( i i ) ACKNOWLEDGEMENTS • ( v i i i ) CHAPTER 1 INTRODUCTION 1 1.1 Intro d u c t i o n 1 1.2 Operations Research i n Plywood Industry .... 6 1.3 Wood Losses i n Plywood. Manufacture 11 CHAPTER 2 FORMULATION OF THE PROBLEM 14 2.1 The Veneer Thickness Problem 14 2.2 Terminology Related to Plywood Designs 16 2.3 The Design and Manufacturing Problem 19 2.4 Formulating the PDM Model 22 2.5 Discussion and Extensions 34 CHAPTER 3 SOLUTION TO THE PROBLEM 38 3.1 S o l u t i o n to the PDM Model 38 3.2 An I m p l i c i t Enumeration Algorithm 40 3.3 Branching Tests and Bounds 48 3.4 Computational Results 60 CHAPTER 4 EXTENSIONS TO MORE SPECIES 63 4.1 Extensions to the PDM Model 63 4.2 O r d e r f i l e Independent of Species 64 4.3 O r d e r f i l e Dependent on Species 81 4.4 Mix of Species w i t h i n a Panel 85 CHAPTER 5 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 (v) CHAPTER 6 NUMBER OF VENEER THICKNESS 101 6.1 The Optimum Number of Veneer Thicknesses 101 6.2 A l t e r n a t e Face Veneer Thickness 110 CHAPTER 7 CONCLUSIONS 113 BIBLIOGRAPHY H 5 APPENDIX I: L i s t i n g of FORTRAN Program 120 APPENDIX I I : Plywood M i l l Data 133 APPENDIX I I I : Results of PDM Model: Four Veneers 136 APPENDIX IV: Results o f PDM Model: Three Veneers 138 APPENDIX V: Results of PDM Model: A l t e r n a t e Face Veneer ... 140 (vi) LIST OF TABLES 1. Number of Veneers f o r Face, Core and Centre 17 2. Example of Plywood Designs with Four Veneers 19 3. P e r m i s s i b l e Construction A l t e r n a t i v e s f o r Three Ply Plywood 27 4. Per m i s s i b l e Construction A l t e r n a t i v e s f o r Five Ply Plywood 28 5. Per m i s s i b l e Construction A l t e r n a t i v e s f o r Seven Ply Plywood 28 6. Per m i s s i b l e Construction A l t e r n a t i v e s f o r Nine Ply Plywood 29 7. Symmetry i n Core/Centre Veneers 54 8. Bounds on x^ and x^ 56 9. PDM Subproblem Example 76 10. Species Lay-up A l t e r n a t i v e s 86 11. Computation of the Upperbound ZQ 107 12. Optimum Number of Veneer Thickness and Design E f f i c i e n c y . . 108 ( v i i ) LIST OF FIGURES 1. A Flowchart of the Algorithm to Solve the PDM Problem 59 2. The PDM Subproblem as a Network Flow Problem 74 3. Feasible Region of Plywood Design Constraint 92 ( v i i i ) ACKNOWLEDGEMENTS I am g r e a t l y indebted to Prof. Derek Atkins and Prof. Daniel Granot, co-chairmen o f my t h e s i s committee, f o r t h e i r valuable guidance, encouragement and as s i s t a n c e during the development of t h i s t h e s i s . I am g r a t e f u l to Prof. G.G. Young and Dr. D.H. Williams of the Faculty of Forestry and Dr. I s r a e l Zang, V i s i t i n g Professor i n Management Science, f o r many h e l p f u l comments and suggestions. Many thanks to Barbara Strouts f o r her prompt and e f f i c i e n t t y p i n g of an otherwise complicated manuscript. I am thankful to my w i f e , Leela and son, Prasad, f o r t h e i r patience understanding and encouragement throughout the w r i t i n g of t h i s t h e s i s . I owe a deep sense o f g r a t i t u d e to my brother Vasu, mother Kamalabai and my l a t e f a t h e r Gururajachar f o r a l l t h e i r help, encouragement and support l e a d i n g to the w r i t i n g o f t h i s t h e s i s . F i n a n c i a l support received from the Canadian Commonwealth Schola r s h i p Committee as well as some support received from NSERC grants A-4181 and A-4743 are g r a t e f u l l y acknowledged. 1 CHAPTER 1 1.1 I n t r o d u c t i o n In the conversion of logs to veneer and plywood, wood l o s s e s occur at several stages due to f a c t o r s such as the s i z e and shape o f logs 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 the l o g input by volume, the remaining being conversion lo s s e s i n the form o f r e s i d u a l s and l o s s e s due to shrinkage and compression. Though much of the r e s i d u a l s i s converted i n t o byproducts such as wood chips and hog fu e l and used elsewhere, t h e i r economic value i s conside r a b l y reduced. In the context o f dwindling f o r e s t resources, i n c r e a s i n g manufacturing costs and f l u c t u a t i n g and competitive market s t r u c t u r e , the 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 timber i s a l l the more greater now than i t was ever before. Though many of the f a c t o r s of wood l o s s are e i t h e r b i o l o g i c a l or t e c h n o l o g i c a l , there are some areas i n plywood manufacture where decision-making or p o l i c y f a c t o r s c o n t r i b u t e 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 the form o f excess thickness i n plywood due to the choice of veneer thicknesses and plywood designs used i n assembly. While the extent o f wood l o s s i n t h i s form may appear to be s m a l l , the savings any improvement i n t h i s area may bring about can prove to be s u b s t a n t i a l . The importance stems from the f a c t t h a t processed wood with an added value from manufacture i s l o s t i n d i r e c t l y . The veneer peel thickness and plywood designs 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 time. There i s no evidence 2 i n the published 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 economical peel thicknesses and plywood designs f o r a m i l l and how to f i n d them. In recent y e a r s , t e c h n o l o g i c a l innovations have lead to improved manufacturing processes i n the plywood i n d u s t r y . Adoption of 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 techniques and computers as aids i n decision-making and process c o n t r o l have brought considerable savings to the i n d u s t r y . Yet, t h i s one aspect o f the wood l o s s and plywood design problem has not been given the a t t e n t i o n i t deserves. The problem of determining the optimum veneer thicknesses and associated plywood designs i s very complex due to the f o l l o w i n g reasons. Many types of plywood are assembled by each plywood m i l l using only a few (three or four) veneer t h i c k n e s s e s . G e n e r a l l y , a l l the plywood so assembled should conform to the 'balanced design', a requirement which regulates t h e d i r e c t i o n o f g r a i n and the order i n which veneers of d i f f e r e n t species and thicknesses can be assembled i n t o plywood. S p e c i f i c a l l y , the assembly of veneers should be such that they are symmetrical about the c e n t r a l p l y ( i e s ) . There should be at l e a s t one balanced design f o r each type of plywood which, i n a d d i t i o n , should 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 of plywood using any design 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 that design a l t e r n a t i v e being f e a s i b l e f o r the balanced design requirement and other s p e c i f i c a t i o n s . When veneer thicknesses are themselves d e c i s i o n v a r i a b l e s , the r e l a t i o n between consumption of logs and the production of veneers i s n o n - l i n e a r . The volumetric wood l o s s i n the form of excess t h i c k n e s s i n plywood, or e q u i v a l e n t l y , the consumption of 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. The economics of the 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 set of veneer thicknesses and 3 plywood designs are u t i l i s e d to form the best product mix subject to c o n s t r a i n t s on resource a v a i l a b i l i t y and product demand. Associated with the veneer thickness and plywood design problem described above i s the problem of determining the optimum number of veneer thicknesses f o r a m i l l . G e n e r a l l y , the higher the number of veneer thicknesses used by a m i l l , the lower i s the wood l o s s and l o g c o s t s . However, a higher number of veneer thicknesses i s associated with higher setup costs i n p e e l i n g , d r y i n g , storage, lay-up, handling and record-keeping. On the other hand, the lower the number of veneer t h i c k n e s s e s , the higher i s the wood l o s s and l o g c o s t s , but the setup costs are reduced c o n s i d e r a b l y . The problem of determining the optimum number of veneer thicknesses 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 of the veneer t h i c k n e s s , plywood design and product mix problem. In t h i s d i s s e r t a t i o n , we address the above aspects of these two problems using a q u a n t i t a t i v e approach. We formulate the problem of determining the veneer t h i c k n e s s , associated plywood designs and product mix as a mathematical 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 of the 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 of the 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 to 0-1 values. The model takes on d i f f e r e n t forms depending on the f a c t o r s considered i n the problem and m i l l p r a c t i c e s . Some of the general problems are considered and, i n each case, s o l u t i o n procedures which e x p l o i t the s t r u c t u r e i n the model are developed. A computer code i s w r i t t e n and data obtained from a plywood m i l l i s analysed to demonstrate the s u i t a b i l i t y o f the model. To our knowledge, there are no l i t e r a t u r e references which d i r e c t l y consider the e v a l u a t i o n of optimum veneer t h i c k n e s s e s , plywood 4 designs and product mix. Whatever l i t t l e work done i n t h i s f i e l d r e l a t e s to the study of strength or s t r u c t u r a l p r o p e r t i e s of known thicknesses and designs (Colebeck and Northcott (1958), N o r r i s , Werren and McKinnon (1961)). However, there i s considerable work done i n the area of optimum product mix f o r plywood using, mostly, l i n e a r programming techniques. We consider t h i s and other Operations Research a p p l i c a t i o n s i n the 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 area of wood los s e s i n plywood production. We b r i e f l y describe t h i s work as w e l l , analyse the f a c t o r s o f wood l o s s and o p t i m i -z a t i o n techniques a s s o c i a t e d with them i n s e c t i o n 1.3. The o r g a n i s a t i o n of the 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 f o l l o w s . In chapter two we present the terminology and f a c t o r s a s s ociated with the plywood manufacturing process and the design problem. We define 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 of these v a r i a b l e s . We f u r t h e r demonstrate how the balanced design requirement can be achieved through the development of a set o f design 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 and we formulate the Plywood Design and Manufacturing (PDM) problem as a no n - l i n e a r mixed i n t e g e r mathematical programming model. We present an i m p l i c i t enumeration algorithm f o r s o l v i n g the PDM problem i n chapter t h r e e . 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 PDM problem, we demonstrate how the s i z e of the search t r e e , employed by the i m p l i c i t enumeration a l g o r i t h m , can be s u b s t a n t i a l l y reduced. We f u r t h e r show t h a t the subproblem to be solved at each node o f t h i s search t r e e . i s a Linear 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 s t r u c t u r e . We describe a computer code w r i t t e n i n FORTRAN f o r the i m p l i c i t 5 enumeration algorithm to solve the PDM problem. Using t h i s code, we analyse data from a plywood m i l l i n B r i t i s h Columbia and show th a t f o r the 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 the problem parameters, the s o l u t i o n obtained from our model i s f a r s u p e r i o r than those c u r r e n t l y used. Extension o f the PDM model to 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 species and varying o r d e r f i l e r e s t r i c t i o n s are considered i n chapter four. The PDM model i s reformulated i n each case and i t i s demonstrated that e s s e n t i a l l y the same t r e e search procedure can be used to solve a l l these 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 to be solved at each node of the t r e e i s a Generalized Network problem. I t i s shown that t h i s Generalized Network problem can be reduced to a Generalized Transportation problem f o l l o w i n g some pro p e r t i e s of the PDM model. When the o r d e r f i l e i s dependent on s p e c i e s , the subproblem decomposes in t o several l i n e a r m u l t i p l e choice knapsack problems whose s o l u t i o n can be given e x p l i c i t l y . I f more than one species of veneer can be mixed w i t h i n a panel, the subproblem i s a l i n e a r program. In chapter f i v e , we show how the PDM model can be n a t u r a l l y cast as a s p e c i a l case of a D i s j u n c t i v e Programming problem. We analyse the c h a r a c t e r i s t i c s of 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 chapter s i x , we consider the problem of determining the optimum number of veneer thicknesses f o r a plywood m i l l . Using computer codes of the i m p l i c i t enumeration algorithm and data from the plywood m i l l we d e r i v e the maximum revenue associated with various number of veneer t h i c k n e s s e s . Together with hypothetical data on setup costs we demonstrate 6 how the optimum number of veneer thicknesses can be determined. 1.2 Operations Research In The Plywood Industry Extensive information on Operations Research (OR) techniques i n f o r e s t r y and the 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 other processing i n d u s t r i e s using wood as the raw m a t e r i a l ) are a v a i l a b l e from the b i b l i o g r a p h i e s of Bare (1971), Martin and Sendak (1973) and F i e l d (1976, 1977). However, much of the work reported i n these sources r e f e r s to a p p l i c a t i o n o f OR techniques i n f o r e s t r y operations 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 other s i m i l a r areas. A general a p p r a i s a l o f p o s s i b l e OR a p p l i c a t i o n s i n the f o r e s t products i n d u s t r y mentioning the areas where these can be e f f e c t i v e l y used i s given by Holmes (1976). Diaz (1974) has prepared a ' b i b l i o g r a p h y ' of OR i n lumber production and 'other f o r e s t r y i n d u s t r i e s ' . However, his b i b l i o g r a p h y s u f f e r s from the drawback that 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 exhaustive of a l l OR a p p l i c a t i o n s i n the i n d u s t r y . In order to complement these sources with recent l i t e r a t u r e and unreported use of OR i n the i n d u s t r y , with p a r t i c u l a r reference to plywood and s a w m i l l i n g , a survey was undertaken i n B r i t i s h Columbia (Raghavendra (1979)). OR work r e l a t e d to the plywood i n d u s t r y as found from published l i t e r a t u r e and i n d u s t r i a l p r a c t i c e s i n B r i t i s h Columbia are b r i e f l y described i n t h i s s e c t i o n . The d e s c r i p t i o n i s by the technique employed f o r s o l u t i o n such as Linear Programming, Dynamic Programming and S i m u l a t i o n . c 7 1.2.1 Linear Programming A s . i s the case i n many other i n d u s t r i e s , the most widely used OR methodology i n plywood i n d u s t r y i s Linear Programming (LP). From simple t r a n s p o r t a t i o n problems to complex opera t i o n a l planning models i n v o l v i n g l o g a l l o c a t i o n , production scheduling and d i s t r i b u t i o n , 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 contexts 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 (1974), Martin and Sendak (1 973) and F i e l d (1976, 1977)]. A p p l i c a t i o n o f Linear Programming to plywood manufacture has been considered by Bethel and Harrel (1957), Koenigsberg (1960), Donnelly (1966), Ramsing (1965, 1968), Everett (1967) and Lee (1968). An extensive 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 production i s described by Kotak (1976). The model, developed f o r the plywood and hardboard d i v i s i o n of Canadian Forest Products, New Westminster, BC and in use s i n c e 1969, b a s i c a l l y determines an optimum balance between the a v a i l a b l e wood mix and o r d e r f i l e requirements so as to maximize the c o n t r i b u t i o n margin of the d i v i s i o n . The model serves as a basis f o r an annual operating plan g i v i n g d e t a i l s of production schedules, keeping t r a c k of 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 and t a r g e t s on a biweekly b a s i s . The s t r a t e g i e s are f u r t h e r r e v i s e d on a day to day basis with schedules f o r log p e e l i n g , veneer d r y i n g , press production and shipping 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 cumulative performance up to the day. A f i n a n c i a l variance a n a l y s i s i s prepared to analyze the e f f e c t of v a r i a t i o n s due to p r i c e and mix of raw-materials on the operating income. The net c o n t r i b u t i o n margin of the d i v i s i o n 8 i s reported to have increased by an average of one m i l l i o n d o l l a r s per year during the period 1969-1975 due to the use of t h i s model. Wellwood (1970) describes how the o r d e r f i l e requirements of d i f f e r e n t types of plywood can be l i n k e d to press, d r y i n g and l a t h e schedules using LP. His a r t i c l e i s s i m i l a r to that of Kotak but gives greater d e t a i l on scheduling m a t e r i a l flow at d i f f e r e n t processing centres and has the o b j e c t i v e of minimizing the penalty associated with unused veneer and downgrading of veneer. Yaptenco and Wylie (1970) consider a h y p o t h e t i c a l 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 production scheduling problem of 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 , patchers and glue spreaders i s brought down to l i n e a r programming formulation through 'algebraic and d i f f e r e n c e equations'. Dobson (1971) describes the use of LP f o r the a l l o c a t i o n of logs to plywood, sawmill and open market s a l e s . There are several other LP models that are a c t u a l l y i n use i n m i l l s but are not reported i n the l i t e r a t u r e (Raghavendra (1979)). Many of these models are meant f o r intermediary stages of manufacture and t h e i r r e s u l t s may hot prove to be optimal when the o v e r a l l problem i s considered. Nevertheless, these serve..theypurpose of a n a l y s i n g e f f i c i e n c y or 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. One such LP i s used to give l a t h e schedules f o r p e e l i n g d i f f e r e n t species and veneer thicknesses f o r each one of the l a t h e s with the 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 e f f i c i e n c y i n d r y i n g s i n c e the d r y i n g process i s a bottleneck i n plywood manufacture. The d r y i n g time of veneer depends on the s i z e and type o f d r y e r , the number of decks, species and t h i c k n e s s of veneer and the o b j e c t i v e of the LP i s to minimize t o t a l drying time. A t h i r d LP model i s 9 designed to give optimum pressing schedules on a weekly basis with c o n s t r a i n t s on pressing time, glue-spreader press c o n f i g u r a t i o n s , o r d e r f i l e requirements and setup time between batches. There are several other areas i n the i n d u s t r y where LP models have been used or recommended f o r use. S i t t e r (1969) describes i n general how 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 of 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 to determine the l e a s t cost energy mix f o r a f o r e s t products complex c o n s i d e r i n g a l t e r n a t i v e sources from m i l l r e s i d u e s , o i l , natural gas and s o l a r energy. Holecek (1975) and M c K i l l o p (1974) describe the use of 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 i n C a l i f o r n i a . 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 of veneer peeling and d r y i n g only. Resch and Scheurman (1977) simulated the softwood veneer drying operation so as to determine the optimum flow o f veneer through two j e t dryers. Simulated computer runs demonstrated the e f f e c t of veneer t h i c k n e s s , d r y i n g temperature, and p r e - s o r t i n g of veneer on d r y i n g time and c o s t s . Tobin and Bethel (1969) describe an a n a l y t i c a l procedure to evaluate the q u a l i t y and q u a n t i t y o f veneers recoverable i n the r o t a r y c u t t i n g process using s i m u l a t i o n . 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 analysing hypothetical problems r e l a t e d to l o g supply from several areas with n o n - l i n e a r c o s t s . Pnevmaticos , and Mann (1972) and Briggs (1978) demonstrate that Dynamic Programming can be used f o r bucking of t r e e - l e n g t h or long logs i n t o small l o g s . When the values of small logs are known, the c u t t i n g process i s viewed as a sequential d e c i s i o n process in which the optimal values from bucking at several 'stages' of the l o g length are determined through 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 to determine the optimum l a t h e s e t t i n g s f o r producing the best q u a l i t y of veneer. Three basic v a r i a b l e s , namely, k n i f e rake angle, r o l l e r bar compression and k n i f e r o l l e r lead i n the veneer peeling process are r e s t r i c t e d by physical c o n s i d e r a t i o n s as l i n e a r c o n s t r a i n t s . The o b j e c t i v e f u n c t i o n i s based on the formation and s e v e r i t y of l a t h e checks.in the veneer and i s expressed as a non-linear f u n c t i o n of the three d e c i s i o n v a r i a b l e s . For various process parameters such as s t r e s s , shear and strength in the c u t t i n g process, the optimum l a t h e s e t t i n g s are derived using a Non-Linear Programming model. 11 1.2.5 Other OR Techniques Tyre and Scr e p e t i s (1978) propose an inventory system based on weight s c a l i n g o f roundwood logs and describe a method f o r c o n t r o l o f veneer, sawntimber and pulpwood volumes. T r a d i t i o n a l inventory c o n t r o l techniques such as EOQ are reported to be used (Raghavendra (1979)) i n the area o f l o g boom c o n t r o l , in-process veneer or lumber inventory and f i n i s h e d products inventory. Ramalingam (1976) i s reported ( i n Briggs (1978)) to have used a Branch and Bound approach to the t r e e bucking problem. Successive reductions of the t r e e stems or long logs 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 d i s c r e t e s t a t e Markov Process by Luken (1978) (as reported i n Luken et a l . (1980)). Economic impact of f o r e s t based i n d u s t r i e s i s analysed using L e o n t i e f input-output models by Reimer (1969) and Raizada and Nautiyal (1974). 1.3 Mood Losses In Plywood Manufacture The foregoing a n a l y s i s o f l i t e r a t u r e sources and actual p r a c t i c e o f OR i n the plywood i n d u s t r y i n d i c a t e the wide v a r i e t y o f problems t h a t can be t a c k l e d through some facet of the OR methodology. Examined c l o s e l y , the success o f many OR techniques l a y i n t h e i r a b i l i t y to give e f f i c i e n t o p e r a t i o n a l 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 the conversion o f tre e s to l o g s , veneer and plywood. Even when the o b j e c t i v e o f an OR model i s the maximization o f value, 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 the p r i n c i p a l raw-material namely, wood. This leads us to the basic questions: why 12 wood l o s s e s occur i n the conversion o f logs to veneer and plywood and how can i t be prevented or reduced. Extensive information on the areas o f wood l o s s i n plywood manufacture, f a c t o r s r esponsible for,them and the extent o f these lo s s e s are a v a i l a b l e i n the l i t e r a t u r e [ Dobie and Hancock (1972), Woodfin (1973), Nagaraju, Raghavendra and Venkataraman (1974), Meriluoto (1965), Heiskanen (1966), B r a c k l e y (1968), Baldwin (1975) and Wood (1962)]. Many of the f a c t o r s o f wood l o s s i n plywood are i n t e r a c t i v e . However, they can be broadly c l a s s i f i e d i n t o three c a t e g o r i e s : i ) B i o l o g i c a l F a ctors: Due to the inherent nature and v a r i a b i l i t y of the raw m a t e r i a l . Losses due to rounding of l o g s , defects i n wood, shrinkage i n dry i n g and compression i n pressing can be c l a s s i f i e d to f a l l i n t o t h i s category; i i ) Technological Factors: Due to processing l i m i t a t i o n s or n o n - a v a i l a b i l i t y of b e t t e r technology. Core losses which occur due to the l i m i t a t i o n of the s p i n d l e i n veneer l a t h e , rounding lo s s e s due to centering e r r o r s , l o s s due to spur t r i m and handling lo s s e s are some of the t e c h n o l o g i c a l f a c t o r s r e s p o n s i b l e f o r wood losses i n plywood manufacture and i i i ) Decision-Making/Design F a c t o r s : Due to manufacturing designs 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 setup. At l e a s t part o f the bucking l o s s e s i n the l o g ya r d , trimming lo s s e s caused by o v e r s i z e allowance f o r veneer and l o s s i n the form o f excess thickness i n plywood are some of the 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 . I n v e s t i g a t i o n of the b i o l o g i c a l or t e c h n o l o g i c a l f a c t o r s of wood l o s s e s i n plywood manufacture .is beyond the scope o f t h i s d i s s e r t a t i o n . Among the decision-making or design f a c t o r s r esponsible f o r wood l o s s e s , e v a l u a t i o n o f some of these f a c t o r s with an economic o b j e c t i v e has receiv e d some a t t e n t i o n i n recent years. The problem o f log bucking f o r maximising value, though not s p e c i f i c a l l y in the context o f plywood manufacture, has been studied by Conway (1978), Pnevamaticos and Mann (1972), Briggs (1978), Ramalingam (1976), Lefebvre (1978) and Western Forest Products Laboratory (1978). Evaluation o f the t r i m allowance i n veneer f o r plywood has been studied to some extent by Hawkins and Clarke (1970) and Raghavendra and Nagaraju (1975). The concept o f e v a l u a t i n g veneer peel thicknesses and plywood designs f o r economic o p t i m a l i t y i s , to our knowledge, not considered i n the 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 the development of a q u a n t i t a t i v e technique to de r i v e the optimum veneer t h i c k n e s s e s , plywood designs and product mix. The c h a r a c t e r i s t i c s of t h i s problem are described i n the next chapter. CHAPTER 2 14 2.1 The Veneer Thickness Problem Most plywood m i l l s manufacture a v a r i e t y of plywood types, each v a r y i n g , apart from species and surface q u a l i t y c h a r a c t e r i s t i c s , i n the number, thickness and order of veneers or p l i e s and the t o t a l t h i c k n e s s of plywood. In any one m i l l logs are peeled to one of t h r e e , four or f i v e basic veneer thicknesses and a l l the plywood types are assembled as some i n t e g r a l m u l t i p l e combination of these veneers. The veneers as well as the 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 , surface q u a l i t y and other f a c t o r s . Most of the plywood so produced should a l s o conform to the "Balanced Design", a requirement which regulates the order i n which veneers of d i f f e r e n t species and thicknesses can be assembled i n t o plywood. The use of a l a r g e number of veneer peel thicknesses tends to increase the cost o f plywood because of the added costs of p e e l i n g , d r y i n g , storage, handling, lay-up and record-keeping involved (Colebeck and Northcott (1958)). A l t e r n a t i v e l y , i n a b i l i t y to manufacture c e r t a i n types w i t h i n plywood thickness 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 o f most plywood types ending up i n higher than required thickness and requirement, of s p e c i f i c customer orders dactate the use of two or more peel t h i c k n e s s e s . For example, the two plywood types 5 p l y 20.5 mm having 5 veneers adding up to 20.5 mm thickness and 7 ply 20.5 mm having 7 veneers adding up to the same thic k n e s s cannot both be assembled using a s i n g l e veneer thickness w i t h i n a t o l e r a n c e of - 0.5 mm. 15 S i m i l a r l y , a 7 p l y 20.5 mm plywood and a 7 ply 23.5 mm plywood cannot both be assembled to w i t h i n + 0.5 mm using a s i n g l e veneer t h i c k n e s s . The economics of plywood manufacture depend not only on the costs of 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 designs used i n the assembly of 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 are being e x t e n s i v e l y used [ Dobson (1971), Kotak (1976), Lee (1968) and Ramsing (1965, 1968)] mainly centre around optimal product mix o f log grades, species and veneer lay-up a l t e r n a t i v e s . However, one basic information which goes i n t o these LP models as input i s the e x i s t i n g set of veneer peel t h i c k n e s s e s , associated y i e l d f a c t o r s and design 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 evidence i n published l i t e r a t u r e to c l a i m that the veneer thicknesses and the associated designs so used are t r u l y the best f o r any p a r t i c u l a r m i l l . Most of the plywood designs c u r r e n t l y used appear to 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 manufacturing requirements" (Colebeck and Northcott (1958)). The foregoing 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 questions: (1) what i s the optimum number of veneer thicknesses f o r a m i l l , and (2) given the number of veneer t h i c k n e s s e s , what should be these thicknesses and what plywood designs 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 are the best f o r a m i l l . The answer to the f i r s t question depends on how best the "best" veneer thicknesses are f o r each number i n the second problem and how responsive the changes i n costs are to the a d d i t i o n of each veneer t h i c k n e s s . Both these, i n t u r n , are 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 other resource r e s t r i c t i o n s . 16 I t i s demonstrated i n t h i s chapter that the problem o f f i n d i n g the optimum veneer thicknesses can be formulated as an o p t i m i z a t i o n model which i s a mixed 0-1 non- l i n e a r programming problem. Some of the terminology r e l a t e d to plywood designs and the manufacturing process are described i n s e c t i o n 2.2. The veneer t h i c k n e s s , plywood design and product mix problem i s described i n terms o f t h i s terminology in s e c t i o n 2.3. The model formulation with d e t a i l s of the d e c i s i o n v a r i a b l e s , the c o n s t r a i n t s and the o b j e c t i v e f u n c t i o n are presented i n s e c t i o n 2.4. In se c t i o n 2.5 a d i s c u s s i o n o f the c h a r a c t e r i s t i c s of the 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 extensions are presented. 2.2 Some Terminology Related to Plywood Design and Manufacture Before presenting the mathematical formulation o f the plywood design problem some of- the terminology associated with the manufacture of plywood are considered. G e n e r a l l y , most plywood sheets c o n s i s t o f an odd number o f l a y e r s or p l i e s o f veneer bonded together by an adhesive i n such a way that the gra i n d i r e c t i o n of adjacent p l i e s i s at r i g h t angles to each other. In recent y e a r s , however, plywood assembled from an even number o f p l i e s i s a l s o being made (Paras i n (1976), COFI (1978)). In a plywood sheet with odd number of p l i e s , counting from the top or bottom veneer, a) The f i r s t and the l a s t veneers are c a l l e d face veneers, b) A l l even numbered veneers with g r a i n d i r e c t i o n perpendicular to that o f the face are cal1ed core veneers or cross-band and 17 c) A l l other odd numbered veneers, i f any, with g r a i n d i r e c t i o n 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 of veneers f o r fa c e , core and centre i n plywood sheets having odd number of 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 even-ply plywood, the fa c e , core and centre veneers are s i m i l a r l y defined r e l a t i v e to t h e i r p o s i t i o n and alignment of grain d i r e c t i o n . # o f p l i e s i n plywood # of veneers rfor Face Core 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) T a b l e ! : Number of Veneers f o r Face, Core and Centre A s p e c i f i c a t i o n which describes the number, t h i c k n e s s , species o f veneer and the order i n which they are assembled i n t o a plywood sheet i s c a l l e d the design or c o n s t r u c t i o n o f plywood. Since the number of veneers f o r f a c e , core and centre p l i e s can be determined once the number of p l i e s i n plywood i s known (Table 1 ) , a design can be s p e c i f i e d by d e s c r i b i n g the thickness and species f o r each one of fa 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 that w i t h i n a plywood panel, the s p e c i e s , thickness and d i r e c t i o n of gra i n of veneers should be symmetrical about the c e n t r a l p l y ( i e s ) . This i s s t i p u l a t e d from 18 c o n s i d e r a t i o n s r e l a t e d to the strength p r o p e r t i e s and warping of the panel. Together with general m i l l p r a c t i c e s , a balanced design i m p l i e s that w i t h i n a plywood panel a) a l l face veneers should be of the same thickness and s p e c i e s , b) a l l core veneers should be of the same thickness and s p e c i e s , c) a l l centre veneers should be of the same thickness and species and d) the species and/or thickness i n any one group a ) , b) or c) above might be the same as those of the o t h e r ( s ) . An unbalanced design or modified c o n s t r u c t i o n r e f e r s to plywood panels which vary from the requirements f o r a balanced design i n th a t the g r a i n d i r e c t i o n , species and/or thickness of inner p l i e s may be unbalanced about the c e n t r a l p l y ( i e s ) . B r i e f l y , the process o f plywood manufacture invol v e s the conversion of logs to veneer, drying the veneer to remove excessive moisture and g l u i n g , assembling and pressing the veneers to form a plywood panel. The thickness of veneer before the drying process i s c a l l e d the green  thickness while that a f t e r d r y i n g i s c a l l e d the dry t h i c k n e s s . Plywood thicknesses g e n e r a l l y r e f e r to the thickness of the f i n i s h e d panel ready f o r market, a f t e r accounting f o r sanding l o s s e s , i f necessary. Throughout our a n a l y s i s veneer thicknesses r e f e r to the green thickness and plywood thicknesses r e f e r to the thickness of 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 type i s normally designated by the number of p l i e s i t contains and the thickness o f plywood. Thus,a 7 p l y 20.5 mm plywood implies that t h i s plywood has seven veneers i n i t and that i t s thickness i s 19 20.5 m i l l i m e t r e s . Using the notations and terminology introduced above, we can now describe the plywood design and manufacturing problem. 2.3 D e s c r i p t i o n o f the Plywood Design and Manufacturing Problem Consider an example of four veneer peel thicknesses and a sample of plywood types and associated designs as presented i n Table 2. These are taken from the actual p r a c t i c e s o f a plywood m i l l i n B r i t i s h Columbia Plywood P l i e s - Design Veneers f o r Plywood Excess(+) Type Thickness(mm) A l t e r n a t i v e Face Core Centre Thickness(mm) or inches Green Dry 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% ( i i ) 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% ( i i ) 1/10 1/7 1/7 17.27 16.24 +4.77% ( i i i ) 1/8 1/8 1/8 16.76 15.76 +1 .68% ( i v ) 1/8 1/7 1/10 17.32 16.28 +5.03% 5 7 p l y 18.5 (D 1/10 1/10 1/8 20.16 18.96 +2.49% ( i i ) 1/10 1/8 1/10 20.83 19.58 +5.84% ( i i i ) 1/8 1/10 1/10 20.16 18.96 +2.49% Table 2: Plywood Designs with Four Veneer Thicknesses; Veneer Thickness i n mm (inches) are: 2.69 (1/10), 3.35 (1/8), 3.96 (1/7) and 4.98 (3/16). . 20 whose design and manufacturing problem w i l l be f u l l y considered i n chapter 3. The veneer thicknesses used were 2.69, 3.35, 3.96 and 4.98 mm corresponding approximately to 1/10, 1/8, 1/7 and 3/16 of an inch r e s p e c t i v e l y . For some plywood types there i s more than one design a l t e r n a t i v e , which i s described i n the form of veneers f o r face, core and centre p l i e s . The 'green' thickness of plywood represents the sum t o t a l o f the thickness o f veneer i n them while the 'dry' thickness r e f e r s to the actual f i n a l t h i c k n e s s o f plywood a f t e r accounting f o r loss e s due to shrinkage i n d r y i n g , compression in pressing and glue-l i n e a d d i t i o n s . The l a s t column, excess or shortage, represents the percentage d e v i a t i o n from the intended thickness o f the plywood type induced by the choice o f design. Within p e r m i s s i b l e t o l e r a n c e s , shortages preceeded by a negative s i g n i n d i c a t e savings i n wood while excesses 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 actual extent o f wood l o s s or gain can be computed by m u l t i p l y i n g the absolute d e v i a t i o n s with the 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 that the volume o f wood l o s s or gain depends not only on the veneer thickness and plywood design but a l s o on how e f f i c i e n t l y the designs are used to meet the o r d e r f i l e under c o n s t r a i n t s of log a v a i l a b i l i t y , machine c a p a c i t i e s and other m i l l r e s t r i c t i o n s . This leads to two aspects o f the problem, namely, (1) the veneer thickness and design problem and (2) the product- mix and manufacturing problem. In the veneer thickness and design problem, the l i s t of plywood types, t h e i r s p e c i f i c a t i o n s and the number of veneer peel thicknesses are known. The o b j e c t i v e i s : 21 ( i ) To determine the thickness and ( i i ) To s p e c i f y how these veneers should be assembled so as to produce a l l types o f plywood w i t h i n s p e c i f i c a t i o n . On the other hand, i n the product mix and manufacturing problem the veneer t h i c k n e s s e s , the plywood designs, the a v a i l a b i l i t y o f l o g s , the demand f o r end products and other m i l l r e s t r i c t i o n s are known. The o b j e c t i v e i s : ( i ) To f i n d the optimum qua n t i t y o f veneers of each species and thickness to be produced and ( i i ) To f i n d the optimum q u a n t i t y o f plywood to be assembled under each design a l t e r n a t i v e . The two aspects o f the problem are i n t e r r e l a t e d as veneer th i c k n e s s and plywood designs are input to the product mix part o f the problem. The o b j e c t i v e f o r both problems combined would be ( i ) to minimize i m p l i c i t wood l o s s i n the form of excess thickness i n plywood which, as a r e s u l t , would minimize t o t a l l o g consumption, or ( i i ) to maximize net revenue. When more than one species with varying l o g costs are used, the 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 give the same r e s u l t s . This can happen, f o r example, when an expensive species has a lower y i e l d compared to an inexpensive species. Recognizing t h a t the u l t i m a t e t r a d e - o f f can be measured i n terms of value, we use maximizing of net revenue as the o b j e c t i v e . The product mix part o f the problem can be solved using l i n e a r programming (Kotak (1976), Dobson (1971), Lee (1968) and Ramsing (1965, 1968)) which i s now an accepted m i l l p r a c t i c e . However, the veneer thickness and plywood design problem as presented above has not been considered to date. Whatever l i t t l e research has been undertaken i n 22 t h i s regard r e l a t e s to stre n g t h or s t r u c t u r a l p r o p e r t i e s o f known thicknesses and designs (Colebeck and Northcott (1958), N o r r i s , Werren and McKinnon (1961) and B i b i i s , Hsu and Chiu (1972)). In t h i s t h e s i s , we consider simultaneously both aspects o f t h i s problem, r e f e r r e d to as the Plywood Design and Manufacturing (PDM) problem. In the PDM problem we seek the veneer t h i c k n e s s e s , associated plywood designs and q u a n t i t i e s o f veneers and plywoods to be produced which w i l l maximize the net revenue f o r a m i l l . 2.4 Formulating the Model To s i m p l i f y the p r e s e n t a t i o n , we do not consider 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 , log grades, surface q u a l i t y of veneers, plywood grades and machine c a p a c i t i e s i n the formulation o f the PDM model i n t h i s chapter. These f a c t o r s can be e a s i l y incorporated i n the model as demonstrated by the e x i s t i n g plywood L.P. models (Lee (1968), Ramsing (1965, 1968), Kotak (1976)). Further, though we consider i n our formulation plywood designs with an odd number of veneers having balanced designs, our model can be extended to even-ply c o n s t r u c t i o n and/or unbalanced designs. We discuss the i m p l i c a t i o n s o f some of these i n s e c t i o n 2.5. 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 types having up to nine p l i e s and three veneer thicknesses but can be extended to any number of p l i e s and any number of veneer thick n e s s e s . Sizes o f veneer and plywood sheets are expressed i n equivalents o f the standard s i z e of plywood (8 1 x 4' or 2.44 m x 1.22 m). Decision V a r i a b l e s Let K be the number of veneer thicknesses i n general and N be the number of plywood types. Define the 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 : t h x^ = k veneer thickness ( 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 the thickness of the face veneers, L^ = Quantity of logs peeled i n t o veneer thickness x^ ( i n cubic metres), = Quantity of veneer sheets produced of thickness x^ ( i n number o f sheets o f s i z e equivalent to 2.44 m x 1.22m s i z e of plywood), P. .= Quantity o f plywood o f type i produced using 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 of standard s i z e 2.44 m x 1.22 m or 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 using c o n s t r u c t i o n a l t e r n a t i v e j k = l , 2 . . . k ; j = l , 2 . . . n . and i = 1,2...N. Genera l l y , the number of veneer t h i c k n e s s e s , K, used i n most m i l l s i s three or four and seldom more than four peel thicknesses are used. The number of design a l t e r n a t i v e s , n., depends on the number o f p l i e s i n plywood. The Constraints In t h i s s e c t i o n we w i l l describe the various c o n s t r a i n t s a r i s i n g i n the PDM problem. a) Veneer_Jhickness Jolerance and_Constraints Tolerances f o r veneer thickness are normally l a i d down i n company standards or s p e c i f i c a t i o n s r e l a t e d to the product. Council of Forest I n d u s t r i e s of B r i t i s h Columbia (COFI (1978)) standards f o r e x t e r i o r plywoods, f o r example, s p e c i f y , tolerances f o r face veneers separately from those f o r core or centre veneers. To e s t a b l i s h tolerances f o r veneer t h i c k n e s s e s , we should f i r s t consider the c a p a b i l i t i e s o f the veneer pe e l i n g l a t h e of a m i l l . Most peeling l a t h e s can produce only a d i s c r e t e l y f i n i t e set of veneer t h i c k n e s s e s . However, some peeling l a t h e s might be capable of producing veneer thicknesses i n a continuous range of values. But, veneer thicknesses beyond a c e r t a i n degree of accuracy may not be p o s s i b l e from p r a c t i c a l c o n s i d e r a t i o n s . Thus, we l e t the veneer thickness take only d i s c r e t e v a l u e s , i f necessary, by transforming the range of peel thicknesses i n t o a d i s c r e t e set i n steps 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 set c o n t a i n i n g a l l p o s s i b l e veneer thicknesses the peeling l a t h e ( s ) of a m i l l can produce. Of t h i s , l e t T^ and T £ be the subset representing a l l thicknesses w i t h i n tolerances s p e c i f i e d f o r face and core/centre veneers r e s p e c t i v e l y . Then x k e T k = U k \ T k 2, T k m ( k ) } , k = 1,2,...K (2. where m(k) i s the number of elements i n T.. i f k = 1 and that i n T f c otherwise. Since x k can take on only one value i n T k, i t can be expressed as f o l l o w s : 25 (2.1 A) m(k) m=l km = 1 (2.IB) i f x m and km (2.1 C) 0 otherwise C l e a r l y , i n a l l the c o n s t r a i n t s of the PDM model i n which x k appears i t can be replaced by the r i g h t hand s i d e o f (2.1A) and i n th a t case the other two c o n s t r a i n t s (2.IB) and (2.1C) are to be included e x p l i c i t l y . 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 that can assume one value from the set as expressed by (2.1). b) P1_ywood_Jhickness_ Con s t r a i n ^ These are perhaps the most complicated c o n s t r a i n t s i n the PDM problem due to the f o l l o w i n g reasons: i ) A l l the N plywood types are to be assembled using some per m i s s i b l e combination o f one or more of. the x^'s, with the number of p l i e s adding to 3, 5, 7 or 9 (or higher, i f the case d i c t a t e s ) ; i i ) There should be at 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 the balancing requirement and thickness t o l e r a n c e f o r each plywood type; , i i i ) There might be more than one plywood type having the same number of 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 i v ) There might be more than one plywood type having the same thickness but d i f f e r i n g i n the number of p l i e s . A l l o f these problems were overcome i n our formulation by a c a r e f u l e v a l u a t i o n o f the balanced c o n s t r u c t i o n requirement. Since a l l the plywood types w i l l have face veneers, the convention that i s the face veneer leads to the f a c t that 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 have at l e a s t two veneers o f x^. A n a l y s i s o f the balancing requirement with the number of veneers required f o r f a c e , core and centre veneers i n d i c a t e that there are only a few pe r m i s s i b l e combinations of veneers i n which a plywood of a given number of p l i e s can be assembled. I f K=3, a 3 p l y plywood can be assembled i n one of 3 a l t e r n a t e ways and there are 9 p o s s i b l e ways f o r each one of higher p l y c o n s t r u c t i o n having odd number o f p l i e s (5, 7, 9 or higher odd). These perm 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 the veneers f o r face, core and c e n t r e , using some or a l l of the three veneer thicknesses are l i s t e d i n Tables 3, 4, 5 and 6 f o r plywood made of 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 f o r any number of veneer thicknesses and/or plywood made with any number of p l i e s . Using Tables 3-6, the 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 the f o l l o w i n g set of 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 using c o n s t r u c t i o n a l t e r n a t i v e j (2.4) 0 Otherwise f o r a l l j = 1,2 , . . . n. and i= 1, 2, ... N, 27 where, a .. = iNumber o f veneers o f thic 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 type i ; The a ^ ^ are taken from Tables 3 - 6, b1-'"(b.^) - Lower (Upper) to l e r a n c e f o r thickness o f plywood type i , M = A l a r g e p o s i t i v e number, n.. = Number of pe 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 the i * * 1 type o f plywood. Construction A l t e r n a t i v e ( j ) x l Number X2 of Veneers of X 3 Face Veneers f o r Core Centre 1. 3 0 0 x l X l 2 2 1 0 x l X2 3 2 0 1 x l X 3 Table 3: Permissible Construction A l t e r n a t i v e s f o r Three Ply Plywood 28 Construction A l t e r n a t i v e ( j ) Number of Veneers of x 2 X 3 Face Veneers f o r Core Centre 1 5 0 0 x l x l x l 2 4 1 0 x l X l x 2 3 4 0 1 x l X l X 3 4 . 3 2 0 x l X2 X l 5 3 0 2 X 3 X l 6 2 3 0 x l X2 X2 7 2 0 3 X-i X 3 X 3 8 2 2 1 x l X2 X 3 9 2 1 2 X , X 3 X2 Table 4: Perm i s s i b l e Construction A l t e r n a t i v e s f o r Five p ly Plywood. Construction Al t e r n a t i v e ( j ) Number of Veneers of x2 X 3 Face Veneers for Core Centre 1 7 0 0 x l x l x l 2 5 2 0 X , x l X2 3 5 0 2 x l x l X 3 4 4 3 0 x l X2 X l 5 4 0 3 x l X 3 X l 6 2 5 0 x l X2 X2 7 2 0 5 x l X 3 X 3 8 2 3 2 x l X2 X 3 9 2 2 3 x l X 3 X2 Table 5: Pe r m i s s i b l e Construction A l t e r n a t i v e s f o r Seven Ply Plywood 29 Construction A l t e r n a t i v e ( j ) V : x l Number of X2 Veneers of X 3 Face Veneers Core f o r Centre 1 9 0 0 X , x l X l 2 6 3 0 x l x l X2 3 6 0' 3 x l x l X 3 4 5 4 0 x l X2 X l 5 5 0 4 X , X 3 X1 6 2 7 0 x l X2 x 2 7 2 0 7 X , X 3 X 3 8 2 4 3 x l X2 X 3 9 2 3 4 X , X 3 X2 Table 6: P e r m i s s i b l e Construction A l t e r n a t i v e s f o r Nine Ply Plywood Observe t h a t i n c o n s t r a i n t (2.2), k|-| a 1 - j | < x | < represents the t o t a l t h ickness o f veneer which i s t r e a t e d as the thickness of plywood. In actual p r a c t i c e , t h i s would not be the same as shrinkage i n d r y i n g , compression i n pressing and spreading o f glue between veneer l a y e r s a f f e c t the f i n a l t h i c k n e s s o f plywood. Appropriate c o r r e c t i o n f a c t o r s can be used i n the actual a p p l i c a t i o n of the model. From Tables 3 - 6 i t can be observed that when there are three veneer t h i c k n e s s e s , n. = 3 f o r three p l y plywood and n. = 9 f o r 5, 7 or 9 p l y plywood. In general, s i n c e x-j i s t r e a t e d as face veneer by convention, the number of p o s s i b l e combinations i n which K veneer thicknesses 2 can be used f o r core and centre veneers would be K x K = K . As three p l y plywood won't have centre p l i e s i n i t , the corresponding number of combinations f o r them would only be K. Thus, i n g e n e r a l , n.. = K f o r 3 p l y plywood and n. = K f o r any higher p l y (odd) plywood. I t should be noted again here t h a t most plywood m i l l s use 3, 4 or at most 5 veneer thicknesses and we need not th e r e f o r e be concerned about l a r g e number of c o n s t r u c t i o n a l t e r n a t i v e s associated with higher values of K. c) Constraint_Linking_Log Consumption_to Veneer Production To r e l a t e the qua n t i t y of veneer produced to the consumption of l o g s , the f o l l o w i n g assumption i s made (we discuss the i m p l i c a t i o n s of 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 the same i r r e s p e c t i v e o f the thickness of veneer i n t o which i t i s converted. This i s equivalent to saying that the number of veneer sheets of a f i x e d s i z e obtainable from a l o g i s i n v e r s e l y proportional, to the veneer thickness i n t o which i t i s peeled. With t h i s assumption, these v a r i a b l e s s a t i s f y the equation yx^v^ = L^, or y x k v k - L k = 0 f o r a 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 logs and f o r un i t s and dimensions o f the three v a r i a b l e s x k , v k and L k > For example, i f x k i s i n m i l l i m e t r e s , v k i s the number of veneers of s i z e 2.6 m x 1.4 m, L-k i s i n cubic metres and the y i e l d o f veneers i s 7 60% o f l o g Volume, then y i s given 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 are t y p i c a l c o n s t r a i n t s of resource a v a i l a b i l i t y . The qu a n t i t y o f logs peeled i n t o d i f f e r e n t veneer thicknesses should be l e s s than or equal to the qua n t i t y o f logs a v a i l a b l e . This i s given by 31 K E L < W (2.6) k=l K where W is the quantity of logs available. Using (2.5) as a definit ion for L^, (2.6) can be replaced by K y Z :x: v « W • (2.7) k=l K 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 1 j f or , P.. - M6.. < 0 for a l l i , j (2.8) 1 J "I J where M is a large positive number as in (2.2). f) 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 K N n. z z 1 . a. .. P. . - v. < 0 for a l l k (2.9) i=l j=l 1 J K 1 J K 32 The slack in this constraint represents the excess quantity of veneer produced, but not used, in any of the construction alternatives. g) P § [ T § D ^ Z P r ^ § r r i l § _ Cons trai nts The quantity of each type of plywood produced should meet the demand or orderf i le requirements. These are specified by typical constraints of the form n. E 1 P.. > d. for a l l i (2.10) j=l 1 J 1 where d. is the demand for product i F inal ly , a l l the decision variables used in the formulation are required to be non-negative, i . e . , V Lk- p i j > 0 — <2-"> for a l l k = l,2...Kj, j = l , 2 . . . n . and i =1,2. The Objective Function The objective is to maximize net revenue for the m i l l . If we assume that revenues and costs are l inear functions, the objective function can be expressed as N n. K Max E E 1 r . P . . - C E L . i=l j=l 1 1 J k=l K where 33 r . = Revenue per plywood sheet of type i C = Cost of l o g per u n i t Again, u t i l i s i n g (2.5) as a d e f i n i t i o n f o r L^, t h i s can be w r i t t e n as N n. K Max E E 1 r . P . . - Cy. E x.v (2.12) i= l j=l 1 1 J k=l K K The O v e r a l l F o r m u l a t i o n of the PDM Problem 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 c o n s t r a i n t s introduced above, the PDM problem can be w r i t t e n as: N n. K Max E E 1 r.P.. - Cy E x.v. 1=1 j=l 1 1 J • k=l K K Subject to x G T = { T 1 T 2 T " i ( k ) }  x k k * 'k ' 'k k ' 1^  b , L - M ( l - 6 j j ) < * • « ! - « , . ) = ( 0 , 1} n. E 1 6. . > 1 K y £ x. v. < w k=l K K P. . - Ms. . < 0 U i J 34 ..N n. n i :-E P. . > d. J - l 1 J 1 P i j ' V 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 the PDM problem as a mathematical programming problem i n which the o b j e c t i v e f u n c t i o n and some of the c o n s t r a i n t s are n o n-linear and, i n a d d i t i o n , some of the 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 to 0-1 values. The s i z e of the problem depends on f a c t o r s such as the number of veneer t h i c k n e s s e s , the number of plywood types and the number of p l i e s i n each of them. 2.5 Discussion and Extensions In the formulation of the PDM model i n areas r e l a t e d to the product mix p a r t , we have deviated from the t r a d i t i o n a l L.P. models (Kotak (1976), Ramsing (1965), Lee (1968)). In p a r t i c u l a r , the q u a n t i t y of veneer and plywood are defined i n terms o f the number o f sheets r a t h e r than volume,and y i e l d f a c t o r s are used as d i r e c t percentages r a t h e r than inverse m u l t i p l i e r s or recovery r a t i o s . This o f f e r s several advantages i n the design problem since the 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 of veneers, the f e a s i b l e designs 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 to production o f plywood through s.. and a... . The model assumes that the volume of veneer from a l o g remains the same i r r e s p e c t i v e of the thickness o f veneer i n t o which i t i s peeled. This assumption i s made i n the absence of r e l e v a n t information 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 of wood peeled 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 lower volumetric y i e l d due to f a c t o r s such as s p l i t s or l a t h e checks i n veneer. In such cases, i f information i s a v a i l a b l e on the r e l a t i o n between veneer thickness and y i e l d , i t can be included i n the model i n the form of 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). The r e s t o f the model formulation or s o l u t i o n procedure w i l l not be a f f e c t e d by t h i s change. The plywood thickness t o l e r a n c e b.'" and b ^ i n c o n s t r a i n t (2.2) r e f e r to design tolerances and not the thickness t o l e r a n c e o f an i n d i v i d u a l plywood sheet at a random po i n t . I f only i n d i v i d u a l panel tolerances are a v a i l a b l e they can be adjusted i n design f o r chance v a r i a t i o n s through s t a t i s t i c a l concepts such as 3 - sigma l i m i t s . The o r d e r f i l e requirements 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 of the 'less than or equal t o ' form o r , may combine both type of i n e q u a l i t i e s . In the formulation of the PDM model we considered only those plywood types with odd number of p l i e s having balanced designs as they c o n s t i t u t e the m a j o r i t y of the panels c u r r e n t l y manufactured. The model can be e a s i l y extended to even p l y and/or unbalanced designs of plywood. For even-ply plywood with balanced designs, design c o e f f i c i e n t s s i m i l a r to the a... of t a b l e s 3 - 6 can be constructed f o r any number of p l i e s . For example, with K = 3, f o r a f o u r - p l y plywood, there would be three design a l t e r n a t i v e s represented by the vectors ( 4 , 0, 0 ) , ( 2 , 2, 0) and (2, 0, 2) f o r ( a n . a - H ^ - ^ plyw°°d can be made with unbalanced 36 designs, the number of design a l t e r n a t i v e s n. w i l l increase c o n s i d e r a b l y , e s p e c i a l l y f o r those plywood types having l a r g e number of p l i e s . However, i f there are r e s t r i c t i o n s on the face veneers to be of the same t h i c k n e s s , the increase i n the number of design a l t e r n a t i v e s w i l l not be s u b s t a n t i a l and they can be handled w i t h i n the framework of our PDM model. We have t r e a t e d as the face veneer by convention since there are more r e s t r i c t i o n s on the thickness and surface q u a l i t y c h a r a c t e r i s t i c s o f face veneer than any other veneer i n a plywood sheet. A d d i t i o n a l l y , i n the manufacturing process, face veneers must be peeled to the f u l l length of plywood sheets while core veneers which go across the face can be peeled i n lengths r e l a t i v e to the width of plywood. Having more than one face veneer thickness would t h e r e f o r e r e s u l t i n more sch e d u l i n g , handling, s o r t i n g and surface preparation c o s t s . However, having a second face veneer thickness might r e s u l t i n b e t t e r designs f o r some plywood types. Such a s i t u a t i o n , i . e . having more than one face veneer t h i c k n e s s , can al s o be included w i t h i n the framework of 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 chapter s i x . E x i s t i n g algorithms f o r l a r g e s c a l e problems can solve e i t h e r i n t e g e r l i n e a r programs or non-linear programs i n continuous v a r i a b l e s . The unique features o f the PDM model i s that i t contains both d i s c r e t e and continuous v a r i a b l e s as well as n o n - l i n e a r i t i e s i n the 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 . Relaxing the in t e g e r v a r i a b l e s o f the PDM problem to continuous values w i l l produce a non - 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 not n e c e s s a r i l y a global one. Thus, to our knowledge, no algorithm or s o l u t i o n procedure that can produce a global s o l u t i o n to the PDM model i s a v a i l a b l e . 37 One of the important 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 the 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 the PDM problem. By 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 model and by s e l e c t i n g appropriate v a r i a b l e s to branch on, we develop an e f f i c i e n t i m p l i c i t enumeration al g o r i t h m to d e r i v e a global s o l u t i o n to the PDM problem. This algorithm i s described i n the next chapter. When f a c t o r s such as s p e c i e s , l o g grades, veneer types and plywood grades are i n c l u d e d , the s i z e o f the problem increases manyfold. In each s i t u a t i o n , however, the essence o f the problem formulation remains the same since the design aspect o f the problem ( c o n s t r a i n t s (2.1) through (2.4)) i s unaffected. Only the product mix part o f the problem changes. A set of optimal veneer thicknesses and optimal plywood designs f o r a PDM problem with one species need not remain the optimal s o l u t i o n when more than one species are included i n the PDM model. We consider some of these extensions to the PDM problem, analyse the corresponding s t r u c t u r e s and s o l u t i o n procedures i n chapter four. CHAPTER 3 38 3.1 S o l u t i o n to the PDM Model From chapter 2 we r e c a l l t h a t the plywood design and manufacturing (PDM) problem can be formulated as the f o l l o w i n g non-linear mixed 0-1 programming problem. N n. K Z Q = Max z E 1 r.P.. - C y z x.v. u i-1 j=l 1 1 J k=l K K s.t. b 1 L - M ( l - 6 1 j ) < L a ^ x ^ b ^ + M C l - ^ j ) n. E 1 6. . > 1 K y E x. v. < W k-1 K K P. . - Ms:. < 0 N n. E E 1 a. .. P. . - v. < 0 1-1 j - l 1 j k 1 J k n. E 1 P. . > d. j - l 1 J 1 6. . G {0,1} x e T = {T 1 T 2 T m ( k ) } x k 'k u k ' '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 the set o f veneer thicknesses f o r x k , k = 1,2,...K. Further, r e c a l l t h a t by convention i s the face veneer thickness and i n most m i l l s K, the number of veneer thicknesses i s g e n e r a l l y three or four. We develop an i m p l i c i t enumeration algorithm f o r s o l v i n g the PDM problem. The e f f i c i e n c y o f our i m p l i c i t enumeration algorithm stems from the f a c t that f o r a given set of veneer thicknesses the PDM problem reduces to a s p e c i a l s t r u c t u r e Linear M u l t i p l e Choice Knapsack (LMCK) problem [ Zemel (1980), Glover and Klingman (1979)]. Moreover, each one of these LMCK problems can be solved e x p l i c i t l y . F urther, we show t h a t due to the s p e c i a l s t r u c t u r e o f the PDM problem, we can reduce s u b s t a n t i a l l y , from the o u t s e t , the number of veneer thicknesses t h a t need to be considered i n the 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 algorithm was coded i n FORTRAN and was used to solve some r e a l world problems. Data obtained from a plywood m i l l i n B r i t i s h Columbia was used to evaluate the PDM model and the algorithm.For the p a r t i c u l a r c o n f i g u r a t i o n of the problem parameters used i n the study our model gave a set of veneer thicknesses and plywood designs which can increase the annual net revenue by more than 6.8 per cent. The plan o f t h i s chapter i s as f o l l o w s . In s e c t i o n 3.2 we describe our i m p l i c i t enumeration algorithm and analyze the LMCK problems. In s e c t i o n 3.3 we show how we can use the s t r u c t u r e o f the PDM problem to a c c e l e r a t e the performance o f the i m p l i c i t enumeration a l g o r i t h m , while i n s e c t i o n 3.4 we present our computational r e s u l t s . 40 Though much of the contents of t h i s chapter i s mathematical, the i m p l i c a t i o n s of the end r e s u l t are s t r a i g h t f o r w a r d . What i s shown here i s that a seemingly complicated non-linear mixed 0-1 mathematical programming problem can be solved e f f i c i e n t l y f o r a global s o l u t i o n by e x p l o i t i n g the s t r u c t u r e s i n the model. In f a c t , the main job of the computer code w r i t t e n f o r our algorithm to solve the PDM problem i s more o f a house-keeping nature than one i n which complicated o p t i m i z a t i o n r o u t i n e s are i n v o l v e d . 3.2 An I m p l i c i t Enumeration Algorithm to Solve the PDM Problem Our i m p l i c i t enumeration algorithm can be, i n a very rudimentary manner, described as f o l l o w s : Algorithm A: (Rudimentary algorithm f o r s o l v i n g the PDM problem) * * Step 1 : Let x^ . = x^ , x t e T^, t = 1, 2, K-1 denote a p r e v i o u s l y unselected set of values f o r the thicknesses x^, x^, ...» x^_^ i n the PDM problem. I f none e x i s t s , terminate. Otherwise, denote by PDM (x-| , ..., x ^ _ i ) the PDM problem i n which x t = x t , t = 1 , ..., K-1, and go to step 2. Step 2: Attempt to fathom PDM (x^ , x^_-| ). I f s u c c e s s f u l , go to step 1. Otherwise, Step 3: Solve PDM (x^ x^_^ ). Store the optimal s o l u t i o n i f b e t t e r than the incumbent, and go to step 1. C l e a r l y , algorithm A w i l l terminate a f t e r a f i n i t e number of i t e r a t i o n s with an optimal s o l u t i o n to the PDM problem. In the d i s c u s s i o n which f o l l o w s we show how to execute e f f i c i e n t l y step 3 o f a l g o r i t h m A. In s e c t i o n 3.3.we develop t e s t s which a s s i s t i n 41 fathoming the PDM ( x 1 , XK_-| ) problem i n step 2 of . algorithm A. Further, we show i n that s e c t i o n how we can e l i m i n a t e , from the * o u t s e t , values o f = x t , t = 1, 2, .... K-1 which are not c o n s i s t e n t w i t h an optimal s o l u t i o n o f our PDM problem. 3.2.1 The Linear M u l t i p l e Choice Knapsack Problem We w i l l consider now the PDM problem in which a l l thicknesses have * * been determined, i . e . x^ = x^ , x£ G T^ f o r k = 1, 2, ... K. For a l l j = 1, 2, ..., n.. and i = 1, 2, N, l e t I • i f b i L < * a i i k x k < b i U <5 . = ( 1 k=l l j k k 1 (3.1) i ) Otherwise fi Note that i f E"1 s*.= 0 f o r any i , then plywood type i cannot be J ' * assembled with the set of thicknesses x^ , k = 1, 2, K, and th e r e f o r e the remaining PDM problem i s i n f e a s i b l e . I f , on the other n. hand, E 1 6*. 1 f o r al 1 i = 1 , 2 , ... N, 1 et j=l 1 J I. = l j | s * . = 1} (3.2) I. represents the index set of f e a s i b l e design a l t e r n a t i v e s f o r plywood type i with x k = x^ , k = 1 , 2, ..., K. Now, the c o n s t r a i n t s P.. - Ms.. < 0 and P.. >0-V-'i,j i n the PDM problem can e q u i v a l e n t l y be replaced 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 K * Max E E r .P. . - Cy I T T J E i=l j G I. 1 1 J k=l K k s . t . y s x. v. < w k=l K K N E E a. .. P. . - v. < 0 1=1 j e I. ^ k !J k E P. . > d. p 1 J - V k > 0 for al l k = 1, 2, . . .K ; j e I. and i = 1, 2, . . . N. Let * * * denote by PDM (x-j , x 2 , . . . x^  ), problem (3.4) - (3.8). Clearly, * * * PDM (x-j , x2 , . . . x^ ) is a l inear program. Notice further that at an optimal solution to this problem,(3.6) would be sat isf ied as an equality. Substituting v^, given by N v, = E E a . . . P . . -V- k — k i = 1 . e i j k u in (3.5) produces N E E b. .P. . 1=1 J e I. ^ ^ 43 where b i j = j/«ijk xk — , ( 3 : n ) Note that b. . > 0 V i , j e I. si n c e y > 0, x* > 0, a. . t > 0, and there e x i s t s at l e a s t one k f o r each j such that a.. > 0. S u b s t i t u t i n g (3.9) i n the o b j e c t i v e f u n c t i o n (3.4) we get that N K * N E E r.P.. - Cy E x, v. = 2 E f..P.. (3".12) i=l j e I i 1 1 J k=l k k 1=1 j e i . U U where K r i j = r i " C y k E = 1 a i J k X k = r . - Cb.. (3.13) Thus, the PDM ( x * , x^ » ..., x£ ) has been reduced to an o p t i m i z a t i o n problem of the form: N Z = Max E E r..P. . 1=1: J G I • s . t . N E .-" E -b,,P,, < W E P. • > d. (3.14) P.j > 0 -V- i = 1, 2 N and j e I. 44 ic - ic ic The PDM (x^ , x 2 , ... x^ ) problem given by (3.14) i s e a s i l y recognized as a l i n e a r m u l t i p l e choice knapsack (LMCK) problem. ( I t d i f f e r s s l i g h t l y from the t r a d i t i o n a l LMCK [Zemel (1980), Glover and Klingman (1 979)], i n t h a t , the l a t t e r problem has ( i ) e q u a l i t y s i g n i n the 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 ) m inimization as the o b j e c t i v e ) . Using p r o p o s i t i o n 3.1 which f o l l o w s , we show that due to the s p e c i a l s t r u c t u r e o f the c o e f f i c i e n t s i n the PDM (x-| , * * x 2 , ...,-x^ ) problem, (3.14) can be solved e x p l i c i t l y , without having to go through any of the LMCK algorithms [such as Zemel (1980)]. 3.2.2 An E x p l i c i t S o l u t i o n f o r the PDM ( x 1 , x^ ) Problem Let b, , = Min {b } ^ i (3.15) 1 J i j G I i 1 J and observe from (3.13) i t f o l l o w s that r . , = r . - Cb. . l J i 1 l J i = Max {r..} ¥ i (3.16) j e I. 1 J P r o p o s i t i o n 3.1: There e x i s t s an optimum s o l u t i o n to (3.14) such that Pij- = 0 V j f J. and ¥ i = 1 ,2,...N (3.17) Proof: Suppose (3.17) does not hold. That i s , there 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) with 1=1' but P' > 0 f o r some IJ ,p"t i = p, t G r \ {Jp}- Consider the new s o l u t i o n given by p' + P ' . pt p j . o • P P I . I J i = p . J = J i = p , J'=t O t h e r w i s e 4 5 ( 3 . 1 8 ) O b s e r v e f i r s t t h a t ( P . . ) i s f e a s i b l e f o r ( 3 . 1 4 ) . I n d e e d , - . ( P ' . . . ) f e a s i b l e l J l J f o r ( 3 . 1 4 ) i m p l i e s : CD * > 0 * U j e i . ( i i ) z j e * P . . I . 1 J I = z P ' . , , > d . -V- i ; J e I . ^ 1 and ( i i i ) N z i = l z N b . . P i . = z Z b . .P ' . . + b , P ! . -U U i = f j (=. j U u p J p P t b . P ' p t p t N < Z z b . . P1. . s i n c e , b . < b . ¥ j 6 I , 1 J i J P J p PJ J P i = l j e i f r o m ( 3 . 1 5 ) N e x t , s u p p o s e b . i s s u c h t h a t b . = b . ; i . e . t h e r e e x i s t s more t h a n p t p t p J p o n e J e I p m i n i m i z i n g b ^ f o r i = p i n ( 3 . 1 5 ) . Then i t f o l l o w s f r o m ( 3 . 1 6 ) t h a t r , = r . a n d t h e r e f o r e z z r . . P . . = z z r . . P ' . . = V P J p P t i j e j _ i j i j i j G T . i j u On t h e o t h e r h a n d , i f J p i s u n i q u e f o r i = p ; i . e . b p j < b ^ . , j e I p , t h e n i t f o l l o w s f r o m ( 3 . 1 6 ) t h a t r , > r a n d o b s e r v e t h a t p p p rP t P p t + r p J p P p J p = r p J p [ P p J p + P p t ] r . P . ' + r . P' + P J p P J p P t P t 46 * Therefore, z z r..P.. >s z r..P'.., c o n t r a d i c t i n g the o p t i m a l i t y o f i j G I. 1 J 1 J i j G I. J J (P1. .) and the proof o f p r o p o s i t i o n 3.1 f o l l o w s . 1 J Following p r o p o s i t i o n 3.1, the LMCK reduces to s o l v i n g N Max Y, r. P. 1=1 K K s.t. N Z b. P. <W i=l n* n' (3.19) P. > d. -V- i = 1 , 2, ... N. Where b. = b. , and r . = r . , = r . - Cb. l . i J. 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 1 Observe that Max (3.20) I f r > 0, that i s , i f there e x i s t s at l e a s t one plywood type with P" p o s i t i v e net p r o f i t , the s o l u t i o n of (3.19) i s given by d. W - Z b.d. i i p (3.21) I f r ^Q, that i s , i f the net p r o f i t o f a l l the plywood types P • i s n o n - p o s i t i v e , the s o l u t i o n o f (3.19) i s given by 47 * P i . = d i ¥ i = 1 , 2 , ... N (3.22) Transforming these r e s u l t s to our LMCK problem, we have t h a t the * * s o l u t i o n to the PDM (x-| ) i s given by <W - E b. . d. 1*> l J i 1 0 i i P, J = J i 1 = P ' j = Jp i f r p J p ^ ° 1 = P- j = Jp i f r P J > 0 P Otherwi se (3.23) From (2.9) i t f o l l o w s that the q u a n t i t y o f veneers to be produced i s given by N = z a i=l i J l k 1 J i •V- k (3.24) 7^ Thus, when the x^'s are assigned values x^. , k = 1 , 2, ... K, we have shown that the PDM problem can be e x p l i c i t l y solved. The s o l u t i o n to s.... P.. and v. would be as given by (3.1), (3.23) and (3.24) r e s p e c t i v e l y . In order to produce (P..-..) as given by (3.23), we need to f i n d the i n d i c i e s J . , i = 1 , 2 , ... N and the index p. The computational d i f f i c u l t y i n generating (P*.) i s t h e r e f o r e 0 C [ I -j I + |I 2 I +•••+ |I NI + N). 48 3 . 3 B r a n c h i n g T e s t s and B o u n d s I n t h e f i r s t p a r t o f t h i s s e c t i o n we d e v e l o p b o u n d s on t h e v a l u e o f x k when v a l u e s o f x a r e f i x e d f o r t = 1 , 2 , . . . K - 1 . U s i n g t h e s e b o u n d s , a n d m a k i n g u s e o f t h e r e s u l t s f r o m t h e p r e v i o u s s e c t i o n , we d e m o n s t r a t e how an u p p e r bound 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 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 e m a n a t i n g *: * 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 ( v) p r o b l e m . I n t h e s e c o n d p a r t , 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 d e v e l o p b r a n c h i n g t e s t s w h i c h , f r o m t h e o u t s e t , c a n s u b s t a n t i a l l y r e d u c e 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 , x ^ t h a t n e e d t o be c o n s i d e r e d i n 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 ^ and an U p p e r b o u n d 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-) , . . . x ^ -|) P r o b l e m . * * T. . . .Assume t h a t x t = x ^ > x t e T ^ , t = 1 , 2 , . . . K-1 and l e t K-1 r. • = E a • -4 . x . U t ^ - | i j t t •V- i , j ( 3 . 2 5 ) F u r t h e r , l e t T ^ m i n and T K " ' a A d e n o t e t h e minimum and maximum p o s s i b l e t h i c k n e s s f o r X | ( , i . e . , T K m i n - M / n { T K m } ] a n d T K m a x = M * x {T^ 1 1 1 }. F o r a l l i , j f o r w h i c h a i - j K > 0 , l e t m x b L - T . . b U - T - . - T — ~ < T K m a X > J ^ - L L > V " n and a „ „ > 0 a i j K K a i j K K l j K ( 3 . 2 6 ) 49 D.j represents the set ;of a l l design a l t e r n a t i v e s with a..j K»0 which * might be f e a s i b l e f o r plywood type i with x^ = x^ , t = 1, 2, ... K-1 and X|< e T|<. Define and max mm Min Max , I bY J i 1 a. hi, max U 1 X. K = Max ..Min , min i j K (3.27) (3.28) i . . v = 0 but b. < T.. < b., then plywood I f f o r some i and j , a min _min v"lcl' type i can be assembled without x K i n which case set X.. K = and xi« max T ™ x . Let K D. l <T.. < b ? ' i j K = 0} and D! U D. (3.29) (3.30) Now, define min XK - ";XtC> — 0 . 3 1 ) and max X, = M j n { X - X } — ( 3 . 3 2 ) Observe t h a t i f % u m n > X I > m a x , the PDM problem i s i n f e a s i b l e f o r t h i s set of x t , t = 1, 2, ... K -1, and the node associated with the PDM (x^ , x K - i ) ^ s fathomed. Further, f o r values of x K such that x K > X^ax 50 or x K < x K m i n ' P D M ( x* » ••• X K - 1 ^ 1 S i n f e a s i b l e . An upperbound on the value o f the o b j e c t i v e f u n c t i o n o f the PDM problem when = , t = 1, 2, ... K-1 can now be computed by a r e l a x a t i o n * * of the PDM (. x-| , . . . x ^ _ i ) problem. Recall from p r o p o s i t i o n 3.1 that i f the LMCK i s f e a s i b l e , then i n an optimal s o l u t i o n to (3.14), f o r each K i , P > 0 only f o r that index j f o r which b. . = z y a...x. i s minimum. IJ l j IJK K Thus, si n c e y i s a constant and when x~ i s known, P.. would be p o s i t i v e fc i j only f o r t h a t j f o r which (T. . + a.. Kx K) i s a minimum, j e D.. Using these r e s u l t s , a r e l a x a t i o n o f the c o n s t r a i n t s r e l a t e d to plywood thickness 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 there e x i s t s x^ such that and 4 6 ( C n ' V A V T K — (3.33) b[ < T.j + a ^ x j . < b^ f o r at l e a s t one j e D. ? — (3.34) The r e l a x a t i o n i s that the value(s) o f x K s a t i s f y i n g (3.33) and (3.34) may d i f f e r f o r each i . I f such an x n K does not e x i s t f o r some i , the * * PDM (x.j ,...xK_.|) problem i s i n f e a s i b l e and the node associated with * i x t = x t » t = ^' 2> ••• -K-1 i s fathomed. Assuming t h a t an x. s a t i s f y i n g K (3.33) and (3.34) e x i s t s f o r each i define b^ = j ^ n . ^T.j + 3 . ^ x ^ ( 3 . 3 3 ) and (.3.34) are f e a s i b l e } (3. b. represents the minimum pos s i b l e t h i c k n e s s f o r plywood type i , given 51 t h a t x. = x. , t = 1, 2, ... K-1 and x„ i s an element of (X ™ l n , X I / m a x ) n t t l\ K K An upperbound on the o b j e c t i v e f u n c t i o n value of the PDM (x-j , X K _ - | > ) problem i s given then by the optimal value, Z, of the o p t i m i z a t i o n problem given below N f 1 = Max Z r'. p. 1-1 1 1 s . t . N ,. z b.P. < W/y 1=1 1 1 ^ — P. > d. l i where r . = r . - Cyb. ( 3 . 3 6 ) i s a Linear Knapsack Problem (LKP) i n bounded v a r i a b l e s . N , s b.d. >W/y, ( 3 . 3 6 ) i s i n f e a s i b l e . Otherwise, i f ( r /b ) = i=l 1 1 P P i i Max{r./b.}, the s o l u t i o n o f ( 3 . 3 6 ) i s given by 1 1 1 I f P i P { w/y z b.d. iri> ' 1 i f P i i = p i f r < 0 v P i = p i f r ^ > 0 and _ N , * Z = z r.P. 1=1 I f Z, as given by ( 3 . 3 8 ) , i s l e s s than the corresponding value of the incumbent s o l u t i o n i n step 3 of a l g o r i t h m A, the node associated with 52 * _ x t = x t > t = 1 ' 2 , ... K-1 c a n be f a t h o m e d . I f Z i s h i g h e r t h a n t h e i n c u m b e n t s o l u t i o n t h e n b r a n c h i n g on t h e v a l u e o f xv e (xJ'n l n, X l / m a x ) n j s h o u l d be m a d e . I f t h e number o f e l e m e n t s i n n ( X K m i n , X K m a x ) i s l a r g e , t h i s i n t e r v a l c a n f u r t h e r be s u b d i v i d e d and bounds on Z c a n be c o m p u t e d i n t h e same way a s a b o v e f o r e a c h s u c h s u b d i v i s i o n . H o w e v e r , s i n c e t h e number o f e l e m e n t s i n i s g e n e r a l l y s m a l l a n d s i n c e o t h e r b r a n c h i n g t e s t s t h a t f o l l o w e l i m i n a t e a s u b s t a n t i a l number o f t h e s e e l e m e n t s , f u r t h e r d i v i s i o n s o f t h e i n t e r v a l n ( X ^ m i n , X ^ m a x ) may n o t be n e c e s s a r y . 3 . 3 . 2 B r a n c h i n g T e s t s f r o m t h e S t r u c t u r e o f t h e PDM Model We u s e 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 p r o b l e m t o s u b s t a n t i a l l y r e d u c e , a t t h e o u t s e t , t h e s i z e o f t h e t r e e t h a t h a s t o be s e a r c h e d by a l g o r i t h m A . R e d u c t i o n 1 : ( D i s t i n c t V e n e e r T h i c k n e s s e s ) S i n c e t h e a . . . ' s 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 f e a s i b l e K 1 J K v a l u e s o f E a . . . x . w i t h 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 k_1 1 J K K S C w i t h x f x t , we c a n a s s u m e t h a t i n t h e PDM p r o b l e m x s f x t 1 < s , t < K ( 3 . F o r e x a m p l e , c o n s i d e r x^ = 2 . 5 , x 2 = 2 . 5 , Xg = 4 . 0 and K = 3 . S u p p o s e t h e d e s i g n a l t e r n a t i v e n i n e , ( a ^ - j , a i 9 2 ' a i 9 3 ^ = ^ ' ^ » i s f e a s i b l e f o r a 5 - p l y p l y w o o d . T h a t i s , E a ^ x ^ = 2x-j + x 2 + 2 x 3 •= 1 5 . 5 mm i s k—1 53 w i t h i n t h e p e r m i s s i b l e l o w e r and u p p e r t o l e r a n c e s . T h e n , p r e c i s e l y t h e same o r d e r i n g o f t h e p l i e s f o r f a c e , c o r e a n d c e n t r e v e n e e r s c a n be R e d u c t i o n 2 : ( S y m m e t r y i n ( C o r e / C e n t r e V e n e e r s ) Due t o t h e s y m m e t r y i n t h e c o n s t r u c t i o n o f t h e d e s i g n c o e f f i c i e n t s a i j k ^ o r t ' i e D a ^ a n c i n 9 r e q u i r e m e n t , i d e n t i c a l d e s i g n s c a n be p r o d u c e d u s i n g d i f f e r e n t b u t s y m m e t r i c s e t o f t h i c k n e s s e s . E x p l i c i t l y , s u p p o s e x = T , x = T , 2 < s , , s 9 < K . Then t h e p l y w o o d p r o d u c e d by S-j S£ I ^ t h e s e t o f t h i c k n e s s e s x , , . . . x , . . . x , . . . x . c a n be p r o d u c e d a s w e l l I S-j S 2 K. by u s i n g t h e s e t o f t h i c k n e s s e s x , , . . . x , . . . x , . . . X , , . F o r e x a m p l e , l e t x^ = 2 . 4 , x 2 = 2 . 7 a n d x^ = 3 . 1 be o n e s e t o f v e n e e r t h i c k n e s s e s d e s i g n a t e d a s s e t I i n T a b l e 7 . L e t t h e s e c o n d s e t o f t h i c k n e s s e s be g i v e n by x-j = 2 . 4 , x 2 = 3 . 1 , x^ = 2 . 7 , d e s i g n a t e d s e t I I . T a b l e 7 shows t h a t f o r a l l p o s s i b l e v a l u e s o f a l t e r n a t i v e d e s i g n s j h a v i n g t h r e e o r f i v e p l i e s t h e s e two s e t s p r o d u c e p l y w o o d s o f t h e same t h i c k n e s s e s . p r o d u c e d 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 ( a 3 ( 3 , 0 , 2 ) . I n d e e d , z a i 5 ( < x k = 3x.j + 2 x 3 = 1 5 . 5 mm. i 5 T a i 5 2 54 Number o f D e s i g n P l y w o o d T h i c k n e s s PI i e s A l t e r n a t i v e , j S e t I S e t I I 3 1 7 . 2 7 . 2 2 7 . 5 7 . 9 3 7 . 9 ^ " - ^ 7 . 5 5 1 1 2 . 0 1 2 . 0 2 1 2 . 3 > 1 2 . 7 3 1 2 . 7 — > 1 2 . 3 4 1 2 . 6 ^ ^ ^ 1 3 . 4 5 1 3 . 4 > 1 2 . 6 6 1 2 . 9 ^ 1 4 . 1 7 1 4 . 1 ~~^>12.9 8 13.; 3 ^ _ ^ » 1 3 . 7 9 1 3 . 7 > 1 3 . 3 T a b l e 7 : S y m m e t r y i n C o r e / C e n t r e V e n e e r s . The a b o v e o b s e r v a t i o n i s v a l i d f o r a n y number o f p l i e s , f o r a n y K > 3 and f o r a l l x , x s u c h t h a t 2 < s , , s , < K. T h e r e f o r e , we 1 2 c a n a s s u m e t h a t o u r s e t o f t h i c k n e s s e s x ^ , x^, . . . x^, i n t h e PDM p r o b l e m i s s u c h t h a t x c > x c f o r a l l 2 < s , < s 0 < K ( 3 . 4 0 ) s ^  s -j x c R e d u c t i o n 3 : ( U p p e r B o u n d on F a c e V e n e e r T h i c k n e s s ) L e t T, be t h e minimum p o s s i b l e v e n e e r t h i c k n e s s , i . e . T, < T . m -V-m, k . 55 I n 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 , t h e f a c e v e n e e r t h i c k n e s s . I f L. i s t h e number o f p l i e s i n p l y w o o d t y p e i , t h e minimum t o t a l t h i c k n e s s t h e r e m a i n i n g ( L . - 2 ) v e n e e r s c a n have i s ( L . - 2 ) T ^ . 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 o f t y p e 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.. - ( L i - 2 ) T ^ ] . T h u s , 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 X.j f o r t h a t p l y w o o d t y p e i s [ b^ - ( - 2 ) T - j ] / 2 . When a l l p l y w o o d t y p e s 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 ^|n[ (bV - ( L . - 2 ) T ^ ) / 2 ] . S u p p o s e T 1 m a x = M a x { T . | m } 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 v e n e e r . Then i n o u r PDM p r o b l e m we m u s t h a v e t h a t x ] < X1 = Mi n m m i rbU. - ( L . - 2 ) T , , i i ' l I max 2 J f ' T l ( 3 . 4 1 ) C o n s i d e r , f o r e x a m p l e , t h e 12 p l y w o o d t y p e s l i s t e d i n T a b l e 8 b e l o w . T h e s e a r e t a k e n f r o m t h e a c t u a l l i s t o f p l y w o o d t y p e s p r o d u c e d by a m i l l w i t h ^ = 2 . 4 mm, T 1 m a x = 3 . 2 mm a n d bY i s + 0 . 5 mm o f t h e s p e c i f i e d p l y w o o d t h i c k n e s s f o r a l l i . From Column 3 o f t h e t a b l e i t f o l l o w s t h a t x l < X 1 M i n m m j b i " ( L i - 2 > T 1 3 . 2 = 2 . 8 0 N o t i c e t h a t 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 s p e c i f i c a t i o n i f X^  e x c e e d s 2 . 8 mm. V a l u e s o f x.| h i g h e r t h a n X 1 w o u l d 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 . 56 P l y w o o d T y p e i P l y L. 1 - T h i c k n e s s mm b? - <V2»Ti 1 b i - 2 X 1 2 L . - 2 i 1 3 - 7 . 5 2 . 8 0 1 . 4 0 2 3 - 9 . 5 3 . 8 0 3 . 4 0 3 5 - 1 2 . 5 2 . 9 0 2 . 1 3 4 5 - 1 5 . 5 4 . 4 0 3 . 1 3 5 7 - 1 8 . 5 3 . 5 0 2 . 4 8 6 7 - 2 0 . 5 4 . 5 0 2 . 8 8 7 7 - 2 2 . 5 5 . 5 0 3 . 2 8 8 9 - 2 3 . 5 3 . 6 0 2 . 4 9 9 9 - 2 5 . 5 4 . 6 0 2 . 7 7 10 9 - 2 7 . 5 5 . 6 0 3 . 2 0 11 9 . .- 2 8 . 5 6 . 1 0 3 . 3 4 12 9 - 3 0 . 5 7 . 1 0 3 . 4 9 T a b l e 8 : B o u n d s on x-| a n d x ^ . t h R e d u c t i o n 4 : ( L o w e r B o u n d on K T h i c k n e s s ) From r e d u c t i o n 3 , a s g i v e n by ( 3 . 4 1 ) i s t h e maximum p o s s i b l e f a c e v e n e e r t h i c k n e s s and f r o m ( 3 . 4 0 ) i t f o l l o w s t h a t x ^ > x ^ _ t f o r a l l t = 1 , 2 , ... K - 2 . S i n c e 2X^ i s t h e maximum p o s s i b l e t h i c k n e s s o f t h e two f a c e v e n e e r s , a l o w e r bound Ij^, on t h e v e n e e r t h i c k n e s s X K , f o r a p l y w o o d w i t h L. p l i e s a n d 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 g i v e n 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 p l y w o o d t y p e s Max L " t h e l o w e r b o u n d o n x ^ i s o b t a i n e d by . [ (b.. - 2 X ^ • ) ' / ( L . - 2 ) ] . I f m i n _.rimnj ~ l m ( 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 h a v e v Max i I L. - 2 f' 'K ( 3 . 4 2 ) 57 The r e d u c t i o n i m p l i e d by ( 3 . 4 2 ) i s i l l u s t r a t e d u s i n g t h e p r e v i o u s e x a m p l e i n t h e l a s t c o l u m n o f T a b l e 8 w i t h T K m 1 n = 2 . 4 0 , ^ = 2 . 8 0 a n d 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 . I n t h a t e x a m p l e O b s e r v e t h a t a 9 p l y 3 0 . 5 mm p l y w o o d c a n n o t be a s s e m b l e d w i t h i n s p e c i -f i c a t i o n i f x „ i s l e s s t h a n 3 . 4 9 mm. V a l u e s o f x „ l o w e r t h a n X „ w o u l d o n l y make t h e PDM p r o b l e m i n f e a s i b l e . To a p p r e c i a t e t h e i m p o r t a n c e o f t h e r e d u c t i o n s ( 3 . 3 9 ) - ( 3 . 4 2 ) , c o n s i d e r a t y p i c a l p r o b l e m w i t h 12 t y p e s o f p l y w o o d a s l i s t e d i n t a b l e 8 w i t h K = 4 , :m(.l) = 9 , and m ( k ) = 27 f o r k = 2 , 3 , 4 . Then t h e number o f s e t s o f v e n e e r t h i c k n e s s e s t h a t n e e d t o be c o n s i d e r e d i n t h e 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 t o 1 2 , 4 0 0 by t h e r e d u c t i o n s s p e c i f i e d by ( 3 . 3 9 ) - ( 3 . 4 2 ) . I f p l y w o o d c a n be made w i t h u n b a l a n c e d d e s i g n s , some o f t h e b r a n c h i n g t e s t s g i v e n a b o v e may h a v e t o be m o d i f i e d . In p a r t i c u l a r , t h e r e d u c t i o n s p e c i f i e d by ( 3 . 4 0 ) w o u l d d e p e n d o n t h e d e s i g n a l t e r n a t i v e s c o n s i d e r e d a n d t h e bounds g i v e n by ( 3 . 4 1 ) a n d ( 3 . 4 2 ) w o u l d n o t be v a l i d i f t h e two f a c e v e n e e r s i n a p l y w o o d s h e e t c a n be o f d i f f e r e n t t h i c k n e s s e s . U s i n g t h e a n a l y s i s d e v e l o p e d 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 s p e c i f i e d by ( 3 . 3 9 ) - ( 3 . 4 2 ) , we c a n now p r e s e n t , i n g r e a t e r d e t a i l , 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 p r o b l e m a s f o l l o w s . 2 . 4 0 = 3 . 4 9 v Al g o r i t h m B : (An a l g o r i t h m f o r s o l v i n g t h e PDM p r o b l e m ) S t e p 0 : I n i t i a l i z e ( x - j , x ^ , . . . x^,) a n d Zg * * S t e p 1 : L e t x^ = x t , x^. e T^., t = 1 , 2 , . . . 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 v a l u e s f o r t h e t h i c k n e s s e s x . j , x^, . . . x ^ o f t h e PDM p r o b l e m w h i c h s a t i s f y ( 3 . 3 9 ) - ( 3 . 4 1 ) . I f n o n e e x i s t s , t e r m i n a t e . O t h e r w i s e , go t o s t e p 2 . S t e p 2 : E v a l u a t e X K M L N a n d X K M A X f r o m ( . 3 . 3 1 ) and ( 3 . 3 2 ) . I f Je Je X K M I N > X K M A X , PDM ( x | , . . . x ^ ) i s i n f e a s i b l e and go t o s t e p 1 . O t h e r w i s e , c o m p u t e Z . ' f r P m ( 3 . 3 8 ) . I f Z i s l e s s t h a n Z Q go t o s t e p 1 . O t h e r w i s e , * Je Jc S t e p 3 : L e t x K e T K n ( X K M I N , X K M A X ) a l o n g w i t h ( 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 . 4 2 ) . I f n o n e e x i s t s , go t o s t e p 1 . O t h e r w i s e , S t e p 4 : S o l v e t h e 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 * * * x-j , x 2 , . . . x ^ u s i n g ( 3 . 2 3 ) . S t o r e t h e s o l u t i o n and u p d a t e Zg i f b e t t e r t h a n t h e i n c u m b e n t . Go t o s t e p 3 . In t h e i m p l e m e n t a t i o n o f t h i s a l g o r i t h m f o r p r a c t i c a l p r o b l e m s , t h e e x i s t i n g v a l u e s o f x ^ , . a n d t h e c o r r e s p o n d i n g v a l u e o f Z Q f o r t h e m i l l c a n be u s e d a t t h e i n i t i a l n o d e . O t h e r w i s e , we c a n s t a r t w i t h ZQ = -°° as t h e i n i t i a l v a l u e a n d u p d a t e i t w h e n e v e r b e t t e r s o l u t i o n s a r e g e n e r a t e d . C l e a r l y , a l g o r i t h m B w i l l t e r m i n a t e i n 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 p r o b l e m . As i s t r u e w i t h a n y o t h e r a l g o r i t h m , t h e c o m p u t a t i o n a l t i m e r e q u i r e d t o s o l v e t h e 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 s i z e i n t e r m s o f t h e number o f v a r i a b l e s a n d c o n s t r a i n t s i n t h e p r o b l e m . A f l o w c h a r t o f A l g o r i t h m B i s p r e s e n t e d i n F i g . 1 . 59 F i g u r e 1 : A F l o w c h a r t o f t h e 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 . 60 3 . 4 C o m p u t a t i o n a l R e s u l t s Our i m p l i c i t e n u m e r a t i o n 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 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 and t h e number o f d e s i g n a l t e r n a t i v e s f o r e a c h p r o d u c t d e p e n d e d on t h e number o f v e n e e r t h i c k n e s s e s , s e p a r a t e c o d e s w e r e w r i t t e n f o r d i f f e r e n t v a l u e s o f K. I n p u t d a t a f o r t h e c o d e w e r e t h e a v a i l a b i l i t y o f l o g s , l o g c o s t , t h e y i e l d f a c t o r , c o r r e c t i o n f a c t o r f o r c o m p r e s s i o n and s h r i n k a g e , s e t o f a l l p o s s i b l e v e n e e r t h i c k n e s s e s o f t h e p e e l i n g l a t h e ( s ) , t h e u p p e r t o l e r a n c e f o r f a c e v e n e e r t h i c k n e s s , l i s t o f p l y w o o d t y p e s , t h e i r r e v e n u e s , o r d e r f i l e r e q u i r e m e n t s and t h e r e s p e c t i v e t h i c k n e s s t o l e r a n c e s . 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 , t h e o p t i m a l v e n e e r t h i c k n e s s e s , q u a n t i t y o f l o g s t o be c o n v e r t e d t o e a c h t h i c k n e s s , maximum n e t r e v e n u e a t t h e o p t i m a l s o l u t i o n , m a r g i n a l v a l u e o f w o o d , a l l t h e 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 e a c h t y p e o f p l y w o o d and t h e q u a n t i t y o f p l y w o o d t o be p r o d u c e d u n d e r e a c h d e s i g n a l t e r n a t i v e . A p p e n d i x I g i v e s a l i s t i n g o f t h e FORTRAN p r o g r a m f o r t h e PDM p r o b l e m w i t h f o u r v e n e e r t h i c k n e s s e s ( K = 4 ) . 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 d e m o n s t r a t e t h a t t h e PDM model 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 e n u m e r a t i o n 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 i n c o d i n g . F o r 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 m o d e l a n d t h e i m p l i c i t e n u m e r a t i o n a l g o r i t h m f o r r e a l - w o r l d s i t u a t i o n s , r e p r e s e n t a t i v e d a t a o b t a i n e d f r o m a p l y w o o d m i l l i n B r i t i s h C o l u m b i a was u s e d . The m i l l was m o s t l y m a n u f a c t u r i n g e x t e r i o r p l y w o o d a d h e r i n g t o t h e s p e c i f i c a t i o n s l a i d b y t h e C o u n c i l o f F o r e s t I n d u s t r i e s d f B r i t i s h C o l u m b i a (COFI ( 1 9 7 8 ) ) . C u r r e n t l y t h e m i l l was u s i n g f o u r v e n e e r t h i c k n e s s e s ( K = 4 ) 61 a s 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 t h i c k n e s s e s t h a t c o u l d be c o n s i d e r e d w i t h i n t h e f r a m e w o r k o f v e n e e r t h i c k n e s s t o l e r a n c e a m o u n t e d t o m ( l ) = 9 a n d m ( k ) = 27 f o r k = 2 , 3 and 4 . The m i l l ' s o r d e r f i l e c o n s i s t e d o f t w e l v e t y p e s o f p l y w o o d . D e t a i l s o f t h e p l y w o o d t y p e s , t h e i r e x i s t i n g 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 , l o g c o s t , y i e l d f a c t o r , r e v e n u e a n d o r d e r f i l e r e q u i r e m e n t s o f p l y w o o d , t h e i r t h i c k n e s s s p e c i f i c a t i o n s a r e g i v e n i n a p p e n d i x I I . When t h e m i l l ' s d a t a w i t h t h e e x i s t i n g s e t o f v e n e e r t h i c k n e s s e s a n d d e s i g n s w e r e u s e d i n a l i n e a r p r o g r a m , t h e maximum p o s s i b l e a n n u a l n e t r e v e n u e f o r t h e m i l l was $ 1 3 , 4 1 6 , 6 9 4 . S t a r t i n g w i t h t h i s as an i n i t i a l s o l u t i o n , o u r i m p l i c i t e n u m e r a t i o n a l g o r i t h m c o d e s f o r t h e PDM 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 o f B r i t i s h C o l u m b i a ' s AMDAHL 4 7 0 V8 c o m p u t e r w i t h a WATFIV c o m p i l e r . When t h e number o f v e n e e r t h i c k n e s s e s , K , was f o u r , t h e c o d e t o o k 2 9 . 3 s e c o n d s o f CPU t i m e and g a v e an o p t i m a l s o l u t i o n t o t h e PDM p r o b l e m w i t h a maximum n e t r e v e n u e o f $ 1 4 , 3 3 7 , 3 7 0 . D e t a i l s o f t h e c o r r e s p o n d i n g o p t i m u m v e n e e r t h i c k n e s s e s , t h e 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 , q u a n t i t i e s o f p l y w o o d t o be p r o d u c e d u n d e r e a c h d e s i g n a l t e r n a t i v e and s u c h o t h e r i n f o r m a t i o n a r e g i v e n i n a p p e n d i x I I I . 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 p r o b l e m p a r a m e t e r s u s e d i n t h e s t u d y , t h e a n n u a l n e t r e v e n u e o b t a i n e d f r o m o u r model was 6 . 8 6 % h i g h e r t h a n t h a t f o r t h e e x i s t i n g s e t o f v e n e e r t h i c k n e s s e s a n d p l y w o o d d e s i g n s u s e d by t h e m i l l . The i n c r e a s e r e s u l t e d f r o m t h e f a c t t h a t t h e wood l o s s i n t h e f o r m o f e x c e s s t h i c k n e s s i n p l y w o o d r e d u c e d f r o m t h e c u r r e n t 7 9 4 4 . 7 c u b i c m e t r e s t o 1 6 4 7 . 4 c u b i c m e t r e s . I n a d d i t i o n t o t h e a b o v e , p o s s i b i l i t i e s o f m a n u f a c t u r i n g a l l t h e m i l l ' s p r o d u c t s f r o m o n l y t h r e e v e n e e r t h i c k n e s s e s i n s t e a d o f f o u r 62 was t e s t e d u s i n g o u r PDM model w i t h K = 3 . I t was f o u n d t h a t a l l t h e p r o d u c t s o f t h e m i l l c o u l d be a s s e m b l e d w i t h i n p l y w o o d t h i c k n e s s s p e c i f i c a t i o n s u s i n g o n l y t h r e e v e n e e r t h i c k n e s s e s , a f a c t w h i c h was n o t known e a r l i e r . D e t a i l s o f t h e c o r r e s p o n d i n g o p t i m u m v e n e e r t h i c k n e s s e s , p l y w o o d d e s i g n s and s u c h o t h e r d a t a a r e g i v e n i n a p p e n d i x I V . The maximum p o s s i b l e n e t r e v e n u e a s s o c i a t e d w i t h t h r e e v e n e e r t h i c k n e s s e s was $ 1 3 , 9 3 0 , 6 7 0 a n d t h e c o r r e s p o n d i n g wood l o s s i n t h e f o r m o f e x c e s s t h i c k n e s s i n p l y w o o d was 3 6 1 2 . 1 c u b i c m e t r e s . T h i s i n d i c a t e s t h a t i n a d d i t i o n t o an i n c r e a s e o f 3 . 8 3 % i n a n n u a l n e t r e v e n u e , t h e r e was s u b s t a n t i a l s a v i n g s p o s s i b l e i n t h e f o r m o f r e d u c e d s e t up c o s t s a s s o c i a t e d w i t h t h e f o u r t h p e e l t h i c k n e s s . A more d e t a i l e d a n a l y s i s o f t h e r e s u l t s o f t h e PDM model f o r d i f f e r e n t number o f t h i c k n e s s e s and t h e i r c o m p a r i s o n i s g i v e n i n c h a p t e r s i x . CHAPTER 4 63 4 . 1 E x t e n s i o n s t o t h e PDM P r o b l e m I n t h e f o r m u l a t i o n o f t h e P l y w o o d D e s i g n a n d M a n u f a c t u r i n g (PDM) p r o b l e m , 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 o n e s p e c i e s i s u s e d i n t h e a s s e m b l y o f t h e end p r o d u c t s . H o w e v e r , m o s t p l y w o o d m i l l s m i g h t 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 l o g c o s t s and y i e l d f a c t o r s . I n a d d i t i o n t o t h i s , t h e o r d e r f i l e r e q u i r e m e n t s may be i n d e p e n d e n t o f s p e c i e s o r s p e c i f i e d s p e c i e s w i s e . As we w i l l show i n t h i s c h a p t e r , t h e p l y w o o d d e s i g n a n d m a n u f a c t u r i n g p r o b l e m c a n be r e f o r m u l a t e d t o i n c o r p o r a t e a l l t h e s e f a c t o r s a n d s o l v e d e f f i c i e n t l y . The i n c l u s i o n o f m o r e t h a n o n e s p e c i e s i n t h e m o d e l d o e s n o t a f f e c t t h e d e s i g n p a r t o f t h e PDM p r o b l e m . C h a n g e s i n g r e e n v e n e e r t h i c k n e s s due t o v a r i a t i o n s i n t h e d e n s i t y o r s p e c i f i c g r a v i t y o f a s p e c i e s a r e 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 and c o m p r e s s i o n d u r i n g d r y i n g a n d h o t p r e s s i n g . H o w e v e r , i f s i g n i f i c a n t d e v i a t i o n s o c c u r i n p l y w o o d t h i c k n e s s f o r a n y p a r t i c u l a r s p e c i e s , i t c a n be a d j u s t e d by m a k i n g a p p r o p r i a t e c o r r e c t i o n s i n t h e v e n e e r . In s u c h a c a s e , t h e v e n e e r 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 t o t h e t h i c k n e s s o f a s t a n d a r d s p e c i e s o f v e n e e r s u c h a s D o u g l a s f i r i n t h e P a c i f i c N o r t h -w e s t r e g i o n . The 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 m o d e l , h o w e v e r , a f f e c t s t h e p r o d u c t m i x p a r t o f t h e p r o b l e m . The l i n e a r s u b p r o b l e m t o be s o l v e d a t e a c h f e a s i b l e n o d e o f t h e i m p l i c i t e n u m e r a t i o n . ' s s e a r c h t r e e a s s u m e s d i f f e r e n t f o r m s d e p e n d i n g on t h e f a c t o r s i n c l u d e d i n t h e m o d e l . 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 a l g o r i t h m 64 d e s c r i b e d i n t h e p r e v i o u s c h a p t e r c a n be u s e d t o s o l v e t h e s e v a r i o u s e x t e n s i o n s o f t h e PDM p r o b l e m . We now c o n s i d e r e x t e n s i o n s t o t h e PDM model i n v o l v i n g some o f t h e s e s i t u a t i o n s . I n t h e f i r s t c a s e , i n s e c t i o n 4 . 2 , we show t h a t when t h e o r d e r f i l e i s i n d e p e n d e n t o f s p e c i e s , t h e l i n e a r s u b p r o b l e m i s a G e n e r a l i z e d N e t w o r k p r o b l e m . We show t h a t , due t o some s p e c i a l s t r u c t u r e i n t h e PDM m o d e l , t h i s G e n e r a l i z e d N e t w o r k p r o b l e m r e d u c e s 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 p r o b l e m w h i c h , i n t u r n , c a n be r e d u c e d t o a s t a n d a r d T r a n s -p o r t a t i o n p r o b l e m . We s o l v e an e x a m p l e o f t h i s s u b p r o b l e m and p r e s e n t m o d i f i c a t i o n s r e q u i r e d i n t h e i m p l i c i t e n u m e r a t i o n a l g o r i t h m . In s e c t i o n 4 . 3 , we c o n s i d e r t h e s i t u a t i o n when t h e o r d e r f i l e i s d e p e n d e n t on s p e c i e s . We d e m o n s t r a t e t h a t , i n t h i s c a s e , a l i n e a r s u b p r o b l e m a s s o c i a t e d w i t h a n o d e o f t h e s e a r c h t r e e d e c o m p o s e s i n t o s e p a r a b l e 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 s . In s e c t i o n 4 . 4 , we c o n s i d e r t h e s i t u a t i o n when v e n e e r s o f d i f f e r e n t 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 p a n e l . 4 . 2 O r d e r f i l e I n d e p e n d e n t o f S p e c i e s 4 . 2 . 1 F o r m u l a t i o n D e f i n e 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 a s f o l l o w s : v^ = Q u a n t i t y o f v e n e e r s h e e t s o f t h i c k n e s s k f r o m s p e c i e s s ( i n number o f s h e e t s o f s t a n d a r d s i z e o r e q u i v a l e n t ) . P . . = Q u a n t i t y o f P l y w o o d o f t y p e i , s p e c i e s s , made 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 s h e e t s o f s t a n d a r d s i z e o r e q u i v a l e n t ) . r . = R e v e n u e f o r P l y w o o d t y p e ' i , s p e c i e s s . ( $ ) w"s = Q u a n t i t y o f l o g s a v a i l a b l e , s p e c i e s s ( c u . m t r s . ) C s = C o s t p e r u n i t o f l o g , s p e c i e s s ( $ / c u . m t r . ) y = Y i e l d f a c t o r f o r v e n e e r f r o m s p e c i e s s . s = 1, 2 , . . . S ; i = 1, 2 , . . . N ; j = 1 , 2 , . . . n . a n d k = 1 , 2 A l l o t h e r v a r i a b l e s a n d p a r a m e t e r s o f t h e e a r l i e r model r e m a i n u n c h a n g e d . F o r m u l a t i n g t h e p r o b l e m i n t h e same way a s i n C h a p t e r 2 , t h e PDM model i n t h i s c a s e w o u l d be N S n . S K Z Q = Max E E E 1 r . P - E C y E x . v . u i = l s = l j = l 1 S 1 J S s = l s s k=l k k s s . t . 1^  k=l n . E 1 6.. >1 j = l 1 J P . . - Mfi . . < 0 i j s °1J K ys k E = 1Vks < W s S n . E E 1 P . . > d . s = l j = l ^ s 1 5 i j = { 0 ' 1 } 66 x k G T k " ' ^ k 1 * Jk> T k m ( k ) } — { 4 - 9 ) P . . , v. > 0 ( 4 . 1 0 ) i j s k s f o r a l l s = 1 , 2 , . . . S ; i = 1 , 2 , ...N; j = 1 , 2 , . . . n . a n d k = 1 , 2 , . . K . A g a i n , t h e a b o v e i s a n o n - l i n e a r m i x e d i n t e g e r p r o g r a m m i n g p r o b l e m . N o t e t h a t i n t h e a b o v e model r e p r e s e n t s t h e q u a n t i t y o f l o g s p e e l e d t o t h i c k n e s s x^ f r o m s p e c i e s s a n d 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 q u a n t i t y o f l o g s o f s p e c i e s s . A l s o , z P . . i s t h e q u a n t i t y o f j = l 1 J S 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 z 1 P . . i s t h e t o t a l q u a n t i t y o f p l y w o o d t y p e i . s = l j = l 1 J S 4 . 2 . 2 The G e n e r a l i z e d N e t w o r k S u b p r o b l e m The 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^ , x^ G T ^ , k = 1 , 2 , , t o t h e e x t e n d e d PDM p r o b l e m g i v e n by ( 4 . 1 ) - ( 4 . 1 0 ) c a n be v e r i f i e d by e v a l u a t i n g t h e i n d i c a t o r v a r i a b l e s 6.. u s i n g ( 3 . 1 ) . L e t s. . . a r i d I . be a s d e f i n e d i n ( 3 . 1 ) a n d ( 3 . 2 ) r e s p e c t i v e l y , o f t h e p r e v i o u s c h a p t e r . T h e n , f o l l o w i n g a r g u m e n t s s i m i l a r t o t h o s e u s e d i n t h a t c h a p t e r , we c a n show t h a t ( 4 . 5 ) r e d u c e s t o o r i j G i . k s 1 J K k 1 J S -z z b . . P . , < W 1 j G I . ^ S 1 J S S w h e r e K b . . = y z a . . , x , ' i j s ^ s k = 1 l j k k ( 4 . 1 1 ) 67 The o b j e c t i v e f u n c t i o n r e d u c e s t o K * l . . s T s r i s P i j s T E z z C s ( y s z a i i k x k > p i i s i ] e I . s l b 1 J S , i j e l i s s s k = l 1 J K K 1 J S T, T, s r . . P . . 1 j I I i s ^ s U s w h e r e K * r i j s = r i s " C s y s k z = 1 a i j k x k r . - C b . . (4 1 2 ) i s s I J S V H ' X C ' ic ic it The l i n e a r s u b p r o b l e m 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 , x 2 , . . . x k , w o u l d t h e n be Z = Max E E E T . . P . . i j e I . s ^ s s . t . E E b . . P . . < W -V- s i j e I . 1 J S 1 J S s E E P > d A/- i ( 4 . 1 3 ) s j e I . 1 J b 1 P i j s > 0 f i , s a n d j e I . ( 4 . 1 3 ) i s known a s a G e n e r a l i z e d N e t w o r k (GN) p r o b l e m and when | I . | = 1 f o r a l l i , ( 4 . 1 3 ) i s s p e c i a l i z e d 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 (GT) p r o b l e m , b . . c a n be i n t e r p r e t e d a s t h e amount o f wood r e q u i r e d i j s t o p r o d u c e o n e u n i t o f p l y w o o d t y p e i , s p e c i e s s , u s i n g d e s i g n a l t e r n a t i v e 68 j a n d r „ s i s t h e c o r r e s p o n d i n g u n i t n e t r e v e n u e . The u n i q u e f e a t u r e o f ( 4 . 1 3 ) i s t h a t e a c h v a r i a b l e i n " i t a p p e a r s a t m o s t t w i c e i n t h e c o n s t r a i n t s . T h i s s p e c i a l s t r u c t u r e i s f u r t h e r r e f l e c t e d by t h e f a c t t h a t we c a n a s s o c i a t e 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 r e p r e s e n t t h e c o n s t r a i n t s a n d 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 e a c h v a r i a b l e 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 i n t h e c o n s t r a i 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 s t r u c t u r e 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 t i m e t h a n a l i n e a r p r o g r a m ( K e n n i n g t o n a n d H e l g a s o n ( 1 9 8 0 ) , E l a m , G l o v e r a n d K l i n g m a n ( 1 9 7 9 ) , P h i l l i p s a n d G a r c i a - D i a z ( 1 9 8 1 ) ) . H o w e v e r , o u r GN p r o b l e m a s g i v e n by ( 4 . 1 3 ) r e d u c e s t o a GT p r o b l e m a s shown i n t h e f o l l o w i n g s e c t i o n . 4 . 2 . 3 The 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 S . u b p r o b l e m R e c a l l t h a t 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 ( 4 . 1 3 ) a r e r e l a t e d by r . . = r . J 1JS 1 i s - C b . . s i j S u p p o s e i i b . i J . s = M i n { b . . . } j e I , J ( 4 . 1 4 ) T h e n , i t f o l l o w s t h a t r . r . - C b . s i J , s i s = Max t r . . } j 6 I, 1 J S ( 4 . 1 5 ) P r o p o s i t i o n 4 . 1 : F o r a n y i = 1 , 2 , . . . N , t h e i n d e x J . ( o r i n d i c i e s , i f more t h a n o n e e x i s t s ) m i n i m i z i n g b . . i n ( 4 . 1 4 ) i s ( a r e ) t h e same I J J f o r a l l s = 1 , 2 , . . . S . P r o o f : The p r o o f f o l l o w s f r o m t h e d e f i n i t i o n o f b . . j s s i n c e = y M i n { z a , . . x } s j e I . k 1 J K K a n d s a . . . x . i s i n d e p e n d e n t o f s . P r o p o s i t i o n 4 . 2 : T h e r e e x i s t s an o p t i m a l s o l u t i o n t o ( 4 . 1 3 ) , i n w h i c h p i j s = 0 • if,. a n d -V- i , s P r o o f : S u p p o s e ( 4 . 1 6 ) d o e s n o t h o l d f o r some i and s . T h a t i s , t h e r e e x i s t s a s o l u t i o n ( P ^ - s ) w h i c h i s o p t i m u m f o r ( 4 . 1 3 ) w i t h Z = V b u t P' > 0 f o r some i = p , s = f and t e I \ { J } . Assume p t f r P P r p J f ^ ^ a n c ' c o n s 1 c * e r t n e n e w s o l u t i o n g i v e n by P * . P- + P t f pi P J p f b P J p f P t f i = P » J = J p » s = f i = p , j = t , s = f P . . O t h e r w i s e 70 O b s e r v e t h a t ( 4 . 1 7 ) i s f e a s i b l e f o r o u r GN s u b p r o b l e m ( 4 . 1 3 ) . S i n c e ( ? ' . . ) i s a f e a s i b l e s o l u t i o n , i t f o l l o w s t h a t ( 1 ) P j j s > 0 ¥ i , j , s ( i i ) z E P * i s s i n c e b . f > b . > 0 , p t f p J p f + h p t f . . - , P i , . ! 7T _ 1 IP' p t f "Ui i / ' J * s i n c e " » « * " p y > d ie ic and ( i i i ) b . _P . . + b . -P-"-. - = 0 + b . -p t f p t f p j p f p J p f p J p f P' + P t f p i . P J p f b p J f P t f P b P 1 + b P 1 P J p f P J p f p t f p t f s o t h a t z z b . . P . . 1 j €= I . ^ S ^ S = E E b . . P'. i j G I . 1JS 1JS < W -V- s . s F u r t h e r , o b s e r v e t h a t i f b . . . i s s u c h t h a t b . = b . r , i . e . T h e r e p t t p t f P ^ p f e x i s t s more t h a n o n e j e I p m i n i m i z i n g b p ^ s i n ( 4 . 1 4 ) . T h e n , i t f o l l o w s f r o m ( 4 . 1 5 ) t h a t = r p j ^ a n d t h e r e f o r e 71 On t h e o t h e r h a n d , i f ( i ) J p i s u n i q u e f o r i = p , i . e . b p J f < b p . . f , j f J , j e i . s o r ( i i ) t h e i n d e x t i s s u c h t h a t b . < b , , . , t h e n P i P J p T P^T i t f o l l o w s f r o m ( 4 . 1 5 ) t h a t r , f > r . f . Now, o b s e r v e t h a t P r +rP*4.r + r , - P * f = r . - | P ' . - + - r ^ f P' ' p t f p t f p J p f p J p f p j p f I p J p f b p J f p t f > V i / p j f + r p J f p P t f f r o m < 4 - 1 4 > p p r p r > r . f P ' + r . . P ' f r o m ( 4 . 1 5 ) p p p J p p p t f * T h e r e f o r e E E Er.. P . . > E E Er.. P 1 . . , c o n t r a d i c t i n g t h e 1 j e I . s 1 J S ^ s 1 j e I . s 1 J S ^ s o p t i m a l i t y o f ( P 1 . . ) . The f a c t t h a t t h i s r e s u l t h o l d s e v e n when r p j < 0 c a n be shown by d r o p p i n g t h e f a c t o r b p t f / b p J ^ i n ( 4 . 1 7 ) and u s i n g s i m i l a r a r g u m e n t s a s a b o v e . The p r o o f o f P r o p o s i t i o n 4 . 2 t h e n f o l l o w s . P r o p o s i t i o n s 4 . 1 a n d 4 . 2 w o u l d t o g e t h e r i m p l y t h a t i n an o p t i m a l s o l u t i o n t o t h e g e n e r a l i z e d n e t w o r k p r o b l e m ( 4 . 1 3 ) , i f a p l y w o o d t y p e i s made w i t h more t h a n o n e s p e c i e s , t h e d e s i g n a l t e r n a t i v e f o r t h e m 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 a n d 4 . 2 i t f o l l o w s t h a t t h e GN p r o b l e m r e d u c e s t o s o l v i n g Z - Max ? ! V S P . . S s . t . Eb. P . < W -\f s . V S V s s z P1.s>d1 ¥ 1 s P > 0 ¥ i , s 1 ' s 72 ( 4 . 1 8 ) w h e r e , b . . - b . and r . „ - r . . T h i s i s 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 1 5 ^ I •> l ' S IJ^S p r o b l e m . A g a i n , b ^ > s i s t h e amount o f wood r e q u i r e d t o p r o d u c e o n e u n i t o f p l y w o o d t y p e i , s p e c i e s s , 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 J ^ , and r . i s t h e c o r r e s p o n d i n g u n i t r e v e n u e . O b s e r v e t h a t ( 4 . 1 8 ) c a n be r e p r e s e n t e d i n t h e f o l l o w i n g t a b u l a r f o r m o f a GT p r o b l e m w i t h t h e s p e c i e s a s ' s o u r c e s ' and p l y w o o d t y p e s a s ' d e s t i n a t i o n s ' . P I y w o o d T y p e ( i ) 1 2 . . . . N ' S u p p l y 1 b M V l r 2 - l . . . . . . V l r N - l P N - 1 • . . . . . . . . . . . . . . . . • s " i-s r l - s b 2 - S r 2 - S . . . . b N - S r N - S •Vs P 2 - S P N - S demand > > d 2 . . . . -( s ) The 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 f r o m t h a t o f a s t a n d a r d T r a n s p o r t a t i o n p r o b l e m i n t h a t t h e b a s i s g r a p h o f a GT c a n h a v e more t h a n o n e m a x i m a l c o n n e c t e d s u b g r a p h o r c o m p o n e n t . The d u a l v a r i a b l e s 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 n o n - d e g e n e r a t e f e a s i b l e s o l u t i o n w i l l h a v e a s many b a s i c v a r i a b l e s as t h e number o f c o n s t r a i n t s . A l g o r i t h m s t o s o l v e a GT p r o b l e m h a v e been d e s c r i b e d i n t h e l i t e r a t u r e [ B a l a s and I v a n e s c u ( 1 9 6 4 ) , E i s e m a n n ( 1 9 6 4 ) , L o u r i e ( 1 9 6 4 ) , B a l a s ( 1 9 6 6 ) , Taha ( . 1 9 7 1 ) , G l o v e r and K l i n g m a n (1 9 7 3 ) ] . The p r o b d e m a s r e p r e s e n t e d by ( 4 . 1 8 ) d e v i a t e s f r o m s t a n d a r d GT p r o b l e m s ( f o r e x a m p l e , B a l a s a n d I v a n e s c u ( 1 9 6 4 ) , Taha ( 1 9 7 1 ) , G l o v e r a n d K l i n g m a n ( 1 9 7 3 ) ) i n t h a t t h e s t a n d a r d GT p r o b l e m s w i l l h a v e e q u a l i t y s i g n i n t h e demand c o n s t r a i n t s . (4".18) c a n be p u t i n t h e s t a n d a r d 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 c o l u m n s , o n e f o r e a c h p r o d u c t and a s l a c k . H o w e v e r , we c a n s o l v e ( 4 . 1 8 ) i n t h e p r e s e n t f o r m n o t i n g t h a t t h e d u a l v a r i a b l e ( s ) w o u l d be z e r o ( i ) f o r a l l c o l u m n s h a v i n g a l l o c a t i o n ( E P - ) more t h a n demand ( d . ) a n d s s ( i i ) f o r a l l r o w s h a v i n g w e i g h e d a l l o c a t i o n (Eb. g P ^ . s ) l e s s t h a n t h e s u p p l y (W s ) . 4 . 2 . 4 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 P r o b l e m A l t e r n a t e t o s o l v i n g ( 4 . 1 8 ) a s a GT p r o b l e m , i t i s p o s s i b l e t o s o l v e i t a s a T r a n s p o r t a t i o n p r o b l e m f o l l o w i n g a s c a l i n g p r o c e d u r e f o r n e t w o r k p r o b l e m s w i t h g a i n s ( T r u e m p e r ( 1 9 7 6 ) ) . O b s e r v e t h a t t h e 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 f l o w p r o b l e m w i t h t h e a r c s h a v i n g g a i n / l o s s f a c t o r s . S p e c i f i c a l l y , i n s u c h n e t w o r k s t h e amount o f 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 t o t h e amount o f f l o w l e a v i n g t h e a r c . F o r e x a m p l e , t h e GT p r o b l e m ( 4 . 1 8 ) w i t h two s p e c i e s (S = 2 ) a n d two p r o d u c t s (N = 2 ) c a n be t r a n s f o r m e d i n t o a n e t w o r k a s shown i n F i g . 2 . 74 F i g . 2 : 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 P r o b l e m In F i g u r e 2 , S-| and 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 a n d 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 . S 1 i s a ' s u p e r s o u r c e ' , a c o n s o l i d a t i o n o f a l l t h e s u p p l y ( o r s p e c i e s ) n o d e s and T 1 i s a ' s u p e r s i n k ' , a c o n s o l i d a t i o n o f a l l t h e demand ( o r p r o d u c t ) n o d e s . The t h r e e numbers 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 a r c c a p a c i t y , u p p e r a r c c a p a c i t y and r e v e n u e p e r u n i t ' f l o w ' r e s p e c t i v e l y . The number 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 a s s o c i a t e d w i t h a u n i t o f ' f l o w ' . In a p u r e n e t w o r k f l o w p r o b l e m t h e s e l o s s / g a i n f a c t o r s w o u l d a l l be e q u a l t o u n i t y . Our p r o b l e m ( 4 . 1 8 ) c a n be v i e w e d a s 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 i n w h i c h b.1 = (1/b. . # s ) , t h e q u a n t i t y o f p l y w o o d o f t y p e i p r o d u c e d p e r o n e u n i t o f s p e c i e s s , i s t h e q a i n f a c t o r , r.' = ( r . / b . ) i s t h e r e v e n u e g e n e r a t e d by t h e 3 l s I • s i • s a l l o c a t i o n o f o n e u n i t o f s p e c i e s s t o p l y w o o d t y p e i . The s u p p l y ( W g ) a n d demand ( 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 a r c c a p a c i t i e s a s shown i n t h e f i g u r e . I f f. . i s t h e ' f l o w ' f r o m s p e c i e s n o d e S g t o p r o d u c t n o d e P.. i n t h e n e t w o r k , i t i s r e l a t e d t o t h e v a r i a b l e p. o f ( 4 . 1 8 ) by f . = P . > s b . # s . Our GT p r o b l e m ( 4 . 1 8 ) 75 w o u l d t h e n be e q u i v a l e n t t o t h e f o l l o w i n g m i n i m a l c o s t ( o r m a x i m a l r e v e n u e ) n e t w o r k f l o w p r o b l e m w i t h g a i n s . Max v y r ! f . i s vs.; i s s . t . f s ' j " f j s - ' V < T 3 3 f r j " f j r f j r > ' F w h e r e , t i s t h e i n d e x o f a l l i n t e r m e d i a r y n o d e s ( i . e . , e x c l u d i n g S 1 a n d T 1 ) i n t h e n e t w o r k , F i s t h e f l o w a v a i l a b l e a t t h e s u p e r s o u r c e S 1 ( = 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 i n k s s V ( - z d . . ) . T r u e m p e r ( 1 9 7 6 ) g i v e s a s c a l i n g p r o c e d u r e by w h i c h a n e t w o r k p r o b l e m w i t h 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 p r o b l e m . S i n c e ( 4 . 1 8 ) i s 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 p r o b l e m , t h e p u r e n e t w o r k p r o b l e m o b t a i n e d by T r u e m p e r ' s s c a l i n g p r o c e d u r e y i e l d s a s t a n d a r d T r a n s p o r t a t i o n p r o b l e m . T h i s i s i l l u s t r a t e d t h r o u g h t h e f o l l o w i n g e x a m p l e o f a PDM s u b p r o b d e m . E x a m p l e 4 . 1 : C o n s i d e r t h e f o l l o w i n g e x a m p l e . ( h y p o t h e t i c a l ) o f a PDM s u b p r o b l e m w i t h two s p e c i e s , F i r ( F ) a n d H e m l o c k ( H ) , f o u r p r o d u c t s (N=4) a n d f o u r v e n e e r 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 , 3 . 3 5 , 3 . 9 6 a n d 4 . 9 8 mm f o r k = 1 , 2 , 3 a n d 4 r e s p e c t i v e l y ; C - = 3 5 . 0 0 , 76 C H = 3 0 . 0 0 , Y F = 0 . 0 0 6 0 6 7 , Y R = 0 . 0 0 6 2 7 6 , W p = 7 0 , 0 0 0 and WH = 8 0 , 0 0 0 . O t h e r d a t a f o r t h e PDM p r o b l e m a r e g i v e n i n T a b l e 9 . P l y w o o d T y p e , i P l y L. l - T h i c k (mm) R e v e n u e ( $ ) r i F r i H Demand d . i D e s i g n j a . . . U k 1 3 - 7 . 5 4 . 3 4 . 1 171107 1 3 0 0 0 2 5 - 1 2 . 5 6 . 1 5 . 8 5 0 2 2 8 9 1 5 0 0 0 2 4 1 0 0 3 5 - 1 5 . 5 7 . 6 7 . 4 350192 1 2." 2 0 1 2 2 1 2 0 4 7 - 1 8 . 5 8 . 9 8 . 6 4 2 3 3 9 4 1 5 2 0 0 2 4 3 0 0 T a b l e 9 : I n p u t d a t a 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 , O r d e r f i l e I n d e p e n d e n t o f S p e c i e s . We w i l l 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 . C o m p u t a t i o n o f t h e b . . -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 f r o m ( 4 . 1 4 ) g i v e s J . = 1 , 1 , 2 a n d 1 f o r i = 1 , 2 , 3 a n d 4 r e s p e c t i v e l y . F o r t h e s e d e s i g n a l t e r n a t i v e s , b . a n d r . a r e p r e s e n t e d i n t h e t a b u l a r i • s l • s f o r m o f a GT p r o b l e m b e l o w . 77 P r o d u c t s S p ^ e c i e > s \ v 1 2 . 3 4 S u p p l y F . 0 4 8 9 6 2 . 5 8 6 4 . 0 8 1 6 0 3 . 2 4 4 0 . 1 0 1 0 1 4 . 0 6 4 5 . 1 2 2 2 5 4 . 6 2 1 3 < 7 0 , 0 0 0 H . 0 5 0 6 5 2 . 5 8 0 6 . 0 8 4 4 1 3 . 2 6 7 7 . 1 0 4 4 9 4 . 2 6 5 1 . 1 2 6 4 6 4 . 8 0 6 2 < 8 0 , 0 0 0 Demand > 171107 5 0 2 2 8 9 3 5 0 1 9 2 > 4 2 3 3 9 4 -The a b o v e 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 known a l g o r i t h m s ( B a l a s a n d I v a n e s c u ( 1 9 6 4 ) , Taha ( 1 9 7 1 ) , G l o v e r a n d K l i n g m a n ( 1 9 7 3 ) ) o r t r a n s f o r m e d i n t o a s t a n d a r d T r a n s p o r t a t i o n p r o b l e m u s i n g t h e s c a l i n g p r o c e d u r e o f T r u e m p e r a n d t h e n s o l v e d . We u s e t h e l a t t e r m e t h o d f o r i l l u s t r a t i o n . O b s e r v e t h a t 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 g i v e n a b o v e 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 P~ ,= + . 1 2 2 2 5 P . _ < 7 0 , 0 0 0 1 • F 2 • F 3 - F 4 - F . 0 5 0 6 5 P, n + . 0 8 4 4 1 P 0 „ + . 1 0 4 4 9 P , . u + . 1 2 6 4 6 P . „ < 8 0 , 0 0 0 1 ' H 2 * H 3 H 4 - H ( 4 . 2 0 ) ( 4 . 2 1 ) P 1 - F + P 1 . H > 1 7 1 1 0 7 — ( 4 . 2 2 ) 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 ) T h e r e e x i s t s a s e t o f m u l t i p l i e r s , one e a c h f o r e a c h o f c o n s t r a i n t s ( 4 . 2 0 ) - ( 4 . 2 5 ) , s u c h t h a t t h e g a i n f a c t o r s ( c o e f f i c i e n t s on t h e LHS 78 o f ( 4 . 2 0 ) and ( 4 . 2 1 ) ) c a n be t r a n s f o r m e d t o u n i t y ( T r u e m p e r ( 1 9 7 6 ) , P h i l l i p s a n d G a r c i a - D i a z ( 1 9 8 1 ) ) . In p a r t i c u l a r , s t a r t i n g w i t h ( 4 . 2 0 ) a s t h e i n i t i a l c o n s t r a i n t , i f we u s e t h e m u l t i p l i e r s 1 , 0 . 9 6 6 7 , 0 . 0 4 8 9 6 , . 0 8 1 6 0 , . 1 0 1 0 1 a n d . 1 2 2 2 5 f o r t h e c o n s t r a i n t s ( 4 . 2 0 ) t o ( 4 . 2 5 ) r e s p e c t i v e l y , we g e t .04896 P, I • F + .08160 P„ F + .10101 P3-F + 712225 P4>P < 7 o , o o o - - . ( 4 . 2 6 ) .04896 P, 1 • H + .08160 P2> H + .10101 P 3 - H ± .12225 P4.H < 77336.0 — - ( 4 . 2 7 ) .04896 P, I • F + .04896 P1 M H 8377.40 — - ( 4 . 2 8 ) .08160 P0 F + .08160 P9 H > 40986.78 — - ( 4 . 2 9 ) .10101 P3. F + .10101 P3> H > 35372.89 — - ( 4 . 3 0 ) .12225 P4> F + .12225 P. H > 51759.92 — - ( 4 . 3 1 ) s e t t i n g P1 I s = = .04896 P1 • s , P' = Y 2 s .08160 P 2 - s ' p , 3 s = = .10101 P, 3 - s a n d P ' 4 s = . 1 2 2 2 5 P 4 > 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 on t h e LHS o f ( 4 . 2 6 ) t o ( 4 . 3 1 ) a r e t r a n s f o r m e d t o u n i t y . The r e s u l t i n g t r a n s p o r t a t i o n p r o b l e m , i n a t a b u l a r f o r m , w o u l d be a s f o l l o w s " " P r o d u c t s S p e c i e s >s 1 2 3 4 S u p p l y F 5 2 . 8 2 6 8 3 9 . 7 5 4 9 4 0 . 2 3 8 6 3 7 . 8 0 2 0 ST70.000 H 5 2 . 7 0 8 3 4 0 . 0 4 5 3 4 2 . 2 2 4 5 3 9 . 3 1 4 5 < 7 7 3 3 6 . 0 Demand > 8 3 7 7 . 4 > 4 0 9 8 6 . 8 3 5 3 7 2 . 9 > 5 1 7 5 9 . 9 _ The n u m b e r s i n e a c h c e l l o f t h i s t r a n s p o r t a t i o n p r o b l e m a r e t h e r e v e n u e s a s s o c i a t e d w i t h t h e t r a n s f o r m e d v a r i a b l e s P ' . j s > s ' = F> H a n d i = 1 , . . . 4 . The a b o v e p r o b l e m i s n o t i n t h e s t a n d a r d f o r m o f a T r a n s p o r t a t i o n p r o b l e m s i n c e ( i ) t h e c o n s t r a i n t s h a v e i n e q u a l i t y s i g n s a n d ( i i ) t o t a l s u p p l y and t o t a l demand a r e n o t b a l a n c e d . C o n v e r t i n g t h i s p r o b l e m t o t h e s t a n d a r d f o r m a n d s o l v i n g y i e l d s t h e s o l u t i o n : P ' 1 F = 1 9 2 1 6 . 4 , P ' 2 p = 4 0 9 8 6 . 8 , P ' 3 H = 3 5 3 7 2 . 9 , P ' 4 p = 9 7 9 6 . 8 , P ' 4 H = 4 1 9 6 3 . 1 a n d P ' - ^ = P ^ = P ' ^ p = u - T r a n s f o r m i n g t h e s e t o t h e o r i g i n a l v a r i a b l e s we g e t P ] . p = 3 9 2 4 ^ 2 . 0 , P 2 - p = 5 0 2 2 8 9 . 0 , P 3 > H = 3 5 0 1 9 2 . 0 , P 4 . p = 8 0 1 3 7 . 5 , P 4 - H = 3 4 3 2 5 6 . 5 and P 1 # R = P 2 - H = P 3 . p = 0 a s t h e s o l u t i o n t o t h e 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 p r o b l e m . The c o r r e s p o n d i n g o b j e c t i v e f u n c t i o n v a l u e i s Z = $ 6 , 1 5 8 , 2 7 0 . 4 . 2 . 5 I m p l i c i t E n u m e r a t i o n f o r t h e O v e r a l l P r o b l e m To s o l v e t h e o v e r a l l PDM p r o b l e m a s g i v e n by ( 4 . 1 ) - ( 4 . 1 0 ) , 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 a l g o r i t h m d e s c r i b e d i n c h a p t e r 3 c a n be u s e d . B r a n c h i n g i s i n i t i a l l y d o n e on t h e v e n e e r t h i c k n e s s e s x ^ , t = 1 , 2 , . . . K - 1 . A l l t h e r e d u c t i o n s i n t h e number o f n o d e s o f t h e s e a r c h t r e e s p e c i f i e d by ( 3 . 3 9 ) - ( 3 . 4 2 ) a n d t h e b o u n d s on x K g i v e n by ( 3 . 3 1 ) a n d ( 3 . 3 2 ) a r e e q u a l l y a p p l i c a b l e h e r e . The u p p e r bound on Z g i v e n by ( ' 3 . 3 8 ) , h o w e v e r , c h a n g e s f o r t h i s p r o b l e m . F o l l o w i n g t h e r e l a x a t i o n s p e c i f i e d by ( 3 . 3 3 ) a n d ( 3 . 3 4 ) , l e t b .^ be a s d e f i n e d i n ( 3 . 3 5 ) . T h e n , s i n c e t h e l i n e a r s u b p r o b l e m i n t h i s c a s e r e d u c e s 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 p r o b l e m , an u p p e r bound on Z f o r a l l b r a n c h e s f r o m t h i s n o d e c a n be o b t a i n e d a s a s o l u t i o n o f ( 4 . 3 2 ) b e l o w : 80 Z = Max z z r 1 . , P . S . t . S b l P . < W / y ^ i r s s / J s s 1 , s 1 P . ^ 0 ¥ i , s i • s ( 4 . 3 2 ) w h e r e r.' = r . - C y b\ i - s i s s ^ s i A g a i n , (4.32) 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 o u t l i n e d * e a r l i e r . The n o d e a s s o c i a t e d w i t h t h e s u b p r o b l e m i n w h i c h x t = x t , t = 1, 2, ... K-1 c a n be f a t h o m e d i f 1 o b t a i n e d i n ( 4 . 3 2 ) i s l e s s t h a n t h e i n c u m b e n t Zg . . I f , h o w e v e r , b r a n c h i n g on t h e l a s t v e n e e r t h i c k n e s s x K i n t h e s e t ( X K m i n , X K M A X ) n T k i s f o u n d n e c e s s a r y , a d d i t i o n a l b o u n d s c a n be c a l c u l a t e d f o r t h e GT p r o b l e m u s i n g t h e s t r u c t u r e i n o u r PDM p r o b l e m . C o n s i d e r t h e d u a l o f (4.18) g i v e n by Z n = M i n EW I + E d . t . 0 s 5 s i 1 1 s . t . b . n + t . > r . ( 4 . 3 3 ) 1 • S S 1 1 • s 1 ' n s S 0, t . < 0 -V- i , s R e c a l l f r o m d u a l i t y t h e o r y t h a t an u p p e r bound on (4.18) c a n be o b t a i n e d u s i n g a n y f e a s i b l e s o l u t i o n t o ( 4 . 3 3 ) . An e f f i c i e n t u p p e r b o u n d c a n be o b t a i n e d u s i n g t h e s t r u c t u r e o f t h e c o n s t r a i n t s i n ( 4 . 3 3 ) . 81 O b s e r v e t h a t i n ( 4 . 3 3 ) , t h e c o n s t r a i n t s l e a d t o n >r. / b . -V- s S T S T S a n d t . > r . - b . n -V- i , s 1 T S T S S L e t n = Max { r . . / b . . } S . T S T S and t . = Max { r . . s - b . . s i i s } T h e n , Z = EW n + Ed . t . w i l l g i v e an u p p e r bound on t h e v a l u e o f Z S s s . 1 1 J S 1 _ i i n ( 4 . 1 8 ) . I n t u i t i v e l y , t h i s bound i s s h a r p s i n c e when S = 1 , Z = Z , 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 a t an o p t i m a l s o l u t i o n o f t h e c o r r e s p o n d i n g LMCK p r o b l e m . F o r t h e e x a m p l e c o n s i d e r e d e a r l i e r , = 5 2 . 8 2 8 6 , = 5 0 . 9 4 9 7 , t-, = 0 , t 2 = - 1 . 0 3 3 0 , t 3 = - 1 . 0 5 8 6 and t 4 = - 1 . 6 3 6 9 . T h i s i g i v e s Z : = $ 6 , 1 9 1 , 2 2 0 , w h i c h d e v i a t e s f r o m t h e a c t u a l Z by h a l f a p e r c e n t o n l y . 4 . 3 O r d e r f i l e D e p e n d e n t on S p e c i e s In t h i s 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 o r d e r f i l e i s d e p e n d e n t on s p e c i e s a n d a n a l y z e i t s s t r u c t u r e . We d e m o n s t r a t e t h a t t h e s u b p r o b l e m o b t a i n e d i n t h i s c a s e d e c o m p o s e s i n t o 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 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 u s i n g r e s u l t s o f c h a p t e r 3 . F o r 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 , t h e same i m p l i c i t e n u m e r a t i o n a l g o r i t h m d e s c r i b e d i n t h e p r e v i o u s c h a p t e r c a n 82 be u s e d , w i t h f e w c h a n g e s i n t h e c o m p u t a t i o n o f t h e bound on t h e o b j e c t i v e f u n c t i o n . When t h e o r d e r f i l e i s d e p e n d e n t on s p e c i e s , c o n s t r a i n t ( 4 . 7 ) c h a n g e s t o z 1 P , , c > d . -V- i , s ( 4 . 3 4 ) 1JS l b w h e r e d . g i s t h e q u a n t i t y o f p l y w o o d r e q u i r e d o f t y p e i , s p e c i e s s . A l l o t h e r 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 o f s e c t i o n 4 . 2 r e m a i n u n c h a n g e d . 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 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 , e T^ f o r k = 1 , 2 , . . . K , 5 . . a n d I . a r e a s g i v e n by ( 3 . 1 ) and ( 3 . 2 ) r e s p e c t i v e l y , we c a n show t h a t t h e l i n e a r s u b p r o b l e m r e d u c e s t o Max z z z r . . P . . s i j e I . ^ ^ s . t . Z Z b. . P. . < W t A ,rx . . T i j s i j s s ( 4 . 3 5 ) i Z P. . > d. J G I . 1 J S 1 5 P i j s > 0 ¥ U j ^ T i a n d s -Where b . . a n d r . . a r e a s d e f i n e d by ( 4 . 1 1 ) and ( 4 . 1 2 ) r e s p e c t i v e l y . 83 ( 4 . 3 5 ) i s s e p a r a b l e i n t o S d i s t i n c t l i n e a r p r o g r a m m i n g p r o b l e m s . F o r e a c h s, t h e p r o b l e m i s o f t h e f o r m : M a x u e i V u s s . t . z z b . . P . . < W i j e I , 1 J S 1 J S 5 ( 4 . 3 6 ) zz P . . > d . j e I. 1 J S 1 5 P i j s > 0 ¥ i j e i . O b s e r v e t h a t , f o r e a c h s , ( 4 . 3 6 ) i s a 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 . A l s o , d e f i n i n g b . , „ a n d r . . c a s i n ( 4 . 1 4 ) a n d ( 4 . 1 5 ) 1 S 1 u.j S r e s p e c t i v e l y we c a n show t h a t p r o p o s i t i o n s 4 . 1 a n d 4 . 2 a r e e q u a l l y a p p l i c a b l e h e r e . From c h a p t e r 3 , we know t h a t t h e s o l u t i o n o f ( 4 . 3 6 ) i s g i v e n by * P . . = u s i s z d . . b . . 1 t P 1 5 1 J i s pJ p s i t P , J = J,-1 = P , j = J p i f r p J p S < 0 1 = p ' j = Jp i f r p j s > ° P O t h e r w i s e ( 4 . 3 7 ) w h e r e r / b = Max { r . / b . c } = Max V p J p S i 1 J i S 1 J i P J S P J S . U.S ' U.S , , c j U S ' 1 J SJ i . j e I . The i n d e x p m a x i m i s i n g ^ r - j j . s / b n - j . s ^ my d i f f e r f r o m s p e c i e s t o s p e c i e s : 4 . 3 . 2 The I m p l i c i t E n u m e r a t i o n A l g o r i t h m 84 To s o l v e t h e o v e r a l l PDM p r o b l e m i n t h i s c a s e , t h e same i m p l i c i t e n u m e r a t i o n a l g o r i t h m o f c h a p t e r 3 c a n be u s e d . B r a n c h i n g i s d o n e o n v e n e e r t h i c k n e s s e s x^. , t = 1 , 2 , . . . K - 1 . A l l t h e b r a n c h i n g t e s t s and b o u n d s s p e c i f i e d by ( 3 . 3 1 ) , ( 3 . 3 2 ) and ( 3 . 3 9 ) - ( 3 . 4 2 ) a r e a p p l i c a b l e h e r e a l s o . The u p p e r bound on Z , g i v e n by 1 i n ( 3 . 3 8 ) i s t o be m o d i f i e d s l i g h t l y f o l l o w i n g t h e i n c l u s i o n o f more s p e c i e s i n t h e PDM m o d e l . I n t h i s c a s e , Z i s o b t a i n e d a s a s o l u t i o n o f Max z Ezr. P . 1 s v s v s s . t . Eb1. P . < WJy ? 1 v s s / J s P . >d. -V- i , s T S IS w h e r e b|. a n d r i a r e a s g i v e n i n ( 4 . 3 2 ) . F o l l o w i n g t h e p r o p e r t i e s o f LMCK i n o u r PDM p r o b l e m , t h e s o l u t i o n o f ( 4 . 3 8 ) : i s g i v e n , f o r e a c h s , by i - s d i s 'PS W s / y - b . d . r s i t p 1 1 S i r/'P i = P i f r p . s < 0 1 = P i f r p - s > 0 0 O t h e r w i s e and 85 * _ The n o d e w i t h x t = x t , t = 1 , 2 , . . . K-1 i s f a t h o m e d i f Z o b t a i n e d f r o m ( 4 . 4 0 ) i s l e s s t h a n t h e i n c u m b e n t v a l u e o f Zg i n t h e i m p l i c i t e n u m e r a t i o n a l g o r i t h m . 4 . 4 M i x o f S p e c i e s w i t h i n a P l y w o o d S h e e t I n some p l y w o o d m i l l s , more t h a n one s p e c i e s m i g h t be u s e d w i t h i n a p l y w o o d s h e e t . T h i s i s n o r m a l l y d o n e when t h e r e a r e t r a d e o f f b e n e f i t s a s s o c i a t e d w i t h t h e c o s t a n d y i e l d f a c t o r s o f d i f f e r e n t s p e c i e s o f v e n e e r . H o w e v e r , t h e s e m i x e d s p e c i e s p a n e l a r e a s s e m b l e d i n s u c h a way t h a t t h e b a l a n c e d d e s i g n r e q u i r e m e n t i s n o t a f f e c t e d by t h e l a y - u p o f s p e c i e s . The v e n e e r s f o r a n y o n e o f f a c e , c o r e o r c e n t r e p l i e s w o u l d be o f t h e 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 s p e c i e s h a v i n g s i m i l a r p h y s i c a l p r o p e r t i e s . In s u c h c a s e s , t h e r e m i g h t be a d d e d r e s t r i c t i o n s on l a y - u p o f v e n e e r s . ; s u c h a s ' f i r o n l y f o r f a c e v e n e e r s ' o r o t h e r r e s t r i c t i o n s i m p o s e d by s p e c i f i c c u s t o m e r o r d e r s . In t h i s s e c t i o n , we i l l u s t r a t e t h e f o r m u l a t i o n o f t h 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 a n d a n a l y z e i t s s t r u c t u r e . We d e f i n e a l a y - u p a l t e r n a t i v e t o 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 o r g r o u p o f s p e c i e s f o r f a c e , c o r e and c e n t r e v e n e e r s i n a p l y w o o d p a n e l . F o r e x a m p l e , i f f i r ( F ) , h e m l o c k (H) a n d o t h e r s p e c i e s ( 0 ) f o r m t h r e e g r o u p s o f s p e c i e s and 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 v e n e e r t o be o f s p e c i e s f i r o n l y , t h e p o s s i b l e l a y - u p a l t e r n a t i v e s ( 1 ) f o r a b a l a n c e d d e s i g n w o u l d be a s s p e c i f i e d i n T a b l e 1 0 . As e a r l i e r , we l e t t h e e d e s i g n a l t e r n a t i v e s ( j ) s p e c i f y t h e number o f v e n e e r s o f e a c h t h i c k n e s s t o be u s e d i n t h e p a n e l . 86 L a y - u p S p e c i e s f o r A l t e r n a t i v e , 1 F a c e C o r e C e n t r e 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 T a b l e 1 0 : S p e c i e s L a y - u p a l t e r n a t i v e s . 4 . 4 . 1 F o r m u l a t i o n o f t h e PDM M o d e l D e f i n e new d e c i s i o n 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 p r o b l e m a s f o l l o w s . P. . j1 = Q u a n t i t y o f p l y w o o d o f t y p e i , c o n s t r u c t i o n a l t e r n a t i v e j and l a y - u p a l t e r n a t i v e 1 ( i n number o f s h e e t s o f s t a n d a r d s i z e o r e q u i v a l e n t ) . r . j i = R e v e n u e f o r p r o d u c t i , l a y - u p a l t e r n a t i v e 1 ( $ ) ; I f r e v e n u e i s i n d e p e n d e n t o f 1 , r ^ = r- f o r a l l 1 . d..-| = Demand f o r p r o d u c t i , l a y - u p a l t e r n a t i v e 1 ( i n number o f s h e e t s o f s t a n d a r d s i z e o r e q u i v a l e n t ) ; I f demand i s i n d e p e n d e n t o f 1 , d n = dn- f o r a l l 1 . 87 a i j k l s = M u m ' : , e r of veneers of species s , thickness k, used in lay-up alternative 1 and construction alternative j for product i . 1 = 1, 2 . . . L; L is the number of lay-up alternatives. All other variables and parameters of the problem remain unchanged. The a - i - j ^ i s - ' a r e known numbers similar to a ^ ^ and are related to them by " i . . i i . " a . . i "V"l— 1 , 2 , . . . L h j k l s l jk and -V- i , j , k The model would be Max z z z r ^ P ^ - zC^ z x ^ s . t . b 1 L - M ( l - 6 i j . ) < z a . . k x k < b U + M ( l - 6 . . ) Z 6. . > 1 j 1 J Z / s x k V k s < W s P. . , - M 6 . . < 0 (4.41) i j l IJ Z Z Z a - . , n P. -n - V. < 0 ? . ^ T jk ls l j l ks z P . > d . , j l j l 1 1 88 6. . = ( 0 , 1} e T = { T 1 T 2 m ( k ) }  x k ' k l l k ' ' k ' ' k * P i j l ' v k s > 0 * i - J > k . 1 a n d s T h i s i s a g a i n a n o n - l i n e a r m i x e d i n t e g e r p r o g r a m m i n g p r o b l e m . 4 . 4 . 2 The L i n e a r S u b p r o b l e m * * When t h e 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 ^ = x ^ , x^ e T^, k = 1 , 2 , ... K t h e r e s u l t i n g p r o b l e m r e d u c e s t o s o l v i n g Max z z s r . . . P . . , i j e I . 1 s . t . ? j l l . ? b U s l P 1 J l < H s * 5 E P >d + 1 , 1 ( 4 . 4 2 ) j G I i 1 J I 1 1 P . > 0 U l w h e r e , b . . s l = j y s a 1 j k l s x * , r . ^ = - S C s b 1 j s l a n d I . i s a s d e f i n e d K S by ( 3 . 2 ) . I n t h e a b o v e model we h a v e a s s u m e d t h e o r d e r f i l e t o be d e p e n d e n t on t h e l a y - u p a l t e r n a t i v e . I f i t i s i n d e p e n d e n t o f t h e l a y - u p a l t e r n a t i v e , t h e demand c o n s t r a i n t i n ( 4 . 4 2 ) i s r e p l a c e d by 89 z z P . . , fed., w h e r e , d . i s t h e demand f o r p r o d u c t i . 1 j e I . 1 J I 1 1 I r r e s p e c t i v e o f w h e t h e r t h e o r d e r f i l e i s d e p e n d e n t on t h e l a y - u p a l t e r n a t i v e o r n o t , 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 l i n e a r p r o g r a m , b . i s t h e q u a n t i t y o f wood o f s p e c i e s s r e q u i r e d t o p r o d u c e o n e u n i t o f p l y w o o d t y p e i u s i n g d e s i g n a l t e r n a t i v e j a n d l a y - u p a l t e r n a t i v e 1 * * when t h e v e n e e r t h i c k n e s s e s a r e ( x , , . . . x „ ) . S i n c e b . . , and r . . - , a r e d e p e n d e n t on b o t h s p e c i e s and t h e l a y - u p a l t e r n a t i v e , t h e s u b p r o b l e m s do n o t r e d u c e t o a n y o t h e r s i m p l e r s t r u c t u r e . C o n s e q u e n t l y , i n an o p t i m a l s o l u t i o n t o t h e l i n e a r s u b p r o b l e m , a p l y w o o d t y p e may have more t h a n o n e l a y - u p a l t e r n a t i v e a n d / o r more t h a n o n e d e s i g n a l t e r n a t i v e . The o p t i m a l d e s i g n and l a y - u p a l t e r n a t i v e f o r a n y p a r t i c u l a r p l y w o o d t y p e w o u l d d e p e n d o n t h e 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 amount o f wood r e q u i r e d o f e a c h s p e c i e s , t h e i r r e l a t i v e c o s t a n d y i e l d f a c t o r s , t h e a v a i l a b i l i t y o f l o g s o f e a c h s p e c i e s a n d t h e o r d e r f i l e . T h i s i s i l l u s t r a t e d by t h e f o l l o w i n g e x a m p l e . E x a m p l e 4 . 2 : C o n s i d e r a PDM s u b p r o b l e m w i t h o n e p l y w o o d t y p e , 7 P l y 22mm, w i t h two l a y - u p a l t e r n a t i v e s ( F , F , H) and ( F , H , F) f o r f a c e , c o r e and c e n t r e r e s p e c t i v e l y . The v e n e e r t h i c k n e s s e s a r e 2 . 5 0 , 3 . 1 0 , 3 . 9 0 and 4 . 8 1 mm a n d t h e d e s i g n a l t e r n a t i v e s a r e ( 5 , 0 , 0 , 2 ) a n d ( 2 , 3 , 2 , 0 ) . O t h e r c o e f f i c i e n t s o f t h e model a r e : C p = 3 5 . 0 0 , C H = 3 0 . 0 0 , Y F = 0 . 0 0 6 0 6 7 , Y^ = 0 . 0 0 7 2 , W p = 8 7 . 7 2 9 , WR = 5 5 . 0 8 0 a n d r . = 8 . 0 . S u p p o s e t h e o r d e r f i l e i s i n d e p e n d e n t o f l a y - u p a l t e r n a t i v e a n d d.j = 1 0 0 0 . Then t h e s o l u t i o n o f t h i s s u b p r o b l e m i s P ^ ^ = P - | 2 2 = 0 , P 1 1 2 = 5 0 0 . 0 a n d P ^ 2 1 = 5 0 0 . 0 . S u p p o s e t h e o r d e r f i l e i s d e p e n d e n t o n t h e ' 1 ayrup . w i t h i - c L y ; = ' 5 0 0 , d.^ = 5 0 0 a n d i s c h a n g e d f r o m 5 5 . 0 8 0 t o 6 0 . 0 0 0 , a l l o t h e r c o e f f i c i e n t s r e m a i n i n g u n c h a n g e d . Then t h e 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 ^ ^ = 0 , P ^ ^ = 2 5 5 . 3 2 , P 1 2 1 = ^ 1 . 1 4 a n c * P-j22 = i n d i c a t i n g t h a t more t h a n o n e d e s i g n 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 c a n be i n t h e f i n a l s o l u t i o n . CHAPTER 5 91 5 . 1 D i s j u n c t i v e P r o g r a m m i n g The PDM model f o r m u l a t e d i n c h a p t e r two i s a n o n . - r l i n e a r m i x e d i n t e g e r ( 0 - 1 ) m a t h e m a t i c a l p r o g r a m m i n g p r o b l e m . The f e a s i b l e r e g i o n f o r m e d by t h e c o n s t r a i n t s o f t h i s model i s n o n - c o n v e x due t o t h e p r e s e n c e o f i n t e g e r v a r i a b l e s 6.. and x. ( t h r o u g h ( 2 . 1 A ) - ( 2 . 1 C ) ) a n d 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 ( 2 . 5 ) . The e f f i c i e n c y o f t h e i m p l i c i t e n u m e r a t i o n a l g o r i t h m i n s e e k i n g a g l o b a l s o l u t i o n t o s u c h a p r o b l e m r e s u l t e d f r o m t h e f a c t t h a t by b r a n c h i n g on x ^ , t h e i n t e g e r v a r i a b l e s <$.. w e r e e x p l i c i t l y e v a l u a t e d 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 a n d o b j e c t i v e f u n c t i o n w e r e r e d u c e d t o l i n e a r i t i e s . In r e c e n t y e a r s , m u c h a t t e n t i o n has b e e n f o c u s s e d on t r e a t i n g i n t e g e r p r o g r a m m i n g a n d a h o s t o f o t h e r n o n - c o n v e x p r o g r a m m i n g p r o b l e m s a s l i n e a r p r o g r a m s w i t h l o g i c a l c o n d i t i o n s . An o u t g r o w t h o f t h i s a p p r o a c h i s d i s j u n c t i v e p r o g r a m m i n g , i n w h i c h an i n t e g e r p r o g r a m m i n g p r o b l e m c a n be t r a n s f o r m e d i n t o an e q u i v a l e n t l i n e a r p r o g r a m w i t h d i s j u n c t i v e c o n s t r a i n t s ( B a l a s ( 1 9 7 9 ) ) . F o r e x a m p l e , t h e 0-1 v a r i a b l e s 5.. i n t r o d u c e d i n t h e f o r m u l a t i o n o f t h e PDM model f o r 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 a n d d e s i g n f e a s i b i l i t y c a n be o v e r c o m e by r e p l a c i n g c o n s t r a i n t s ( 2 . 2 ) - ( 2 . 4 ) by t h e l o g i c a l c o n d i t i o n b^ " < z a . . . x . < bV f o r a t l e a s t o n e j , f o r a l l i . ( 5 . 1 ) 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 92 b i * V "njk*k * » i ¥ 1 — < 5 - 2 ' j k Where t h e s y m b o l V s t a n d s f o r d i s j u n c t i o n i m p l y i n g t h a t t h e c o n s t r a i n t s h o u l d be s a t i s f i e d f o r a t l e a s t o n e j . The n o n - c o n v e x i t y i m p l i e d by t h e s e c o n s t r a i n t s 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 o f o n e p l y w o o d t y p e w i t h two v e n e e r t h i c k n e s s e s . S u p p o s e t h e p l y w o o d t y p e has 3 p l i e s and t h e d e s i g n a l t e r n a t i v e s a r e ( 3 , 0 ) a n d ( 2 , 1 ) f o r t h e two v e n e e r t h i c k n e s s e s x-j a n d x , , . T h e n , a s s u m i n g t h a t x-j and x 2 a r e c o n t i n u o u s v a r i a b l e s , t h e f e a s i b l e r e g i o n i n t h e x-| - x 2 p l a n e s a t i s f y i n g c o n s t r a i n t ( 5 . 1 ) o r ( 5 . 2 ) w o u l d be s t a r - s h a p e d a s i l l u s t r a t e d i n f i g u r e 3 . When more t h a n one p l y w o o d t y p e i s i n c l u d e d , t h e f e a s i b l e r e g i o n w o u l d be t h e i n t e r s e c t i o n o f s u c h s t a r - s h a p e d r e g i o n o f e a c h p l y w o o d t y p e . 93 The a l t e r n a t e a p p r o a c h o f e l i m i n a t i n g t h e i n t e g e r v a r i a b l e s d o e s n o t o v e r c o m e p r o b l e m s a s s o c i a t e d w i t h n o n - c o n v e x i t y as i t i s i m p l i e d by d i s j u n c t i o n . H o w e v e r , t h e t r e a t m e n t o f some i n t e g e r p r o g r a m s a n d o t h e r n o n - c o n v e x p r o g r a m m i n g p r o b l e m s a s d i s j u n c t i v e p r o g r a m m i n g o r l i n e a r p r o g r a m m i n g p r o b l e m s w i t h l o g i c a l c o n d i t i o n s h a v e l e a d t o some i n t e r e s t i n g p r o p e r t i e s ( B a l a s ( 1 9 7 9 ) ) . I n t h i s c h a p t e r , we e x p l o r e t h e p o s s i b i l i t i e s o f t r e a t i n g t h e PDM p r o b l e m a s a d i s j u n c t i v e p r o g r a m m i n g p r o b l e m . We s t u d y t h e i m p l i c a t i o n s o f some o f i t s p r o p e r t i e s on t h e PDM m o d e l . 5 . 2 The PDM a s a D i s j u n c t i v e P r o g r a m m i n g P r o b l e m E x p r e s s i n g t h e 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 a n d d e s i g n f e a s i b i l i t y c o n s t r a i n t s by ( 5 . 2 ) i n s t e a d o f ( 2 . 2 ) - ( 2 . 4 ) d o e s n o t i t s e l f l e a d t h e PDM p r o b l e m t o a d i s j u n c t i v e p r o g r a m m i n g p r o b l e m s i n c e i d e n t i f y i n g t h e f e a s i b l e d e s i g n c o n s t r a i n t ( 2 . 8 ) w i t h o u t 8.. becomes c o m p l i c a t e d a n d t h e n o n - l i n e a r i t i e s i n ( 2 . 5 ) c o n t i n u e t o e x i s t . H o w e v e r , f r o m an a l t e r n a t i v e a p p r o a c h , t h e o v e r a l l PDM p r o b l e m c a n be c a s t a s a d i s j u n c t i v e p r o g r a m m i n g p r o b l e m . R e c a l l f r o m c h a p t e r 3 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 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 l i n e a r p r o g r a m a n d , i n p a r t i c u l a r , * an LMCK g i v e n by ( 3 . 1 4 ) . S i n c e <$.. and h e n c e I . a r e d e p e n d e n t on t h e 1 J 1 s e t o f v e n e e r t h i c k n e s s e s t h r o u g h ( 3 . 1 ) and ( 3 . 2 ) , t h e c o n s t r a i n t s o f t h e LMCK a r e d e p e n d e n t on t h e s e t o f v e n e e r t h i c k n e s s e s . S u p p o s e -k ~k "k "k "k "k X = ( x , , x 0 , . . . x „ ) a n d P = ('P. , , .) i s an o p t i m a l s o l u t i o n t o t h e I c. I\ 1 J PDM p r o b l e m o b t a i n e d f r o m t h e i m p l i c i t e n u m e r a t i o n a l g o r i t h m . O b s e r v e I I I i t h e n t h a t f o r a n y o t h e r s e t o f v e n e e r t h i c k n e s s e s X = ( x - p Xg. ... x^), ' * ' I * * X f X , and a s s o c i a t e d 6.. a n d I . , P = ( P . . ) n e e d n o t be f e a s i b l e f o r t h e c o r r e s p o n d i n g LMCK g i v e n by i Max E E , r . . P . . s . t . E E , b . . P . . < W i j e In. 1 J 1 J E , P . . > d . j e l , " 1 P K O V i , j e l j w h e r e , b . . = y E a . . . x . and r . . = r . - C b . . . T h u s , t h e ( P . . ) a r e a l s o d e p e n d e n t on t h e s e t o f v e n e e r t h i c k n e s s e s and i n an o p t i m u m s o l u t i o n * t o t h e PDM p r o b l e m , w h a t m a t t e r s i s t h a t ( P . . ) m u s t be f e a s i b l e f o r t h e •k "k ie LMCK a s s o c i a t e d w i t h X = (x-| , . . . x ^ ) . From t h e s e o b s e r v a t i o n s , t h e PDM p r o b l e m c a n be f o r m u l a t e d a s a d i s j u n c t i v e p r o g r a m m i n g p r o b l e m as f o l l o w s . S u p p o s e X = (x-| , . . . x K ) , f o r v a r i o u s v a l u e s o f h , r e p r e s e n t d i s t i n c t s e t s o f v e n e e r t h i c k n e s s e s f e a s i b l e f o r t h e d e s i g n c o n s t r a i n t s ( 2 . 1 ) - ( 2 . 4 ) . L e t Q be t h e i n d e x s e t o f a l l s u c h h . The PDM model c a n e q u i v a l e n t l y 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 p r o g r a m m i n g p r o b l e m . 95 s . t . r Z Q = Max Z 1 ~ ? • Z Th riV?j < ° i J e I . I J I J V < h e Q E E h b P < U i j e l j 1 J 1 J ( 5 . 3 ) w h e r e . i f = ( j | b f < z a . . . x j < b U a n d X h f e a s i b l e } ( 5 . 4 ) y I 9ijk Xk IJ r. - Cb'.. ( 5 . 5 ) ( 5 . 6 ) The d i s j u n c t i o n V i n d i c a t e s t h a t t h e c o n s t r a i n t s i n b r a c k e t s s h o u l d h o l d f o r a t l e a s t o n e h . The number o f e l e m e n t s i n Q i s t h e number o f d i s t i n c t f e a s i b l e s e t s o f v e n e e r t h i c k n e s s e s , f e a s i b l e f o r t h e d e s i g n K c o n s t r a i n t s ( 2 . 1 ) - ( 2 . 4 ) a n d i s b o u n d f r o m a b o v e by n m ( k ) . The k=l d i s j u n c t i o n r e s u l t s i n t h e c o n s t r a i n t s e t o f ( 5 . 3 ) b e i n g n o n - c o n v e x . Our d i s j u n c t i v e p r o g r a m ( . 5 . 3 ) d e v i a t e s f r o m s t a n d a r d p r o b l e m s ( s e e , f o r e x a m p l e , B a l a s (1 9 7 9 ) ) i n t h a t i n ( 5 . 3 ) t h e v a r i a b l e s ( P ^ . ) a r e d e p e n d e n t on h . H o w e v e r , ( 5 . 3 ) c a n be b r o u g h t i n t o t h e s t a n d a r d f o r m a t by d e f i n i n g a v a r i a b l e s e t c o n s i s t i n g o f Z a n d (P\ .) f o r a l l h a n d a d j u s t i n g t h e c o e f f i c i e n t s a p p r o p r i a t e l y . A u n i q u e f e a t u r e o f 96 ( 5 . 3 ) i s t h a t t h e RH!S o f ( 5 . 3 ) i s t h e same f o r a l l h w h i l e i n s t a n d a r d d i s j u n c t i v e p r o g r a m m i n g p r o b l e m s t h e y w o u l d be d e p e n d e n t on h . 5 . 2 . 1 The Dual o f t h e D i s j u n c t i v e P r o g r a m B a l a s ( 1 9 7 9 ) d e f i n e s a ' d u a l ' f o r t h e d i s j u n c t i v e p r o g r a m m i n g p r o b l e m and s t u d i e s some o f t h e r e l a t i o n s b e t w e e n t h e o r i g i n a l p r o b l e m a n d t h e d u a l s o d e f i n e d . ( T h i s i s n o t t o be c o n f u s e d w i t h t h e c l a s s i c a l d u a l o f L P ) . I n t e r e s t i n g l y , t h o u g h t h e o r i g i n a l p r o b l e m i s n o n - c o n v e x , t h e ' d u a l ' ' ' o f i t i s a c o n v e x l i n e a r p r o g r a m m i n g p r o b l e m . F o l l o w i n g B a l a s , t h e ' d u a l ' o f ( 5 . 3 ) i s g i v e n by LL = M i n U s . t . A < he q U - Wnh + 2 d . t ^ > 0 • r . .e + b i j n t } > 0 •v- i , j G i : > I h . h h ^ „ n , t i , e > 0 •V- i ( 5 . 7 ) w h e r e t h e s y m b o l A s t a n d s f o r c o n j u n c t i o n i m p l y i n g t h a t t h e c o n s t r a i n t s i n b r a c k e t s s h o u l d be s a t i s f i e d f o r a l l h G Q. ( 5 . 7 ) i s a c o n v e x l i n e a r p r o g r a m m i n g p r o b l e m . O b s e r v e t h a t a s o l u t i o n t o ( 5 . 7 ) c a n be o b t a i n e d by s o l v i n g , f o r e a c h h , t h e p r o b l e m 97 U h = M i n Wnh - z d . t j i s . t . b . .n - t . ^ r . . U i i j -V- i , j e l ! ( 5 . 8 ) n \ t } > 0 Ar i a n d i t f o l l o w s t h a t LL = Max [ U h ] ( 5 . 9 ) u h ( 5 . 8 ) i s p r e c i s e l y t h e d u a l o f an LMCK s u b p r o b l e m o f t h e PDM p r o b l e m w i t h x^ = x ^ , k = 1 , 2 , . . . K. From t h e s o l u t i o n o f t h e LMCK d e r i v e d i n c h a p t e r 3 a n d u s i n g c o m p l e m e n t a r y s l a c k n e s s c o n d i t i o n s , t h e s o l u t i o n o f ( 5 . 8 ) i s g i v e n by p . i . . h , h h h / r n \ t , = b. n - r . ( 5 . 1 1 ) w h e r e , a n d l l . b ^ / ^ M b ^ } — ( 5 . 1 2 ) h _ Max . CJ\ , r , h , c , ~ v r i . - j e * l f < r i j > = r i " C b i . — ( 5 . 1 3 ) 5 . 2 . 2 R e l a t i o n b e t w e e n t h e D i s j u n c t i v e P r o g r a m and i t s Dual S u p p o s e P h = { ^ P i j ) I '(-PlV s a t 1 s f 1 e s c o n s t r a i n t s o f ( 5 . 3 ) } a n d 98 h h h = {(n , t p e ) | c o n s t r a i n t s o f ( . 5 . 7 ) a r e s a t i s f i e d f o r h > . Assume t h a t t h e f o l l o w i n g r e g u l a r i t y c o n d i t i o n h o l d s . R e g u l a r i t y C o n d i t i o n ( B a l a s ) : I f ( 5 . 3 ) i s f e a s i b l e and ( 5 . 7 ) i s i n f e a s i b l e , t h e n t h e r e e x i s t s h e Q s u c h t h a t f and = <j>. T h e n , t h e r e l a t i o n b e t w e e n t h e d i s j u n c t i v e p r o g r a m a n d i t s d u a l c a n be c h a r a c t e r i z e d by t h e f o l l o w i n g t h e o r e m w h i c h we s t a t e w i t h o u t p r o o f ( f o r p r o o f , s e e B a l a s ( 1 9 7 9 ) ) . T h e o r e m 5 . 1 ( B a l a s ) : Assume t h a t t h e d i s j u n c t i v e p r o g r a m ( 5 . 3 ) a n d i t s d u a l ( 5 . 7 ) s a t i s f y t h e r e g u l a r i t y c o n d i t i o n . Then e x a c t l y o n e o f t h e f o l l o w i n g two s i t u a t i o n s . h o l d : a ) B o t h p r o b l e m s a r e f e a s i b l e ; e a c h h a s an o p t i m a l s o l u t i o n a n d b) One o f t h e p r o b l e m s i s i n f e a s i b l e ; t h e o t h e r o n e i s e i t h e r i n f e a s i b l e o r h a s no f i n i t e o p t i m u m . We a s s u m e i n t h e s e q u e l t h a t b o t h p r o b l e m s a r e f e a s i b l e , t h a t an o p t i m u m s o l u t i o n e x i s t s and t h a t Zg = U g . Now, s i n c e ( 5 . 7 ) i s a l i n e a r p r o g r a m i t s t r a d i t i o n a l d u a l i s g i v e n by 99 s . t . A h e Q n n = Max En u h n " * • Z T h r i j f l i j < 0 E h U i j + d . X < 0 J i and E x = 1 . h G Q h h h > n j » n , X > 0 h V i , j G ( 5 . 1 4 ) The d u a l o f t h e ' d u a l ' a s g i v e n by ( 5 . 1 4 ) i s a l i n e a r p r o g r a m w i t h a b l o c k - a n g u l a r s t r u c t u r e , w i t h o n e b l o c k o f c o n s t r a i n t s f o r e a c h h , l i n k e d t h r o u g h t h e common c o n s t r a i n t E x = 1 . I t f o l l o w s f r o m h G Q d u a l i t y t h e o r y o f l i n e a r p r o g r a m s t h a t when ( 5 . 7 ) a n d ( 5 . 1 4 ) a r e u f e a s i b l e ,UQ = r\Q. O b s e r v e t h a t i f P.., f o r some H e Q i s f e a s i b l e H H W f o r ( 5 . 3 ) t h e n t h e c o r r e s p o n d i n g s o l u t i o n w i t h y^. = P.. a n d x = 1 i s f e a s i b l e f o r ( 5 . 1 4 ) . I n t h e b l o c k - a n g u l a r s t r u c t u r e o f t h e c o n s t r a i n t s o f ( 5 . 1 4 ) , W a n d d . o c c u r i n a l l t h e b l o c k s o n c e e a c h f o r e a c h h . x^ i n ( 5 . 1 4 ) a c t s a s a s c a l i n g f a c t o r f o r d i f f e r e n t s e t s o f v e n e e r t h i c k n e s s e s X* 1 . S i n c e t h e f a c t o r s t h e y s c a l e a r e W and d . , b o t h i n d e p e n d e n t o f h , i t h' 1 f o l l o w s t h a t ( 5 . 1 4 ) w i l l h a v e 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 f o r ' h 1 some h ; G Q a n d \ = 0 f o r a l l h e Q\"{h f }. E x c e p t i o n t o 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 t h a n o n e s e t o f v e n e e r 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 f u n c t i o n v a l u e . B a r r i n g s i t u a t i o n s o f m u l t i p l e s o l u t i o n s t o ( 5 . 1 4 ) , i t f o l l o w s t h a t an o p t i m a l s o l u t i o n o f ( 5 . 1 4 ) w i l l a l s o be an o p t i m a l s o l u t i o n t o o u r o r i g i n a l d i s j u n c t i v e p r o g r a m m i n g p r o b l e m ( 5 . 1 3 ) . ( I n c i d e n t a l l y , t h i s s i t u a t i o n n e e d n o t h o l d good f o r a l l d i s j u n c t i v e p r o g r a m s s i n c e t h e RHS o f many d i s j u n c t i v e p r o g r a m s a r e d e p e n d e n t on h . The d u a l o f t h e ' d u a l ' f o r s u c h d i s j u n c t i v e p r o g r a m s may o n l y be a r e l a x a t i o n o f t h e o r i g i n a l p r o b l e m ) . Though t h e PDM model c a n be f o r m u l a t e d a s a d i s j u n c t i v e p r o g r a m a n d t r a n s f o r m e d t o an e q u i v a l e n t l i n e a r p r o g r a m a s i n ( 5 . 1 4 ) , i t d o e s n o t o f f e r e a s i e r s o l u t i o n s f o r p r a c t i c a l s i t u a t i o n s d u e . t o s e v e r a l r e a s o n s . F i r s t , t h e number o f p o s s i b l e e l e m e n t s i n Q i s g e n e r a l l y v e r y h h l a r g e . S e c o n d l y , f o r e a c h s u c h f e a s i b l e s e t o f t h i c k n e s s , b ^ . a n d r.^ h h s h o u l d be c o m p u t e d e x p l i c i t l y a n d P . . f o r j e I . s h o u l d be i d e n t i f i e d . T h i r d , t h e number o f r o w s and c o l u m n s o f t h e LP g i v e n by ( 5 . 1 4 ) w o u l d be s u b s t a n t i a l l y l a r g e . F o r e x a m p l e , f o r t h e p l y w o o d m i l l d a t a o f c h a p t e r 3 , w i t h K = 4 , t h e r e c a n be a maximum o f 177147 b l o c k s o f h , e a c h b l o c k h a v i n g a n y w h e r e b e t w e e n 14 t o 170 v a r i a b l e s and 14 c o n s t r a i n t s . The d i s j u n c t i v e p r o g r a m a p p r o a c h d o e s n o t t h e r e f o r e o f f e r a n y c o m p u t a t i o n a l a d v a n t a g e s o v e r t h e i m p l i c i t e n u m e r a t i o n a l g o r i t h m o f c h a p t e r 3 . N e v e r t h e l e s s , i t shows 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 c a s e o f a d i s j u n c t i v e p r o g r a m m i n g p r o b l e m a n d g i v e s an i n s i g h t i n t o i t s p r o p e r t i e s . 101 CHAPTER 6 6 . 1 The Optimum Number o f V e n e e r T h i c k n e s s e s The PDM model p r e s e n t e d i n t h e e a r l i e r c h a p t e r s d e t e r m i n e s t h e 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 c o r r e s p o n d i n g p r o d u c t m i x f o r 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 . The maximum n e t r e v e n u e s o o b t a i n e d i s d e p e n d e n t on t h e number o f p e e l t h i c k n e s s e s , K, a n d d o e s n o t i n c l u d e t h e s e t u p o r o v e r h e a d c o s t s a s s o c i a t e d w i t h i t . G e n e r a l l y , t h e l o w e r t h e number o f v e n e e r t h i c k n e s s e s , t h e l o w e r i s t h e o p e r a t i o n a l c o s t s a s s o c i a t e d w i t h p e e l i n g , d r y i n g , s t o r a g e , a s s e m b l y a n d r e c o r d - k e e p i n g b u t t h e y a l s o r e s u l t i n h i g h e r wood l o s s e s i n t h e f o r m o f e x c e s s t h i c k n e s s i n p l y w o o d a n d l o w e r r e v e n u e . The h i g h e r t h e number o f v e n e e r t h i c k n e s s e s , t h e h i g h e r i s t h e o p e r a t i o n a l c o s t s b u t t h e y a l s o r e s u l t i n h i g h e r • • r e v e n u e . F o r a n y p a r t i c u l a r m i l l t h e r e f o r e , t h e o v e r a l l b e n e f i t s a r e d e t e r m i n e d by t h e t r a d e - o f f b e t w e e n t h e t w o . 6 . 1 . 1 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 T h i c k n e s s e s E v e r y t i m e a v e n e e r t h i c k n e s s i s c h a n g e d a t t h e p e e l i n g l a t h e , t h e r e w o u l d be a s e t t i n g t i m e r e q u i r e d t o c h a n g e t h e l a t h e s e t t i n g s s u c h a s t h e h o r i z o n t a l g a p , v e r t i c a l g a p o r e x i t 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 t o t h e a p p r o p r i a t e t h i c k n e s s . T h i s i s f o l l o w e d by c h a n g e s i n s o r t i n g a t t h e c l i p p e r w h e r e t h e v e n e e r s a r e c l i p p e d t o d i f f e r e n t s i z e s and s o r t e d by t h e t h i c k n e s s a n d s i z e o f t h e v e n e e r . The h i g h e r t h e number o f p e e l t h i c k n e s s e s , t h e h i g h e r w o u l d be t h e s e t u p c o s t s i n p e e l i n g and 102 c l i p p i n g o f v e n e e r s . b) PryiD9_2f_yeD?§!T§ When v e n e e r s a r e d r i e d t o r e d u c e t h e m o i s t u r e c o n t e n t , t h e d r i e r s a r e t o be a d j u s t e d t o p r o p e r s e t t i n g s o f t h e t e m p e r a t u r e a n d d r y i n g s p e e d / t i m e i n c o n v e n t i o n a l d r y e r s . T h e s e s e t t i n g s d e p e n d on s p e c i e s , t h i c k n e s s and t h e i n i t i a l m o i s t u r e c o n t e n t o f v e n e e r s . T h e r e w o u l d be a t i m e l o s s a s s o c i a t e d w i t h t h e c h a n g e s . i n d r y e r s e t u p , e a c h t i m e a c h a n g e i n v e n e e r t h i c k n e s s i s m a d e . A g a i n , more v e n e e r t h i c k n e s s e s i m p l y h i g h e r s e t u p c o s t s i n d r y i n g . c ) H a n d ! i n g _ a n d _ S t o r a g e _ o f _ V e n e e r s T h i s p r o b l e m i s m u l t i p l i e d s e v e r a l t i m e s s i n c e v e n e e r s a r e c l a s s i f i e d on t h e b a s i s o f s p e c i e s , s i z e s ( f u l l , h a l f , s t r i p s a n d f i s h t a i l s ) a n d s u r f a c e q u a l i t y 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 , s u r f a c e s m o o t h n e s s , l a t h e c h e c k s a n d o t h e r f a c t o r s ) . The s t o r a g e a n d h a n d l i n g c o s t s d o u b l e s i n c e v e n e e r s a r e s t o r e d b o t h i n t h e g r e e n end ( a f t e r p e e l i n g b u t b e f o r e d r y i n g ) a s w e l l a s t h e d r y end ( a f t e r d r y i n g b u t b e f o r e a s s e m b l y ) . d ) A s s e m b l i n g _ V e n e e r s I n t h e g l u i n g a n d a s s e m b l y o f v e n e e r s , h i g h e r K i m p l i e s more s p a c e and m a t e r i a l h a n d l i n g f a c i l i t i e s . More v e n e e r t h i c k n e s s e s l e a d t o more d e s i g n a l t e r n a t i v e s . C h a n g e s i n d e s i g n a l t e r n a t i v e s r e s u l t i n h i g h e r s c h e d u l i n g c o s t s and h i g h e r s e t u p t i m e b e t w e e n b a t c h e s . e ) R e c o r d k e e p i n g I n c r e a s e d number o f p e e l t h i c k n e s s e s r e s u l t i n i n c r e a s e d c o s t s o f 103 r e c o r d - k e e p i n g . R o u t i n e s c h e d u l i n g p r o b l e m s r e l a t e d t o c o n v e r s i o n o f d i f f e r e n t s p e c i e s a n d t y p e s o f l o g s t o d i f f e r e n t v e n e e r t h i c k n e s s e s , a s s i g n i n g v e n e e r s o f d i f f e r e n t s p e c i e s , s i z e , g r a d e and t h i c k n e s s t o d i f f e r e n t d r y e r 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 t h e s e i n t o s e v e r a l t y p e s o f p l y w o o d 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 e a c h p e e l t h i c k n e s s . A l l t h e a b o v e f a c t o r s i n d i c a t e t h a t t h e s e t u p c o s t s i n c r e a s e , p e r h a p s e x p o n e n t i a l l y , w i t h i n c r e a s e i n t h e number o f p e e l t h i c k n e s s e s . 6 . 1 . 2 B e n e f i t s A s s o c i a t e d w i t h More P e e l T h i c k n e s s e s G e n e r a l l y , more p e e l t h i c k n e s s e s r e s u l t i n more d e s i g n a l t e r n a t i v e s p e r p l y w o o d t y p e . C o n s e q u e n t l y , t h e y r e s u l t i n l o w e r wood l o s s i n t h e f o r m o f e x c e s s t h i c k n e s s i n p l y w o o d a n d h i g h e r r e v e n u e s . S u p p o s e Z^ . i s t h e maximum p o s s i b l e n e t r e v e n u e o b t a i n a b l e f r o m t h e PDM model when t h e number o f v e n e e r t h i c k n e s s e s i s K. T h a t i s , s u b j e c t t o t h e u s u a l c o n s t r a i n t s a s d e s c r i b e d i n c h a p t e r t w o . [ ( 6 . 1 ) a p p l i e s t o s i n g l e s p e c i e s m o d e l ; I f more t h a n o n e s p e c i e s i s " u s e d , . t h e a p p r o p r i a t e o b j e c t i v e f u n c t i o n v a l u e s h o u l d be u s e d ] . L e t Z^ = i f t h e o p t i m i z a t i o n model i s i n f e a s i b l e f o r some K. When t h e p r o b l e m i s f e a s i b l e a n d lv i s f i n i t e , i t s a t i s f i e s t h e r e l a t i o n K Z „ = Max s E r . P . . - Cy E X . V i j 1 1 J k=l k ( 6 . 1 ) > 1 K f o r a l l K ( 6 . 2 ) 104 ( 6 . 2 ) f o l l o w s f r o m t h e f a c t t h a t w i t h K + 1 t h i c k n e s s e s , o n e w i l l h a v e a t l e a s t a s many d e s i g n o p t i o n s a s w i t h K p l u s t h e a d d e d b e n e f i t o f t h e e x t r a p e e l t h i c k n e s s . 6 . 1 . 3 U p p e r B o u n d on Though t h e maximum n e t r e v e n u e i n c r e a s e s w i t h i n c r e a s e ! i n t h e number o f p e e l t h i c k n e s s e s , b e y o n d a c e r t a i n v a l u e o f K , t h e r e w o u l d n o t be a n y i m p r o v e m e n t i n 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 . In f a c t , i t c a n be p o s t u l a t e d t h a t t h e m a r g i n a l r a t e o f i n c r e a s e i n Z^ d e c r e a s e s a s 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 some v a l u e o f K. Though i t i s d i f f i c u l t t o d e t e r m i n e s u c h a v a l u e o f K , t h e e x a c t v a l u e o f t h e u p p e r b o u n d , -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 Z K, f o r a l l K , c a n be c a l c u l a t e d f o r t h e s i n g l e s p e c i e s model o f t h e PDM p r o b l e m . R e c a l l f r o m c h a p t e r 3 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 - | , x 2 , • • • x K , 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 . F u r t h e r , i n an o p t i m a l s o l u t i o n t o t h i s LMCK p r o b l e m , f o r e a c h i , P . . w o u l d be n o n - n e g a t i v e o n l y f o r t h a t j f o r w h i c h b . . i s m i n i m u m . O b s e r v e * * 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 = x^ , x^ e T^ , k = 1 , 2 , . . . K , i t m u s t s a t i s f y I K * U b . < z a . . 1.x. < b f o r a t l e a s t o n e j a n d f o r a l l i 1 IJK K 1 K * I T h u s , f o r e a c h i , z a . . . x . i s bound f r o m b e l o w by b . , w h i c h i s i n d e p e n d e n t o f K. T h e n , i t f o l l o w s t h a t , f o r e a c h i , * L L b ^ = y J a i j . k x k >y b, a n d r . j = r . - Cb. . . . < r . . - C y b . . . C o n s e q u e n t l y , 105 f o r a n y i and j , { r . . / b . . } i s bound f r o m a b o v e by "10 "I J r„ - Cyb*" M a v f r . - C y b L From ( 6 . 3 ) a n d t h e s o l u t i o n t o t h e 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 3 , i t f o l l o w s t h a t t h e u p p e r b o u n d o n 2^, f o r a n y K, i s g i v e n by W - E b.d.X V . ^ F . d . + | i&LzLAj r p — ( 6 . 4 ) J . / w h e r e , 77 = (r. - C y b j : ) , b . = yb\ a n d (F^ /b^ ) = M a x {FT / b . } . I f a c o r r e c t i o n f a c t o r , C^, i s u s e d f o r s h r i n k a g e a n d c o m p r e s s i o n , t h e n K * L t h e p l y w o o d t h i c k n e s s i s g i v e n by C_ z a . - . x . a n d i n t h a t c a s e b . t k = 1 I J K K i i n ( 6 . 3 ) and ( 6 . 4 ) i s r e p l a c e d by (bir / C f ) . I n t u i t i v e l y , t h e u p p e r -* L bound b e i n g o b t a i n e d when C , Ea..|X, = b . f o r a l l i i s m e a n i n g f u l T K i j K K 1 s i n c e i t i m p l i e s t h a t a l l p l y w o o d t y p e s a r e a s s e m b l e d t o t h e minimum p e r m r s s i b l e t h i c k n e s s a n d t h e r e i s no l o s s i n t h e f o r m o f e x c e s s t h i c k n e s s i n p l y w o o d . 6 . 1 . 4 D e s i g n E f f i c i e n c y S u p p o s e ZJJ r e p r e s e n t s t h e maximum n e t r e v e n u e f o r a s e t o f v e n e e r t h i c k n e s s e s a n d p l y w o o d d e s i g n s a n d ZQ i s t h e u p p e r b o u n d on t h e v a l u e o f Z^ f o r a l l K. Then t h e r a t i o E n = x 100 ( 6 . 5 ) A 0 106 c a n be t e r m e d t h e d e s i g n e f f i c i e n c y , a p e r c e n t a g e r a t i n g f o r t h i s s e t o f p e e l t h i c k n e s s e s and p l y w o o d d e s i g n s . V a l u e s o f Eg c l o s e r t o 100% i n d i c a t e b e t t e r p e e l t h i c k n e s s e s and d e s i g n s . S i n c e Zg i s i n d e p e n d e n t o f K b u t ZQ i s d e p e n d e n t on i t , E g m e a s u r e s t h e e f f i c i e n c y i g n o r i n g t h e s e t u p c o s t s a s s o c i a t e d w i t h K. N e v e r t h e l e s s , i t g i v e s an i n d i c a t i o n a s t o how good a s e t o f v e n e e r t h i c k n e s s e s a n d p l y w o o d d e s i g n s a r e a n d t h e e x t e n t o f f u r t h e r i m p r o v e m e n t s , i f a n y , f r o m h i g h e r number o f p e e l t h i c k n e s s e s . 6.1.5 D e t e r m i n i n g t h e Optimum Number o f T h i c k n e s s e s F o r a n y p a r t i c u l a r m i l l , l e t 0„ be t h e t o t a l s e t up c o s t a s s o c i a t e d w i t h a l l t h e f a c t o r s 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 number o f v e n e e r t h i c k n e s s e s i s K. L e t lv be t h e c o r r e s p o n d i n g maximum n e t r e v e n u e o b t a i n a b l e f r o m t h e PDM m o d e l . Then = Z^ - 0^ g i v e s t h e n e t b e n e f i t a s s o c i a t e d w i t h K. The o p t i m u m number o f t h i c k n e s s e s f o r t h e m i l l i s g i v e n by t h a t K f o r w h i c h t h e n e t b e n e f i t R^ i s maximum. O b s e r v e t h a t Z^ i s bound f r o m a b o v e by Zg w h i l e 0^ , i s u n r e s t r i c t e d . The d e r i v a t i o n o f t h e u p p e r bound Z g , t h e d e s i g n e f f i c i e n c y a n d t h e o p t i m u m number o f v e n e e r t h i c k n e s s e s i s i l l u s t r a t e d t h r o u g h t h e f o l l o w i n g e x a m p l e . E x a m p l e 6.1 : The p l y w o o d m i l l d a t a u s e d i n c h a p t e r 3 i s a g a i n c o n s i d e r e d i n t h i s e x a m p l e . The p l y w o o d t y p e s (N = 1 2 ) , t h e l o w e r t o l e r a n c e on p l y w o o d t h i c k n e s s e s ( b ! r ) , t h e r e v e n u e ( r . . ) a n d t h e o r d e r f i l e ( d . ) a r e a s l i s t e d i n a p p e n d i x I I . F o r t h e s e p l y w o o d t y p e s , b ^ , r. a n d ( / 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 i n T a b l e 1 1 . P l y w o o d p l i e s -Type T h i c k b . u i r . l (r./b.) 3 _ 7 . 5 . 0 4 5 1 8 2 . 7 1 8 8 6 0 . 1 7 9 1 3 - 9 . 5 . 0 5 8 0 9 2 . 7 6 7 0 4 7 . 6 3 6 1 5 - 1 2 . 5 . 0 7 7 4 5 3 . 3 8 9 3 4 3 . 7 6 2 5 5 - 1 5 . 5 . 0 9 6 8 1 4 . 2 1 1 7 4 3 . 5 0 4 3 7 - 1 8 . 5 . 1 1 6 1 7 4 . 8 3 4 0 41 . 6 1 0 5 7 - 2 0 . 5 . 1 2 9 0 8 5 . 2 8 2 2 4 0 . 9 2 1 9 7 - 2 2 . 5 . 1 4 1 9 9 5 . 5 3 0 4 3 8 . 9 4 9 9 9 - 2 3 . 5 . 1 4 8 4 4 6 . 1 0 4 5 41 . 1 2 4 0 9 - 2 5 . 5 . 1 6 1 3 5 6 . 4 5 2 8 3 9 . 9 9 2 3 9 - 2 7 . 5 . 1 7 4 2 6 7 . 0 0 1 0 4 0 . 1 7 5 9 9 - 2 8 . 5 . 1 8 0 7 1 7 . 2 7 5 1 4 0 . 2 5 7 9 9 - 3 0 . 5 . 1 9 3 6 2 7 . 8 2 3 3 4 0 . 4 0 5 4 Tab! e l l : C o m p u t a t i o n s f o r t h e U p p e r b o u n d Z. From t a b l e 11 , we h a v e t h a t = M a x { F T / b . } = 6 0 . 1 7 9 1 s o t h a t p = 1 . T h e n , f r o m ( 6 . 4 ) we h a v e 1 _ / 3 0 0 0 0 0 - E b . d . Z N = E r . d . + ifl— "* l f l \ ~ 0 4 5 1 8 = $ 1 4 , 6 7 9 , 7 6 0 T h u s , f o r t h e p l y w o o d m i l l d a t a o f c h a p t e r 3 ( a p p e n d i x I I ) , t h e maximum n e t r e v e n u e c a n n e v e r e x c e e d $ 1 4 , 6 7 9 , 7 6 0 , i r r e s p e c t i v e o f t h e number o f v e n e e r t h i c k n e s s e s u s e d . 2 . 7 1 8 8 108 C o m p u t e r c o d e s o f t h e i m p l i c i t e n u m e r a t i o n a l g o r i t h m o f t h e PDM model w e r e u s e d w i t h t h e m i l l ' s d a t a t o d e t e r m i n e lv f o r v a l u e s o f K f r o m 1 t o 5 . The maximum n e t r e v e n u e Z^, and t h e q u a n t i t y o f wood l o s s i n t h e f o r m o f e x c e s s t h i c k n e s s d e r i v e d f r o m t h e PDM model a r e g i v e n i n T a b l e 1 2 . F o r c o m p a r i s o n p u r p o s e s , t h e c o r r e s p o n d i n g v a l u e s f o r t h e e x i s t i n g s e t o f v e n e e r 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 p r e s e n t e d . 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 w e r e c o m p u t e d u s i n g t h e v a l u e o f Z Q = 1 4 , 6 7 9 , 7 6 0 o b t a i n e d a b o v e a n d a r e p r e s e n t e d i n c o l u m n 4 o f t a b l e 1 2 . K Maximum R e v e n u e ( $ ) , h Wood L o s s i n E x c e s s T h i c k n e s s ( c u . m t r . ) D e s i g n E f f i c i e n c y ( % ) , E D E s t i m a t e d S e t u p C o s t * ( $ ) , 6 K N e t B e n e f i t ( $ ) R K i x i s t i n ( 4 ) g 1 3 , 4 1 6 , 6 9 4 7 , 9 4 4 . 7 91 . 4 0 3 , 7 1 4 , 7 7 0 9 , 7 0 1 , 9 2 4 1 2 3 4 5 I n f e a s i b l e I n f e a s i b l e 1 3 , 9 3 0 , 6 7 0 1 4 , 3 3 7 , 3 7 0 1 4 , 5 6 2 , 5 3 0 3 , 6 1 2 . 1 1 , 6 4 7 . 4 5 5 8 . 3 9 4 . 9 0 9 7 . 6 7 9 9 . 2 0 1 , 1 1 8 , 8 7 0 1 , 6 6 9 , 1 60 2 , 4 9 0 , 0 9 0 3 , 7 1 4 , 7 7 0 5 , 5 4 1 , 7 9 0 11 , 4 4 0 , 5 8 0 1 0 , 6 2 2 , 6 0 0 9 , 0 2 0 , 7 4 0 * E s t i m a t e d f r o m 0 K = 7 5 0 , 0 0 0 e x V T a b l e 1 2 : Optimum Number o f V e n e e r T h i c k n e s s e s and D e s i g n E f f i c i e n c y F o r 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 t h e p r o d u c t s o f t h e m i l l c a n n o t be a s s e m b l e d w i t h i n s p e c i f i c a t i o n s u s i n g o n e o r two p e e l , t h i c k n e s s e s . F o r K = 3 , t h e o p t i m a l v e n e e r t h i c k n e s s e s w e r e 2 . 6 , 3 . 2 a n d 4.4mm and t h e maximum n e t r e v e n u e was $ 1 3 , 9 3 0 , 6 7 0 ( a p p e n d i x I V ) . T h i s i s h i g h e r t h a n t h e c o r r e s p o n d i n g v a l u e f o r t h e 1 0 9 e x i s t i n g s e t o f f o u r t h i c k n e s s e s by 3 . 8 3 % . T h i s i n d i c a t e s t h a t n o t o n l y b e t t e r d e s i g n s e x i s t e d f o r t h e m i l l b u t s u b s t a n t i a l f u r t h e r s a v i n g s i n t h e s e t u p c o s t s a s s o c i a t e d w i t h t h e f o u r t h p e e l t h i c k n e s s w e r e a l s o p o s s i b l e . When K = 4 , t h e o p t i m u m v e n e e r t h i c k n e s s e s w e r e 2 . 5 , 3 . 1 , 3 . 9 a n d 4 . 8 mm and t h e c o r r e s p o n d i n g n e t r e v e n u e was $ 1 4 , 3 3 7 , 3 7 0 ( a p p e n d i x I I I ) . T h u s , i f t h e m i l l i n t e n d s k e e p i n g f o u r v e n e e r p e e l t h i c k n e s s e s f o r a n y r e a s o n , a b e t t e r s e t o f t h i c k n e s s e s a n d p l y w o o d d e s i g n s r e s u l t i n g i n a d d i t i o n a l r e v e n u e o f $ 9 2 0 , 6 7 6 ( e q u a l t o 6 . 8 6 % h i g h e r t h a n t h a t f o r t h e c u r r e n t s e t ) e x i s t s . When K = 5 , t h e o p t i m a l v e n e e r t h i c k n e s s e s w e r e 2 . 4 , 2 . 7 , 3 . 2 , 3 . 7 a n d 4 . 8 mm a n d t h e a s s o c i a t e d n e t r e v e n u e , Z^, was $ 1 4 , 5 6 2 , 5 3 0 . S i n c e t h e r e w e r e 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 f o r t h e 12 p l y w o o d t y p e s o f t h e m i l l , d e t a i l e d r e s u l t s o f t h e model a r e n o t p r e s e n t e d . The d e s i g n e f f i c i e n c y a t t a i n e d was 9 9 . 2 0 % i n d i c a t i n g t h a t f u r t h e r b e n e f i t s f r o m a n y h i g h e r number o f p e e l t h i c k n e s s e s w o u l d n o t be s i g n i f i c a n t . The wood l o s s i n t h e f o r m o f e x c e s s t h i c k n e s s i n p l y w o o d was 5 5 8 . 3 c u b i c m e t r e s . T h i s s e t o f f i v e v e n e e r t h i c k n e s s e s r e s u l t s i n a d d i t i o n a l r e v e n u e o f $ 2 2 5 , 1 6 0 o v e r t h e o n e f o r f o u r v e n e e r t h i c k n e s s e s o b t a i n e d f r o m t h e PDM m o d e l . H o w e v e r , t h i s i n c r e a s e i n r e v e n u e s h o u l d be c o m p a r e d w i t h t h e a d d i t i o n a l s e t u p c o s t s a s s o c i a t e d w i t h t h e f i f t h p e e l t h i c k n e s s . D a t a o n 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 number o f p e e l t h i c k n e s s e s w e r e n o t a v a i l a b l e f o r t h e m i l l . F o r c o m p a r i s o n p u r p o s e s , some h y p o t h e t i c a l d a t a on s e t u p c o s t s w e r e u s e d a l o n g w i t h t h e e x i s t i n g d a t a on r e v e n u e s . I t was p o s t u l a t e d t h a t t h e s e t u p c o s t s w e r e e x p o n e n t i a l l y r e l a t e d w i t h t h e number o f p e e l t h i c k n e s s e s by 0 K = 7 5 0 , 0 0 0 e x p ° ' ^ K . . The e s t i m a t e d s e t u p c o s t s o b t a i n e d u n d e r t h i s a s s u m p t i o n a r e g i v e n i n c o l u m n 5 o f T a b l e 1 2 . The l a s t c o l u m n o f t h i s t a b l e g i v e s R K = Z K " \ > t n e e s t i m a t e d 110 n e t b e n e f i t s . F o r t h i s s e t o f d a t a , t h e maximum n e t b e n e f i t o c c u r s a t K = 3 i m p l y i n g t h a t t h e o p t i m u m number o f v e n e e r t h i c k n e s s e s i s t h r e e . 6 . 2 A l t e r n a t e F a c e V e n e e r T h i c k n e s 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 c h a p t e r t w o , we h a d t r e a t e d x-j t o be t h e f a c e v e n e e r t h i c k n e s s by c o n v e n t i o n . T h i s c o n v e n t i o n f o l l o w s f r o m t h e f a c t t h a t t h e r e 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 s u r f a c e q u a l i t y c h a r a c t e r i s t i c s o f a f a c e v e n e e r t h a n t h o s e on t h e i n n e r p l i e s i n a p l y w o o d s h e e t . The f a c e v e n e e r s h o u l d n o r m a l l y be a f u l l s i z e s h e e t w i t h b l e m i s h - f r e e s u r f a c e w h e r e a s s m a l l s i z e d v e n e e r s f r o m s t r i p s 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 p l i e s o n l y . From m a n u f a c t u r i n g c o n s i d e r a t i o n s , h a v i n g t h i n v e n e e r s a s f a c e v e n e e r i s a d v a n t a g e o u s s i n c e more f u l l s i z e v e n e e r s h e e t s c a n be o b t a i n e d f r o m a g i v e n l o g f o r a t h i n v e n e e r r a t h e r t h a n t h a t f o r a t h i c k v e n e e r . F u r t h e r , r e p a i r s o r r e w o r k c a u s e d by f a c t o r s s u c h a s k n o t h o l e s , p i t c h p o c k e t s o r s p l i t s a r e e a s i e r w i t h t h i n v e n e e r s t h a n w i t h t h i c k v e n e e r s . In a d d i t i o n t o t h i s , f o r s t a n d a r d c o n s t r u c t i o n (COFI ( 1 9 7 8 ) ) , t h e f a c e v e n e e r s m u s t have t h e g r a i n d i r e c t i o n a l o n g t h e l e n g t h o f t h e p a n e l i m p l y i n g t h a t f o r f a c e v e n e e r s t h e l o g s m u s t be p e e l e d t o t h e f u l l l e n g t h o f t h e p a n e l w h i l e t h a t f o r c o r e v e n e e r s t h e y c a n be 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 t h e p a n e l . The c o n v e n t i o n t h a t x-j i s t h e f a c e v e n e e r t h i c k n e s s i s n o t a s e r i o u s l i m i t a t i o n o f t h e c a p a b i l i t i e s o f t h e PDM model o r t h e i m p l i c i t e n u m e r a t i o n a l g o r i t h m t o s o l v e i t . I f more t h a n o n e t h i c k n e s s c a n be u s e d f o r f a c e - v e n e e r , s u b j e c t t o a l l o f t h e m m e e t i n g t h e r e l e v a n t I l l s p e c i f i c a t i o n s , s u c h a s i t u a t i o n c a n a l s o be i n c o r p o r a t e d w i t h i n t h e f r a m e w o r k o f t h e PDM m o d e l . The i m p l i c i t e n u m e r a t i o n a l g o r i t h m r e q u i r e s a few m o d i f i c a t i o n s i n t h a t c a s e b u t i t s e f f e c t i v e n e s s w o u l d v i r t u a l l y be t h e s a m e . We i l l u s t r a t e t h e s e i n t h e f o l l o w i n g p a r a g r a p h s . R e c a l l t h a t m o s t p l y w o o d m i l l s u s e t h r e e o r f o u r v e n e e r t h i c k n e s s e s . S u p p o s e t h e r e c a n be two f a c e v e n e e r s , x-j and x 2 , t h e n i t l e a d s t o few more d e s i g n a l t e r n a t i v e s w h i c h c a n be o b t a i n e d by i n t e r c h a n g i n g t h e c o l u m n s a s s o c i a t e d w i t h a.^ a n d a. . ^ i n t a b l e s 3 - 6 . H o w e v e r , t h i s c a n r e s u l t i n some o f t h e d e s i g n a l t e r n a t i v e s a l r e a d y i n t a b l e s 3 - 6 . b e i n g d u p l i c a t e d a n d s u c h a l t e r n a t i v e s may be e x c l u d e d . F o r e x a m p l e , f o r a f i v e - p l y p l y w o o d w i t h K = 3 , when a d d i t i o n a l d e s i g n a l t e r n a t i v e s a r e g e n e r a t e d by i n t e r c h a n g i n g a_..p and a ^ 2 i n t a b l e 4 , t h e d e s i g n s r e p r e s e n t e d by t h e v e c t o r s ( 2 , 3 , 0 ) , ( 3 , 2 , 0 ) a n d ( 2 , 2 , 1) f o r ( a . . j i , a i j 2 ' a i j 3 ^ would be r e p e a t e d . E l i m i n a t i n g s u c h d u p l i c a t i o n s , i t c a n be v e r i f i e d t h a t f o r K = 3 t h e r e s u l t a n t number o f d e s i g n a l t e r n a t i v e s ( n ^ ) w o u l d be 6 , 1 5 , 15 a n d 18 f o r p l y w o o d made w i t h 3 , 5 , 7 a n d 9 p l i e s r e s p e c t i v e l y . In s o l v i n g t h e PDM model u s i n g t h e i m p l i c i t e n u m e r a t i o n a l g o r i t h m o f c h a p t e r 3 , some o f t h e b r a n c h i n g t e s t s n e e d m o d i f i c a t i o n . T r e a t i n g x^. t o be t h e f a c e v e n e e r t h i c k n e s s f o r f = 1 , 2 , t h e b o u n d s d e r i v e d i n ( 3 . 4 1 ) a n d ( 3 . 4 2 ) w o u l d c h a n g e , r e s p e c t i v e l y t o W h e r e , Tp i s t h e maximum p e r m i s s i b l e f a c e v e n e e r t h i c k n e s s . F o l l o w i n g a r g u m e n t s s i m i l a r t o t h a t i n c h a p t e r 3 , we c a n show t h a t t h e b r a n c h i n g t e s t ( 3 . 4 0 ) w o u l d be r e p l a c e d by t h e f o l l o w i n g two t e s t s : x , < x 2 ( 6 . 8 ) and f o r 3 "< s , < s „ < K ( 6 . 9 ) W i t h t h e m o d i f i c a t i o n s d e s c r i b e d a b o v e , t h e p l y w o o d m i l l d a t a o f c h a p t e r 3 ( a p p e n d i x I I ) was u s e d t o d e m o n s t r a t e t h e e f f e c t i v e n e s s o f t h e PDM m o d e l w i t h a l t e r n a t e f a c e v e n e e r t h i c k n e s s e s f o r K = 3 . The r e s u l t s o b t a i n e d f r o m t h i s m o d i f i e d model a r e p r e s e n t e d i n a p p e n d i x V . The o p t i m u m v e n e e r t h i c k n e s s e s i n t h i s c a s e w e r e 2 . 5 , 3 . 1 a n d 3 . 8 mm w i t h b o t h 2 . 5 a n d 3 . 1 mm v e n e e r s b e i n g u s e d a s f a c e v e n e e r s . I t c a n be o b s e r v e d f r o m t h e s e r e s u l t s t h a t t h e p l y w o o d t y p e s 3 p l y 9 . 5 mm, 9 p l y 2 8 . 5 mm and 9 p l y 3 0 . 5 mm m u s t be a s s e m b l e d u s i n g 3 . 1 mm o n l y a s t h e f a c e v e n e e r a n d t h a t t h e p l y w o o d t y p e s 3 p l y 7 . 5 mm and 5 p l y 1 2 . 5 mm m u s t be a s s e m b l e d u s i n g 2 . 5 mm a s t h e f a c e v e n e e r s . F o r a l l o t h e r t y p e s o f p l y w o o d , a l t e r n a t e d e s i g n s h a v i n g e i t h e r 2 . 5 mm o r 3 . 1 mm a s f a c e v e n e e r s e x i s t e d . H o w e v e r , a l l t h e s e d e s i g n s a r e v a r i a b l e s f o r t h e LMCK p r o b l e m a n d a s shown i n c h a p t e r 3 , t h e s o l u t i o n o f t h e LMCK w o u l d be s u c h t h a t o n l y o n e d e s i g n , a l t e r n a t i v e i s u s e d f o r e a c h p l y w o o d t y p e . The maximum n e t r e v e n u e o b t a i n e d f o r t h i s model was $ 1 4 , 3 8 7 , 2 8 0 . H o w e v e r , i t s h o u l d be n o t e d t h a t t h i s f i g u r e d o e s n o t i n c l u d e t h e c o s t s a s s o c i a t e d w i t h k e e p i n g two v e n e e r t h i c k n e s s e s a s f a c e v e n e e r s . T h e s e r e s u l t s a r e p r e s e n t e d h e r e o n l y t o d e m o n s t r a t e t h a t t h e PDM m o d e l and t h e i m p l i c i t e n u m e r a t i o n a l g o r i t h m t o s o l v e i t c a n be u s e d u n d e r v a r y i n g c i r c u m s t a n c e s . CHAPTER 7 113 7 . 1 C o n c l u s i o n s In t h i s d i s s e r t a t i o n , a r e a l - w o r l d p r o b l e m o f p r a c t i c a l s i g n i f i c a n c e w h i c h was h i t h e r t o n o t c o n s i d e r e d due t o i t s c o m p l e x i t y has been f o r m u l a t e d 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 a p p r o a c h . The i m p o r t a n c e o f t h e p r o b l e m s t e m s f r o m t h e f a c t t h a t a p r o c e s s e d m a t e r i a l w i t h v a l u e a d d e d f r o m m a n u f a c t u r e i s l o s t i m p l i c i t l y due t o t h e n o n - a v a i l a b i l i t y o f b e t t e r m e t h o d s o f e v a l u a t i o n . In t h e wake o f d w i n d l i n g s u p p l y o f t i m b e r r e s o u r c e s and f l u c t u a t i n g a n d c o m p e t i t i v e m a r k e t s t r u c t u r e t h e n e e d f o r e f f i c i e n t u t i l i z a t i o n o f t h e r a w m a t e r i a l i s a l l t h e more g r e a t e r now t h a n i t was e v e r b e f o r e . The p r o b l e m o f d e t e r m i n i n g a s e t o f 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 a n d p r o d u c t m i x w h i c h m a x i m i z e s t h e n e t r e v e n u e f o r a p l y w o o d m i l l has been f o r m u l a t e d a s a m a t h e m a t i c a l p r o g r a m m i n g m o d e l . A m e t h o d o f e v a l u a t i n g a l l f e a s i b l e p l y w o o d d e s ' i g n s f o r a s e t o f v e n e e r t h i c k n e s s e s i s d e v e l o p e d . The n o n - l i n e a r m i x e d i n t e g e r ( 0 - 1 ) p r o g r a m m i n g p r o b l e m s o f o r m u l a t e d i s s o l v e d f o r a g l o b a l s o l u t i o n u s i n g an i m p l i c i t e n u m e r a t i o n a l g o r i t h m . The e f f i c i e n c y , o f t h i s a l g o r i t h m a r i s e s f r o m i t s a b i l i t y t o e x p l o i t t h e s t r u c t u r e s i n t h e m o d e l . A c o m p u t e r c o d e i s w r i t t e n and d a t a f r o m a p l y w o o d m i l l i s a n a l y z e d t o d e m o n s t r a t e t h e p r a c t i c a l i t y o f t h e m o d e l . V a r i a t i o n s and e x t e n s i o n s o f t h e model u n d e r d i f f e r e n t c i r c u m s t a n c e s h a v e been c o n s i d e r e d and t h e i r s o l u t i o n p r o c e d u r e s h a v e been a n a l y s e d . I t i s shown t h a t t h e PDM p r o b l e m i s a n o n - c o n v e x p r o g r a m m i n g p r o b l e m w h i c h c a n be c a s t a s a s p e c i a l c a s e o f a d i s j u n c t i v e p r o g r a m . F o l l o w i n g 114 t h e d e v e l o p m e n t o f t h e m a t h e m a t i c a l p r o g r a m m i n g model and i t s s o l u t i o n p r o c e d u r e , e t h o d s t o d e t e r m i n e t h e e f f i c i e n c y o f p l y w o o d d e s i g n s a n d t h e o p t i m u m number o f v e n e e r t h i c k n e s s e s f o r a m i l l a r e d e v e l o p e d . The i m p l i c a t i o n o f t h i s d i s s e r t a t i o n a r e s t r a i g h t - f o r w a r d . A p l y w o o d m i l l 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 , c o s t o f l o g s , y i e l d f a c t o r l i s t o f p l y w o o d t y p e s , o r d e r f i l e and s u c h o t h e r f a c t o r s i n t h e PDM model t o d e r i v e t h e o p t i m a l 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 p r o d u c e m i x . I f i n c l u s i o n o f f a c t o r s s u c h a s l o g g r a d e s , v e n e e r s i z e s and p l y w o o d g r a d e s p r o v e t o be c o m p u t a t i o n a l l y e x p e n s i v e , an a b r i d g e d v e r s i o n o f t h e PDM model may be u s e d . The v e n e e r t h i c k n e s s e s and p l y w o o d d e s i g n s o b t a i n e d a s a s o l u t i o n o f t h i s a b r i d g e d PDM m o d e l c a n be u s e d as i n p u t i n a d e t a i l e d l i n e a r p r o g r a m m i n g model t o v e r i f y i f t h e y a r e i n d e e d b e t t e r t h a n t h e e x i s t i n g o n e s . In t h i s w a y , t h e model p r o v i d e s a l t e r n a t i v e s w h i c h n e v e r e x i s t e d b e f o r e . In t h e a b s e n c e o f m i l l LP m o d e l s , t h e PDM model c a n be u s e d a s a b a s i s f o r d e c i s i o n s on t h e c h o i c e o f v e n e e r t h i c k n e s s e s , p l y w o o d d e s i g n s and t h e p r o d u c t m i x . When a l l r e l e v a n t f a c t o r s a r e c o n s i d e r e d , t h e PDM model c a n , a t w o r s t , end up w i t h t h e s e t o f v e n e e r s and d e s i g n s c u r r e n t l y u s e d by a m i l l . The c o m p u t a t i o n a l t i m e a n d money i n v o l v e d i n g e n e r a t i n g a s o l u t i o n t o t h e PDM model i s r e l a t i v e l y 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 t h e p o t e n t i a l b e n e f i t s i t c a n b r i n g a b o u t . BIBLIOGRAPHY 115 B a i l e y , G . R . 1 9 7 0 . Log A l l o c a t i o n by D y n a m i c P r o g r a m m i n g . P h . D T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , B r i t i s h C o l u m b i a . B a i l e y , G . R . 1 9 7 2 . A New A p p r o a c h t o an o l d P r o b l e m - Log A l l o c a t i o n by D y n a m i c P r o g r a m m i n g . P u l p a n d P a p e r M a g a z i n e o f C a n a d a , 73 ( 9 ) T247 - T 2 4 9 . B a i l e y , G . R . 1 9 7 3 . Wood A l l o c a t i o n by D y n a m i c P r o g r a m m i n g . C a n a d i a n F o r e s t r y S e r v i c e , P u b l i c a t i o n # 1 3 2 1 , O t t a w a , O n t a r i o . B a l a s , E. a n d P . L . I v a n e s c u ( H a m m e r ) . 1 9 6 4 . On t h e 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 P r o b l e m . Management S c i e n c e , 11 ( 1 ) , 1 8 8 - 2 0 2 . B a l a s , E. 1 9 6 6 . The Dual M e t h o d f o r t h e 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 P r o b l e m . Management S c i e n c e , 12 ( 7 ) , 5 5 5 - 5 6 8 . B a l a s , E. 1 9 7 9 . D i s j u n c t i v e P r o g r a m m i n g . A n n a l s o f D i s c r e t e M a t h e m a t i c s , 5 , 3 - 5 1 . B a l d w i n , R . F . 1 9 7 5 . P l y w o o d M a n u f a c t u r i n g P r a c t i c e s . M i l 1 e r - F r e e m a n P u b l i c a t i o n s , S a n F r a n c i s c o , C a l i f o r n i a . B a r e , B . B . 1 9 7 1 . A p p l i c a t i o n s o f O p e r a t i o n s R e s e a r c h i n F o r e s t M a n a g e m e n t ; A S u r v e y . Q u a n t i t a t i v e S c i e n c e P a p e r # 2 6 , C e n t r e f o r Q u a n t i t a t i v e S c i e n c e i n F o r e s t r y , F i s h e r i e s and W i l d l i f e , U n i v e r s i t y o f W a s h i n g t o n , S e a t t l e , W a s h i n g t o n . B e t h e l , J . S . and C. H a r r e l . 1 9 5 7 . The A p p l i c a t i o n o f L i n e a r P r o g r a m m i n g t o P l y w o o d P r o d u c t i o n and D i s t r i b u 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 , 7 ( 7 ) , 2 2 1 - 2 2 7 . B i b l i s , E . J . , S - T „ Hsu and Y - M . C h i u . 1 9 7 2 . C o m p a r i s o n o f C e r t a i n S t r u c t u r a l P r o p e r t i e s among 3 - P l y , 4 - P l y a n d 5 - P l y % i n c h S o u t h e r n P l y w o o d S h e a t h i n g . Wood a n d F i b e r , 4 ( 1 ) , 1 3 - 1 9 . B r a c k e l y , A . M . 1 9 6 8 . An I n v e s t i g a t i o n o f t h e Y i e l d o f R o t a r y Cut V e n e e r and P l y w o o d f r o m E a s t e r n S p r u c e u n d e r S e l e c t e d C o n d i t i o n s o f M a n u f a c t u r e . M . S . T h e s i s , U n i v e r s i t y o f M a i n e , O r o n o , M a i n e . B r i g g s , D . G . 1 9 7 8 . A D y n a m i c P r o g r a m m i n g Model f o r B u c k i n g T r e e S t e m s i n t o L o g s . T r o p i c a l F o r e s t U t i l i z a t i o n S y s t e m , C o n t r i b u t i o n # 3 0 , I n s t i t u t e o f F o r e s t P r o d u c t s , U n i v e r s i t y o f W a s h i n g t o n , S e a t t l e , W a s h i n g t o n . C O F I . 1 9 7 8 . E x t e r i o r P l y w o o d M e t r i c M a n u f a c t u r i n g S t a n d a r d s a n d M i l l G u i d e . C o u n c i l o f F o r e s t I n d u s t r i e s o f B r i t i s h C o l u m b i a , V a n c o u v e r , B r i t i s h C o l u m b i a . C o l e b e c k , H . G . M . a n d P . L . N o r t h c o t t . 1 9 5 8 . Some F a c t o r s I n f l u e n c i n g t h e D e s i g n o f D o u g l a s - f i r P l y w o o d P a n e l s . F o r e s t P r o d u c t s J o u r n a l , 8 , 2 2 6 - 2 2 9 . 116 C o n w a y , S . 1 9 7 8 . T i m b e r C u t t i n g P r a c t i c e s ; ( 3 r d E d i t i o n ) . M i l l e r -Freeman P u b l i c a t i o n s , San F r a n c i s c o , C a l i f o r n i a . D a n t z i g , G . B . 1 9 6 3 . L i n e a r P r o g r a m m i n g a n d E x t e n s i o n s . P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , New J e r s e y . D i a z , R . E . 1 9 7 4 . B i b l i o g r a p h y o f O p e r a t i o n s R e s e a r c h i n F o r e s t r y I n d u s t r i e s . C o l l e g e o f F o r e s t R e s o u r c e s , U n i v e r s i t y o f W a s h i n g t o n , S e a t t l e , W a s h i n g t o n . D o b i e , J . and W . V . H a n c o c k . 1 9 7 2 . V e n e e r Y i e l d s f r o m B . C . I n t e r i o r D o u g l a s - f i r and W h i t e S p r u c e . C a n a d i a n F o r e s t I n d u s t r i e s , 9 2 ( 7 ) , 3 2 - 3 3 . D o b s o n , P . O . 1 9 7 1 . How MB A l l o c a t e s Logs by L i n e a r P r o g r a m m i n g . C a n a d i a n F o r e s t I n d u s t r i e s , 91 ( 2 ) , 4 4 - 4 5 . D o n n e l l e y , R . H . 1 9 6 6 . L i n e a r P r o g r a m m i n g i n P l y w o o d M a n u f a c t u r i n g . S e m i n a r on OR i n F o r e s t P r o d u c t s I n d u s t r y , IBM C o r p o r a t i o n , Los A n g e l e s , 2 2 - 3 9 . E i s e m a n n , K. 1 9 6 4 . The G e n e r a l i z e d S t e p p i n g S t o n e M e t h o d f o r t h e M a c h i n e L o a d i n g M o d e l . Management S c i e n c e , 11 ( 1 ) , 1 5 4 - 1 7 6 . E l a m , J . , F. G l o v e r a n d D. K l i n g m a n . 1 9 7 9 . A S t r o n g l y C o n v e r g e n t P r i m a l S i m p l e x A l g o r i t h m f o r G e n e r a l i z e d N e t w o r k s . M a t h e m a t i c s  o f O p e r a t i o n s R e s e a r c h , 4 ( 1 ) , 3 9 - 5 9 . E v e r e t t , J . P . 1 9 6 7 . An E x p e r i m e n t a l A p p l i c a t i o n o f L i n e a r P r o g r a m m i n g t o P l y w o o d P r o d u c t i o n . M . F . T h e s i s , U n i v e r s i t y o f W a s h i n g t o n , S e a t t l e , W a s h i n g t o n . F i e l d , D . B . 1 9 7 6 . A p p l i c a t i o n s o f O p e r a t i o n s R e s e a r c h t o Q u a n t i t a t i v e D e c i s i o n P r o b l e m s i n F o r e s t r y a n d t h e F o r e s t P r o d u c t s I n d u s t r i e s -A B i b l i o g r a p h y ( F o u r t h R e v i s i o n ) . S c h o o l o f F o r e s t r y and E n v i r o n m e n t a l S t u d i e s , Y a l e U n i v e r s i t y , New H a v e n , C o n n e c t i c u t . F i e l d , D . B . 1 9 7 7 . L i n e a r P r o g r a m m i n g ; Out o f t h e C l a s s r o o m a n d i n t o t h e W o o d s . J o u r n a l o f F o r e s t r y , 75 ( 6 ) , 3 3 0 - 3 3 4 . G a r f i n k e l , R . S . and G . L . N e m h a u s e r . 1 9 7 2 . I n t e g e r P r o g r a m m i n g . J o h n W i l e y a n d S o n s , New Y o r k . G l o v e r , F. a n d D. K l i n g m a n . 1 9 7 3 . A N o t e on C o m p u t a t i o n a l S i m p l i f i c a t i o n s i n S o l v i n g 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 P r o b l e m s . T r a n s p o r t a t i o n  S c i e n c e , 7 , 3 5 1 - 3 6 1 . G l o v e r , F. a n d D. K l i n g m a n . 1 9 7 9 . A 0(_n l o g n ) A l g o r i t h m f o r LP K n a p s a c k s w i t h GUB C o n s t r a i n t s . M a t h e m a t i c a l P r o g r a m m i n g , 17 ( 3 ) , 3 4 5 - 3 6 1 . 117 H a w k i n s , B . T . a n d L . N . C l a r k e . 1 9 7 0 . A l l o w a n c e f o r T r i m When M a k i n g P l y w o o d . The A u s t r a l i a n T i m b e r J o u r n a l , 36 ( 9 ) , 3 9 . H e i s k a n e n , V . 1 9 6 6 . S t u d i e s on t h e D e f e c t s o f B i r c h , T h e i r I n f l u e n c e on t h e Q u a l i t y a n d Q u a n t i t y o f R o t a r y c u t V e n e e r and T h e i r C o n s i d e r a t i o n s i n V e n e e r B i r c h G r a d i n g . A c t a F o r e s t a l i a F e n n i c a , 80 ( 3 ) , 1 - 1 2 8 . H o l e c e k , D . F . 1 9 7 5 . A S y s t e m s Model o f an I n t e g r a t e d F o r e s t P r o d u c t s F i r m w i t h D e c i s i o n A p p l i c a t i o n s . P h . D D i s s e r t a t i o n , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , C a l i f o r n i a . H o l m e s , S . 1 9 7 6 . I n t r o d u c t i o n t o O p e r a t i o n s R e s e a r c h A p p l i e d t o F o r e s t P r o d u c t s I n d u s t r i e s . F o r e s t P r o d u c t s J o u r n a l , 26 ( 1 ) , 1 7 - 2 2 . K e n n i n g t o n , J . L . a n d R . V . H e l g a s o n . 1 9 8 0 . A l g o r i t h m s f o r N e t w o r k  P r o g r a m m i n g . J o h n W i l e y a n d S o n s , New Y o r k . K l a m e c k i , B . E . 1 9 7 8 . U t i l i s i n g S o l a r E n e r g y i n t h e F o r e s t P r o d u c t s I n d u s t r y . F o r e s t P r o d u c t s J o u r n a l , 28 ( 1 ) , 1 4 - 2 0 . K l a m e c k i , B . E . 1 9 7 8 . O p t i m i z a t i o n o f V e n e e r L a t h e S e t t i n g s - a N o n -l i n e a r P r o g r a m m i n g A p p r o a c h . Wood S c i e n c e , 10 ( 4 ) , 2 2 0 - 2 2 4 . K o e n i g s b e r g , E. 1 9 6 0 . A p p l y i n g L i n e a r P r o g r a m m i n g t o t h e P l y w o o d I n d u s t r y . F o r e s t P r o d u c t s J o u r n a l , 10 ( 9 ) , 481 - 4 8 6 . K o t a k , D . B . 1 9 7 6 . A p p l i c a t i o n o f L i n e a r P r o g r a m m i n g t o P l y w o o d M a n u f a c t u r e . I n t e r f a c e s , 7 ( 1 ) , 5 6 - 6 8 . L a s d o n , L . S . 1 9 7 0 . O p t i m i z a t i o n T h e o r y f o r L a r g e S y s t e m s . M c M i l l a n P u b l i s h i n g C o m p a n y , New Y o r k . L e e , M . H . 1 9 6 8 . L i n e a r P r o g r a m m i n g A n a l y s i s A p p l i e d t o a S e l e c t e d P l y w o o d M a n u f a c t u r i n g F i r m . M . B . A . T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , B r i t i s h C o l u m b i a . L e f e b v r e , E. 1 9 7 8 . How I m p r o p e r Log B u c k i n g R e d u c e s Lumber R e v e n u e s . C a n a d i a n F o r e s t I n d u s t r i e s , 98 ( 9 ) , 2 1 - 2 5 . L o u r i e , J . R . 1 9 6 4 . T o p o l o g y a n d C o m p u t a t i o n o f t h e 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 P r o b l e m . Management S c i e n c e , 11 ( 1 ) , 1 7 7 - 1 8 7 . L u k e n , T . P . 1 9 7 8 . M a r k o v C h a i n s a s P l a n n i n g T o o l s f o r Long Log M e r c h a n d i s i n g and S a w m i l l i n g . P a p e r P r e s e n t e d a t 0RSA/TIMS m e e t i n g , Los A n g e l e s , C a l i f o r n i a ( I n L u k e n e t a l . ( 1 9 8 0 ) , R e f e r e n c e C i t e d ) . L u k e n , T . P . , A . O t t e n a n d P . S . H a u g e n . 1 9 8 0 . Q u i c k and C l e a n ; An A s p e c t o f IE/OR I n t e r f a c e . I n t e r f a c e s , 10 ( 5 ) , 1 - 6 . M a r t i n , A . J . a n d P . E . S e n d a k . 1 9 7 3 . O p e r a t i o n s R e s e a r c h i n F o r e s t r y -A B i b l i o g r a p h y . USDA F o r e s t S e r v i c e G e n e r a l T e c h n i c a l R e p o r t N E - 8 , N o r t h e a s t F o r e s t a n d Range E x p e r i m e n t S t a t i o n , U p p e r D a r b y , P e n n s y l v a n i a . 118 M c K i l l o p , W. 1 9 7 4 . An O p e r a t i o n s R e s e a r c h S t u d y o f an I n t e g r a t e d Wood P r o c e s s i n g F i r m i n C a l i f o r n i a . P a p e r P r e s e n t e d a t FPRS M e e t i n g , N o r t h e r n C a l i f o r n i a S e c t i o n , # N C - 7 4 - S 2 6 , R e d d i n g , C a l i f o r n i a . M e r i l u o t o , J . 1 9 6 5 . The I n f l u e n c e o f Raw M a t e r i a l F a c t o r s on t h e Q u a n t i t y and Q u a l i t y o f R o t a r y c u t B i r c h V e n e e r . . A c t a F o r e s t a l l ' a  Fenn i c a , 80 (1 ) , 1 - 1 5 5 . N a g a r a j u , S . , B . G . R a g h a v e n d r a a n d J . V e n k a t a r a m a n . 1 9 7 4 . G r e e n V e n e e r Y i e l d i n R o t a r y C u t t i n g . I P I R I J o u r n a l , J o u r n a l o f t h e I n d i a n P l y w o o d I n d u s t r i e s R e s e a r c h I n s t i t u t e , 4 ( 4 ) , 1 4 1 - 1 5 1 . N o r r i s , C . B . , F. W e r r e n and F . R . M c K i n n o n . 1 9 6 1 . The E f f e c t o f V e n e e r T h i c k n e s s a n d G r a i n D i r e c t i o n on t h e S h e a r S t r e n g t h o f P l y w o o d . U . S . F o r e s t P r o d u c t s L a b o r a t o r y , R e p o r t # F P L - 1 8 0 1 , M a d i s o n , W i s c o n s i n . P a r a s i n , A . V . 1 9 7 6 . E x p e r i m e n t a l a n d T h e o r e t i c a l C o m p a r i s o n o f 4 v s . 5 P l y and 6 v s . 7 P l y P l y w o o d C o n s t r u c t i o n by S t i f f n e s s and Moment C a r r y i n g C a p a c i t y . R e p o r t # 1 0 8 , C o u n c i l o f F o r e s t I n d u s t r i e s o f B r i t i s h C o l u m b i a , N o r t h V a n c o u v e r , B r i t i s h C o l u m b i a . P h i l l i p s , D . T . and A . G a r c i a - D i a z . 1 9 8 1 . F u n d a m e n t a l s o f N e t w o r k  A n a l y s i s . P r e n t i c e - H a l l , E n g l e w o o d C l i f f s , New J e r s e y . P n e v m a t i c o s , S . M . a n d S . H . M a n n . 1 9 7 2 . D y n a m i c P r o g r a m m i n g i n T r e e B u c k i n g . F o r e s t P r o d u c t s J o u r n a l , 22 ( 2 ) , 2 6 - 3 0 . R a g h a v e n d r a , B . G . a n d S . N a g a r a j u . 1 9 7 5 . O p t i m i s i n g t h e T r i m A l l o w a n c e i n P l y w o o d M a n u f a c t u r e . J o u r n a l o f t h e I n d i a n A s s o c i a t i o n f o r  Q u a l i t y and R e l i a b i l i t y , 3 , 1 1 9 - 1 2 4 . R a g h a v e n d r a , B . G . 1 9 7 9 . A S u r v e y o f O p e r a t i o n s R e s e a r c h A p p l i c a t i o n s i n Wood P r o d u c t s I n d u s t r i e s . Term P a p e r S u b m i t t e d t o F o r e s t r y 5 5 9 : OR i n F o r e s t r y , F a c u l t y o f F o r e s t r y , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r ( u n p u b l i s h e d ) . R a m a l i n g a m , P. 1 9 7 6 . A B r a n c h a n d B o u n d A p p r o a c h t o t h e T r e e B u c k i n g P r o b l e m . P a p e r P r e s e n t e d a t t h e ORSA/TIMS M e e t i n g , A n a h e i m , C a l i f o r n i a ( I n B r i g g s ( 1 9 7 8 ) , R e f e r e n c e C i t e d ) . R a m s i n g , K . D . 1 9 6 5 . A L i n e a r P r o g r a m m i n g M o d e l 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 O r e g o n , E u g e n e , O r e g o n . R a m s i n g , K . D . 1 9 6 8 . L i n e a r P r o g r a m m i n g f o r P l y w o o d M i x P r o b l e m . F o r e s t P r o d u c t s J o u r n a l , ! 8 ( . 4 ) , 9 8 - 1 0 1 . R a i z a d a , H . C . a n d J . C . N a u t i y a l . 1 9 7 4 . An I n p u t - O u t p u t M o d e l o f O n t a r i o F o r e s t B a s e d I n d u s t r i e s . C a n a d i a n J o u r n a l o f F o r e s t  R e s e a r c h 4 ( 3 ) , 3 7 2 - 3 8 0 . 119 R e i m e r , D . R . 1 9 6 9 . 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 M o d e l 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 E c o n o m i c I m p a c t o f a New F o r e s t I n d u s t r y . P h . D D i s s e r t a t i o n , P u r d u e U n i v e r s i t y , L a f a y e t t e , I n d i a n a . R e s c h , H. a n d H . L . S c h e u r m a n . 1 9 7 7 . C o m p u t e r S i m u l a t i o n o f t h e S o f t w o o d V e n e e r D r y i n g O p e r a 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 , 27 ( 2 ) , 3 2 - 3 5 . S i t t e r , R . M . 1 9 6 9 . LP f o r B e t t e r P r o f i t L e v e l s . BC L u m b e r m a n , 53 ( 5 ) , 3 1 - 3 5 . T a h a , H. 1 9 7 1 . O p e r a t i o n s R e s e a r c h . M a c M i l l a n P u b l i s h i n g C o m p a n y , New Y o r k . T o b i n , L . R . a n d J . S . B e t h e l . 1 9 6 9 . V e n e e r 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 T h r o u g h C o m p u t e r S i m u l a t i o n . Wood and F i b e r , 1 ( 2 ) , 9 7 - 1 0 7 . T r u e m p e r , K. 1 9 7 6 . An E f f i c i e n t S c a l i n g P r o c e d u r e f o r G a i n s N e t w o r k s . N e t w o r k s , 6 , 1 5 1 - 1 6 0 . T y r e , G . L . a n d G . D . S c r e p e t i s . 1 9 7 8 . S y s t e m f o r Roundwood I n v e n t o r y a n d 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 . F o r e s t  P r o d u c t s J o u r n a l , 2 8 ( 1 ) , 4 0 - 4 1 . W a g n e r , H . M . 1 9 7 5 . 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 ; ( 2 n d E d i t i o n ) . P r e n t i c e - H a l l , E n g l e w o o d C l i f f s , New J e r s e y . W e l l w o o d , E .W. 1 9 7 1 . S c h e d u l i n g and C o n t r o l o f a D o u g l a s F i r P l y w o o d 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 . F o r e s t P r o d u c t s J o u r n a l , 21 ( 1 1 ) , 4 2 - 5 0 . W e s t e r n F o r e s t s P r o d u c t s L a b o r a t o r y . 1 9 7 8 . P r o g r a m R e v i e w 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 (Now, F o r i n t e k Canada C o r p o r a t i o n ) , V a n c o u v e r , B r i t i s h C o l u m b i a . W o o d , A . D . 1 9 6 3 . P l y w o o d s o f t h e W o r l d ; T h e i r D e v e l o p m e n t , M a n u f a c t u r e  a n d A p p l i c a t i o n . J o h n s t o n a n d B a c o n 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 . J r . 1 9 7 3 . 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 u r S p e c i e s . 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 . Y a p t e n c o , R . Q . a n d A . E . W y l i e . 1 9 7 0 . A Q u a n t i t a t i v e A p p r o a c h t o P l y w o o d 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 ) , 5 4 - 5 9 . Z e m e l , E. 1 9 8 0 . 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 . O p e r a t i o n s  R e s e a r c h , 28 ( 6 ) , 1 4 1 2 - 1 4 2 3 . 120 APPENDIX I L i s t i n g o f t h e FORTRAN P r o g r a m f o r t h e I m p l i c i t  E n u m e r a t i o n A l g o r i t h m ; F o u r V e n e e r T h i c k n e s s c c 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 THE FOLLOWING SECTION PRINTS THE INPUT DATA AS READ C BY THE COMPILER FOR VERIFICATION PURPOSES. C PRINT220 220 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 FORMAT!10X. '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 FORMAT!10X,'RESULTS OF THE BRANCHING TESTS') PRINT184 184 FORMAT! ' ' , 10X ,' . ') DO 20 1=1,NUMBER IF(IPLY(I).E0.5)G0 TO 21 122 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. GO TO 25 TMIN=(BL(I)-2. GO TO 25 TMIN=(BL(I ) -2. GO TO 25 23 TMIN=(BL( I )-2 . IFfTMIN.GT.TKMIN)TKMIN=TMIN PR I NT 29.TKMIN '.5X.'MINIMUM PERMISSIBl MM ' ) 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 26 27 25 29 FORMAT( C6X.F8.2 •TKMAX ) 'TKMAX )/3. 'TKMAX)/5. ' TKMAX)/7. E THICKNESS FOR X4 FROM (3.42)= THIS SECTION IS INTENDED TO PRINT AN ITERATIVE SUMMARY AS AND WHEN IMPROVED VENEER THICKNESSES ARE FOUND. 199 PRINT195 195 FORMAT('-PRINT 194 194 FORMAT(' PRINT196 196 FORMAT('-,5X,'RUN STATISTICS: . 5X . . 5X.'IMPROVED VENEER SETS FOUND SO FAR AND CORRESPONDING C THICKNESS AND OBJECTIVE' ) PRINT 1960 ' . 5X FUNCTION VALUE ARE AS FOLLOWS:') , 5X, 'NUMBER' , 10X, 'VENEER THICKNESS' ,5X, 'OBJ. FN. VALUE ( 1960 FORMAT(' PRINT 197 197 FORMAT('-C$ ) ' ) PRINT74.(XSTAR(K).K=1.4),ZSTAR 74 FORMAT(2X.'EXISTING SET'.3X,4F6.2,2X.F15.2) DO 30 L1=1,MAX1 X(1)=T(L1) DO 31 L2=1,N2 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 ) C C COMPUTATION OF THE BOUNDS ON X4 FROM ( 3.31)-(3.32 ) C AND THE INFEASIBILITY TESTS ASSOCIATED WITH THEM. C 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=1.3 402 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 404 403 IF(TH( I ,J ) . LT . EL ( I ) .OR.THf I , J .) .GT . BU( I ) )G0 TO 401 AMIN = T(MIN4 ) AMAX = T(N2 ) 404 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 401 CONTINUE IF(I DENT(I) .EO.0)GO TO 311 GO TO 400 4 10 DO 4 11 J= 1 , 16 ID(I ,d)=0 SUM=0. DO 4 12 K=1,3 412 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 14 413 IF(TH(I,0) .LT.BL(I).OR.TH(I,J) .GT.BU(I ) )G0 TO 411 AMIN = T(MIN4 ) AMAX = T(N2) 4 14 ID(I,J)=1 IDENT(I) = IDENT(I )+1 124 IF(AMIN.LT.XMIN(I))XMIN(I)=AMIN IF(AMAX.GT.XMAX(I))XMAX(I ) = AMAX 411 CONTINUE IFfIDENT(I).EO.O)GO TO 311 GO TO 400 420 DO 421 J=1,16 ID(I,J)=0 SUM=0. DO 422 K=1,3 422 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 424 423 IF(TH( I .d) .LT.BL(I ) .OR.THI I .d> .GT .BUI I) )G0 TO 421 AMIN=T(MIN4) AMAX = T(N2) 424 ID( I .J ) =1 IDENT(I ) = IDENT(I )+ 1 IFfAMIN.LT.XMINI I ) )XMIN( I )=AMIN IF(AMAX.GT.XMAX(I))XMAX(I)=AMAX 421 CONTINUE IF(IDENT(I).EO.0)G0 TO 311 GO TO 400 430 DO 43 1 d=1.16 ID( I .d ) =0 5UM=0. DO 432 K=1,3 432 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 434 433 IF(TH(I.d ) .LT .BL( I ) OR.TH(I,d ) .GT.BU(I ) )G0 TO 431 AMIN=T(MIN4) AMAX = T(N2) 434 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 431 CONTINUE IF(IDENTfI ) .EO.O)GO TO 311 400 CONTINUE X4MIN=TKMIN X4MAX=T(N2) DO 440 1 = 1 ,NUMBER IF(XMIN(I).GT.X4MIN)X4MIN=XMIN(I) 440 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 COMPUTATION OF THE UPPER BOUND ON Z FROM (3.38): C '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 4 5 0 C O N T I N U E T S M A X = 0 . DO 4 6 0 1 = 1 , N U M B E R T S ( I ) = ( R ( I ) - ( C * Y * B D ( I ) ) / C F ) / ( ( B D ( I ) > Y ) / C F ) I F ( T S ( I ) . L E . T S M A X )G0 TO 4 6 0 T S M A X = T S ( I ) I B E 5 T = I 4 6 0 C O N T I N U E S W 0 0 D = O . DO 46 1 1 = 1 , N U M B E R I F ( I . E O . I B E S T l G O TO 461 P D ( I ) = D E M ( 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 ) I F ( P D ( I B E S T ) . L T , D E M ( I B E S T ) )G0 TO 311 Z B 0 U N D = O . DO 4 6 2 1 = 1 , N U M B E R 4 6 2 Z B O U N D = 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 E N D =IBND+1 GO TO 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 : ' Z N O D E ' I S 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 FROM ( 3 . 1 4 ) . 198 DO 32 L 4 = M I N X 4 , M A X X 4 X ( 4 ) = T ( L 4 ) I F ( L 4 . L E . L 3 )G0 TO 32 I F ( X ( 4 ) . E 0 . X ( 2 ) )G0 TO 32 I F ( X ( 4 ) . E Q . X ( 1 ) ) G 0 TO 32 NODE =NODE+ 1 DO 4 0 1 = 1 . N U M B E R I T E S T ( 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 TO 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 T H I C K ( I , J ) = T H ( I , J ) + X ( 4 ) * P L Y 3 ( J , 4 ) » C F 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 ) ) G 0 TO 44 I D E L T A ( I . J ) = 1 I T E S T ( I ) = I T E S T ( I ) + I D E L T A ( I . J ) 44 C O N T I N U E I F ( I T E S T ( I ) . G E . 1 )G0 TO 4 0 GO TO 32 4 1 DO 4 5 J= 1 , 16 I D E L T A ( I . J ) = 0 T H I C K ( I , J ) = T H ( I . J ) + X ( 4 ) * P L Y 5 ( J , 4 ) * C F 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 ) ) G 0 TO 4 5 I D E L T A ( I , J ) = 1 I T E S T ( I ) = I T E S T ( I ) + I D E L T A ( I , J ) 4 5 C O N T I N U E I F ( I T E S T ( I ) . G E . 1 ) G 0 TO 4 0 GO T O 32 42 DO 46 J= 1 . 16 I D E L T A ( I , J ) = 0 T H I C K ( I , J ) = T H ( I , d ) + X ( 4 ) * P L Y 7 ( J , 4 ) * C F 127 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 ) ) G O TO 46 I D E L T A ( I , J ) = 1 I T E S T ( I ) = I T E S T ( I ) + I D E L T A ( I , J ) 46 C O N T I N U E I F ( I T E S T f I ) . G E . 1 ) G O T O 4 0 GO T O 32 4 3 DO 47 J = 1 , 1 6 I D E L T A ( I , J ) = 0 T H I C K ! I . J ) = T H ( I . J ) I F ( T H I C K ( I . J ) . L T . B L ( I ) . OR . T H I C K ! I , J ) . GT .BUI. I ) )G0 TO 47 I D E L T A ( I . J ) = 1 I T E S T ( I ) = I T E S T ( I ) + I D E L T A ( I . J ) 47 C O N T I N U E I F ( I T E S T l I ) . G E . 1 ) G 0 TO 4 0 GO TO 32 4 0 C O N T I N U E C C S E T T I N G L I M I T S ON C O M P U T A T I O N S ; ' K S E T ' IS T H E C M A X I M U M NUMBER OF LMCK P R O B L E M S TO B E S O L V E D . C K S E T = K S E T + 1 I F ( K S E T . G T . 5 0 0 0 ) G O TO 2 0 0 DO 5 0 K = 1 . 4 5 0 B ( K ) = Y * X ( K ) T W M A X = 0 . DO 51 1 = 1 . N U M B E R T P( I )=0 . I F ( I P L Y l I ) . E O . 5 ) G 0 TO 52 I F I I P L Y l I ) . E Q . 7 ) G 0 TO 53 I F ( I P L Y l I ) . E O . 9 ) G 0 TO 54 DO 55 J= 1 . 4 T W I I . J ) = 0 . R S U M ( I , J ) = 0 . B S U M I I , J ) = 0 . I F I I D E L T A ( I . J ) . E Q 0 ) G 0 TO 55 DO 6 11 K=1 , 4 611 B S U M I I , J ) = B S U M ( I , J ) + B ( K ) * P L Y 3 ( J , K ) RSUMI I , J ) = R ( I ) - B S U M I I , J ) * C T W I I . J ) = R S U M ( I . J ) / B S U M ( I . J ) IF ITWI I . J ) . L T . T P ! I ) )G0 TO 55 T P ( I )= TW( I . J ) J B E S T I I ) = J I F I T P I I l . L T . T W M A X ) G O TO 55 TWMAX = T P ( I ) I B E S T = I 5 5 C O N T I N U E GO TO 5 1 52 DO 56 J= 1 , 16 T W I I , 0 ) = 0 . R S U M t l , J ) = 0 . B S U M t I . J ) = 0 . I F ( I D E L T A ( I , J ) . E O . O ) G O TO 56 DO 6 1 2 K = 1 , 4 6 1 2 B S U M ( I , J ) = B S U M ( I , J ) + B ( K ) * P L Y 5 ( J , K ) R S U M ( I , J ) = R ( I ) - C * B S U M ( I . d ) T W ( I , J ) = R S U M ( I , J ) / B S U M ( I . J ) I F ( T W ( I , J ) . L T . T P ( I ) )G0 TO 56 T P ( I ) = T W ( I , J ) J B E S T ( I ) = J I F ( T P ( I ) . L T . T W M A X ) G 0 TO 56 128 T W M A X = T P ( I ) I B E S T = I 5 G C O N T I N U E GO TO 51 5 3 DO 57 J = 1 , 1 6 T W ( I . J ) = 0 . R S U M ( I , J ) = 0 . B S U M ( 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 ) R S U M ( I , J ) = R ( I ) - C * B S U M ( I . J ) T W ( I . J ) = R S U M ( I , J ) / B S U M ( I . J ) I F ( T W ( I . J ) . L T . T P ( I ) )GO TO 57 T P ( I ) = T W ( I . J ) J B E S T ( I ) = J I F ( T P < I ) . L T . T W M A X )GO TO 57 T W M A X = T P ( I ) I B E S T = I 57 C O N T I N U E GO TO 51 54 DO 58 J = 1 . 16 T W ( I , J > = 0 . R S U M ( I , J ) = 0 . B S U M ( 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 ) T W ( I . J ) = R S U M I I . J ) / B S U M ( I . J ) I F ( T W ( I . J ) . L T . T P ( I ) ) G 0 TO 58 T P ( I ) = T W ( I . J ) J B E S T ( I ) = J I F f T P ( I ) . L T . T W M A X ) G 0 TO 58 T W M A X = T P ( I ) I B E S T = I 58 C O N T I N U E 51 C O N T I N U E S U M B = 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 NI =4 I F ( I P L Y ( I ) . G T . 3 )NI = 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 ) = D E M ( I ) . W O O D ( I ) = B S U M ( I , J ) * P ( I . J ) R E V ( I ) = R S U M ( I . J ) * P ( I . J ) 61 C O N T I N U E S U M B = S U M B + W O O D ( I ) GO TO 6 0 6 9 I P = I 6 0 C O N T I N U E N I = 4 I F ( I P L Y ( I P ) . G T . 3 ) N I = 16 DO 7 0 J = 1 , N I P ( I P . J ) = 0 . I F ( J . N E . J B E S T ( I P ) ) G 0 TO 7 0 P ( I P , d ) = ( W - S U M B ) / B S U M ( I P . J ) 1 2 9 I F ( P ( I P . J ) . L T . D E M ( I P ) ) G O TO 32 R E V ( I P ) = R S U M ( I P , J ) * P ( I P . J ) 7 0 C O N T I N U E Z N O D E = 0 . 0 0 7 1 1 = 1 , N U M B E R 7 1 Z N O D E = Z N O D E + R E V ( I ) C C U P D A T I N G T H E I N C U M B E N T S O L U T I O N AND P R I N T I N G C IT FOR I T E R A T I V E S U M M A R Y . C I F ( Z N O D E . L E . Z S T A R ) G O TO 32 I M P R = I M P R + 1 Z S T AR = Z N O D E WOODVL=TWMAX DO 72 K= 1 . 4 72 X S T A R ( K ) = X ( K ) P R I N T 7 5 . I M P R . ( X S T A R ( K ) , K = 1 . 4 ) . Z 5 T A R 7 5 F O R M A T ( ' O ' . 5 X . I 3 . 9 X . 4 F 6 . 2 . 2 X . F 1 5 . 2 ) DO 73 1 = 1 . N U M B E R NI =4 I F ( I P L Y ( I ) . G T . 3 ) N I = 16 DO 73 J = 1 . N I P S T A R ! I . J ) = P ( I . J ) R ST A R ( I . J ) = R S U M ( I , J ) T S T A R ( I , J ) = T H I C K ( I . J ) TWSTARf. I . J I = TW ( I . J ) 73 I S T A R f I . J ) = I D E L T A ! I . J ) 32 C O N T I N U E 31 1 C O N T I N U E 31 C O N T I N U E 3 0 C O N T I N U E 1 F ( K S E T . G T . 0 )G0 TO 100 C C M E S S A G E I F T H E P R O B L E M I S I N F E A S I B L E . C P R I N T 101 101 F O R M A T ! ' 1 ' , 5 X . ' T H E P R O B L E M IS I N F E A S I B L E : V E N E E R T H I C K N E S S S E T SAT C I S F Y I N G C O N S T R A I N T S ( 2 . 2 ) - ( 2 . 4 ) D O E S NOT E X I S T ' ) GO TO 9 9 9 C C M E S S A G E I F T H E NUMBER OF LMCK P R O B L E M S TO BE C S O L V E D E X C E E D S A P R E S E T L I M I T . C 2 0 0 P R I N T 2 0 1 201 F O R M A T ( ' 1 ' , 5 X , ' N U M B E R OF F E A S I B L E V E N E E R T H I C K N E S S S E T S E X C E E D S 5 0 C O O : P R O G R A M T E R M I N A T E D P R E M A T U R E L Y . C U R R E N T R E S U L T S A R E P R E S E N T E D ' C ) C C T H I S S E C T I O N G I V E S A SUMMARY OF T H E P E R F O R M A N C E C OF T H E I M P L I C I T E N U M E R A T I O N A L G O R I T H M . C 1 0 0 PR I NT 1 9 3 , M X N O D E 193 F O R M A T ! ' - ' , 5 X , ' M A X I M U M P O S S I B L E S E T S OF V E N E E R T H I C K N E S S . O R I G I N A L C P R O B L E M = ' , 5 X , 1 6 ) P R I N T 4 7 0 . I B N D 4 7 0 F O R M A T ! ' - ' , 5 X . ' N U M B E R OF T I M E S U P P E R BOUND ON Z ( 3 . 3 8 ) WAS E F F E C T I C V E = ' , 8 X , 1 6 ) P R I N T 1 9 2 , N O D E 192 F O R M A T ( ' - ' , 5 X , ' N U M B E R OF V E N E E R S E T S E V A L U A T E D A F T E R B R A N C H I N G T E S C T S = ' , 9 X , 1 6 ) 130 P R I N T 1 9 1 , K S E T 191 F O R M A T ( 5 X . ' N U M B E R OF V E N E E R S E T S E V A L U A T E D FOR LMCK P R O B L E M = ' C 1 2 X . 1 6 ) DO 77 K = 1 . 4 V ( K ) = 0 . DO 78 1 = 1 , N U M B E R I F ( I P L Y l I ) . E 0 . 5 )GO T O 1 7 0 I F ( I P L Y l I ) . E O . 7 ) G 0 TO 171 I F ( I P L Y ( I ) . E Q . 9 ) G 0 TO 172 DO 173 J= 1 . 4 I F ( I S T A R U , J ) . E O . O J G O T O 173 V ( K ) = V ( K ) + P L Y 3 ( J , K ) * P S T A R ( I . J ) 173 C O N T I N U E GO TO 78 170 DO 174 J=1 , 16 I F ( I S T A R ( I , J ) . E O . 0 ) G O TO 174 V ( K ) = V ( K ) + P L Y 5 ( J . K ) * P S T A R ( I . J ) 174 C O N T I N U E GO TO 78 17 1 DO 175 J = 1 . 1 6 I F I I S T A R f I . J ) . E O . O I G O TO 175 V ( K ) = V ( K ) + P L Y 7 ( J , K ) * P S T A R ( I . J ) 175 C O N T I N U E GO T O 78 N 172 DO 176 J= 1 . 16 I F I I S T A R I I . J ) . E O . 0 ) G O TO 176 V ( K > = V ( K ) + P L Y 9 ( J . K ) * P S T A R ( I . J > 176 C O N T I N U E 78 C O N T I N U E B S T A R I K ) = Y * X S T A R ( K ) L S T A R I K ) = V ( K ) * B S T A R I K ) 77 C O N T I N U E C C T H I S S E C T I O N G I V E S T H E S O L U T I O N TO T H E PDM P R O B L E M . C P R I N T 102 102 F O R M A T I ' 1 ' , 5 X . ' R E S U L T S : ' ) P R I N T 103 103 F O R M A T ( ' ' . 5 X . ' ' ) P R I N T 1 0 4 , ( X S T A R ( K ) . K = 1 , 4 ) 104 F O R M A T ! ' 0 ' . 5 X , ' O P T I M A L V E N E E R T H I C K N E S S E S ARE ( M M ) : ' , 4 F 8 . 2 ) P R I N T 1 7 7 177 F O R M A T ! ' O ' , 5 X , ' Q U A N T I T Y OF L O G S FOR C O R R E S P O N D I N G ' ) P R I N T 1 7 7 0 . ( L S T A R I K ) ,K=1 , 4 ) 1 7 7 0 F O R M A T ( ' ' , 5 X . ' T H I C K N E S S ( C U . M T R S . ) : ' . 1 5 X . 4 F 8 . 0 ) P R I N T 1 0 5 . Z S T A R 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 ( $ ) : ' , F 2 0 . 2 ) P R I N T 1 9 0 , W O O D V L 190 F O R M A T ! ' O ' . 5X , ' 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 MIX C A R E A S F O L L O W S : ' ) P R I N T 107 107 F O R M A T ! ' - ' , 5 X . ' P L Y W O O D T Y P E ' . 8 X . ' D E S I G N A L T E R N A T I V E ' , 5 X , ' T H I C K N E S C S ' . 5 X . ' Q U A N T I T Y ' , 1 0 X , ' N E T R E V E N U E ' ) P R I N T 1 0 8 108 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 P R I N T 1 1 1 , I P L Y ( I ) , T P L Y ( I ) 111 F O R M A T ( ' - ' , 5 X , 1 2 , ' P L Y ' , 2 X , F 4 . 1, ' M M ' ) 131 I F ( I P L Y ( I ) . E O . 5 ) G 0 T O 112 I F ( I P L Y ( I ) . E O . 7 ) G 0 TO 113 I F ( I P L Y ( I ) . E 0 . 9 ) G 0 TO 114 DO 115 d=1 , 4 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 ) 116 F O R M A T ( ' ' . 2 2 X . 4 F 5 . 0 . 7 X . F 5 . 2 . 5 X . F 1 2 . 1 . 8 X . F 8 . 4 ) 115 C O N T I N U E GO TO 1 10 112 DO 117 d= 1 , 16 I F ( I S T A R ( I , d ) . E Q . O ) G O TO 117 P R I N T 1 1 8 . ( P L Y 5 ( 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 ) 118 F O R M A T ( ' ' . 2 2 X . 4 F 5 . 0 . 7 X , F 5 . 2 . 5 X . F 1 2 . 1 , 8 X , F 8 . 4 ) 117 C O N T I N U E GO TO 1 10 113 DO 119 d= 1 , 16 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 ) . T S T A R ( I , d ) . P S T A R ( I , d ) , R S T A R ( I , d ) 120 F O R M A T ( ' ' , 2 2 X , 4 F 5 . 0 , 7 X , F 5 . 2 , 5 X . F 1 2 . 1 . 8 X . F 8 . 4 ) 119 C O N T I N U E GO TO 110 114 DO 122 d= 1 . 16 I F ( I S T A R ( I , d ) . E O . O ) G O TO 122 P R I N T 1 2 1 . ( P L Y 9 ( 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 ) 121 F O R M A T ( ' ' . 2 2 X . 4 F 5 . 0 . 7 X . F 5 . 2 . 5 X . F 1 2 . 1 . 8 X . F 8 . 4 ) 122 C O N T I N U E 110 C O N T I N U E 9 9 9 S T O P END C C T H I S L A S T S E C T I O N I S T H E I N P U T D A T A . T H E F I R S T C 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 . T H E S E C A R E F O L L O W E D BY S P E C I F I C M I L L D A T A U S E D FOR C I L L U S T R A T I O N I N C H A P T E R T H R E E . $ D A T A 3 0 0 0 2 1 o 0 2 0 1 0 2 0 0 1 5 0 0 0 4 1 0 0 4 0 1 0 4 0 0 1 3 2 0 0 3 0 2 0 3 0 0 2 2 3 0 0 2 0 3 0 2 0 0 3 2 2 1 0 2 2 0 1 2 1 2 0 2 1 0 2 2 0 2 1 2 0 1 2 7 0 0 0 5 2 0 0 5 0 2 0 5 0 0 2 132 4 3 0 0 4 0 3 0 4 0 0 3 2 5 0 0 2 0 5 0 2 0 0 5 2 3 2 0 2 3 0 2 2 2 3 0 2 2 0 3 2 0 3 2 2 0 2 3 9 0 0 0 6 3 0 0 6 0 3 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 1 2 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 502289.0 6.1 5 15.5 15.0 16.0 350192.0 7.6 7 18.5 18.0 19.0 423394.0 8.9 7 20.5 20.0 21.0 443442.0 9.8 7 22.5 22.0 23.O 2952.0 10.5 9 23.5 23.0 24.0 6135.0 11.3 9 25.5 25.0 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 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 300000.0 35.0 0.006067 0.94 3.2 2.64 3.35 3.96 4.98 13416690. 133 APPENDIX I I  D a t a f r o m a P l y w o o d M i l l i n B . C . 1 . Number o f v e n e e r t h i c k n e s s e s ( K ) : 4 2 . V e n e e r t h i c k n e s s e s ( x k ) : 2 . 6 9 , 3 . 3 5 , 3 . 9 6 a n d 4 . 9 8 mm 3 . L o g C o s t ( C ) : 3 5 . 0 0 $ / c u . m t r . 4 . Log a v a i l a b i l i t y ( W ) : 3 0 0 0 0 0 c u . m t r . 5 . V e n e e r y i e l d f a c t o r ( y ) : 0 . 0 0 6 0 6 7 ( c o r r e s p o n d s t o a 60% y i e l d by v o l u m e ) 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 ( C f ) : 0 . 9 4 7 . U p p e r l i m i t on f a c e v e n e e r t h i c k n e s s ( T ^ ) : 3 . 2 0 mm 8 . Number o f p l y w o o d t y p e s ( N ) : 12 134 9 . P l y w o o d t y p e s a n d d e s i g n s : P l y w o o d Type P l i e s - T h i c k n e s s (mm) D e s i g n A l t e r n a t i v e Number o f v e n e e r s o f 2 . 6 9 3 . 3 5 3 . 9 6 4 . 9 8 3 - 7 . 5 1) 3 0 0 0 3 - 9 . 5 i ) 2 0 0 1 5 - 1 2 . 5 i) 5 0 0 0 i i ) 4 1 0 0 5 - 1 5 . 5 1) 2 2 0 1 i i ) 2 1 2 0 7 - 1 8 . 5 i) 5 2 0 0 i i ) 4 3 0 0 7 - 2 0 . 5 i) 2 5 0 0 i i ) 4 0 3 0 7 - 2 2 . 5 i) 2 0 5 0 i i ) 2 2 3 0 9 - 2 3 . 5 D 6 3 0 0 9 - 2 5 . 5 i) 6 0 3 0 i i ) 5 4 0 0 9 - 2 7 . 5 i) 5 0 4 0 i i ) 2 7 0 0 9 - 2 8 . 5 i) 6 0 0 3 i i ) 2 4 3 0 9 - 3 0 . 5 i) 2 0 7 0 i i ) 5 0 0 4 135 1 0 . 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 T h i c k n e s s (mm ) R e v e n u e O r d e r f i l e PI i e s S p e c i f i e d Lower L i m i t U p p e r L i m i t ( $ / P a n e l ) ('# 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 1 0 . 0 4 . 8 1 0 6 3 7 8 5 1 2 . 5 1 2 . 0 1 3 . 0 6 . 1 5 0 2 2 8 9 5 1 5 . 5 1 5 . 0 1 6 . 0 7 . 6 3501 92 7 1 8 . 5 1 8 . 0 1 9 . 0 8 . 9 4 2 3 3 9 4 7 2 0 . 5 2 0 . 0 2 1 . 0 9 . 8 4 4 3 4 4 2 7 2 2 . 5 2 2 . 0 2 3 . 0 1 0 . 5 2952 9 2 3 . 5 2 3 . 0 2 4 . 0 1 1 . 3 6135 9 2 5 . 5 2 5 . 0 2 6 . 0 1 2 . 1 :111 72 9 2 7 . 5 2 7 . 0 2 8 . 0 1 3 . 1 2 7 3 8 9 2 8 . 5 2 8 . 0 2 9 . 0 1 3 . 6 1278 9 3 0 . 5 3 0 . 0 3 1 . 0 1 4 . 6 38 1 1 . Maximum p o s s i b l e Net R e v e n u e ( $ ) : 1 3 4 1 6 6 9 4 1 2 . Wood l o s s i n t h e f o r m o f e x c e s s t h i c k n e s s i n p l y w o o d ( c u . m t r . ) : 7 9 4 4 . 7 136 APPENDIX I I I R e s u l t s f r o m t h e PDM Model w i t h F o u r V e n e e r T h i c k n e s s e s 1 . O p t i m a l v e n e e r t h i c k n e s s e s (mm): 2 . Q u a n t i t y o f l o g s f o r c o r r e s p o n d i n g t h i c k n e s s e s ( c u b i c m e t r e s ) : 3 . O b j e c t i v e f u n c t i o n v a l u e a t o p t i m a l s o l u t i o n ( $ ) : 4 . M a r g i n a l v a l u e o f wood ( $ / c u b i c m e t r e ) : 5 . Wood l o s s i n t h e f o r m o f e x c e s s t h i c k n e s s i n p l y w o o d ( c u b i c m e t r e s ) 2 . 5 3 . 1 3 . 9 4 . 8 2 0 5 8 4 8 4 8 8 8 6 3 1 7 9 9 1 3 4 6 8 1 4 3 3 7 3 7 0 . 0 0 5 9 . 5 0 0 3 1 6 4 7 . 4 137 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 m i x : P l y w o o d T y p e p l y - T h i c k n e s s (mm) D e s i g n A l t e r n a t i v e Number 2 . 5 o f 3 . 1 V e n e e r s 3 . 9 o f 4 . 8 Q u a n t i t y o f P l y w o o d , # o f P a n e l s , 2 . 4 4 m x 1 . 2 2 m 3 - 7 . 5 i) 3 0 0 0 2 3 7 1 6 5 3 i i ) 2 1 0 0 -3 - 9 . 5 i) 2 0 0 1 1 0 6 3 7 8 5 - 1 2 . 5 1) 4 1 0 0 5 0 2 2 8 9 i i ) 3 2 0 0 -5 - 1 5 . 5 i) 2 0 3 0 _ i i ) 2 2 0 1 3501 92 7 - 1 8 . 5 1) 4 3 0 0 4 2 3 3 9 4 7 - 2 0 . 5 i) 5 0 0 2 i i ) 4 0 3 0 4 4 3 4 4 2 i i i T 2 3 2 0 -7 - 2 2 . 5 1) 4 0 0 3 i i ) 2 3 0 2 2 9 5 2 9 - 2 3 . 5 1) 5 4 0 0 6135 9 - 2 5 . 5 i) 6 0 3 0 _ i i ) 2 7 0 0 ' i l l 172 9 - 2 7 . 5 D 2 4 3 0 2738 i i ) 6 0 0 3 -9 - 2 8 . 5 1) 2 3 4 0 1278 9 - 3 0 . 5 i) 2 0 7 0 38 138 APPENDIX IV  R e s u l t s f r o m t h e PDM M o d e l w i t h T h r e e V e n e e r T h i c k n e s s e s 1 . O p t i m a l v e n e e r t h i c k n e s s e s (mm): 2 . 6 3 . 2 4 . 4 2 . Q u a n t i t y o f l o g s f o r c o r r e s p o n d i n g t h i c k n e s s ( c u b i c m e t r e s ) : 3 . O b j e c t i v e f u n c t i o n v a l u e a t o p t i m a l s o l u t i o n ( $ ) : 4 . M a r g i n a l v a l u e o f wood ( $ / c u b i c m e t r e ) : 2 3 1 5 3 3 3 2 0 8 7 3 6 3 8 0 1 3 9 3 0 6 7 0 5 5 . 8 6 5 7 Wood l o s s i n t h e f o r m o f e x c e s s t h i c k n e s s i n p l y w o o d ( c u b i c m e t r e s ) : 3 6 1 2 . 1 139 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 a n d p r o d u c t m i x : P l y w o o d T y p e P l y - T h i c k n e s s (mm) D e s i g n A l t e r n a t i v e Number 2 . 6 o f 3 . 2 v e n e e r s o f 4 . 4 Q u a n t i t y o f P l y w o o d , # o f p a n e l s , 2 . 4 4 m x 1 . 2 2 m 3 7 . 5 i ) 3 0 0 2 2 7 7 0 7 2 i i ) 2 1 0 -3 - 9 . 5 1) 2 0 1 1 0 6 3 7 8 5 - 1 2 . 5 1) 5 0 0 5 0 2 2 8 9 i i ) 4 1 0 -5 - 1 5 . 5 i ) 3 0 2 _ i i ) 2 2 1 3 5 0 1 9 2 7 - 1 8 . 5 1) 5 2 0 4 2 3 3 9 4 i i ) 4 3 0 -7 - 2 0 . 5 1) 5 0 2 4 4 3 4 4 2 7 - 2 2 . 5 i ) 4 0 3 _ i i ) 2 3 2 2952 9 - 2 3 . 5 i ) 6 3 0 6135 9 - 2 5 . 5 1) 2 7 0 11172 9 - 2 7 . 5 i ) 6 0 3 2 7 3 8 9 - 2 8 . 5 1) 5 0 4 1278 9 - 3 0 . 5 1) 2 3 4 38 140 APPENDIX V R e s u l t s f r o m t h e PDM M o d e l ; A l t e r n a t e F a c e V e n e e r s 1 . O p t i m a l v e n e e r t h i c k n e s s e s (mm): 2 . 5 3 . 1 3 . 8 2 . Q u a n t i t y o f l o g s f o r c o r r e s p o n d i n g t h i c k n e s s ( c u b i c m e t r e s ) : 1 8 5 6 7 0 8 2 8 6 9 31461 3 . F a c e v e n e e r s (mm): 2 . 5 3 . 1 4 . O b j e c t i v e f u n c t i o n v a l u e a t o p t i m a l s o l u t i o n ( $ ) : 1 4 , 3 8 7 , 2 8 0 5 . M a r g i n a l v a l u e o f wood ( $ ) : 5 9 . 5 0 0 3 6 . Wood l o s s i n t h e f o r m o f e x c e s s t h i c k n e s s i n p l y w o o d ( c u b i c m e t r e s ) : 1 4 0 3 . 0 / 141 7 . 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 a n d p r o d u c t m i x : P l y w o o d Type D e s i g n Number o f V e n e e r s o f Q u a n t i t y o f P l y w o o d , P l i e s - T h i c k (mm) A l t e r n a t i v e * 2 . 5 3 . 1 3 . 8 # o f P a n e l s 3 - 7 . 5 3 0 0 2 3 8 3 2 5 9 i i ) 2 1 0 -3 - 9 . 5 D A 0 2 1 1 0 6 3 7 8 5 - 1 2 . 5 i ) 4 1 0 5 0 2 2 8 9 i i ) 4 0 1 -i i i ) 3 2 0 -5 - 1 5 . 5 i ) 2 0 3 _ i i ) A 0 4 1 350192 i i i ) A 0 3 2 -i v ) A 1 2 2 -7 - 1 8 . 5 i ) 5 0 2 _ i i ) 4 3 0 4 2 3 3 9 4 i i i ) A 3 4 0 -7 - 2 0 . 5 i ) 4 0 3 _ i i ) 2 3 2 -i i i ) A 0 7 0 — i v ) A 3 2 2 4 4 3 4 4 2 7 - 2 2 . 5 i ) 2 0 5 _ i i ) A 0 4 3 2 9 5 2 9 - 2 3 . 5 i ) 5 4 0 6135 i i •) A 4 •5 0 -9 - 2 5 . 5 i ) 2 7 0 11172 i i ) A 4 2 3 -9 - 2 7 . 5 i ) 2 4 3 2 7 3 8 i i ) 2 3 4 -i i i ) A 3 2 4 -9 - 2 8 . 5 i ) A 0 6 3 1278 i i ) A 0 5 4 -9 - 3 0 . 5 i ) A 0 2 7 38 ' * ' A ' i n d i c a t e s a l t e r n a t i v e f a c e v e n e e r t h i c k n e s s o f 3 . 1 mm 

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