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A comparison of twenty simulated strategies for achieving maximum parts value recovery in optimizing… Chow, Gordon Ray 1998

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A COMPARISON OF TWENTY SIMULATED STRATEGIES FOR ACHIEVING MAXIMUM PARTS V A L U E R E C O V E R Y IN OPTIMIZING CHOP SAW OPERATIONS. by GORDON RAY CHOW B.Sc. (Forestry), The University of British Columbia, 1992 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF MASTERS OF SCIENCE in T H E F A C U L T Y OF GRADUATE STUDIES (Department of Wood Science) We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA April 1998  © Gordon Ray Chow, 1998  In  presenting  degree freely  this  thesis  in  partial  fulfilment  at the University  of  British  Columbia,  available for reference  copying  of this  department publication  or  thesis by  of this  and study.  for scholarly  his thesis  or  her  purposes  gain  of  \A/00/}  The University of British Vancouver, Canada  Date  DE-6 (2/88)  SC Columbia  agree  that that  It  is  permission  advanced make  it  for extensive  by the head  understood  shall not be allowed  /£A/C<T  for an  the Library shall  may be granted  representatives.  for financial  the requirements  I agree  I further  permission.  Department  of  that without  of my  copying  or  my written  ABSTRACT A simulation model, using a dynamic program and other functions, successfully emulated an automated chop saw that cuts, from a sample m i l l , full length lumber into usable parts according to a cutting b i l l . The program determined i f value recovery per board foot (value/fbm) from a m i l l run could be increased by modifying the chop saw's priority settings which forced parts into a cutting solution. Twenty different cutting algorithms were employed o f which 18 prioritized up to three different parts, one was based on the sample m i l l ' s priority settings and the remaining one was a control or regular optimization. Each algorithm scanned the cutting bill to dynamically update priority settings, using the value o f pieces remaining to be cut as a deciding factor each time a full length board was processed. Production was stopped on parts which had reached a certain production point or cutoff percentage. Data, consisting o f grades and lengths o f all lumber sections from two dimensions, 3" width and 3.75" width, were put into a simulation model for processing. Students-Newman-Keuls tests showed that there was no single algorithm that performed decisively better than others. The highest rated cutting algorithm prioritized three different parts and forced one o f each part into a cutting solution with an 80% cutoff percentage. Further data analysis proved that good value/fbm results were not obtained by using more infeed lumber or sacrificing parts recovery percentage.  K e y W o r d s : chop saw, cutting algorithms, dynamic program, optimization, priority settings, simulation.  '  il  9  T A B L E OF CONTENTS A Comparison Of Twenty Simulated Strategies For Achieving Maximum Parts Value Recovery In Optimizing Chop Saw Operations. ABSTRACT  j  •Vl  LIST O F T A B L E S .  f• T [O V/ / /  LIST O FFIGURES ACKNOWLEDGEMENTS  V  UNIVERSITY O F BRITISH COLUMBIA ( U B C ) :  V / /'/  Committee M e m b e r s  V// / /  Other Contributors  \y' J i '  INDUSTRY:  ^  1.0 I N T R O D U C T I O N  j j j 1  1.1 BACKGROUND AND OBJECTIVES  1  1.2 CHOP SAW TECHNOLOGY AND LITERATURE REVIEW  2  1.2.1 Grades and Dimensions o f L u m b e r Parts  3  1.2.2 Prices O f L u m b e r Parts  4  1.2.3 Input L u m b e r  5  1.2.4 L u m b e r Defect Detection  6  1.2.5 Infeed  8  1.2.6 Cutting M e c h a n i s m  8  1.2.6.1 Manual  8  1.2.6.2 Semi-Automated  9  1.2.6.3 Automated  9  1.2.7 Sorting  10  1.2.7.1 Completely Manual Sort 1.2.7.2 Kicker Sorting  11 11  1.2.8 C h o p Saw Software Options  12  1.2.8.1 Simulation 1.2.8.2 Modify Priority Settings  12 12  1.3 MACHINERY SETUP  15  2.0 M A T E R I A L S A N D M E T H O D S  18  2.1 DESCRIPTION O F T H E TEST M I L L  18  2.1.1 M i l l Operation  18  2.1.2 R a w Material  20  2.1.2.1 3" Width Lumber  20  2.1.2.2 3.75" Width Lumber  21  2.1.3 Products 2.1.4 Costs  21 =  21  2.2 D A T A COLLECTION  22  2.2.1 L u m b e r Data  23  2.2.2 T i m e and M o t i o n Study  23  2.2.3 Data F r o m M i l l E m p l o y e e s  24  2.2.4 Y i e l d Data  25  2.2.4.1 Basic Structure  25  2.2.4.2 Modifications To Demand  26  2.2.4.2.1 Actual Cut 2.2.4.2.2 Actual Demand 2.2.4.2.3 Pseudo Demand 2.3 DEVELOPMENT O F CHOP SAW ALGORITHMS  27 27 28 29  «  C  I /  i  9  2.4 SIMULATION PROCEDURES  30  2.4.1 Graphical Elements O f T h e C h o p Saw Simulation M o d e l  32  2.4.1.1 Lumber Movement  32  2.4.1.2 Downtime  33  2.4.2 C Program Inserts O f C h o p Saw Simulation M o d e l  35  2.4.2.1 Assigning Lumber Data  35  2.4.2.1.1 Get Board Number/Selection Information  36  2.4.2.1.2 Select Board  37  2.4.2.1.3 Select Interarrival Time  39  2.4.2.2 Making Cutting Decisions  39  2.4.2.2.2 Get The Best Cutting Pattern  42  2.4.2.2.3 Dynamic Program  44  2.4.2.2.3.1 Features Common T o Both Cutting Modes 2.4.2.2.3.1.1 Accuracy  44 •  44  2.4.2.2.3.1.2 Fingerjointing  44  2.4.2.2.3.1.3 Cutting Considerations  45  2.4.2.2.3.2 Features Applicable Only To Actual Cutting 2.4.2.2.3.2.1 Statistics From Each Board  46 46  2.4.2.2.3.2.2 Kicker  47  2.4.2.2.3.2.3 Intervention  47  2.5 D A T A ANALYSIS 2.5.1 A n a l y s i s Variables 2.5.1.1 Value Recovery Per Fbm  49 49 49  2.5.1.2 Number O f 20' Clears Added  50  2.5.1.3 Percent Recovery O f Parts  50  2.5.1.4 Number O f Boards Processed 2.5.2 Setup O f A n a l y s i s 2.5.2.1 2 Way Anova Description 2.5.2.2 Student-Newman-Keuls Range Test 2.5.3 A N O V A Assumptions  51 51 51 53 53  2.5.3.1 Normal Data  53  2.5.3.2 Repeatable Process  54  2.5.3.3 Homogeneous Variances 2.5.4 C o m p a r i n g A n a l y s i s Variables  3.0 RESULTS AND DISCUSSION  54 54  56  3 . 1 3 . 7 5 " WIDTH LUMBER  56  3.2 3" WIDTH L U M B E R  57  3.2.1 V a l u e / F b m M e a n s 3.2.1.1 Differences  57 58  3.2.1.2 Number O f Parts Prioritized  60  3.2.1.3 Cutoff Percentages  61  3.2.2 Interaction Results 3.3 3" WIDTH EXTERNAL VARIABLES  61 62  3.3.1 N u m b e r O f 2 0 ' C l e a r s A d d e d  63  3.3.2 N u m b e r o f Boards Processed  66  3.3.3 Percent Recovery O f Parts 3.4 3" WIDTH FINANCIAL BENEFITS O F T H E IDEAL ALGORITHM  4.0 SUMMARY AND RECOMMENDATIONS  70 73  75  4.1 SUMMARY  75  4.2 RECOMMENDATIONS  76  5.0 LITERATURE CITED  78  APPENDIX I.  DATA F R O M B O T H GRADES O F LUMBER  80  APPENDIX II.  PROJECT CUTTING BILLS  81  APPENDIX HI.  DOWNTIME DATA  86  APPENDIX IV.  FINGERJOINTING RULES  87  APPENDIX V.  TWO WAY ANOVA PROCEDURES  88  APPENDIX VI.  BARTLETT'S TEST PROCEDURES/EXAMPLE  90  APPENDIX VII.  SNK PROCEDURES/EXAMPLE  92  APPENDIX VIII. 3.75" WIDTH DATA MATRIX  93  APPENDIX IX.  3" WIDTH V A L U E / F B M MEANS DATA MATRIX  94  APPENDIX X.  3" WIDTH V A L U E / F B M - B A R T L E T T ' S TEST  95  APPENDIX XI.  20' CLEARS ADDED DATA MATRIX  96  APPENDIX XII.  3" WIDTH NUMBER OF 20' CLEARS ADDED - BARTLETT'S TEST  97  APPENDIX XIII. 3" WIDTH 20' CLEARS ADDED (INTERVENTION) - SNK  98  APPENDIX XIV.  3" WIDTH NUMBER OF BOARDS PROCESSED DATA MATRIX  99  APPENDIX XV.  3" WIDTH NUMBER OF BOARDS PROCESSED - BARTLETT'S TEST  100  APPENDIX XVI.  3" WIDTH NUMBER OF BOARDS PROCESSED - SNK  101  APPENDIX XVII. 3" WIDTH PERCENT PARTS R E C O V E R Y DATA MATRIX  102  APPENDIX XVIII. 3" WIDTH PERCENT PARTS R E C O V E R Y - BARTLETT'S TEST  103  APPENDIX XIX.  3" WIDTH PERCENT PARTS R E C O V E R Y - SNK  104  APPENDIX XX.  3" WIDTH Y E A R L Y V A L U E DIFFERENCES F R O M A L G O R I T H M 14  105  APPENDIX XXI.  E X A M P L E OF 3 WAY V A L U E / F B M ANOVA SETUP  106  V  LIST OF TABLES Table 1. E x a m p l e o f A Cutting B i l l  13  Table 2. Production Results F r o m A 180" Usable Section  14  Table 3. 3" W i d t h L u m b e r :  20  Table 4. 3.75" W i d t h L u m b e r :  21  Table 5. A c t u a l C h o p Saw Cutting B i l l  25  Table 6. "Actual C u t " Cutting B i l l D e m a n d  27  T a b l e 7. Pseudo D e m a n d E x a m p l e  29  Table 8. Cutting A l g o r i t h m s  30  Table 9. C Program Steps T o Obtain L u m b e r Information  36  Table 10.  Sample Q f L u m b e r Identification File  36  Table 11. L u m b e r Package Length Composition Rules Table 12.  37  E x a m p l e O f B o a r d Data File  38  Table 13. Cumulative Distribution O f Interarrival T i m e s F o r 10' L u m b e r  39  Table 14. E x a m p l e O f Fingerjoint Rules (3" Width - C u t b i l l #6)  45  T a b l e 15. E x a m p l e O f Simulation R u n Summary Report  46  Table 16. E x a m p l e O f B o a r d Data File  47  Table 17. E x a m p l e O f 3" W i d t h V a l u e / F b m Table  52  Table 18. E x a m p l e O f A v e r a g e V a l u e / f b m versus Average N u m b e r O f Boards Processed Data  55  Table 19. A N O V A Table F o r 3.75" W i d t h L u m b e r  56  Table 20.  A N O V A Table F o r 3" Width L u m b e r  57  T a b l e 21.  3" W i d t h V a l u e / F b m Bartlett's Test Results  58  Table 22.  3" W i d t h N u m b e r O f 20' Clears A d d e d A N O V A  63  Table 23.  3" W i d t h N u m b e r O f 20' Clears A d d e d Bartlett's Test Results  63  Table 24.  N u m b e r o f Boards Processed A N O V A  67  Table 25.  3" W i d t h N u m b e r o f Boards Processed Bartlett's Test Results  67  Table 26.  Percent R e c o v e r y O f Parts A N O V A  70  Table 27.  3" W i d t h Percent R e c o v e r y O f Parts Bartlett's Test Results  70  vi  LIST O F FIGURES Figure 1. C h o p Saw Input ( A ) and Outputs (B)  3  Figure 2. Sequence O f C o m m o n C h o p Saw Features  5  Figure 3. E x a m p l e O f B o a r d G r a d i n g System  19  Figure 4.  23  Steps T o A c q u i r e L u m b e r Data  Figure 5. Illustration O f L u m b e r G r a d i n g Station  24  Figure 6.  31  Discrete Simulation E x a m p l e U s i n g N o r m a l Distribution  Figure 7. F u l l Length L u m b e r and Parts F l o w Networks Figure 8. Downtime Networks  33 :  Figure 9. Diagram O f B o a r d F r o m R o w 207 In Table 12  34 38  Figure 10. Flowchart O f Procedures F o r Cutting Parts F r o m F u l l Length L u m b e r  40  Figure 11. Cutting A l g o r i t h m and Cutting Pattern  41  Figure 12. E x a m p l e O f B o a r d Scanning  42  Figure 13. Cutting Patterns F o r Test M o d e  43  Figure 14. Rankings G r a p h E x a m p l e - V a l u e / f b m Versus N u m b e r O f Boards Processed  55  Figure 15. 3.75" Width V a l u e / f b m  57  Figure 16. Student-Newman-Keuls C o m p a r i s o n O f Means Table  59  Figure 17. 3" W i d t h V a l u e / F B M Means  59  Figure 18. Cutting Prioritized Parts W i t h A l g o r i t h m s 10, 11, and 12  60  Figure 19. 3" Width Interaction Plot  62  Figure 20.  Interaction - Average N u m b e r O f 20' Clears A d d e d  64  Figure 21.  Average N u m b e r O f Clears  65  Figure 22. Rank Comparisons - N u m b e r O f 20' Clears A d d e d V s . V a l u e / f b m  66  Figure 23.  Interaction - A v e r a g e N u m b e r O f Boards Processed  68  Figure 24.  Average N u m b e r O f Boards Processed  68  Figure 25. Rank Comparisons - A v e r a g e N u m b e r O f Boards Processed V s . $/fbm  69  Figure 26.  71  Interaction - A v e r a g e Percent Recovery O f Parts  Figure 27. Percent Recovery O f Parts  72  Figure 28. R a n k Comparisons - Percent R e c o v e r y O f Parts V s . $/fbm  73  Figure 29.  74  Y e a r l y V a l u e Differences F r o m A l g o r i t h m 14  ACKNOWLEDGEMENTS There are many individuals i n both the academic and industrial fields o f forestry who should receive full recognition for their contributions to this thesis project.  University Of British Columbia (UBC): Committee Members Project advisor was D r . Thomas Maness who not only gave the graduate student permission to start his project but also made sure that proper funding and procedures were always in place. D r . Peter Marshall regularly assisted with statistical and simulation work while other supplemental help was provided by Dr. Robert Kozak, D r . Farrokh Sassani and M r . Jan Aune.  Other Contributors Other individuals at U B C also made the project possible. Dr. Antal K o z a k provided advice and publications on data analysis. modifications.  M r . Darrell Wong assisted with model testing and algorithm  M r . Dallas Foley rectified some o f the C programming problems.  M s . Peggy  Grahame and M s . Dayna Furst took care o f purchasing and reimbursements.  Industry: Due to a secrecy agreement, details about the participating m i l l are confidential. Unfortunately, this also means that public recognition cannot be made o f the many m i l l workers and managers who donated their expertise to the project. M r . Ian Harvey , Manager - Education and Training Programs - B C W S G , helped coordinate communications with the participating m i l l at the early stages o f the project and allowed his organization to cover a percentage o f the expenses that were incurred. M r . Harvey also regularly checked i n on the project and made sure that all work was  V lii  applicable to his company's members.  M r s . Jean O ' R e i l l y , Product Support and University  Relations - Pritsker Corporation, gave complete professional customer support when simulation model development and preliminary C programming work were done.  1.0  INTRODUCTION  1.1 B a c k g r o u n d a n d Objectives For as long as furniture and homes have been built there has always been demand for attractive and durable lumber parts. Derived from full-length lumber after undesirable characteristics have been crosscut out by machines known as optimizing chop saws, lumber parts are subject to processes such as trimming, sanding, moulding, and gluing before contributing to a finished product. Grading rules restrict the characteristics o f allowable lumber defects while a cutting bill or list o f parts wanted by a customer influences chop saw cutting decisions.  The objective o f this thesis project was to find the cutting algorithm among the twenty proposed that achieved the highest value per board foot (value/fbm) figure. Three properties o f a cutting b i l l , priority settings, prices, and quantities o f parts demanded were the main criteria under study in this project, although many other aspects o f chop sawing could have been included. Data taken from a sample m i l l from several cutting bills from two dimensions o f lumber was used to confirm the validity o f the chosen algorithm.  A simulation program was developed to analyze modifications made to conventional methods o f changing priority settings i n an optimizing chop saw using methods such as remaining value to select parts and updating parameters each time a board is processed. The model was based on actual m i l l data and emulated the process from lumber grading to parts tallying.  1  Once all information was gathered and organized, established statistical tests determined the best algorithm. Further analysis was done on three other chop saw run variables (external variables) to see i f high value/fbm figures came at the expense o f other chop saw run properties. Calculations, based on lumber volume data from the project m i l l , derived the financial benefits o f the recommended algorithm.  A literature search was conducted to complement the thesis and ensure that the work differed from previous projects. Therefore, references to previous work have been made throughout the current chop saw technology section where it applied directly to a certain aspect o f chop sawing.  1.2 C h o p Saw Technology and L i t e r a t u r e Review Figure 1 shows examples o f items that enter ( A ) and leave (B) a chop saw i n which waste or unacceptable sections o f the board are separated before any cutting is done (unless defects are irrelevant). Depending on the type o f saw and/or the cutting bill (list o f parts required by the mill), there may be only one part (Figure 1 - Sections 1 and 2) or more (Figure 1 - Section 3) cut from each usable section. If the defect marking station miscalculated the minimum allowable length, the entire usable section w i l l be treated as waste (Figure 1 - Section 4).  2  Figure 1. Chop Saw Input ( A ) and Outputs (B)  Usable  Usable  Section 1  Section 2  Usable  Usable  Section 3  Section 4 Waste  (A)  Waste (Does Not  (B)  c  Meet M i n i m u m Length)  1.2.1 G r a d e s a n d Dimensions o f L u m b e r Parts L i k e saw mills producing lumber from logs, chop saw plants employ grading rules for parts that vary significantly between products and customers.  A n example o f grading discrepancy is the  tendency for parts destined for use in furniture to require more rings per inch (tighter grain) cross section than input stock allocated for making doors.  G i v i n g even a general definition o f grading rules would not provide a realistic view o f industry standards. The only generalization that can be made about grading is the tendency for full-length lumber qualities as raw material for secondary manufacturing mills to be at least equivalent to "Shop" grade. Using lumber lower than Shop grade may require excessive use o f m i l l resources and significantly reduce recovery.  3  Dimensions o f parts contributing to certain finished products such as doors and windows are, however, based on established industry standards. Top rails for doors, for example, have widths between 4" and 6" with accompanying lengths between 25" and 47". Parts destined for making window supports have even less dimensional restrictions with a m i n i m u m o f 3 feet being needed for a cutting.  1.2.2 Prices O f L u m b e r Parts Each lumber part in a cutting b i l l has a price associated with it which represents costs incurred in its production and margins for the m i l l . Depending on the chop saw m i l l ' s layout and its system of accounting, costs can be incurred from the following processing stages:  • • • • • • • •  raw material cost lumber drying ripping (cut lumber/parts lengthwise) chopping moulding and/or planing handling (labor) inventory (cost o f keeping part in warehouse) additional overhead or fixed/variable costs associated with production  It was deemed inappropriate to use 2 or 3 examples o f chop saw setups to generalize existing technology given the large range o f lumber inputs and finished products that exist. M a k i n g the discussion o f chop saws even more complicated is the tendency for basic or simple saws to have selected properties from a sophisticated saw. Therefore, only features commonly seen in chop saws are described here. Figure 2 shows the order in which they often appear. With computer technology significantly changing on an almost annual basis, listed capabilities and limitations should only be seen as examples o f what is currently available.  4  Figure 2. Sequence O f C o m m o n Chop Saw Features 1) INPUT LUMBER  2) L U M B E R DEFECT  3) I N F E E D  5) S O R T I N G  DETECTION  CHOP SAW SOFTWARE OPTIONS  1.2.3 Input L u m b e r F u l l length input lumber is almost always k i l n dried stock as under or over sized lumber can be easily identified and green or wet lumber is more difficult to cut.  Some mills purchase wide  faced lumber and rip it into narrower strips which are then cross cut by an optimizing chop saw while others cross cut first.  In some cases, mills only apply a minimal amount o f planing or  edging before the lumber is subjected to cross or rip sawing. There is still no definitive method for processing full length lumber.  K l i n e et al (1993) showed that significant improvements i n recovery were possible when furniture parts were produced from unedged and untrimmed lumber.  Defect and dimensional data were  collected from 120 red oak boards (3 mills) to get objective values on volume o f lumber lost to edging and trimming. A l l data was entered into a "crosscut then rip" based yield optimizing computer program to generate random-width cuttings from numerous edging and trimming scenarios.  A s expected, volume yields from unedged and untrimmed lumber were the highest,  being 2 5 % and 18% higher than those from actual and optimally cut parts from unprocessed lumber, respectively.  5  Huber et al (1983) showed that certain species o f "residue" logs or those too short to qualify as standard logs (minimum 8-foot length) and/or too small i n diameter to be economically processed in conventional sawmills can be a valuable source o f material for resource strapped furniture mills.  Using common industry data, a computer program calculated the point at which the cost o f producing a certain volume o f lumber parts from short bolts equalled expenses associated with getting the same results with standard lumber grade material break even point. It was found that species producing large yields o f lumber parts and those having high value were the most suitable for furniture production as their high break even points allowed for fluctuations i n stumpage, harvesting, and bolt sawing costs.  Steele and Gazo (1995) calculated the benefits o f sorting lumber prior to rough m i l l processing or cutting to lumber parts. A simulation program called R A M used rough m i l l data and six cutting bills to recreate results that would have been yielded from two different m i l l configurations; crosscut-first (crosscut, straight-line rip, salvage crosscut) and rip-first (gang-rip, crosscut, salvage straight-line rip).  Sorting lumber prior to crosscut-first and rip-first sawing resulted in yield increases o f 1 and 0.3 percent, respectively. For crosscut-first sawing, total processing time was not influenced by sorting while rip-first sawing showed a decrease o f 2 percent.  1.2.4 L u m b e r Defect Detection The value o f the eventual end use o f the lumber parts dictates the way a m i l l  handles  unacceptable lumber defects. If only low grade items such as pallets are being made, then there 6  may be no need to spend time noting defects. For items with significant value, locations o f grade determining flaws must be "marked" out with fluorescent lumber crayons, scanned by cameras, or considered i n the board grade.  Systems employing lumber crayons require a worker to observe a board and draw lines perpendicular to the edges to separate defects from usable clear sections. When the crayon marks are detected by a computer vision system, signals are sent to the computer to indicate the starting and ending points.  Lengths o f sections or boards are derived from calculations that consider  elapsed time between two signals and feed speed o f the conveyor belt. A less commonly used setup has the lumber grader observing a board and entering locations o f defects directly into a computer which i n turn stops and conveys the board into the saw to make cuts.  A more expensive and less established defect detection system utilizes an array o f color line cameras, each taking pictures o f an edge or face to feed video signals back to the chop saw computer.  Using digital image processing, the chop saw software processes board picture  information before an optimization program calculates the cutting patterns. This system, despite being more accurate and less labor intensive, requires extensive calibration work to correlate colors with lumber properties.  A third technique is to assign a grade to each face o f full-length lumber.  Crayon marks  symbolizing a grade are applied by a worker with the expectation that all the pieces cut from the board are o f the same grade. A m i l l producing furniture parts is an example o f an operation that would use only grading or combine it with other defect noting systems.  7  1.2.5  Infeed  F u l l length lumber can enter a chop saw either automatically or manually. In a manual setup, the worker removes a board from a lumber queue and positions it against stops or rails before cross cuts are made. M i l l s which do not require defecting often use a "pack" type o f saw where a stack of boards can be cut each time.  Automated infeed saws have a table that moves lumber closer to the cutting area v i a chains or other means. Once a board is ready to be scanned or cut, top or bottom feed rollers press against its face to provide support during subsequent activities. Positioning accuracy levels o f +-.004" (.1 mm) have been reported by companies selling chop saws equipped with automatic board infeeds. Stating a typical feed speed figure is difficult due to the tendency for input lumber to slow down when cuts are being made.  1.2.6 C u t t i n g M e c h a n i s m In most mills, the lumber parts are chopped out by upward or downward moving saw blades. Automatic saws can reportedly cut with an accuracy between .031" (.8 mm) and .063" (1.6 mm) while the preciseness o f manual saws is difficult to gauge.  Blade diameters depend on input  material but commonly range between 14 and 22 inches. M i l l owners have the choice o f manual, semi-automated, or completely automated cutting systems, which are described below.  1.2.6.1 M a n u a l M i l l s with manual sawing have a number o f cutting stations set up next to the bin or queue containing full-length lumber. A t each station, a worker manually applies cross cuts to lumber at locations where crayon marks exist.  N o optimization occurs here - decisions made at the  8  marking station dictate the size o f the parts unless the m i l l has a trimming station for modifying part lengths.  1.2.6.2 Semi-Automated Preset cutting sequences or patterns that always cut the same number and dimensions o f parts from a given length o f lumber are often used by mills that make products where defect location is not a significant factor.  A n example o f preset cutting would be an order having the chop saw  cutting five 24" long parts from each 10' board. Benefits from this system include being able to accurately predict when an order quota w i l l be met, minimized waste o f input materials, and relative simplicity o f the computer software.  1.2.6.3 Automated Computer optimized cutting where the chop saw makes the "best" cutting decision is required i f certain types or dimensions o f lumber defects are unacceptable for finished products.  After  initial scanning to get lengths o f usable lumber sections, data is forwarded to a chop saw optimizing program that uses a proprietary algorithm to decide on the resulting parts.  There are two commonly known methods o f optimization; exhaustive search and deterministic dynamic programming.  Exhaustive search uses loops i n a computer program to examine a l l  possible combinations o f cuts until the best set is found .  Dynamic programming makes  decisions recursively using the best solution at the final point. Using these methods, the m i l l can make longer or higher priced parts a top priority unless user intervention has taken place. Brochures released by chop saw companies claim that yield improvements ranging between 4  9  and 15 percent can be achieved i f a computer based automated cutting system is used instead o f a manual one.  A method that is still i n its experimental stages is laser cutting where each board is scanned and then subjected to a hot beam capable o f producing parts from any location within a board, not just in a cross-cut or rip pattern.  M i l l s adopting this system, known as Automated Lumber  Processing System ( A L P S ) , not only eliminate the need for more than one type o f saw but also lose less kerf (wood volume equal to the width o f the saw blade). The optimization procedures remain the same as those described above.  Klinkhachorn et al (1989) applied A L P S to 100 simulated N o . 1 and N o . 2 C o m m o n boards using four cutting bills and 16 algorithms to find that recovery results from the best algorithm ranged from 70.57% to 76.89%. Conventional chop sawing, according to a recent W o o d Components Manufacturers study (Wiedenbeck and Scheerer (1996)), obtains an average recovery figure o f 61%.  Huber (1989) found, based on a comparative analysis, that A L P S was not economically  feasible for mills yielding small-sized production levels such as 5 M B F / d a y . A L P S has still not been fully implemented commercially despite the promising results due to the high cost o f its technology and the speed that has been sacrificed to accommodate yield.  1.2.7 S o r t i n g Once parts have been cut, they are either conveyed directly to stations such as rip or cross-cut saws for further processing or are kept in inventory. M i l l s that immediately process parts either use sensors or laborers to determine the right path for parts. Operations which store parts rarely use completely automated sorting due to the volume o f work involved and the many dimensions 10  and grades associated with a cutting b i l l . B e l o w are descriptions o f two approaches often used to move parts to pallets for storage.  1.2.7.1 C o m p l e t e l y M a n u a l Sort Lumber parts are conveyed to a table at which workers grade items and put them i n the appropriate piles or only transport/organize the parts.  M i l l s with this type o f setup often have  separate tables or areas for long and short parts to make the job easier for the workers. Even though extra time is spent on categorizing parts according to length, mills with a completely manual sorting chop saw often require less space to install their equipment compared to more automated operations.  1.2.7.2 K i c k e r S o r t i n g A n alternative setup to a completely manual sort is to have a series o f kickers, each responsible for pushing a certain part into a bin, installed along the conveyor belt at the outfeed area o f the chop saw. When a part is cut, computer programs combine conveyor belt speed with the distance of the required bin from the cutting area to schedule a time at which the appropriate kicker is to be activated. For example, i f a part is cut that belongs to a bin 20' away from the saw and it takes 1.5 seconds to move 1' along the conveyor belt, the designated kicker is programmed to push the part 30 seconds later.  Depending on the m i l l , kicker sorting still needs manual labor for grading parts and to ensure that cut stock is properly placed on pallets or in containers. In situations where the number o f items in a cutting list outnumbers the amount o f kickers, additional labor is mandatory for organizing parts at the default bin.  11  1.2.8 Chop Saw Software Options Besides making cutting decisions, software from the chop saw manufacturer is often capable o f simulating runs and allowing users to modify priority settings.  1.2.8.1 Simulation Chop saws with this option allow users to estimate the total financial value that could be incurred by a cutting bill before any cutting occurs. Lumber data from a previous run consisting o f the dimensions (length and width), crayon mark locations, and positions where cuts were made is incorporated with the regular cutting algorithm and new cutting bill.  The number o f boards  observed by the data collecting mechanism varies between equipment manufacturers.  1.2.8.2 Modify Priority Settings If uninterrupted, a chop saw selects either the longest or most valuable part o f the cutting list to have the highest priority when production decisions are made.  The prioritized part is usually  included in the optimal solution unless its length is longer than the observed clear section. Once the production quota o f a prioritized part has been met, attention is shifted to the item that previously had the second highest value. This sequence continues i n a way that ignores parts with satisfied production requirements until a l l quotas are filled. There are no limits to the number o f parts that can be prioritized i n a chop saw cutting bill although observations from three mills indicates that up to three parts are usually prioritized at a time.  There may be times, however, when a less valuable part is cut at a rate that is limiting production or the mill immediately needs items that are i n short supply. In these or other instances where conventional cutting algorithms are not fully compatible with what is needed, 12  chop saw  operators may be able to force a part into the top priority position.  However, most mill  supervisors are too busy with other responsibilities and w i l l not adjust settings unless they are seriously dissatisfied with production results.  Table 1 demonstrates an example o f a chop saw cutting bill that allows users to override the prioritization process. Under normal circumstances, item number 12 would be the most sought after part as its quota has not been completely filled and it has the highest value ($8.56) o f all parts.  However, a closer look at the bill reveals that item 7 has completed only 611 out its  needed 7000 items while all other cutting requirements are close to completion. In table 1, the user has decided to speed up production o f part number 7 by entering a 1 i n its "priority o f part" column. In this case, the 1 prioritizes part 7. Item number 3 has a 2 i n its priority column as all 2500 o f its needed parts have been cut. In this case, the 2 removes part 3 from all optimization and prioritization decisions.  Table 1. Example o f A Cutting B i l l . item #  length (inch)  part price ($)  # parts demanded  # parts cut  priority o f part  1  6.5  0.14  20000  19231  0  2  26.75  1.05  7500  6210  0  3  26.75  1.44  2500  2503  2  4  32  1.88  2000  1182  0  5  36  1.54  5000  3318  0  6  36  2.12  6000  4722  0  7  49  4.47  7000  611  1  8  51  4.65  1100  892  0  9  55  5.02  2000  1017  0  10  59  5.38  1100  936  0  11  61  5.56  3500  2443  0  12  72  8.56  1800  1211  0  13  Table 2 shows three solutions for chop saw production results based on the cutting bill from Table 1: a 180" long usable section, and three different cutting algorithms; 1) normal cutting, 2) forcing 1 prioritized part into the solution, 3) forcing as many prioritized parts as possible into the solution. The normal solution (1) is based on trying to maximize yield o f the most valuable part (eg. part 12 in Table 2). When only 1 prioritized part is forced into production (2), lumber remaining after the first part is processed according to normal optimizing procedures. Forcing as many prioritized parts as possible into the solution (3) maximizes yield o f the highlighted part and only applies normal cutting procedures to whatever may be left over.  According to the  supervisor at the data collection m i l l , algorithms (1) and (3) emulate a chop saw employing unmodified and user imposed priority settings, respectively.  Table 2. Production Results From A 180" Usable Section, (part number 7 = prioritized part, algorithm numbers i n brackets) N o r m a l Solution (1)  Force 1 Part 7 Into Solution (2)  Force A s M a n y O f Part 7 A s Possible Into Solution (3)  part  part  number  length  part price  part  part  part price  part  part  part  number  length  number  length  price ($)  6  36  ($) 2.12  7  49  ($) 4.47  4  32  1.88  12  72  8.56  10  59  5.38  7  49  4.47  12  72  8.56  12  72  8.56  7  49  4.47  7  49  4.47  total value  $19.24  total value  $18.41  total value  $15.29  total waste  0"  total waste  0"  total waste  1"  Table 2 shows that algorithm 1 or normal cutting excluded the prioritized part but achieved the highest financial value ($19.24) o f all algorithms. Algorithm 2 fit one prioritized part into its solution but ended up with a total value that was 4.5% lower than algorithm 1. Algorithm 3 cut three prioritized parts (maximum number that could fit) but accumulated a financial value that was 25.8% lower than algorithm 1 and 20.4% lower than algorithm 2. This demonstrates that user imposed restrictions to force prioritized parts can reduce total value from a board.  Based on preliminary calculations, the profitability o f a m i l l can be improved by limiting the number o f parts that are forced into a usable section o f lumber.  The 20.4% value difference  between algorithms 2 and 3 in Table 2 is an example o f what can be achieved. Under the current system, once a priority setting has been imposed, there are minimal opportunities for optimization as all efforts are put into cutting the prioritized part.  1.3 M a c h i n e r y Setup  It is virtually impossible to find two secondary manufacturing plants with an identical setup as there are many factors such as customer demands and limitations o f input lumber that influence operations. Determining the ideal setup for a given plant can also be difficult as market demands can change quickly once production has started. M a k i n g changes to an existing facility is often difficult due to limited resources such as plant space and money for upgraded equipment.  However, Araman (1977) showed that significant improvements in secondary manufacturing production performances can be achieved by applying minor modifications to the plant set-up. A yellow poplar m i l l i n which full length lumber went through a planer, gang ripsaw, marker station, defect saw, and automated cut to length saw, respectively, was simulated using data from an existing m i l l . A t least 4000 board feet o f interior furniture parts from N o . 1 or N o . 2 A Common 4/4 lumber had to be produced over an eight hour shift to satisfy business obligations.  Unfortunately, only 2400 and 2200 board feet o f finished parts were yielded from the higher and lower grades, respectively. The main bottleneck for both grades o f lumber was the marking station  15  (marker and defect saw) which operated 100% o f the time while the planer and gang ripsaws operated only 19 to 24% of the time.  A second set o f simulation runs was done using a machinery set-up similar to the initial one but with an extra defect saw and automated cut to length saw. The new configuration fully satisfied production goals as finished parts yields o f 4400 and 3900 board feet were derived from N o . 1 C o m m o n and N o . 2 A C o m m o n grades, respectively. This example shows how simulation can be a useful tool to improve productivity and yield in a secondary manufacturing plant.  This project w i l l examine and analyze data from algorithms having up to three different prioritized parts to find the most profitable combination. A n example o f an algorithm would be one that prioritizes two parts in a cutting bill and forces, i f possible, one o f each length into a clear section.  Questions about the transferability o f project findings to potential users w i l l , hopefully, be answered by the wide range o f input variables. The range o f input variables include three cutoff percentages (70%, 80%, 90% - complete) to indicate the point at which a part becomes removed from prioritization, nine different cutting bills (7 for 3" width, 2 for 3.75" width), and twenty unique priority settings rules.  Algorithms having minimal user intervention w i l l most likely yield the highest dollar values. However, it is still necessary to do a full scale comparative analysis to objectively calculate the extent o f value lost to forced cutting. Results from each data analysis w i l l hopefully achieve the following main goals: 16  1. To find the cutting algorithm that achieves the highest value recovery. 2. To create a practical chop saw simulation model to test different optimization algorithms. 3. To determine, objectively, the financial benefits o f using the recommended algorithm.  17  2.0 MATERIALS AND METHODS  2.1 Description Of The Test Mill Conducting a project based on actual changes to the system would have been too expensive, time consuming, and restricted to a few algorithms. Therefore, the most logical choice for generating m i l l production data was computer simulation, an operations research technique that involves building and validating a model to represent the system. A l l attempts were made to minimize model assumptions by collecting large volumes o f data from the project m i l l .  B e l o w are  descriptions o f the project m i l l , data collected, simulation development, development o f algorithms, simulation procedures, and data analysis, respectively.  2.1.1 Mill Operation Only basic features o f the chop saw i n the test m i l l can be described because confidentiality was a condition for collecting data. The m i l l studied for this project is a secondary manufacturing plant in western Canada that initially buys green lumber from sawmills. A l l full length lumber is kiln dried to a moisture content that is acceptable for cutting and gluing.  Dried full length  lumber is planed or edged, i f necessary, to make its cross sectional dimensions closer to those o f the parts required.  Lumber piles, each containing up to three different lengths, are positioned next to a conveyor belt that moves graded boards closer to the chop saw entrance area. responsible for the same task, can work simultaneously.  U p to three graders, all  Each grader isolates unacceptable  defects by drawing, on the face o f the board, crayon lines perpendicular to the edge.  18  Grading  decisions are based on all four sides o f a board and crayon marks on one edge indicate the grade o f a usable section. Figure 3 below shows an example o f the grading system.  Figure 3. Example O f Board Grading System  GRADING M A R K FOR C L E A R SECTION EDGE  UNACCEPTABLE LUMBER DEFECT  - L U M B E R C R A Y O N LINE  CLEAR SECTION  Boards are guided into the scanning area by rollers that apply downward pressure onto the top face. Crayon lines on the face o f a board are detected by an overhead sensor that is about 2.5 inches away from the lumber surface. Profile data pertaining to the location o f grading marks is collected by a side sensor or one that gets its signals from the specimen's edge. A n additional light based sensor collects board length data by noting when a light beam shining across the infeed belt is blocked. Both side based sensors are about 1.5 inches away from the edge o f moving lumber.  A software algorithm in the chop saw analyzes data from usable lumber sections and calculates, according to cutting bill parameters,  the most valuable combination o f parts that can be  recovered. A set o f side sensors located adjacent to the chop saw blades indicate when a board is ready for processing.  One upward moving blade makes cross cuts and the board is moved  forward by a conveyor belt until the next cutting location is reached.  19  Lumber parts are conveyed into a sorting area where six or seven automated kickers push items into their appropriate bins. Parts that do not fall into any o f the categories covered by the kickers flow forward into either a waste bin for excessively short cutouts or a second conveyor area. N o direct processing occurs unless a part is i n very heavy demand by another station in the plant.  2.1.2  Raw Material  O f the five different dimensions (thickness and width) o f full-length lumber that are processed by the participating m i l l , only two o f them, 1.875'72.000" x 3.000" (3" width) and 1.875" x 3.750" (3.75" width), were studied in this project.  Grades o f input lumber were not disclosed by the  m i l l . Properties o f both lumber types are described below.  2.1.2.1 3" W i d t h L u m b e r Even though this kind o f lumber had two possible thicknesses, 1.875" and 2.000", its width was always 3.000".  Seven cutting bills, described in Table 3 below, were filled during two data  collecting sessions.  Table 3. 3" Width Lumber: cutting bill number 1 2 3 4 5 6 7  thickness  dates run  2.000 2.000 1.875 2.000 2.000 1.875 1.875  6/6/97 6/3/97 6/3/97 - 6/4/97 6/20/97 & 6/23/97 6/23/97 6/23/97 - 6/24/97 6/5/97 - 6/6/97  20  number of items 9 16 16 16 15 15 18  2.1.2.2 3.75" Width Lumber Thickness and width were always 1.875" and 3.75", respectively for 3.75" Width lumber. Only two cutting bills were covered but both, as Table 4 shows, contained more parts than 3" width bills. Table 4. 3.75" Width Lumber: Cutting bill number  thickness  dates run  number of items  1 2  1.875 1.875  6/10/97-6/13/97 7/3/97 - 7/4/97  23 19  2.1.3 Products Parts cut from full length lumber were either packaged and shipped out to customers or were laminated on-site to form products for future processing.  The participating m i l l was very  cautious about revealing exactly what was eventually created from the parts but one o f the managers said that doors, ladders, cribs, and shelves were examples o f end products.  For both dimensions o f full-length lumber, characteristics o f the parts varied significantly. Ranges o f lengths and prices o f 3" width lumber parts were 6.5" to 192" and $.015 to $19.88, respectively. Lumber parts with a 3.75" width had the same range o f lengths as the o f 3" width, but had a slightly smaller price range o f $.04 to $19.15.  2.1.4 Costs For this project, value refers to the difference between what was paid for a board and its revenue. Actual m i l l profits w i l l be lower than project figures because other costs associated with parts production such as planing, handling, and drying were excluded as m i l l personnel considered this  21  information to be confidential. Therefore, using the term value/fbm was more appropriate than profit/fbm.  Again, the m i l l did not reveal the grades o f their input lumber, so, a price per 1000 board feet o f full-length lumber was obtained by averaging data from all 1995 copies o f Madison's Canadian Lumber Reporter (Anonymous(1995)), a weekly publication that regularly listed two grades o f 2x4 k i l n dried coastal hemlock lumber, Standard/#2&Better and Utility/#3Common. Appendix I shows all data from both grades. After a 40% U . S . exchange rate (November 4, 1997 figure) was multiplied by the average Standard/#2&Better price, a lumber cost o f $417.37/1000 fbm (Canadian) was calculated.  The main reason for applying a price or cost factor to each full-length board was to provide more realistic financial figures, as it was possible for a low grade piece, without costs, to generate a "profit" (actually revenue) o f $3.00 from part prices even though a big loss should have been incurred after lumber price was considered.  2.2  D a t a Collection  Most o f the work i n this research was non-intrusive to production.  Board information was  obtained from the chop saw computer and time data were collected manually. Other data were provided by m i l l employees.  A total o f four separate m i l l runs, each one consisting o f about  eight shifts or four days o f production, had to be examined to collect a large enough data base.  22  2.2.1 L u m b e r D a t a A l l full length lumber in the simulation model was from actual board data, nothing was randomly generated. Board data files, consisting o f the locations o f crayon marks made by the grader and total lumber length, were read into the simulation program at the beginning o f a run. Figure 4 shows the steps that were taken to transform lumber crayon marks into useful model data. F o r each dimension o f lumber, chop saw computer data files were downloaded once or twice per shift to even out the distribution o f lengths collected. A l l information from computer files was printed out, put through optical character recognition software (converts graphics to text), checked for errors (manually and by a spreadsheet macro), and subject to calculations which derived the length o f each section.  Figure 4. Steps To Acquire Lumber Data 1) A C T U A L BOARD GR 1  0.000"  ^  /  I  I  31.415"  WASTE ^ / G R 3 ^  I  53.164"  /  WASTE  ^  /  GR 1  I  67.663"  94.245"  120.827"  CHOP SAW DOWNLOADS DATA OCR, CHECK, & 2) CHOP SAW D A T A F I L E  3) BOARD D A T A FOR M O D E L MAKE CALCULATIONS  grade  location  grade  1  0 31.415  1 W  W  53.164  3  67.663  W 1  94.245 120.827  1  part length 31.415 < 21.749 <  (31.415-0) (53.164-31.415)  3  14.499 ^ —  (67.663-53.164)  W  26.582  < —  (94.245-67.663)  26.582  <:  (120.827-94.245)  2.2.2 T i m e a n d M o t i o n Study None o f the simulation model features could function properly unless time data were added to the graphics network. Lumber interarrival times to the chop saw queue or area where full length  23  lumber waited until it was ready to be scanned was gathered manually by a stop watch. Only one set o f time data, from the first run, was gathered for each dimension o f lumber. Figure 5 shows the grading station layout where two measurements, one for the first board and a second for the board immediately following, were taken to get one interarrival time.  Figure 5. Illustration O f Lumber Grading Station CHOP SAW SCANNING AREA v  CHOP  SAW  QUEUE  The initial time value in Figure 5 (6.25 seconds) was based on the board 1 arriving at 0 seconds or when collection started and board 2 arriving 6.25 seconds later. A t time 8.38 seconds, board 3 reached the queue, completing requirements for the second time value o f 2.13 seconds.  2.2.3 D a t a F r o m M i l l Employees Grading station workers wrote down when lumber packages arrived and the lengths contained within them. Extra care was taken to ensure that only information pertaining to project cutting bills was recorded as runs were often interrupted to work on different products.  A s was done  with stop watch data, graders were only asked to record data from the first run o f each dimension as the lumber population had to be consistent in data analysis.  24  A n y changes made to chop saw priority settings during a run were recorded by m i l l managers on a notepad placed near the chop saw control panel. Most modifications to cutting bills involved adding and/or removing parts on a permanent or temporary basis. Fortunately, all cutting bill priorities were implemented before runs commenced and did not change during the run.  2.2.4 Y i e l d D a t a 2.2.4.1 B a s i c Structure Parameters from cutting bills were obtained from summary reports that were printed at the end o f each run. I f nothing was cut for a particular part, the item i n question was not included i n the simulation. Table 5 shows an example o f a printed chop saw cutting b i l l . Table 5. Actual Chop Saw Cutting B i l l Item = item number  Pr. = priority setting (2 = finished, 1 = prioritized)  Length = part length (inches)  price = price per part  Req. = number o f parts required  p r i c e / M B F = price ($) per 1000  Cut = number o f parts actually cut  fact = cut factor (not used in project)  board feet  Rest = number o f parts remaining to be cut  kicker = kicker number GRADE 1  Item  Length  Req.  Cut  Rest  Pr  Price  price/MBF  fact  kicker  1  24.803  1000  0  1000  2  1.570  1519.2  1.00  6  2  36.000  5000  63  4937  0  1.870  1246.7  1.05  3  3  48.425  1500  90  1410  0  3.060  1516.6  1.10  6  4  60.236  1500  95  1405  0  3.800  1514.0  1.15  7  5  72.047  1500  100  1400  0  4.550  1515.7  1.20  7  6  96.000  500  51  449  0  6.500  1625.0  1.25  0  7  120.000  1100  0  1100  2  12.420  2484.0  1.30  0  8  144.000  3250  8  3242  1  14.910  2485.0  1.35  0  9  168.000  250  0  250  1  17.390  2484.3  1.45  0  10  192.000  500  0  500  1  19.880  2485.0  1.50  0  Item  Length  Req.  Cut  Rest  Price  price/MBF  fact  kicker  GRADE 2 Pr  1  36.000  2500  203  2297  0  1.870  1246.7  0.85  3  2  96.000  1500  18  1482  0  5.200  1300.0  0.86  0  3  120.000  500  59  441  0  12.410  2482.0  0.87  0  4  144.000  1500  28  1472  1  14.890  2481.7  0.88  0  5  168.000  250  0  250  0  17.370  2481.4  0.89  0  6  192.000  250  0  250  0  19.880  2485.0  0.90  0  GRADE 3  25  Item  Length  Req.  Cut  Rest  Pr  Price  price/MBF  fact  kicker  1  6.500  10000  140  9860  0  0.140  516.9  0.50  1  2  36.000  2500  467  2033  0  1.480  986.7  0.75  5  GRADE 4 Item  Length  Req.  Cut  Rest  Pr  Price  price/MBF  fact  kicker  1  35.000  250  22  228  0  0.025  17.1  0.50  7  2  84.000  250  2  248  0  0.070  20.0  0.75  7  2.2.4.2 Modifications T o D e m a n d A preset demand was provided by each cutting b i l l , however, part quotas were rarely filled and no consistent completion percentage existed. M i l l personnel said that taking exactly what was requested by customers and entering it into the cutting b i l l was not always done as some parts could be overproduced without being a burden on inventory levels. I f a part with a high grade was long enough to be reprocessed into another usable item, then no limits were necessary. Another reason for large demand figures was that i f a particular part was required by three or four cutting bills, demand for the entire run was used, not the individual cutting b i l l . The m i l l also sometimes judged demand for a part by entering a large number and stopping production after inventory levels were examined or i f the manager felt that the next run could easily fill the void.  A s expected, the subjective nature o f the m i l l ' s techniques for deriving demand made them nontransferable to project work.  However, attempts still had to be made to estimate actual  demand to make simulated data comparable to those from the m i l l . B e l o w are descriptions o f the three attempts that were made to find suitable demand levels for each cutting bill.  26  2.2.4.2.1 Actual Cut Basing cutting b i l l demand on what was actually cut resulted i n lumber recovery figures that were almost identical to actual data from the mills.  Table 6 shows how demand data i n the  simulation model cutting b i l l (B) was transferred from a m i l l printout (A).  Table 6. "Actual Cut" Cutting B i l l Demand (A) = Cutting Bill Downloaded From Chop Saw, (B) = Cutting Bill From Chop Saw Model GRADE 2 Item  Length  Req.  Cut  Rest  Pr  Price  Price/MBF  Fact.  Kicker  1 2 3  36.000 96.000 120.000  2500 1500 500  203 18 59  2297 1482 441  0 0 0  1.870 5.200 12.410  1246.7 1300.0 2482.0  0.85 0.86 0.87  3 0 0  Grade  Length  Kicker  2 2 2  36 96 120  3 0 0  Req.  Cut  203 18 59  0 0 0  Price  Pr  1.870 5.200 12.410  0 0 0  Expectations o f deriving financial and yield results resembling actual data were rarely met as the priority settings formula did not give attention, until later i n a run, to parts which were highly valued but had l o w demand levels. Consequently, large volumes o f long usable lumber were cut into shorter parts and the simulation runs ended up being much longer than they should have been.  2.2.4.2.2 Actual Demand A second approach was taken to see i f emulating m i l l results was possible or worthwhile. U s i n g three 3" width cutting bills with demand from m i l l runs, trials using m i l l settings (2 or 3 parts that were prioritized by the mill) were done by tripling the prices o f prioritized parts. Each run  27  was stopped once simulated time equaled the duration o f the actual run. Multiple printouts o f all cutting bills confirmed that prioritized parts selected by m i l l personnel at the beginning o f a run did not change.  Again, failure was encountered when comparisons were made with actual data.  Slight  improvements were seen in contrasts between highly valued parts but differences between other items were so large that conducting statistical tests for significance would have been a waste o f time.  2.2.4.2.3 Pseudo D e m a n d W i t h little or no chances o f the project programs being able to reproduce exactly what was done by the m i l l , the only alternative was to create a "pseudo" demand to be used for comparing performance o f the algorithms. W i t h pseudo demand, no direct comparisons could be made with actual data, however, practicality and consistency were improved.  Pseudo demand was formulated i n a way that transformed what was actually cut into a format that was not only more compatible with the priority settings formula but was also similar to, but less than, actual demand. B e l o w are the rules for creating pseudo demand:  •  For each cutting bill item, i f the actual number o f parts cut is less than 10, demand equals 10, otherwise, multiply actual number o f parts cut by 1.5:  •  Result from above are then categorized according to the following format: If If If If  (actual (actual (actual (actual  cut)* 1.5 cut)* 1.5 cut)* 1.5 cut)* 1.5  <= 10, < 100, >= 100, >= 1000,  demand demand demand demand  = 10 rounded to upper 10 increment. rounded to upper 10 increment. rounded to upper 100 increment. th  th  th  28  Table 7 below shows an example o f how number o f parts cut data from an actual chop saw run cutting bill ( A ) was converted to demand or number o f parts needed for a simulation model cutting bill (B).  Table 7. Pseudo Demand Example. (A)  (B)  N u m b e r O f Parts C u t  Demand  *7Q  1100  ?Q  50  ?8S  430  ^  ?1?  320  178  270  • •  3  •  37  10 50 60 60  Appendix II shows all cutting bills that were employed in the project.  2.3 Development O f C h o p Saw A l g o r i t h m s Cutting algorithms used the priority settings to govern how parts were cut from a section o f lumber. A n example o f a cutting algorithm using one item would be one that forced as many prioritized parts as possible into a section and cut remaining lumber according to normal optimization. A total o f 20 algorithms (Table 8) were tested during the project.  29  Table 8. Cutting Algorithms  Algorithm #  1 =  # Prioritized Parts  # Forced In  Cutoff %  1  1  1  70  2  1  1  80  3  1  1  90  4  1  UNLIMITED  70  5  1  UNLIMITED  80  6  1  UNLIMITED  90  7  2  1 OF EACH  70  8  2  1 OF EACH  80  9  2  1 OF E A C H  90  10  2  2 OF EACH  70  11  2  2 OF EACH  80  12  2  2 OF EACH  90  13  3  1 OF EACH  70  14  3  1 OF EACH  80  15  3  1 OF EACH  90 70  1  16  3  2 OF EACH  17  3  2 OF EACH  80  18  3  2 OF EACH  90  19  MILL SETTINGS  NA  NA  20  N O PRIORITY SETTINGS  NA  NA  Part(s) were prioritized according to remaining value or (number o f parts needed - number o f parts cut)* price per part  2 =  A part became ineligible for prioritization once a certain percentage o f its demand was satisfied.  C programs emulating algorithm 19, m i l l settings, only considered the 2 to 4 items that had been given preference i n the real run. N o new parts were prioritized once the initially emphasized ones were completed. Selection o f parts was done by part value, not remaining value.  2.4 Simulation Procedures Simulation can be either continuous or discrete, the former having one or more sections o f the model changing continuously over time while changes only occur in the latter at a discrete set o f time points.  For this thesis project, only properties and rules o f discrete simulation was  considered as events commonplace in lumber processing such as lumber arriving at a rip saw happen at exact points i n time.  30  It must be noted here that simulation programs included i n automated chop saw software are different from discrete simulation. Chop saw software uses deterministic simulation or event generation based on board data downloaded from previous m i l l runs. There is no randomness in deterministic simulation - a typical run would apply a new set o f cutting rules to clear section values from a previous run.  Discrete simulation creates events or numerical values by combining random numbers with statistical parameters from previous m i l l runs. Data collected from an actual event are analyzed to get the distribution and any relevant statistical values. Figure 6 shows graphically how discrete simulation has derived a board length o f 92.45" by taking a random number and applying it to a normal distribution with a mean o f 112.45".  Figure 6. Discrete Simulation Example Using N o r m a l Distribution  Mean O f  92.45"  112.45"  (clear length in inches)  Discrete simulation also allows direct or unbiased comparisons between two different algorithms i f the seeds or starting points of the random numbers are consistent in all runs.  31  2.4.1 Graphical Elements Of The Chop Saw Simulation Model A l l simulation work was done using A w e s i m , a software package that enables users to create models graphically with icons that represent common simulation elements such as queues and entity paths. Program modules can be inserted into models when necessary.  For this project,  each simulation run had a distinct set o f program inserts (written i n C ) but the same graphical model was used in all runs or replications.  It should be made clear that the term "graphical" did not necessarily mean "animation" as the former refers to a windows interface work area that simplified model development by allowing users to "drag and drop" icons. Graphical models were needed regardless o f the model output and consisted o f two sections, lumber movement and downtime.  2.4.1.1 Lumber Movement Figure 7 shows the network that facilitated lumber entity flow with the help o f more complicated C program inserts, symbolized by rectangular boxes. A t the beginning o f a run, 16,000 entities representing full length lumber were put into a queue from which boards were selected, assigned characteristics, and forwarded to another waiting area where they remained until the chop saw was ready for cutting. Each time a lumber part was cut, a new entity was created, added to the network at point (A) in Figure 7, and tallied.  32  Figure 7. F u l l Length Lumber and Parts F l o w Networks.  [1]  [3] 2.4.1.2 D o w n t i m e A l s o present in the graphical model were networks to add downtime or times during a run when m i l l machinery was not operating. Both forms o f industrial downtime, regular (shift change, coffee/meal breaks) and random (in which work was halted due to unexpected factors such as mechanical failures) were managed by networks i n Figure 8.  33  Figure 8. Downtime Networks Regular Downtime Network  120 MINUTES OF WORK  Random Downtime Network  Using the equation below, 4.4 (4) and 4.95 (5) minutes o f random downtime per hour were calculated for 3" width and 3.75" width lumber, respectively.  random downtime per hour = runtime - work time - waiting time where: runtime work time waiting time  [Equation 1]  = length o f the work shift (minutes). = time spent by chop saw on cutting, scanning, and optimizing (minutes), = time spent by chop saw waiting for full length lumber (minutes).  Although equation 1 is a fixed figure, it is referred to as "random" as it occurs at randomly (uniform distribution) selected points within each hour o f simulated time. Data pertaining to run time and work time were copied from reports that were printed either at the end o f a shift or when work had to be done on a new cutting bill. Each observation in the calculation o f average 34  random downtime per hour came from one shift/run report. M a n y evening shift totals were lost as the chop saw computer clock reset at midnight. Overall, twenty production intervals were successfully recorded and appear in Appendix III.  Waiting time or time spent by the chop saw waiting for full length lumber was the result o f combining time data with some very subjective assumptions. For both lumber dimensions, two assumptions were made for waiting downtime; boards entered the scanning area every two seconds as queue sizes o f zero were seen often and 250 boards were processed each hour. A n average o f 8.33 minutes ((2/60)*250) o f waiting downtime occurred each hour.  2.4.2 C P r o g r a m Inserts O f C h o p Saw S i m u l a t i o n M o d e l  Figures 7 and 8 show only four stations at which C programs were called upon to process lumber or check downtime. Program [3] in Figure 7 (kicker) and [4] in Figure 8 (downtime) were easy to implement.  However, the other two, assigning lumber data ([2] in Figure 7) and cutting  lumber parts ([3] i n Figure 7), were much more complex. B e l o w are descriptions o f functions that assigned lumber data and made cutting decisions, respectively.  2.4.2.1 A s s i g n i n g L u m b e r D a t a In actual production, lumber was carried by a forklift to a grading area where boards were handled and assigned grades by one or two graders before they were conveyed to the chop saw queue. One grader was present most o f the time as a second grader usually was not added unless previously cut lumber was re entered into the grading area.  35  For the model, only one grader was used at all times and lumber properties were not assigned to entities until they were released or selected from the initial lumber queue. Table 9 below shows all the steps taken to give entities all the information they needed to mirror actual lumber.  Table 9. C Program Steps To Obtain Lumber Information 1) 2) 3) 4)  Get Board Number Get Board Selection Information Select Board Get Interarrival Time  (eg-3) (eg. choose board between #'s 201 and 300, 10' length) (eg. select board # 207 and get all section information) (eg. based on 10' length and random #, 5 seconds)  2.4.2.1.1 Get Board Number/Selection Information After selection, an entity was given a board number equal to the total number o f processed boards plus one. A data file designated two interval numbers (for selecting board properties) and an integer length (for interarrival time selection) to the entity according to its board number. A n example o f data assignment is shown in Table 10 where a 3.75" width entity having a board number o f 3 receives interval values o f 201 and 300 or the interval covering 10' long lumber.  Table 10. Sample O f Lumber Identification File board #  board selection interval (first)  board selection interval (second)  assigned length in feet  1-96  201  300  10  97 - 167  101  200  9  1  100  8  168  -238  Data i n the lumber identification file was provided by grading station workers who, as was mentioned earlier, wrote down lumber package information. A s Table 9 shows, the sequence o f lumber arrivals, not their times, were transferred to the simulation model.  36  A l l packages always consisted o f 238 pieces, however, getting accurate length tallies was impossible as random or inconsistent length distributions were always present.  Numerous  packages i n the storage yard were examined to help make reasonable assumptions about length data. I f three different lengths were enclosed in a package, for example, 96 boards or 40.336% o f the contents were o f the longest length and leftover stock was split equally among the other two lengths. Table 11 shows a complete list o f the package composition rules for both dimensions.  Table 11. Lumber Package Length Composition Rules (Applies To Both Dimensions, Rank - 1 = Longest Length) Length/Package  Rank  # Boards  Percentage  2  1  119  50.000 %  2  119  50.000 %  1  96  40.336 %  2  71  29.832 %  3  71  29.832 %  1  71  29.832 %  2  71  29.832 %  3  48  20.168%  4  48  20.168%  3  4  5  1  54  22.689 %  2  54  22.689 %  3  54  22.689 %  4  38  15.966%  5  38  15.966%  2.4.2.1.2 Select B o a r d A uniformly distributed random number, limited by the entity's two interval values (eg. 201 and 300 from the previous step), was drawn. A large board data file consisting o f information from about 1300 full length boards or 100 for each length (8' to 20') was scanned until the row having a value equal to the random number was located. More boards could have been added to the information base but some o f the m i l l ' s lumber lengths (such as 18 footers) were not cut regularly. Each row i n the file was created from actual chop saw computer data and consisted o f grades and lengths o f all sections, usable or waste, from one particular board.  37  For demonstrative purposes, an example is shown below where random number 207 is chosen. Table 12 shows the resulting information taken from the board data file i n which the selected board characteristics appear on line 207 (bold font).  Table 12. Example O f Board Data File. #  board length  number o f  section 1  section 1  section 2  section 2  section 3  section 3  (inches)  sections  length (in.)  grade  length (in.)  grade  length (in.)  grade  205  120.157  3  69.646  3  2.283  17  48.228  1  206  120.236  1  120.236  1  0  0  0  0  207  120.236  2  102.598  1  17.638  3  0  0  Figure 9 shows the 3 dimensional view o f the board selected at row 207 o f the data file.  Figure 9. Diagram O f Board From R o w 207 In Table 12 SECTION 1  SECTION 2  i  i  102.598" GRADE 1  - 17.638" GRADE 3  120.236"  Sampling with replacement, a technique which selected a board from a batch and returned it for possible reselection, was the sampling method used. For example, a 96.215" board chosen from 100 other boards having similar lengths could have been the very next sample or never been selected again.  Alternatively, new boards could have been created by generating sections  (grade/waste) from statistical distributions and summing them until the full length o f the sample was reached. Despite its higher level o f randomness, generating new boards was abandoned as too much time would have been required to accommodate all the different lengths.  38  2.4.2.1.3 Select Interarrival Time A n entity could not advance until it was granted an interarrival time (time between lumber arrivals) to the chop saw queue.  A C program function drew a uniformly distributed random  number between 0 and 1 which helped get the entity's length based interarrival time from a cumulative frequency distribution. Using Table 13 below, i f a random number o f .72 was drawn and the frequency distribution for 10' lumber (10 - integer length from the example) was utilized, the entity's interarrival time would be 9 seconds.  Table 13. Cumulative Distribution O f Interarrival Times For 10' Lumber interarrival time (seconds)  interval  5  0.066265  6  0.216867  7  0.228916  8  0.710843  9  0.740964  10  0.746988  11  0.921687  13  0.951807  14  0.96988  16  0.993976  18  2.4.2.2 Making Cutting Decisions Once a lumber entity entered the chop saw, procedures shown i n Figure 10 transformed lumber sections into parts. Detailed descriptions o f points i n the network follow:  39  Figure 10. Flowchart O f Procedures For Cutting Parts From F u l l Length Lumber  The terms "cutting algorithm" and "pattern" are explained here to prevent further confusion. Figure 11 shows how a board was cut with a pattern that created two sections from three by combining sections b) and c) and used a cutting algorithm that forced in as many priority 1 parts as possible.  40  Figure 11. Cutting Algorithm and Cutting Pattern (cutting pattern - initial sections = a), b), and c), final sections = (1), (2) )  (2) WASTE  (1) ^  WASTE  2.4.2.2.1 Get C u t t i n g B i l l P r i o r i t y Settings Each time a full length board entered the chop saw, scanning o f the cutting bill was done to get identities o f part(s) to be forced into the combination o f parts removed from a section. A l s o known as priority settings, forced parts were based on highest remaining value or:  remaining value = (number o f parts needed - number o f parts cut)*price o f part  [Equation 2]  Cross sectional dimensions o f the parts i n question were constant, hence, they were not included in equation 2. I f only one grade is present in a cutting b i l l , formulas based on length are fine but up to four different grades o f lumber were cut during a m i l l run. Part length was therefore excluded as it was possible for a long part with low value to override a shorter but more valuable item.  41  Previously published length based formulas include Priority = Length * Width (from Wodzinski 2  and Hahm; (1966)), Priority = L e n g t h  Wei8ht  * Width (from Maristany et al. (1990)), and Length * 2  W i d t h (from Steele and Gazo (1995)). Thomas (1996) introduced weighting factors ( W F ) with 2  a Priority = L e n g t h ^ , , ^ * W i d t h ^ ^ , , formula.  Biases relating to parts quantities which  existed in older priority formulas were overcome by W F s but part value was still being ignored. A complete description o f how priority settings were utilized in cutting algorithms is explained in the subsequent "Development O f Algorithms" section.  2.4.2.2.2 Get The Best Cutting Pattern U p to three consecutive lumber sections were scanned before a series o f tests determined the most valuable cutting pattern or sequence by which sections were cut.  Actual cutting was  applied to the most profitable pattern after all possibilities had been tested. Figure 12 shows an example o f scanning where the remaining third section o f the first scan became the first section o f the second scan.  Figure 12. Example O f Board Scanning.  SECOND SCAN  Depending on the grades present or number o f sections, up to 14 possible cutting patterns were examined by the test mode section o f the program (area enclosed by a dotted line in Figure 10). Demotion o f a grade occurred i f a section was not long enough to meet grading standards. Figure 13 shows diagrams o f all the patterns. 42  Figure 13. Cutting Patterns For Test Mode 1 = h i g h e s t g r a d e , 2 = 2 n d h i g h e s t g r a d e , 3 = l o w e s t g r a d e , d m = d e m o t i o n , bef. = b e f o r e b l a n k s e c t i o n s = i f v a l u e f r o m n o n b l a n k s e c t i o n s are best, c u t b l a n k s e c t i o n s o n l y ( a l l g r a d e s b a s e d o n after d e m o t i o n results u n l e s s s p e c i f i e d )  example o f explanation:  pattern (2): a)  C u t parts f r o m the s e c t i o n h a v i n g the h i g h e s t g r a d e a n d a d d its r e m a i n d e r to the s e c t i o n h a v i n g the  b)  C u t parts f r o m the s e c t i o n h a v i n g the s e c o n d h i g h e s t g r a d e a n d a d d its r e m a i n d e r to the s e c t i o n  lowest grade.  h a v i n g the l o w e s t g r a d e . c)  C u t parts f r o m the s e c t i o n w i t h h a v i n g the l o w e s t g r a d e - r e m a i n d e r is w a s t e .  43  2.4.2.2.3 Dynamic Program A s was shown earlier, the chop saw program obtained, through a series o f trial runs, the most valuable cutting pattern before any actual cutting commenced. Even though trial runs and actual cutting used the same dynamic program, outputs from each cutting mode were treated differently.  In actual cutting, tallied parts were incorporated into the cutting bill and remaining lumber was retained for further use or discarded. Trial runs had no influence on the cutting bill and returned a dollar value that was kept for comparisons with other cutting patterns.  2.4.2.2.3.1 Features Common To Both Cutting Modes 2.4.2.2.3.1.1 Accuracy Three levels o f cutting accuracy (1", 1/5", and 1 mm) were thoroughly tested to find the one that best satisfied the criteria o f relatively quick processing time and minimal number o f errors or incorrect cutting decisions due to measuring units which were incompatible with part lengths. N o errors were found with 1 m m accuracy, however, it took an hour to complete a mere 857 minutes o f simulated time. R u n times were significantly shorter with 1" accuracy but errors were common even after adjustments were made.  One fifths o f an inch accuracy or about 5.08 m m  was chosen because o f its compatibility with most o f the cutting bill parts and respectable run execution time o f one hour to 5400 minutes o f simulated time.  2.4.2.2.3.1.2 Fingerjointing Waste from parts production was minimized by fingerjointing, a feature o f the dynamic program that examined leftover lumber and made a length based decision to either call it waste or stock 44  that could be used later to create laminated lumber. If higher value was achievable, usable lumber was cut to fingerjointing stock instead o f cutting bill parts.  Rules for fingerjointing varied between cutting bills and were contained i n small data files which were imported into the C program before any simulation work started. Table 14 below shows an example o f fingerjointing rules and Appendix I V shows rules from all nine cutting bills ( 7 - 3 " width, 2 - 3 . 7 5 " width).  Table 14. Example O f Fingerjoint Rules (3" Width - Cutbill #6) grade  min.  $/inch o f  length  length  kicker  1  8  0.04  2  2  8  0.02  2  3  10  0.015  4  2.4.2.2.3.1.3 Cutting Considerations Parts and leftover lumber were not the only components o f a board.  Small amounts o f wood  removed by a saw blade or kerf also had to be incorporated into cutting calculations. A l l m i l l runs used a kerf value o f .197" (5 mm); therefore the same constant was implemented i n project models.  After a cutting bill or fingerjoint part was cut, kerf was always subtracted from  remaining lumber before any subsequent work commenced. Even though it was possible to get two parts from one section, kerf was applied to all parts as the one additional cut compensated for any end trim requirements.  For lead trim (kerf applied to the front end o f a board before  scanning) a value o f .187" was always used.  45  2.4.2.2.3.2 Features Applicable Only To Actual Cutting 2.4.2.2.3.2.1 Statistics From Each Board In addition to influencing the cutting b i l l , actual cutting added data to the array that kept track o f yield from each board. Each time a part was cut, selected information was recorded to an array cell (named according to the board number - eg. board[2][x]) which was later used for generating post-run reports. Examples o f summary reports and board data files from a simulation run are shown i n Tables 15 and 16, respectively:  Table 15. Example O f Simulation Run Summary Report 1 cutting bill 2  thickness  3  width  955  completed boards  957  boards in system  211.31 mins completion time 0.56  $/fbm (overall)  22.94 linear inches (average waste/board) 85016 inches (total lin.parts cut 50.94 pet rec(parts) 59980.06 inches (total lin. fj cut) 35.94  pctrec(fj)  166905.50 inches (total lin.overall) 86.87 pet rec(total) 3924.78 $ total value -1 from completed boards 6968.99 $ total value -2 from kickers 891  boards were profitable  64  boards lost money  63  clears (inactivity)  0 clears (shorts) 63  clears total  number of 20 ft. clear boards for inactivity = 63 percent of 20 ft. clear boards for inactivity = 6.597 number of 20 ft. clear boards for inadequate length = 0 percent of 20 ft. clear boards for inadequate length = 0 random # = 1, cutoff pct= 0.70, algorithm = la  46  Table 16. Example O f Board Data File (totalS = total value from board,  partslen = total length o f parts cut, fjlen = total length o f fingerjoint stock cut,  fintime = time when board completed)  #  length  totalS  waste  partslen  fjlen  fintime  cost  1  168.622  0.1403  57.43  90.5  20.69  0.29  2.93  2  168.346  2.9484  19.2  121  28.15  0.47  2.93  3  168.543  2.0347  30.26  85  53.28  0.61  2.93  4  170.039  12.613  2.04  168  0  0.84  2.96  5  168.583  4.5566  9.12  108  51.46  1.11  2.93  2.4.2.2.3.2.2 Kicker After cutting and information collecting, parts were directed to a kicker at which they were tallied.  Use o f kickers was implemented primarily for animated models as the cutting bill  provided enough useful data for model users.  2.4.2.2.3.2.3 Intervention Modifying priority settings on a regular basis was not the only aspect o f the model where automated intervention was allowed to override natural cutting. When testing o f the chop saw simulation model was done, it was revealed that there were times when input lumber lengths were less than the shortest part required by the cutting bill. Another weakness with the initial model was the tendency for the chop saw to run for long periods o f time without producing a cutting b i l l part.  Printouts o f test runs showed that when nearing completion, the chop saw cutting bill sometimes went unchanged for up to 40 minutes even i f input lumber was longer than the required parts. Under no circumstances would any secondary manufacturing plant tolerate excessive production of fingerjoint grade parts.  47  When faced with poor production results of any kind, mill managers have two relatively economically feasible options of enhancing yields of scarce stock; certain cutting bill parts can be removed and/or priority settings can be adjusted to emphasize needed parts. A last resort method is to introduce long high grade lumber into the chop saw run, a practice that fills parts quotas but increases costs. For the project model, adding clear lumber when it was appropriate was the only logical choice as removing parts from the cutting bill was too subjective for C programs and priority settings were closely monitored by the model.  Functions that constantly kept track of production activities added 20' grade 1 or clear lumber if the chop saw went 5 minutes without adding items to the cutting bill or if lumber at the grading station was too short for creating non-fingerjoint parts. When mill workers were asked about when they added long or clear lumber, they had trouble giving definite answers as many circumstantial factors go into such a decision. A five minute inactivity period was chosen and found to be relatively non-dictative of production.  A premium or penalty multiplication factor of 1.50 to increase the cost of 20' clear lumber was calculated by taking the average of ratios based on dividing lower grade prices into higher ones or: penalty for using 20' clears = [Standard/#2&Better price($)]/[Utility/#3Common price ($)] [Equation 3] Each 20' clear; therefore, cost the mill $5.87 instead of $3.91. Without higher costs for long clears, it would have been impossible to distinguish between the value recovery results of poorly and well performing algorithms.  48  2.5 Data Analysis  2.5.1 Analysis Variables 2.5.1.1 Value Recovery Per Fbm The main goal was to find an algorithm which yielded the best performance results i n terms o f value recovery or profits per board footage (value/fbm), a variable that encompassed both the economical and productive aspects o f sawmilling. Foot board measure or fbm is a commonly 1  used forestry industry term to present lumber volume. Value recovery per fbm was calculated by the formula:  value per fbm =  total profit from all parts($) —— — total fbm of input lumber  [equation 4]  A s an example, i f a total o f 178212.2 linear inches o f lumber was processed during a 3.75" width run, its volume equivalency was 8701.77 fbm or:  thickness width length  = 47.625 m m (1.875 inches) =3.75 inches = 178212.2 inches  fbm  = ((1.875*3.75)/12 )*(178212.2/12) = 8701.77  Assuming that the total value from all parts from a board equaled $6362.91 then:  value per fbm = y  $6362.91 8701.77 fbm  =  .7312 fbm  Algorithm averages, each one consisting o f 21 observations (3 replications X 7 cutting bills), from three other simulation run variables (number o f 20' clears, number o f boards processed, and  1  One foot board measure is equal to 1' x 12" x 1" o f lumber 49  percent recovery o f parts) were examined to ensure that high value/fbm figures were not obtained at the expense o f other production factors. Furthermore, analysis o f the three external variables was necessary to see i f cutting algorithm was the only factor that determined value/fbm results.  2.5.1.2 Number Of 20' Clears Added One o f the most powerful influences on a chop saw run is the option o f adding long, clear full length lumber when parts are cut at an unacceptable rate.  I f high quality stock is used as a  corrective factor, caution has to be taken as it is possible for the worst chop saw setup to produce respectable results i f large volumes o f clears have been implemented. Analysis o f the average number o f grade 1 - 2 0 ' clears used by each cutting algorithm was therefore necessary to reveal i f any biases were present.  2.5.1.3 Percent Recovery Of Parts The ability o f a chop saw to complete an order with minimal waste and fingerjoint stock was measured by its parts recovery or the percentage o f input lumber that contributed to producing cutting bill parts. Formulated as:  % Parts Recovery =  Total Linear Length Of Parts ProducedQnches) —— ——-— lotal Linear Lengtn{inches)  [equation 5]  Linear length was used instead o f fbm because all full length lumber in a run had the same cross sectional dimensions. Fingerjoint graded parts were excluded from the equation as their presence inflated recovery percentages, especially i f all grades o f lumber could be salvaged.  50  2.5.1.4 N u m b e r Of B o a r d s Processed Secondary manufacturing mills are often under pressure from their customers to complete orders by specified dates. Meeting deadlines can be difficult due to the heterogeneous nature o f full length lumber and the lack o f available information about how chop saws work.  Mills  encountering chronic problems with run times would probably be interested in a cutting algorithm that completes a cutting bill with the least amount o f input lumber.  It can be said with confidence that normal run conditions were present when both dimensions were cut as twenty foot lumber (longest possible) was evenly distributed in all package arrival data. Total number o f boards processed, unlike many o f the other variables, was dependent on cutting bill structure. Therefore, comments pertaining to this measure o f performance w i l l only refer to average number o f boards processed per algorithm.  2.5.2 Setup Of Analysis 2.5.2.1 2 W a y A n o v a Description Two way analysis o f variance ( A N O V A ) was used with three replications for each combination of factors (cutting bills and algorithms). Table 17 shows an abbreviated version o f a 3" width value/fbm matrix in which the cell containing " * " is the third replication o f a simulation run using algorithm 2 and cutting bill 1. T w o large matrices, one for each lumber dimension, were generated.  51  Table 17. Example O f 3" Width Value/Fbm Table (each replication or observation is a mean or total value/total input fbm)  cutting bill 1  algorithm 1  algorithm 2  algorithm 3  algorithm 20  .67074  .67074  .62074  .70174  .65234  .68314  .63314  .71966  .68193  .68235 *  .64723  .74116 •  cutting bill 7  .87323  .88132  .81249  .84664  .84286  .86359  .82203  .87981  .86006  .85279  .82315  .88371  Two way A N O V A was chosen for its ability to objectively determine i f significant differences existed between the effects o f a cutting bill or algorithm on value/fbm.  Interactions or  differences between how cutting bill (or algorithm) effected value/fbm at a given cutting algorithm (or bill) could also be measured.  Appendix V provides a detailed description o f the  formulas and procedures associated with two way A N O V A .  Bartlett's test (see Appendix V I for details), a technique based on a statistic whose sampling distribution is approximated very closely by the chi-square distribution when k random samples are drawn from independent normal populations, tested the hypothesis: H :<i?=ol 0  =-  =a  2 k  against: H : the variances are not all equal x  52  A three factor table i n which cutoff percentage was added to cutting bill and algorithm would have enhanced data analysis. Under the ideal setup, significant differences, i f present, between cutoff percentages could have been objectively determined. However, such a setup would have excluded algorithms 20 (no priority settings) and 19 (mill settings).  2.5.2.2 Student-Newman-Keuls Range Test If cutting algorithm or interaction means were found to be statistically different, a range test was used to examine all possible pairs o f means and ascertain which ones were significantly different from each other.  Student-Newman-Keuls ( S N K ) test was chosen out o f the several that were  available for its ability to handle large numbers o f means and compatibility with Excel macros. Comparisons were made only between the 20 cutting algorithm means. Appendix V I I shows the procedures involved in an S N K test and an example o f it, respectively.  2.5.3 ANOVA Assumptions A N O V A cannot be considered valid until all o f three o f its assumptions, listed below, have been fully satisfied:  • • •  The population distribution being sampled is normal. The process is in control, that is, it is repeatable The variance o f errors within all k levels o f the factor is homogeneous  2.5.3.1 Normal Data Normality o f project data was confirmed by the central limit theorem which, according to Mandel (1984) is defined as follows:  53  " G i v e n a population o f values with a finite (non-infinite) variance, i f we take independent samples from this population, all o f size N , then the population formed by the averages o f these samples w i l l tend to have a Gaussian (normal) distribution, regardless o f what the distribution is of the original population; the larger N , the greater w i l l be this tendency towards normality."  According to Sternstein (1996) and Spiegel (1992), normality status could be granted to samples sizes larger than 30. A total o f 420 observations were contained within the 3" width A N O V A table, making all data normal even though it was skewed.  Sample size could have also been  interpreted as the number o f replications per algorithm/cutting bill combination.  I f such an  approach had been taken, additional data would have had to have been collected.  2.5.3.2 Repeatable Process A l l data generation could easily be repeated by either changing random seeds in the simulation software or gathering information from future m i l l runs. Both dimensions o f lumber are cut at least once a month at the participating m i l l .  2.5.3.3 Homogeneous Variances Using Bartlett's test with an a o f 5%, all variances from A N O V A work were found to be homogeneous.  2.5.4 Comparing Analysis Variables Comparisons between analysis variables were limited to correlation coefficient (p) calculations and plotting rankings due to the different units o f measurement. Correlation coefficients for each  54  comparison between the main variable, value/fbm, and the other three variables were done using data from all algorithms.  Comparisons between analysis variables were done graphically by first ranking the data and then plotting the rankings on a bar graph. Rules for ranking data are discussed in subsequent sections o f this thesis. A s an example, Table 18 and Figure 14 show the ranked and plotted data o f a four sample average value/fbm versus the average number o f boards processed.  Table 18. Example O f Average Value/fbm versus Average Number O f Boards Processed Data Algorithm  |  Average Value/fbm  Rank  1.276 1.245 1.294 1.356  3 4 2  1  2 3 4  |  Average Number O f Boards Processed  Rank  472 543 386 365  2  1  1  3 4  Figure 14. Rankings Graph Example - Value/fbm Versus Number O f Boards Processed  1  2  3 ALGORITHM  55  4  3.0 R E S U L T S A N D D I S C U S S I O N 3.1 3.75" W i d t h L u m b e r  Appendix VIII and Table 19 show the 3.75" width value/fbm data matrix and F tests, respectively. 3.75" width value/fbm means were found to be homogeneous, therefore, Bartlett's test and other subsequent work was n o w considered irrelevant even though not much o f a difference existed between the calculated and critical F values. Generating a second A N O V A table without cutting bills might have shown that 3.75" width value/fbm means were different; however, the statistical procedures had to be consistent for both lumber dimensions.  Table 19. A N O V A Table For 3.75" Width Lumber. Source o f  SS  Variation  degrees o f  MS  F  P-value  F critical  1  0.13944  3282.326  1.03E-66  3.960352  freedom  Cutting Bills  0.13944  Algorithms  0.000924  19  4.86E-05  1.144302  0.32613  1.718025  Interaction  0.00024  19  1.27E-05  0.297811  0.997848  1.718025  Within  0.003399  80  4.25E-05  Total  0.144002  119  While one might say that using only two cutting bills for 3.75" width A N O V A analysis was a setup destined for statistical "failure" unless more data had been generated, all attempts had to be made to show that project findings were not only applicable to one dimension o f lumber. Too many assumptions about the effects o f grading rules and width would have been made i f the project used one data matrix with nine cutting bills, 7 from 3" width and 2 from 3.75" width. Cutting bills from previous runs were readily available but information about when the priority settings were changed could not be found.  Data from 3.75" width runs were presented for  reference purposes only.  56  Figure 15 below shows 3.75" width value/fbm means. In algorithms which used priority settings, a 70% cutoff percentage produced the highest value/fbm results. The only exception to this trend was seen among algorithms 10, 11, and 12 which fared poorly in the 3" width analysis.  Figure 15. 3.75" Width Value/fbm  2 p r . - 1/2  1 pr. - unlim.  1 pr. - 1 only  70% 1  80% 2  2 pr. - 1/2/1/2  3 pr. - 1/2/3  r  90% 3  70% 4  80% 5  90% 6  70% 7  80% 8  90% 9  70% 80% 10 11 ALGORITHM  90% 12  70% 13  80% 14  90% IS  3 pr. - 1/2/3/1/2/3  70% 16  80% 17  n  90% 18  mill 19  none 20  3.2 3 " W i d t h L u m b e r 3.2.1 V a l u e / F b m M e a n s Appendix I X and Table 20 show the 3" width data matrix and F test results, respectively. The F value o f 20.90272 was decisively higher than the 5% critical value o f 1.623819, algorithm means were significantly different.  Table 20. A N O V A Table For 3" Width Lumber. Source o f  SS  Variation  degrees o f  MS  F  P-value  F critical  6  1.879937  22774.36  0  2.131028  freedom  Cutting B i l l s  11.27962  Algorithms  0.032783  19  0.001725  20.90272  4.4E-43  1.623819  Interaction  0.096539  114  0.000847  10.25892  3.79E-56  1.28545  Within  0.023113  280  8.25E-05  Total  11.43206  419  57  therefore,  Table 21 shows that variances from the 3" width matrix produced a Bartlett's test value (see Appendix X for complete data) o f 1.404812 which was well below the 5% chi-square critical value o f 30.143505. Table 21. 3" Width Value/Fbm Bartlett's Test Results  n  21  (# obs./cell)  N  420  (total # obs.) (total # cells)  k  20  s2p  0.028498  q  0.620775  h  1.017500  S(l/(n-l))  1  l/(N-k)  0.0025  b  1.404812  5% a  30.143505  3.2.1.1 Differences Expectations o f finding algorithms which were decisively good disappeared quickly once it was found that a mere .0295/fbm difference existed between the highest and lowest ranked means. The S N K range test identified exactly which means were statistically similar or different from each other. Figure 16 below shows, graphically, all possible combinations o f means. T o make comparisons easier, mean values, not algorithm names, increase along the X or horizontal axis o f the table from left to right while Y or vertical axis values decrease from the top to bottom row. I f two means are similar, a "  " appears in the matrix cell or intersection o f the algorithms.  Cells  containing blanks and gray shading symbolize statistical differences and irrelevant comparisons, respectively.  58  Figure 16. Student-Newman-Keuls Comparison O f Means Table (labels on X and Y coordinates = algorithm) 12  10  11  6  20  5  19  7  4  8  3  16  9  2  1  18  17  13  15  14 14 15 17 13 1 18 2  fijltilili  9 16 3 8 4 7 5 19  lif^&lliiii 1HI1I  ~ - r  .'i  20  iililii WSmf.  »;/...;  6 11 10 12  ¥ " "\„  •':;<•• '  ;  3 * - •  Before any conclusions can be made about value/fbm means, it must be noted that significant interactions occurred in algorithms 10 to 12 as Figure 19 shows.  When all algorithms were  considered, significant differences existed primarily between two groups; algorithms 10, 11, 12, 6, 19, 20 and all the others. If algorithms 13, 14, 15, and 17 were excluded from comparisons, only algorithms 10 through 12 were different from the rest. Figure 17 below shows, i n graph format, how all value/fbm means compare to each other. Figure 17. 3" Width V a l u e / F B M Means  1  m  |  • m  ---  if  llf  1 pr. - 1 only  -  -  ---  ! p  ...  -  _  1  I I 1 p r . - unlim.  in  1 3  -  1  ---  - --  - 1/2  -  81  2  I  •  —  |  -- ---  g§  1/2/3  3p  2 p r . - 1/2/1/2  m  •  1  --  l  3 Pi" " 1/2/3/1/2/3  1  •-  1  ...  11  |  1=1 70%  80%  90%  70%  80%  90%  70%  80%  90%  70%  80%  90%  70%  80%  90%  70%  80%  90%  mill  none  1  2  3  4  5  6  7  8  9  10  II  12  13  14  IS  16  17  18  19  20  ALGORITHM  59  3.2.1.2 Number Of Parts Prioritized Statistically, there was no single algorithm that could have been called the best. However, three o f the top four "better" algorithms prioritized three different parts and forced one o f each, i f possible, into a section o f lumber. Algorithm 14 which used an 80% cutoff point had the best value/fbm result at .7585/fbm while algorithms 15 (90% cutoff) and 13 (70% cutoff) ranked second and fourth, respectively.  Poor value/fbm results were always yielded when two items were prioritized and forced into a lumber section according to the diagram in Figure 18.  Figure 18. Cutting Prioritized Parts W i t h Algorithms 10, 11, and 12 (alg. 10 cutoff = 70%, alg. 11 cutoff = 80%, alg. 12 cutoff = 90%)  Values from algorithms 10, 11, and 12 were so bad that the best performer o f the three, algorithm 11, was 1.5 cents/fbm lower than the fourth worst performing algorithm. Large discrepancies between the 1/2/1/2 priority setup and 1 part algorithms were caused by the formers inability to quickly complete a part the way the latter did. Three part algorithms outperformed the 1/2/1/2 setup as the latter probably forced i n more prioritized parts, hence fewer opportunities for "natural" optimization were available. When three different parts were prioritized, there were probably many instances where the third and subsequent parts were too long to be included in the cutting solution.  60  3.2.1.3 Cutoff Percentages Figure 17 also shows that when one part was prioritized, maximum value/fbm values were achieved with a 70% cutoff percentage.  M a x i m u m two part prioritization values came from a  90% cutoff mark for algorithm 9 (1/2) while 80% was the optimal point for algorithm 11 (1/2/1/2). Three part prioritization was most compatible with an 80% cutoff percentage.  These results contradict the often mentioned "evils" o f imposing restrictions on natural or optimal cutting patterns.  Before any simulation data was generated, it was expected that  algorithms with a 70% cutoff point would have rated highly as less time was spent on cutting prioritized parts. While data from runs involving only one prioritized part, algorithms 1 through 6, confirmed the benefits o f a lower cutoff percentage, better value/fbm results were produced with an 80% or 90% mark when two or three parts were involved.  Ideally, a larger range o f cutoff percentages would have been examined to possibly find a better algorithm.  In most cases, peak values were found for each combination o f parts and cutoff  percentages. A notable exception occurred in algorithms 4 through 6 as a continuous decrease in value/fbm occurred as the cutoff mark increased. This pattern indicates that an algorithm based on unlimited cutting o f one prioritized part and a cutoff percentage o f 60% or 50% could have possibly outperformed algorithm 14.  3.2.2 Interaction Results Interaction or when a change in one factor produces a different change in the response variable at one level o f another factor than at other levels o f this factor was significant i n the 3" width  61  ANOVA.  A s was shown in Table 20, 3" width interaction produced a calculated F value o f  10.25892 while F critical was 1.28545.  Figure 19 below shows that the most obvious signs o f interaction occurred when algorithms 10, 11, and 12 were applied to cutting bill 1, an order that had only nine parts while the other six bills had at least 15 items. Other noticeable deviations appear when cutting bill 3 and algorithms 8 through 20 are combined. Figure 19. 3" Width Interaction Plot ( C B = cutting bill)  1  2  3  4  5  6  7  8  9  10  11  12  13  14  IS  16  17  18  19  20  ALGORITHM  3.3 3" W i d t h E x t e r n a l V a r i a b l e s Correlation coefficients were calculated and data ranking was done to determine i f good value/fbm results were associated with poor performances o f external variables. A complete data analysis was conducted to provide additional information for comparing cutting algorithms.  62  3.3.1 N u m b e r O f 2 0 ' C l e a r s A d d e d Appendix X I shows 2 0 ' clears data from all 3 " width simulation runs. Table 22 below shows that algorithm means were different while Bartlett's test, see Table 23 for results and Appendix X I I for calculations, successfully validated variances. Table 22. 3" Width Number O f 20' Clears Added A N O V A  Source o f  SS  df  MS  F  P-value  F crit  4732.42  Variation Cutting Bills  13114279  6  2185713.1  3.5E-278  2.131028  Algorithms  37110.152  19  1953.1659 4.228918 3.22E-08  1.623819  Interaction  99886.248  114  876.19515  1.28545  Within  129320.67  280  461.85952  Total  13380596  419  1.897103 1.07E-05  Table 23. 3" Width Number O f 20' Clears A d d e d Bartlett's Test Results  n  21  (# obs./cell)  N  271  (total # obs.)  k  20  (total # cells)  sP 2  42385.175  q  1.157374  h  1.027904  S(l/(n-l))  1.59449  l/(N-k)  0.00398  b  2.59262  5% a  30.14351  Caution must be exerted before any official statements can be made about number o f 20' clears added means as numerous sources o f interaction appeared after a plot o f means according to cutting bill (Figure 20) was done.  Figure 20. Interaction - Average Number O f 20' Clears Added  600  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  ALGORITHM  S N K tests on "number o f 20' clears added" means (data in Appendix XIII) proved that algorithms 19 and 20, m i l l settings and no priorities respectively, were significantly different from the rest. Figure 21 below shows a bar chart o f average percentage o f clears for each cutting algorithm.  64  Figure 21. Average Number O f Clears cl - short =  clears added because the input lumber was shorter than the shortest part in the cutting bill,  cl - inact. =  clears added because no cutting bill parts had been cut in a five minute time period.  g e l - short Qd  70% 1  80% 2  90% 3  70% 4  80% 5  90% 6  70% 7  80% 8  90% 9  70% 10  80% II  90% 12  70% 13  80% 14  90% 15  70% 16  80% 17  90% 18  mill 19  - i n act.  none 20  ALGORITHM  Twenty foot clears did not dominate the composition o f completed boards in any o f the 420 simulation runs as a maximum clears composition percentage o f 13.77% was seen in six cutting b i l l 1 runs. In terms o f maximum volume, 673 o f the 8722 input boards for the cutting bill 7 run consisted o f clears. A n average o f 103.39 clears or 2.41% were added to each run.  Algorithms with an 80% cutoff percentage tended to require more clears than others while minimal intervention was required for setups based on m i l l settings and no priorities. A l l algorithms had to "resort" to using clears and the range was a mere 39 boards. When average number o f clears and value/fbm data was ranked in a way that gave higher ratings to larger values, Figure 22 below was derived.  65  Figure 22. Rank Comparisons - Number O f 20' Clears Added V s . Value/fbm  When one part was prioritized, an inverse relationship existed between % clears and value/fbm. In all other algorithms, except 10 to 12, both variables o f interest appeared to be correlated. A correlation coefficient o f 0.29306 proved that no strong relationship existed between value/fbm and average number o f clears when all algorithms were considered.  3.3.2 Number of Boards Processed Appendix X I V shows the data matrix for this performance variable. Table 24 below shows that number o f boards processed means were significantly different when A N O V A was applied to them while the Bartlett's test b value (0.134190 - see Appendix X V for data and Table 25 for results) was w e l l under the 5% alpha chi-square limit.  66  Table 24. Number o f Boards Processed A N O V A Source of Variation Cutting Bills Algorithms Interaction Within  SS  df  MS  3.35E+09 371233.3 2835193 1565806  6 19 114 280  Total  3.36E+09  419  F  P-value  F crit  0 5.59E+08 99919.77 2.131028 19538.59 3.493923 2.37E-06 1.623819 24870.12 4.447315 2.01E-24 1.28545 5592.164  Table 25. 3" Width Number o f Boards Processed Bartlett's Test Results  n  21  (#obs./cell)  N  420  (total # obs.)  k  20  (total # cells)  sp 2  q h S(l/(n-l)) l/(N-k)  8392518.910000 0.059249 1.016665 0.952380952 0.0025  b  0.134190  5% a  30.143505  Statistically, number o f boards processed interaction results were significant. However, a plot o f the results in Figure 23 shows that the degree o f severity was evenly spread out among the algorithms.  67  Figure 23. Interaction - Average Number O f Boards Processed  -CB 1 -I  1  1  1-  X,  CB 4  o  > <  3000 Jh m  m—m  •  m  •  m  •  •  9  •  a-—m  •  •  •  •  m  •»  •  a  10  11  13  14  15  16  17  18  19  20  12  ALGORITHM  Figure 24 below shows a graph o f the number o f boards processed means.  According to S N K  results shown i n Appendix X V I , algorithms 1, 2, 3, 8, 11, and 14 had lumber consumption means which were lower than the rest.  Figure 24. Average Number O f Boards Processed  3pr. - 1/2/3/1/2/3  g 3870 I pr. - 1 only  X  2 pr. - 1/2  2 pr. - 1/2/1/2  3 pr. - 1/2/3  g 3860  < O CO 3850 3t  O > 3840  70% 1  n  80% 90% 70% 80% 90% 70% 80% 90% 70% 80% 90% 70% 80% '90% 70% 80% 90% mill none 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ALGORITHM  68  A s was shown earlier, algorithms 19 and 20 had the least amount o f intervention and, as expected, ended up processing larger quantities o f boards than most o f the other setups. Algorithm 6, which allowed the least amount o f optimization among one priority setups, required the most infeed boards. Although preliminary A N O V A proved that lumber consumption means were different, the range between the means was a respectable 88 boards or 37% o f a package.  Figure 24 also shows that algorithms which rated highly i n value/fbm analysis needed the fewest amount o f full length lumber. A graphical comparison, Figure 25 below, was done by plotting the ratings or rankings o f the two variables. A rating o f 1 was given to the highest value o f each variable although the lowest lumber consumption figure could have been interpreted as the best.  Figure 25. Rank Comparisons - Average Number O f Boards Processed V s . $/fbm  . _||-  70% 1  80% 2  90%  70%  81)%  90%  7(1%  811%  90%  J  4  5  6  7  8  9  71)% 10  80%  90%  70%  80%  90%  70%  80%  90%  11  12  13  14  15  16  17  18  mill  none  19  20  -val./fbm # boards pr.  ALGORITHM  Figure 25 clearly shows that when 1 part is prioritized, value/fbm is inversely related to number of boards processed, a relationship which was also seen i n algorithms 13 to 16 and the two setups  69  that d i d not use priority settings.  Value/fbm and number o f boards processed were almost  completely independent as a -0.14559 correlation coefficient was derived from relevant data.  3.3.3 Percent Recovery O f Parts Each cutting algorithm's level o f efficiency was best measured by its ability to obtain a large yield o f cutting bill parts relative to the amount o f full length lumber that was fed into the chop saw.  Appendix X V I I shows the data matrix, A N O V A results appear i n Table 26, Bartlett's  results and data appear in Table 27 and Appendix X V I I I , respectively.  Table 26. Percent Recovery O f Parts A N O V A Source of Variation Cutting Bills Algorithms Interaction Within  SS 34284.027 104.95303 881.07362 701.3777  df 6 19 114 280  Total  35971.431  419  MS 5714.0045 5.5238436 7.728716 2.5049203  F 2281.112 2.205197 3.085414  P-value 1.9E-234 0.003071 1.49E-14  F crit 2.131028 1.623819 1.28545  Table 27. 3" Width Percent Recovery O f Parts Bartlett's Test Results n  21  (# obs./cell)  N  420  (total # obs.)  k  20  (total # cells)  sp 2  89.666196  q  1.279695  h  1.016665  S(l/(n-l))  0.952381  l/(N-k) b 5% a  0.0025 2.898325 30.143505  70  A l l A N O V A criteria was satisfied as means from all four 3" width analysis variables were found to be heterogeneous and relevant variances were homogeneous.  A plot o f average percent recovery (Figure 26) showed that interactions  were common,  especially in algorithms 13 to 18.  Figure 26. Interaction - Average Percent Recovery O f Parts  a. LU 70  o > o  a. 65 in cc 60 < 0.  > <  55 50 45 40 9  10  11  12  13  14  15  16  17  18  19  20  ALGORITHM  Complete S N K data appears in Appendix X I X . Algorithm 6's parts recovery percentage o f 71.196 was found to be significantly lower than those from other algorithms even though Figure 27 below shows that it was not much different from algorithms 19 and 20.  71  Figure 27. Percent Recovery O f Parts  I pr. - 1 only  70% 1  80% 2  90% 3  1 pr. - unlim  7(1% 4  80% 5  90% 6  70% 7  80% 8  90% 9  70% 10  80% 11  90% 12  Jpr.1/2/3/1/2/3  3 nr.-1/2/3  2 pr. - 1/2/1/2  2 nr.- 1/2  |  70% 13  80% 14  90% IS  70% 16  80% 17  90% 18  milt 19  none 20  ALGORITHM  Most o f the algorithms which used priority settings posted parts recovery results that were about .5 to 1 percent better than m i l l settings and natural optimization. Algorithm 6, which yielded a disappointing 71.2%, was the only notable exception and reinforced the need for some optimization. L o w recovery percentages for algorithms 19 and 20 can be explained by the lower amounts o f 20' clears that they consumed.  After percent recovery o f parts and value/fbm were ranked, the data were plotted to create Figure 28 below. Except for algorithms 10 to 12, a direct relationship appears to exist between the two variables, although the correlation coefficient was only 0.25489.  72  Figure 28. Rank Comparisons - Percent Recovery O f Parts V s . $/fbm  3.4 3 " W i d t h F i n a n c i a l Benefits O f T h e Ideal A l g o r i t h m Total value lost in a year by using algorithm X instead o f the highest rated algorithm (14) was calculated by the following formula:  (Algorithm 14 Value/Fbm - Algorithm X Value/Fbm)* (Total F b m Processed In One Year) where: Total F b m Processed In One Year = (Average fbm/hour from m i l l data) * (Total Number O f Operational Hours/Year) = (1658.53 fbm/hour) * (220 days/year) * (16 hours/day) = 5,838,041.53 fbm processed per year After a call was made to the m i l l to get operational parameters, it was found that a total o f $63,459.51 was gained by employing algorithm 14 instead o f m i l l settings or algorithm 19. A table showing comparisons between algorithm 14 and all other setups appears in Appendix X X and Figure 29 below shows a bar chart o f yearly value differences.  73  Figure 29. Yearly Value Differences From Algorithm 14. SIMM) -si"i,>0M< .m . -S20,IMM).0» . -S3I(,lMMUH> . -S4(0 )I)I).<XI -S50,l)l>0.«0 S -6O0,IMMM> . -S70D , WMH) _ -S8Uu ,T)IMH> -S9»,(1W).W) _  LT  TU  (  lpr.-l only  I pr.iinlini  2pr.1/2/1/2  2 pr. - 1/2  -S100,OI)I|.I>(»  •sin,i>on.w> _ -S12<MH)IMM> _ -SI3»,IH)IMr(> J  -S14»,OI>0.0» . -S1SO,l)l)l).l)(> _ -S16U,W)IU>»  -S17O),H)0U . <> _ -S180,)M)l).im ALGORITHM  74  3pr.1/2/3  3pr.1/2/3/1/2/3  4.0 S U M M A R Y A N D R E C O M M E N D A T I O N S 4.1 S u m m a r y A discrete simulation chop saw model capable o f providing accurate estimates o f m i l l production was successfully created.  Time and production data from an existing chop saw m i l l were  collected and used to create the model parameters.  Extensive C programming was done to  facilitate cutting decisions while lumber flow and tallying procedures were taken care o f by the model's graphical network.  Major limits to the model were the number o f full length lumber  sections that could be considered in a cutting decision and cutting accuracy.  Modifications  would have to be made to the current simulation model to make it compatible with other optimizing chop saws.  For 3" width lumber, data from simulation runs proved that algorithm 14 or the one that prioritized three different parts and forced one o f each into a usable section o f lumber with an 80% cutoff mark achieved the highest value/fbm figure. Statistically, this algorithm cannot be deemed the best as A N O V A analysis found two distinct groups o f value/fbm means.  No  comparisons could be done among the algorithms with the 3.75" width data.  We can confidentially say that high value/fbm figures are obtained without sacrificing the performances o f other important chop saw run variables. When three variables, number o f 20' clears added, number o f boards processed, and percent recovery o f parts were compared to value/fbm, none o f the correlation coefficients showed any relationships. Graphs o f the ranked results showed that some trends did exist but only when one particular arrangement o f priority settings was considered (example - 1 prioritized part, 1 forced in).  75  Financial gains were possible i f new cutting algorithms were implemented. Yearly increases in value o f $63,459.51 and $64,276.84 were obtainable over m i l l settings and no priority settings, respectively with algorithm 14 or the one with the highest value/fbm figure. If algorithm 12 or the worst performing one was used instead o f m i l l settings, a financial loss o f $108,762.71 was incurred.  4.2 Recommendations This project objectively proved that using priority settings in chop saw production is not as detrimental as was initially thought as setups based on mills settings and no prioritized parts were included with the lower ranked algorithms. Although project personnel were satisfied with the project results, there are definitely many opportunities for improvement over what was achieved. B e l o w are some o f the areas that can be expanded.  Data analysis for 3" width stock worked out well, however, most o f its cutting bills consisted o f 16 to 18 parts. A larger variety o f bills would have been more desirable but none o f the m i l l runs during the month o f July included any large bills consisting o f more than 20 parts.  For 3.75"  width lumber, only one cutting bill was worked on each time a m i l l visit was made.  In addition to providing more realistic data, adding more cutting bills to a project contributes to the statistical power o f the tests.  Future projects with more cutting bills can be conducted  without excessive expenses by having m i l l staff monitor cutting bill data when a run is on and notifying the responsible individual when everything is completed.  76  There were no obvious signs during the project that a cutoff percentage between 70% and 90% was detrimental to value/fbm results. However, what i f a 60% or 100% mark was present? It is not known i f more clears w i l l have to be added i f a 100% point is used or what the possible added economic benefits from a 60% limit are. A project with more than the current three cutoff percentages would answer these questions.  Only one form o f intervention, add 20' clears after 5 minutes o f cutting bill inactivity, was present in the simulation model. B y how much would costs go down i f a 10, 15, or 20 minute inactivity level was used? policy was followed?  W o u l d costs increase significantly i f a 2 or 3 minute intervention  A very useful A N O V A analysis would be a three factor setup, see  Appendix X X I , i n which cutting algorithms are further categorized according to intervention time.  Algorithms which prioritized three different parts produced high value/fbm values i f only one o f each was forced into a lumber section.  A project with algorithms including four or more  different parts would determine i f a "saturation point" or one where value decreases i f too many parts are forced in exists.  77  5.0 L I T E R A T U R E C I T E D Anonymous. 1995. Madison's Canadian Lumber Reporter (all 1995 issues). Vancouver, B . C . Anderson, D . R . 1994. Introduction To Statistics: Concepts A n d Applications. 3 ed. West Publishing Company, St. Paul, M N . rd  Araman, P. A . 1977. Use O f Computer Simulation In Designing and Evaluating A Proposed Rough M i l l For Furniture Interior Parts. U S D A Forest Service N . E . Forest Experimental Station. Upper Darby, N e w York. Gazo, R. and P . H . Steele. 1995. A Procedure For Determining The Benefits O f Sorting Lumber B y Grade Prior To Rough M i l l Processing. Forest Prod. J. 45:51-53. Hicks, C . R . 1993. Fundamental Concepts In The Design O f Experiments. 4 ed. Saunders College Publishing. Montreal. th  Huber, H . A . , H . N . Rosen, H . A . Stewart, and S.B. Harsh. 1983. A Financial Analysis O f Furniture Parts From Short Bolts. Forest Prod. J. 33:55-58. K l i n e , D . Earle, C . Regalado, E . M . Wengert, F . M . Lamb, and P . A . Araman. 1993. Effect O f Hardwood Sawmill Edging A n d Trimming Practices O n Furniture Parts Production. Forest Prod. J. 43:22-26. Klinkhachorn, P., J.P. Franklin, C . W . M c M i l l i n , and H . A . Huber. 1989. A L P S : Y i e l d Optimization Cutting Program. Forest Prod. J. 39:53-56. Mandel, J. 1984. The Statistical Analysis O f Experimental Data. Dover Publications, Inc., N e w York. Maristany, A . G . , C . C . Brunner, and J.D. Anderson. 1990. Effect O f A n Exponential Weighting Function O n Random Width Dimension Yields. Oregon State University, Corvalis, Oregon. (Unpublished). N o w a k o w s k i , K . L . 1984. Random Sampling Process For Improving Lumber Cut-up Recovery. Forest Prod. J. 34:23-24. Spiegel, M . R . 1992. Schaum's Outline O f Theory A n d Problems O f Statistics. 2 M c G r a w - H i l l , Inc. Toronto, Ontario.  nd  ed.  Sternstein, M . 1996. Statistics. Barron's Educational Series, Inc. Hauppauge, N e w York. Thomas, E . R. 1996. Prioritizing Parts From Cutting B i l l s When Gang-Ripping First. Forest Products Journal. In press. Walpole, R . E . 1974. Introduction To Statistics. 2 Toronto, Ontario. 78  nd  ed. Collier M a c M i l l a n Canada, L t d . ,  Wiedenbeck J. and C . Scheerer. 1996. A Report O n Rough M i l l Y i e l d Practices A n d Performance - H o w W e l l A r e Y o u Doing? In Hardwood Symposium Proceedings, 8-11 M a y , Cashiers, N . C . Wodzinski, C . and E . Hahm. 1966. A Computer Program To Determine Yields O f Lumber. U D S A Forest Service., F P L Unnumbered Publ., For. Prod. Lab., Madison, W i s .  79  A P P E N D I X I. DATA FROM BOTH GRADES OF LUMBER  (all prices in dollars/1000 board feet, ratio = std/#2 btr./ util/#3 ) week  std/#2 btr.  util/#3  ratio  week  std/#2 btr.  util/#3  ratio  1/6/95  330  235  1.40  6/30/95  275  185  1.49  1/13/95  325  220  1.48  7/7/95  290  185  1.57  1/20/95  320  220  1.45  7/14/95  309  190  1.63  1/27/95  315  215  1.47  7/21/95  295  180  1.64  2/3/95  310  215  1.44  7/28/95  290  175  1.66  2/10/95  315  215  1.47  8/4/95  283  180  1.57  2/17/95  320  215  1.49  8/11/95  285  185  1.54  2/24/95  320  215  1.49  8/18/95  305  200  1.53  3/3/95  315  210  1.50  8/25/95  300  200  1.50  3/10/95  315  210  1.50  9/1/95  315  195  1.62  3/17/95  310  210  1.48  9/8/95  320  200  1.60  3/24/95  308  210  1.47  9/15/95  320  195  1.64  3/31/95  290  210  1.38  9/22/95  320  200  1.60  4/7/95  290  210  1.38  9/29/95  325  200  1.63  4/13/95  285  210  1.36  10/6/95  320  195  1.64  4/21/95  290  210  1.38  10/13/95  315  180  1.75  4/28/95  285  205  1.39  10/20/95  290  180  1.61  5/5/95  290  205  1.41  10/27/95  290  190  1.53  5/12/95  285  205  1.39  11/3/95  300  200  1.50  5/19/95  290  200  1.45  11/10/95  305  200  1.53  5/26/95  280  200  1.40  11/17/95  280  185  1.51  6/2/95  278  195  1.43  11/24/95  268  180  1.49  6/9/95  260  185  1.41  12/1/95  260  190  1.37  6/16/95  260  185  1.41  12/8/95  290  190  1.53  6/23/95  270  185  1.46  12/15/95  300  200  1.50  Canadians  U.S. $  average std/#2 btr. price (A)  298.12  417.37  average util/#3 price (B)  199.10  278.74  average ratio (A/B)  1.50  80  A P P E N D I X II. PROJECT CUTTING BILLS 3" Width Cutting B i l l 1 Grade  Length  3 4  Kicker  Demand  Cut  Complete  6.5  1  1100  0  0  0.14  35  7  50  0  0  0.015  Price  1  36  3  430  0  0  1.81  3  36  5  320  0  0  0.85  1  72  7  270  0  0  3.34  4  84  7  10  0  0  0.07  1  144  0  50  0  0  12.49  1  168  0  60  0  0  15.57  1  192  0  60  0  0  19.24  Price  3" Width Cutting B i l l 2 Grade  Length  Kicker  Demand  Cut  Complete  3  6.5  1  310  0  0  0.14  4  35  7  150  0  0  0.025  1  36  3  970  0  0  1.87  2  36  3  550  0  0  1.87  3  36  5  1800  0  0  1.48  1  48.425  6  220  0  0  3.06  1  60.236  7  220  0  0  3.8  1  72.047  7  590  0  0  4.55  4  84  7  10  0  0  0.07  1  96  0  510  0  0  6.5  2  96  0  60  0  0  5.2  2  120  0  200  0  0  12.41  1  144  0  120  0  0  14.91  2  144  0  120  0  0  14.89  1  168  0  40  0  0  17.39  1  192  0  40  0  0  19.88  81  3" W i d t h C u t t i n g B i l l 3 Grade  Length  Kicker  Demand  Cut  Complete  Price  3  6.5  1  2300  0  0  0.14  1  24.803  6  1000  0  0  1.57  4  35  7  80  0  0  0.025  1  36  3  650  0  0  1.82  2  36  3  470  0  0  1.82  3  36  5  1800  0  0  1.5  1  48.425  6  490  0  0  3.07  1  60.236  7  390  0  0  3.82  1  72.047  7  980  0  0  4.56  4  84  7  20  0  0  0.07  1  96  0  700  0  0  7.4  2  96  0  200  0  0  7.4  1  144  0  210  0  0  12.75  2  144  0  180  0  0  12.75  1  168  0  10  0  0  15.47  1  192  0  10  0  0  19.27  3" W i d t h C u t t i n g B i l l 4 Grade  Length  Kicker  Demand  Cut  Complete  Price  3  6.5  1  1200  0  0  0.14  1  36  5  4600  0  0  1.81  2  36  5  3500  0  0  1.81  3  36  3  5000  0  0  1.49  1  48.425  6  1500  0  0  3.01  1  60.236  6  1500  0  0  3.75  1  72.047  7  1300  0  0  4.48  4  84  7  60  0  0  0.07  1  96  0  120  0  0  6.46  2  96  0  430  0  0  4.96  1  144  0  120  0  0  12.49  2  144  0  170  0  0  12.49  1  168  0  40  0  0  15.52  2  168  0  30  0  0  15.52  1  192  0  30  0  0  19.24  2  192  0  10  0  0  19.24  82  3" W i d t h Cutting Bill 5 Grade  Length  Kicker  Demand  Cut  Complete  Price  3  6.5  1  2200  0  0  0.14  1  36  5  1300  0  0  1.81  2  36  5  460  0  0  1.81  3  36  3  920  0  0  1.49  4  48  7  140  0  0  0.02  1  48.425  6  50  0  0  3.01  1  60.236  6  50  0  0  3.75  1  72.047  7  40  0  0  4.48  4  84  7  120  0  0  0.07  1  96  0  200  0  0  6.46  2  96  0  50  0  0  4.96  1  144  0  40  0  0  12.49  2  144  0  30  0  0  12.49  1  168  0  10  0  0  15.52  1  192  0  10  0  0  19.24  3" Width Cutting Bill 6 Grade  Length  Kicker  Demand  Cut  Complete  Price  3  6.5  1  1900  0  0  0.14  1  36  5  460  0  0  1.82  2  36  5  1300  0  0  1.6  3  36  3  5000  0  0  1.5  1  37.008  4  920  0  0  2.01  4  48  7  160  0  0  0.019  1  48.425  6  300  0  0  3.07  1  56.693  6  410  0  0  3.08  1  60.236  6  190  0  0  3.82  1  72.047  7  150  0  0  4.56  1  76.378  7  280  0  0  6.12  4  84  7  190  0  0  0.07  2  96  0  50  0  0  4.98  1  144  0  20  0  0  12.75  2  144  0  40  0  0  12.51  83  3" Width Cutting Bill 7 Grade  Length  Kicker  Demand  Cut  Complete  3  6.5  1  4400  0  0  0.14  4  35  7  500  0  0  0.025  Price  1  36  3  2900  0  0  1.81  2  36  3  2800  0  0  1.81  3  36  5  5200  0  0  1.49  3  40  5  750  0  0  1.655  1  48.425  6  1300  0  0  3.01  1  60.236  7  1400  0  0  3.75  1  72.047  7  1850  0  0  4.48  4  84  7  120  0  0  0.07  1  96  0  50  0  0  6.46  2  96  0  340  0  0  6.46  1  144  0  880  0  0  12.49  2  144  0  500  0  0  12.49  1  168  0  150  0  0  15.52  2  168  0  50  0  0  15.52  1  192  0  320  0  0  19.24  2  192  0  60  0  0  19.24  3.75" Width Cutting Bill 1 Grade  Length  Kicker  Demand  Cut  Complete  Price  3  6.5  1  10000  0  0  0.14  1  36  3  4000  0  0  2.59  2  36  3  1600  0  0  2.59  3  36  5  3900  0  0  1.4  4  48  7  110  0  0  0.04  1  49  3  1600  0  0  4.96  1  51  6  400  0  0  5.16  1  55  6  560  0  0  5.56  1  59  6  2700  0  0  5.97  1  72  7  243  0  0  8.5  1  76.378  7  670  0  0  9.58  1  78.74  0  2400  0  0  9.29  2  78.74  0  100  0  0  8.334  1  82.677  0  2100  0  0  9.76  4  84  7  60  0  0  0.08  2  96  0  300  0  0  10.17  1  120  0  600  0  0  13.41  1  144  0  180  0  0  15.25  2  144  0  450  0  0  15.25  1  157.48  0  20  0  0  17.88  1  168  0  110  0  0  15.45  2  168  0  10  0  0  17.79  1  192  0  70  0  0  19.15  84  3.75" W i d t h C u t t i n g B i l l 2 Grade  Length  Kicker  Demand  Cut  Complete  Price  3  6.5  1  8800  0  0  0.14  1  36  3  2200  0  0  2.46  2  36  3  2500  0  0  2.59  3  36  5  3700  0  0  1.4  1  37.008  6  1200  0  0  3.17  4  48  7  80  0  0  0.04  1  49  3  2100  0  0  4.96  1  56.693  7  210  0  0  5.83  1  59  6  500  0  0  5.97  1  61  6  890  0  0  6.18  1  63  7  700  0  0  6.38  1  72  7  500  0  0  8.5  1  76.74  7  1250  0  0  11.34  1  78.74  0  1800  0  0  9.25  4  84  7  40  0  0  0.08  2  96  0  240  0  0  7.3  1  120  0  90  0  0  13.59  2  144  0  350  0  0  12.75  1  157.48  0  220  0  0  18.12  85  A P P E N D I X III. DOWNTIME DATA (** - R e g u l a r breaks were subtracted f r o m downtime) Item  Date  Shift Runtime (d/n)  3" Width  6/3/97  d  3" W i d t h  6/3/97  n  3" Width  6/4/97  3" W i d t h  Adjusted Runtime  Worktime  Waiting Downtime  14:38:31  3:29:05  2:19:00  25.00  1:10:05  0:25:05  17:47:35  3:09:04  2:27:43  25.00  0:41:21  0:26:21  d  11:58:47  8:00:00  3:01:18  66.67  4:58:42  0:28:42  6/5/97  d  8:01:23  8:01:23  5:45:12  66.67  2:16:11  1:16:11  3" W i d t h  6/6/97  d  8:38:26  8:00:00  4:51:17  66.67  3:08:43  2:08:43  3" Width  6/6/97  n  2:17:58  2:17:58  1:04:01  16.67  1:13:57  0:28:57  3" W i d t h  6/6/97  n  2:37:00  2:37:00  0:57:35  16.67  1:39:25  0:24:25  3.75" Width  6/11/97  d  8:39:24  8:00:00  5:18:09  66.67  2:41:51  1:41:51  3.75" Width  6/13/97  d  8:32:52  8:00:00  5:23:19  66.67  2:36:41  1:36:41  3.75" Width  6/13/97  n  0:47:09  0:47:09  0:24:36  0.00  0:22:33  0:07:33  3" Width  6/20/97  d  8:29:46  8:00:00  4:30:48  66.67  3:29:12  2:29:12  3" Width  6/23/97  d  8:24:35  8:00:00  6:20:16  66.67  1:39:44  0:39:44  3" W i d t h  6/23/97  n  10:17:05  9:47:05  7:45:54  75.00  2:01:11  1:01:11  3" W i d t h  6/23/97  n  4:51:48  4:51:48  3:12:28  33.33  1:39:20  0:54:20  3" W i d t h  6/24/97  d  8:35:01  8:00:00  5:22:04  66.67  2:37:56  1:37:56  3" Width  6/24/97  n  5:12:05  5:12:05  3:07:29  41.67  2:04:36  1:19:36  3" Width  6/24/97  n  2:06:22  2:06:22  1:30:55  16.67  0:35:27  0:20:27  3.75" Width  7/3/97  d  8:39:36  8:00:00  5:12:02  66.67  2:47:58  1:47:58  3.75" Width  7/4/97  d  15:34:53  8:00:00  5:33:22  66.67  2:26:38  1:26:38  3.75" Width  7/4/97  n  6:55:10  6:55:10  4:12:13  50.00  2:42:57  1:27:57  86  Random Downtime  **Adjusted Ran Downtime  APPENDIX IV. FINGERJOINTING RULES Dimension  Cutting Bill  Grade  Minimum  Price/Inch  Kicker  Length 3" Width  3" Width  3" Width  3" Width  3" Width  3" Width  3" Width  3.75" Width  3.75" Width  1  2  3  4  5  6  7  1  2  1  8  0.04  2  2  8  0.02  4  1  8  0.05  2  2  8  0.03  2  3  8  0.015  4  1  8  0.05  2  2  8  0.02  2  3  8  0.01  4  1  8  0.04  2  2  8  0.021  2  3  8  0.012  4  1  8  0.04  2  2  8  0.01  2  1  8  0.04  2  2  8  0.02  2  3  10  0.015  4 2  1  8  0.05  2  8  0.02  2  3  8  0.01  4  1  8  0.06  2  2  8  0.02  2  3  8  0.01  4  1  8  0.06  2  2  8  0.02  2  3  8  0.01  4  87  APPENDIX V.  TWO W A Y ANOVA PROCEDURES Spreadsheet (Excel) macros were used to open text files containing results from simulation runs and copy relevant information to the A N O V A matrix. Formulas programmed into spreadsheet cells calculated statistics which helped identify components o f the A N O V A model, shown below: Yjjk M- + ij + Ek(ij) ij = Aj + Bj + A B y =  T  T  where: i = 1, 20 for the 20 cutting algorithms 1, 6 for the 6 cutting bills j = 1,2,3 for the three observations i n each i , j treatment combination = cell value or observation = population mean T:: treatment effects: cutting algorithm effect cutting b i l l effect = AB, = interaction error within each o f the 120 treatments  -k(ij)  A l l coordinates o f an A N O V A table were covered by the spreadsheet data analysis program. Locations o f totals from columns ( T ) , rows ( T j ) , and the entire table (T ) are shown below. L  Locations O f T w o W a y A N O V A Data Y = one replication/observation ijk  CUTTING  ALGORITHM  1  2  3  4  20  1  (Y )*3  (Y )*3  (Y )*3  (Y )*3  (Y )*3  T.i.  CUTTING  2  (Y )*3  (Y )*3  (Y )*3  (Y )*3  (Y )*3  T.2.  BILL  3  (Y )*3  (Y )*3  (Y )*3  (Y )*3  (Y )*3  T.3.  4  (Y )*3  (Y )*3  (Y )*3  (Y )*3  (Y )*3  5  (Y )*3  (Y )*3  (Y )*3  (Y )*3  (Y )*3  6  (Y )*3  (Y )*3  (Y )*3  (Y )*3  (Y )*3  T .  T20.  T...  ijk  ijk  ijk  ijk  ijk  ijk  ijk  ijk  ijk  ijk  ijk  ijk  T„ 2  ijk  ijk  ijk  ijk  ijk  ijk  ijk  ijk  ijk  ijk  ijk  ijk  T ..  T„  4  3  88  ijk  ljk  ijk  ijk  ijk  ijk  T.4. T.5. 6  Sums o f squares (SS) and mean squares ( M S ) data were then generated from the statistics by formulas in the table below:  Sums O f Squares and M e a n Squares Formulas Source  degrees o f freedom  Sums O f Squares (SS)  M e a n Squares (MS)  Cutting Algorithms  T  A,  Ai  T  2  Cutting B i l l s Bj  ss  y ' i nb nab bT T •y 1- '.. j na nab a bT a rp2 b T  a-l  b- 1  a-l  2  ss  Bi  1  A X B Interaction  r  2  r  b-\ 2  n->2  ( a - l ) ( b - 1)  Error k(ij)  E  Total  ijK abn  j nb j na a b T  i j k abn  i j rp2  r  SS .  A Bj  nab  2  a b ( n - 1) n  Hij) ab{n -1) E  abn - 1  F values for all sources (cutting bills, cutting algorithms, and interaction) were derived by dividing mean square error (MSe  k{ij)  ) into the M S values. For each source, critical F values at a =  5% (significance level) were extracted from an F table by using degrees o f freedom from error and source. Differences within an A N O V A source (interaction, factors) were considered to be statistically significant i f calculated F values exceeded critical values.  89  APPENDIX VI.  BARTLETT'S TEST PROCEDURES/EXAMPLE Excel spreadsheets, using the steps below, facilitated Bartlett's calculations (courtesy o f Walpole (1974)): •  compute k sample variances s ,, s , 2  ,s\  2  2  from samples o f size n,, n ,  ,n  2  k  with ^ ] « . = N (  ;=i  •  combine the sample variances to give the pooled estimate  I=I  N-k •  calculate the b statistic  b = 2.3026f h where: k q = (N-k)\og  s  ^ - l ) l o g sf  2 p  and  h= l+  1=1  Z/=i '  •  choose an a (eg. 5%) level o f significance and a critical chi-square value (B)  •  i f b>B reject hypothesis; otherwise fail to reject  1  1  -1  N-k  Example: Take four value/fbm variances from an A N O V A table as shown in the table below. A N O V A Table For Bartlett's Example - each o  2  based on 21 (3 per cell * 7 cutting bills) algorithm observations  ALGORITHM 1  2  3  4  1  3 obs 3 obs 3 obs 3 obs  2  3 obs 3 obs 3 obs 3 obs  CUTTING  3  3 obs 3 obs 3 obs 3 obs  BILL  :  3 obs 3 obs 3 obs 3 obs  :  3 obs 3 obs 3 obs 3 obs  7  3 obs 3 obs 3 obs 3 obs  o\  90  o\  *\  a  2 4  hypothesis:  o'<*\ = 2 3 4 H : The variances are not all equal  H  a  =  C  T  =  C  f  ]  n, = n = n = n = 2 1 , N = 84, k = 4 a = 0.05, therefore B = 7.81 a , = 8.469E-05, a = 1.344E-05, a 2  3  2  4  2  2  3  = 1.236E-05, a  2 4  = 3.999E-05  (21)(8.469£ - 05) + (21)(1.344£ - 05) + (21)(1.236£ - 05) + (21)(3.999£ - 05)  2 5  2  " "  80  = 0.000038  q  = (80)log(0.000038) - [(21)log(8.469E-05) + (21)log(1.344E-05) + (21)log(1.236E-05) + (21)log(3.999E-05) ] = 11.029478  h  = 1.020833  _ (2.3026X11-029478) 1.020833 = 24.878180 b > B or 24.878180 > 7.81  therefore, reject H , variances are not equal 0  91  A P P E N D I X VII.  SNK PROCEDURES/EXAMPLE B e l o w are the steps that were taken each time a S N K test was done (courtesy o f Hicks (1993)): •  Arrange the 20 means i n order from lowest to highest.  •  Calculate standard error o f the mean (  I Y-i •  )  for each treatment:  error mean  V number of  square  observations iny  Using A N O V A table parameters, get significant ranges from a Studentized range table. B e l o w is a table showing the 19 (number o f means minus one) ranges that were chosen by using an a o f 5% and 240 degrees o f freedom from error mean square. Studentized Range T a b l e # Of Means Compared Range @ 5% a  2  3  4  5  6  7  8  9  10  11  12  13  14  2.77 3.32 3.63 3.86 4.03 4.17 4.29 4.39 4.47 4.55 4.62 4.68 4.74  15 4.8  16  17  18  19  20  4.84 4.89 4.93 4.97 5.01  •  Create a group o f least significant ranges by multiplying by  each range from step 3.  •  Test all 190 possible ranges or combinations o f means. T w o given means are significantly different i f their difference is greater than the least significant range value for the interval under consideration.  •  Draw up a diagram or table showing all the means i n order and their relationships (significant/not significant) to each other.  Example O f S N K Test: Value/fbm means in order: #1 - 0.7585 #2 = 0.7581 #3 = 0.7302 Standard error o f the mean from A N O V A table: 0.00198 Significant ranges based on a = 5% and 30 degrees o f freedom: 2 = 2.89 3 = 3.48 Least significant ranges ( L S R ) : 2 = (0.00198)(2.89) = 0.00573 Test all ranges:  #1 & #2 #1 & #3 #2&#3  3 = (0.00198)(3.48) = 0.00690 difference = 0.0004 difference = 0.0283 difference = 0.0279 92  < L S R (.00573) > L S R (.00690) > L S R (.00573)  not different different different  rS  in  —  —  (N  (N  ©  \o  >n  OO ©  rs  rs  rs  in oo o rs  rs  oo — m rs  —  ON  vD rs  ON  r—  TJ-  vo rs  ON  r-»  r-  r- r > N O m ON OO NO v~» N O rs rs rs  rs m  ON  «n O  «n rs  o  TT  m  NO —  in © o rs  cn  ON  oo  CN  m o rs  O  rON  oo r-» oo  rr-  ON  OO  ON  r-  (N  in oo  p- ©  OS NO  rs  rs  r-  —  «n  —  ON  oo  ON  ON ON  co  ON ON  Ko r-  cn  ©  >n rs  r-  O  ON  ON  rs  rs cn  o  r-  o  o  rs  ON NO  to  rs  — NO  NO  ©  ci rrs  *0  x -  O  NO  o  ON  NO  NO  2  ©  (S  rs  o rs  rs  ON  CO  m  rs  r-» rs  ©  ON  ©  rs  —  rs  rs  o oo oo —  — cn o rs  oo —  © rs  rs  cn  OO  T^-  rs  rs  rs  rr —  rs r-»  ro  rs  rs  o rs Irs  oo r-  rs oo  «n  rs  H  >  H  Q Z W  H  X  PH PN  a H Q  5  o  NO  rs  ON NO NO  Vi  O NO ^O  V I O NO  rs  rs  rs o  rrs  ON  NO  cn  r-»  m  rs  —  in  «n cn  ON  cn oo  <N  ON ~  ON —  rs  o  rn  Tt  m I rs  rs  —  ON  —  rNO  rs  r-  ,  in rs  prs  o  © «n —  rs  m rs oo  NO ON  i NO  cn  ON  1*^ -  I  I  ON  rs  NO  m  Irs  r-  —  co  "  rs  —  rs  NO  in "n rs  TJ-  co  OO  rs  OS  —  oo  O  O ON NO  o  i—  rs  rs  rs  O «n oo oo oo O N  m r-  m  NO  rs  -  rs  oo —  rs  ,  NO NO  rs  NO  ©  rs  V I ON  NO  ON  NO  rs  oo rs  ON  Vi  rs  rs — ©  —  -  in  rs  —  oo  —  —  OO  ©  —  ~ Irs  oo —  —  ON  rs —  © rs  O  2  z  U  h  S3  0.5980  0.9446  0.6914  0.6938  0.6921  0.5995  0.5917  0.5900  0.6291  0.6323  0.6282  0.9444  0.6907  0.6964  0.6923  0.6036  0.5992  0.5944  0.6278  0.6316  0.6264  0.9287  0.9398  0.9290  0.6267  0.6329  0.6276  0.5878  0.5869  0.9544  0.9347  0.9301  0.5460  0.75567 0.75482 0.75423  0.9287  0.6923  0.9544  0.9516  0.9287  0.6940  0.9462  0.9468  0.9290  0.6905  0.9289  0.9303  0.9398  0.9446  0.9355  0.9386  0.9290  0.6050  0.9462  0.9301  0.9314  0.9398  0.6854  0.9286  0.5460  0.9292  0.9292  0.9292  0.9402  0.9294  0.6093  0.6155  0.6122  0.5853  0.5828  0.5975  0.6642  0.6677  0.6582  0.9391  0.9496  0.5648  0.9415 0.9293  0.9304  0.9313  0.6280  0.6322  0.6288  0.5905  0.5874  0.6042  0.6710  0.6759  0.6859  0.9575  0.9664  0.9503  0.9291  0.9355  0.9266  0.5460  0.5556  0.9397  0.9315  0.6249  0.6313  0.6273  0.5959  0.6005  0.6034  0.6787  0.6854  0.6709  0.9448  0.9526  0.9289  0.9289  0.9386  0.9298  0.5460  0.5556  0.5648  0.9293  0.9415  0.9313  0.6298  0.6331  0.6301  0.5863  0.5867  0.5991  0.6652  0.6676  0.6830  0.9634  0.9797  0.9594  0.9290  0.9346  0.9264  0.5460  0.5556  0.5648  0.9052  0.9272  0.9167  0.6210  0.6261  0.6212  0.5939  0.5972  0.6014  0.6648  0.6852  0.6681  0.9228  0.9307  0.8992  0.9158  0.9296  0.9208  0.4553  0.4550  0.4763  0.9157  0.9374  0.9164  0.6206  0.6294  0.6220  0.5883  0.5721  0.6018  0.6568  0.6831  0.6798  0.9415  0.9469  0.9290  0.9224  0.9271  0.9166  0.4553  0.4550  0.4763  0.9157  0.9359  0.9164  0.6084  0.6206  0.6066  0.5850  0.5701  0.6005  0.6490  0.6543  0.6440  0.9464  0.9652  0.9405  0.9206  0.9265  0.9160  0.4553  0.4550  0.4763  0.9588  0.9324  0.9336  0.9431  0.9307  0.6257  0.6319  0.6278  0.5952  0.5989  0.6042  0.6783  0.6892  0.6879  0.9625  0.9504  0.9338  0.9416  0.9326  0.6264  0.6325  0.6283  0.5898  0.5899  0.6057  0.6718  0.6904  0.6860  0.9492  0.9183  0.9327  0.9209  0.6148  0.5661  0.5757  0.9296  0.9115  0.9370  0.9322  0.6148  0.5661  0.5757 0.5757  0.9338  0.9423  0.9326  0.6246  0.6305  0.6247  0.5873  0.5873  0.5998  0.6659  0.6691  0.6616  0.9759  0.9816  0.9752  0.9228  0.9297  0.9196  0.6148  0.5661  0.9100  0.9312  0.9357  0.6264  0.6320  0.6287  0.5941  0.5954  0.6023  0.6779  0.6855  0.6708  0.9370  0.9495  0.9196  0.9202  0.9355  0.9314  0.6148  0.5661  0.5757  0.9103  0.9326  0.9290  0.6271  0.6336  0.6292  0.5960  0.5874  0.6026  0.6708  0.6898  0.6837  0.9594  0.9623  0.9482  0.9245  0.9355  0.9224  0.6148  0.5661  0.5757  0.9350  0.9242  0.9104  0.9331  0.9290  0.6212  0.6255  0.6199  0.5869  0.5846  0.5987  0.6651  0.6699  0.6603  0.9759  0.9815  0.74760  0.9322  0.9382  0.9359  0.6123  0.6189  0.6153  0.5880  0.5897  0.5961  0.6908  0.6945  0.6911  0.9009  0.9073  0.9103  0.9407 0.9745  0.9364 0.9227  0.5460  0.5556  0.5644  0.9135  0.6148  0.5661  0.5757  0.75416 0.75027 0.74685 0.75285 0.75418 0.75438 0.73017 0.73302 0.72897 0.75696 0.75847 0.75814 0.75428 0.75719 0.75493  0.9402  0.9294  0.6132  0.6189  0.6155  0.5895  0.5829  0.6048  0.6662  0.6881  0.6825  0.9391  0.9496  0.9461  0.9288  0.9294 0.9461  0.9344  0.9283  0.5460  0.5556  0.5644  0.9367  0.9283  0.5460  0.5556  0.5644  0.9402  0.9294  0.6138  0.6322  0.6252  0.5952  0.5981  0.6942  0.6881  0.9391  0.9497  0.9464  0.9313  0.9379  0.9310  0.5460  0.5556  0.5644  0.5460  0.5644 0.5556  0.5556  o  0.5644  5  0.5556  H  0.5644  s X ~  z <  S  0.5644  0.74746  0.9318  0.9385  0.9358  0.6123  0.6189  0.6153  0.5880  0.5897  0.5961  0.6909  0.6944  0.6910  0.9027  0.9061  0.9079  0.9347  0.9404  0.9362  0.5460  0.5556  APPENDIX X. WIDTH V A L U E / F B M - BARTLETT'S TEST  #  s  2  s *(# obs -1)  (log(s ))*(# obs -1)  2  2  1  0.027682967  0.55366  -31.15575  2  0.027845926  0.55692  -31.10477  3  0.027999301  0.55999  -31.05706  4  0.027814767  0.55630  -31.11449  5  0.028569014  0.57138  -30.88209  6  0.029185644  0.58371  -30.69661  7  0.027690377  0.55381  -31.15342  8  0.028756861  0.57514  -30.82517  9  0.029578732  0.59157  -30.58041  10  0.031701786  0.63404  -29.97833  11  0.033742136  0.67484  -29.43655  12  0.035403935  0.70808  -29.01897  13  0.025506045  0.51012  -31.86714  14  0.026696524  0.53393  -31.47091  15  0.028706661  0.57413  -30.84035  16  0.025219877  0.50440  -31.96514  17  0.026233894  0.52468  -31.62274  18  0.028136308  0.56273  -31.01466  0.026769945  0.53540  -31.44705  0.53446  -31.46228  19 20  0.026723061  95  m „ NO v> „ ,  n  >t  ifl  lOO  X  3 z w OH OH  H  o o  Q Q W Q Q  -J <  I  "  I x  T.  _  2 H  T.  o  ON NO m I m r; — |  If  r  oo  I NO  < in  Oi <  -  I  <-i oo u i ON r s  I  — m oo <r> ON (N I -  ON  •*  U o  ON  ION ON  OO "TN f N OO ( N i n •/-> »n  oo V I (N 1 oo rN ui v i ini  oo v i oo Ul  VI  rs Ull  Z  3  u  A P P E N D I X XII. 3 " W I D T H N U M B E R O F 20' C L E A R S A D D E D ( I N T E R V E N T I O N ) - B A R T L E T T ' S TEST  #  n  1/n  n-l  1  s  2  s*(# obs -1)  (log(s))*(# obs -1)  2  2  12  0.083333333  11  5.00E+04  549998.917  51.68866  2  13  0.076923077  12  4.77E+04  572712.769  56.14507  3  13  0.076923077  12  4.77E+04  571959.231  56.13821  4  16  0.0625  15  4.00E+04  600471.750  69.03602  5  14  0.071428571  13  3.84E+04  499089.882  59.59506  6  15  0.066666667  14  4.33E+04  605977.733  64.90860  7  14  0.071428571  13  4.14E+04  538654.000  60.02576  8  14  0.071428571  13  3.86E+04  502438.857  59.63282  9  13  0.076923077  12  4.12E+04  494373.077  55.37848  10  14  0.071428571  13  3.76E+04  488592.929  59.47505  11  17  0.058823529  16  3.32E+04  531270.471  72.33913  12  15  0.066666667  14  3.74E+04  523470.933  64.01870  13  14  0.071428571  13  4.36E+04  566563.714  60.31097  14  16  0.0625  15  3.61E+04  541313.938  68.36037  15  15  0.066666667  14  3.87E+04  541965.333  64.22981  16  13  0.076923077  12  5.78E+04  694153.692  57.14729  17  16  0.0625  15  4.07E+04  610467.938  69.14357  18  15  0.066666667  14  4.40E+04  615985.333  65.00819  19  6  0.166666667  5  5.47E+04  273477.500  23.68976  20  6  0.166666667  5  6.31E+04  315740.833  24.00180  97  APPENDIX XIII.  3" WIDTH 20' CLEARS ADDED (INTERVENTION) - SNK  19  20  7  10  12  13  15  16  1  2  18  9  — — — — — — — — — .... .... .... — .... .... — .... .... .... — .... .... — .... .... .... .... .... — .... — .... .... .... .... .... — — — .... .... — — — — .... .... .... .... .... .... — .... .... — — — — .... — .... .... — .... — .... — — .... .... .... — — .... .... .... .... .... — .... — .... — .... .... — — .... — .... — .... .... .... — .... .... .... .... — .... .... .... — .... .... — ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  . . . .  ....  3  6  11  8  .... .... .... ....  —  ....  — -  >.  ....  17  5  — — — — — — — — — — — — — — — .... 'Bill — .... .... — .... — .... — — — — — — .... — .... .... — .... .... .... iiiiiiiiis — lltlsl — — — ....  ....  5  17 4  14 8  11 6 3 2  18 9 1  ....  /  4  ....  ....  ....  14  ....  16 15 13 •  12 10 7  20 19  98  rss oo rs r rss r rs rs oooooo  iifin oo po-\  ON  ON  ON  o  o oor- rs rso rsrsrs Oo o o or» ON  ON V") ON ON  —  ON C N ON  00  —  CN  Ci  v int  rss oo r rss r rs rs ON  ON  NO  Cl Cl Cl  NO NO P -  p-  r-  OO  rs o rs rrss rrss  — — 'n p- p- pr-s rs m rs rs rsrs rs rsrsr rs r- rs rs ro ci ci s  ON  ON O N ©O O N ON ON P —  NO  NO NO — C l NO  NO NO —  ON ON ON NO  CO  ON  —  OO ON  V) VI ON  in m oO o v-» ci oo r rss r rss rsin ci p"- r-rs ci  •m n ON  ON ON  rf  o  r--  ON  O  —  ON — ON OO  rf  ON  _  ON  rf  ci ci  im n  v ~i m  ON NO  ON O N OO ON P-  ON  —  O  —  ON ON OO ON P-  NO ON ON OO NO P -  V)  ON  ON  ON  r-  ON  ON  ON  OO V£5  —  rf  ON  OO ON  p-  ON  5  o o X X  Q Z •W PH  <  ON  ON  ON  ON  ON  ON  ON  ON  ON  ON  P -O rf ON  rf ci  — GO Cl Cl  o  o  ON  ON  ON  ON  P - rf ON  o >n m O©— i•n n o oo o oo ON NO  P-  —  Cl Cl Cl  ON ON NO •  ON  —  ON  ON c i r f c i ON — rf ci c i c i ct  v -» oor- o o ci •n r rss rss r rss r —  ON  ON  ON  m  ON  VI ON  OO ON  ON ON ON  >n  ON OO ON  I— ON ON  V> ON  —  ON  CI  CO  ON  ci  ON  ON  ON  ON  —  ci  ON  O  O  ON rf  O ON  p- p-  ON  ON  ON  ON  ON  ON NO  ci  CO  NO  O0  OO CO  in — oo—  ON NO  NO  CO  —  Cl Cl Cl  P-  NO  OO OO  CO  P^  p- rr S i•n oo s— n oor- r r ssciso •n p- rcirs ci rss r rss r rr r- p- pON  ON  ON  C l Cl Cl  ON  rf  POO  Oin©— —  ON  ON  ON  ON  ON  P-  — —- rs  o im n oor- r soo rss or ci r rss r rs rss  ON  co oo  Cl Cl Cl  r s rss rs r rss r rs rs rf ©  —  ON  OO CO  m rs m r rss r rss r rss  P -—  NO  ON  co rs o p- r- p—inrs — •n O — rs — rs rs rs p- p- p-  ON  < n oor-rs o o m r rss r rss r rss ON  ON ON NO  ON  ON  OS OS  m " n cpcc oo o oo oo  NO  co  r-  CO  ON  r s ro so rs m r — rssr cis ci ors p- p- p— - rs — o < n cor-o r s oso rsin rsco o in ci "n r ci rs o rss r rs rs « rss inn oo r rss r rs  ON  P-  NO  Cl Cl Cl  rf  NO  ci  OO NO ' CO OO '  ci ci  NO NO  rf  ON  ON  ci ci  ON  OO  m p- ci irns•n rs ' O©—  rf  ON  PON  vO  C l C| C i  ts oo — s— m socisrrr «n oo r r cis rss r rs r rss r p- pv -n oo rs rs rs— o — «> rs rs rs ON  CO  c o o— —  ci  —o  ON  OO  OO CO  ON r f vO NO  ci ci  CO  oo  P-  in r- isc irsc t rs•rns o oo ac r •n oo cr rs rs rsc i C i—c i rs H ON  Cl  —o —  CI C l C l  rf  ON  rf  ON  rs «n oo oo oo  —  rf  r s rss ci r rss r rs rs ci ci ct m cis rs— o — vi oor*~ ci r rss r rs rs ON  ON  Cl  rf  ON  rs  OO ON OO O0  —o —  P-  NO  (N  rf  ON  ON NO NO CO ON r f  OO  ci ci c i c i c>  r p- o «nssr-rs r rss r rs rsin in r- in p- r-» p•rns r — • rrs— s o rs s ci o ors rs  NO ON ON OO ON P-  ON  ON — ON O N CO OO OO  NO  ON  V-J ON  ON  ON  OO  c i CI C l  ON  OO 0 0  oo  ON  ci  oo oo oo <  u  hi  APPENDIX XV.  WIDTH N U M B E R OF BOARDS PROCESSED - B A R T L E T T ' S TEST  #  n  1/n  n-l  s  2  s *(# obs -1)  (Iog(s ))*(# obs -1)  2  2  1  21  0.04762  20  8.22E+06  164369831.23810  138.29584  2  21  0.04762  20  8.22E+06  164359946.95238  138.29532  3  21  0.04762  20  8.22E+06  164300617.23810  138.29218  4  21  0.04762  20  8.21E+06  164285385.80952  138.29138  5  21  0.04762  20  8.32E+06  166479409.23810  138.40661  6  21  0.04762  20  8.52E+06  170456502.66667  138.61167  7  21  0.04762  20  8.36E+06  167133171.80952  138.44065  8  21  0.04762  20  8.30E+06  165962409.14286  138.37959  9  21  0.04762  20  8.46E+06  169154259.14286  138.54506  10  21  0.04762  20  8.41E+06  168121589.23810  138.49187  11  21  0.04762  20  8.27E+06  165344894.28571  138.34722  12  21  0.04762  20  8.51E+06  170175287.80952  138.59733  13  21  0.04762  20  8.23E+06  164539283.23810  138.30479  14  21  0.04762  20  8.28E+06  165618660.57143  138.36159  15  21  0.04762  20  8.54E+06  170761949.23810  138.62722  16  21  0.04762  20  8.71E+06  174253085.14286  138.80301  17  21  0.04762  20  8.79E+06  175749306.95238  138.87727  18  21  0.04762  20  8.99E+06  179792857.23810  139.07485  19  21  0.04762  20  8.17E+06  163356347.80952  138.24212  20  21  0.04762  20  8.14E+06  162792769.23810  138.21210  100  APPENDIX XVI. 3" W I D T H N U M B E R O F B O A R D S P R O C E S S E D - S N K  11  14  1  2  3  8  15  4  9  — — ....  — — — — .... — .... .... — — — .... — — — — .... — .... .... — .... .... — ....  ....  ....  ....  ....  ....  ....  ....  —  ....  ....  ....  ....  —  — — —  ....  ....  ....  ....  ....  — — — — — .... —  —  ....  ....  .... ....  ....  .... .... ....  ....  ....  13  10  12  7  17  — — — — — — — — .... — — .... — — — — — .... — .... — — — — — .... .... — — — — .... .... — .... .... — — — .... — .... — .... .... .... .... — — .... .... — — — .... .... .... .... — — — — — .... — .... .... — — .... — .... — — .... — .... .... .... — — .... .... .... — — .... .... — — .... — .... .... — — — — — ....  ....  ....  ....  ....  ....  ....  ....  5  18  16  20  ....  :<  T  6 19 20 16 IS 5 17 7  <•.<  12 10 13 9  .4 ' "  4 15  ....  ....  6  — — — — — — — — — — — .... .... — — — — — — —  ....  ....  19  ^§111^  8 3 2 1 14  <  ....  11  101  I— i n oo Os so »n fN rco r-- Tf Tf Os oo so O N T J - oo rs fN so Os rd Os oo od od i n T T Tf r- r» SD so  ro fN O r-^ SO  fN so so Os Os oo OS " n so so OS so sq i n Os Tjv i " n «n" r- CO Os Os rr~-  ro Tf co ro ro oo Os sO r- m Os ro m CO Os oo T T ' Tf ro i n i n TT' r-~r- r- r-  TT Tf m  r- »n ro rO N Tf d Os Tf  rr~sd so  Os Os o Os Os oo Os r- so Os SO r~- •^r m Os so i n <n i n as Os r- r- r- oo r-~ r-  fO oo oo TT" r-  oo Tf m  NO  o  oo Tf CO od Tf  Tt r» oo roo oo ro rs fN so CN ro od od od r- r- r- so so  «n  Tt so o Os O "n «n o fN OO fN CO O >r, Tt <n s6 od r- r~- r~- r- r- r»  as m p m Tt Os d Tf so  SO o «n Tt Os ro in O m Tt fN Os d so m Tf rSO  Os ro so m sO fN <n Os ro ro Os ro d d d od rr~OO  fN so O TT" r-  0O  rT | - oo fN m m CO CO m ro r- rso Os r- OO sO OS fN T T o CO Os o ^> ro ro o O oo fN « n v~~ sq o m CO m r-^ sd TT" i n od d t< sd rr- rr- r- r- r- r~- oo r— OO  fN so O TT' r-  as CO Tf TT" r-  so O m Os ro i n Tt o Os d <n Tf so  oo SO o SO so r-Tt ^ CN in S r- O ro  so O i n Os ro p i n Tt Os d i n Tf so  Os ro OO m r-  so O m Os rO O m Tf OS d m so TT  i— Os oo rfN so fN fN oo fN fN so Os Os O fN Ti- S O SO fN o O O O OO SO oo oo cs fN r- p »n" in od Os od ^r" sd rr- r rCO r- rr-  H  5  o o  fN o oo ro r~- i n ro CO o • n r - »n SO o CO o i n sd sd od d r-- rr- rr~-  Os Os m rs m Os r~- oo CO d d od od rr- r~OO  r- o Os o o oo i n oo sO fN so so fN oo o ro r- m o S O ro sO fN oo ro OO Os ro CO fN rSO so i n r-^ >n d d fN Os d r~- r- rrOO r- oo rr-  m Os d  Os oo oo oo Tt  o  •n  rN Os Tf Os Tt  >  m Os d m  fN Os Tf Os Tf  »n Os Os oo • n Os o CO «n fN Os oo i n Os oo 00 »n p sq sd CO od ro ro ro m rr- r- rTt rr-  X X  Q Z W  o-  C/2  H OH  H U  w a, H Q  CO '3o  oo * n oo fN Ov OO Tf r- Tt r- fN Os Tt oq Os ro p d Os od sd m Tt r- r~- r~Tt oo Tf Os d  m rTt Os Tf  rrt T T fN ro <n oo m Os SD Tf fN r- CO ro sq o oo r- r~ sq fN ro ^r' fN Os d ^f' i n «n rr- rOO r- oo ro  ro '^f m oo >/-.sq fN fN 00 oo  r- Os o »n Os ro OS o ro so fN rOS Os o ro sq SO CO r- CO Os T T ' sd •^t <n r- r- r- r- rr- r— r-  oo «n 00 (N so o m r- SO o oo Os o Tf r - Tf SO ro »n O Os fN oo <n Tf 00 rO N OS ro o Os Tt co Os so ro" ro fN SO d Os CO ro d r- r- r~- roo i n Tt Tf rOO  <r.  r-- «n OO oo ^1fN sO m ro rfN »n 00 OO fN O Os Tf Tf oo ro ro Os so ro d Os CO r-^ sd ro ro fN rm r- r- rTt r-  rro Os d in  m r~Tf Os Tf  Tt so Os r- SO o oo oo ro Os o ro Ti- fN »n CO fN ro ro ON so r-^ SO i n sd in" fN r~- r - rrr~- r - rTT r- i n sO Ti- Tf r- o fN fN ro O in sd so i n ro rrr-  as m TT m r-  oo so ro r-" r-  SO fN o ro Os »n Tf" rr-  m oo sd r-  TT m r~~ Os OO o roo ro r-; »n r- Os Os r-^ od so Tf sd 1— r- r- rr-  fN Tf r~Tf r-  oo fN fN r-  fN r Tf r- oo Tf' fN r- rfN fN m fN r~-  fN so so so fN Os Os o oo CO o fN O0 S O Tf o CO ro ro ro so ro o Os N O O r-- ro OS ro p i n so CO o ro oo fN »n i n in in TT" SO Tt' fN d d CO od CO r- r~r~CO rr~- rrOs oo fN Os r-  r» <n oo ro r- Tf Os Tt oo d Os 00 i n Tf Tt  Os i n so fN so r-d fN r-  r~~ Tf TT as oo oo so oo Os r- ro ro so o o OO sq NO TT" od r-^ r- in" r- ro CO rrr-  so m o r~- TT 00 CO [—  •n  CO Tf oo CO Tt  SO oo fN fN r-  o o r- fN CO i n 00 ON SO OS fN o o r- CO O o O ro ro sq Os SO SO TT Tt Os fN d d r-^ sd TT" or-" i n fN OO oo rr- rOO r- r~-  Os oo CO CO Tf  fN O ro r-^ r- r--  CO m TT* d I— r»  r~-  so fN so Tf fN so Os T T o Os •^r m • n o ro i n SO <n fN o « n o Os fN fN o p o o Tf sq m Tf ro fN i n ro Os ro sd m TT" d fN fN ^r" ro fN d d r- rrr- OO oo r- r- r~- rr-~- rOO r-  fN OS Os T T d Os • n Tt •n  ro fN ro so r~- r- o CO ro m ro" in" in" T T ' r- r- r-- r-  Os o Os r-- m so so oo i n fN od i n SO in" r~rr-  SO o i n fN ro co «n m Os ro r-- m ro Os o m >n O m Tf r- or- ro ro ro ro p sq p OS d »n ro ro i n Tf so r- r» r- r- r- r- r- r-  s  Tf Os Os Tf' r-  r-  r-  co co o fN co OS ro o Os N O o OO OO fN i Tn' N O r-^ r-^ Tt SO TrfN rrrr-  fN o i n fN Os fN n ro ro sq Tf CO r*^ sd TT* d rOO r-  ro O fN fO Os ri n Tf' rr-  oo ro ro CN r-  r~~ S O o so Tf fN i n so r» Os o fN m o • n TT Os ro <n Os m TT Os TT d d od od ro ro ro Tf i n Tf r- r- rrrCO rr-  '^f ro so fN ro d r-  Os OO so r-^  o  SO  oo SD (N so • n <n oo o so ro fN co Tt fN • n fN Os CO fN o o ro i n r- fN o in Tf CO ro sq p sq ro O ro so ro fN fN m TT sq T T Os Tf ro ro" fN i n Tf fN od Os Tt ro ^r' r- rrr- r - r - r CO r r - r~- r r-  oo o fN ro o in «n o Tt <n SO so Os o o SO o fN oo sq 00 fN sq fN CO ro ro fN i n »n ^r" rrr~- rTf rrr-  o  oo fN fN OO  Tf ro oo d oo  Tf T T SO OS Tf fN i n T T o ro • n TT T T ro TT m r- Os O N Tj- m d sd r-^ ro Tf i n Tf fN r- r- r- r- i — r— CO r-  r- • n oo r- so o Os oo ro r-» Tf so S O Os Os Tt oo ro Os sq OS CO d Os od ro ro fN so i n Tf Tt rr- rrr-  sO Os Os fN ro CO o ro fN ro SO so Os »n O Os ro fN so fN p ro ON fO sq sq • n sq i n *r. i n Os m 00 CO Tf" Tf fN r- rrrCO rrrr— r- r-  rro Os d  Tt T T r~- r- ro fN O ro OS so ro ro >n oo Os fN m m O OS fN CO m sq Os p ro Os ro ri n T ] - Tf p • n <n m Os Os od od Tt i n Tt fN r- rCO r- r~- r- r- r - rr- r-  «n  m r— Tt Os Tf  oo Os «n Os Tt oo fN O co p sq Os ro" od ro rTt rr~r-  r- i n 00 o oo r- Os ro ro r- Tf >n so ro m Os Os Tt 00 so CO fN rro" ro d Os od r-" r~«n Tf Tt  00 so oo fN r-  oo fN fN Ti- r~- o r- o o o ro m oo S O ro OS OO ro fN i n o Os oo sq Os sq ro r- Os p ro Os ro in" T T ' fN >n «n r- fN d d Tf r- m rrCO oo CO  rm CO fN  Z  < 3  (N O  APPENDIX XVIII. 3" W I D T H P E R C E N T P A R T S R E C O V E R Y - B A R T L E T T ' S T E S T  #  n 1/n n-l  s  2  s *(#obs-l)  (log(s ))*(#obs-l)  2  2  1  21  0  20  9.92E+01  1983.66058  39.92875  2  21  0  20  9.70E+01  1939.15163  39.73164  3  21  0  20  9.58E+01  1916.63503  39.63019  4  21  0  20  9.77E+01  1954.18030  39.79869  5  21  0  20  9.13E+01  1826.66249  39.21257  6  21  0  20  8.80E+01  1760.40022  38.89163  7  21  0  20  9.70E+01  1939.68685  39.73403  8  21  0  20  9.58E+01  1916.34061  39.62885  9  21  0  20  9.64E+01  1927.46844  39.67915  10  21  0  20  9.94E+01  1988.16097  39.94843  11  21  0  20  9.85E+01  1970.15514  39.86941  12  21  0  20  9.69E+01  1938.09801  39.72691  13  21  0  20  7.59E+01  1517.22625  37.60041  14  21  0  20  7.35E+01  1469.73421  37.32418  15  21  0  20  7.51E+01  1501.98058  37.51269  16  21  0  20  7.48E+01  1496.34968  37.48006  17  21  0  20  7.33E+01  1465.55455  37.29944  18  21  0  20  7.20E+01  1440.70480  37.15090  19  21  0  20  9.78E+01  1956.13215  39.80736  20  21  0  20  9.79E+01  1958.19593  39.81652  103  APPENDIX XIX. 3" W I D T H P E R C E N T P A R T S R E C O V E R Y - S N K  20  19  5  12  18  7  16  9  10  4  8  3  2  17  11  — — —  —  — —  — —  —  —  —  ....  ....  ....  ....  —  — —  — —  ....  ....  —  ....  ....  ....  ....  ....  ....  — —- — — ....  —  ....  ....  ....  —  ....  —  1  ....  ....  ....  ....  ....  ....  ....  — —  —  ....  — —  — — — —  — —  ....  — — — — —  — —  ....  — — — — —  ....  11  ....  ....  ....  —  —  ....  ....  ....  —  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  —  —  —  ....  —  ....  —  —  ....  —  ....  — —  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  —  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  —  ....  ....  ....  ....  —  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  ....  — — —  ....  ....  ....  —  ....  —  ....  ....  ....  ....  —  ....  ....  —  ....  ....  ....  ....  ....  —  ....  ....  ....  ....  6  ....  ....  ....  ....  1  ....  13  15  14  — — —  — —  —  14 15 13  ....  17 2 3 8  ;- i * \ .  •  f  1  4  iiiifc  10 9 '  16  •>  7 18 iSllil  i ... .  K  illllB ,. y :  12 5 19 20  ....  6  104  APPENDIX XX. WIDTH Y E A R L Y V A L U E DIFFERENCES F R O M A L G O R I T H M 14 Algorithm Number  Value/Fbm  Yearly Value Difference From Algorithm 14  1  0.75567  -$16,346.52  2  0.75482  -$21,308.85  3 4  0.75423 0.75416  5  0.75027  -$24,753.30 -$25,161.96 -$47,871.94  6 7 8  0.74685 0.75285 0.75418 0.75438  -$67,838.04  0.73017 0.73302  -$165,216.58 -$148,578.16 -$172,222.23 -$8,815.44  15  0.72897 0.75696 0.75847 0.75814  16 17 18 19 20  0.75428 0.75719 0.75493 0.7476 0.74746  -$24,461.39 -$7,472.69 -$20,666.67 -$63,459.51 -$64,276.84  9 10 11 12 13 14  105  -$32,809.79 -$25,045.20 -$23,877.59  0.00 -$1,926.55  APPENDIX XXI.  E X A M P L E O F 3 W A Y V A L U E / F B M ANOVA SETUP A L G O R I T H M 1 (1 part - 70%)  A L G O R I T H M 2 (1 part - 80%)  A L G O R I T H M 3 (1 part - 90%)  repl.  5 mins.  10 mins.  15 mins.  5 mins.  10 mins.  15 mins.  5 mins.  10 mins.  15 mins.  1  1.346  1.725  1.713  1.237  1.463  1.116  1.361  1.376  1.424  2  1.278  1.021  1.367  1.551  1.370  1.306  1.692  1.279  1.666  3  1.370  1.587  1.486  1.500  1.880  1.610  1.560  1.727  1.771  CUTTING BILL  1  2  3  I  1.418  1.809  1.488  1.521  1.573  2.139  1.657  1.811  1.777  2  1.285  1.835  1.887  2.032  1.267  1.291  1.363  1.054  1.554  3  1.248  1.311  1.561  1.527  1.467  1.655  1.429  1.578  1.752  1  1.522  1.469  1.344  1.349  1.892  1.876  1.741  1.669  1.588  2  1.372  1.385  2.051  1.463  1.722  1.397  1.354  1.477  1.226  3  1.401  1.298  1.905  1.899  1.901  1.599  1.425  1.612  2.099  106  

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