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Determination of economically marginal tree size through the application of conventional and linear programming… Valg, Leonid 1962

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DETERMINATION OF ECONOMICALLY MARGINAL TREE SIZE THROUGH THE APPLICATION OF CONVENTIONAL AND LINEAR PROGRAMMING TECHNIQUES by Leonid Valg B.S.F., University of B r i t i s h Columbia, 1957 A Thesis Submitted i n P a r t i a l Fulfilment of the Requirements for the Degree of MASTER OF FORESTRY in the Faculty of Graduate Studies We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1962 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the l i b r a r y s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or p u b l i -cation of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Faculty of Graduate Studies at the University of B r i t i s h Columbia Vancouver 8, B r i t i s h Columbia. A p r i l 19, 1962. ABSTRACT Various investigators of logging operation efficiency-have stated that the harvesting of small trees is inevitably associated with higher operating costs. A comprehensive survey of literature has been presented to substantiate this fact. The cited information was supplemented, for the purpose of this thesis, by a time study conducted at the University Research Forest, near Haney, B. C , in June, 1961. During this study f e l l i n g , bucking, yarding and loading of timber was studied at two different operations in that Forest. These studies supplied basic data for the computation of the size of the zero marginal tree. It was found that, under existing conditions, the indicated sizes were 12 and 14 i n . d.b.h. for Douglas f i r and hemlock trees, respectively. Further i t was shown that the milling operation constituted the largest cost component, especially penalizing the small dimensions. A new schedule, with certain proposed improvements in operating efficiency, was established. Under this schedule the milling operation was omitted, and the logs were assumed to be the f i n a l , marketable product. The solution of this computation revealed that, under the assumed conditions, the i i i zero marginal limit in terms of d.b.h. for Douglas f i r and hemlock was lowered to 7 and 8 i n . , respectively, provided the logs from such small trees could be sold at the same price as # 3 sawlogs. The shape of the net return function suggests, however, that only around and above 15 in. d.b.h. could the operation be regarded as safely paying i t s way, under current market conditions and restrictions as to minimum log size and length. The technique of linear programming (LP) has been success-f u l l y employed in other sectors of manufacturing and transpor-tation. It i s demonstrated in this thesis that the LP technique may be applied to certain forest harvesting situations. Progressing through three problem situations of increasing complexity, i t is shown how an optimum strategy of action may be established in terms of the economically marginal tree size. The d i f f i c u l t y of obtaining precise time and cost values in sufficient quantity was encountered throughout this work. Consequently, the main purpose of these computations is to ill u s t r a t e the underlying principles of the application of LP, and to demonstrate i t s applicability to certain aspects of forest harvesting problems. This area offers wide scope for future investigation and for improvement of techniques. iv TABLE OF CONTENTS PAGE INTRODUCTION 1 REVIEW OF LITERATURE 4 METHODS AND DEFINITIONS 9 I. PRODUCTION FUNCTION 10 I I . MARGINAL PRODUCTIVITY 11 I I I . ISOQUANTS AND COST FUNCTION 11 IV. PROFIT MAXIMIZATION 12 DETERMINATION OF CONVERSION RETURNS 14 SELLING VALUE OF LUMBER OF VARYING GRADES AND DIMENSIONS 16 LUMBER GRADE OUTPUT FROM LOGS OF VARYING SIZE AND QUALITY 20 MILLING COSTS FOR LOGS OF VARYING SIZES 28 DETERMINATION OF NET VALUE OF LOGS IN THE MILL POND 33 TRANSPORTATION AND LOGGING COSTS BY LOG SIZE 35 A. THE EFFECT OF TYPE OF ROAD ON HAULING COST 35 B. THE EFFECT OF LOG SIZE ON HAULING COST 37 C. LOADING COST 39 D. YARDING COSTS 44 E. HORSE SKIDDING 58 F. NET VALUE OF LOGS IN THE WOODS 60 G. FELLING AND BUCKING 62 H. DETERMINATION OF NET VALUE OF STANDING TREES 75 V PAGE ZERO MARGINAL TREE SIZE FOR AN OPERATION OF IMPROVED PERFORMANCE (PROGRAM II) 82 A. MARKET PRICES OF LOGS 83 B. THE UTILIZED VOLUME PER TREE 85 C. GROSS VALUES OF TREES 85 D. FELLING AND BUCKING 87 E. LOADING COST 87 F. COST OF YARDING 89 G. OTHER COSTS ASSOCIATED WITH PROGRAM II 91 H. COMPUTATION OF NET REVENUES PER TREE 92 I. RESULTS 93 LINEAR PROGRAMMING 96 A. INTRODUCTION 96 B. PRINCIPLES 96 PROBLEM AND SOLUTION NO. 1 97 PROBLEM AND SOLUTION NO. 2 104 SOLUTION OF PROBLEM NO. 2 ' 107 PROBLEM AND SOLUTION NO. 3 109 SOLUTION OF PROBLEM NO. 3 114 DISCUSSION 121 SUMMARY AND CONCLUSIONS 125 BIBLIOGRAPHY 127 APPENDIX I. VANCOUVER LOG PRICES SEPTEMBER 1961 131 vx PAGE APPENDIX I I . SIMPLEX TABLEAU. PROBLEM NO. 2 I 3 3 APPENDIX I I I . ALWAC III-E WORKSHEET. PROBLEM NO. 2 1 3 5 APPENDIX IV. ALWAC III-E WORKSHEETS. PROBLEM NO. 3 1 3 7 APPENDIX V. THE USE OF ALWAC III-E FOR SIMPLEX METHOD 1 4 4 COMPUTATION v i i LIST OF TABLES TABLE PAGE 1. Average Value of Coast Douglas F i r Lumber per Thousand (M) Board Feet (f.b.m.) 16 2. Relative Lumber Prices by Grades for Small Douglas F i r and Young Hemlock 17 3. Average Dressed-Lumber Prices for October-December 1960. Interior of B. C. 18 4. Proposed Price Schedule for Douglas F i r and Hemlock Lumber 18 5. Lumber Recovery Values per M f.b.m. by Log Diameter for Small Douglas F i r and Young Hemlock 20 6. Grade Recovery from Second-Growth Douglas F i r Logs in Per Cent of Green Chain Tally 21 7. Average Percentage of Lumber Grade Recovery by Diameter Classes (Rough Green Basis) for Interior Spruce 22 8. Average Percentage of Lumber Grade Recovery by Diameter Classes for Second-Growth Douglas F i r 23 9. Average Percentage of Lumber Grade Recovery by Diameter Classes for Second-Growth Hemlock 24 10. Lumber Grade Recovery in Per Cent of Green Chain Tally - Young Douglas F i r 24 TABLE PAGE 11. Grade Recovery i n Small Douglas F i r (Unpublished V.F.P.L. Data) 25 12. Lumber Realization Value per M f.b.m. and Per Log of Various Sizes 27 13. Sawing Times i n Minutes per M f.b.m. for Ponderosa Pine Logs of Various Diameters 28 14. E f f e c t of Log Diameter on the Time Required to Saw M f.b.m. of Lumber 29 15. Net Sawing Time of 16-foot Logs per M f.b.m. of Lumber T a l l y 30 16. Lumber Manufacturing Cost i n the Southern I n t e r i o r S e l l i n g Pine Zone (Spring 1960) 31 17. Cost of Sawing per M f.b.m. and per Log of Various Sizes 32 18. Conversion Return per M f.b.m. and per Log of Various Sizes, at the M i l l Pond 34 19. Hauling Cost per M f.b.m. (Reynolds) 39 20. Loading and Hauling Costs per M f.b.m. and per Log of Various Sizes 41 21. Net Value of Logs at the Woods Landing S i t e 43 22. Yarding Times of Various Turn Volumes 46 23. Breakdown of Yarding Time Requirements 46 24. Haulback Times at University Research Forest Salvage Logging Operation, June 1961. 47 TABLE 25. High-lead Yarding Cost per M f.b.m. over an Average Distance of 400 Yards 26. Tractor Skidding Time - Machine Minutes per 100 Cu. Ft. 27. Horse-Skidding Times per Cord of 100-inch Bolts, by Log Diameter Classes and Distance 28. Hourly Production (cu. f t . ) for Horse-Skidding of 8-foot Logs, by Skidding Distance and Log Diameter 29. Cost for Horse-Skidding - Distance 400 Feet 30. Net Value of Logs i n the Woods after F e l l i n g and Bucking 31. Average Daily Productive and Non-Productive Time for 3-man F a l l i n g Crew 32. The E f f e c t of Stump Height Diameter on F e l l i n g and Bucking Time and Cost 33. E f f e c t of Tree Size on F e l l i n g and Bucking Times - I n t e r i o r B. C. 34. Hourly Production per Man, and Cost per M f.b.m., on F e l l i n g and Bucking of I n t e r i o r F i r , Larch and Spruce 35. F e l l i n g and Bucking Costs for Logs of Various Diameters (Worthington & Shaw) PAGE 52 55 58 59 60 61 64 65 65 66 67 X TABLE PAGE 36. Cutting Times i n F e l l i n g and Bucking Operations at the University Research Forest 70 37. Determination of F e l l i n g and Bucking Cost for Douglas F i r Trees of Different Sizes 72 38. Bucking Schedule of Trees of Different Sizes 73. 39. Net Values of Standing Trees 76 40. Summary of Program I Cost Items and Returns 81 41. Approximate Stand Quality 83 42. Bucking Schedule for Program II 84 43. Gross Values of Standing Trees 85 44. F e l l i n g and Bucking Cost per Tree 87 45. Loading Cost per Tree 89 46. Miscellaneous Costs of Program II . 92 47. Program II Costs and Net Revenues per Tree 93 48. Stand Composition and Volume 97 49. Stand Composition and the Expected Percentage No. 2 Logs 104 50. F e l l i n g , Bucking, Yarding and Loading Cost Associated with Problem No. 2 105 51. Direct Charges per Tree i n Problem No. 2 105 52. The Solution for Problem No. 2 108 53. Stand Composition for Problem No. 3 110 54. Yarding Time per Tree for Various Yarding Distances 111 x i TABLE PAGE 55. F e l l i n g , Bucking and Loading Time Requirements per Tree 111 56. Revenue from Log Prices per Tree - Program No. 3 112 57. Summary of Solutions for Problem No. 3 115 58. Summary of Problem No. 4 Solution 120 x i i LIST OF FIGURES FIGURE PAGE 1. E f f e c t of Road Type on Hauling Cost 36 2. E f f e c t of Mean Log Volume on Loading Rate 40 3. E f f e c t of Yarding Distance on High-lead Yarding Time 49 4. E f f e c t of Yarding Distance and Volume on Round-t r i p Time; D-2 Tractor 56 5. E f f e c t of Tree Size on Time Required for F e l l i n g , Limbing and Bucking 63 6. E f f e c t of Tree D.b.h. on F e l l i n g Times 68 7. E f f e c t of Log Diameter on Bucking Time 69 8. Net Value of Trees aft e r F e l l i n g and Bucking 77 9. Net Revenue per Tree; Program I 80 10. F e l l i n g and Bucking Cost; Program II 86 11. Loading Cost; Program II 88 12. Net Revenue per Tree; Program II 94 13. Graphical Solution of Two-dimensional Problem No. 1 101 14. Extrapolation of Net Revenues for Operations of Various Durations - Problem No. 3 117 15. The Most P r o f i t a b l e Time Expenditure on the Operation - Problem No. 3. 118 ACKNOWLEDGEMENTS Award of the University Forest Fellowship i s g r a t e f u l l y acknowledged. In p a r t i c u l a r , my thanks are due to Dr. J . H. G. Smith and Professor F. M. Knapp for t h e i r encouragement and advice. I have appreciated the many comments and new ideas received from them during my work and on the completion of the f i r s t d r a f t , which has helped greatly i t s f i n a l r e v i s i o n . At the University Research Forest, Mr. J . P. Tessier has been of great help i n advising me on the time studies i n the f i e l d . Mr. J . Csizmazia has, as always, given his time and energy i n processing the Linear Programming data on Alwac I I I E Electr o n i c Computer, for which he deserves special thanks. DETERMINATION OF ECONOMICALLY MARGINAL TREE SIZE THROUGH THE APPLICATION OF CONVENTIONAL AND LINEAR PROGRAMMING TECHNIQUES INTRODUCTION The Coastal Region of Br i t i s h Columbia has for decades produced lumber products in sizes and quantities matched only in a few locations elsewhere in the world. The wealth of old-growth stands has created a tradition of logging on a gigantic scale. Both the equipment and methods in the coastal operations have been adapted for handling of large dimensions and high volumes per acre. Accessible locations on tidewater and low stumpage prices created a favourable economic climate allowing a generous profit margin in spite of the relative isolation of this region from the major con-suming centres of the world. As the old-growth stands on the most accessible areas became depleted and loggers moved up the h i l l s and farther inland, distances from tidewater or railroad spur lines increased. Growing transportation problems raised new econo mical implications: in order to make a profit, which trees should be taken and which should be ignored because they would not pay their way? The loggers f e l t intuitively that 2 the largest trees are the best money-earners whereas the handling of smaller trees is less profitable i f not entirely undesirable. In the early days of forest operations in this province this kind of reasoning, together with a relative abundance of unexploited forest areas, and a lack of governmental enforce-ment of a firm forest policy, gave ri s e to a period of indis-criminate removal of old-growth trees. In many instances this was accompanied by destruction of the residual stand. However, with the dwindling ava i l a b i l i t y of virgin stands and increased public knowledge about the situation in the forests, these conditions obviously could not continue to prevail. Logging practices have changed considerably since the early days of lumbering. Changed economic conditions have imposed new problems on the operators in the woods and saw-mills. One basic fact that has remained unchanged i s : the removal and subsequent manufacture of a small tree is s t i l l far more expensive, per unit volume, than an equivalent volume manufactured from a larger tree. It has become apparent that each tree has a unique value under specified conditions such as i t s location, available harvesting and milling technology, and the state of markets for the particular products extracted from that tree. Harvesting of timber is a business enterprise and i f 3 conducted under the "free enterprise" system, i s subject to the profit maximization principle. This principle i s equally valid i f the forest is operated as a public enterprise where in addition to declared monetary profits a number of intan-gible returns may be included. Under the "free enterprise" system these returns may also be met by legislation controlling the operation.of companies for the "public good". In general, the determination of marginal values of trees of various sizes becomes the basis for subsequent planning of an operation. The determination of these values, using various computational approaches, is the chief purpose of this thesis. Although these computations strive to approach actual conditions encountered in the f i e l d , the d i f f i c u l t y in ob-taining accurate data often made i t necessary to resort to certain assumptions. Thus an element of approximation and compromise was introduced wherever no other avenue of approach was possible. Because the author was mainly concerned with a develop-ment of system and method, this approach should not diminish the value of the work. At this time only a reasonable model was looked for, which could easily be improved as more detailed basic data become available. 4 REVIEW OF LITERATURE One of the earlier s c i e n t i f i c investigations into the effects of tree size on logging and manufacturing costs was conducted by Ashe (1916) in Tennessee, Virginia and North Carolina. This study was intended to lend strength to the argument in favor of leaving the small, unprofitable trees to grow to larger sizes for future cutting. The results of this investigation are in principle quite comparable to the operations of later days, allowing for the changed tech-nology. Ashe found that for f e l l i n g and bucking the optimum size for trees was between 30 and 38 inches in diameter and that the cost increased rapidly for trees below 18 inches in d.b.h. In skidding, conducted with teams of horses, logs averaging 8 inches in diameter were nearly three times as costly as logs averaging 24 inches in diameter. Similar trends were also noted in sawing logs of different diameters. In the Douglas F i r Region, one of the f i r s t comprehensive studies on the cost of f e l l i n g and bucking by different tree sizes was published by Rapraeger (1931). In his study, which primarily was concerned with the establishment of r e a l i s t i c rates of pay for the workers in the woods, Rapraeger established production rate schedules for Douglas f i r and hemlock trees of various stump diameters and also investigated the effect of 5 different wage systems on the rate of output. In a separate study carried out in Eastern Oregon, Rapraeger (1932) investi-gated the efficiency in sawmilling and the most economic use of timber. Within this broader framework of study, considera-tion was given also to the effect of log size on sawing time. It was found that the cost of making lumber from a 6-inch log was more than three times the unit cost for a 30-inch log. In a later study in Idaho, Rapraeger (1936) compared the output of fa l l e r s and buckers in the Idaho White Pine Area with loggers working in the Douglas F i r Region. He found that, whereas in the latter region a daily output of two fallers and two buckers was 30,000 to 40,000 f.b.m. of logs, giving a per-man-day output of 7,500 to 10,000 f.b.m., at that time in western white pine camps a daily production of 5,000 to 6,000 f.b.m. was considered an excellent day's work. This difference was shown to arise directly from the difference in tree sizes of the two species. Several time studies have been reported for the Southern Pine Region. Garver and Cuno (1932), working in loblolly pine forests of North Carolina, concluded that i t is five times more costly per thousand f.b.m. gross log scale to log 9-inch trees than 21-inch trees. A trend of decreasing costs with increasing log size was noted also for milling of loblolly pine trees. By combining the value of lumber in each tree and production cost for a particular d.b.h. class, Garver was able to tabulate 6 the size at which the trees began paying their way. In that particular study, a 13-inch d.b.h. loblolly pine tree proved to be marginal, i f a 20 per cent profit was to be earned on the operation. A comprehensive treatment of logging and milling costs, as affected by tree and log size, was given by Reynolds et a l . (1944) for the second growth pine-hardwood forests in Arkansas, Louisiana, Texas and Oklahoma. Guttenberg and Duerr (1949) illustrated the application of conversion surplus methods as a guide for determining the most profitable products obtainable from a tree. In the "Inland Empire" region, Bradner et a l . (1933) conducted extensive studies over a period from 1919 to 1928. The data were collected to show the effects of species, size of timber, slope and season of the year, on logging cost. In f e l l i n g and bucking operations the effect of season was found to be minor. Species, however, had considerable effect on the rate of output. Also the incentive created through contract payment, as compared to daily wage system, was aptly illustrated in their study. According to the surveys, the contract crew exceeded the day crew by 300 f.b.m. per hour when sawing 20-log-per-thousand ponderosa pine. As the timber increased in size, the difference in output between the two crews also increased. The results of skidding studies indicated consistently that the output, regardless of methods employed, 7 increased with decreasing skidding distance and increasing log size. In general, as the size of material decreased, the smaller output was attributed to the increased handling time in making up and unhooking the load. With a tractor, a limit in the number of pieces which could be skidded per load was reached in the smaller material before the weight had an appre-ciable effect. Therefore, a greater difference was found between the output for small and large timber in tractor skidding than in horse skidding. In handling of logs with trucks, the bulk rather than the weight was found to be the limiting factor. It precluded the handling of enough small logs to equal the scale of larger logs that could be carried. Long distance also increased the loss in output of small, as compared with larger, timber. The gross output per hour for a l l sizes of timber decreases as the handling distance increases: regardless of the scale per load, fewer trips are made per hour or per day. The effect of road type on pulpwood and log production cost has been investigated by Reynolds (1936) in the pine-hardwood regions of Louisiana, Arkansas and Texas. His results showed that the handling costs per M f.b.m. decreased with increasing quality of road surface and increasing log size. McClary (1953), also working in the Southern Pine Region, conducted time studies in pulpwood cutting and log making. His chief interest was the effect of tree size on the production rate, using a chainsaw instead of the then conventional hand 8 cross-cut saws. McClary found that, i f i t takes 100 man-hours to cut a given amount of pulpwood from 10-inch, 7-bolt (bolt = 5\ ft.) trees, i t would take 145 man-hours to cut the same amount from 7-inch 5-bolt trees. High-lead yarding costs in a 100-year-old Douglas f i r , sitka spruce and hemlock stand in Oregon were investigated by Tennas et a l . (1955). In their work they observed the effects of crew size, haul-in distance, volume per turn, slope, length of yarding roads and number of logs per acre on production rate and cost. In a cost analysis comparing two areas of pulpwood opera-tions in Eastern Canada, Holt (1949) demonstrated that a selec-tive cutting, where small immature trees are l e f t to grow to maturity, was a more profitable method of harvesting than the accepted clearcutting method. More recent work in logging and milling studies has also been done in Eastern Canada by Doyle (1957) and Doyle and Calvert (1961). In these investigations the whole range of conversion operations from stump through to the sawn board has been investigated for spruce, balsam f i r and jack pine. In both reports the main object of the research has been the effect of tree (and log) size on the cost of the various stages in harvesting and milling operations. Again small logs and trees were much more expensive to log and saw. 9 METHODS AND DEFINITIONS Chapman and Meyer (1947) outlined in considerable detail the computational procedures used in determination of tree stump values for various sizes, grades and species. This method, which w i l l be used later in this thesis, approaches the estimated value of a tree in steps from the direction of the f i n a l product. It considers methodically a l l inter-vening operational steps. After allowing for the various costs involved, i t arrives at a schedule of values which indicates the p r o f i t a b i l i t y of a given logging chance and determines the tree of zero margin. The profit of an operation is obviously the difference between the total revenue from the sale of a l l outputs, in this case the lumber products, and the expenditure upon a l l inputs such as labor, operating cost of equipment, taxes and so forth. As w i l l be shown, profit is a function of the variable inputs and is maximized with respect to these variables alone. Consequently, the evaluation of the functional relationships between the variable inputs and the outputs associated with them becomes of primary importance in the profit maximization analysis. Often the required information is not available, or i t may be scanty and not sufficiently reliable. Some underlying principles of production function and the 10 nature of profit maximization w i l l be brie f l y outlined in the following sections. I. PRODUCTION FUNCTION The entrepreneur's production function gives mathematical expression to the relationships between the quantities of inputs he employs and the quantity of output he produces. Thus in a simple production process of two variable inputs (X^ and X 2 ) , and one or more fixed inputs producing a single output Q, the production function states the quantity of output (q) as a funct-ion of the quantities (X^) and ( X 2 ) : q = f ( X 1 ? X 2) (1) Here (1) is assumed to be a single-valued continuous function with continuous f i r s t and second-order partial deriva-tives. It should be emphasized that a production function pre-supposes technical efficiency and states the maximum output obtainable under every possible input combination. Since the function (1) is continuous, the number of possible combinations of X^ and to the entrepreneur i s in f i n i t e . The best u t i l i z a t i o n of any particular input combi-nation is a technical, not an economic, problem. The selection of the best input combination for the production of a particular output level depends upon input and output prices and is the subject of economic analysis. I I . MARGINAL PRODUCTIVITY 11 The t o t a l productivity of variable i n the production of output Q i s defined as that quantity of Q that can be secured from the input of X^ i f X2 i s held f i x e d at an assigned l e v e l q = f ( X L X°) (2) The average productivity of X-^  i s i t s t o t a l productivity divided by i t s quantity: a f <X1 X2> X l X l F i n a l l y , the marginal productivity of X^ i s the rate of change of i t s t o t a l p roductivity with respect to variatio n s of i t s quantity, i . e . , the p a r t i a l derivative of (1) with respect to X^: OT=%= f l ( X l X 2 > < * > Families of AP and MP curves can be constructed by assigning d i f f e r e n t values to X^. I I I . ISOQUANTS AND COST FUNCTION For a fix e d output l e v e l of q° the production function becomes , 0 _ q° - f ( X l f X 2) (5) 12 The locus of a l l the combinations of X^ and X 2 which satisfy (2) forms an isoquant. It is apparent that along an isoquant the ratio of X-^  to X 2 changes continuously, whereas the output remains constant. Consequently the income remains constant but the costs of the inputs are varying. Since the entrepreneur wishes to maximize the difference between his income and expense, he attempts to choose X-^  and X 2 at such a point that the cost function should attain an absolute minimum relative to the isoquant passing through that point. The cost function is expressed as: C - + P 2X 2 + B (6) where: C i s the total cost of production, P^ and P 2 are the respective prices of X^ and X 2 and B is the cost of fixed inputs. IV. PROFIT MAXIMIZATION The profit (ST) of the entrepreneur i s the number of units sold (q) multiplied by the fixed unit price (p) less the total cost of production: ft- pq-c (7) The following substitutions are made: q = f(X^ X 2) from (1) and C = P X + P X + B from (3). Then 13 * ~ P £< X1 V -\ h - *2 X2 " B < 8 > Thus p r o f i t i s a function of X^ and X^ and i s maximized with respect to these vari a b l e s . Setting p a r t i a l derivatives of ( 8 ) with respect of X^ and X^ equal to zero: ^ - p f i - p i • ° Moving the input p r i c e terms to the r i g h t side of the equations -(9) The f i r s t - o r d e r conditions for p r o f i t maximization require that each input be u t i l i z e d up to a point at which the value of  i t s marginal product equals i t s p r i c e . 14 DETERMINATION OF CONVERSION RETURNS The above statement implies that the entrepreneur can increase his p r o f i t as long as the addition to his revenue from the employment of an additional u n i t of exceeds i t s cost. This p r i n c i p l e underlines the marginal analysis of p r o f i t maximization. L i t e r a l a pplication of t h i s formula i n a r e a l s i t u a t i o n gives r i s e to many severe complications. F i r s t l y , the inputs and t h e i r i n t e r r e l a t i o n s are so complex that even a reasonable approximation necessarily e n t a i l s severe o v e r - e i m p l i f i c a t i o n . This, however, diminishes the sig n i f i c a n c e of the p a r t i a l derivatives and also of the marginal values of the various inputs. The above i s es p e c i a l l y true for logging operations where wide v a r i a b i l i t y i s the rul e rather than the exception. Chapman and Meyer (1947) circumvented t h i s d i f f i c u l t y by analyzing one s p e c i f i c s i t u a t i o n at a time, using the best available time study and cost accounting data a/ailable. Their analysis i s a r b i t r a r i l y divided into steps which follow and complement each other i n a l o g i c a l sequence. These steps, somewhat modified, are enumerated below and discussed i n d e t a i l i n the following chapters: Step 1. Determination of s e l l i n g value of lumber of varying grades and dimensions. 15 Step 2. Lumber grade output of logs of varying size and quality. Step 3. Milling costs for logs of varying sizes. Step 4. Combination of Steps 1, 2 and 3 w i l l give net value of logs by varying size as they arrive at the m i l l . Step 5. Determination of logging and transportation costs by size of log. Step 6. Combination of Steps 4 and 5 to obtain net value of logs by size as they l i e on the ground after f e l l i n g and bucking. Step 7. Felling and bucking costs by size of tree. Step 8. Combination of Steps 6 and 7 and part of Step 5 to arrive at net value of single standing trees of diffe-rent sizes. A more recent technique in handling complex problems, where some profit function must be maximized subject to a set of economic restrictions, has been developed under the designation of linear programming (LP). Although LP has been used with remarkable success in agricultural, industrial and even military problems, i t s application to forestry and forest industry has been only slight. It w i l l be shown in this thesis that LP methods are suitable for the solution of certain problems in forest harvesting operations. 16 SELLING VALUE OF LUMBER OF VARYING GRADES AND DIMENSIONS The p r i c e of lumber has a decisive e f f e c t on the entire subsequent analysis. Minor va r i a t i o n s i n p r i c e take place constantly over short periods of time and, over long periods, the value of products may change considerably. As an i l l u s t r a -t i o n of t h i s point, i n Table 1 i s shown the changes i n p r i c e of coast Douglas f i r lumber between the years 1944 to 1955. TABLE 1 AVERAGE VALUE OF COAST DOUGLAS FIR LUMBER PER THOUSAND (M) BOARD FEET (f.b.m.) Year Dollars per M f.b.m. 1944 33.59 1947 63.63 1950 70.94 1953 75.39 1955 78.75 Whereas the p r i c e l e v e l s may change quite s u b s t a n t i a l l y , r e l a t i v e p r i c e l e v e l s between various grades and dimensions tend to remain more constant. This should not imply that d i f f e r e n -t i a l p r i c e changes do not take place, because l o c a l l y they may be highly s i g n i f i c a n t . There i s considerable merit, a f t e r defining a base-price, to establishing a r e l a t i v e p r i c e schedule which then may be modified for s p e c i f i c cases. McBride has presented such p r i c e schedules for small Douglas f i r (1949) and 17 young hemlock (1951) which can be shown i n Table 2. TABLE 2 RELATIVE LUMBER PRICES BY GRADES FOR SMALL DOUGLAS FIR AND YOUNG HEMLOCK Douglas F i r Hemlock C i r c u l a r C i r c u l a r Grade Gangmill M i l l M i l l P r i c e as a percentage of 1" No. 1 Common Lumber Clears 201 199 205 Select Com. 1" 107 107 107 111 111 111 Se l . Com; Timbers 147 139 120 No. 1 Common 1" 100 100 100 II I I I I 2" 104 104 104 No. 1 Conu Timbers 135 130 115 No. 2 Common 93 93 98 No. 3 Common 71 71 69 No. 4 Common 49 49 40 Average s e l l i n g prices for dressed lumber from the I n t e r i o r of B r i t i s h Columbia were reported by the B. C. Forest Service i n i t s Annual Report for 1960. For the l a s t quarter of 1960 the Report quoted the following prices by various species: 18 TABLE 3 AVERAGE DRESSED-LUMBER PRICES FOR OCTOBER - DECEMBER 1960 - INTERIOR B. C. Species Basis M f.b.m. Average Price F i r - l a r c h 144,706 $51.29 Spruce 185,821 50.72 Cedar 4,451 49.70 White pine 5,328 92.39 Yellow pine 2,044 49.93 Accepting $51 per M as the p r i c e of 1 inch Common No. 1 stock, then the following p r i c e schedule may be set up. A m i l l with a c i r c u l a r saw i s contemplated (Table 2). TABLE 4 PROPOSED PRICE SCHEDULE FOR DOUGLAS FIR AND HEMLOCK LUMBER Lumber Grade Douglas F i r Hemlock Clears $101 $94 Select Common 1" 55 49 2" 57 51 " '•' Timbers 71 55 Noi 1 Common 1" 51 46 I I I I I I 2 " 53 48 " '•' " Timbers 66 53 No. 2 Common 47 45 No. 3 Common 36 41 No. 4 Common 25 18 As the prices change from week to week, these values should be replaced with the l a t e s t and most accurate estimates a v a i l a b l e . In t h i s thesis a l l calculations w i l l be c a r r i e d out with t h i s 19 m i l l price schedule. Increasing width increases the price in higher grades, but in lower grades (No. 3 and No. 4 Common) the price f a l l s for wider boards, mainly due to lack of strength and greater amount of defects present. The base price of hemlock was set 10 per cent below that of Douglas f i r , v i z . .90(51) = $46. 20 LUMBER GRADE OUTPUT FROM LOGS OF VARYING SIZE AND QUALITY The grade output of lumber varies with log q u a l i t y and size and with the equipment and practices used i n any p a r t i c u l a r m i l l where the studies are conducted. Because regional d i f -ferences make i t inadvisable to r e l y on values too far away from the l o c a l i t y where the analysis i s to be applied, the large number of U. S. studies w i l l not be drawn upon. McBride has given lumber recovery values for small Douglas f i r and young hemlock which can be shown as i n Table 5. TABLE 5 LUMBER RECOVERY VALUES (PER M f.b.m.) BY LOG DIAMETER FOR SMALL DOUGLAS FIR AND YOUNG HEMLOCK Douglas F i r Hemlock Log Gangmill C i r c u l a r M i l l C i r c . M i l l Diam. Dollars per M f.b.m. l b r . t a l l y as per cent of In. that from 12" logs  Log # 2 # 3 # 2 # 3 # 3 6 - - _ — 90 9 - 97 - 98 92 12 100 100 100 100 100 15 103 100 104 102 108 18 106 96 107 102 113 21 109 94 111 101 117 24 112 93 114 99. -27 115 98 117 98 -For second-growth Douglas f i r i n the P a c i f i c Northwest, Matson (1952) presented the following grade recovery d i s t r i b u t i o n 21 for various diameter classes. TABLE 6 GRADE RECOVERY FROM SECOND-GROWTH DOUGLAS FIR LOGS IN PER CENT OF GREEN-CHAIN TALLY Log Lumber Grade Recovery (%) Dia. Select -In. Structural No. 1 No. 2 No. 3 8 24.9 67.8 5.6 1.7 10 29.5 59.2 9.5 1.8 12 32.9 52.2 12.8 3.1 14 35.4 46.8 15.5 2.3 16 37.0 42.9 17.2 2.9 18 37.4 40.9 18.1 3.6 20 36.7 40.5 18.4 4.4 Average: 31.9 53.8 12.1 2.2 For spruce i n Prince George d i s t r i c t , McBride (1956) found that the percentage of No. 2 and Better Common decreases with increase i n diameter, and the percentage of Clears, No. 3 Common, No. 4 Common and No. 5 Common increases as the diameter increases. The average lumber grade recovery values are given i n Table 7. 22 TABLE 7 AVERAGE PERCENTAGE OF LUMBER GRADE RECOVERY BY DIAMETER CLASSES (ROUGH GREEN BASIS) FOR INTERIOR SPRUCE Log Dia. Lumber Grade Recovery, (%) Value D & Btr No. 2 & No. 3 No. 4 No. 5 In. Clear Btr .Com. Com. Com. Com. $/M f.b.m. 6 _ 46 45 9 — 78.00 9 1 45 46 8 - 83.75 12 2 43 47 8 - 84.55 15 3 38 49 10 - 84.05 18 4 33 51 11 1 83.10 21 4 28 54 13 1 81.75 24 5 21 58 14 2 80.35 Matson and Rapraeger (1950), working in second-growth Douglas f i r region of the Willamette National Forest, Oregon, reported on the grade recovery percentages by log diameters. In their study, a l l logs under 12 inches in diameter were classed as No. 3 logs and those of larger diameters were a l l No. 2 sawmill quality logs. Their results are summarized in Table 8. 23 TABLE 8 AVERAGE PERCENTAGE OF LUMBER GRADE RECOVERY BY DIAMETER CLASSES FOR SECOND-GROWTH DOUGLAS FIR Log Lumber Grade Recovery (%) Dia. Select - -In. Structural No. 1 No. 2 No. 3 6 10.6 79.4 5.5 4.5 8 21.8 69.6 5.2 3.4 10 29.8 61.8 5.9 2.5 12 34.7 55.8 7.6 1.9 14 36.4 51.8 10.4 1.4 16 34.9 49.9 13.8 1.4 18 30.3 50.0 18.6 1.1 20 22.5 52.5 23.6 1.4 Average: 30.0 59.2 8.9 1.9 A similar evaluation of grade recovery by diameter classes for second-growth hemlock has been reported by McBride (1951). The sawmill studied was a small portable type, with circular headsaw cutting 10 M f.b.m. per eight-hour shift. The values have been summarized in Table 9. A l l logs belonged to Grade No. 3. 24 TABLE 9 AVERAGE PERCENTAGE OF LUMBER GRADE RECOVERY BY DIAMETER CLASSES FOR SECOND-GROWTH HEMLOCK Lumber Grade Recovery (%,) Selected Log Comm. Selected Dia. Boards Com. No. 1 No. 2 No. 3 In. Clear & Dim. Timber Com. Com. Com. 6 - 9 _ 63 25 3 9 2 16 - 57 22 3 12 6 23 18 35 16 2 15 11 5 46 26 10 2 18 16 5 49 20 8 2 21 22 5 50 15 7 1 Average: 12 8 43 24 11 2 Worthington (1955) reported grade recovery for young Douglas f i r i n Washington, cut i n a m i l l using a 30-inch round-log Swedish gang. These values are given i n Table 10. TABLE 10 LUMBER GRADE RECOVERY IN PER CENT OF GREEN-CHAIN TALLY - YOUNG DOUGLAS FIR Dia. Grade Recovery In. No. 1 & Better No. 2 No. 3 No. 4 6 28 40 27 5 8 45 38 13 4 10 53 35 9 3 12 56 33 8 3 14 57 32 8 3 16 56 32 9 3 18 54 32 11 3 25 Grade recovery values were tabulated for about 300 second-growth Douglas f i r logs from some unpublished data gathered by the Vancouver Forest Products Laboratory, Canada Department of Forestry. The re s u l t s are given i n Table 11. From these r e s u l t s i t may be seen that smaller logs produce a r e l a t i v e l y larger percentage of No. 1 Common lumber than do larger logs. No. 2 Common Grade i s r e l a t i v e l y constant over the various diameter classes and constitutes about 15 per cent of the volume recovered. No. 3 Common i s low f o r smaller logs but increases r a p i d l y with increasing log s i z e . TABLE 11 GRADE RECOVERY IN SMALL DOUGLAS FIR (UNPUBLISHED DATA FROM V.F.P.L.) Log Top Diameter Classes - In.  6-9 9-12 12-15 15-18 Grade Per Cent of Total Recovered Volume Clears 0.8 1.2 2.7 5.1 No. 1 77.7 66.3 57.2 35.1 No. 2 13.9 16.6 14.3 17.1 No. 3 7.5 13.5 24.4 37.6 No. 4 0.1 2.4 1.4 5.1 Kirkland and Brandstrom (1936) studied the recovery of lumber grades. Their r e s u l t s , based on a sample of 1,336 logs, show that the percentage of No. 1 Common decreases with increas-ing log diameter, whereas Select grade and No. 3 Common lumber increase over the same diameter range. 26 The various grade recovery tables presented so far indicate the great v a r i a b i l i t y which i s found between the d i f f e r e n t investigations. A comparison of the studies shows that, although they a l l follow si m i l a r patterns, i n d i v i d u a l values exhibit very wide v a r i a t i o n . Obviously there are no two i d e n t i c a l t r a c t s of forest i n respect of recovery p o s s i b i l i t i e s . In t h i s thesis a reasonable compromise i s attempted between the various studies. For lumber recovery values, Table 6 by Matson (1952) has been used. The over-run values for Douglas f i r have been adopted from McBride (1949) and for hemlock from McBride (1951). Using the p r i c e schedule i n Table 4, the lumber recovery value for a 16-foot, 12-inch Douglas f i r log i s found to be $70.63 per M f.b.m., and the •k corresponding value for hemlock was taken 107o lower, or $63.57. In Table 12, the lumber r e a l i z a t i o n values per M f.b.m. and the value of single logs have been computed for logs of various lengths and top diameters. * suggested by L. B. Dixon, Chief Inspector, B r i t i s h Columbia Lumber Manufacturers Association (B.C.L.M.A.). TABLE 12 LUMBER REALIZATION VALUE PER M f.b.m. AND PER LOG OF VARIOUS SIZES DOUGLAS FIR (F) HEMLOCK (H) Log Vol. of Lumber t a l l y No. of logs Lumber r e a l i z . Value per Vol. of Lumber t a l l y No. of logs Value per Vol. of Lumber t a l l y No. of logs Value per D.i.b. 16-ft. log % Overrun f.b.m. f.b.m. per M Value $/M 16-ft. log 24-ft. log f.b.m. f.b.m. per M 24-ft . log 32-ft. log f.b.m. f.b.m. per M 32-ft. log i n . B.C. f.b.m. F i H2 F H F H F H F H B.C. f.b.m. F H F H F H B.C. f.b.m. F H F H F H 6 15 85 77 28 27 35.7 37.0 64.17 57.75 1.80 1.56 23 43 41 23.2 24.4 2.76 2.37 31 57 55 17.5 18.2 3.67 3.17 7 23 60 55 37 36 27.0 27.8 65.36 58.82 2.42 2.12 35 56 54 17.9 18.5 3.65 3.18 46 74 71 13.5 14.1 4.84 4.17 8 32 50 41 48 45 20.8 22.2 65.47 58.92 3.15 2.65 ' 48 }72 68 13.9 14.7 4.71 4.01 64 96 90 10.4 .11.1 6.30 5.31 9 43 43 29 61 55 16.4 18.2 67.81 61.03 4.13 3.35 64 92 83 10.9 12.0 6.22 5.09 86 123 111 8.1 9.00 8.37 6.78 10 55 33 25 73 69 13.7 14.5 68.94 62.05 5.03 4.28 83 110 104 9.1 9.6 7.57 6.46 110 146 137 6.8 7.30 10.14 8.50 11 69 27 22 88 84 11.4 11.9 69.83 62.85 6.13 5.28 103 131 126 7.6 7.9 9.19 7.96 137 174 167 5.75 5.99 12.14 10.49 12 84 23 19 103 100 9.7 10.0 70.63 63.57 7.28 6.36 126 155 150 6.5 6.7 10.87 9.49 168 207 200 4.83 5.00 14.62 12.71 13 101 21 17 122 118 8.2 8.5 71.32 64.19 8.70 7.55 151 183 177 5.5 5.6 12.97 11.46 201 243 235 4.12 4.26 17.31 15.07 14 119 19 15 142 137 7.0 7.3 71.86 64.67 10.27 8.86 178 212 205 4.7 4.9 15.29 13.20 238 283 274 3.53 3.65 20.36 17.72 15 139 18 14 164 158 6.1 6.3 72.33 65.10 11.86 10.33 208 245 237 4.1 4.2 17.64 15.50 278 328 317 3.05 3.15 23.71 20.67 16 160 17 12 187 179 5.3 5.5 72.56 65.30 13.69 11.87 240 281 269 3.6 3.7 20.16 17.65 320 374 358 2.67 2.79 27.18 23.41 17 183 16 11 212 203 4.7 4.9 72.59 65.33 15.44 13.33 274 318 304 3.1 3.3 23.42 19.80 366 424 406 2.36 2.46 30.76 26.56 18 207 15 9 238 226 4.2 4.4 72.62 65.36 17.29 14.85 311 358 339 2.8 2.9 25.94 22.54 415 477 452 2.10 2.21 34.58 29.57 19 233 15 8 268 252 3.7 4.0 72.41 65.17 19.57 16.29 350 403 378 2.5 2.6 28.96 25.06 466 536 503 1.87 1.99 38.72 32.75 20 261 14 6 297 277 3.4 3.6 72.07 64.86 21.20 18.02 391 446 414 2.2 2.4 32.76 27.02 521 594 552 1;68 1.81 42.90 35.83 21 290 14 5 331 304 3.0 3.3 71.47 64.32 23.82 19.49 434 495 455 2.0 2.2 35.73 29.24 579 660 608 1.52 1.64 47.02 39.22 22 320 14 4 365 333 2.7 3.0 70.91 63.82 26.26 21.27 480 548 499 1.8 2.0 39.39 31.91 640 730 666 1.37 1.50 51.76 42.55 23 352 13 4 398 366 2.5 2.7 70.72 63.65 28.29 23.57 528 597 549 1.7 1.8 41.60 35.36 704 796 732 1.26 1.36 56.13 46.80 24 386 13 3 436 397 2.3 2.5 70.45 63.41 30.63 25.36 578 653 595 1.5 1.7 46.97 37.30 771 871 794 1.15 1.26 61.26 50.33 25 421 13 3 476 434 2.1 2.3 70.08 63.07 33.37 27.42 631 713 650 1.4 1.5 50.06 42.05 841 950 978 1.05 1.02 66.74 61.83 1 McBride (1949) 2 McBride (1951) 28 MILLING COSTS FOR LOGS OF VARYING SIZES I t has been demonstrated through numerous investigations that smaller logs are more co s t l y to handle than larger logs. Ashe (1916), i n his investigation i n Tennessee, V i r g i n i a , and North Carolina, showed that sawing time increased r a p i d l y for logs 16 inches i n diameter and l e s s , and also that species and log length a f f e c t the rate of production. A study by Rapraeger (1932) i n ponderosa pine region i n Eastern and Central Oregon showed that the making of lumber from 6-inch logs was more than three-fold of the cost for 30-inch logs. In Table 13 are summarized the sawing times per M f.b.m., rough green lumber t a l l y basis. TABLE 13 SAWING TIME IN MINUTES PER M f.b.m. OF LUMBER FOR PONDEROSA PINE LOGS OF VARIOUS DIAMETERS Log Dia. In. Min. per M  f.b.m. % of 12 i n . times 6 8 10 12 14 16 18 20 24 28 32 20.4 17.5 12.6 10.9 9.9 9.0 8.4 8.0 7.3 6.8 6.6 187.1 160.5 115.6 100.0 90.8 82.6 77.1 73.4 67.0 62.4 60.6 29 A similar trend was found by Doyle and Calvert (1961) in their m i l l study on jack pine in Northern Ontario. In Table 14 are shown the average times in man-minutes required to produce one M f.b.m. of lumber. TABLE 14 EFFECT OF LOG DIAMETER ON THE TIME REQUIRED TO SAW M f.b.m. OF LUMBER Log Dia. Man-Min. % of 12" In. M f.b.m. times • 5 460 191.6 6 320 133.3 7 230 95.8 8 200 83.3 9 180 75.0 10 180 75.0 11 200 83.3 12 240 100 Doyle and Calvert's results indicate that the sawmill used was specifically designed for small material, but even in this case, sawing of logs less than 7 inches in diameter becomes rapidly more time-consuming and consequently more expensive. For the Douglas f i r region, Matson and Rapraeger (1950) have given the times required to produce a specified amount of lumber from logs of various sizes. Sawing times at the headrig of a 24-inch Swedish gang-saw for 16-foot logs are shown in Table 15. TABLE 15 NET SAWING TIME OF 16-FOOT LOGS PER M f.b.m. OF LUMBER TALLY Log Dia. In. Minutes (Fir) % of 12 in.  times 6 8 10 12 14 16 18 20 53.1 30.7 20.1 14.6 12.7 11.5 10.4 10.0 363.6 210.2 137.6 100 86.9 78.7 71.2 71.2 The setting of cost figures requires that the analyst must have access to the accounting figures of a sawmill. Because such information i s usually kept secret, i t is d i f f i -cult to deal here with specific figures. Only approximate figures w i l l be used in the present instance. In establishing the price schedule for lumber products, the prices from the B. C. Interior for autumn 1960 (Table 4) were used. For the same region and the same period of time, average sawmilling costs figures were suggested by Mr. L. B. Dixon of B. C. L. M. A. as reasonable approximations. These cost values are presented in Table 16. 31 TABLE 16 LUMBER MANUFACTURING COST IN THE SOUTHERN INTERIOR SELLING PRICE ZONE (SPRING 1960) (STATIONARY MILLS) Cost of Cost of Cost of K i l n T otal Sawing Planing Drying Cost Per Per Per Per Species M f.b.m. M f.b.m. M f.b.m. M f.b.m. F i r & Other $14.00 $ 8.70 $ - $22.70 Spruce 15.30 11.70 2.90 29.90 Cedar 15.00 11.20 - 26.30 W. Pine 15.60 12.60 3.60 31.80 Y. Pine 15.10 11.50 2.50 29.10 Assuming that no planing i s considered and that the average value i n the above tables ref e r s to a log with a * diameter of 18 inches, the cost schedule shown i n Table 17 may be set up, based on d i s t r i b u t i o n of sawing times as given by Matson and Rapraeger (1950) i n Table 15. This may overestimate the average log s i z e . TABLE 17 COST OF SAWING PER M f.b.m. LUMBER TALLY &F 16, 24 and 32 FOOT LOGS Log  Dia. In. 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Sawing time per M - minutes $ Cost/M f.b.m. 16-ft.-Logs F H 1007. 24-ft. Logs F H 79.27. 32-ft. Logs F H 65.67. 16-ft. 24-ft. 32-ft. 16-ft. $ Cost per Log 24-ft. 32-ft. 53.1 39.5 30.7 24.5 20.1 16.9 14.6 12.9 12.7 12.0 11.5 10.5 10.4 10.4 9.8 9.4 9.3 9.1 8.9 8.8 55.0 40.6 32.8 27.2 21.3 17.6 15.1 13.4 13.1 12.5 11.9 10.9 10.8 10.5 10.2 10.1 9.8 9.6 9.5 9.5 42.1 31.3 24.3 19.4 15.9 13.4 11.6 10.2 10.1 9.5 9.1 8.3 8.2 8.2 7.8 7.4 7.4 7.2 7.0 7.0 43.6 32.2 26.0 21.5 16.9 13.9 12.0 10.6 10.4 9.9 9.4 8.6 8.6 8.3 8.1 8.0 7.8 7.6 7.5 7.5 34.8 25.9 20.1 16.1 13.2 11.1 9.6 8.5 8.3 7.9 7.5 6,9 6.9 6.9 6.4 6.2 6.1 6.0 5.8 5.8 36.1 26.6 21.5 17.8 13.9 11.5 9.9 8.8 8.6 8.2 7.8 7.2 7.1 6.9 6.7 6.6 6.4 6.3 6.2 6.2 84.76 63.50 49.00 39.11 32.07 26.97 23.31 20.58 20.26 19.14 18.00 17.20 16.60 16.60 15.60 15.00 14.90 14.60 14.20 14.00 H 87.80 64.81 52.36 43.42 34.00 28.09 24.10 21.39 20.91 19.50 18.50 17.60 17.00 16.60 16.10 15.95 15.50 15.15 15.00 14.90 67.13 50.29 38.81 30.98 25.40 21.36 18.46 16.30 16.05 15.16 14.26 13.62 13.15 13.15 12.36 11.88 11.80 11.56 11.25 11.09 H 69.54 51.33 41.47 34.39 26.93 22.25 19.09 16.94 16.56 15.44 14.65 13.94 13.46 13.15 12.75 12.63 12.28 12.00 11.88 11.80 55.60 41.66 32.14 25.66 21.04 17.69 15.29 13.50 13.29 12.56 11.81 11.28 10.89 10.89 10.23 9.84 9.77 9.58 9.32 9.18 H 57.60 42.52 34.35 28.48 22.30 18.43 15.81 14.03 13.72 12.79 12.14 11.54 11.15 10.89 10.56 10.46 10.17 9.94 9.84 9.77 2.37 2.35 2.36 2.38 2.34 2.37 2.40 2.51 2.89 3.14 3.40 3.66 3.95 4.49 4.59 5.00 5.52 5.84 6.17 6.67 H 2.37 2.33 2.36 2.39 2.34 2.36 2.41 2.52 2.86 3.09 3.36 3.59 3.86 4.15 4.47 4.83 5.17 5.61 6.00 6.48 2.89 2.81 2.79 2.84 2.79 2.81 2.84 2.96 3.41 3.70 3.96 4.39 4.70 5.26 5.62 5.94 6.56 6.80 7.50 7.92 H 2.85 2.77 2.82 2.87 2.81 2.82 2.85 3.03 3.38 3.68 3.96 4.22 4.64 5.06 5.31 5.74 6.14 6.67 6.99 7.87 3.18 3.08 3.09 3.17 3.09 3.08 3.17 3.28 3.76 4.12 4.42 4.78 5.19 5.82 6.09 6.47 7.13 7.60 8.10 8.74 H 3.16 3.02 3.09 3.16 3.05 3.08 3.16 3.29 3.76 4.06 4.35 4.69 5.05 5.47 5.83 6.38 6.78 7.31 7.81 9.58 33 DETERMINATION OF NET VALUE OF LOGS IN THE MILL POND Manufacturing costs shown i n Table 17 are subtracted from the lumber r e a l i z a t i o n values of Table 12. The net values per M f.b.m. (lumber t a l l y basis) are shown i n Table 18. TABLE 18 CONVERSION RETURN PER M f.b.m. AND PER LOG Loft Top  D.i.b. In. OF VARIOUS LENGTHS AT THE MILL POND $ Value per M f.b.m. 16 f t . H 24 f t . H 32 f t . H Net Value of One Log $ 16 f t . H 24 f t . H 32 f t . H 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 -20.59 -30.05 -2.96 -11.79 8.57 0.15 1.86 16.47 28.70 36.87 42.86 47.32 50.74 51.60 53.19 54.56 55.39 56.02 55.81 56.47 56.47 56.01 56.12 56.25 56.08 - 5.99 6.56 17.61 28.05 34.76 39.47 42.80 43.76 45.60 46.80 47.73 48.36 48.57 48.76 48.37 48.32 48.50 48.41 48.17 15.07 26.66 36.83 43.54 48.47 52.17 55.02 55.81 57.17 58.30 58.97 59.47 59.26 59.71 59.59 59.11 59.16 59.20 58.99 7.49 17.45 26.64 35.12 40.60 44.48 47.25 48.11 49.66 50.65 51.39 51.90 52.02 52.11 51.69 51.54 51.65 51.53 51.27 23.70 33.33 42.15 47.90 52.14 55.34 57.82 58.57 59.77 60.75 61.31 61.73 61.52 61.84 61.63 61.14 61.14 61.13 60.90 16.30 24.57 32.55 39.75 44.42 47.76 50.16 50.95 52.31 53.16 53.79 54.21 54.28 54.30 53.86 53.65 53.71 53.57 53.30 -0.57 0.07 0.79 1.75 2.69 3.76 4.88 6.19 7.38 8.72 10.29 11.78 13.34 15.08 16.61 18.82 20.74 22.45 24.46 26.70 -.81 -.21 0.29 0.96 1.94 2.92 3.95 5.03 6.00 7.24 8.51 9.74 10.99 12.14 13.55 14.66 16.10 19.96 19.36 20.94 -.13 .84 1.92 3.38 4.78 6.38 8.03 10.01 11.88 13.94 16.20 19.03 21.24 23.70 27.14 29.79 32.83 34.80 39.47 42.14 -.48 .41 1.19 2.22 3.65 5.14 6.64 8.43 9.82 11.82 13.69 15.58 17.90 20.00 21.71 23.50 25.77 28.69 30.31 34.18 .49 1.76 3.21 5.20 7.05 9.06 11.45 14.03 16.60 19.59 22.76 25.98 29.39 32.90 36.81 40.55 44.63 48.53 53.16 58.00 .01 1.15 2.22 3.62 5.45 7.41 9.55 11.78 13.96 16.61 19.06 21.87 24.52 27.28 30.00 32.84 35.77 39.49 42.52 52.25 4> 35 TRANSPORTATION AND LOGGING COSTS BY LOG SIZE Logging is conducted under operating conditions which vary constantly. Each individual logging unit, however small, presents a different problem to the operator. Fortunately, a large number of systematic logging studies have been carried out throughout the years and these records afford a means of estimating the performance for a defined set of logging condi-tions. In addition to published data which w i l l be presented in subsequent paragraphs, the author has conducted some original time studies in the University Research Forest near Haney, B. C. These various sources w i l l be used to establish costs for hauling, loading, yarding, bucking and f a l l i n g operations. A. THE EFFECT OF TYPE OF ROAD ON HAULING COST The effect of road type on pulpwood and log production costs has been investigated by Reynolds (1936) in the pine-hardwood region of Louisiana, Arkansas and Texas. He demon-strated in this study that the hauling cost per M f.b.m. decreased with increasing quality of road surface and increasing average log size. The costs per M f.b.m. of hauling logs for one mile over three types of roads are shown in Figure 1. Cost data of this nature may be used as a basis for determining whether construction of a certain type of road is j u s t i f i e d , 36 •IOh 1060 Woods Road Graded Earth Road Gravel or Hard Surface Road 1160 1265 1370 Ave- Load F b-rrv 1470 60 80 100 120 Ave- Log Size F-b-m-140 Fig- I Effect of Road Type on Hauling Cost 37 provided the volume to be hauled over that road i s known approximately. Tessier and Knapp (1961) reported a road construction cost of $16,452 per mile for the University Research Forest. The hauling on the same road was l a t e r done by contract at a rate of $6.25 per M f.b.m. The t o t a l hauling distance was 18.1 miles, with 14.1 miles on surfaced highway and the remain-ing 4 miles on a Forest logging truck road. This contract hauling represents a cost of $0.35 per M f.b.m. per mile. B. THE EFFECT OF LOG SIZE ON HAULING COST In addition to hauling distance and road surface, hauling cost depends on log s i z e . Doyle and Calvert (1961) reported that the hauling cost of jack pine logs i n Ontario decreased with increasing log s i z e , and that the rate of decrease was most rapid i n the smallest log s i z e s . Thus the cost of hauling f i v e - i n c h logs was more than twice that for logs eleven inches i n diameter. These costs included both hauling and loading. During June, 1961 the loading of 18 truckloads of logs was observed and timed by the author i n the University Research Forest. The mean load was 2,589 f.b.m. with a standard devia-t i o n of 248 f.b.m., although the number of logs i n these loads ranged from 6 to 36. The r e l a t i v e consistency of the load volume meant that the hauling cost per load and per mile did not d i f f e r much with the varying log s i z e . Using the previously 38 quoted figure of 35 cents per M f.b.m. per mile, the cost of an 18-mile tr i p would have ranged from $12.19 to $29.85 per load, varying directly with the load volume. Bradner et a l . (1933) reported hauling studies for ponderosa pine logs with a 7%-ton truck. It was found in hauling that the bulk, rather than weight, was the limiting factor. This precludes the handling of enough small logs to equal the scale of larger logs that could be carried. Hauling distance also increases the loss in output of small as compared with large timber. The gross output per hour for a l l sizes of timber decreases as the hauling distance increases; regardless of the scale per load, fewer trips are made per hour or day. Reynolds et a l . (1944) presented the number of man-hours required for a 4-mile truck haul in the second-growth pine-hardwood forest in the Southern U. S. A. The current cost figure may be obtained from that information by multiplying the number of man-hours by going hourly rate of pay (e.g. $2), shown in Table 19. 39 TABLE 19 HAULING COST PER M f.b.m. ACCORDING TO REYNOLDS et a l . (1944) D.b.h. Man-hours Cost per  4 miles Cost per 18 miles 12 14 16 18 20 24 30 2.917 2.148 1.740 1.564 1.433 1.156 .984 $5,834 4.296 3.480 3.128 2.866 2.312 1.968 $26.25 19.33 15.66 14.08 12.90 10.40 8.86 Reynolds' higher costs, as compared to today's rates, are the result of technological improvements in truck and road Although the hauling cost of f u l l truckloads was found not to be influenced much by log diameters, the loading operation studied by the author was significantly influenced by log size. The performance of a 4-man crew at the University Research Forest i s shown in Figure 2. This curve shows the time in minutes needed to load one M f.b.m. of logs of various lengths and diameters. Tessier and Knapp (1961) reported an average loading cost $2.90 per M f.b.m. The size of the average log was 400 f.b.m. Assuming machine rental as $2.00 per hour and labourers' rate of pay as $2.00 per hour, a cost schedule may be set up for loading of logs of various sizes as shown in Table 20. design. C. LOADING COST 24' 16' F- bm-60 120 180 240 300 . 0 10 20 30 40 50 Log Length Mean Log Volume in Cu- ft-32' « !—! , 1 , , , , , ( , r 6 7 8 9 10 M 12 13 14 15 16 17 Log Diameter in In * I I I I I l I I I I I I I 1 6 7 8 9 10 II 12 13 14 15 16 17 18 19 Log Diameter in In-i i i i i i i i i I i i i i i i i r~ 67 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 Log Diameter in In Fig 2 Effect of Mean Log Volume on Loading Rate University Research Forest, June 1361 TABLE 20 LOADING AND HAULING COSTS PER M f.b.m. AND LOG FOR 16, 24 AND 32-FOOT LOGS Top D.i.b. Time to Load $ Cost of Loading Hauling Total Loading & Total Loading & Hauling Cost per Log $ M f. b.m. min. per M f.b.m. * Cost Hauling Cost per M 16 f t . i:c. 24 f t . 32 f t . In. 16 f t . 24 f t . 32 f t . 16 f t . 24 f t . 32 f t . 18 mi. 16 f t . 24 f t . 32 f t . F H F H F H 6 33.2 29.5 26.5 5^53 4.92 4.42 6.30 11.83 11.22 10.72 .33 .32 .48 .46 .62 .59 7 29.5 25.5 22.6 4.92 4.25 3.77 n 11.22 10.55 10.07 .42 .40 .59 .57 .75 .71 8 26.3 22.2 19.0 4.38 3.70 3.17 10.68 10.00 9.47 .51 .48 .72 .68 .91 .85 9 23.4 19.0 15.2 3.90 3.17 2.53 10.20 9.47 8.83 .62 .56 .87 .79 1.09 .98 10 20.8 15.8 12.4 3.47 2.63 2.07 i i 9.77 8.93 8.37 .71 .67 .98 .93 1.23 1.15 11 18.0 13.2 9.6 3.00 2.20 1.60 9.30 8.50 7.90 .82 .78 1.12 1.08 1.37 1.32 12 15.6 10.6 7.4 2.60 1.77 1.23 i i 8.90 8.07 7.53 .92 .89 1.24 1.20 1.56 1.51 13 13.4 8.6 5.8 2.23 1.43 .97 n 8.53 7.73 7.27 1.04 1.00 1.41 1.38 1.76 1.71 14 11.3 6.9 4.8 1.88 1.15 .80 8.18 7.45 7.10 1.17 1.12 1.59 1.52 2.01 1.95 15 9.6 5.6 4.2 1.60 .93 .70 7.90 7.23 7.00 1.29 1.29 1.76 1.72 2.29 2.22 16 7.9 4.8 3.8 1.32 .80 .63 7.62 7.10 6.93 1.44 1.39 1.97 1.92 2.60 2.48 17 6.7 4.2 3.7 1.12 .70 .62 7.42 7.00 6.92 1.58 1.51 2.26 2.12 2.93 2.81 18 5.6 4.0 3.7 .93 .67 .62 7.23 6.97 6.92 1.72 1.64 2.49 2.40 3.30 3.13 19 5.0 3.8 3.7 .83 .63 .62 7.13 6.93 6.92 1.93 1.78 2.77 2.67 3.71 3.48 20 4.5 3.7 3.7 .75 .62 .62 7.05 6.92 6.92 2.07 1.96 3.15 2.88 4.12 3.82 21 4.1 3.7 3.7 .68 .62 .62 6.98 6.92 6.92 2.33 2.11 3.46 3.14 4.55 4.22 22 3.9 3.7 3.7 .65 .62 .62 6.95 6.92 6.92 2.57 2.32 3.84 3.46 5.05 4.61 23 3.7 3.7 3.7 .62 .62 .62 6.92 6.92 6.92 2.77 2.56 4.07 3.84 5.49 5.09 24 3.7 3.7 3.7 .62 .62 .62 6.92 6.92 6.92 3.01 2.77 4.61 4.07 6.02 5.49 25 3.6 3.7 3.7 .60 .62 .62 6.90 6.92 6.93 3.29 3.00 4.94 4.61 6.59 6.78 * Hourly Cost: 4-man Crew - $8 per hr. Machine - $2 per hr. Tota l $10 per hr. (or 16.67 cents per min.) 42 If the total cost per M f.b.m. of loading and hauling i s subtracted from the net log value at the m i l l pond, net value at the woods landing is obtained. These values are presented in Table 21. TABLE 21 LOP, Top D.i.b.  In. 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 $ Value per M f.b.m. Scale 16 f t . 24 f t . 32 f t . H H H -32.42 -41.88 -14.18 -23.01 -2.15 -10.57 - 9.36 5.79 18.50 27.10 33.56 38.42 42.21 43.42 45.29 46.94 47.97 48.79 48.68 49.42 49.49 49.06 49.20 49.33 49.18 -17.21 - 4.12 7.51 18.28 25.46 30.57 34.27 35.58 37.70 39.18 40.31 41.13 41.44 41.71 41.39 41.37 41.58 41.49 41.27 4.52 16.66 27.36 34.61 39.97 44.10 47.29 48.36 49.94 51.20 51.97 52.50 52.33 52.79 52.67 52.19 52.24 52.28 52.07 • 3.06 7.45 17.17 26.19 32.10 36.41 39.52 40.66 42.43 43.55 44.39 44.93 45.09 45.19 44.77 44.62 44.73 44.61 44.35 13.63 23.86 33.32 39.53 44.24 47.81 50.55 51.47 52.77 53.82 54.39 54.81 54.60 54.92 54.71 54.22 54.22 54.21 53.98 6.23 15.10 23.72 31.38 36.52 40.23 42.89 43.85 45.31 46.23 46.87 47.29 47.36 47.38 46.94 46.73 46.79 46.65 46.38 WOODS LANDING SITE Net Value of One Log $ 16 f t . 24 f t . 32 f t . F H F H F H -.90 -1.13 -.61 -.94 -.13 -.58 -.35 T .60 .25 -.16 1.01 .44 .28 - .19 1.20 .51 2.30 1.37 1.13 .40 2.51 1.43 4.11 2.64 1.98 1.27 3.80 2.72 5.82 4.30 2.94 2.14 5.26 4.06 7.69 6.09 3.96 3.06 6.79 5.44 9.89 8.04 5.15 4.03 8.60 7.05 12.27 10.07 6.21 4.88 10.29 8.30 14.59 12.01 7.43 5.95 12.18 10.10 17.30 14.39 8.85 7.12 14.23 11.77 20.16 16.58 10.20 8.23 16.77 13.46 23.05 19.05 11.62 9.35 18.75 15.50 26.09 21.39 13.15 10.36 20.93 17.33 29.19 23.80 14.54 11.59 23.99 18.83 32.69 26.18 16.49 12.55 26.33 20.36 36.00 28.62 18.17 13.78 28.99 22.31 39.58 31.16 19.68 15.40 30.73 24.85 43.04 34.40 21.45 16.59 34.86 26.24 47.14 37.03 23.41 17.94 37.20 29.57 51.41 45.47 44 D. YARDING COSTS Many time studies of yarding by log sizes have been conducted over the years. A comprehensive study was carried out by Bradner ejt a l . (1933) in the Inland Empire. In this study skidding output data were analysed for horse, tractor and donkey (groundline) skidding. Results of their studies indicate consistently that output in skidding, for the various methods, increases with decreasing skidding distance and increasing log size. Thus when skidding with horses in the ponderosa pine type on 0 to 15 per cent slopes, the output in timber with 3 to 5 logs per thousand, skidded a distance of 100 feet, is three times as high as in timber with 18 to 25logs per thousand. In general, i t was found that a smaller output, as the size of material decreased, was attributable to the increased handling time in making up and unhooking the load. With the tractor, a limit in the number of pieces which can be skidded per load is reached in the smaller material before the weight has become an apprecia-ble effect. Therefore, a greater difference is found between the output for small and large timber in tractor skidding than in horse skidding. Tennas et a l . (1955) studied high lead yarding costs in Western Oregon. They found that haul-in distance and volume per turn were the most significant factors on haul-in time. The haulback time, for a given speed, is determined by haul-in 45 distance. Choker set time increased as the distance from the spar tree increased and obviously decreased when several choker setters were used. Finally, unhooking time was found to increase with number of logs yarded per turn. Tennas et a l . summarized their findings in the following regression equation: Y = 366.43 + .451D + .000265D2 +.0000485DV + .00072VS - 49.25C + 8.60N (10) Where Y = Time per turn in 1/100 minute D = Haul-in distance in feet V = Volume per turn in board feet by the Scribner rule S - Slope in per cent C = Number of choker setters (including head rigger) N = Number of logs per turn No provision was made in their report to study the effect of individual log size on operating time and cost. Stewart (1961) in his salvage study at the University Research Forest reported a yarding cost of $16.16 per M f.b.m. as compared to the average cost of yarding of $3.64 on West Coast Vancouver Island. Mcintosh and Gunn (1960) studied the performance of high-lead yarding in a pre-logging operation with a portable steel spar. Yarding time per 100 cubic feet, as related to net turn volume and yarding distance, is shown in Table 22. 46 TABLE 22 YARDING TIMES OF VARIOUS TURN VOLUMES Turn Volume Man--Minutes Cu. F t . Per Cu. F t . )istance , .505 f t , 365 f t . 20 331.5 205.0 40 170.9 111.5 60 117.0 77.5 80 90.3 58.5 100 70.2 46.0 120 55.2 35.5 140 45.5 28.5 160 39.0 22.5 180 33.1 17.5 200 29.9 -220 26.6 -240 23.4 -Time requirements for the various phases of yarding opera tions were d i s t r i b u t e d according to the time breakdown i n Table 23. TABLE 23 BREAK-DOWN OF YARDING TIME REQUIREMENTS Operation Per Cent of Total Time Yarding Distance 505 f t . 365 f t Haulback 5.7 6.1 Stop Chokers 4.7 5.8 Hook-up 26.1 24.6 Haul 19.0 15.0 Unhook Chokers 7.8 9.0 Hang-ups 11.3 3.3 Landing Delays 3.9 1.2 Other Delays 21.5 35.0 47 Tessier and Knapp (1961) reported on the operation of a Madill Mobile s t e e l spar. The average cost was $4.76 per M f.b.m. and the average log size in th i s operation was 413 f .b .m. The time studies which the author conducted at the University Research Forest also included a yarding study. The equipment used was a 10-10 Lawrence donkey, a 70-foot spar tree and a 4-man crew. The high-lead method of yarding was used. The haulback time was found to vary with the skidding distance as in Table 24. TABLE 24 HAULBACK TIMES AT UNIVERSITY RESEARCH FOREST SALVAGE LOGGING OPERATION, JUNE 1961 Distance, Feet Time, Minutes 60 .193 100 .215 200 .390 250 .370 300 .361 350 .450 400 .588 450 .716 500 .836 An attempt was made to correlate yarding time and log volume yarded. No c o r r e l a t i o n was detected between these variables. However, a d e f i n i t e r e l a t i o n s h i p existed between yarding time (Y) and yarding distance (X). This r e l a t i o n s h i p may be described by the following regression equation: 48 Y = 1.229 - 0.633X + 0.193X2 (11) This relationship is also shown in Figure 3. The lack of correlation between yarding time and log volume seems to contradict the results of Tennas et a l . (1955), who based their findings on a sample of 2,304 logs. The University Forest study included the yarding times of only 173 logs and consequently failed to show the effect of log volumes due to the much smaller sample. If skidding distance is the major factor, skidding cost becomes a function of the mean length of yarding road. The relationship i s not s t r i c t l y linear, because with increasing yarding road length haulback time increases, and also the choker-set time increases, as shown by Tennas et al. At the University Research Forest, the mean choker-set time was 1.28 minutes and i t did not vary significantly with haul-in distance. During the same operation, the mean skidding distance was 380 feet. At that distance the haul-in time equalled 1.60 minutes. Added to this are 1.28 minutes for choker-set, 0.5 minutes for unhooking, and 0.58 minutes for haulback, for a total of 3.96 minutes. The mean turn volume during the investigation was 26.66 cu. f t . so that the mean rate of production was 404 cu. f t . per hour. If the size of the log is varied, a whole schedule of production rates may be computed. These production rates f a l l under the following direct costs: 49 Fig 3 Effect of Yarding Distance on Highlead Yarding Time-University Research Forest,June 1961-50 Labor: 4 men @ $2.00 per hr. = $ 8.00/hr. Machine Rental: $ 2.00/hr. Total = $10.00/hr. In addition to directly productive time, the yarding operation requires time for changing yarding roads, changing corner blocks, swinging blocks on the spar tree and tightening guy lines. Times for changing yarding roads and corner blocks are fixed times per road. It is assumed here that i t takes two days to move the donkey and erect a new spar tree. The average cost of yarding may be based on following considerations. Assuming a setting with 600 f t . radius, the area thus covered equals 25.7 acres. If one acre contains 606 cu. f t . of yardable material (Stewart, 1961) then the total accessible material on this setting is 15,733 cu. f t . or 96,286 f.b.m. Assuming a working rate of 2,150 f.b.m. per hour, the total effective yarding time necessary is 96.286 m 2,470 J y n r ' Assuming further that there are 40 roads to the 360° of the setting and that each change of road requires one-half-hour, an additional 20 hours w i l l be required. The total working time w i l l be 59 hours, or approximately 60 hours. Under ideal working conditions, where effective working time in an 8-;hour day is 6 hours, the setting should be finished in 60 + 2 = 12 days. The daily cost of a crew of four and the 6 51 donkey was assumed to be $80.00. The total cost per setting then would be 12 x 80 = $960 or 960 = $10.00 per M f.b.m. 96.286 The real situation at the University Research Forest showed an effective working time (yarding time and loading time) •k of only 3% hours per 8-hour working day. Then, instead of $10.00, the apparent hourly cost w i l l be 8 * 1 0 = $22.86, i f the productive time alone is charged with the daily cost of $80.00. 60 At this rate, -rj- + 2 - 19 days are required to finish a setting. The cost for this time w i l l be 19 x 80 = $1,520 or Qp" 1^ = $15.79 per M f.b.m. This hypothetical figure is f u l l y comparable to Stewart's cost, quoted as $16.16. As a rule, two or more logs are hauled simultaneously, thereby increasing productivity and lowering unit cost. Thus the cost per M f.b.m. becomes a function of volume per turn, rather than the size of an individual log. Consequently the yarding cost w i l l vary with the size of a load but remains constant i f considered on a "per log" basis. It is assumed that the mainline has two chokers and that the logs hauled have equal diameters. In Table 25 the volumes of such pairs of logs have been assembled and the cost of yarding per M f.b.m. The cost per log, which is constant for the mean yarding distance of 400 feet, is $0.80 in this case. * A l l operations studied were "gyppo", working on a production contract basis only. They were not directly employed and super-vised by the University Research Forest. TABLE 25 HIGHLEAD YARDING COST PER M f . b . m . OVER A MEAN DISTANCE OF 400 F E E T L o g T o p V o l u m e o f Two L o g s , Lumbe r T a l l y Y a r d i n g C o s t p e r M f . b .m . * D . i . b . 16 f t . 24 f t . 32 f t . 16 f t . 24 f t . 32 f t . i n . ' F H F H F H F H F H F H 6 56 54 86 82 114 110 2 8 . 4 4 2 9 . 4 9 1 8 . 5 2 1 9 . 4 2 1 3 . 9 7 1 4 . 4 8 7 74 72 112 108 148 142 2 1 . 5 2 2 2 . 1 2 1 4 . 2 2 1 4 . 7 4 8 . 8 0 1 1 . 2 1 8 96 90 144 136 192 180 1 6 . 5 9 1 7 . 6 9 1 1 . 0 6 1 1 . 7 1 8 . 2 9 8 . 8 5 9 122 110 184 166 246 222 1 3 . 0 5 1 4 . 4 8 8 . 6 5 9 . 5 9 6 . 4 7 7 .17 10 146 138 220 208 292 274 1 0 . 9 1 1 1 . 5 4 7 .24 7 .66 5 .45 5 .81 11 176 168 262 252 348 334 9 . 0 5 9 . 4 8 6 . 0 8 6 .32 4 . 5 8 4 . 7 7 12 206 200 310 300 414 400 7 .73 7 .96 5 .14 5.31 3 . 8 5 3 . 9 8 13 244 236 366 354 486 470 6 . 5 3 6 . 7 5 4 . 3 5 4 . 5 0 3 . 2 8 3 . 3 9 14 284 274 424 410 566 548 5 .61 5 .81 3 .76 3 .88 2 .81 2 .91 15 328 316 490 474 656 634 4 . 8 5 5 .04 3 .25 3 .36 2 . 4 3 2 .51 16 374 358 562 538 748 716 4 . 2 6 4 . 4 5 2 . 8 3 2 . 9 6 2 . 1 3 2 . 2 2 17 424 406 636 608 848 812 3 . 7 6 3 .92 2 . 5 0 2 .62 1 .88 1.96 18 476 452 716 678 954 904 3 .35 3 .52 2 . 2 2 2 .35 1 .67 1 .76 19 536 504 806 756 1072 1006 2 .97 3 .16 1 .98 2 .11 1 .48 1 .58 20 594 554 892 828 1188 1104 2 . 6 8 2 .87 1 .79 1.92 1 .34 1 .44 21 662 608 990 910 1320 1216 2 .41 2 .62 1.61 1.75 1.21 1.31 22 730 666 1096 998 1460 1332 2 . 1 8 2 . 3 9 1.45 1 .60 1 .09 1 .19 23 796 732 1194 1098 1592 1464 2 . 0 0 . 2 . 1 8 1.33 1.45 1 .00 1 .09 24 872 794 1306 1190 1742 1588 1.83 2 .01 1.22 1 .34 .91 1 .00 25 952 868 1426 1300 1902 1956 1.67 1.83 1.12 1.22 . 8 4 .81 * F u l l t u r n 1 m i n . c o s t 4 . 1 8 4 . 1 8 m i n u t e s $ . 3 8 0 9 5 $ 1 . 5 9 2 4 53 I f these values are applied to the table of net log values at landing (Table 21), i t becomes immediately apparent that the minimum f i r log to be yarded i s 10 inches i n top diameter. For hemlock th i s size i s 11 inches. Before net values a f t e r f e l l i n g and bucking are computed, yarding costs for t r a c t o r -and horse-skidding w i l l also be explored. Gunn and Guernsey (1958) reported the r e s u l t s on f i v e t ractor operations i n the B. C. I n t e r i o r , i n which i t was pointed out that a notably long time was used for the hook-up phase of the work. Also, a relationship was found between turn volume and skidding distance. Thus i t takes approximately the same time to skid 250 cu. f t . a distance of 280 feet as i t does to skid 550 cu. f t . a distance of 2,720 feet, per unit volume. Consequently, per unit volume, i t i s more economical to skid larger volumes greater distances than to skid small volumes short distances. The distance factor could often be m u l t i p l i e d , or i t s e f f e c t reduced, by more e f f e c t i v e supervision i n regard to turn volume, according to the authors. For these f i v e opera-tions, the time was d i s t r i b u t e d between the various phases of yarding as follows: 1. Make road and swamp 6.8% 2. Return 15.1 3. Hook-up 33.0 4. Haul 17.1 5. Unhook 6.6 6. Hang-up 1.1 7. Non-productive time 20.3 100.0% The effects of skidding distance and turn volume on skidding time in the five operations investigated are shown in Table 26. TABLE 26 SKIDDING TIME - MACHINE MINUTES PER 100 CU. FT. (GUNN AND GUERNSEY) Turn Operation A B C D 2 E Vol. Crew Size - 3 3 3 2 Cu. Skidding Distance s - feet Ft. 280 660 1640 2720 180 620 715 190 370 190 1280 50 — 17.7 -100 - 15.6 17.4 19.0 13.8 - 11.9 -150 10.9 12.8 14.3 11.8 14.9 8.5 -200 12.9 7.4 9.2 10.5 10.2 12.9 - -250 10.7 5.3 7.7 8.8 9.2 11.5 - 18.5 300 8.8 10.6 14.6 4.3 6.9 8.2 8.4 10.3 - 16.0 350 7.4 9.2 12.8 3.7 6.4 7.7 9.1 - 13.7 400 6.4 8.0 11.2 14.1 3.2 6.0 7.1 7.9 - 11.7 450 5.8 7.3 10.0 12.5 5.6 6.6 - 10.3 500 5.2 6.7 9.2 11.3 5.4 6.0 - 9.6 550 4.8 6.2 8.5 10.5 - 9.0 600 4.5 5.7 7.8 9.8 - 8.6 650 - - - 9.4 - -700 - - - 8.9 U i U i 56 Fig- 4 200 400 600 800 Distance in Feet 1000 1200 1400 Effect of Yarding Distance and Volume on Roundtrip Time-, D-2 Tractor Operation,University Research Forestj March, 1961-57 Tractor yarding was studied during alder logging in March 1961 at the University Research Forest. The volumes of single trips varied between 145 and 840 f.b.m. and the yarding distances varies between 150 and 1,400 feet. In Figure 4 is shown the effects of these two factors on the times of a roundtrip. Cost of tractor yarding. Machine rate - $ 3.00 per hour Driver " - $ 2.00 per hour Chokerman 11 - % x $2.00 per hour (used half the time) Total Yarding Cost - $ 6.00 per hour. If i t is assumed that out of the 8-hour working day only 5% hours are spent in active yarding, the apparent hourly rate C o is r i - $8.73. From Figure 4 the mean roundtrip for a 5^ 600 f t . skidding distance is 15 minutes. Assuming that three logs are hauled out simultaneously, the cost per turn would be 60 x^8.73 = $ 3 > 4 9 a n d t h e C Q S t p e r l o g i 8 $ 1 > 1 6 > In this case the log size was ignored mainly because the original f i e l d measurements were not sufficiently numerous and showed a considerable variation about the mean value. Thus, the production rate for 12 i n . d.i.b. logs 16 feet long would be 3 0 9 * 6 0 = 1235 f.b.m. per hour and the cost 8 - 7 3 * 1 0 0 0 « 15 12.35 $7.07 per M f.b.m. Computation shows that D-2 yarding is somewhat more econo-mical than high-lead yarding (Table 25). This i s -true only i f the high-lead works with the present low efficiency and that the D-2 w i l l yard out at least three logs per turn. That, of course, 58 i s not always the case. E. HORSE SKIDDING Doyle (1957) and Doyle and Calvert (1961) have studied the performance of horse skidding i n Eastern Canada. The re-ported average skidding cost varied from $3.26 to $5.50 per M f.b.m., and decreased with increasing log s i z e . S i l v e r s i d e s (1960) gave the time study data i n Table 27 on skidding pulpwood by bolt s i z e and distance yarded. TABLE 27 HORSE-SKIDDING TIMES PER CORD OF 100 IN. (OR 8 FT.) BOLTS, BY LOG DIAMETER CLASS AND DISTANCE Log Dia. Time Required per Cord with Skidding Distance of: In. 100 f t . 200 f t . 300 f t . 6 0.78 1.22 1.40 8 0.78 1.22 1.40 10 0.62 0.97 1.10 12 0.53 0.82 0.93 14 0.43 0.67 0.77 Worthington and Staebler (1961) showed that for products whose volume i s less than 10 cu. I f t . , horses are most s a t i s -factory for skidding, provided that the t e r r a i n i s favourable and skidding distances are less than 600 feet. The time study r e s u l t s by these authors are given i n Table 28. 59 TABLE 28 HOURLY PRODUCTION (CU. FT.) FOR HORSE-SKIDDING OF 8-FOOT LOGS, BY SKIDDING DISTANCE AND LOG DIAMETER Log Dia. Skidding Distance i n Feet In. 100 200 400 600 800 7 32 26 20 16 13 8 41 33 26 20 16 10 62 50 39 30 24 12 82 65 48 36 29 14 105 80 57 42 33 The cost of owning and using a horse as given by Worthington (1957) i s : Fixed Cost per horse-hour $0,119 Variable " " " 11 0.277 Tota l " " " " $0,396 $.40 This cost i s based on an i n i t i a l cost for a horse of $100 to $175, less salvage $25 and a working l i f e of 4 years. The annual working time was assumed to be 1,600 hours per 200 days (8-hr. day). I f the hourly wage of the teamster i s $2.00 and i t i s assumed that a team produces 5% hours of active yarding each day, then the cost for horse skidding can be estimated as shown in Table 29. 60 TABLE 29 COST OF HORSE-SKIDDING - DISTANCE 400 FEET Log Volume per hour (8 -foot log) Yarding Cost/M M-f .b.m. Dia. Adi. for (Hourly In. cu. f t . * Fu l l Eff. 68% Eff. Cost $2.40) 10 39 0.245 0.168 $14.28 11 46 0.299 0.206 11.65 12 48 0.319 0.219 10.96 13 49 0.332 0.228 10.53 14 57 0.393 0.270 8.89 * from Table 28. It is evident, by comparison with D-2 Caterpillar tractor, that yarding with a horse becomes somewhat more expensive. F. NET VALUE OF LOGS IN THE WOODS The foregoing cost figures have shown that, under the prevailing conditions in the University Research Forest in June, 1961, the yarding operation was carried out more economi-cally with a D-2 Caterpillar tractor. The author f e l t that this result was not due to an inherent superiority on the part of the tractor, but rather due to inefficiency in the high-lead opera-tion. It became apparent that, with some improvement in the high-lead yarding equipment, the efficiency of that operation could be raised sufficiently to make i t comparable to or even better than tractor skidding. Consequently, in the following computations, the cost figures of a high-lead system w i l l be used in the calculation of log values in the woods after f a l l i n g and bucking. These values are given in Table 30. TABLE 30 NET VALUE OF LOGS IN THE WOODS AFTER FELL ING AND BUCKING L o g T o p $ V a l u e p e r M f . b . m . $ N e t V a l u e o f One L o g D . i . b . 16 f t . 24 f t . 32 f t . 16 f t . 24 f t . 32 f t . I n . F H F H F H F H F H F H 6 - 6 0 . 8 4 - 7 1 . 3 7 - 3 2 . 7 0 - 4 2 . 4 3 - 1 6 . 1 2 - 2 5 . 0 5 - 1 , 7 0 - 1 . 9 3 -1 .41 - 1 . 7 4 - .93 - 1 . 3 8 7 - 3 0 . 8 8 - 3 9 . 3 3 - 9 . 7 0 - 1 7 , 8 0 * 4 . 8 3 - 4 . 9 8 - 1 . 1 5 - 1 . 4 0 - .65 - .96 0 . 2 1 - . 36 8 - 1 0 . 8 0 - 2 1 . 8 1 5 . 6 0 - 4 . 2 6 1 5 . 5 7 6 . 2 5 - .52 - .99 . 40 - .31 1 .50 .57 9 5 .45 - 7 .07 1 8 . 7 1 7 . 5 8 2 6 . 8 5 1 6 . 5 5 .33 - . 4 0 1.71 .83 3 .31 1 .84 10 1 6 . 1 9 6 . 7 4 2 7 . 3 7 1 8 . 5 3 3 4 . 0 8 2 5 . 5 7 1 .18 .47 3 . 0 0 1.92 5 .02 3 . 5 0 11 2 4 . 5 1 1 5 . 9 8 3 3 . 8 9 2 5 . 7 8 3 9 . 6 6 3 1 . 7 5 2 . 1 4 1 .34 4 . 4 6 3 .26 6 . 8 9 5 .29 12 3 0 . 6 9 2 2 . 6 1 3 8 . 9 6 3 1 . 1 0 4 3 . 9 6 3 6 . 2 5 3 .16 2 .26 5 .99 4 . 6 4 9 . 0 9 7 . 2 4 13 3 5 . 6 8 2 7 . 5 2 4 2 . 9 4 3 5 . 0 2 4 7 . 2 7 3 9 . 5 0 4 . 3 5 3 .23 7 . 8 0 6 . 2 5 1 1 . 4 7 9 . 2 7 14 3 7 . 8 1 2 9 . 7 7 4 4 . 6 0 3 6 . 7 8 4 8 . 6 6 4 0 . 9 4 5.41 4 . 0 8 9 . 4 9 7 . 5 0 1 3 . 7 9 1 1 . 2 1 15 4 0 . 4 4 3 2 . 6 6 4 6 . 6 9 3 9 . 0 7 5 0 . 3 4 4 2 . 8 0 6 . 6 3 5 .15 1 1 . 3 8 9 . 3 0 1 6 . 5 0 1 3 . 5 9 16 4 2 . 6 8 3 4 . 7 3 4 8 . 3 7 4 0 . 5 9 5 1 . 6 9 4 4 . 0 1 8 .05 6 .32 1 3 . 4 3 1 0 . 9 7 1 9 . 3 6 1 5 . 7 8 17 4 4 . 2 1 3 6 . 3 9 4 9 . 4 7 4 1 . 7 7 5 2 . 5 1 4 4 . 9 1 9 . 4 0 7 .43 1 5 . 9 7 1 2 . 6 6 2 2 . 2 5 1 8 . 2 6 18. 4 5 . 4 4 3 7 . 6 1 5 0 . 2 8 4 2 . 5 8 5 3 . 1 4 4 5 . 5 3 1 0 . 8 2 8 .55 1 7 . 9 5 1 4 . 7 0 2 5 . 2 9 2 0 . 5 9 19 4 5 . 7 1 3 8 . 2 8 5 0 . 3 5 4 2 . 9 8 5 3 . 1 2 4 5 . 7 8 1 2 . 3 5 9 . 5 6 2 0 . 1 3 1 6 . 5 3 2 8 . 3 9 2 3 . 0 0 20 4 6 . 7 4 3 8 . 8 4 5 1 . 0 0 4 3 . 2 7 5 3 . 5 8 4 5 . 9 4 1 3 . 7 4 1 0 . 7 9 2 3 . 1 9 1 8 . 0 3 3 1 . 8 9 2 5 . 3 8 21 4 7 . 0 8 3 8 . 7 7 5 1 . 0 6 4 3 . 0 2 5 3 . 5 0 4 5 . 6 3 1 5 . 6 9 1 1 . 7 5 2 5 . 5 3 1 9 . 5 6 3 5 . 2 0 2 7 . 8 2 22 4 6 . 8 8 3 8 . 9 8 5 0 . 7 4 4 3 . 0 2 5 3 . 1 3 4 5 . 5 4 1 7 . 3 7 1 2 . 9 8 2 8 . 1 9 2 1 . 5 1 3 8 . 7 8 3 0 . 3 6 23 4 7 . 2 0 3 9 . 4 0 5 0 . 9 1 4 3 . 2 8 5 3 . 2 2 4 5 . 7 0 1 8 . 8 8 1 4 . 6 0 2 9 . 9 3 2 4 . 0 5 4 2 . 2 4 3 3 . 6 0 24 4 7 . 5 0 3 9 . 4 8 5 1 . 0 6 4 3 . 2 7 5 3 . 3 0 4 5 . 6 5 2 0 . 6 5 1 5 . 7 9 3 4 . 0 6 2 5 . 4 4 4 6 . 3 4 3 6 . 2 3 25 4 7 . 5 1 3 9 . 4 4 5 0 . 9 5 4 3 . 1 3 5 3 . 1 4 4 5 . 5 7 2 2 . 6 1 1 7 . 1 4 3 6 . 4 0 2 8 . 7 7 5 0 . 8 1 4 5 . 6 7 62 G. FELLING AND BUCKING I t i s desirable to evaluate f e l l i n g and bucking as d i s -t i n c t operations. In many otherwise complete time studies, however, f e l l i n g and bucking have been lumped together and the quoted costs cover both operations. A great number of time studies on f e l l i n g and bucking had been conducted p r i o r to the introduction of power-driven chain saws. Ashe (1916), Rapraeger (1931) (1936), Bradner et a l . (1933), Reynolds et a l . (1944), McClay (1953), Koroleff (1947) and Doyle (1957) have reported on the effects that the size of a log and tree have on f a l l e r s ' and buckers' rate of output. Although the values from these studies are no longer applicable to present logging conditions, they neverthe less confirm the fact that handling of a small tree i s r e l a -t i v e l y more c o s t l y than handling of a large one. Performance values on f e l l i n g , limbing and bucking are given by Doyle and Calvert (1961). The r e s u l t s of t h e i r studies on the e f f e c t of power tools on the above-mentioned operations are shown i n Figure 5, where the si z e of the jack pine tree, by d.b.h. classes, shows a d e f i n i t e e f f e c t on the time that i s required to f e l l , buck and limb that tree. Kurta (1961) investigated the performance of f a l l e r s and buckers i n the University Research Forest. His r e s u l t s are shown i n Table 32. Kurta found that the rate of production was highest for Douglas f i r , followed by hemlock and cedar. 63 20 18 16 inures per Tr 14 12 inures per Tr 10 8 Man- 6 4 2 0. -i i i i i i i 7 8 9 10 1 1 12 13 14 15 16 17 D-B-H- Class - inches Fig- 5 Effect of Tree Size on Actual Time Required for Felling, Limbing,and Bucking (Doyle and Calvert, 1961) 64 Consequently, the f a l l i n g and bucking cost per M f.b.m. was lowest for Douglas f i r and highest for cedar by $1.00 and $1.50 respectively for the 18-inch d.b.h. class trees. During the time studies, Kurta found that the eight-hour day was u t i l i z e d by the 3-man working crews as in Table 31. TABLE 31 AVERAGE DAILY PRODUCTIVE AND NON-PRODUCTIVE TIME FOR 3-MAN FALLING CREW Productive Non-Productive Time Time  Per Cent Faller 68 32 Marker 66 34 Bucker _81 19 Average: 72 28 The mean effective working time was 5% hours per 8-hour day, or 72 per cent of the total time available. If a f a l l e r and bucker receive an hourly pay of $5.00 or $40.00 per day, the hourly rate over the effective working time would be 40 r r = $7.28. Applying this value to the man-minutes required to f e l l and buck trees of different sizes, a cost schedule may be compiled as shown in Table 32. 65 TABLE 32 THE EFFECT OF STUMP HEIGHT DIAMETER ON FELLING AND BUCKING TIME AND COST (KURTA 1961) Dia. @ Douglas F i r W. Hemlock W. R. Cedar Stump Man-min. Cost Man-min. Cost Man-min. Cost Ht. In. per M $/M per M T 7 M per M 3/M 14 36.9 4.46 16 - - 28.9 3.50 30.9 3.74 18 16.5 2.00 21.8 2.64 24.8 3.00 20 13.9 1.68 16.5 2.00 19.7 2.38 22 11.7 1.42 12 i 8 1.56 16.0 1.94 24 10.2 1.24 10.9 1.32 13.4 1.62 26 9.0 1.10 9.7 1.18 11.4 1.38 28 8.0 0.96 8.8 1.06 10.0 1.32 30 7.3 0.88 8.3 1.00 8.8 1.06 32 6.6 0.80 7.8 0.94 8.0 0.98 34 6.1 0.74 7.5 0.90 7.3 0.88 Nixon and Gunn (1957) determined the productivity values shown i n Table 33, for In t e r i o r Douglas f i r , l a r c h , and spruce. TABLE 33 EFFECT OF TREE SIZE (D.B.H.) ON FELLING AND BUCKING TIMES PER M f.b.m. GROSS VOLUME - LOGS 12 TO 22 FEET LONG (NIXON AND GUNN) Time i n Man-Minutes per M f.b.m. D.b.h. D. F i r Larch Spruce 12 92.3 70.6 56.4 16 76.2 59.2 45.7 20 62.9 49.3 37.4 24 54.4 43.0 32.5 28 49.0 40.3 30.1 32 44.9 39.1 28.9 36 40.8 - -40 37.4 - -66 The cost of production per M f.b.m. may be computed from the foregoing table. The worker's wage rate i s assumed to be $2.00 per hour and the t o t a l cost of operating and owning a power saw i s $0.50 per hour (Worthington and Staebler, 1961). Results of t h i s computation are shown i n Table 34. TABLE 34 HOURLY PRODUCTION PER MAN, AND COST PER M f.b.m., OF FELLING AND BUCKING OF INTERIOR FIR, LARCH AND SPRUCE (NIXON AND GUNN) D.b.h. Douglas F i r Larch Sprue e Prodn. Cost Prodn. Cost Prodn. Cost In. Mfbm/h. $/Mfbm Mfbm/h. $/Mfbm Mfbm/h. $/Mfbm 12 .650 3.85 .849 2.94 1.064 2.35 16 .787 3.18 1.013 2.47 1.313 1.90 20 .954 2.62 1.217 2.05 1.604 1.56 24 1.103 2.27 1.395 1.79 1.846 1.35 28 1.224 2.04 1.489 1.68 1.993 1.25 32 1.336 1.87 1.534 1.63 2.076 1.20 36 1.470 1.70 - - - -40 1.604 1.56 - - - -It i s immediately obvious that the In t e r i o r costs, as presented i n the above table, baseid on the values of Nixon and Gunn, are higher than the values arr i v e d at by Kurta. The chief reason for t h i s difference i s the much larger tree volume on the Coast as compared to a tree of sim i l a r d.b.h. i n the In t e r i o r of t h i s Province. F e l l i n g and bucking of 8-foot logs of various sizes was investigated by Worthington and Shaw (1952). Their cost figures are presented here as percentages of the cost of f e l l i n g and 67 bucking 12-inch logs. I f t h i s cost i s assumed to be $3.50, then the cost schedule of Table 35 may be set up. TABLE 35 FELLING AND BUCKING COSTS FOR LOGS OF VARIOUS DIAMETERS (WORTHINGTON AND SHAW) Log Top Cost as Cost D.i.b. Percentage of In. 12-inch logs $/M f.b.m. 7 160 5.60 8 148 5.18 9 130 4.55 10 118 4.13 11 109 3.82 12 100 3.50 13 91 3.18 14 82 2.87 These values are not comparable to the cost figures of Gunn and Dixon, which were based on tree sizes rather than log s i z e s . A time study of li m i t e d extent on f a l l i n g and bucking performance was conducted by the author at the University Research Forest. The 8-hour-day of the f a l l e r and bucker was found to consist of following component times: Preparing to f a l l and/or buck 13.5% F a l l i n g 18.2% 56.7% Bucking, Limbing 25.0%> Saw Service 17.9% Walking 14.7% Resting .10.7% 100.0% 68 0 5 10 15 20 25 D- B- H- in Inches Fig- 6 Effect of Tree D-B-H- on Felling Time-University Research Forest, June 1961-69 Log Diameter in Inches Fig- 7 Effect of Log Diameter on Bucking Time-University Research Forest, June 1961-70 Cutting times with a Pioneer 5.5 H.P. chain saw were recorded and plotted i n Figure 6 and Figure 7. The curved data are given i n Table 36. TABLE 36 CUTTING TIMES IN FELLING AND BUCKING OPERATIONS AT THE U. B. C. RESEARCH FOREST D.b.h. F e l l i n g Time Diameter at Bucking Time In. Min. Cutting P o s i t i o n In. i . b. Min. 6 0.40 6 0.14 8 0.50 8 0.22 10 0.63 10 0.32 12 0.90 12 0.42 14 1.40 14 0.52 16 2.10 16 0.68 18 3.00 18 0.82 20 3.9 20 1.00 22 4.9 22 1.20 24 5.7 24 1.41 26 6.3 26 1.70 28 6.9 30 7.4 32 7.8 34 8.2 During the f e l l i n g and bucking operation, the observed prac-t i c e of the f a l l e r was sometimes to buck i n the woods. On other occasions i t was only to top a tree and to buck i t l a t e r at the landing. Because the l a t t e r procedure produces fewer logs to be skidded, t h i s i s a more economical way to operate. Bucking at the landing i s safer and more convenient, although i t requires more saw maintenance. This i s due mainly to sand and rock imbedded i n the bark of the logs, which d u l l s the 71 saw faster than i n the woods. In the following computations i t w i l l be assumed that a l l f e l l e d trees are bucked i n the woods into 16, 24 and/or 32-foot lengths. On the Coast, the f a l l e r and bucker are usually paid at a certain rate per M f.b.m. produced. This rate i s adjusted according to the size of the timber i n the stand to assure the workers a f a i r minimum d a i l y wage. An introduction of changing rates of pay i n t h i s study, however, would serve no usefu l purpose. I t w i l l be assumed that the one-man crew, operating both as a f a l l e r and a bucker, receives a f l a t d a i l y rate. When thi s amount i s divided by the production rates r e s u l t i n g from d i f f e r e n t tree sizes, the costs associated with the various d.b.h. classes may be d i r e c t l y obtained. In keeping with the t r a d i t i o n a l l y high rates of pay that the coast f a l l e r s receive, i t i s assumed that the d a i l y pay for the f a l l e r equals $40.00. Further, i t i s assumed that 72 per cent of his time may be c l a s s i f i e d as productive, as found by Kurta's i n v e s t i g a t i o n . F e l l i n g and bucking costs are calculated and presented i n Table 37. * Information supplied by J . A. Mcintosh, Vancouver Laboratory, Forest Products Laboratories of Canada. TABLE 37 DETERMINATION OF FELLING AND BUCKING COST FOR DOUGLAS FIR TREES OF DIFFERENT SIZES (Curved) Utilized D.b.h. Volume Felling Bucking Necessary Total Production Adiusted Daily Cost Cost in a Tree Time Unprod. Time Output Lbr. t a l l y Time (per tree) Time per Tree Rate for 727o (8 hr.) per M per Tree In. f ,b .m. min. min. min. min. fbm/hour Efficiency f.b.m. $ $ 6 7 28 .43 .14 3.0 3.57 471 339 2772 14.43 0.40 8 37 .50 .19 3.2 3.89 571 411 3288 12.17 0.45 9 42 .56 .22 3.5 4.28 589 424 3392 11.79 0.50 10 50 .63 .19 3.8 4.62 649 467 3736 10.71 0.54 11 60 .75 .22 4.0 4.97 724 521 4168 9.60 0.58 12 80 .90 .19 4.3 5.39 890 641 5128 7.80 0.62 13 100 1.10 .22 4.5 5.82 1031 742 5936 6.74 0.67 14 120 1.40 .22 4.8 6.42 1121 807 6456 6.20 0.74 15 150 1.73 .64 5.0 7.37 1221 879 7032 5.69 0.85 16 180 2.10 .74 5.3 8.14 1327 955 7640 5.24 0.94 17 220 2.50 .74 5.5 8.74 1510 1087 8696 4.60 1.01 18 260 3.00 .82 5.8 9.62 1622 1168 9344 4.28 1.11 19 310 3.4 .71 6.1 10.21 1822 1312 10496 . 3.81 1.18 20 377 3.9 .90 6.3 11.10 2038 1467 11736 3.41 1.29 21 430 4.4 .74 6.6 11.74 2198 1583 12664 3.16 1.36 22 505 4.9 U64~ 6.8 13.34 2271 1635 13080 3.06 1.55 23 600 5.3 1.64 7.1 14.04 2564 1846 14768 2.71 1.63 24 699 5.7 1.91 7.4 15.01 2794 2012 16096 2.49 1.74 25 798 6.0 1.74 7.6 15.34 3121 2247 17976 2.23 1.78 26 897 6.3 1.64 7.9 15.84 3398 2447 19576 2.04 1.83 27 1020 6.6 2.99 8.1 17.69 3460 2491 19928 2.01 2.05 28 1140 6.9 3.24 8.4 18.54 3689 2656 21248 1.88 2.14 29 1260 7.2 2.87 8.6 18.67 4049 2915 23320 1.72 2.17 30 1390 7.4 2.87 8.9 19.17 4351 3133 25064 1.60 2.22 31 1520 7.6 2.93 9.1 19.63 4646 3345 26760 1.49 2.26 32 1660 7.8 3.29 9.4 20.49 4861 3500 28000 1.43 2.37 TABLE 38 BUCKING SCHEDULE OF TREES OF D IFFERENT S IZES D . b . h . Top D i a m e t e r o f L o g s Ob ta i ned ' ' " U t i l i z e d Top T o t a l 2 1 6 - f t . 2 4 - f t . 3 2 - f t . L e n g t h L e f t T r e e I n . L o g s L o g s L o g s f t . f t . L e n g t h f t . 6 ( 5 " ) 16 37 53 7 ( 6 " ) 16 45 61 8 ( 7 " ) 16 52 68 9 ( 8 " ) 16 59 75 10 ( 7 " ) 24 58 82 11 8 24 65 89 12 7 32 63 95 13 8 32 69 101 14 8 32 75 107 15 12 8 40 73 113 16 14 8 40 79 119 17 14 8 40 84 124 18 15 8 48 81 129 19 14 7 56 78 134 20 16 8 56 83 139 21 8 , 14 64 80 144 22 19 14 8 72 76 148 23 20 8 , 12 80 73 153 24 22 8 , 13 80 77 157 25 21 8 , 12 88 73 161 26 8 , 1 2 , 20 96 69 165 27 25 18 8 , 12 104 65 169 28 25 19 8 , 12 104 68 172 29 1 8 , 24 8 , 12 112 64 176 30 1 8 , 24 8 , 12 112 67 179 31 1 4 , 26 8 , 13 112 69 181 32 27 8 , 1 2 , 18 120 64 184 74 TABLE 38 (Continued) 1 Based on taper data supplied by Mr. J . A. Mcintosh from V. F. P. L. 2 Based on Local Volume Tables of U. B. C. Research Forest. Species: Immature Douglas F i r . Maximum Height: 200 feet. 75 H. DETERMINATION OF NET VALUE OF STANDING TREES I t was assumed that the f e l l e d trees were bucked into 16, 24 or 32-foot lengths. In order to e s t a b l i s h the " l o g -content" of trees of various d.b.h. si z e s , the author consulted l o c a l volume tables, compiled for Douglas f i r and hemlock trees of the University Research Forest. In addition to these tables, Douglas f i r taper curves were constructed from some unpublished Vancouver Laboratory data. The bucking schedule (Table 38) was assembled by applying the outlined sources of information. This table indicates the number of logs of 16, 24 and 32-foot length which may be obtained, on the average, from trees of various s i z e s . E a r l i e r , i n Table 30, the net values of various logs a f t e r f e l l i n g and bucking were assembled. The information from that source and from that i n Table 38 can be combined to assign a net value for each tree a f t e r f e l l i n g and bucking. This calcu-l a t i o n , which involves the summation of the net values of the appropriate logs, has been shown i n Columns 1 and 2 of Table 39. This new schedule of values does not progress smoothly because of the stepwise increase of u t i l i z e d volumes i n the trees. Therefore, the values of Columns 1 and 2 have been smoothed out as shown i n Figure 8. The curved values have been * Information supplied by J . A. Mcintosh, Vancouver Laboratory, Forest Products Laboratories of Canada. TABLE 39 NET VALUES OF STANDING TREES D . b . h . I n . N e t V a l u e o f  A l l U t i l i z e d  L o g s i n a T r e e  a f t e r F e l l i n g  a n d B u c k i n g F e l l i n g &  B u c k i n g C o s t P e r T r e e C u r v e d V a l u e s N e t V a l u e o f  S t a n d i n g T r e e F H F H F H 6 7 - 1 .70 - 1 .93 - 1 .70 - 1 .93 . 4 0 - 2 . 1 0 - 2 .33 8 - 1 .15 - 1 .40 - 1 .50 - 1 .40 .45 - 1.95 - 1.85 9 - .52 - . 99 - 1 .00 - 1 .20 . 50 - 1 .50 - 1 .70 10 - . 65 - . 9 6 - . 5 0 - 1 .00 .54 - 1 .04 - 1 .54 11 . 4 0 - .31 0 - .70 .58 - .58 - 1 .28 12 .21 - . 36 . 75 - . 10 .62 .13 - .72 : 13 1 .50 .57 1 .60 . 40 .67 .93 - .27 14 1 .50 .57 2 . 5 0 1 .10 .74 1 .76 .36 : 15 3 . 5 6 1 .95 3 . 5 6 2 . 0 0 .85 2 .71 1.15 16 5 .81 3 . 7 7 5 . 0 0 3 . 0 0 .94 4 . 0 4 2 . 0 6 17 5 .81 3 . 7 7 6 . 6 0 4 . 2 5 1.01 5 .59 3 . 2 4 18 8 .13 5 . 7 2 8 . 2 0 5 .90 1.11 7 .09 4 . 7 9 19 9 . 7 0 7 . 1 4 1 0 . 0 8 7 . 8 0 1 .18 8 . 9 0 6 .62 20 1 4 . 9 3 1 1 . 5 4 1 3 . 6 0 1 0 . 0 0 1.29 1 2 . 3 1 8 .71 21 1 5 . 2 9 1 1 . 7 8 1 7 . 0 0 1 2 . 8 0 1.36 1 5 . 6 4 1 1 . 4 4 22 2 3 . 3 4 1 9 . 4 3 2 0 . 8 0 1 5 . 7 5 1.55 1 9 . 2 5 1 4 . 2 0 23 2 4 . 3 3 1 8 . 6 0 2 5 . 0 0 1 9 . 2 0 1.63 2 3 . 3 7 1 7 . 5 7 24 3 0 . 3 4 2 2 . 8 2 2 9 . 8 0 2 3 . 0 0 1 .74 2 8 . 0 6 2 1 . 2 6 25 3 6 . 1 2 2 7 . 3 7 3 6 . 0 0 2 7 . 5 0 1 .78 3 4 . 2 2 2 5 . 7 2 26 4 2 . 4 8 3 3 . 1 9 4 2 . 0 0 3 2 . 2 0 1.83 4 0 . 1 7 3 0 . 3 7 27 5 1 . 1 5 3 9 . 6 5 4 8 . 0 0 3 6 . 9 0 2 .05 4 5 . 9 5 3 4 . 8 5 28 5 3 . 3 3 4 1 . 4 8 5 4 . 0 0 4 1 . 6 0 2 . 1 4 5 1 . 8 6 3 9 . 4 6 29 6 2 . 6 0 4 7 . 9 5 6 0 . 0 0 4 6 . 1 0 2 .17 5 7 . 8 3 4 3 . 9 3 30 6 2 . 6 0 4 7 . 9 5 6 5 . 6 0 5 0 . 9 0 2 .22 6 3 . 3 8 4 8 . 6 8 D o u g l a s F i r H e m l o c k ON 77 DBH- in Inches Fig- 8 Net Value of Trees after Felling and Bucking 78 entered i n Columns 3 and 4 of Table 39. Next, the cost of f e l l i n g and bucking per tree (Table 37, entered into Column 5 of Table 39) i s subtracted from the values i n Columns 3 and 4, giving the net values of standing trees. These f i n a l values are shown in Columns 6 and 7 of Table 39 and the net revenue values per tree i n graph form i n Figure 9. It can be seen that 12-inch d.b.h. Douglas f i r and 14-inch hemlock trees are the smallest sizes paying t h e i r way. They constitute the zero marginal tree s i z e , provided the s p e c i f i e d conditions hold and provided the natural v a r i a b i l i t y i n the estimates from the o r i g i n a l data i s ignored. Results of the computations, leading to the quoted tree si z e s , have been summarized i n Table 40. The various cost items are shown on a per tree basis. M i l l i n g cost has been given a separate p o s i t i o n because of i t s r e l a t i v e magnitude i n respect of the other costs. This tabulation emphasizes the fact that the smaller logs are unprofitable to saw and the conversion process becomes p r o f i t a b l e only for the larger s i z e s . The various r e s u l t s obtained up to t h i s stage may be summarized as follows: 1. The zero marginal size for Douglas f i r i s 12 i n . d.b.h. 2. The zero marginal size for hemlock i s 14 i n . d.b.h. 3. I t i s more p r o f i t a b l e to s e l l the logs obtained from trees of d.b.h. < 21 i n . 4. I t i s more p r o f i t a b l e to m i l l the logs from trees with d.b.h. > 21 i n . , and to market the products. This p a r t i c u l a r set of r e s u l t s , with the associated condi-tions and pr i c e schedules, w i l l henceforth be referred to as Program I. 80 DBH in Inches Fig- 9 Net Revenue per Tree-, Program I-TABLE 40 SUMMARY OF PROGRAM I - COST ITEMS AND RETURNS V a r i o u s C o s t I t e m s , D o l l a r s P e r T r e e D . b . b : . F e l l i n g Y a r d i n g L o a d i n g G o v ' t . - G r a n d I n . B u c k i n g H i g h l e a d H a u l i n g R o y a l t y Towing S c a l i n g T o t a l M i l l i n g T o t a l 12 0 . 6 2 0 . 4 0 0 . 7 5 0 . 1 0 0 . 0 6 0 . 0 1 1 .94 3 . 0 8 5 .02 13 0 . 6 7 0 . 4 0 0 . 9 1 0 . 1 2 0 . 0 8 0 . 0 2 2 . 2 0 3 . 0 9 5 . 2 9 14 0 . 7 4 0 . 4 0 0 . 9 1 0 . 1 4 0 . 1 0 0 . 0 2 2 .31 3 . 0 9 5 . 4 0 15 0 . 8 5 0 . 8 0 1 .64 0 . 1 8 0 . 1 2 0 . 0 3 3 .62 5 . 1 9 8 .81 16 0 . 9 4 0 . 8 0 1 .89 0 . 2 2 0 . 1 4 0 . 0 3 4 . 0 2 - 5 . 6 8 9 . 7 0 17 1 .01 0 . 8 0 1 .89 0 . 2 6 0 . 1 8 0 . 0 4 4 . 1 8 5 . 6 8 9 . 8 6 18 1.11 0 . 8 0 2 . 2 0 0 . 3 1 0 . 2 1 0 . 0 5 4 . 6 8 6 . 2 3 1 0 . 9 1 19 1 .18 0 . 8 0 2 . 3 4 0 . 3 7 0 . 2 5 0 . 0 6 5 . 0 0 6 . 4 9 1 1 . 4 9 20 1 .29 0 . 8 0 2 . 8 8 0 . 4 5 0 . 3 0 0 . 0 7 5 .79 7 . 0 5 1 2 . 8 4 21 1 .36 0 . 8 0 2 . 9 2 0 . 5 2 0 . 3 4 0 . 0 8 6 . 0 2 6 . 8 5 1 2 . 8 7 22 1 .55 1 .20 4 . 4 3 0 . 6 1 0 . 4 0 0 . 0 9 8 . 2 8 1 0 . 9 9 1 9 . 2 7 23 1 .63 1 .20 4 . 7 4 0 . 7 2 0 . 4 8 0 . 1 1 8 . 8 8 1 0 . 9 6 1 9 . 8 4 24 1 .74 1 .20 5 . 2 4 0 . 8 4 0 . 5 6 0 . 1 3 9 .71 1 1 . 8 9 2 1 . 6 0 25 1 .78 1 .20 5 . 9 3 0 . 9 6 0 . 6 4 0 . 1 4 1 0 . 6 5 1 2 . 2 0 2 2 . 8 5 L o g P r i c e s P e r T r e e $ N e t R e v e n u e $ # 3 # 3 D . b . h . # 2 # 3 # 2 # 3 F i r H e m l o c k B a l s a m I n . F i r F i r H e m l o c k H e m l o c k B a l s a m L o g g i n g M i l l i n g L o g g i n g M i l l i n g L o g g i n g 12 4 . 1 8 3 .31 3 . 1 4 2 . 2 4 0 . 1 3 1.37 - 0 . 7 2 1 .20 13 - 5 .23 - 4 . 1 4 3 .92 3 .03 0 . 9 3 1 .94 - 0 . 2 7 1 .72 14 7 . 7 0 6 . 2 7 - 4 . 9 6 4 . 7 1 3 .96 1 .76 2 . 6 5 0 . 3 6 2 . 4 0 15 9 . 7 1 7 . 8 4 - 6 . 2 0 5 .89 4 . 2 2 2 .71 2 . 5 8 1 .15 2 .22 16 1 1 . 6 5 9 . 4 1 - 7 . 4 4 7 . 0 6 5 .39 4 . 0 4 3 .42 2 . 0 6 3 . 0 4 17 1 4 . 2 4 1 1 . 5 0 - 9 . 1 0 8 .63 7 .32 5 .59 4 . 9 2 3 . 2 4 4 . 4 5 18 1 6 . 8 3 1 3 . 5 9 - 1 0 . 7 5 1 0 . 2 0 8 .91 7 . 0 9 6 . 0 7 4 . 7 9 5 .52 19 2 0 . 0 7 1 6 . 2 1 - 1 2 . 8 2 1 2 . 1 6 1 1 . 2 1 8 . 0 9 7 .82 6 . 6 2 7 . 1 6 20 2 4 . 4 1 1 9 . 7 1 1 7 . 7 2 1 5 . 5 9 1 4 . 7 9 1 3 . 9 2 1 2 . 3 1 9 . 8 0 8 .71 9 . 0 0 21 2 7 . 8 4 2 2 . 4 8 2 0 . 2 1 1 7 . 7 8 1 6 . 8 7 1 6 . 4 6 1 5 . 6 4 1 1 . 7 6 1 1 . 4 4 1 0 . 8 5 22 3 2 . 6 9 2 6 . 4 0 2 3 . 7 4 2 0 . 8 9 1 9 . 8 2 1 8 . 1 2 1 9 . 2 5 1 2 . 6 1 1 4 . 2 0 1 1 . 5 4 23 3 8 . 8 4 3 1 . 3 7 2 8 . 2 1 2 4 . 8 2 2 3 . 5 4 2 2 . 4 9 2 3 . 3 7 1 5 . 9 4 1 7 . 5 7 1 4 . 6 6 24 4 5 . 2 5 3 6 . 5 4 3 2 . 8 6 2 8 . 9 1 2 7 . 4 3 2 6 . 8 3 2 8 . 0 6 1 9 . 2 0 2 1 . 2 6 1 7 . 7 2 25 5 1 . 6 6 4 1 . 7 2 3 7 . 5 1 3 3 . 0 1 3 1 . 3 1 3 1 . 0 7 3 4 . 2 2 2 2 . 3 6 2 5 . 7 2 2 0 . 6 6 82 ZERO MARGINAL TREE SIZE FOR AN OPERATION WITH IMPROVED PERFORMANCE (PROGRAM II) I t appears from the r e s u l t s of Program I that the operations at the University Research Forest at the time of t h i s study were not e f f i c i e n t . Although the t h e o r e t i c a l zero margin l i e s at 12 inches d.b.h. for Douglas f i r and 14 inches d.b.h. for hemlock, there i s no allowance for a "safety factor" to counter the ine v i t a b l e inaccuracies i n the various measurements and the normal v a r i a b i l i t y within the machine performance. The curves of Figure 9 show that the increase i n net revenue, with increas-ing d.b.h. s i z e s , i s small for the lower ranges, and increases more r a p i d l y when larger trees are handled. This suggests the uncertainty associated with a precise determination of zero marginal trees. A safety margin of $4.00 per tree would increase the zero marginal si z e by approximately four inches, as shown i n Figure 9. Afte r considering various ways i n which the same operation could be improved, i t was decided to r e c a l c u l a t e the zero margin for an operation which i s more e f f i c i e n t yet s t i l l r e a l i s t i c . The s e l l i n g p rice of logs, rather than that of sawn lumber, was taken as the basis of the analysis. The following computations and solutions w i l l henceforth be r e f e r r e d to as Program II and w i l l occupy the next part of t h i s t h e s i s . 83 A. MARKET PRICES OF LOGS Log p r i c e quotations are available and form a convenient basis for computations. The prices used i n the following analysis are the Vancouver Log Prices, September 1961. The attached statement i n the Appendixldoes not take into consideration the size of logs. I t i s assumed here that the "small logs", e.g. below size 8 inches by 32 feet, may be sold at the same p r i c e as the large ones.provided the percentage of such logs i s not excessive. Log grades are indicated by the following table, which shows the quality of the stand being harvested. TABLE 41 APPROXIMATE STAND QUALITY D.b.h.  In. Douglas F i r Hemlock Percentage No. 2 Sawlogs 7 to 16 0 20 34 41 41 42 42 43 44 0 0 0 0 0 12 23 27 30 16 18 20 22 24 26 28 30 * Data supplied by Dr. J . H. G. Smith, Faculty of Forestry, University of B r i t i s h Columbia. TABLE 42 BUCKING SCHEDULE FOR PROGRAM II Number Volume Y i e l d Per Tree, f.b.m. D.b.h. of Logs Bucked at Landing Adjusted for Overrun In. Hauled Top - i n s . Length - f t . Douglas F i r Hemlock 7 I 6 X 16 24 23 8 1 7 X 16 37 36 9 1 7% X 16 40 38 10 1 8 X 16 48 45 11 1 6% X 32 58 56 12 1 7 X 32 74 71 13 1 7% X 32 85 81 14 1 8 X 32 96 90 15 1 9 X 30 123 111 16 1 6% X 16 10 X 32 175 165 17 1 7 X 16 10% X 32 190 181 18 1 8 X 16 11 X 32 222 212 19 2 6% X 32 11% X 32 251 241 20 2 7 X 32 13% X 32 338 326 21 2 6 X 16 8 X 32 14% x 32 424 407 22 2 6% X 16 • 8% X. 32 15% x 32 492 461 23 2 8 X 32 12 X 32 20 x 16 600 555 24 2 6% X 16 8 X 32 13 x 32 21% X 16 716 670 25 2 7 X 32 11 X 32 19 x 32 784 741 26 2 8 X 32 12 X 32 20 x 32 897 842 27 2 6 X 16 8% X 32 12% x 32 21 X 32 1017 949 28 2 7 X 16 9 X 32 13 x 32 21% X 32 1098 1016 29 2 7% X 16 10 X 32 14 x 32 22% X 32 1231 1151 30 2 8 X 16 10% X 32 15 x 32 23% X 32 1374 1273 CO 85 B. THE UTILIZED VOLUME PER TREE I t i s assumed that f e l l e d trees are bucked at a 6-inch top. They are yarded out f u l l length, the l i m i t for handling being set at 60 feet. The trees are bucked into logs at the landing. A bucking schedule has been prepared which shows the u t i l i z e d volumes per tree for the various size classes (Table 42). C. GROSS VALUES OF TREES A schedule of gross values per tree can be set up as shown in Table 43, by combining the information of log market p r i c e s , stand quality i n respect of # 2 and # 3 sawlog d i s t r i b u t i o n , and the available volume per tree. TABLE 43 GROSS VALUES OF STANDING TREES D.b.h. Gross Value Per Tree, Dollars  In. Douglas F i r Hemlock 7 1.25 0.95 10 2.51 1.86 12 3.87 2.94 14 5.02 3.72 16 9.58 6.82 18 12.55 8.77 20 19.40 13.48 25 45.09 31.36 30 79.36 54.82 86 Fig 10 Felling and Bucking Cost per Tree; Douglas Fir and Hemlock-, Program 2-87 D. FELLING AND BUCKING F e l l i n g and bucking costs are based on the time studies of June, 1961 at the University Research Forest and have been assembled i n Table 44 for the proposed new bucking schedule of Program I I . The f i n a l per tree costs have been smoothed graphically to minimize v a r i a b i l i t y i n the measured data. (Figure 10.) TABLE 44 FELLING AND BUCKING COST PER TREE D.b.h. Cost Per Tree, Dollars In. Douglas F i r Hemlock 7 0.39 0.41 10 0.52 0.54 12 0.65 0.65 14 0.77 0.77 16 0.92 0.91 18 1.11 1.09 20 1.31 1.29 25 1.86 1.83 30 2.29 2.26 E. LOADING COST This cost i s based on the performance shown by the 4-man crew at a salvage operation i n the University Research Forest i n June, 1961. The computed values are shown i n Table 45 and Figure 11. 88 D-B-H- in Inches Fig- II Loading Costj Program 2 89 TABLE 45 LOADING COST PER TREE B.b.h. In. Cost Per Tree, Dollars Douglas F i r Hemlock 7 10 12 14 16 18 20 25 30 0.15 0.21 0.27 0.33 0.40 0.49 0.60 0.93 1.27 0.15 0.21 0.26 0.32 0.39 0.47 0.56 0.86 1.20 F. COST OF YARDING As a r e s u l t of low e f f i c i e n c y i n the yarding operation at the University Research Forest salvage operation, the apparent yarding cost per hour was $22.86, or $15.79 per M f.b.m. This cost figure i s extremely high by any a v a i l a b l e comparison and a reduction of cost at t h i s stage of operation i s of great importance. In s e t t i n g up Program I I , i t was assumed that the improvement would be achieved by a more e f f e c t i v e use of the available time. Because that problem i s regarded as purely a mechanical and organizational one, no s p e c i f i c solution w i l l be offered within t h i s work. I t w i l l be merely assumed that the yarding crew w i l l be working eff e c -t i v e l y 5 hours out of 8 and that the various time requirements remain unchanged from Program I. 90 The t o t a l d a i l y cost of $80.00 w i l l then give an hourly an 60 cost of — = $16.00. The se t t i n g w i l l be fi n i s h e d i n ~ + 2 = 14 days, with a t o t a l cost of 14 x 80 = $1120. This i n turn w i l l give a unit cost = $11.64. This value implies that, yb.Zoo on the average, $4 per M f.b.m. w i l l be saved by t h i s new working rate. Within the framework of Program II the logs are yarded out i n f u l l length up to 60 feet. This procedure w i l l increase the production s u b s t a n t i a l l y and further lower the per unit cost. I f i t i s assumed that a f u l l turn for an average yarding distance of 400 feet w i l l take 4.18 minutes, then the yarding cost per tree w i l l depend only on the number of turns necessary to haul i t to the landing. The bucking schedule (Table 42) shows that trees of the size 7 to 18 inches d.b.h. are hauled out i n one piece whereas trees of size 19 to 30 inches d.b.h. y i e l d 2 logs. Consequently the lumped yarding cost for the f i r s t group (d.b.h. 7 to 18) i s $0.49 per tree and for the second group $0.98. These values are based on an e f f e c t i v e 80 hourly cost of —— = $14. or 95$ per 4.18 minutes. The high-lead mainline i s assumed to be equipped with two chokers. ] 91 G. OTHER COSTS ASSOCIATED WITH PROGRAM II 1. Hauling Hauling is done by contract and the set price for hauling one M f.b.m. 18 miles is $6.25. This haul w i l l transfer the logs from the University Research Forest to a booming ground near the P i t t River Bridge. 2. Scaling and Royalty The established fee for a government scaler is 18c per M f.b.m. The Royalty depends on log grade and is $1.00 per M f.b.m. for # 3 logs and $2.00 per M f.b.m. for # 2 logs. 3. Towing The price quoted by a local towing company indicates that the cost for towing from P i t t River Bridge to Eburne Saw Mills would be 80<? per M f.b.m. The above costs have been recalculated on a per tree basis and are shown in Table 46. 92 TABLE 46 MISCELLANEOUS COSTS OF PROGRAM II $ PER TREE D.b.h. Scaling Royalty Towing Hauling In. F H F H F H F H 7 0.01 0.01 0.02 0.02 0.01 0.15 0.14 10 0.01 0.01 0.05 0.04 0.03 0.30 0.28 12 0.01 0.01 0.07 0.07 0.04 0.46 0.44 14 0.02 0.02 0.10 0.09 0.05 0.60 0.56 16 0.03 0.03 0.21 0.16 0.10 1.09 1.03 18 0.04 0.04 0.30 0.21 0.14 1.39 1.32 20 0.06 0.06 0.48 0.33 0.21 2.11 2.04 25 0.14 0.13 1.11 0.87 0.52 4.90 4.63 30 0.25 0.23 1.98 1.65 0.94 8.59 7.95 H. COMPUTATION OF NET REVENUES PER TREE The various computed cost items are assembled in Table 47. For each d.b.h. class a l l cost items have been totalled and then subtracted from the revenue which each tree i s expected to yield, according to i t s content of saleable logs. The difference, or the net revenue, has been shown in Figure 12. TABLE 47 PROGRAM II - COSTS AND NET REVENUES PER TREE 1 Total Cost Revenue Net Revenue D.b.h. Per Tree $ Log Prices $ Per Tree $ In. F H F H F H 7 1.22 1.23 1.25 0.95 0.03 -0.28 8 1.39 1.41 1.93 1.43 0.54 0.02 9 1.47 1.49 2.09 1.57 0.64 0.08 10 1.61 1.60 2.51 1.86 0.90 0.26 11 1.76 1.78 3.03 2.32 1.27 0.54 12 1.99 1.96 3.87 2.94 1.88 0.98 13 2.16 2.13 4.44 3.35 2.28 1.22 14 2.36 2.30 5.02 3.72 2.66 1.42 15 2.68 2.56 6.43 4.59 3.75 2.03 16 3.24 3.11 9.58 6.82 6.34 3.71 17 3.52 3.36 10.57 • 7.49 7.05 4.13 18 3.96 3.76 12.55 8.77 8.59 5.01 19 4.84 4.62 14.28 9.97 9.44 5.35 20 5.75 5.47 19.40 13.48 13.65 8.01 21 6.69 6.29 24.33 16.83 17.64 10.54 22 7.43 6.89 28.24 19.07 20.81 12.18 23 8.53 7.85 34.43 22.95 25.90 15.10 24 9.67 9.03 41.18 28.17 31.51 19.14 25 10.44 9.82 45.09 31.36 34.65 21.54 26 11.54 10.87 51.59 35.92 40.05 25.05 27 12.73 11.90 58.74 40.59 46.01 28.69 28 13.58 12.68 63.42 43.58 49.84 30.90 29 14.87 13.99 71.10 49.49 56.23 35.50 30 16.30 15.21 79.36 54.82 63.06 39.61 I. RESULTS Program II computations suggest that the zero marginal Douglas f i r has a d.b.h. of 7 inches and hemlock of 8 inches. The r e l a t i v e l y f l a t net revenue curve implies, for the smaller 1 Assuming no penalty for logs under 8 inches top d.i.b. 94 DB H in Inches Fig- 12 Net Revenue per Treei Program 2-95 siz e s , that the increase i n net revenue i s small, and over the range of 7 to 15 inches, the exact value of net revenue i s somewhat i n doubt. I t should be r e a l i z e d that the curve shown i s only a best estimate of highly variable factors and probably has a wide range of v a r i a t i o n which does not appear i n the graph. Around 15 inches d.b.h. the slope of the curve becomes steeper and i t may be assumed that the p r o f i t a b i l i t y of the operation i s no longer subject to an excessive amount of uncertainty. The preceding i n t e r p r e t a t i o n of the revenue curves i s far too subjective. Undoubtedly, with more basic data a v a i l a b l e , the estimates may be made more precise and both the t h e o r e t i c a l and economically safe zero marginal tree si z e may be determined with a greater degree of certainty. 96 LINEAR PROGRAMMING A. INTRODUCTION In the foregoing discussion, the determination of zero marginal tree has been approached in a very general manner. No emphasis was given to the actual size of the available timber stand, available number of machine-hours or capital. A consideration of this aspect is obviously important when a specific operation is to be evaluated. The introduction of material restrictions or capital restrictions w i l l limit the number of possible choices of action on the part of the operator. The problem w i l l become that of optimization under a given set of restrictive factors. The technique of linear programming (LP) i s often useful in solving complex problems of this nature. B. PRINCIPLES LP encompasses problems in which the quantity to be maxi-mized (e.g. profit) or minimized (e.g. cost) is stated as a linear function of the independent variables, and is subject to a system of linear inequalities stated in terms of these variables. Thus, an LP problem has three quantitive components: 1. An objective; 2. Alternative methods or processes for attaining the 97 objective, and 3. Resource or other r e s t r i c t i o n s . A problem which has these three components can always be expressed as a LP problem. Thus i t i s easy to construct a logging s i t u a t i o n , expressed i n terms of l i n e a r equations, and maximized according to LP methods. PROBLEM AND SOLUTION NO. 1 An area of one acre i s to be logged. Stand composition for t h i s area i s given i n Table 48. TABLE 48 STAND COMPOSITION AND VOLUME F.b.m. V o l .  Vol. Per D.b.h. No. of Trees Per Acre Per D.b.h. Group In. D. F i r Hemlock Cedar Total Tree Class/ac . No. 6 24 26 21 71 20 1420 ' 8 21 17 10 47 37 1739 r T 10 12 19 4 25 48 1200 1 12 12 5 3 21 74 1554 J 14 8 4 2 14 96 1344 " ) 16 4 3 1 8 175 1400 > II 18+ 5 1 1 7 250 1750 . I t was demonstrated i n the e a r l i e r chapters that a l l the various tree sizes have d i f f e r e n t unit costs which decrease with increasing tree diameter. I f i t were possible to carry out a computation whereby each tree was treated as a d i s t i n c t u n i t , and provided that s u f f i c i e n t l y d e tailed information were 98 a v a i l a b l e t o charge a p r o p e r amount o f c o s t a g a i n s t each i n d i v i -d u a l t r e e , t h e n i t w o u l d be p o s s i b l e t o a r r i v e a t a c o m p l e t e l y c o r r e c t e s t i m a t e o f t h e n e t v a l u e s o f t h e t r e e s i n t h a t a r e a . Such a c a l c u l a t i o n o b v i o u s l y i s q u i t e i m p r a c t i c a l . A t t h e o t h e r extreme, i t i s p o s s i b l e t o t r e a t t h e whole s t a n d i n terms o f averages - v i z . average s i z e t r e e s and average h a u l i n g d i s t a n c e s . A c a l c u l a t i o n b a sed on t h i s a p p r o a c h i s q u i t e f e a s i b l e and v e r y o f t e n t h e one used i n p r a c t i c e . I t i s apparent t h a t , due t o o v e r s i m p l i f i c a t i o n , much d e t a i l i s l o s t and t h e r e s u l t i s a t b e s t an a p p r o x i m a t i o n . T h i s s o l u t i o n cannot f i n d t h e optimum c o u r s e o f a c t i o n u n l e s s t h r o u g h p u r e chance. O b v i o u s l y a compromise must e x i s t between t h e s e two o p p o s i t e a p p r o a c h e s . I t i s s u g g e s t e d t h a t t h e s t a n d s h o u l d be d i v i d e d i n t o a number o f d.b.h. c l a s s e s . The average t r e e o f each such c l a s s w i l l be used as an e n t i t y , o r a " p r o c e s s " as such u n i t s a r e o f t e n d e s i g n a t e d i n LP c o n t e x t . A f t e r t h e v a r i o u s t e c h n i c a l c o e f f i c i e n t s and r e s t r i c t i o n s a r e d e t e r m i n e d f o r each p r o c e s s , t h e r e g u l a r LP t e c h n i q u e s a r e a p p l i e d . The r e s u l t i n g s o l u t i o n s s u ggest how much o f each " p r o c e s s " mufct be t a k e n i n o r d e r t o maximize t h e o v e r a l l p r o f i t f u n c t i o n . As an i n i t i a l a p proach t o a s o l u t i o n o f t h e h a r v e s t i n g i n t e n s i t y o f t h e i n d i c a t e d one a c r e , t h e t r e e d i a m e t e r s a r e d i v i d e d i n t o two g r o u p s , l a r g e and s m a l l . W i t h i n each group t h e mean d.b.h. v a l u e ( w e i g h t e d by volumes and numbers o f t r e e s ) 99 is calculated to represent that group: Mean Trees per Vol. per Vol. per Group d.b.h. Acre Acre Tree 1 8 164 5913 36 2 16 29 4494 155 As a further simplification, i t is assumed that a l l trees are Douglas f i r . The problem i s to determine how many of the available trees in each group should be cut in order to maximize the returns from the operation. The profit equation i s : C - p l X ] L + p 2x 2- ( x 1 s 1 + x 2 s 2 ) where p^ = the price of 8-in. tree = $1.93/tree p 2 = the price of 16-in. tree = $9.58/tree x^ = No. of trees cut in Group I x 2 = No. of trees cut in Group II s^ and s 2 = miscellaneous charges (towing, royalty, etc.) per respective group. The restrictions in this problem are the number of available machine-hours per day for the various phases of operation, as well as the required number of machine-hours per tree to execute the various operations. These values are available from time study data which have been discussed earlier in this thesis. Assume the following available number of machine-hours per day: 100 Felling and bucking 7 hr. ( = 420 min.) Yarding (500 ft.) 5 hr. ( = 300 min.) Loading.. 5 hr. ( = 300 min.) The machine-hour requirements per tree are: Felling Yarding Loading (min.) Group I 3.89 3.00 0.9 Group II 8.14 3.00 1.24 Contract hauling per tree: Group I - $0.23 Group II - 1.09 Miscellaneous charges per tree are: Royalty Scaling Towing Hauling Total Group I 0.04 0.01 0.02 0.23 0.30 Group II 0.21 0.03 0.10 1.09 1.43 The profit equation becomes: C = 1.93XX + 9.58X2 - 0.30X1 - 1.43X2 or C = 1.63X1 + 8.15X2 subject to the following restrictions: 3.89X1 + 8.14X2 n< 420 3.0QXl + 3.00X2 ^ 300 0.9Xl + 1.24X2 £ 300 X x £ 164 X 2 ^ 29 X± >/ 0 x 2 >, 0 Because there are only two variables in the problem, a graphic solution may be carried out as in Figure 13. The seven inequalities are represented by a number of lines and the area 102 of feasible solutions is determined. To find the optimum solution among the feasible solutions, the profit function C = l.63X± + 8.15X2 is plotted for various values of C, e.g. C = 200. This line i s then moved, parallel to i t s original position, away from the origin u n t i l i t is tangent to the farthest point of the con-strained region. From Figure 13 i t may be seen that this point is located at the crossing of the lines of f e l l i n g and supply restriction of Group II. The solution for the optimum strategy may thus be obtained by solving the following equations: 3.89X1 + 8.14X2 = 420 X 2 = 29 then X x = 47 X 2 = 29. This gives the profit equation the following value: C = (1.63 x 47) + (8.15 x 29) = $312.96. The real net returns are obtained after fallers wages, yarding and loading costs have been subtracted from this value. The solution shows that out of 164 available trees in Group I, only 47 should be taken. The solution does not indicate which these trees are, since a l l trees within a group are treated as equals. It seems most lik e l y that the largest ones are taken and the smaller ones l e f t behind. The stand composition table (Table 48) l i s t s 21 and 25 trees 103 in the 12 and 10-inch d.b.h. groups respectively. Consequently a l l trees 10-inch d.b.h. and larger should be harvested and a l l others l e f t behind. The graphical solution for this Problem No. 1 was possible because the inequalities were given for two variables. Two-dimensional representation has the great advantage in making the nature of LP problems better understood by the reader. The solution may be made more accurate by extending this two-dimensional problem into the nth dimension. In the present instance, n would indicate the number of size classes into which the stand has been divided. Obviously, the larger n i s , the closer the zero marginal tree can be determined, although the need for more detailed information w i l l grow and hence also the cost of the whole project. The solution of a n-dimensional LP problem is usually obtained by the Simplex Method. This method, which was developed by G. B. Dantzig (Koopmans, 1951), is an iterative procedure, which approaches the optimum solution in a systematic manner by testing and rejecting feasible solutions u n t i l the maximum (or minimum) value has been reached. Because the computations are long and repetitive, the use of electronic computers f a c i l i t a t e s the application of this technique to a very considerable degree. 104 PROBLEM AND SOLUTION NO. 2 Once more one acre of forest i s considered for harvesting. The composition of the stand i s given i n Table 49. TABLE 49 STAND COMPOSITION AND THE EXPECTED PERCENTAGE OF # 2 No. of 7. Price D.b.h. Trees # 2 Per Tree In. Per Ac. Loss (DF) 6 71 0 1.00 88 47 0 1.93 10 25 0 2.51 12 21 0 3.87 14 14 0 5.02 16 8 20 9.58 18+ 7 40 19.40 The avai l a b l e number of machine-hours i s unchanged from the previous problem (No. 1) and the machine-hour requirements per tree for each d.b.h. class are given i n Table 50. A l l values are based on time study data outlined i n the f i r s t parts of th i s t h e s i s . 105 TABLE 50 FELLING, BUCKING, YARDING AND LOADING TIMES ASSOCIATED WITH PROBLEM NO. 2 D.b.h. F e l l i n g & 500' Yarding Loading In. Bucking min. min. min. 6 3.50 1.5 0.6 8 3.89 1.5 0.9 10 4.62 1.5 0.95 12 5.39 1.5 1.2 14 6.42 1.5 1.3 16 8.14 3 1.3 18+ 12.0 3 1.5 TABLE 51 MISCELLANEOUS CHARGES PER TREE PROBLEM NO. 2 D.b.h. Contract In. Royalty Scaling Towing Hauling To t a l 6 $0.01 $0.01 $0.01 $0.13 $0.16 8 0.04 0.01 0.02 0.23 0.30 10 0.05 0.01 0.03 0.30 0.39 12 0.07 0.01 0.04 0.46 0.58 14 0.10 0.02 0.05 0.60 0.77 16 0.21 0.03 0.10 1.09 1.43 18+ 0.48 0.06 0.21 2.11 2.86 The objective function i s : C = 1.00X, + 1.93X. + 2.51X0 + 3.87X. + 5.02XC + 9.58X, + 19.40X, 1 2 3 4 5 6 7 - 0.16X, - 0.30Xo - 0.39Xo - 0.58X. - 0.77XC - 1.43X, -1 2 3 4 5 6 2.86X? ; C = 0.84X, + 1.63X_ + 2.12X0 + 3.29X. + 4.25XC + 8.15X, + 16.54X.,. i L J "t D D / Subject to the following r e s t r i c t i o n s : 107 By adding disposal a c t i v i t i e s or so-called slack v a r i a b l e s , the above i n e q u a l i t i e s are converted to a system of e q u a l i t i e s , shown i n Appendix I I . This Simplex Tableau or Computational table i s the s t a r t i n g point for the computation steps which comprise the Simplex Method. The solution for Problem No. 2 was obtained using the Alwac III E ele c t r o n i c computer at the Computing Centre of the University of B r i t i s h Columbia. A standard program for the solution of Simplex problems i s available at the Centre's tape l i b r a r y . In Appendix V are outlined b r i e f l y the procedures followed i n using the computer for the solution of a Simplex problem. SOLUTION OF PROBLEM NO. 2 A summary of the information i n the problem and the o p t i -mum solution i s shown i n the attached copy of the o r i g i n a l worksheet from the computer (Appendix I I I ) . Beneath the o r i g i n a l matrix, which for technical reasons has a d i f f e r e n t order as compared to the standard Simplex Tableau, are given the net return values for the objective function. I t can be seen that the optimum value §340.7) was reached i n f i v e i t e r a -tions . The l a s t column on the worksheet indicates the input l e v e l s for the optimum solution. The "p" designates a factor to be used and the "q" stands for a discarded value. Thus the f i r s t f igure, p03-14.67748856 means that the t h i r d input i n the objective function ( = d.b.h. class 10 inches) should be 108 u t i l i z e d at a l e v e l of 14.68. These r e s u l t s are summarized i n Table 52. TABLE 52 THE SOLUTION FOR PROBLEM NO. 2 Number Total D.b.h. Trees Program Price Gross In. Available Solution $ Revenue 6 71 0 0.84 0 8 47 0 1.63 0 10 25 14.7 2.12 31.16 12 21 21 3.29 69.09 14 14 14 4.25 59.50 16 8 8 8.15 65.20 18+ 7 7 16.54 115.78 $340.73 Consequently the gross returns per one acre and one day's work were $340.73. The net return for t h i s time and area i s obtained by subtracting the pertinent costs from t h i s value: Gross return $340.73 F a l l e r & bucker wages 40.00 Highlead & crew cost 80.00 Loader & crew cost 80.00 Net revenue $140.73 I t i s obvious that a change i n the siz e of the area or time c o n s t r i c t i o n w i l l a l t e r a t e the r e s u l t s . Thus, i f a large area i s considered but the t o t a l time allowed i s short, the zero marginal l i m i t w i l l not be reached; the optimum program sp e c i f i e s that the largest trees be harvested i l i r s t . A further refinement i n the formulation of the problem can consider t h i s 109 t i m e f a c t o r a l o n g w i t h t h e o t h e r s . P r o b l e m N o . 3 i l l u s t r a t e s s u c h an a p p l i c a t i o n . PROBLEM AND SOLUTION NO. 3 A c c o r d i n g t o t h e s o l u t i o n o f P r o b l e m N o . 2 , t h e z e r o m a r g i n a l s i z e r e f e r s t o a t r e e o f 10 i n c h e s d . b . h . T h e p r o -g ram s u g g e s t s t h a t a b o u t h a l f o f t h e a v a i l a b l e t r e e s i n t h a t s i z e g r o u p s h o u l d be u s e d . A l t h o u g h i t i s n o t made e x p l i c i t , i t i s s a f e t o assume t h a t t h o s e 10 i n . t r e e s s h o u l d b e t a k e n , w h i c h r e q u i r e s h o r t e r y a r d i n g . T h e t r e e s s i t u a t e d f a r t h e r away f r o m t h e s p a r t r e e become s u b - m a r g i n a l . T h i s c o n s i d e r a -t i o n i m m e d i a t e l y s u g g e s t s t h e i n c o r p o r a t i o n o f f u r t h e r f a c t o r s i n t o t h e LP m a t r i x . I t mus t b e r e m e m b e r e d , h o w e v e r , t h a t t h e a d d i t i o n o f new f a c t o r s i n c r e a s e s t h e number o f " p r o c e s s e s " a n d c o n s e q u e n t l y a l s o t h e s i z e o f t h e m a t r i x . The l i m i t i s o b v i o u s l y s e t b y t h e c o m p u t e r ' s c a p a c i t y . A more d e t a i l e d s o l u t i o n f o r t h e d e t e r m i n a t i o n o f t h e z e r o m a r g i n a l t r e e may b e c o n s t r u e d w h e r e i n a d d i t i o n t o t r e e s i z e , t h e e f f e c t o f v a r i o u s s p e c i e s , y a r d i n g d i s t a n c e s a n d number o f d a y s o p e r a t e d w i l l b e i n c l u d e d . ' . In p r i n c i p l e t h e r e i s no d i f f e r e n c e b e t w e e n t h i s a n d t h e e a r l i e r p r o b l e m s : t h e v a r i o u s s i z e s , s p e c i e s , y a r d i n g d i s t a n c e s a n d t i m e l i m i t s w i l l o n l y m u l t i p l y t h e number o f v a r i a b l e s . T h e s o l u t i o n o f t h e new m a t r i x w i l l o n l y be l o n g e r a n d more l a b o r i o u s . B e c a u s e y a r d i n g d i s t a n c e s a r e t r e a t e d a s v a r i a b l e s . 110 i n Problem No. 3, i t i s suggested at t h i s stage that tr a c t o r skidding, rather than highlead yarding, be considered. During the time studies at the University Research Forest, the author obtained some performance values for the D-2 C a t e r p i l l a r t r a c t o r . These res u l t s had been featured e a r l i e r i n Figure 4. In t h i s program, three yarding distances w i l l be examined: 200, 600 and 1,000 feet. The species are Douglas f i r and hemlock and the stand composition i s assumed to be as shown in Table 53. TABLE 53 STAND COMPOSITION FOR PROBLEM NO. 3* D.b.h. No. of trees on 20 acres In. Douglas f i r Hemlock 8 410 522 10 246 256 12 256 170 14 150 118 The roundtrip times for a D-2 for the three distances are shown i n Figure 4. Because of considerable v a r i a b i l i t y i n those data, the e f f e c t of load s i z e i s subject to considerable uncer-t a i n t y . This e f f e c t w i l l nevertheless be incorporated into the equations because future work i s most l i k e l y to produce better time study data on t h i s factor. * Source: Smith, Ker, Csizmazia, 1961, p. 22. I l l The t r a c t o r i s equipped with 3 chokers and the cost of machine operation i s $48.00 per day. I f the machine works 5% hours out of 8, then the hourly cost becomes $8.75. TABLE 54 YARDING TIME PER TREE FOR VARIOUS YARDING DISTANCES Distance D.b.h. Logs 200 f t . 600 f t . 1,000 f t . Per Time , minutes, per: In. Tree Turn Tree Turn Tree Turn Tree 8 1 2.3 13 4.3 19 6.3 10 1 7 2.3 13 4.3 19 6.3 12 1 9 3 15 5 21 7 14 1 9 3 15 5 21 7 The time requirements for f e l l i n g and bucking are taken from Table 38. Because of i n s u f f i c i e n t information, the time requirements for hemlock are assumed to be 10% higher than for Douglas f i r ; bath are shown i n Table 55. TABLE 55 FELLING, BUCKING AND LOADING - TIME REQUIREMENTS PER TREE Machine-minutes D.b.h. F e l l i n g & Bucking Loading In. Douglas F i r Hemlock A l l Species 8 3.89 4.28 0.9 10 4.62 5.08 0.95 12 5.39 5.93 1.2 14 6.42 7.05 1.3 112 The d i r e c t charges against the trees such as Royalty, scaler's fee, towing and contract hauling are assumed to be equal for a l l species and have been adopted from Table 51 without change. The revenues, shown i n Table 52, are based on the log pr i c e schedule i n Appendix I. This schedule i s extended i n t h i s problem with a s p e c i f i c a t i o n that the p r i c e for "small" logs be $25.00 per M f.b.m. for hemlock and $27.50 for Douglas f i r . These are a l l logs obtained from trees with 13 inch d.b.h. or l e s s . TABLE 56 REVENUES FROM LOG SALES DOLLARS PER TREE - PROBLEM NO. 3 D.b.h. In. Douglas F i r Hemlock 8 1.02 0.90 10 1.32 1.12 12 2.02 1.77 14 5.02 3.66 Whereas i n Problems 1 and 2 one day was the allowable time, an additional dimension w i l l be added to Problem No. 3 by making t h i s time l i m i t a v a r i a b l e . While the additional operating days show diminishing gross * Modification suggested by Dr. J . H. G. Smith. 113 income because the best trees are removed f i r s t , the daily cost figure w i l l s t i l l remain constant. It is of special interest, therefore, to establish for an operation the number of days at which the income i s maximized. To follow this lead, three operations were considered on that 20-acre area, each of them lasting 1, 5 and 20 days, respectively. The machine-hours available for these operations are: 1 day 5 days 20 days Loading - 5 hours per day 300 min. 1500 min. 6000 min. Yarding - 7 " 11 " 420 min. 2100 min. 8400 min. Felling & bucking -7 hours per day 420 min. 2100 min. 8400 min. The objective function i s : C = 1.02(X + X + X ) + 1.32(X + X + X ) + 2.02(X + X + X ) v 1 2 y 4 5 6' 7 8 9 4- 5.02(X + 10 X 11 + X ) + 0.90(X + X 12 13 14 + x ) 15 + 1.12(X + 16 X 17 + X ) + 1.77(X + X 18 19 20 + x ) 21 + 3.66(X + 22 X 23 + X ) 24 where X^ = no. of D. f i r trees yarded 200 f t . - of 8 in. d.b.h. V — " " " 2 " IT " " 600 f t . - of 8 in. I I X3 ' i t " " 1000 f t . - of .8 • in . ii v — " •> " II " " 200 f t . - of 10 in. n V _ I ' 1 ! f t X5 " i t " 600 f t . - of 10 in. i t I t 114 X^^ = no. of hemlock trees yarded 600 f t . - of 14 i n . d.b.h. class X 2 4 - " " " " " 1000 f t . - of 14 i n . " The system of i n e q u a l i t i e s s t a t i n g the r e s t r i c t i o n s of t h i s problem becomes an extensive one and the subsequent addition of slack variables w i l l extend i t even more. Because of the capa-c i t y l i m i t of Alwac III E, which cannot handle Simplex matrices larger than 32 x 160, the number of factors must be kept down. This i s p a r t l y the reason why only two species and four d.b.h. classes were included i n t h i s analysis. The system of i n e q u a l i t i e s has not been given i n the Appendices because of i t s s i z e . I t may be seen coded on the computer working sheet of Appendix IV. SOLUTION OF PROBLEM NO. 3 From the appended worksheets, the following r e s u l t s were obtained as summarized i n Table 57. Owing to a shortcoming i n the programming, where the stand density had not been con-sidered, the computer could not d i s t i n g u i s h between the trees designated for short hauls and long hauls. A c t u a l l y , the number of avail a b l e trees within a c e r t a i n size class grows with the increasing radius of an operating area provided the stocking i s uniform. TABLE 57 SUMMARY OF SOLUTIONS FOR PROBLEM NO. 3 1. One-day Operation No. trees Progr. D.b.h. available Solution Price In. F H F H F H 8 ;4iL0 522 10 246 256 - - -12 256 170 - - -14 150 118 65 5.02 2. Five- day Operation No. trees Progr. D.b.h. available Solution Price In. F H F H F H 8 410 522 10 246 256 - - -12 256 170 56% - 2.02 14 150 118 150 118 5.02 3.66 3. Twenty-day Operation NQB trees Progr.  D.b.h. available Solution Price In. F H F H F H 8 410 522 410 43 1.02 0.90 10 246 256 246 256 1.32 1.12 12 256 170 256 170 2.02 1.77 14 150 118 150 118 5.02 3.66 T o t a l Gross  Revenue  F H A l l 328 $ 328 T o t a l Gross  Revenue  F H A l l 114 - 114 754 432 1186 $1300 T o t a l Gross Revenue F H A l l 418 39 457 325 284 609 517 301 818 754 432 1186 $3070 328 116 It may be seen from Table 57 that as the time limit is increased, more small trees may be hauled out. As this size decreases the net revenue diminishes because the daily operating costs remain constant. In other words, a series of such programs allows the operator to determine the longest time he should spend on a tract of land and also indicates the lowest size which should be cut during that operation. Under the given situation, the daily cost was assumed to be: Tractor operation $ 48 Faller's wages ..$ 40 Driver's wages $ 18 Loader & Crew cost.....$_80 Total $186 Consequently, the net revenue from one day's operation is 328 - 186 = $142; but for the five-day operation 1300 -5(186) = $360 or •3-|°- = $72 per day; and for the twenty days' operation the net revenue is 3070 - 20(186) = $650 loss or = $32 loss per day. A graphical extrapolation (Figure 14) shows that the average daily net revenue f a l l s continuously as the time spent on the area increases. At the same time, the cumulative net revenue (Figure 15) increases, reaching a maximum at the 5th day and then decreasing to become zero on the 13th day of operation. The five-day operation consequently constitutes the most 117 Number of Days Operated Fig- 14 Extrapolation of Net Revenues for Operations of various Durations- Problem No- 3-118 Fig- 15 Most Profitable Time-expenditure on the Operation- Problem No- 3-119 p r o f i t a b l e duration for t h i s area and i t i s seen that the margi-nal tree sizes are 12 inches d.b.h. for Douglas f i r and 14 inches d.b.h. for hemlock. A maximum of thirteen days could be spent on the area i f for some reason a l l p r o f i t i n the larger trees i s to be dissipated i n logging the smaller trees. To obtain a precise s i z e - l i m i t on the trees for that period, a fourth set of Simplex solutions was solved, substituting the times avai l a b l e at the 12-day l e v e l s . This means that the avail a b l e number of hours for the three operations are as follows: Loading 3900 hours Yarding 5460 " F a l l i n g & Bucking 5460 " With the modified program now available for the U. B. C. Alwac III E elec t r o n i c computer , i t i s possible to process several such systems i n rapid succession. The fourth part of Problem No. 3 was solved and the re s u l t s are summarized i n Table 58. For the worksheet, see Appendix IV. * Modified by J . Csizmazia TABLE 58 FOURTH SOLUTION OF PROBLEM NO. 3 4. Twelve-day Operation D.b.h. No. trees Progr. Price Total Gross av a i l a b l e Solution Revenue In. F H F H F H F H A l l 8 410 522 (36) 1.02 (37) mm (37) 10 246 256 246 - 1.32 325- - 325 12 256 170 256 170 2.02 1.77 517 301 818 14 150 118 150 118 5.02 3.66 753 432 1185 $2365 fO o 121 The net revenue from t h i s operation i s very close to zero: 2365 - 12(186) = $133 or ^ = $6.65 per day. The solution shows, that at t h i s rate of operation, the zero marginal tree size for Douglas f i r i s 10 i n . d.b.h. (the 36 trees i n 8 inch class are ignored) and for hemlock i s 12 inches, two inches below the most p r o f i t a b l e operating l e v e l . These r e s u l t s confirm those ar r i v e d at previously i n t h i s t h e s i s . DISCUSSION The c o l l e c t i o n and analysis of data on logging can be used to a great advantage i n planning the p r o f i t a b i l i t y of new harvesting operations. Such information may serve as a basis for a comparison of the r e l a t i v e advantages of several methods of conducting the various operations i n the woods and during log transportation. Often the major transportation methods are adopted without adequate consideration of t h e i r e f f e c t on the cost from stump to landing. The e f f e c t of i n d i v i d u a l tree s i z e i s seldom con-sidered as a factor i n a r e a l logging s i t u a t i o n . P r a c t i c a l considerations often must overrule the t h e o r e t i c a l l y optimal solutions and a compromise i s usually achieved through the i n t u i t i v e experience of the p r a c t i c a l logger. However, a wider appreciation of the underlying basic factors i s an area which 122 should not be ignored by the practical woods operator as i t has a direct bearing on the profit potentiality of a given logging system. Logging is f u l l of uncertainty. New situations arise continually; which must be solved on the spot and which seem to defy clearcut classification and evaluation in terms of simple time-motion studies. Nevertheless, i f the operating conditions were already so well standardized that l i t t l e or no new knowledge could be added over that acquired through cost accounting, then output studies would become entirely unnecessary. Therefore i t is the purpose of an investigator to distinguish the major factors affecting the output and to bring out their significance by studying their effects under a wide range of conditions. Data must be collected, segregated and compiled on the basis of different natural factors which may be identified and classified. Variation in the data should be recognized and allowed for in the subsequent computations. The extent of the time studies in the University Research Forest, in the summer of 1961, was such as to warrant no sweeping conclusions nor recommendations. Rather, i t emphasized the extent of problems and shed some light on the shortcomings in the present state of operations. The subsequent computations, which occupied the main part of this thesis, further showed the d i f f i c u l t i e s in economic decision-making in connection with the 123 harvesting and marketing of small trees. Although many earlier studies have conclusively proven the higher cost associated with the harvesting of smaller trees, the actual values for any specific condition must be obtained through an independent investigation. The time studies at the University Research Forest indi-cated that the 10 to 12 inch d.b.h. size class was the lower limit for trees s t i l l paying their way. It also showed that some improvement in yarding efficiency could, lower this size limit for Douglas f i r down to 7 inches, at least in theory. This low diameter limit is unrealistic from many points of view. It must be assumed that the market absorbs these small logs at f u l l market price and that the estimates of operating parameters are subject to very l i t t l e variation. The general "safe" lower limit, established rather empirically from the solutions of the three Programs, is around 14 to 15 inches d.b.h. for Douglas f i r and 16 to 17 inches for hemlock. Further-more, in light of the available cost data on sawmilling, which was only a rough estimate obtained from the B.C.L.M.A. person-nel, i t seemed to be unwise to convert the small logs into lumber products. Throughout this investigation, the lack of precise time and cost data has been the most serious problem. However, this shortcoming does not diminish the value of the general approach to the problem on hand. It is particularly true 124 with regard to the application of LP methods in a f i c t i t i o u s logging operation. Whereas the LP techniques have undergone considerable sophistication and refinement in such areas as a i r l i n e routing and the solution of refinery and el e c t r i c a l networks, the inherent uncertainty of a logging situation poses a serious challenge to the user of LP techniques in this f i e l d . According to Lussier (1961), the great complexity of large-scale logging calls for the application of a variety of Operating Research methods, of which LP i s only one of the available techniques. As a rule, the topics of Operation Research in general and LP in particular must be approached as complete and independent subjects in their own rights. It is not possible to explore the many subtle p o s s i b i l i t i e s of this new research tool in a work of this general nature. 125 SUMMARY AND CONCLUSIONS A survey of the literature dealing with logging and milling efficiency studies shows that, regardless of methods employed, the smaller trees are associated with higher harvest-ing and processing costs. Improved logging techniques have made i t possible to lower the size limit for profitable opera-tion, but even presently this marginal size is sufficiently high on the Pacific Coast to seriously cur t a i l efficient u t i l i z a t i o n of trees smaller than a d.b.h. of 12 inches. A time study conducted on the University Research Forest estimated the performances of f e l l i n g , bucking, yarding and loading operations. It was found that especially the high-lead yarding was inefficient and r e a l i s t i c improvements were recom-mended in subsequent calculations (Program II). The application of linear programming technique to some harvesting problems has been shown to be feasible. During the exposition and discussion of this methodology, three problem situations were thoroughly covered. A two-dimensional problem was solved by graphical technique and two multi-dimensional problems were solved by the use of the Alwac III E electronic computer. The results of the time study and the computations may be summarized as follows: 126 1. The present working e f f i c i e n c y , as e x i s t i n g i n the University Research Forest i n July 1961, i s subject to some shortcomings. E s p e c i a l l y serious was the lack of e f f i c i e n c y i n the high-lead yarding operation. 2. The t h e o r e t i c a l zero marginal size for Douglas f i r and hemlock was found to be 12 and 14 inches d.b.h. respectively. A reasonable improvement i n operating e f f i c i e n c y could lower t h i s value by several inches; however the present market would probably not be w i l l i n g to absorb t h i s small material at the p r i c e of No. 3 logs. 3. The application of LP may provide a faster method for analysing various harvesting s i t u a t i o n s . More r e f i n e d programs may show new approaches to the study of harvesting p r o f i t a b i l i t y . 4. Because of the large v a r i a b i l i t y of t y p i c a l time study data from logging operations, the f i e l d sampling must be r e l a -t i v e l y extensive. Only i n that way may f i n a l r e s u l t s be secured within reasonably narrow confidence l i m i t s . The time studies c a r r i e d out at the University Research Forest were i n the opinion of the author not of s u f f i c i e n t s i z e to give the r e s u l t s s a t i s f a c t o r y p r e c i s i o n . 127 BIBLIOGRAPHY Ashe, W. W. 1916. Cost of Logging Large and Small Timber. Forestry Quart., 14, 441-452. Bradner, M., and S. V. Fullaway, J r . 1927, 1928. Size of Timber, Amount of Defect, Important Factors i n Lumbering: An Analysis of Their E f f e c t upon Production Costs and Values and Their Subsequent Influence on P r o f i t a b l e Tree and Log U t i l i z a t i o n i n the Inland Empire. Timberman, 29 (2), 38-40, 44, 46, 48; (3), 40-42, 44, 46; (4), 62-63; (6), 162-166, 174. Bradner, M., F. J . Klobucher, J . W. Girard and S, V. Fullaway, J r . 1933. An Analysis of Log Production i n the "Inland Empire" Region. U. S. Dept. Agric. Tech. B u l l . No. 355. Chapman, H. H. and W. H. Meyer. 1947. Forest Valuation. McGraw-Hill Book Co., Inc. New York, N.Y. Doyle, J . A. 1957. E f f e c t of Tree Size of Spruce and Balsam F i r on Harvesting and Conversion to Lumber i n Nova Scotia. Dept. of North. A f f . and Nat. Res. For. Branch. F.P.L. Technical Note No. 5. Doyle, J . A. and W. W. Calvert, 1961. E f f e c t of Tree Size of Jack Pine on Harvesting and Conversion to Lumber i n Northern Ontario. Dept. of Forestry. Forest Products Research Branch Technical Note No. 19. Garver, R. D. and J . B. Cuno. 1932. The Portable Band Sawmill and Selective Logging i n the L o b l o l l y Pine Forests of North Carolina. U.S.D.A. Tech. B u l l . No. 337. Washington, D. C. Garver, R. D. and R. H. M i l l e r . 1933. Selective Logging i n the Shortleaf and L o b l o l l y Pine Forests of the Gulf States Region. U. S. Dept. Ag r i c . Tech. B u l l . 375, 53 pp. Gunn, D. C. and F. W. Guernsey. 1958. Skidding Time Studies i n the B. C. Southern I n t e r i o r . F.P.L. of Canada. Vancouver Laboratory. Reprint from B. C. Lumberman. 128 Guttenberg, S. and W. A. Duerr. 1949. A Guide to P r o f i t a b l e Tree U t i l i z a t i o n . Occasional Paper No. 114. Southern Forest Experiment Sta., New Orleans, La. Heiberg, S. 0. and P. G. Haddock. 1955. A Method of Thinning and Forecast of Y i e l d i n Douglas F i r . Jour, of For. Vol . 53 (1). Holt, L. 1949. Cutting Costs i n Relation to Tree Size. Pulp and Paper Research I n s t i t u t e of Canada. Woodlands Research Index No. 56 (F-2). Jackson, N. D. and G. W. Smith. 1961. Linear Programming i n Lumber Production. For. Prod. Jour. V o l . XI, No. 6: 272-274. Ker, J . W. 1959. A Study of Log Scaling and Lumber Recovery i n the Prince George Area of B r i t i s h Columbia. Northern In t e r i o r Lumbermen's Association. Prince George, B. C. Kilander, K. 1961. Time Consumption Variations for F e l l i n g of Unbarked Timber i n Northern Sweden (Swedish) Forst-mingsstiftelsen SDA, Meddelande nr 71. Stockholm. Kirkland, Burt P. and A. J . F. Brandstrom. 1936. Selective Timber Management i n the Douglas F i r Region, Pack Founda-t i o n , Washington, D. C. 122 pp. i l l u s . Kirkland, B. P. and A. J . F. Brandstrom. 1936. Selective Timber Management i n the Douglas F i r Region. U. S. D. A. Div. of Forest Economics. Forest Service. Koroleff, A. 1947. Pulpwood Cutting, E f f i c i e n c y of Technique. CP.P.A. Woodlands Section Index No. 630 (B-7-a). Koopmans, T. C. (ed.). 1951. A c t i v i t y Analysis of Production and A l l o c a t i o n , Cowles Commission for Research i n Economics, No. 13. New York: John Wiley & Sons, Inc. Kurta, J . 1961. An Analysis of F e l l i n g and Bucking Time Study on the University of B r i t i s h Columbia Research Forest. A B.S.F. Thesis. University of B r i t i s h Columbia. McBride, C. F. 1951. Lumber Recovery from Second Growth Western Hemlock. B. C. Lumberman. June 1951.. 129 McBride, C. F. 1949. Lumber Recovery from Douglas F i r Logs in British Columbia. Proceedings Vol. 3. Forest Prod. Res. Soc. Madison, Wise. McCloy, T. A. 1953. Relation of Tree Size to Production Rates when Cutting Pine Pulpwood with a Chain Saw. Southeastern Forest Expt. Sta. Research Notes No. 28. Mcintosh, J. A. and D. C. Gunn. 1961. Economics of Pre-Logging and Relogging in Cedar-Hemlock-Balsam. B. C. Lumberman, Vol. 45 No. 8. August 1961. pp. 10-14. Mcintosh, J. A. and D. C. Gunn. 1960. Pre-Logging with a Portable Steel Spar. British Columbia Lumberman, Vol. 44 No. 8. August 1960. Rapraeger, E. F. and H. R. Spelman. 1931. The Effect of Tree and Log Size on Felling and Bucking Costs in the Douglas Fi r Region. W. C. Lumberman. 58 (13): 20-23. Rapraeger, E. F. 1932. The Influence of Ponderosa Pine Log Size and Quality on Overrun, Lumber Grades and Conversion Values. W. C. Lumberman. 59 (8): 12-14, 20. Rapraeger, E. F. 1936. Comparative Cost of Making Logs from Small and Large Western White Pine Trees. W. C. Lumberman. 63 (6): 42, 48. Rapraeger, E. F. 1936. Relation of Tree Size in Western White Pine to Log-Making Cost. Appl. For. Note No. 74. Northern Rocky Mountain Forest and Range Experiment St. Rapraeger, E. F. 1938. Results and Application of a Logging and Milling Study in the Western White Pine Type of Northern Idaho, Univ. Idaho Bull., Vol. XXXIII, No. 16, 55 pp. i l l u s . Reynolds, R. R. 1936. Pulpwood and Log Production Costs as Affected by Type of Road. Occasional Paper No. 96. Southern Forest Experiment Station, New Orleans, La. Reynolds, R. R., W. E. Bond and Burt P. Kirkland. 1944. Financial Aspects of Selective Cutting in the Management of Second-Growth Pine-Hardwood Forests West of the Missis-sippi River. U. S. Dept. Agr. Tech. Bull. 861, 118 pp. i l l u s . 130 Silversides, C. R. 1960. The Influence of Tree Character-i s t i c s on Logging Efficiency and Cost. Pulp and Paper Magazine of Canada. Vol. 61 (8): 132-39. Smith, G. W. and C. Harrell. 1961. Linear Programming in Log Production. For. Prod. Jour. Vol. XI, No. 1: 8-11. Smith, J. H. G., J. W. Ker and J. Csizmazia. 1961. Economics of Reforestation of Douglas F i r , Western Hemlock and Western Red Cedar in the Vancouver Forest D i s t r i c t . Forestry Bulletin No. 3. Faculty of Forestry, University of B r i t i s h Columbia. Stewart, D. F. 1961. A Study of a Salvage Logging Operation on the University of B r i t i s h Columbia Research Forest, Haney, B. C. A B.S.F. Thesis. University of Bri t i s h Columbia. Tennas, M. E., R. H. Ruth and C. M. Berntsen. 1955. A Cost Analysis of Production and Costs in High-lead Yarding. U. S. D. A. Forest Service. Pacific Northwest Forest and Range Experiment Station Research Paper No. 11. Tessier, J. P. and F. M. Knapp. 1961. Cost Analysis of a Mobile Logging Operation on the U. B. C. Research Forest. U. B. C. Fac. of For. Res. Pap. No. 41, 16 pp. Tessier, J. P. and J. H. G. Smith. 1961. Effect of Tree Size of Red Alder on Harvesting and Conversion to Lumber. U. B. C. Fac. of For. Res. Pap. No. 45, 8 pp. Worthington, N. P. 1957. Skidding with Horses to Thin Young Stands in Western Washington. U. S. Forest Serv. Pacific Northwest Forest and Range Expt. Sta. Res. Note 138. 7 pp. i l l u s . Worthington, N. P. and G. R. Staebler. 1961. Commercial Thinning of Douglas F i r in the Pacific Northwest. Tech-nical Bull. No. 1230. U. S. D. A. Forest Service. Worthington, N. P. and E. W. Shaw. 1952. Cost of Thinning Young Douglas F i r . The Timberman. 53: 136-138. 131 APPENDIX I T H E UNIVERSITY OF BRITISH C O L U M B I A I N T E R D E P A R T M E N T A L MEMORANDUM 132 TO . ..Pr. .J.v.H.. G. Smith Mr. J. P. Tessier PROM Prof. F. M. Knapp Faculty of Forestry October 25, 195 1 Vancouver Log Prices September 1961 Fir Peelers #1 - $109.92 2 - 100.71 3 - 90.70 4 - 75.39 Av. $90.31 Sawlogs #1 - $75.38 2 - 64.74 \ 3 - 52.28 / Av. $58.89 Av.All Grades #1 - $94.41 2 - 69.50 3 - 52.28 Av. $63.26 Cedar Lumber #1 - $58.99 2 - 48.66 Av. $52.78 Shingle #1 - $51.92 2 - 41.43 3 - 30.89 Av.$38.61 Merch.Cedar #1 - $46.03 2 - 39.78 ? 3 - 30.11 / Av. $31.88 Av.All Grades #1 - $56.75 2 - 42.60 3 - 30.36 Av. $38.99 Hemlock #1 - $52.17 2 - 47.01 ! 3 - 41.36 / Av. $42.91 Balsam #1 Peeler - $50.00 2 Lumber - 39.24 / 3 Pulp - 32.507 Av. $39.38 Pine #1 - $65.00 2 - 53.06 3 - 41.15 Av.$46.70 Spruce #1 - $47.17 2 - 44.76 3 - 36.51 Av. $39.73 133 APPENDIX II SIMPLEX TABLEAU FOR PROBLEM NO. 2 A c t i v i t y D i s p o s a l A c t i v i t i e s R e a l A c t i v i t i e s L e v e l 111 p hi P. . 84 P 0 1 . 6 3 P , 2 . 1 2 P . 3 . 2 9 P c 4 . 2 5 P , 8 . 1 5 P , 1 6 . 5 4 _ 8 10 —11 12 13 1 2 J 4 5 D 7 420 / 0 0 0 0 0 0 0 0 0 3 . 5 0 3 .89 4 . 6 2 5 .39 6 .42 8 . 1 4 1 2 . 0 300 0 / 0 0 0 0 0 0 0 0 1.5 1.5 1.5 1.5 1.5 3 . 0 3 . 0 300 0 0 / 0 0 0 0 0 0 0 0 . 6 0 . 9 0 . 9 5 1.2 1.3 1.3 1.5 71 0 0 0 / 0 0 0 0 0 0 / 0 0 0 0 0 0 47 0 0 0 0 / 0 0 0 0 0 0 / 0 0 0 0 0 25 0 0 0 0 0 / 0 0 0 0 0 0 / 0 0 0 0 21 0 0 0 0 0 0 / 0 0 0 0 0 0 / 0 0 0 14 0 0 0 0 0 0 0 / 0 0 0 0 0 0 / 0 0 8 0 0 0 0 0 0 0 0 / 0 0 0 0 0 0 / 0 7 0 0 0 0 0 0 0 0 0 / 0 0 0 0 0 0 / LO 135 APPENDIX III 136 8020 1007 420 300 300 71 47 25 21 14 8 7 0 0 0 0 0 0 0 0 0 0 .84 3.5 1.5 . 6 l o 0 o o o o 1.63 3.89 1.5 . 9 0 1 0 0 0 0 0 2.12 4.62 1.5 .95 0 0 1 0 0 0 0 3.29 5.39 1.5 1.2 0 0 0 1 0 0 0 4.25 6.42 1.5 1.3 0 0 0 0 1 0 0 8.15 8.14 3 .0 1.3 0 0 0 0 0 1 0 16.54 12.0 3 .0 1.5 0 0 0 0 0 0 1 $ 0.000000000 p07q0a $ 115.7794685 p06q09 $ 180.9791984 p05q08 $ 240.4792175 p04q07 $ 309.5690841 p03q01 $ 340.6852951 - 2 - $ 3 4 0 .7 P 0 3 14.67748856 q02 180.4837646 q03 221.7563781 q04 71.00000000 q05 47.00000000 q06 10.32251119 po4 21.00000000 P 0 5 i4.oooooooo po6 8.000000000 p07 7.000000000 ALWAC III-E WORKSHEET - PROBLEM NO. 2 137 APPENDIX I V 1124 8400 6000 8400 410 246 256 150 522 256 170 118 0 0 0 0 0 0 0 0 0 0 0 1.02 3.89 0.90 2.30 1 0 0 0 0 0 0 0 1.02 3.89 0.90 4.30 1 0 0 0 0 0 0 0 1.02 3.89 0.90 6.30 1 0 0 0 0 0 0 0 1.32 4.62 0.95 2.30 0 10 0 0 0 0 0 1.32 4.62 0.95 4.30 0 1 0 0 0 0 0 0 1.32 4.62 0.95 6.30 0 1 0 0 0 0 0 0 2.02 5.39 1.20 3.00 0 0 1 0 0 0 0 0 2.02 5.39 1.20 5.00 0 0 1 0 0 0 0 0 2.02 5.39 1.20 7.00 0 0 1 0 0 0 0 0 5.02 6.42 1.30 3.00 0 0 0 1 0 0 0 0 5.02 6.42 1.30 5.00 0 0 0 1 0 0 0 0 5.02 6.42 1.30 7.00 0 0 0 1 0 0 0 0 0.90 4.28 0.90 2.30 0 0 0 0 1 0 0 0 0.90 4.28 0.90 4.30 0 0 0 0 1 0 0 0 0.90 4.28 0.90 6.30 0 0 0 0 1 0 0 0 1.12 5.08 0.95 2.30 0 0 0 0 0 1 0 0 1.12 5.08 0.95 4.30 0 0 0 0 0 1 0 0 1.12 5.08 0.95 6.3O 0 0 0 0 0 1 0 0 1.77 5.93 1.20 3.00 0 0 0 0 0 0 10 1.77 5.93 1.20 5.00 0 0 0 0 0 0 1 0 1.77 5.93 1.20 7.00 0 0 0 0 0 0 10 3.66 7.05 1.30 3.00 0 0 0 0 0 0 0 1 3.66 7.05 1.30 5.00 0 0 0 0 0 0 0 1 3.66 7.05 1.30 7.00 0 0 0 0 0 0 0 1 4 Z £uy oin c, I<=* 10 3 / z&fiia So 0 g <2l ;0 C 1 « cwr o r 2.8 00 o, 0<+ ALWAC III-E WORKSHEET - PROBLEM NO. 3 139 8020 1124 64oo 6000 84oo 4 l 0 246 256 150 522 256 170 118 $ 0.000000000 pOcq07 752.9987945 pl8qOb 1184.877746 p09q06 $ 1701.995239 pl5qOa $ 2002.89^897 p06q05 $ 2327.614868 pl2q09 2^14.333923 P 0 3 q 0 3 $ 2675.760375 P l i p l 5 v 2785.855285 p l O p l 2 $ 2951.645202 p04q04 3032.532531 pOdqOl $ 3071.488952 p l ^ p l j $ 3071.488952 pl3pl4 ? 3071.488952 p l ^ p l j $ 3071.488952 p ! 3 p l 4 3071.488952 J pOd 43.28499889 q02 4255.542968 J p03 409.9999923 j po4 149.7888641 J po6 96.21113014 ^ p09 256.0000000 - pOc 150.0000000 q08 478.7149963 - plO 256.0000000 ' pi 3 170.0000000 * pl8 118.0000000 $ 3071.488952 ALWAC III-E WORKSHEET - PROBLEM NO. 3 $ 0.000000000 pocq.07 752.9987945 pl8q0b 1184.877746 P 0 9 q 0 3 1249.517456 pl6q01 $ 1299.218933 P17P16 $ 1299.218933 P l 6 p l 7 9 1299.218933 P 1 7 p l 6 ? 1299.218933 P l 6 p l 7 1299.218933 V pi 6 43.05844116 q02 1083.674133 J P09 56.6o482311 q04 410.0000000 q05 246.0000000 q06 199.3951759 J pOc 150.0000000 q08 522.0000000 q09 256.0000000 qOa 170.0000000 J P18 74.94155693 $ 1299.218933 p l 6 4 ALWAC III-E WORKSHEET - PROBLEM NO. 3 2 f c / 141 e.ooooooooc pOcq03 pOaqOl pObpGa pOapOb pObpOa "riOn nob $ 301.1995162 v 328.4106826 I) 328.4106826 $ 328.4106750 J28.4106750 v pOb I8.97IC96388 q02 214 .9532699 •J pOc 4 6 . 4 4 8 5 9 6 0 0 qo4 4 1 0 . 0 0 0 0 0 0 0 q05 2 4 6 . 0 0 0 0 0 0 0 q06 2 5 6 . 0 0 0 0 0 0 0 q07 8 4 . 5 7 9 4 3 7 2 5 q 0 8 5 2 2 . 0 0 0 0 0 0 0 q09 2 5 6 . 0 0 0 0 0 0 0 qOa 1 7 0 . 0 0 0 0 0 0 0 qOb 1 1 8 . 0 0 0 0 0 0 0 $ 3 2 8 . 4 1 0 6 7 5 0 pOb ALWAC III-E WORKSHEET - PROBLEM NO. 3 1124 546o 3900 546o t t t t 1124 5460 3900 5460 410 246 256 150 522 256 170 118 0 0 0 0 0 0 0 0 0 0 0 1.02 3.89 0.90 2.30 1 0 0 0 0 0 0 0 1.02 3.89 0.90 4.30 1 0 0 0 0 0 0 0 1.02 3.89 0.90 6.30 1 0 0 0 0 0 0 0 1.32 4.62 0.95 2.30 0 1 0 0 0 0 0 0 1.32 4.62 0.95 4.30 0 1 0 0 0 0 0 0 1.32 4.62 0.95 6.30 0 1 0 0 0 0 0 0 2.02 5-39 1.20 3.00 0 0 1 0 0 0 0 0 2.02 5.39 1.20 5.00 0 0 1 0 0 0 0 0 2.02 5.39 1.20 7.00 0 0 1 0 0 0 0 0 5.02 6.42 1.30 3.00 0 0 0 1 0 0 0 0 5.02 6.42 1.30 5.00 0 0 0 1 0 0 0 0 5.02 6.42 1.30 7.00 0 0 0 1 0 0 0 0 0.90 4.28 0.90 2.30 0 0 0 0 1 0 0 0 0.90 4.28 0.90 4.30 0 0 0 0 1 0 0 0 0.90 4.28 0.90 6.30 0 0 0 0 1 0 0 0 1.12 5.08 0.95 2.30 0 0 0 0 0 1 0 0 1.12 5.08 0.95 4.30 0 0 0 0 0 1 0 0 1.12 5.08 0.95 6.30 0 0 0 0 0 1 0 0 1.77 5.93 1.20 3.00 0 0 0 0 0 0 1 0 1.77 5.93 1.20 5.00 0 0 0 0 0 0 1 0 1.77 5.93 1.20 7.00 0 0 0 0 0 0 1 0 3.66 7.05 1.30 3.00 0 0 0 0 0 0 0 1 3.66 7.05 1.30 5.00 0 0 0 0 0 0 0 1 3.66 7.05 1.30 7.00 0 0 0 0 0 0 0 1 80 ALWAC III-E WORKSHEET - PROBLEM NO. 3 8020 $ 0.000000000 p0cq07 $ 752.99879^5 pl8q0b $ 1184.877746 p09q06 $ 1701.995239 pl5q0a $ 2002.894897 p06q03 $ 2129.028198 P13P15 $ 2271.504394 p04q05 ? 2327.614868 pl2q01 $ 2358.621948 p03pl2 $ 2364.492065 p l4pl3 $ 2364.492065 pl3pl4 $ 2364.492065 * P03 36.15423011 q02 2774.161010 rf po6 122.1070766 q04 373.8457641 po4 v 123.8929100 • p09 256.0000000 y poc 150.0000000 q08 522.0000000 q09 255.9999961 Pl3 170.0000000 Pl8 118.0000000 $ 2364.492065 P03 ffoi A/**. 143 ALWAC III-E WORKSHEET 144 APPENDIX V 145 THE USE OF ALWAC III-E FOR SIMPLEX METHOD COMPUTATION THE PROCEDURE - GENERAL The Program i s i n the Computing Center's Program Library under Code No. 03. To use the program: The program can be loaded through the highspeed reader by typing 0E00 CR on the Flexowriter. At the same time on the Control Panel a l l switches have to be i n Normal p o s i t i o n , and Isolation-Free switch on the top of the panel should be on Free. The Data Tape should be loaded through the Flexowriter. Preparation of the Data Tape The data should be stated i n the following manner as out-lined: ; MMM MM M CR (for A. ) 0 0 0 CR ° C-L C 2 CR nn n.n -n CR (for A ^ ) n nnn n CR (for A-2) • • • * • • • • • • • • n.nn -n nn CR Note: MM stands for c o e f f i c i e n t s i n the Requirements Vector. May contain any number of d i g i t s . In row 2, the number of zeros, separated by spaces, equals the number of (MM). 146 C-^  are the net p r i c e s . May be any d i g i t , with decimals, p o s i t i v e or negative. nn are the c o e f f i c i e n t s i n the Structural Column vectors (A^, A.2 A n ) . May be any d i g i t , with decimals, p o s i t i v e or negative. Loading the Data Tape Insert the data tape into Flexowriter's reading devices. Set J.S. No. 2 (Jump Switch) on Jump p o s i t i o n . Press Clear button. Type 8020 CR ( = keyword) Type yyzz CR Note: yy —> no. of equations (here 10). zz -»• no. of variables (here 20). Press Start Read button. Now the machine w i l l s t a r t reading i n data and stops at lb command. After the lb occurs on the control panel, switch J . S. No. 3 to Jump and back to Normal. The machine proceeds to compute and w i l l out-put the re s u l t s on the Flexowriter i n a form as i l l u s t r a t e d on the attached sheets. Note: Because the program uses the Isolated Memory, No. 2's and No. 3 1s have to be reloaded to the Main Memory through the highspeed reader. Press the Clear-Normal Switch (located on side of highspeed reader) to Clear and back to 147 Normal. Now the tape i s read. To test that the program has gone into the Main Memory properly, type on Flex. 0400 . I f the machine types out OKAY , everytning i s i n order. Also check the Free-Isolated  Switch which now must be at Isolated p o s i t i o n . Fig- I Effect of Road Type on Hauling Cost 25-F bm-60 120 180 240 300 Log Length 32' « 10 20 30 40 Mean Log Volume in Cu- ft-'T-IS 50 T 17 24' 16' i i r 6 7 8 10 12 i 14 15 Log Diameter in In I i i — i — i — r 6 7 8 9 10 II 16 18 19 n i i i r 12 13 14 15 16 Log Diameter in In 17 I I i i I—l—I—l 1 1 1 1 1 1 1 1 1 1 -67 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 Log Diameter in In Fig- 2 Effect of Mean Log Volume on Loading Rate- University Research Forest, June 1961 100 200 300 400 500 Distance in Feet Fig- 3 Effect of Yarding Distance on Highlead Yarding Time-University Research Forest, June 1961* 5 0«r 1 0 200 400 600 800 1000 1200 1400 Distance in Feet Fig- 4 Effect of Yarding Distance and Volume on Roundtrip Timej D-2 Tractor Operation,University Research Forestj March, 1961-Fig- 5 Effect of Tree Size on Actual Time Required for Felling, Limbing, and Bucking- (Doyle and Calvert, 1961) 0 5 10 15 20 25 DBH- in Inches Fig- 6 Effect of Tree DBH- on Felling Time-University Research Forest, June 1961-10 15 20 Log Diameter in Inches 25 30 Fig- 7 Effect of Log Diameter on Bucking Time-University Research Forest, June 1961-- i o L / \ M 1 1 1 1 1— V5 10 15 20 25 30 D-BH- in Inches Fig- 8 Net Value of Trees after Felling and Bucking DBH- in Inches Fig- 9 Net Revenue per Tree Program I-0 3 DBH- in Inches Fig- 10 Felling and Bucking Cost per Treej Douglas Fir and Hemlocki Program 2-D-B-H- in Inches Fig- II Loading Costi Program 2-DBH- in Inches Fig- 12 Net Revenue per Tree* Program 2-x 2 Fig- 13 Graphical solution of two-dimensional Problem No-1 — 50' i i i i I ' • • i I i i • • i • • • ' I 0 5 10 15 20 Number of Days Operated Fig- 14 Extrapolation of Net Revenues for Operations of various Durations- Problem No- 3-400 Number of Days Operated Fig- 15 Most Profitable Time-expenditure on the Operation- Problem No- 3-THE UNIVERSITY OF BRITISH COLUMBIA I N T E R D E P A R T M E N T A L MEMORANDUM TO Dr. J. H. G. Smith Mr. J. P. Tessier PROM Prof. F. M. Knapp Faculty of Forestry October 25, ^ g l Vancouver Log Prices September 1961 Fi r Peelers #1 - $109.92 2 - 100.71 3 - 90.70 4 - 75.39 Av. $90.31 Sawlogs #1 - $75.38 2 - 64.741 3 - 52.28J Av. $58.89 Av.All Grades #1 - $94.41 2 - 69.50 3 - 52.28 Av. $63.26 Cedar Lumber #1 - $58.99 2 - 48.66 Av. $52.78 Shingle #1 - $51.92 2 - 41.43 3 - 30.89 Av.$38.61 Merch.Cedar #1 - $46.03 2 - 39.78 ) 3 - 30.11 J Av. $31.88 Av.All Grades #1 - $56.75 2 - 42.60 3 - 30.36 Av. $38.99 Hemlock #1 - $52.17 2 - 47.01 \ 3 - 41.36 / Av. $42.91 Balsam #1 Peeler - $50.00 2 Lumber - 39.24 / 3 Pulp - 32.50 7 Av. $39.38 Pine #1 - $65.00 2 - 53.06 3 - 41.15 Av.$46.70 Spruce #1 - $47.17 2 - 44.76 3 - 36.51 Av. $39.73 8020 1007 ' 420 300 300 71 47 25 21 14 8 7 0 0 0 0 0 0 0 0 0 0 .84 3-5 1.5 . 6 1 0 0 0 0 0 0 1.63 3.89 1.5 . 9 0 1 0 0 0 0 0 2.12 4.62 1.5 .95 0 0 1 0 0 0 0 3.29 5.39 1.5 1.2 0 0 0 1 0 0 0 4.25 6.42 1.5 1.3 0 0 0 0 1 0 0 8.15 8.i4 3 . 0 1.3 0 0 0 0 0 1 0 16.54 12.0 3.0 1.5 0 0 0 0 0 0 1 $ 0.000000000 p07q0a $ 115.7794685 p06q09 $ 180.9791984 p05q08 $ 240.4792175 p04q07 $ 309.5690841 p03q01 $ 340.6852951 340.7 p03 14 .677^856 q02 180.4837646 q03 221.7563781 q04 71.00000000 q05 47.00000000 q06 10.32251119 po4 21.00000000 P05 i4.oooooooo po6 8.000000000 P 0 7 7.000000000 8020 1124 8400 6000 8400 410 246 256^150 522 256 170 118 $ 0.000000000 p0cq07 $ 752.9987945 pl8q0b $ 1184.8777^ p09q06 $ 1701.995239 pi5q0a $ 2002.894897 p06q05 $ 2327.614868 pl2q09 $ 2614.333923 P03q03 $ 2675.760375 P13P15 $ 2785.855285 p l 0 p l 2 $ 2951.645202 p04qC-4 $ 3032.532531 pOdqOl $ 3071.488952 P14 P 1 3 $ 3071.488952 pl3pl4 $ 3071.488952 pl4 P 1 3 $ 3071.488952 P13pl4 $ 3071.488952 J pOd 43.28499889 q02 4255.542968 J P03 409.9999923 J po4 149.7888641 J p06 96.21113014 J p09 256.OOOOOOO J pOc 150.0000000 q08 478.7149963 * plO 256.0000000 J pi 3 170.0000000 \ p l 8 118.0000000 $ 3071.488952 1124 8400 6000 8400 410 246 256 150 522 256 170 118 0 0 0 0 0 0 0 0 0 0 0 1.02 3 .89 0 . 9 0 2.30 1 0 0 0 0 0 0 0 1.02 3-89 0 .90 4 . 3 0 1 0 0 0 0 0 0 0 1.02 3 .89 0 .90 6.30 1 0 0 0 0 0 0 0 1.32 4.62 0.95 2.30 0 1 0 0 0 0 0 6 1.32 4.62 0.95 4 . 3 0 0 1 0 0 0 0 0 0 1.32 4.62 0.95 6 .30 0 1 0 0 0 0 0 0 2.02 5.39 1.20 3 .00 0 0 1 0 0 0 0 0 2.02 5.39 1.20 5.00 0 0 1 0 0 0 0 0 2.02 5-39 1.20 7 .00 0 0 1 0 0 0 0 0 5.02 6.42 1.30 3 .00 0 0 0 1 0 0 0 0 5.02 6.42 1.30 5.00 0 0 0 1 0 0 0 0 5.02 6.42 1.30 7.00 0 0 0 1 0 0 0 0 0.90 4 .28 0 .90 2.30 0 0 0 0 1 0 0 0 0 .90 4 . 2 8 0 . 9 0 4 . 3 0 0 0 0 0 1 0 0 0 0 .90 4 . 2 8 0 .90 6.30 0 0 0 0 1 0 0 0 1.12 5.08 0.95 2.30 0 0 0 0 0 1 0 0 1.12 5 .08 0.95 4 . 3 0 0 0 0 0 0 1 0 0 1.12 5.08 0.95 6.30 0 0 0 0 0 1 0 0 1.77 5.93 1.20 3 . 0 0 0 0 0 0 0 0 1 0 1.77 5.93 1.20 5 .00 0 0 0 0 0 0 1 0 1.77 5-93 1.20 7 .00 0 0 0 0 0 0 1 0 3.66 7.05 1.30 3 . 0 0 0 0 0 0 0 0 0 1 3 .66 7-05 1.30 5.00 0 0 0 0 0 0 0 1 3 .66 7.05 1.30 7.00 0 0 0 0 0 0 0 1 4 Z £uy om C ' *2- <=* #0 04 Oc « 7 o f l f ! ^ 2so Lj. O ST 2.Z °° o) 8DO t 0 I ~ 0 i t $ 0.000000000 pOcg.07 $ 752.9987945 pl8qOb $ 1184.877746 P 0 9 q 0 3 $ 1249,517456 pl6q01 $ 1299.218933 Pl7pl6 $ 1299*218933 p l 6 p l 7 $ 1299o2l8933 P 1 7 p l 6 $ 1299-218933 P l 6 p l 7 $ 1299.218933 7 p l 6 43.05844116 q02 1083.674133 J p09 56.60482311 q04 410.0000000 q05 246.0000000 q06 / 199.3951759 J pOc 150.0000000 q08 522.0000000 q09 256.0000000 qOa , 170.0000000 J pl8 74.94155693 $ 1299.218933 pi 6 4 8020 $ Oo000000000 p0cq03 pOaqOl pObpOa pOapOb pObpOa jnna.-nOb. $ 3 0 1 . 1 9 9 5 1 6 2 $ 3 2 8 o 4 1 0 6 8 2 6 $ 328.4106826 $ 3 2 8 . 4 1 0 6 7 5 0 $ 3 2 8 o 4 l 0 6 7 5 0 J pOb 18.97196388 q02 2 I 4 . 9 5 3 2 6 9 9 pOc 46o44859600 q04 410.0000000 q 0 5 246.0000000 q 0 6 25600000000 q 0 7 84.57943725 q 0 8 522.0000000 q 0 9 25600000000 qOa 170.0000000 qOb 118.0000000 $ 3 2 8 . 4 1 0 6 7 5 0 pOb 1124 546o 3900 5^60 t t t t 1124 5460 3900 5460 410 246 256 150 522 256 170 118 0 0 0 0 0 0 0 0 0 0 0 1.02 3.89 0.90 2.30 1 0 0 0 0 0 0 0 1.02 3.89 0.90 4.30 1 0 0 0 0 0 0 0 1.02 3-89 0.90 6.30 l o o o o o o o 1.32 4.62 0.95 2.30 0 1 0 0 0 0 0 0 1.32 4.62 0.95 4.30 0 1 0 0 0 0 0 0 1.32 4.62 0.95 6.30 0 1 0 0 0 0 0 0 2.02 5.39 1.20 3-oo 0 0 1 0 0 0 0 0 2.02 5.39 1.20 5.00 0 0 1 0 0 0 0 0 2.02 5-39 1.20 7.00 0 0 1 0 0 0 0 0 5.02 6.42 1.30 3-00 0 0 0 1 0 0 0 0 5.02 6.42 1.30 5.00 0 0 0 1 0 0 0 0 5.02 6.42 1.30 7.00 0 0 0 1 0 0 0 0 0.90 4.28 0.90 2.30 0 0 0 0 1 0 0 0 0.90 4.28 0.90 4.30 0 0 0 0 1 0 0 0 0.90 4.28 0.90 6.30 0 0 0 0 1 0 0 0 1.12 5.08 0.95 2.30 0 0 0 0 0 1 0 0 1.12 5.08 0.95 4.30 0 0 0 0 0 1 0 0 1.12 5-08 0.95 6.30 0 0 0 0 0 1 0 0 1.77 5-93 1.20 3.00 0 0 0 0 0 0 1 0 1.77 5.93 1.20 5.00 0 0 0 0 0 0 1 0 1-77 5.93 1.20 7.00 0 0 0 0 0 0 1 0 3.66 7.05 1.30 3.00 0 0 0 0 0 0 0 1 3.66 7.05 1.30 5.00 0 0 0 0 0 0 0 1 3.66 7.05 1.30 7.00 0 0 0 0 0 0 0 1 80 81 8020 $ 0.000000000 p0cq07 $ 752.9987945 pl8q0b $ 1184.877746 p09q.06 $ 1701.995239 p i 5q.0a $ 2002.894897 p06q03 $ 2129.028198 P13P15 $ 2271.504394 p04q05 $ 2327.614868 pl2q01 $ 2358.621948 P03P12 $ 2364.492065 pl4pl3 $ 2364.492065 P13P14 . $ 2364.492065 < P03 36.15423011 q02 2774.161010 ^ po6 122.1070766 qo4-373-8457641 po4 v 123.8929100 / p09 256.OOOOOOO y pOc 150.0000000 q08 522.0000000 q09 255.9999961 Pl3 170.0000000 p l 8 118.0000000 $ 2364.492065 P03 

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