{"Affiliation":[{"label":"Affiliation","value":"Business, Sauder School of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Rapoport, Alex","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2009-05-26T18:58:47Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1998","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Science in Business - MScB","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"The problem bucking of trees into logs is solved using dynamic programming. The state of\r\nthe model is described by the physical dimensions of the tree, such as length, diameters and\r\nsweep. This thesis incorporates decay as a description of the state.\r\nThis thesis begins with a detailed analysis of decay morphology, obtaining the expected\r\nvalues and dispersion of key variables. The main goal of this thesis is to evaluate the decay\r\ninformation that could be gathered; this is done by incorporating different levels of information\r\ninto the bucking algorithms. The first level assumes perfect information; in this case the algorithm\r\nis provided with the observed diameter of decay and its actual length. The second level uses an\r\nobserved decay diameter and a length estimated based on the decay model. The third level also\r\nuses the observed decay diameter, but now the length is assumed to be a random variable, whose\r\ndistribution parameters are based on the decay model. Furthermore, the models will be compared\r\nto the normal practices of forestry companies of British Columbia. A test of four hundred trees in\r\nfour species, Spruce, Lodgepole Pine, Balsam Fir and Western Red Cedar was performed. Clear\r\nimprovements were found when bucking with algorithms that consider decay. The value increased\r\nan average of 11.9%. One of the most relevant findings of this thesis is that this significant lift\r\ndoes not result from lower stumpage and transportation costs, but from higher recoveries. The\r\nstumpage and transportation costs are on average higher in the proposed algorithms as compared\r\nto the normal practices. This means that the use of bucking algorithms that incorporate decay\r\ntranslates into gains for the lumber companies as well as for the Provincial Government and the\r\ntransportation industry.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/8232?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"Extent":[{"label":"Extent","value":"5191941 bytes","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/extent","classmap":"dpla:SourceResource","property":"dcterms:extent"},"iri":"http:\/\/purl.org\/dc\/terms\/extent","explain":"A Dublin Core Terms Property; The size or duration of the resource."}],"FileFormat":[{"label":"FileFormat","value":"application\/pdf","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/elements\/1.1\/format","classmap":"edm:WebResource","property":"dc:format"},"iri":"http:\/\/purl.org\/dc\/elements\/1.1\/format","explain":"A Dublin Core Elements Property; The file format, physical medium, or dimensions of the resource.; Examples of dimensions include size and duration. Recommended best practice is to use a controlled vocabulary such as the list of Internet Media Types [MIME]."}],"FullText":[{"label":"FullText","value":"B U C K I N G T R E E S W I T H D E C A Y by A L E X RAPOPORT B.Sc , Instituto Tecnologico y de Estudios Superiores de Monterey, 1993 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF SCIENCE (BUSINESS ADMINISTRATION) in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Commerce and Business Administration) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A September 1998 \u00a9 Alex Rapoport, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Cowmgftg A(Asi Base Volumetric Proportion = 1 \/ 3L Note: Assume triangular decay. The purpose is to show that the longitudinal proportion (real loses) will be larger than the volumetric (actual decay proportion). It should be clear that if one were to trim at nominal lengths the difference would be even larger. Figure 3-2: Comparison of length and volumetric proportions. 18 3.2 Infection and Defense Mechanisms Shingo (1979) explains the main mechanisms of decay growth. According to Shingo, the infected tree tries to encapsulate the decay. The tree is capable of building chemical barriers around the decay, which will prevent further expansion. There are 4 types of barriers: 1. Along the outmost layer of the sapwood. This layer will keep the decay from expanding to future rings, so if the tree is to survive the invasion, it will be able to grow healthy new layers. This is the most effective of all 4 barriers and the first one to develop. 2. Along the rays of the tree. This protection will prevent the decay to expand to a whole cross-section of the tree. Decayed wood is incapable of transporting nutrients, so i f any one cross-section were completely infected throughout its sapwood, the tree would be unable of feeding above that section and would die. This is the second barrier to develop. 19 3. The third barrier is along the rings. Each ring in a tree consists of a xylem layer and another layer of protective tissue. The latest can form chemical compounds that can prevent the decay to expand to other rings. 4. The last barrier is formed by blocking the xylem of infected sections to prevent the decay from traveling up or down. By these mechanisms the tree tries to encapsulate the decay in such a way that will prevent further invasion and will enable the tree to survive. It should be noted that the capability to develop each of these barriers could vary significantly from species to species, but all species are capable of developing these four mechanisms. Another important factor in the spread of decay is the zone in which it develops. In a tree, the heartwood is dead tissue in the sense that it is incapable of transporting nutrients. Consisting of the oldest rings, this section contains high concentrations of chemicals that can prevent decay. Nevertheless, the tree is incapable of reacting in this zone; it cannot form any further defense mechanisms and must rely on the substances that it has deposited. So if the invading organism is capable of defeating the static barriers, it will spread with ease. Given that the heartwood is not involved in the nutrition of the tree, even if its is completely invaded, the tree can survive. For these two reasons, lack of resistance and no live threatening danger, if a tree has significant amounts of decay, chances are it will be in its center. Within the sapwood, the tree can dynamically react to the spread of decay with the four mechanisms described above. The tissue itself does not contain the concentrated chemicals levels as the heartwood, but its reactivity allows the tree to fight better in this zone. This raises the important concept of point of entry. If the invading organism enters from a scar, broken branch or from any section of the tree where its first contact is with sapwood, the tree will have a good 20 chance of containing it. I f the entry point is a root, a deep frost crack, or any route that will give immediate access to the heartwood, the decay will be harder to contain. We define butt decay as one that has entered from the lowest section of the tree, and expands mainly upward. Most butt decays are root entries and invade the heartwood. Middle decay by definition is one that has invaded from somewhere along the stem. The entry point is commonly a broken branch. This thesis focuses on decay that is exposed when the tree is felled, so most of the time we will be talking about butt decays. 3.3 Variables The key variables to be used throughout this thesis are defined below. For convenience, all variables related to the decay will be lower case and those related to the tree will be upper case. Decay Variables Variable Description Units do Diameter of the decay at the butt end of the tree cm d x Diameter of the decay at a distance x from the butt of the tree cm ho Total length of the decay, from the butt of the tree to the decay vertex cm h x Remaining length of the decay measured from a distance x from the butt to the decay vertex cm Table 3-1: Decay variables 21 Tree Variables Variable Description Units (Do, D 1 ; . . . , D T } Set of diameters of the tree in T Total length of the tree ft Table 3-2: Tree variables Note that the variables {Do, Du..., DT} are sufficient to describe the shape of the tree. The initial set of diameters will be with those measured by the BCFS database. If a diameter is required at a point other than a measured section, it will be calculated by interpolating between its two adjacent sections assuming constant taper. In every case the trees have been assumed to have circular cross-sections. There will also be a set of variables related to the bucking itself; these will be defined when the algorithms are described. Notice that the decay variables are linked to a variable x measured from the butt end of the tree, while the tree variables are linked to a variable L measured from the top of the tree. The reason for the latter should be clear once the bucking algorithms are described. The reason for not measuring the decay using the same reference is simply that it expands in an upward direction (butt decays), so it is natural to describe decay using the butt end as reference. The mixture of metric and imperial units in this thesis is due the nature of the variables. Tree variables are measured in imperial units because it is easier to relate them to lumber dimensions. Decay variables are measured in metric units to be consistent with the BCFS database. 22 3.4 Bucking Models Looking ahead to the bucking problem, this thesis will analyze four different models. In all four models the diameter of the decay at the butt (do) is observed. Furthermore, the first model assumes that the length of the decay (ho) is known; I will call this the complete information model. The second model will estimate the length of the decay (ho) and will proceed as if this estimated length was correct, without updating the information after the bucking has commenced; we will call this the estimation model. The third model will estimate a probability distribution for the expected decay length (ho); I will call this the probability model. The fourth model will simulate, as close as possible, the common practice of British Columbia logging companies, based on the procedures outlined by the Ministry of Forests; I will call this the actual procedures model. Detailed description of the models will follow. Since two of the models use an estimate of the decay length, a procedure to predict this length must be developed. Common sense tells us that there should be a relation between the observed decay diameter (d) and its length (h); that is, h should be a function of d. It should be noted that when no explicit distinction is made in the decay variables d and h to identify a butt observation (subindex 0) or a mid-stem observation (subindex x) then these refer to either (butt or mid-stem). The d - h dependence should be closely linked to the way decay spreads in the tree (and of course the way the tree defends itself), so one can expect to see certain diameter-length patterns per species. In order to find a model for this relation the data was analyzed in a scatter plot, 1 for each species, of the diameter-length pairs. Such plots were too congested to devise any possible trends, so with the sole purpose of uncovering the nature of the diameter-length relation, i f any, the decay diameters were grouped in 2-cm diameter classes (starting at 2.0 cm) and the 23 average length for each group was calculated. These plots, which are shown below, uncover a definite diameter-length relation. Average Length by Diameter Class: Lodgepole Pine 500 \u2014 400 1 300 i 200 5 100 oi < 0 * \u2022 10 \u2014 I \u2014 \u2022 20 30 Diameter Class [cm] Average Length by Diameter Class: Balsam 500 _\u2022 400 |> 300 oi 1 200 2 100 OI 3 0 * \u2022 \u2022 10 20 30 Diameter Class [cm] Average Length by Diameter Class: Spruce \u2022g\" 300 & 250 f, 200 S 150 g, 100 S 50 I 0 \u2022 * \u2022 10 20 Diameter Class [cm] 30 Average Length by Diameter Class: Red Cedar \u2022_ 700 i i 600 \u00a3 500 g4CO J 300 o ro 2 0 0 \u00ab 100 5 0 10 20 Diameter Class [cm] 30 Figures 3-4 a,b,c,d: Average decay length by decay diameter class for Lodgepole Pine, Balsam, Spruce and Red Cedar For a given species, suppose that the distribution of the length of the decay, h, for a given diameter d can be represented by the form: h = A d B L 7 Equation 1 where U is a random variable with a distribution which does not depend on d, and A and B are constants which also do not depend on d. Of course, A, B and U will be species dependent. 24 The most convenient distribution for U is lognormal. A brief description of the lognormal distribution can be found in Appendix 4. To fit the lognormal model to the data set, h was regressed on d in the equation ln(h) = ln(A) + B ln(d) + ln(U) Equation 2 The units for d and h were centimeters. The regression provided the estimates for ln(A) and B . The following table shows the results of the regression, with the t-statistic of the coefficients. Species ln(A) B t-stat ln(A) t-stat B R 2 Lodgepole Pine 3.524 0.596 39.86 15.89 0.211 Balsam 3.048 0.851 48.10 33.51 0.286 Spruce 2.991 0.727 65.83 44.34 0.405 Red Cedar 4.197 0.565 60.82 23.14 0.287 Table 3-3: Results of the regression The high t-statistic shows that the coefficients have a significant role in the regression. The error term of the regression is represented by ln(U). If U is assumed to have a lognormal distribution then ln(U) is expected to have a normal distribution. To learn if such assumption holds, the error term was plotted and compared to a normal distribution by performing a Chi-squared test. Devore (1982) gives an excellent description of this test. To perform a Chi-squared test to test if a sample comes from a specified family of continuous distributions the data is divided into class intervals (cells). The number of data points that fall into each of these cells is compared to the number of data points that would fall into each cell for the theoretical distribution. The choice of cells is arbitrary; but to ensure that the chi-squared test is valid these have to be chosen independently of the sample observations. Once the cells are chosen, it is almost always quite difficult to estimate unspecified parameters (such as ju 25 or a in the normal case) from the observed cell counts, so instead these are calculated from the full sample. For this test the probability space was divided into seven equiprobable class intervals. The seven class intervals for the standard normal distribution are (-co, -1.07), (-1.07, -0.57), (-0.57, -0.18), (-0.18, 0.18), (0.18, 0.57), (0.57, 1.07), (1.07, oo). In our case, the sample has mean zero but the standard deviation is different than one, so the endpoints of these class intervals must be multiplied by the standard deviation of the sample in question. The table with the standard deviation of ln(U) and sample count, for each species, will follow. Species a Sample size Lodgepole Pine 0.827 948 Balsam 0.874 1847 Spruce 0.620 2866 Red Cedar 0.602 1310 Table 3-4: Key statistics of ln(U) The value of %2 f \u00b0 r t n e t e s t i s where the observed is the number of data points that fall into each cell, and the expected, given that we use seven a priori equiprobable cells, is s a m P ' e s * Z y \/ . The null hypothesis for the test is that the sample follows a normal distribution. The test will be as follows: x 2 = Z all cells (observed - expected)2 expected if X2 ^XIXA .reject H\u201e , Do not reject H o if Xlx-x-n, , m tw. 3-\u00ab> * * Diameter [cm] Error Plot - Red Cedar ln(U) -3.0 *, \u2022\u00bb\u00bb* \u2022 \u00ab * \u2022 120 #0 Diameter [cm] Figures 3-6 a,b,c,d: Error plots vs. decay diameter for Lodgepole Pine, Balsam, Spruce and Red Cedar 30 The expected value of a lognormal distribution, whose correspondent normal distribution has ju = 0 and a > 0, is greater than one2. This means that when calculating the expected length of a decay, given its diameter, the following equation will be used: E[h\\d] = AdBE[U] Equation 3 This should not be a problem given the multiplicative nature of the model. This value will be the one used for the estimation model. The values for the mean and standard deviation of U, for each species, are shown in the following table. Species M a Lodgepole Pine 1.381 1.261 Balsam 1.431 1.371 Spruce 1.279 1.116 Red Cedar 1.228 0.855 Table 3-7: Parameters for the lognormal distributions Even though formulas which relate the mean and standard deviation of the lognormal distribution to its correspondent normal exist, the values in Table 3-7 were calculated directly from the sample by calculating the mean and standard deviation of the exponential of the error terms of the regression. Plots of the distribution of the exponential of the error terms of the regression and the assumed lognormal distribution for each species are shown below. These plots have the sole purpose of illustrating the distributions; there will be no statistical analysis on the 2 Mathematically the expected value of a random variable U that follows a lognormal distribution, and whose correspondent normal distribution has parameters \/u and a can be expressed as E(U) = e This is greater than one when ju + a A > \u00b0- Since in our case ~ 0 and a >0, E(U) > 1. 31 accuracy of the fit. The lognormal distributions have been scaled to match the count of the error terms. Distibution of eA(error) vs. lognormal distribution: Lodgepole Pine \u2022Exp[error] \u2022lognormal 5.0 6.0 Figure 3-6a: Distribution of eerror vs. lognormal distribution for Lodgepole Pine Distibution of eA(error) vs. lognormal distribution: Balsam Exp[error] lognormal e A (error) Figure 3-6b: Distribution of eerror vs. lognormal distribution for Balsam 32 Distibution of eA(error) vs. lognormal distribution: Spruce 250 II 0.0 1.0 2.0 3.0 4.0 5.0 6.0 e A (error) Figure 3-6c: Distribution of eemr vs. lognormal distribution for Spruce Distibution of eA(error) vs. lognormal distribution: Cedar 120 -T ~ \u201e e A (error) Figure 3-6d: Distribution of eerror vs. lognormal distribution for Red Cedar 3.5 Morphology As mentioned in Section 3.1, not only the length and diameter of the decay is necessary to calculate the value of a decayed log, but also its shape plays an important role. It is unrealistic to assume that all decay occurrences will follow the same geometric shape, but it should be clear that there is a need of finding an appropriate shape to define the contour of the decay with calculated dimensions d and h. In order to search for such a geometric shape, the BCFS database was used. For a given species, all the occurrences of butt decay were identified. The data obtained described 33 i the transversal dimensions of the decay from a given distance from the butt of the tree; this was enough information to rebuild the shape of the decay. For each species, about 2,000 butt decays were reconstructed. Plots for a few of these instances can be found in Appendix 1. Even though common wisdom is that the shape of the decay follows the shape of a cone, the only author that explicitly mentions this is Basham (1991). After observing several decay shapes, as reconstructed from the BCFS database, the cone shape was questioned; nevertheless, for simplicity purposes, a well defined shape will be assumed throughout this thesis. A paraboloid (a parabola rotated around its axis) was considered. To determine which shape would best fit the data, a regression was performed for each reconstructed decay, in which the base diameter and the length of the decay were known (read from the database), and the intermediate points were estimated using the equations for the cone and the paraboloid. Cone: dx= d 0 f x^ 1 - - (x_ho) Equation 4 Paraboloid: d x = do (1 - \u2014 )m (x pL-Z [ 1 Figure 4-1: Scheme of a tree without decay 40 Given a tree with known diameters {D0, Di,... , Dx} and length T: V(L) = max { R(L, Z) + V(L - Z) } z = s,2s L Equation 6 V(0) = 0 Where: L is the distance from the top of the tree. Notice that L is divided into segments of length S. Z is the length of cut, from L and in direction to the top of the tree. V(L) is the value of a segment with and length L and diameters {D0, Di,... , DL}. This segment might or might not be a single log. R(L , Z ) is the reward of a segment of length Z, from a distance L from the top. When calculating the reward of a segment, the taper has been defined as a constant, that is, a segment to be evaluated will be assumed to be a truncated cone with large-end and small-end diameters DL and DL.Z. 4.2.2 The Backward Induction Algorithm: Notice that the value of a segment where L=0 is zero. This is the starting point of the backward induction algorithm. In the next period, increase L by S and calculate the optimal policy for this stage. Here Z could only take the value of S, this would be the optimal policy and we would move back to 2S from the end of the tree. Now Z has 2 possible actions: Z=S and Z=2S. In the first case the total value of the segment (the last 2S of the tree) is the sum of 2 independent S's. In the second case the total value of the segment would be that of one unique 2S piece. Whichever results to be the optimal action is recorded and the algorithm moves back to 3S. Equation 6 can be interpreted as follows: the value of a segment with a set of diameters {Do, Di,..., DL} and length L equals the sum of the reward obtained from cutting a segment of 41 length Z and the value of the remaining log, with length L - Z. The first reward is calculated through the procedure described in Appendix 2, while the later value is calculated at an earlier stage of the backward induction algorithm. Notice that the value of the second segment is zero when L = Z. 4.3 Models with Decay This section presents new algorithms for solving the decayed-tree bucking problem. These models differ from the simple model without decay in that it has a more complex state definition. In addition to the variables declared above as determinant for the description of a state, decay will now be considered. At each stage, a visible decay might appear on the exposed cross-section, its diameter will be recorded and utilized as an additional state variable. Note that we are only concerned with the decay spreading from the exposed section to the top of the tree. This will comply with the memory-less property of the dynamic programming models; the actions depend merely on the actual conditions of the system, so if a decayed section is observed at a cross-section, it will not matter if there is any other decay exposed anywhere else. As mentioned in the introduction, the main purpose of this work is to evaluate the information that can be gathered about the decay before implementing the bucking algorithm. The diameter of decay at the butt, and then at each exposed section after cutting off a log, will always be known. As mentioned in Chapter 3, the different levels of information will be related to the way the decay length is calculated. Hereinafter, the decay shape is assumed to be parabolic. 42 4.3.1 Complete Information Model This model assumes that the actual length of the decay is known. Physically this can be determined by inserting a probe into the decay. This model requires intensive data collection and relatively simple calculation procedures for the bucking, because there is no decay estimation and the model is deterministic. See Figure 4-2. Figure 4-2: Scheme of a tree with decay The equations for this model are the same as Equation 6, except that the initial parameters include the decay diameter do and decay length ho apart from the diameters and length of the tree. The decay parameters will not be explicitly defined in the dynamic equation. This is because given the initial parameters of the tree (T, { D 0 , D i , . . . , D T } , d 0, and ho), and the variables involved in the dynamics of the backward induction (L and Z), d x and h x can be calculated at any point. Note that L + x = T. Namely, dx+z can be estimated from h x, d x and Z by the parabolid equation. Mathematically, S L=0 Otherwise Equation 7 43 In this case, the reward of the segment of length Z will account for the decay encountered in this segment. It must be clear that the result of this, and the following models, is a set of lengths to be cut from the tree. Once these lengths are determined, the actual tree is bucked accordingly and each segment is evaluated by the procedure described in Appendix 2. In the case of the complete information model, given that all parameters are known, the decay values used by the algorithm will coincide with those observed once the tree is bucked. That might not be the case for the estimation or the probabilistic models. In these, the algorithm might be performed assuming a certain decay length that might not to be right; so when the bucking is performed, and the real decay parameters uncovered, the value of a segment can differ from the predicted one. 4.3.2 Estimation Model The length of decay will be assumed to be equal to the mean decay given by Equation 3 of Chapter 3: E[h\\d]=AdBEp] This length will assumed to be correct and no updating will be done once the bucking has commenced. This model has simple data collection and simple calculation procedures, with an estimation stage and a deterministic algorithm. Apart from the tree variables, only the decay diameter at the butt (do) is required. However, the performance of this model will depend strongly on how well one can estimate the decay length. The equations for this model are the same as for the previous model but with a calculated h 0 instead of a given ho, calculated by Equation 3. For 44 this model not only the tree length T, diameters D, but also the value of A and B (from Chapter 3), and ju (TJ) will be needed as initial parameters. 4.3.3 Probability Model The length of decay, and its distribution, will be calculated through the procedure described in Chapter 3. This model has simple data collection (same as previous model) and fairly complex calculation procedures. The diagram is basically the same as Figure 4-2. The equations differ in that there is now a probabilistic element representing the distribution of the length of the decay given an observed decay diameter. The dynamics of the algorithm are governed by this probability, so solutions are somehow weighted to obtain one that best solves the uncertain problem. Given a tree with diameters D, decay parameters A, B, and \/\/ and a of the lognormal distribution, as described in Chapter 3, and initial decay diameter and length d 0 and ho: V(L,d x)=max i \u00a3 [ R(L ,Z ,d x , h x )+V(L-Z ,d x + z ) ]p(h x | d x ) Z = S.2S....JL[ h ) i = 0 V(0,-) = 0 There has been no attempt to nullify the effect of the heteroscedascity, but to do so one would have to add another dimension to the d - h relation. Given that Equation 8 already accounts for such relation, the addition would only mean a change in the probability function, but not on the structure of the equation itself. It must be mentioned that the probabilistic model is not entirely consistent. The model used in the probabilistic case is not entirely consistent with the parabolic shape introduced in Equation 8 45 Chapter 3. Note that the equation recalculates the parabolic shape at each stage where d x is greater than zero, yielding a shape that can be thought as a succession of parabolic segments instead of a single parabola. For example, a decay with diameter do with expected distribution of lengths ho as described by Equation 1 ( h = A d B t 7 ) ; will have an expected diameter d x at x from X n do calculated from the parabolic equation d x = do (1 - \u2014 ) . This d x will have its own h o corresponding distribution of hx, again described by Equation 1. The distribution of h x should coincide with the distribution of (ho-x), but this is not the case. Nevertheless, the model presented in this section show clear improvements over the actual procedures model, as it will be shown in Chapter 5; this should be enough merit to prove its performance. As explained before, the value of d x +z as predicted by the paraboloid equation will be assumed by d x + z = [o Otherwise Equation 8 will span all the possible decay lengths from zero up to the remaining length L . For simplicity purposes the spanning will be done in discrete steps, instead of through a continuous method (integral). The step of this summation was set so that each interval would cover a probability of 0.10. This means that there will be 10 steps for the summation. Let these steps be identified as k; where i = 1 to 10, then the expected decay length (hx) associated with each step is: h x (ki) = Lognormal\"^ ( i-\u00b0-5)\/io, ft, a) * E[length of decay] Where, 46 Lognormal\"1 is the inverse of the cumulative lognormal probability density function \/\/, a are the parameters of the lognormal distribution as defined in Chapter 3, and E[Length of decay] = A d x B E[U] The i - 0.5 will make the length hx(kT) correspond to that calculated at the midpoint of each interval (i, i+1). For example, let the Eflength of decay]=l the fourth step of the summation (k=4) will assume a decay length hxequal to Lognormal\"^ 0.35 , ju, a). 4.3.4 Actual Procedures Model In this case, the common practice of stem bucking was simulated. First, longbutting was performed according to the Ministry of Forest standards; there was no questioning whether or not to longbutt, if the observed parameters allowed it, the longbutting was performed. The longbutting step length was 2 feet. Then, the remaining length to a 10cm top was calculated and the bucking solution was found in a table. Such table can be found in Appendix 3. 4.4 Model Evaluation The four models differ in data requirements and computational effort. The following table summarizes these characteristics: Model Data Requirements Computational Effort Complete Information High Medium Estimation Low Medium Probabilistic Low High Actual Procedures Low Low Table 4-1: Model evaluation 47 5 RESULTS To analyze the performance of the four models, a test on one hundred stems in each of four different species was performed. The species were Lodegepole Pine, Balsam Fir, Spruce and Red Cedar. Stems with butt decay were randomly selected from the BCFS database. The stems were reconstructed down to a 3-inch top and the details of their correspondent decay (diameter at base and length) were also recorded. The stems were then longbutted and bucked using each of the four models. For each resulting segment, there was the choice of leaving it behind (in the forest), or hauling it into the mill; the appropriate stumpage and hauling costs were assumed for each case. Those logs that made it to the mill were simulated to be sawn in a mill configured by a Canter Twin, an Edger and a Trimmer. The details of the breakdown, as well as the stumpage, hauling and bucking costs, together with the lumber and chip prices used for the test, can be found in Appendix 2. Hauling was only permitted for segments of at least four feet in length. Segments left behind that had a large-end diameter smaller than 10 cm were not accounted for stumpage purposes. Those segments left behind were evaluated according to the Provincial Logging and Waste Procedures Manual. In the case of the actual procedures model, where the longbutting was performed following the same manual guidelines, the evaluation was simple. In the case of the three other models, the 2-feet rule was modified with the following criteria: if a longbutted segment measured more than 2 feet, then it was not accounted (for stumpage purposes) if it had less than 50% sound fibre, or if the rind rule applied in both ends of the segment. In these three models, if the longbutted segment measured 2 feet, or less, and provided that the longbutting specifications were met, then it was not accounted for stumpage purposes. Given that the primary breakdown system assumed (Canter Twin) is normally used to process small to medium logs, care was taken to choose logs with a butt diameter between 8.0 48 and 13.0 inches; furthermore, for each butt diameter inch interval (8.0 - 9.0 in, 9.0 - 10.0 in, etc.) exactly 20 logs were selected. The evaluation of the logs was performed by a Visual Basic application written specifically for this thesis. The program communicated with Microsoft Excel which was used as front-end. Because the processing of trees is a practice that involves different players with different interests, the evaluation of the models is not a simple task. The most obvious objective is to maximize the revenue of the operations, that is to maximize the value of lumber and chips,minus the harvesting, hauling and stumpage costs. This criteria is of interest to the companies exploiting the resources. On the other hand, several parties, like the Provincial Government or environmental groups, would like to see as little waste as possible. It should be noted that even though the reduction of waste was analyzed throughout the models, the main objective of this thesis was to maximize the revenues of the operations. A table summarizing the dimensions of each tree, and its correspondent decay, used for this test can be found in Appendix 5. 5.1 Comparison of Value The results of the runs have been summarized in the following tables. Here, the average value per log in each of the five diameter classes studied for each species is presented for each model. The value of a segment equals the revenues from lumber and chips minus the stumpage, transport and cutting costs. To provide an insight on where the gains or losses come from, similar tables for all the components of income, except for cutting cost which is practically irrelevant, are presented. Al l prices are in Canadian Dollars. 49 Total Value (Averaae Der Log) Lodgepole Pine Balsam Spruce Red Cedar Complete Information 29.11 24.31 24.08 11.09 Probabilistic 28.21 23.39 23.27 10.13 Estimation 27.18 22.36 22.88 8.80 Actual Procedures 26.52 21.67 22.04 8.93 Table 5-1: Average total value per log Lumber Value (Average per Loa) Lodgepole Pine Balsam Spruce Red Cedar Complete Information 39.66 33.38 33.27 16.40 Probabilistic 39.04 32.84 32.89 16.20 Estimation 37.81 31.52 32.13 14.35 Actual Procedures 37.11 30.98 31.23 14.75 Table 5-2: Average value of lumber per log Chip Value (Average per Log) Lodgepole Pine Balsam Spruce Red Cedar Complete Information 8.55 7.90 7.52 6.33 Probabilistic 8.51 7.52 7.82 6.68 Estimation 8.71 8.15 7.73 6.72 Actual Procedures 8.21 7.79 7.20 5.77 Table 5-3: Average value of chips per log Stumpage Cost (Average per Log) Lodgepole Pine Balsam Spruce Red Cedar Complete Information 11.69 10.36 10.24 6.71 Probabilistic 12.01 10.44 10.84 7.73 Estimation 11.95 10.64 10.44 7.28 Actual Procedures 11.69 10.57 10.05 6.96 Table 5-4: Average stumpage cost per log 50 Transport Cost (Average per Log) Lodgepole Pine Balsam Spruce Red Cedar Complete Information 6.28 5.60 5.42 4.01 Probabilistic 6.20 5.34 5.59 4.18 Estimation 6.22 5.61 5.48 4.08 Actual Procedures 5.95 5.43 5.11 3.59 Table 5-5: Average transport cost per log The results will be analyzed in two ways. First, the results of the actual, probabilistic and estimation models will be compared to the complete information model. These results will provide an insight on what percentage of the potential maximum revenue can be obtained by using non-optimum models. The second analysis will compare the results of the probabilistic, estimation and complete information models to those of the actual procedures model. The results will show how much the revenue can be increased by using models that use more complex bucking algorithms. In essence, both analyses are equivalent, but each provides a different insight. Total Value ; (as % of Complete Information) Lodgepole Pine Balsam Spruce Red Cedar Probabilistic 96.9% 96.2% 96.7% 91.3% Estimation 93.4% 92.0% 95.0% 79.3% Actual Procedures 91.1% 89.1% 91.5% 80.5% Table 5-6: Average total value per log as percentage of Complete Information model Lumber Value i (as % of Complete Information) Lodgepole Pine Balsam Spruce Red Cedar Probabilistic 98.4% 98.4%o 98.8% 98.8% Estimation 95.3% 94.4% 96.6% 87.5% Actual Procedures 93.6% 92.8% 93.9% 90.0% Table 5-7: Average lumber value per log as percentage of Complete Information model 51 Chip Value (as % of Complete Information) Probabilistic Lodgepole Pine 99.6% Balsam 95.1% Spruce 104.0% Red Cedar 105.6% Estimation 101.9% 103.2% 102.8% 106.1% Actual Procedures 96.1% 98.6% 95.7% 91.2% Table 5-8: Average chip value per log as percentage of Complete Information model Stumpage Cost (as % of Complete Information) Probabilistic Estimation Lodgepole Pine 102.7% 102.2% Balsam 100.8% 102.7% Spruce 105.8% 101.9% Red Cedar 115.1% 108.4% Actual Procedures 100.0% 102.1% 98.1% 103.6% Table 5-9: Average stumpage cost per log as percentage of Complete Information model Transport Cost (as % of Complete Information) Lodgepole Pine Balsam Spruce Red Cedar Probabilistic 98.7% 95.3% 103.2% 104.3% Estimation 99.0% 100.3% 101.0% 101.6% Actual Procedures 94.8% 97.1% 94.3% 89.6% Table 5-10: Average transport cost per log as percentage of Complete Information model Comparing to the actual procedures model: Total Value (as % of Actual Procedures) Lodgepole Pine Balsam Spruce Red Cedar Complete Information 109.7% 112.2% 109.2% 124.3% Probabilistic 106.4% 108.0% 105.6% 113.5% Estimation 102.5% 103.2% 103.8% 98.6% Table 5-11: Average total value per log as percentage of Actual Procedures model 52 Lumber Value (as % of Actual Procedures) Lodgepole Pine Balsam Spruce Red Cedar Complete Information 106.9% 107.8% 106.5% 111.2% Probabilistic 105.2% 106.0% 105.3% 109.8% Estimation 101.9% 101.8% 102.9% 97.3% Table 5-12: Average lumber value per log as percentage of Actual Procedures model Chip Value (as % of Actual Procedures) Lodgepole Pine Balsam Spruce Red Cedar Complete Information 104.1%o 101.4% 104.5% 109.7% Probabilistic 103.7% 96.5% 108.6% 115.8% Estimation 106.1% 104.6% 107.4% 116.4% Table 5-13: Average chip va ue per log as percentage of Actual Procedures model Stumpage Cost (as % of Actual Procedures) Lodgepole Pine Balsam Spruce Red Cedar Complete Information 100.0% 98.0% 101.9% 96.5% Probabilistic 102.7% 98.8% 107.9% 111.1% Estimation 102.2%) 100.7% 103.9% 104.6% Table 5-14: Average stumpage cost per log as percentage of Actual Procedures model Transport Cost (as % of Actual Procedures) Lodgepole Pine Balsam Spruce Red Cedar Complete Information 105.5% 103.0% 106.1% 111.6% Probabilistic 104.2% 98.2% 109.5% 116.4% Estimation 104.5% 103.3% 107.2% 113.4% Table 5-15: Average transport cost per log as percentage of Actual Procedures model 53 Interpreting the results, comparing the actual procedures model with the complete information model, there is an average potential gain of 11.9% in value, for all diameter classes and species. This gain comes from a 7.6% increase in lumber revenues, 4.6% increase in revenues from chips, and a 0.7% decrease in stumpage. The transportation costs increase by 6.1%. When using the probabilistic model, there is an average gain of 7.4% in value, for all diameter classes and species compared to the actual procedures model. This gain comes from a 6.0% increase in lumber revenues, 5.4% increase in revenues from chips. Stumpage and transportation costs increase by 4.5% and 6.1% respectively. When using the estimation model, there is an average gain of 2.6% in value, for all diameter classes and species compared to the actual procedures model. This gain comes from a 1.5% increase in lumber revenues, 8.1% increase in revenues from chips. Stumpage and transportation costs increase by 2.6% and 6.5% respectively. Compared to the complete information model, the Probabilistic model shows the next best performance recovering 96.0% of the potential maximum gain, for all diameter classes and all species. Next is the Estimation model, recovering 91.7% of the potential. Finally, the Actual model shows an average value of 89.4% of the potential maximum. This thesis is focused on the potential value increase of small logs with butt decay. To best interpret the magnitude of the potential gains, it is important to determine what percentage of the tree population, of the same diameter range, presents butt decay. The following table shows the percentage of trees with butt decay, by species, on the diameter range of 8.0 to 13.0 inches at the butt, as calculated from the BCFS database. 54 Species Fraction of 8-13\" Butt Diameter trees with Butt Decay Logepole Pine 0.093 Balsam 0.157 Spruce 0.208 Red Cedar 0.267 Table 5-16: Fraction of trees with 8\" to 13\" Butt Diameter and Butt Decay By weighting the results of table 5-1 by the fractions of trees with butt decay, the average potential gain in value, for the four species over all trees in the diameter range of 8.0 to 13.0 inches at the butt, is approximately 2.8%. To emphasize this result, assume arc average value per tree of $25.00, then, the average gain per tree would be about $0.70. Assuming now a company with an annual allowable cut of 1,000,000 (one million) cubic meters, from which 200,000 comes from trees within the 8.0 - 13.0 inch diameter range. Assuming an average volume per tree of 0.250 m 3, this company would process 800,000 trees within the 8.0 - 13.0 inch diameter range per year. This translates into a potential gain of $560,000 per year. Applying the same logic to the probabilistic model, the expected gain per year would be about $350,000. As for the estimation model, the expected gain would be about $120,000 per year. There is an interesting result that must not be overlooked. The results show a clear revenue improvement when bucking with algorithms that consider decay. However, these gains do not come at the expense of lower stumpage or transport costs, they are a consequence of higher lumber recoveries. The stumpage and transportation costs are higher on average for all the three models that include decay than for the actual procedures model. A higher value per tree, and higher stumpage and transportation payoffs mean gains for all the parties involved in the forestry 55 practice (forest products industry, Provincial Government and transport industry); so it is in the best interest of all parties involved to implement these improved algorithms. 5.2 Comparison of Waste As mentioned in Chapter 2, waste is defined as the sound fiber that is left behind in the woods. This fiber is usually accompanied by substantial amounts of decay, to the extent that it is not economically feasible to haul it for processing. At least that is the common wisdom. The waste was recorded for each of the trees processed by each of the models. The following table shows the average waste per tree, in dm3, for all species and all diameters. Model Waste [dm3] (Average per tree) Complete Information 7.42 Probabilistic 9.40 Estimation 4.49 Actual Procedures 16.45 Table 5-17: Average waste by model Table 5-17 shows that the actual procedures model leads to considerably more waste than the other three models. 5.3 Longbutting The percentage of trees, of all species and diameter classes, that were longbutted in each of the four models is shown in the following table. 56 Model % of trees longbutted Complete Information 20.3% Probabilistic 19.8% Estimation 11.5% Actual Procedures 46.3% Table 5-18: Percentage of trees longbutted by model Table 5-18 shows that the actual procedures model tends to be more aggressive in longbutting. The whole set of trees was analyzed to look for possible patterns in the longbutting. These patterns would ideally provide heuristics for better longbutting procedures. Nevertheless, there was no apparent pattern; some trees were longbutted even with a moderate amount of decay while others that had considerably amounts of decay were bucked for lumber. Further discussion will follow. 5.4 Conclusions Section 5-2 indicates that there is a clear improvement to be made by applying bucking models that consider decay. The actual procedures model tends to haul less wood (and consequently pay less stumpage and transport), and has the poorest performance in value recovered. This indicates that the longbutting rules, together with the grading procedures, stipulated by the Provincial Government of British Columbia, result in losses for all the parties involved. The better performance of the probabilistic model, compared to the estimation model, indicates that the procedure developed to estimate the expected length of decays is not accurate enough, and that the allowance for errors (as the probabilistic model does), translates in better results. 57 The discussion of how each of these models would be implemented is left open. The requirements of each of these models should be clear from Table 4-1. With the increasing power of computers, the tendency is to move towards solutions that rely more on heavy computations and away from those relying on manual or mechanical procedures. With this in mind, the insertion of a probe might someday be economically infeasible for the time that has to be spent, as compared to the time needed to solve a probabilistic dynamic problem. With the technology available today it would be possible to apply the estimation and the complete information models in the field. The probabilistic model takes longer to solve, but with more powerful computers, or a logistic system that can recognize a log after it has been scanned (bar codes are an option), this model is definitely applicable. It is obvious that longbutting on a log-by-log basis will always yield better solutions than fixed sets of rules, which are not even clear and whose application may result in considerable losses. 58 Bibliography B A S H A M , J.T. 1991. Stem Decay in Living Trees in Ontario's Forests: A user's Compendium and Guide. Information Report O-X-408, Forestry Canada. British Columbia Ministry of Forests. 1994. Provincial Logging Residue and Waste Measurement Procedures Manual. British Columbia Ministry of Forests. 1986. Forest Service Scaling Manual. COBB, B. C. 1958. Bucking trees for top value. Southern Lumberman. 196 : 31-34 DEVORE, J.L. 1982. Probability and Statistics for Engineering and the Sciences. Brooks\/Cole Publishing Company. H A M I L T O N , D. A. , and B R I C K E L L , J. E. 1983. Modeling methods for a two-state system with continuous responses. Canadian Journal of Forest Research. 13 : 1117-1121 L E M A Y , V. M . , K O Z A K , A., and M A R S H A L , P. L . 1993. Using limited dependent variable estimators for estimating percent decay. Canadian Journal of Forest Research. 23 : 266-274 MADISON'S Canadian Lumber Reporter. 1996. Volumes 4 - 8 - 12 - ... - 52 MANESS, T. C , and A D A M S , D. M . 1991. The combined optimization of log bucking and sawing strategies. Wood and Fiber Science. 23(2): 296-314 NLELSON, R.W., DOBLE, J., and WRIGHT, D . M . 1985. Conversion Factors for the Forest Products Industry in Western Canada. Special Publication No. SP-24R, Foreintek Canada Corp. OLSEN, E., PLLKERTON, S., G A R L A N D , J., and SESSIONS, J. 1990. Extending Strategies for Optimal Bucking to Harvested and Site Preparation. Western Journal of Applied Forestry. 5(1): 12-15 PNEVMATICOS, S. M . , and M A N N , S. H. 1972. Dynamic programming in tree bucking. Forest Products Journal. 22(2): 26-30 SESSIONS, J. 1988. Making Better Tree-Bucking Decisions in the Woods. An introduction to optimal bucking. Journal of Forestry. 86 : 43-45 SESSIONS, J., OLSEN, E., and G A R L A N D , J. 1989. Tree Bucking for Optimal Stand Value with Log Allocation Constraints. Forest Science. 35(1) : 271-276 SHTNGO, A.L . 1979. Tree Decay: An Expanded Concept. Agriculture Information Bulletin Number 419. Forest Service, United States Department of Agriculture. W A C K E R M A N , A. E. 1942. Bucking, or Log Making. Southern Lumberman. 164 : 49-52. 59 Appendix 1: Decay Shapes S t e m : 94Q38BA Top P I t in 60 Sten: 352Z4BA Top Plan 61 Sten: 45758BA Top Plan 62 Stem: 4839ZBA lop Plan 63 64 Sten: 54Z15BA Top Plan 65 Sten'. 53714BA lop Plan 66 Appendix 2: Sawing Details The logs have been assumed to be processed in a mill consisting of a Canter-Twin chip-n-saw , a gang, an edger and a trimmer. For those not familiar with this machinery, the Canter-Twin receives a log and cuts a cant out of its center and chips the outmost face of the sides. Figure A2-1 shows the process seen from the end-view of the log. CUTS: Chipped 2 Side-Flitches 1 Cant Figure A2-1: Input and Output of the Canter-twin The cant width is cut at a pre-determined size to produce lumber of a given width. The side-flitches thickness is also set to produce lumber of a given thickness. The cant is later \"sliced\" in a gang. The gang has saws spaced at intervals equal to the desired lumber thickness. Each individual piece, from the gang or direct from the Twin (the flitches) is then passed on to the edger which squares the sides, see Figure A2-2. The edger is also set to cut at predetermined widths. The pieces are then passed to a trimmer that cuts the ends off in order to produce lumber of nominal lengths. All the nominal lengths, widths and thickness can be found in Table A2-1 later in this appendix. 67 Figure A2-1: Input and Output of the Edger Depending on the grade of lumber being produced, the pieces are allowed to have round (non-square) edges up to a certain degree, this is known as wane allowance. For the purpose of this thesis the pieces will have to be square (i.e. no wane allowed). For this thesis, there were five lumber widths considered: 3, 4, 6, 8 and 10-inches, seven lengths: 8, 10, 12, 14, 16, 18 and 20-foot, and one thickness: 2-inches. The actual dimensions of lumber often differ from their nominal sizes, the following tables shows the actual dimensions of lumber. Lumber r fhickness Nominal [inches] Actual [inches] 2 1.5625 Lumber Width Nominal [inches] Actual [inches] 3 2.5625 4 3.5625 6 5.6250 8 7.5000 10 9.5000 Table A2-1: Nominal and Actual lumber sizes 68 In the case of lengths, the actual dimensions were assumed to equal the nominals. The saw kerf was assumed to be 0.150 inches, for all machines. No kerf was assumed for the bucking cuts. No allowance for shrinkage was assumed. To calculate the lumber prices to be used, the Madison's Canadian Lumber Report3 was used. The prices for the last week of each month for 1996 were averaged and this numbers were used for the sawing calculations. Table A2-2 shows these prices, all in Canadian Dollars. Lumber Prices [$\/Mbf] Thickness = 2in 3 4 Lumber Width 6 8 10 8 215.90 287.90 308.70 313.10 318.10 f n 10 231.80 309.00 342.00 312.50 299.40 OJj \u00a7 12 252.30 336.30 335.00 383.90 418.70 v- 14 282.50 376.70 356.00 354.90 517.40 __ 20 6^ 1 16 70 66 20 68 20 70 20 72 20 74 20 76 20 78 20 80 20 82 20 84 20 86 20 88 20 90 20 For example, a 80-foot stem (reconstructed to a 3-inch top) that measures 74 feet to a 4-inch top would first cut a 20-foot segment, leaving 54 feet, it would then cut another 20-foot segment, leaving 34 feet; it would then cut a 18-foot segment, leaving 16 feet. This last segment would be cut as long as possible, that is it will either be a 20-foot segment or a segment measuring the resulting length of subtracting the lengths of the previously bucked segments from the total length. In our example, after bucking two 20-footers and one 18-foot log, the remaining length (to 80 feet) is 22 feet, so the last segment would messure 20 feet. 71 Appendix 4: Lognormal distribution A nonnegative random variable X is said to have a lognormal distribution i f the random variable Y = ln(X) has a normal distribution. The resulting probability density function of a lognormal random variable when ln(X) is normally distributed with parameters ju and a is 1 f(x; ju, a) = < 42TVCT x 0 --L[ln(x)-^f 2 c t x > 0 x<0 72 Appendix 5: Details of tress processed Herewith is a brief summary of the trees processed. The first column, TreeLD corresponds to the Identification number of the BCFS database. The second column is the code given for easier identification, the first two letters of this code identify the species: B A for Balsam, PL for Lodgepole Pine, SP for Spruce, and R C for Red Cedar. The last column shows the value of the length of decay as calculated by Equation 3. Tree ID Tree Code Length Butt Diameter decay length decay diameter Estimated decay length - r _ i r ~ 1 [ft] [in] [cm] [cm] [cm]54215 BA-8-1 30.0 8.15 355.0 10.3 218.2 8176 BA-8-2 40.0 8.11 75.0 10.2 217.3 46943 BA-8-3 38.0 8.70 215.0 6.0 138.4 49807 BA-8-4 28.0 8.11 100.0 12.4 256.6 924259 BA-8-5 32.0 8.19 136.0 15.2 305.1 931600 BA-8-6 32.0 8.19 167.0 7.6 169.2 53307 BA-8-7 50.0 8.82 144.0 6.5 148.1 45743 BA-8-8 36.0 8.70 60.0 2.7 69.1 58610 BA-8-9 32.0 8.35 140.0 7.5 166.4 44234 BA-8-10 44.0 8.82 416.0 12.3 254.8 46602 BA-9-1 36.0 9.80 99.0 11.4 238.9 44178 BA-9-2 36.0 9.61 502.0 15.6 311.9 43846 BA-9-3 46.0 9.80 27.0 2.6 66.8 933510 BA-9-4 48.0 9.69 167.0 5.1 120.5 51625 BA-9-5 50.0 9.45 95.0 11.0 231.7 914926 BA-9-6 44.0 9.49 136.0 10.2 217.3 50204 BA-9-7 26.0 9.02 279.0 5.5 128.5 935979 BA-9-8 48.0 9.80 105.0 15.2 305.1 924481 BA-9-9 36.0 9.21 197.0 2.5 65.7 17727 BA-9-10 30.0 9.29 93.0 7.6 169.2 35283 BA-10-1 66.0 10.98 328.0 10.2 217.3 35129 BA-10-2 34.0 10.12 167.0 11.4 238.9 9093 BA-10-3 50.0 10.00 133.0 10.2 217.3 56404 BA-10-4 42.0 10.20 60.0 6.0 138.4 931064 BA-10-5 48.0 10.79 75.0 10.2 217.3 48457 BA-10-6 48.0 10.20 60.0 2.8 71.3 51947 BA-10-7 42.0 10.20 60.0 4.3 103.2 922179 BA-10-8 28.0 10.20 75.0 2.5 65.7 45670 BA-10-9 52.0 10.98 51.0 4.6 110.4 948378 BA-10-10 58.0 10.98 761.0 10.2 217.3 931023 BA-11-1 54.0 11.81 151.0 5.1 120.5 54414 BA-11-2 70.0 11.26 75.0 9.5 204.6 924543 BA-11-3 30.0 11.61 45.0 5.1 120.5 800906 B A-11-4 74.0 11.42 109.0 17.8 349.0 48018 BA-11-5 54.0 11.93 66.0 10.2 216.4 6263 BA-11-6 78.0 11.42 69.0 7.6 169.2 73 TreelD Tree Code Length Butt M Diameter decay length [in] [cm] decay diameter [cm] Estimated decay JcmJ length 74 TreelD Tree Code Length Bull Diameter decay length decay diameter Estimated decay length E L \u2014 I I L \u2014 E E L _ | C M L 3 0 5 , 35927 36005 BA-11-14 BA-11-15 42.U 56.0 I L S I 11.10 A 4 A O 54.0 74.fi fl 7.6 22.9 169.2 432.4 22187 51058 800913 BA-11-16 BA-11-17 BA-11-18 58.0 32.0 60.0 1 1 A2. 11.30 11.61 30.0 502.0 3.0 7.6 76.7 169.2 924574 56434 BA-11-19 BA-11-20 38.0 28.0 11.18 11.06 136.0 30.0 10.2 4.9 5.1 217.3 115.5 120.5 4602 35202 BA-12-11 BA-12-12 70.0 64.0 iz.uy 12.72 99.0 8.9 193.5 \u00bb\u00abN A ^ O 29295 BA-12-13 46.0 12.91 87.0 10.2 217.3 61820 BA-12-14 88.0 12.13 60.0 3.6 89.6 58678 BA-12-15 48.0 12.48 301.0 75 0 14.0 5.1 284.5 120.5 904716 43928 BA-12-16 BA-12-17 84.0 48.0 12.yy 12.01 36.0 16.6 328.0 948381 BA-12-18 48.0 12.01 700.0 10.2 217.3 <-> A \"7 O 806434 BA-12-19 74.0 12.99 319.0 10.2 217.3 53714 BA-12-20 66.0 12.01 480.0 9.8 209.1 807120 PL-8-1 44.0 8.11 105.0 7.6 157.0 807761 PL-8-2 46.0 8.19 349.0 5.1 123.7 807775 PL-8-3 40.0 8.31 Q C Q 105.0 5.1 10.2 123.7 187.1 1241 807454 PL-8-4 PL-8-5 50.0 48.0 O . O O 8.58 O I 5 7 . U 121.0 f^) 0 2.5 2.5 80.9 80.9 1794 31690 PL-8-6 PL-8-7 28.0 72.0 O . IV 8.70 O u . u 203.0 17.5 258.1 922798 PL-8-8 46.0 8.70 Q O O 196.0 17Q 0 2.5 2.5 80.9 80.9 1827 1273 PL-8-9 PL-8-10 30.0 48.0 0.0\/ 8.82 I I 5 7 . U 130.0 5.1 123.7 922818 PL-9-1 46.0 9.02 45.0 2.5 80.9 A ^ 2357 PL-9-2 44.0 9.09 33.0 5.1 123.7 1140 PL-9-3 50.0 9.09 57.0 12.7 213.2 922880 PL-9-4 46.0 9.09 136.0 12.7 213.2 47491 PL-9-5 50.0 9.29 Ci O Q 30.0 Qfi 0 12.7 16.4 213.2 248.3 42918 39879 PL-9-6 PL-9-7 46.0 42.0 y.^ y 9.41 145.0 10.7 192.5 925356 PL-9-8 40.0 9.41 319.0 17.8 260.7 A \"7 922861 PL-9-9 50.0 9.41 136.0 5.1 123.7 45259 49351 933738 PL-9-10 PL-10-1 PL-10-2 52.0 66.0 60.0 9.69 10.28 10.39 413.0 195.0 167.0 16.3 8.0 17.8 246.9 161.8 260.7 A \u2014j r\\ o 49583 2097 807093 PL-10-3 PL-10-4 PL-10-5 52.0 48.0 48.0 10.55 10.59 10.59 30.0 63.0 105.0 9.5 22.9 5.1 179.3 303.0 123.7 A C \"7 1197 PL-10-6 64.0 10.79 69.0 7.6 157.0 O A 1 O 904026 PL-10-7 52.0 10.79 75.0 12.7 213.2 807092 PL-10-8 60.0 10.79 868.0 2.5 80.9 A O T A 1513 PL-10-9 42.0 10.91 66.0 10.2 187.1 A O ^ A 933723 PL-10-10 62.0 10.91 75.0 10.2 187.1 75 TrieTrT\u2014Tree Code Length Butt Diameter decay length decay diameter Estimated decay length [ft] [in] [cm] [cm] [cm] 807080 PL-11-1 48.0 11.10 105.0 12.7 213.2 1650 PL-11-2 50.0 11.10 69.0 2.5 80.9 45220 PL-11-3 64.0 11.18 416.0 12.5 210.7 807448 PL-11-4 58.0 11.18 105.0 10.2 187.1 807802 PL-11-5 50.0 11.30 105.0 7.6 157.0 807472 PL-11-6 56.0 11.50 75.0 15.2 237.3 49293 PL-11-7 68.0 11.57 60.0 7.4 154.5 806013 PL-11-8 80.0 11.61 136.0 12.7 213.2 1642 PL-11-9 36.0 11.81 243.0 7.6 157.0 807440 PL-11-10 48.0 11.89 121.0 15.2 237.3 932397 PL-12-1 82.0 12.01 212.0 20.3 282.0 53194 PL-12-2 72.0 12.28 141.0 18.0 262.4 1689 PL-12-3 40.0 12.28 93.0 12.7 213.2 34758 PL-12-4 66.0 12.40 273.0 8.9 172.5 30300 PL-12-5 42.0 12.40 81.0 8.1 163.0 806138 PL-12-6 52.0 12.40 228.0 17.8 260.7 806143 PL-12-7 70.0 12.52 410.0 7.6 157.0 922944 PL-12-8 52.0 12.60 1093.0 20.3 282.0 919521 PL-12-9 46.0 12.72 197.0 22.9 303.0 49398 PL-12-10 80.0 12.76 60.0 4.1 108.6 933887 PL-8-11 44.0 8.11 136.0 2.5 80.9 1766 PL-8-12 36.0 8.70 96.0 5.1 123.7 807646 PL-8-13 44.0 8.31 167.0 10.2 187.1 914963 PL-8-14 44.0 8.11 380.0 7.6 157.0 933828 PL-8-15 42.0 8.11 105.0 12.7 213.2 34273 PL-8-16 34.0 8.11 197.0 2.5 80.9 49134 PL-8-17 42.0 8.27 349.0 13.6 221.6 1765 PL-8-18 32.0 8.50 380.0 10.2 187.1 807688 PL-8-19 42.0 8.70 75.0 5.1 123.7 1795 PL-8-20 34.0 8.90 151.0 2.5 80.9 906328 PL-9-11 48.0 9.61 167.0 12.7 213.2 31959 PL-9-12 58.0 9.41 45.0 3.3 95.5 807256 PL-9-13 42.0 9.29 121.0 10.2 187.1 807077 PL-9-14 30.0 9.21 121.0 12.7 213.2 906364 PL-9-15 40.0 9.41 45.0 5.1 123.7 933916 PL-9-16 50.0 9.49 167.0 10.2 187.1 49375 PL-9-17 70.0 9.57 95.0 14.5 230.7 1514 PL-9-18 36.0 9.61 441.0 7.6 157.0 41663 PL-9-19 36.0 9.61 57.0 18.8 269.3 904022 PL-9-20 54.0 9.69 45.0 5.1 123.7 807464 PL-10-11 48.0 10.20 167.0 10.2 187.1 933842 PL-10-12 48.0 10.20 167.0 15.2 237.3 1844 PL-10-13 46.0 10.20 185.0 10.2 187.1 806222 PL-10-14 78.0 10.31 136.0 17.8 260.7 807145 PL-10-15 44.0 10.51 105.0 7.6 157.0 1505 PL-10-16 44.0 10.71 69.0 10.2 187.1 807262 PL-10-17 38.0 10.71 121.0 12.7 213.2 76 TreelD Tree Code Length Butt Diameter decay length decay diameter Estimated decay length mi finl rcml fcml [cm] . 908728 PL-10-18 66.0 10.79 212.0 20.3 282.0 807782 PL-10-19 48.0 10.91 837.0 15.2 237.3 1657 PL-10-10 42.0 10.98 96.0 10.2 187.1 935356 PL-11-11 88.0 11.10 75.0 2.5 80.9 1698 PL-11-12 50.0 11.42 33.0 7.6 157.0 36512 PL-11-13 56.0 11.42 225.0 8.9 172.5 29521 PL-11-14 76.0 11.50 60.0 7.6 157.0 1702 PL-11-15 54.0 11.61 624.0 10.2 187.1 29528 PL-11-16 70.0 11.61 66.0 10.2 187.1 1288 PL-11-17 56.0 11.81 410.0 24.9 318.5 1687 PL-11-18 54.0 11.81 30.0 5.1 123.7 1574 PL-11-19 38.0 11.89 . 340.0 17.8 260.7 531\"58 PL-11-20 78.0 11.85 99.0 20.3 282.0 800162 PL-12-11 86.0 12.01 45.0 5.1 123.7 2255 PL-12-12 34.0 12.01 155.0 17.8 260.7 1515 PL-12-13 46.0 12.20 286.0 7.6 157.0 55544 PL-12-14 68.0 12.24 30.0 7.5 155.7 1658 PL-12-15 46.0 12.28 155.0 12.7 213.2 1633 PL-12-16 38.0 12.28 182.0 7.6 157.0 1674 PL-12-17 56.0 12.52 27.0 5.1 123.7 54686 PL-12-18 76.0 12.60 295.0 15.0 235.4 50003 PL-12-19 66.0 12.64 30.0 1.2 50.9 49342 PL-12-20 68.0 12.68 248.0 20.2 281.1 932013 SP-8-1 46.0 8.82 136.0 2.5 49.6 43349 SP-8-2 28.0 8.39 21.0 2.6 50.3 938435 SP-8-3 48.0 8.58 228.0 12.7 161.6 40524 SP-8-4 28.0 8.58 194.0 6.4 98.2 908015 SP-8-5 28.0 8.82 167.0 15.2 184.1 41104 SP-8-6 44.0 8.50 30.0 3.3 60.7 30930 SP-8-7 42.0 8.19 90.0 6.6 100.4 31558 SP-8-8 34.0 8.19 167.0 15.2 184.1 42762 SP-8-9 48.0 8.19 209.0 14.6 178.8 41094 SP-8-10 32.0 8.11 33.0 3.3 60.7 42385 SP-9-1 32.0 9.80 96.0 18.2 209.5 51062 SP-9-2 40.0 9.57 60.0 7.1 105.9 42367 SP-9-3 32.0 9.21 60.0 18.5 212.4 30650 SP-9-4 42.0 9.49 90.0 10.2 137.8 47844 SP-9-5 40.0 9.69 94.0 12.4 158.8 39044 SP-9-6 50.0 9.61 36.0 3.5 62.7 936445 SP-9-7 40.0 9.69 167.0 12.7 161.6 43302 SP-9-8 44.0 9.41 148.0 6.5 99.3 39437 SP-9-9 38.0 9.49 158.0 12.7 161.6 41908 SP-9-10 26.0 9.09 139.0 14.4 176.6 34440 SP-10-1 36.0 10.79 45.0 7.6 111.3 39458 SP-10-2 34.0 10.20 63.0 10.2 137.8 39682 SP-10-3 66.0 10.39 60.0 3.8 67.2 39374 SP-10-4 58.0 10.79 145.0 25.7 269.7 77 TreelD ' Tree Code Length Butt Diameter decay length decay diameter Estimated decay length mi nn! Jem] [cm]_ _ icmj 42769 SP-10-5 52.0 10.51 237.0 17.4 39799 SP-10-6 44.0 10.98 93.0 2.0 42.2 932139 SP-10-7 46.0 10.98 258.0 17.8 206.5 905689 SP-10-8 44.0 10.39 197.0 17.8 206.5 39617 SP-10-9 56.0 10.12 81.0 5.1 83.3 40093 SP-10-10 56.0 10.59 60.0 7.0 104.8 922473 SP-11-1 36.0 11.42 136.0 17.3 202.3 39669 SP-11-2 66.0 11.69 179.0 7.7 111.8 924887 SP-11-3 78.0 11.30 75.0 7.6 111.3 908316 SP-11-4 78.0 11.10 440.0 22.9 248.0 29875 SP-11-5 56.0 11.81 63.0 5.1 83.3 41625 SP-11-6 28.0 11.30 161.0 10.4 139.7 905739 SP-11-7 60.0 11.61 258.0 22.9 248.0 29335 SP-11-8 60.0 11.10 191.0 17.8 206.5 934960 SP-11-9 88.0 11.42 258.0 15.2 184.1 41880 SP-11-10 50.0 11.61 54.0 18.7 213.7 35119 SP-12-1 44.0 12.72 45.0 6.4 98.2 917976 SP-12-2 82.0 12.91 517.0 30.5 305.5 41953 SP-12-3 48.0 12.01 158.0 19.2 218.2 A f\\ C 29543 SP-12-4 56.0 12.20 30.0 2.5 49.6 948949 SP-12-5 58.0 12.28 121.0 12.7 161.6 A O A A 41743 SP-12-6 52.0 12.80 462.0 15.2 184.1 41431 SP-12-7 36.0 12.28 231.0 25.4 267.5 803477 SP-12-8 90.0 12.09 75.0 7.6 111.3 948968 SP-12-9 72.0 12.52 121.0 12.7 161.6 39625 SP-12-10 70.0 12.91 258.0 17.5 204.0 908509 SP-8-11 44.0 8.82 410.0 7.6 111.3 41096 SP-8-12 36.0 8.11 105.0 8.0 115.5 A O A O 35819 SP-8-13 28.0 8.50 111.0 8.9 124.8 931715 SP-8-14 42.0 8.50 167.0 7.6 111.3 43335 SP-8-15 40.0 8.82 148.0 13.5 168.9 922388 SP-8-16 48.0 8.90 105.0 15.2 184.1 40566 SP-8-17 32.0 8.50 182.0 15.5 186.8 42658 SP-8-18 46.0 8.90 99.0 17.5 204.0 40668 SP-8-19 40.0 8.70 60.0 12.7 161.6 908511 SP-8-20 22.0 8.19 319.0 12.7 161.6 908513 SP-9-11 38.0 9.80 136.0 12.7 161.6 938576 SP-9-12 48.0 9.21 75.0 2.5 49.6 911868 SP-9-13 50.0 9.80 151.0 10.2 137.8 924736 SP-9-14 40.0 9.29 471.0 17.8 206.5 38094 SP-9-15 42.0 9.09 112.0 6.4 98.2 29733 SP-9-16 48.0 9.80 60.0 7.6 111.3 34873 SP-9-17 48.0 9.88 105.0 3.6 64.6 39406 SP-9-18 42.0 9.41 60.0 4.1 71.0 40428 SP-9-19 36.0 9.02 30.0 5.3 85.6 43364 SP-9-20 36.0 9.02 130.0 7.5 110.2 924726 SP-10-11 40.0 10.51 228.0 17.8 206.5 78 decay diameter Estimated decay length 79 TreelD Tree Code Length Butt Diameter decay length decay diameter [cm] [cm] Estimated decay Ipml length 80 TreelD Tree Code Length Butt Diameter decay length decay diameter Estimated decay length [ft] [in] [cm] [cm] [cm] 43734 RC-11-6 40.0 11.18 185.2 13.4 352.6 907680 RC-11-7 36.0 11.30 151.2 7.6 256.5 33715 RC-11-8 36.0 11.81 364.2 15.2 379.4 21829 RC-11-9 32.0 11.61 303.2 12.7 342.8 915603 RC-11-10 44.0 ' 11.30 243.2 12.7 342.8 915409 RC-11-11 42.0 11.89 243.2 20.3 446.7 905387 RC-11-12 46.0 11.42 151.2 12.7 342.8 915427 RC-11-13 46.0 11.61 120.2 12.7 342.8 915445 RC-11-14 46.0 11.18 182.2 15.2 379.4 804440 RC-11-15 38.0 11.10 240.2 15.2 379.4 903358 RC-11-16 48.0 11.81 182.2 10.2 302.9 52588 RC-11-17 50.0 11.89 339.2 12.0 332.0 905381 RC-11-18 38.0 11.89 212.2 22.9 478.2 907469 RC-11-19 42.0 11.81 151.2 2.5 136.9 947208 RC-12-1 54.0 11.42 304.2 10.2 302.9 913183 RC-12-2 34.0 12.52 288.2 22.9 478.2 37382 RC-12-3 62.0 12.72 234.2 11.9 330.4 5084 RC-12-4 56.0 12.52 694.2 20.3 446.7 54872 RC-12-5 46.0 12.13 172.2 14.0 362.2 24036 RC-12-6 40.0 12.28 425.2 22.9 478.2 913255 RC-12-7 36.0 12.80 243.2 20.3 446.7 915410 RC-12-8 48.0 12.72 273.2 12.7 342.8 928342 RC-12-9 44.0 12.01 105.2 2.5 136.9 9922 RC-12-10 64.0 12.52 243.2 20.3 446.7 910195 RC-11-20 36.0 12.72 624.2 20.3 446.7 947205 RC-12-11 62.0 12.99 426.2 22.9 478.2 6689 RC-12-12 60.0 12.80 352.2 12.7 342.8 19317 RC-12-13 48.0 12.80 310.2 5.1 204.8 8997 RC-12-14 38.0 12.01 273.2 10.2 302.9 32690 RC-12-15 30.0 12.40 264.2 10.2 302.9 907565 RC-12-16 42.0 12.40 395.2 17.8 414.8 915365 RC-12-17 40.0 12.80 90.2 2.5 136.9 947182 RC-12-18 40.0 12.99 334.2 17.8 414.8 22592 RC-12-19 46.0 12.91 303.2 7.6 256.5 22544 RC-12-20 52.0 12.80 395.2 12.7 342.8 81 ","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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