‘ANALY’E‘EML METEQDS 303 §REEE€TENG mmmawm m (11!“ 5.066%}? MAW ARE A MGEEEEJ Ffiifi WEEKS mm PRGBUCE‘EVITY figsertatian for aha Degree afi mg}. MICHIGAE STATE mm mecgfi me; 2375 This is to certify that the thesis entitled _ ANALYTICAL METHODS FOR PREDICTING NON-PRODUCTIVE . TIMES OF LOGGING MACHINES AND A MODEL FOR ESTIMATING MACHINE PRODUCTIVITY presented by Chao-ho Meng has been accepted towards fulfillment of the requirements for ._Ell_._D_._degree in ML Mdmmmfluxu December 13, 1974 Date 1‘ 0-7639 ABSTRACT ANALYTICAL METHODS FOR PREDICTING NON-PRODUCTIVE TIMES OF LOGGING MACHINES AND A MODEL FOR ESTIMATING MACHINE PRODUCTIVITY By Chao-ho Meng Recent trends in logging mechanization in eastern Canada have been towards two concepts of logging machines-- processor and harvester. Each of them possesses two major devices: processing (device) and input (device). "Input device" is a term used for convenience to denote the pick-up device in the case of a processor or the felling device in the case of a harvester. The major characteristic of the type of machines (either conceptual machines or existing machines) involved in this research is that the input device and processing device can be operated simultane- ously. The processor can pick up a tree while processing another and similarly the harvester can fell a tree while processing another. If the input time for the next tree (1 + ith) is less than the processing time of the previous tree (ith), a tree waiting time will occur. Conversely, an idle time will occur. Ordinarily, two stages--feasibility and prototype-- are required before a commercial type of logging machine can be produced. During these stages of machine studies, Chao-ho Meng waiting times and idle times are the basis for either selecting available devices, modifying existing ones, or designing new ones. The principal objective of this research is to develop analytical solutions to the distri- butions, means and variances of waiting times and/or idle times. One of the advantages of analytical solutions is that once the input times and processing times are known or assumed, the waiting times and/or idle times can be computed for any parameters of the input times and processing times. However, analytical solutions may not be obtainable due to the difficulty involved in integrations and also due to the type of distributions suggested for the input times and processing times. To overcome these difficulties, the method of enumeration is also presented. With these two methods, distributions, means and variances of waiting times and/or idle times can be conveniently computed. Basic theorems used to derive the analytical solutions and the method of enumeration are proven. Analytical solutions for many common types of distributions are derived and illustrative examples given to demonstrate how these analytical solutions can be applied and how the method of enumeration can be used. The secondary objective of this research is to propose a model for estimating or predicting potential productivity of a conceptual or an existing machine (prototype or commercial type). The major characteristic of the model proposed is that time elements such as input, Chao-ho Meng processing, waiting, etc. can be "estimated" or predicted individually so that the design engineer can decide which devices of the machine should be modified to increase the productivity of the machine. "Estimate" here means to approximate the true value either by guessing or as the out- come of careful consideration or expert knowledge. In addi- tion, formulas for estimating the variance of productivity are provided. An example is given to demonstrate how field- trial data obtained from one region for a specific machine can be used in estimating potential machine productivity for another region. . It must be emphasized that this dissertation research is not concerned with productivity of a particular machine. It is rather a micro research on methods of deriving the waiting times and/or idle times and predicting productivities of those machines in which the input device and the processing device can be operated simultaneously. ANALYTICAL METHODS FOR PREDICTING NON-PRODUCTIVE TIMES OF LOGGING MACHINES AND A MODEL FOR ESTIMATING MACHINE PRODUCTIVITY By Chao-ho Meng A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements .for the degree of DOCTOR OF PHILOSOPHY Department of Forestry 1975 ACKNOWLEDGEMENTS The author would like to thank Dr. Wayne L. Myers, Chairman of my guidance committee for his arrangement of my course work and direction of this research. Special credit is also due to other members of the guidance committee, Dr. Daniel E. Chappelle, Dr. Robert S. Manthy and Dr. Victor J. Rudolph, for their advice to my Ph.D program, and valuable comments to the draft of this dissertation. The author wishes to express his grateful appreciation to the Logging Division of the Pulp and Paper Research Institute of Canada for its supplying field-trial data for the Koehring Harvester and Arbomatik Processor. Finally, thanks are due to Professor T.C. Bjerkelund, Chairman of the Forest Engineering Group and Dr. J.W. Ker, Dean of the Faculty of Forestry, University of New Brunswick for their granting a special leave. ii LIST OF TABLES IOOOOOOOOOOOOOOOOOOOOOOOO... ..... .90.. LIST OF FIGURES OOOOOOOOOOOOOOIOOOOOO OOOOOOOOOOOOOOOO Chapter 1 INTRODUCTION .. ......... .. ....... .. ............ 1.1 Logging Systems in Eastern Canada and the Logging Machine Concepts Involved in this Research ..... ........... 1.2 Definition of Terms ...... ........... ..... 1.3 The Problem ............ ......... ........ 1.4 Development of Logging Machines .......... 1.5 Units of Measurement ..... ..... . .......... 1.6 State of the Art ...... .... ............. 1.6.1 Arithmetic Methods ....... ......... 1.6.2 Regression Methods . ............... 1.6.3 Simulation Methods ....... ......... 1.6.4 Queuing Theory ......... ........... 1.7 Objectives and Clients of Present Research 2 JOINT DENSITY FUNCTIONS AND BASIC THEOREMS 2.1 Random Variables and Distributions ....... 2.2 The Density Function of w = Y - X ........ 2.3 Expected Values ..................... ..... 2.4 Basic Theorems ..... ....... .............. 2.5 Interpretations of Theorems 1,2 and 3 3 TABLE OF CONTENTS ANALYTICAL DERIVATIONS OF NON- PRODUCTIVE TIME DISTRIBUTIONS ................... ..... . ....... 3.1 Non-productive Time Distributions Derived from Uniformly Distributed Felling Times and Exponentially Distributed Tree Heights OIO‘OIIOOI... OOOOOOOOOOOOOOOOOOOOOO 3.1.1 The Probability Density Function of W ......... 3.1. 2 Distribution of Waiting Time ...... 3.1. 3 Distribution of Idle Times ........ 3.1.4 Graphical Illustrations ........... iii 10 36 36 '40 41 43 v.x~_— .. 3.2 Non-productive Time Distribution of Uniform Input Times and Normal Processing Times ... ....... . ................... 45 3.3 Non-productive Time Distribution of Normal Input and Normal Processing Times ...... 47 3.4 Non-productive Time Distribution of Exponential Input and Normal Processing Times ............................ .. 50 3.5 Non—productive Time Distribution of Negative Binomially Distributed Input Times and Normally Distributed Processing Times ............................. 52 3.6 Non- -productive Time Distribution of Negative Binomial Input Times and Poisson Processing Times .... .................. 54 4 APPLICATIONS ....................................... 56 4.1 Applications of Analytical Solutions-- Data Based ................................... 57 4.2 Applications of Analytical Solutions-- Judgement Based ............................... 65 4.3 Methods of Enumeration ........................ 68 4.4 Sensitivity Analysis and the Choice of Distributions ................................. 83 4.5 Verification of Unproductive-time Distributions .............................. ... 85 4.6 Comparison of Methods ......................... 89 S MODELS FOR PREDICTING OR ESTIMATING THE POTENTIAL MACHINE PRODUCTIVITIES ................... 95 5.1 Models ....................................... 96 5.2 Estimating Parameters and Associated Errors ............... . ............ 101 5.2.1 Error of ”Estimation" ................. 101 5.2.2 Error of Prediction .. ................. 103 5.3 Combined Errors ....... ....... ................ 104 5.4 Applications ......... ..... ................... 105 0 SUMMARY, CONCLUDING REMARKS AND FUTURE RESEARCH ... llO APPENDICES APPENDIX 1 PROOF OF EQUATIONS ................... 114 APPENDIX 11 COMPUTER PROGRAM FOR ENUMERATING LIST op DENSITY FUNCTION OF NON-PRODUCTIVE TIMES ................................ 124 REFERENCES ................................... 135 iv Table 1.1.1 1.6.1 4.1.1 4.1.2 4.2.1 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6 4.3.7 4.3.8 4.3.9 4.6.1 4.6.2 4.6.3 4.6.4 LIST OF TABLES Devices of Logging Machines and Their Functions Illustrative Times per Tree in Cmin. . Felling Times for the Koehring Harvester ...... Processing Time for the Koehring Harvester .... Time Per Tree at Different Processing Speeds .. Density Function of Processing Times .......;.. Pick-up Times for the Arbomatik Processor ..... Density Function of W . Y - X ........ Arbomatik Pick-up Device Waiting Times ........ Arbomatik Processing Device Idle Times ... Means and Standard Deviations for Different Time Intervals eeeeeeeeeeeeeeeeeeeeeeeeee l Processing Times with Gamma Density Function .. Felling Times with Normal Density Function .... Density Function of Non-productive Times ...... Simulated Idle Time Distributions . Simulated Waiting Time Distributions .......... Comparison of Methods in Deriving Waiting Times Comparison of Methods in Deriving Idle Times ‘ Page 17 60 61 67 70 71 72 73 74 77 81 82 90 91 LIST OF FIGURES Page Figures 1.1.1 Operating Sequences of a Processor Concept .................... ..... . ........ ... 4 1.1.2 Operating Sequences of a Harvester concept .......... O..........OOIOOOOOOIOOOOOO 5 1.4.1 Decisions, Feasibility and Prototype Studies of Logging Machine Concepts ................. 11 3.1.1 Mapping from the XY Plane to the XW Plane ... 38 3.1.2 The Transformation of fw(w) into g(u) and h(v) ........................................ 43 4.1.1 Histograms of Felling Times and Processing Times ........................... ...... ...... 62 4.3.1 Density Functions of Non-productive Times ... 76 vi 1. INTRODUCTION 1.1 Logging Systems in Eastern Canada and the Logging Machine Concepts Involved in this Research The present logging systems in eastern Canada can be classified roughly as short wood, tree length and full tree systems. In the former two systems, trees are delimbed at the felling site and then skidded or forwarded to a landing or roadside. The difference between the short wood and tree length systems is that in the former system, trees are cut into desired length at the felling site, whereas in the latter system, trees are not cut into length at the felling site. In the last system--full tree system--trees are felled and no limbing and bucking is done at the felling site. In order to mechanize the operations of these systems, the forest industry has placed great emphasis on the harvester concept for the short wood and tree length systems and the processor concept for the full tree system. The processor concept consists of a road-side processor which will pick up one tree at a time from a pile of pre-skidded trees, delimb it and produce pulpwood logs. The harvester concept is to operate a tree harvester in the forest stands. With this harvester concept, trees are felled, processed and transported to collecting points whenever a load is made. 1 I .1... I ...-A: on”. 0". lI4| Ionii "eve I. I. ". '*'«4! Lb! 2 In order to discuss the two concepts simultaneously, let us define "input device" as the felling device if the machine concept is a harvester and as the pick-up device if a machine is a processor. Both machine concepts have four major devices; they are: input, pick-up, processing and moving. Their functions are shown in Table 1.1.1. The operating sequences for the processor and for the harvester are expressed respectively in Figure 1.1.1 and Figure 1.1.2. It should be noted that many of the devices can be designed to Operate simultaneously. Consequently, the sequences such as input and process or process and moving may occur concurrently. The operating sequences produce six time elements; they are: input, processing, moving, unloading, idle and waiting. Obviously the first four time elements result directly from operating the major devices. However, the occurrence of waiting time or idle time requires the following explanations: Take the harvester concept as an example. The felling device fells a tree and brings this tree (ith) to a position ready to be processed. These motions for the next tree (1 + ith) will start again as soon as the processing of the previous (ith) tree has started. If the felling and bringing time of the next tree (1 + ith) is less than the processing time of the previous tree (ith), a tree waiting time will result; if otherwise, a processing idle time will result. The similar situation will occur when operating a processor, except that the felling time should be replaced by picking-up time. -. l ' I-"1' ...“ DI'HA ‘.'~ Table 1.1.1: Devices of Logging Machines and Their Functions Machine Devices Functions Harvester Felling Felling and bringing a tree to the processing device Processing Delimbing and producing pulpwood or sawlogs Moving Moving in forest stands and transporting the processed pulpwood logs to a collecting point whenever a load is made Unloading Unloading the processed pulpwood or sawlogs Processor Picking-up Picking up a tree from a pile of trees and bringing this tree to the processor Processing Delimbing and producing pulpwood logs Moving Moving along a pile of pre-skidded trees Unloading Unioading the processed pu pwood logs Can a tree be. reached ? move Pick up a tree and bring this tree to the processing device Waiting time for processing device occurs here Is processing no wait device free "yes 'T“J§~T“ Processing device idle time occurs here Process this tree and pick up another tree _ l Figure 1.1.1 Operating Sequences of a Processor Concept. no move reached ? Fell a tree and bring this tree to the processing device Waiting time for processing device ’//j;// occurred here yes \ device free 7 - Processing device Eiggeifidtgfiil ' idle time occurred another tree here Transport to collecting yes point unload k—no cumulated ? Figure 1.1.2 Operating Sequences of a Harvester Concept. u ”ea 43:. be... . o. M M3. 1.2 Definition of Terms Most of the terms used in this dissertation are either statistical terms or terms of time elements in logging research. Statistical terms are generally very well established and need not be defined here, except the term for distribution. Ordinarily, the distribution of a variable x is the total frequency of members with variate values less than or equal to x (Kendall and Buckland,197l), whereas in this research the word distribution is used interchangeably with the phrase probability density function . Terms of time elements in logging research are used by different authors in different ways. It is necessary that the following terms be defined: Pick-up device - A major device of a type of processor. This device is not only used to pick up one tree at a time from the road side, but also to bring this tree to the processing device. Pick-up time - Time required by the pick-up device to pick up a tree and bring this tree to the processing device. A major device of a type of harvester. This device fells a tree and brings this tree to the processing device. Felling device Felling time - Time required by the felling device to fall a tree and bring this tree to the processing device. The felling device of a harvester or Input device the pick-up device of a processor. Input time - Pick-up time or felling time. Input actions 7 - The actions of felling a tree and bringing this tree to the processing device by the felling device or the actions of picking up a tree and bringing this tree to the processing device by the pick-up device. Non-productive time elements Cycle time Waiting time Idle time Waiting times and idle times. Processing time + processing idle time, or Input time + waiting time Time spent by the input device waiting for the previous tree (ith) to be processed by the processing device. It begins when input actions for a new tree (1 + ith) have just been completed but the processing device is still busy processing the previous tree. The period during which the processing device is idle, waiting for a new tree ( 1 + ith) to be input for processing. It occurs when the input actions for a new tree (1 + ith) have not been completed at the time that the process- ing of the previous tree (ith) is finished. 1.3 The Problem Ordinarily, twd.stages of studieS«-feasibility and prototype~«are'required.before a commercial type of logging machine can be produced. One way to study the feasibility of a machine concept is on the basis of knowledge about the behavior of familiar machines and of the expected resemblance of the former to the latter. When the machine concept is being developed, the design engineer may analyze the feasibility by considering historical data on the behavior of the devices or components he intends to use. On the basis of the feasibility, a prototype machine will be built for field trials. During the two stages of studies of the logging machine concept, the design engineer will most likely ask questions, such as: (1) What is the machine's productivity? (2) How long is the unproductive time, that is, waiting time and idle time, and can it be reduced? and (3) How can the input and processing devices best be coordinated so that the cost of production per unit of wood can be minimized? For example, consider two types of processing devices: one type is capable of processing a tree at a speed of 100 feet per minute, the other type is capable of processing a tree at a speed of 200 feet per minute. The input-device waiting times experienced by a tree harvester equipped with the latter type of processing device is certainly less than those experienced by a tree harvester equipped with the former type of.device.. On the.other.hand, capital, operating and. maintenance'costs for a machine equipped with the latter type of device are higher than these for a machine equipped with the former type'o£.dev1ce. Thus, if one Can express the expected waiting times in monetary values, it is possible to select the type of processing device which.minimizes the cost of producing one unit of wood. Based on repeated trials and modifications, it is hoped that a commercial type of machine can be produced. It is also hoped that the wood-producing industries will make a sufficient number of advance orders, without which a machine manufacturer is unlikely to risk production of the machine. However, decisions can not be made by the wood producing industries without answers to questions, such as: (1) Is this machine suitable to their stands? (2) If not, what modifications can be made? It is rather obvious that in order to answer these questions, forest researchers will face the following problems: (1) How to derive the waiting time and idle time density functions and their expected means and variances. (2) Whether a proper model for predicting production in a particular region can be developed by using the limited amount of data obtained from field trials in another region. 10 1.4 Development of Logging Machines Development of logging machines for a given system and forest environment may call for analyses of many logging machines of different functions. The process of selecting or designing suitable logging machines to fit the forest harvesting systems can be aided by experience, imagination and system analysis. The seleCtion usually results in a number of alternatives. The formalized method of systems analysis is available in a number of references: Machol, gt. 31. (1965) and MacFarland (1964) produced a general compendium applicable to many fields of engineering. There are quite a few articles describing logging systems (e.g. Lachance, 1966). However, they can hardly be called system analysis. McColl (1969) was perhaps the only one who analyzed some of the logging systems by a systems approach. There is no limit to the detail which may be included in the analysis of a system. Before this detail can be usefully integrated in a final operational analysis, it should be limited to defining those candidate logging- .machine concepts that have sound engineering, mechanical efficiency, and basic economy. The process of deveIOping logging-machine concepts is shown in Figure 1.4.1. Any new idea or requirement is first subjected to a feasibility study (light hatched area) 11 .mumouaou ocfiauma mnfiwmoa mo mofipSpm omxuouoam pee Audaflnfimmom .mGOAmwuen H.e.H-on:ufim \Auflfinefimufima\ 5.23.122 \ xufi>fiuo=voum mmHnaem mo :ofiumaae>o \\ //J//// ///// // \ESSEEE\ e5 \ cm .I oueaflpmo umou mzofi<3<>m 0:3“me i232»: \ mm»eoao¢m uewmmoa .Auflwnmcweufima / A \ wee .xpflafinm ucoaouwacou munoucou -MHOH .xufl>wuo=v Ir ommfimuMMQ oua>ou ecwnuea \\ -oum oueaae>o 1 new we ucwmuoH \\ ou mflweuu vaowm ca. m compeaaawom \\\\ wewmoao>on \\ /.// \ \ \\\ x \\ \ \ \ mwmxaedu mmoc //// -o>fluuomwo umoo ”7r campusum>o maeumhm mqwmuoq A .woanuomm % -scaa on waaonm .eoomam on wasonm ongu wauuoaaou muecno .oocm>ve u:oflou«:¢ou Hugues: mowwouu Lennon: meumuop enwnuua Housuoemacdz, Academe mummawooz udwuuoa 12 which leads to the formulation of basic devices and com- ponents. At this stage evaluations are made of preliminary cost, productivity, reliability and maintainability within the feasibility requirement. These, in turn, provide data for the design, engineering and evaluation of prototypes (dark hatched area). Only at this stage of evaluation can cost-effectiveness analysis and operations research based on design, construction, maintainability, reliability and tests lead to the selection of a specific prototype for production. It can be seen that the evaluation of logging- machine concepts requires complex procedures. All procedures involved could not be addressed in this dissertation research. Nevertheless, any attempt to see where the problem is embedded would hardly be possible without brief discussion of some of the procedures. Reliability ' The reliability of many standard components can be predicted with considerable accuracy on the basis of historical data. However, analysis of reliability is almost pure guess work if the concept is original and uses special components which must be developed. No literature concerned specifically with reliability of logging-machine concepts was found during the literature 13 review. For general information.regarding reliability, the reader is referred to standard text books (ChOrfas, 1960; Bazovsky, 1961; Calabro, 1962; Pieruschka, 1963). Maintainability It is extremely difficult to determine the details of the maintainability at the stage of concept formulation. It is quite unlikely that an engineer or mechanic can visualize this at the drawing or concept stage. However, a rough estimate of the cost of maintenance can be made on the basis of known devices and on attempts to reduce maintenance requirements. Effectiveness Generally, the effectiveness of a concept is a measure of its ability to fulfill its intended role. If, in the final analysis, initial cost is assumed to be the primary factor, effectiveness must be close to a second. The theoretical work for effectiveness analysis is a widely discussed part of operations research not included here. (.1 . D (D 14 1.5 Units of Measurement Linear measurements such as tree height and distance are expressed in feet. For convenience, tree height is also expressed in number of bolts. A bolt is 100 inches in length. Volume measurements are expressed in cubic foot or cunit. A cunit is 100 cubic feet. All time elements are expressed in cmin., which equals 1/100 minute. 15 1.6 State of the Art. There are three broad methods used in studying the logging machines --arithmetic, regression and simulation. 1.6.1 Arithmetic Methods Using the arithmetic method, the total time per tree can be estimated by simply adding up all the rel- evant time elements. For example, a design engineer wants to know the total time per tree of a felling machine. He figures out that the moving time must be the distance (S) moved divided by the machine travelling speed (V). He further reasons that a machine seldom travels in a straight- line pattern; therefore, the distance must be multiplied by a ratio K, the ratio of straight-line distance and the actual distance travelled. The product will be divided by the number of trees felled to obtain the moving time per tree. Thus, he has: T - KS/V (1) 1 To estimate the total time per tree, the engineer uses the following formula: T -,T1 + r2 + . . . Tn (2) where T is the total time per tree, T2 . . . Tn are other 16 relevant time elements. It is assumed that the time elements such as T2 to Tn can be either estimated as T1, that is, by common sense, or from previous studies, films and machine specifications (Hedbring, 33. 51., 1968). All time elements and factors involved in this method are estimated "on the average". This method is algebrically correct; however, models such as Bq. (1) may be useless in reality. The so-called "wander factor", which is something like the problem of random walk, is very difficult to estimate in most cases. In reality, stand factors, such as density and terrain conditions, may offer better opportunities to estimate the moving times. According to the arithmetic method, the waiting time is simply the average input time subtracted from the average processing time. For example, if the former is 20 cmin. per tree on the average and the latter is also 20 cmin., then the waiting time is zero, whereas in reality this may not be true. In Table 1.6.1, both the input time and processing time are 20 cmin. The waiting time is 9 cmin. per tree, not zero as the arithmetic method indicates. Another major weakness of this method is that every time element and every factor are assumed deterministic in a stochastic environment. Consequently, nothing can be said about the precision of the model used. Hedbring and Akesson (1966) and Hedbring, 33. a1. 17 Table 1.6.1: Illustrative Times per Tree* in Cmin. Input Time Processing Time Waiting Time Idle Time 1 + ith tree ith tree 20 30 10 O S 35 30 O 30 10 O 20 10 15 5 0 35 10 0 25 Averages: 20 20 9 9 * Input times and processing times of a conceptual processor or harvester are uSed for the purpose of illustration. 18 (1968) developed formulas.usingthe arithmetic method in order to study the concepts of thinning machines. McCraw and Silversides (1970) used the same method for performance analysis of logging machines and machine devices. 19 1.6.2 Regression Methods Regression models have been widely used in logging- machine studies (e.g. Powell, 1974; Axelsson, 1971). In evaluating the potential productivities of some logging- machine prototypes, the Pulp and Paper Research Institute of Canada has been using regression analysis, and several such regression equations have been established (Axelsson, 1971; Bredberg, 1970; Powell, 1974). However, owing to the lack of data, regression methods are not suitable. For example, to estimate the waiting time, tw, of a processor, which can only process at a rate of 100 feet per minute, the following regression model can be suggested: tw = f (DBH, tree height, etc.) This model can not provide an answer to the type of question such as: What will tw be if the machine's processing rate is improved to 200 feet per minute? This is because no speed other than 100 feet per minute can be observed. Therefore, the type of regression models established by Pulp and Paper Research Institute of Canada may not be able to predict the total time changes before an improved machine can be built. 20 1.6.3 Simulation methods Many authors have used the simulation approach in studying the efficiencies of logging systems. Skarr (1966) applied simulation technique to waiting line problems in wood handling. Osburn (1970) also used the same technique with pulpwood production problems. Bonita (1972) examined problems associated with the management and control of forest harvesting operations by the simulation approach. The problems studied by these authors are complex in the sense that many interactions among machines within a system may occur. For similar types of harvesting machine concepts described in Section 1.3, Newnham and Sjunnesson (1969) and Newnham (1972) used the simulation approach for evaluating machine productivities. Meng (1971) in his comments on Newnham's approach to simulating harvesting machines pointed out that simulation is not necessary when the system is "small”. Meng (1971), in a separate paper, also studied the problems involved in simulation. 21 1.6.4 Queuing Theory Ordinarily, a.queuing situation is characterized by a flow of customers arriving randomly at one or more ' service facilities. The customer upon arrival at the facility may be serviced immediately, or if willing, may have to experience waiting time until the facility or the server is made available. In this situation there can be any finite number of customers waiting in the queue. Operating the input-device and processing device of a tree harvester or processor can be considered as a queuing situation where the input device is the "customer" and the Processing device is the service facility. However, this can only be considered as a special type, since there is only one or no "customer" waiting in the queue. Although analytical SOlutions for many ordinary queuing systems are available in many references (e.g., Saaty, 1961 and Gross and Harris, 1974) no analytical solutions for the special type were found during th e literature review . 22 1.7 Objectives and Clients of the Present Research The failure of present techniques of evaluating the productivity of logging-machine concepts or logging machines under development has been caused by: (l) the persistent seeking for deterministic fixed values where the answer can Only be produced in probabilistic values; and, (2) the limitations of regression analysis. Although most forward-looking forest researchers and forest engineers see the rationality of a probabilistic approach in deriving answers to many of their problems, the number of forest researchers and forest engineers trained in probabilistic theory have been limited. As a result, analytical and enumeration methods for solving waiting times and/or idle times have not been developed. The primary objective of the present research is to fill this gap. Probabilistic approaches are used to develop basic theorems and to derive formulas for the solutions. The secondary objective of this research is to propose a model for predicting potential productivities of processor or harvester prototypes. Methods of estimating the parameters of the proposed model are also discussed. It should be noted that this dissertation research. is not concerned with the study of environmental factors in relation to machine productivity, nor can every aspect of evaluation and prediction during machine deveIOpment be studied and described. This dissertation research can be 23 considered as micro-research on methods of estimating the non-productive times and predicting machine productivities. To the best of my knowledge, no one has yet proposed analytical solutions and numerical approximations by probablistic enumeration for solving these typesof problems. The clients for the results of this research are machine designers, forest researchers and woodlands managers. 2. JOINT DENSITY. FUNCTIONS AND BASIC. THEOREMS 2.1 Random Variables and Distributions Most forest engineering measurements, such as input times and processing times, are quantitative in nature. Very often, the variable under consideration can ' be taking on any one of an infinite number of possible values. A variable whose specific value can not be 1 The predicted with certainty is called a random variable. character of a random variable is described by its probability law, which in turn may be governed in a number of ways. The most common way is through the probability distribution of the random variable. Generally speaking, a probability distribution can be classified as either a discrete probability density function or a continuous probability density function.2 When the number of values that a random variable can take on is restricted to a distinct number, say, the values 1, 2, 3, . . ., the random variable is discrete. Although the discrete random variable is 1 More precisely, a random variable is a function defined on the sample space of experiment. It assigns a numerical value to every possible outcome. (See, for example, Parzen, 1960) 2 Many text books in mathematical statistics use the terms probability mass function and probability density function to describe respectively the discrete variables and continuous random variables. 24 25 appropriate in many situations in describing the input times or processing times, the continuous random variable is more frequently adopted as the mathematical model for physical phenomena of interest to forest engineering. The continuous random variable can take on any value of the real axis. This does not imply that the random variable must take on values over the entire axis, i.e. from -w to m. Intervals, such as the negative range, for example, can be excluded by simply assigning zero probability. The details of specifying the probability density functions, which can be found from many text books in mathematical statistics (for example, Mood, 1950), will not be discussed here. The most general type of random variable that will be encountered in this research is of mixed random variables. A part of the total probability is spread out "continuously" over some interval, while the remaining part is assigned in "lumps" to a finite or infinite discrete set of values. For example, both the input times and processing times may be continuous, with the waiting time distributed as a mixed variable, which is shown in Figure 3.1.1. 26 2.2 The Density Function of W a Y - X. Suppose X and Y are independent variables having respectively the density function fx(x) and fY(y). The problem is to find the density function of W a Y - X. It is well known that the joint density function of any two independent random variables X and Y can always be found by multiplying their probability density functions (Blum and Rosenblatt, 1972). fXXCX.y) = fx(X)fY(y) [2.2.1] Since W = Y - X, substituting Y by W + X, we have fXY(x,w+x) = fx(x)fY(w+x) [2.2.2] the distribution of W follows upon the integration over all values of X, if both X and Y are continuous fw(w) = -£ fx(x)fY(w+x)dx [2.2.3] By the symmetry of the argument, it follows that it is also true that fw(w) = -i fy(Y)fx(Y‘W)dy [2'2°4] In the case where X and Y are independent discrete variables, formulae [2.2.3] and [2.2.4] can be expressed respectively as fw(w) . i fXCx)fY(w+x) [2.2.5] fw(w) = z fY(y)fx(y-w) [2.2.6] '-< 27 2.3 Expected Values A random variable can be represented by an entire function or a list of values the variable can take on and their respective probabilities. It is often that concise descriptions by mean and standard deviation of a variable are sufficient for the purpose at hand. The mean of a discrete variable is E[X] - I: x fx(x) [2.3.1] all x or, for a continuous random variable, as E[X] = f x fx(x) dx [2.3.2] -w The mean says nothing about the risk and dispersion. Several such measures are possible. The most common and most useful such measure of a random variable is the variance. It is defined as Var [X] . alI x (x-E[X])2fx(x), if x is discrete, or Var [X] = f (x-E[X])2fx(x)dx, if x is continuous. -w It can be shown that Var [x1 . E[Xz] - (E[X])2 [2.3.3] where E[Xz] . 2 x2 fx(x) [2.3.4] all x or E[XZ] - f x2 fx(x) dx [2.3.5] , PVT. in.” “‘v‘. 28 2.4 Basic Theorems Theorem 1 Assume X and Y are two random independent variables with probability density functions fx(x) and fY(y) respec- tively. If two new variables are defined as U - Y - X , if Y - X > 0 - O , if Y - X‘i 0 and W - Y - X , -o < N < m and distributed as g(u) and fw(w) respectively, then, if U is continuous 0 g(u) - Ifwht) dw , u- 0 fw(w) , u > 0 - 0 , otherwise [2.4.1] if U is discrete 8(0) ' Z fw(w) u = 0 all W30 , . fw(u) , u = 1,2,... - 0 , otherwise [2.4.2] Proof Since U and W are identical when Y - X > 0, i.e. w - u > 0, thus, g(u) - fWCu). The probability of U - 0 is the same as the probability of U - X 5.0, since U - 0 if Y - X g 0 as defined. 29 That is Pr{U = 0} - Pr{W : 0} - Pr{Y - X :_0} 0 thus, Pr{U - 0} - I fw(w) dw, if W is continuous, -Q or Pr{U - 0} - 2 fw(w), if W is discrete. all Wfp Thus equations [2.4.1] and [2.4.2] follow immediately. Theorem 2 Again assume that if X and Y are two random indepen- dent variables with probability density functions fx(x) and fY(y) respectively. If two new variables are defined as ' V-Y-X,ifY-X 0, thus 0 0 E[Uz] - J u2g(u) du I w2g(w) dw E[vz] . l” v2h(v) dv e l” w2g(w) dw It follows that 0 E[UZ] + E[Vzl w2g(w) dw * I w2g(w) dw - me2g(w) dw Therefore, E[WZ] - E[Uz] + E[VZ]. [2.4.3] From equations [2.4.7] and [2.4.8], we have E[UZ] + E[VZ] - Var [X] + Var[Y] + (E[Y - X])2 or E[VZ] - Var[X] + Var[Y] + (E[Y - X])2 ' E[UZ] 32 By definition, Var m - .Elvzl - (ENDZ Consequently, Var[V] - Var[X] + Var[Y] + (E[Y-x1)2 - E[U2] - (E[VJ)2 Theorem 3 is thus proven. is o‘.‘ Us. I .1. 33 2.5 Interpretations of Theorem 1, Theorem.2 and Theorem 3 If X and Y are defined as random variables repre- senting respectively the input times and processing times, then U - Y - X , if Y > X - 0 , if Y 3.x can be interpreted as the input-device waiting times according to the terms defined (Section 1.2), and g(u) can be interpreted as the probability density function of waiting times. Similarly, variable Y - Y - X , if Y < X - 0 , if Y 3_X can be considered as the processing device idle times, and g(v) is the probability density function of idle times. Note that if X and Y are interchanged (X represents the processing times and Y represents the input times), then U and V can be considered as the processing-device idle times and the input‘device waiting times, respectively. A Theorems 1 and 2 are the foundations of deriving the probability density functions of g(u) and h(v). Theorem 3 can be used to estimate the expected mean and variance for the random variable V once the eXpected values of U and U2 have been estimated. 3. ANALYTICAL DERIVATIONS OP NON-PRODUCTIVE TIME DISTRIBUTIONS In general, the'distributions of waiting times or idle times can be derived either deductively--by analysis, or inductively-«by numerical enumerations or simulation. This chapter, which repreSents the most important, as well as the most difficult, phase of this research, is devoted mainly to the development of analytical solutions to the waiting-time or idle-time distributions and their means and variances. The method of enumeration is demonstrated by way of example in Section 4.3. The method of simulation, which has been dealt with elsewhere (e.g. Gordon, 1969), need not be discussed here. The advantage of an analytical solution is that once the equation is derived, a solution can be easily computed. For example, assume that we want to find the sum of all integers from one to ten. One way is to add up all the integers one by one; another is to derive an analytical formula such as l + 2 + 3 + . . . + n I n(n + 1)/2 for the summation of integers from 1 to n. Derivation of this general solution, of course, requires much more effort and skill than simply adding up the ten numbers from one to ten. However, assume that such general solutions are non-existent and one is required to calculate the sum of integers from one to, say, one hundred million, it would be much simpler to derive an analytical solution than to add up all these 34 35 numbers. To.consider.another example, a well-known analytical method in economics and operations research is the classical methdd of Lagrange multipliers. This method offers general solutions to many optimization problems with a few constraintsT waever, when the number of constraints in a problem becomes large, the solution by the method of Lagrangian Multipliers becomes very complicated. Numerical methods such as mathematical programming may be best suited. Similarly, there are cases where analytical solutions may not be obtained since some of the integrations may prove difficult and sometimes not even feasible. However, this can be approximated by the method of enumeration. The distributions selected for representing the input times and processing times are some of the most commonly adopted for empirical use by the researchers as well as decision makers. This is simply because they are well known, well tabulated and easy to work with. This chapter is written to provide not only the formulas for estimating the non-productive times, but also an under- standing of how the basic theorems can be used. Note that.detailed proofs of many of the equations in this chapter are presented in Appendix I. 36 3.1 Non-productive Time Distribution Derived from Uniformly Distributed Input Times and Exponentially Distributed Tree Lengths 3.1.1 The Probability Density Function of W Assuming the input time X (ranges from a to b units of time) per tree of a Processor or a harvester concept is uniformly distributed as 1/(b ' a), a < X < b fXCX) 0, otherwise [3.1.1] and the tree length 2 of trees piled along the roadside follows a truncated exponential distribution fz(z) = Beach - z), z > h I 0, otherwise [3.1.2] where h is the minimum merchantable tree height and B is the distribution parameter. It can be verified that E[Z] - h'+ l/B. Further, assume the processing time Y has the following relationship with tree length 2. Y I c + rZ where c is the fixed component time per tree and r is the unit time required for processing a unit length of tree. The problem is to find the density function waW). To find the density function fw(w), we must find 37 the density function of Y which requires the Jacobian l 0 l/r 1/r IJI I I l/r (assuming r > 0). By applying the Jacobian transformation (Anderson and Bancroft, 1952), we have the density function of processing times. fY(y) . % ehe ' 8(y ' c)/r’ y': c + rh I 0, otherwise [3.1.4] with E[Y] I c + rh + r/B [3.1.5] Var[Y] - rz/sz. [3.1.6] Since the input time is independent of the processing time we can apply equation [2.2.2] to find the density function f (x,y) I f (x,w + x) for some region B. X,Y X,Y As shown in Figure 3.1.1, the region A where fw(x,y) is greater than zero is bordered by three lines x I a, x I b and z I h. Mapping A in the (w,x) system of coordinates, we would obtain region B bordered by three lines y I a, y I b and w I c + rh I y. We have fx ch,w + X) . 8 98h ' B(w+x-c)/r, ' r .- a if a < x < b and c+rh-x §_w I 0, otherwise. However, we are only interested in the marginal distribution of W. To derive it, we must apply equation [2.2.3]. From .i '0 “.7 .t‘ s¢ b’ 6+- I .s_, -‘ a... . 3 g .3; uppedenR Region 3 indicates the lint Begin-Ah \zi“ 0 ll X >¢ from the XY plane to the XW plane. 3.9 Figure 3.1.1 we can see that the limits of integration are: (1) from c + rh - w'to b when w is between c + rh - b and c + rh - a and (2) from a to b when w > c + rh - a. Now, b fw(w) . J fw(w,x) dx c+rh-w b . __£__ Jane/refit) e-Bx/r dx r(b ' a) c+r -w b 15 since 1 e-Bx/r dx . - £_e-x/r c+r -w B c+rh~w . - £.e'Bb/r _ e-B(c+rh-w)/r , B we have f (w) . __1.__ [1 - 93(h+C/r-b/r)e-BW/r] W b - a For the region where w > c+rh-a, we have b b a er ' a) a . ;_l___[e8(h+CIr-a/r) _ e8(h+c/r-b/r)]e-Bw/r . Consequently, the density function of W is fw(w) I 1/(b - a) - Kze-Bw/r , c+rh-b i w‘: c+rh-a 40 -Bw/r . K1 e c + rh - a < w - 0, otherwise where K2 - ¢‘(h*C/r'b/r)/(b-a) K1 _ [.B(h+c/r-a/r) _ eB(h+c/r-b/r)]/(b_a) [3.1.7] 3.1.2 Distribution of Input-device Waiting Times By applying equation [2.1.4] and the definition of mathematical expectations, it can be shown that the distri- bution of input-device waiting time per tree, h(u), and its expected values for three cases are shown as: Case (A): c + rh - a < 0 8(0) ' 1 - Klr/B . [0 fw(w)dw , u = 0 - K e-Bu/r u 0 1 9 - 0, otherwise [3.1.8] . 2 2 Wlth E[U] - Klr IB [3.1.9] E[UZ] - ZKIrS/BS [3.1.10] Case [3): c + rh - a > O and c + rh - b < 0 II 0 ECU) - (b-c-rh)/(b-a) + (l - e'B(C+rh'b)/T)KZr/B, if u . l/(b - a) - Kze'Bu/r , 0 < u < c+rh-a 41 offiu/r - K , u < c+rh-a 1 .- 0 , otherwise [3.1.11] 2 with E[U] - 19*rh'“) + 19*rh'a)/r * 1 - K2](r2/82) [3.1.12] 2(b-a) b - a E[UZ] - [(c+rh-a)2r/B + 2(c+rh-a)r2/82 + 2r3/83 + (c+rh-a)3/3]/(b-a) - ZKer/B3 [3.1.13] Case [g]; c + rh - b > 0 The density function g(u) is identically distributed as fw(w) if c + rh - a > 0 consequently E[U] = E[W] and E[Uz] - E[WZ]. 3.1.3 Distribution of Processing-device Idle Times. Similarly, by applying equation [2.4.3] and the definition of mathematical expectations, the distribution of processing device waiting time per tree and its expected values can be found as: Case (A): c + rh - a < 0 h(V) - Klr/B - I: fw(w) dw , v = o - Kle'av/r , c+rh-a < v < 0 - l/(b - a) - Kze'ewr , v < c+rh-a - 0 , otherwise [3.1.14] with iaitj 42 with E[V] - c+rh+r/B - (b+a)/2 - Klrz/Bz [3.1.15] E[VZ] . rz/s2 + (b-a)2/12 - 2K1r3/s3 + [c+rh+r/B - (b+a)/2]2 [3.1.16] Case (3): c + rh - a > 0 and c + rh - b > O h(v) - 1/(b - a) - xze‘BV/r , c+rh-b < v < o 1 - (1 - e‘5(c*rh‘b)/’)x2r/B + (c+rh-b)/(b-a) , v = 0 - 0 , otherwise [3.1.17] with E[V] = K2(1 - [(c+rh-b)B/r + 1]e'(c*rh'b)B/r)r2/82 - (c+rh-b)2/[2(b-a)] [3.1.18] E[Vz] = 2x2r3/B3 - (c+rh-b)3/3 - [(c+rh-b)zr/B + 2(c+rh-b)r2/82 + 2r3/e31/(b-a) . [3.1.19] Case (C): c + rh - b > 0 This implies that the processing time is always longer than the input time. Therefore no processing-device waiting time has ever occurred. 43 3.1.4 Graphical Illustrations To demonstrate.how the distribution g(u) and h(v) were transformed from fWCK), equations [3.1.7], [3.1.8] and [3.1.14] are depicted in Figure 3.1.2. Values of the parameters are assumed as: a = 20 cmin., b = 40 cmin., c = S cmin. and r = .5 cmin./ft.. The parameter 8 = is found to be .0667 by assuming E[Z] = 40 ft. and h = 20 ft. (see Sub-section 3.1.1). Both U and V are mixed random variables. When u > 0, U is distributed continuously as fw(w) or g(u) = f(w). At u = 0, g(o) is discrete with probability If” fw(w)dw = .75 (the dark hatched area). Similarly, h(v) is continuous and identical with fw(w), if v < 0. At v = 0, h(o) is discrete with probability {“fw(w)dw = .25 (the light hatched area). hp} 3m [11. h . ‘ I“. 4_4 .05 S [wlwldw ..i::::. :‘tzz 1.0 00$. .031- .0:3: ’9’9‘? S. .$?:. walwldw= probobiliiy lhol :z:::? ¢ o no idle fine occurred 9:39: '9’ :00 9.999; f w -20 -l5 -l0 -.'lA 0 5 l0 I5 20 25 The Difference in Felling Tinee ond Pick-up Time: in Cmin. —probobilily lhol no wailing liune occurred_ glolJ 7 Deneily funclion of woiiing lines '-'- glul .03 .02 .Ol o 5 no is. 20 25 Pickup-device Welling Tile: in Cain. .25 ‘ Moi Deneily funclion oi idle limes = hlvl T T l I : V ~20 -l5 -l0 -5 0 Felling-device ldle Times in Cmin. Figure 3.1.2 The Transformation of fw(w) into g(u) and h(v). 45 3.2 Non-productive Time Distribution of Uniform Input Times and Normal Processing Times Suppose that X and Y are two independent random variables distributed as fx(x) = l/a , 0 < x < a - O , otherwise 2 and fY(y) . exp[7 LX_L7E1_] , - m < y < w n o 20 where a, u and o are parameters. Letting W - Y - X and applying equation [2.2.2], we have ~ 1 x + w - u f(x)f (w+x)-———-exp{- ]. X Y b/Zn o l 202 To find the probability density function of W , we must integrate a 2 fw(w) - I 1 exp - [Fx + ;~- u) ] dx 0 ao/TF 20 8+W'u 2 1 t or f (w) a I ex [- -—] dt W W agar p where t = x + w - u. To find the waiting time distribution, we apply equation [2.4.1] and obtain a+u~u 2 g(u) ' 1 expl- 5—7] dt , u > 0 ao/TF l 20 11"“ O a+u-u 2 l t = -———— expL- _—Zl dt dw , u = 0 [3.2.1] _1 “In ao/Zn Zo 46 It can be shown (Appendix I) that I3W] - u - — + _12; . 2.361.ng -. all} 2 20 + 0'1»l exp[" i2.) + [Lt—L. ... E, _ u]F[a - p] Z/Fa 20 2a 2 _‘G 2 + 2 _ - [a JEFF.) [3.2.2] 23 o ' u/o ' 2 where p[1‘.} . I _1_ exp {3.) dz ° -0. 1r 2 3' U T 2 F{é—;_£Ja I ———-eXp[- 2.) dz ° a. ’7? 2 and E[UZ] a { 0 (Zn-a) _ O (“2+202))8Xp[— a-“ 2) 3 7' 33/717 20 2 2 3 - 2 2 * (n+20)) [“)+[p_q_+.u_-o_u [Sal-2? 2:7 a 3a 32 - 2 3 _ 2 + an - —]F[a_._}‘_) - [L49— + L]F[__u] + o 3 ° 8 3a a 2 + “2 - an + 2.. [3.2.3] 3 47 3. 3 Non- -productive Time Distribution of Normal Input and Normal Processing Times SuppoSe X and Y are statistically independent variables with (X " “1)2 fx(x) ‘ exp - 2 -m < x < m no 201 (y " “2)2 fYCY) = exp - 2 -m < y < m n02 202 Let W = Y - X, then from equation [2.2.4] fWCw) - I fx(x)fY(w + x) dx', -w < w < w on it 2 (x - u ) (X+w-u ) ' 1 J°XP'—‘z'1——‘——T2‘ d" 2"0102.m 201 202 It can be shown that (x - u1)z (x + w - uZJZ 2 * 2 201 202 = [x + ('”1°22 * “12" ' °12“2)/(°12 * "223]2 2 2 2 2 20 o2 /(o1 + 02 ) 1 ("1 * W ' "2)2012022 2 2 2 2 201 02 /(o1 + 02 ) ... BY utilizing this result we have 48 expt-cw +u1 - 1:2)2/(2cal2 + 022m fwcw) - /*' 2 21r(o1 _+ a2 ) [3.3.1] By applying equation [2.4.1], the density function of U can be written as 0 1 (“""‘1"‘2)2 g(u) ' exp - 2 2 dw , u = 0 /—_2'—_ -~ 2n(ol + 022) 2("1 *“2 3 1 (um1m2)2 = exp - 2- 2 u > 0 J2"(012+ 022) 2(o1 +02 ) [3.3.2] The expected value can be written as w 2 (u + u - u) E[U] = u expl- 1 2 du /—T—_T ul-uz 21r(o1 + 02 ) 2(012 + 022) ”2‘“1 o -o 2 1 2 2 (n+u -u ) E[UZ] = u exp - 1 2 du J 2 2 2(0 2+0 2] w 21r(o1 + a2 ) l 2 Let t . u + “1 - “2’ it can be verified (Appendix I) that l; 2 + a 2’ (H _u )2 1 2 1 2 6X " 2 2 2(o1 +02 ) E[U] . n “1 ‘ u2 + Cu2-u1)P [3-3-3] “1 * “2 F7—7 - ~ 2] 0‘“ + .0“ . . -.(p - u ) E[Uzl - 1“ 2 c-uz-ul) exp ———2———2—1 2 f2? 2(o1 + 02 )J 1 . l1 " 11 +(oz+oz+(u-u)2F 2 1 [3.3.4] 1 2 2 l 2 2 :51 + oz] 112'111 a +0 Ll ' U 1 2 2 where F -2 1 I - J exp(- 5 ) dz . [of]? + oz7 -°° 2 59 3.4 Non-productive Time Distribution of Exponential Input and Normal Processing Times Consider the two statistically independent random variables X and Y with probability density functions respectively equal X fx(x) - l—-exp(-A), x . 0,1,2, ... [3.4.1] x! 2 . . 1 - u f (y) - exp (- LZ——z—l— ) -~ < y < s [3-4-2] Y ' JZwo Zo ’ where A, u and o are parameters. Let W - Y - X and applying equation [2.2.5], then we have f - f f w(W) x20 XCX) Y(y) ' E —-—}‘ exPC-A - (x+w -AfiL x-O xl/fib' 202 Applying equation [2.4.1], we have the waiting time density function ( 0 E 4x £3 + W 'LAlz g u) a —— exp(-A - ) dw, u = 0 I x-O XI/IWo 202' oo x 2 a _1_—— exp(-), - Q‘ + I; - A) ), u > 0 x-O x1/7Fo Zo ' 0: otherwise [3.4.3] By the definition of expected values, we can find 51 so ' ‘ 2 l x (3 f u ' A) E[U - u A—-——-). ex (-A - d 1 i xi. l2nox! p 202 ) u "’ z" .. u -(u+x-x)2 u nun.- OX (n1) ex (1 x20 x! p i {Zoo PC Zo2 ) u o 2 and E[UZ] - l u2 z ———l——-Ax exp[-x - (x + u2- X) ] du . x-O J2Fox! 20 E[u] and E[uzl can be simplified as ° 1x a - x - u 2 [ ] Z °XPC )( v exp( L—Ezj—l-') x-O xi + (u - x)Fc'3‘-—*—‘-‘-)) [3.4.41 0‘ m 2 2 _ 1 x -x + u (x - u x20 x! p { /2fl p 20 ) + (a7- + (-x + u)2)F('1‘—1—"—)} [3.4.5] 0 52 3. 5 Non- -productive Time Distribution of Negative Binomally Distributed Input Times and Normally Distributed Processing Times Suppose that X and Y are two random independent variables. X has a negative binomal distribution with a parameters p and k fxcx) - (iii) (1 - p)x ‘ k pk . x - k. k+1. and Y is distributed normally with parameters u and o 2 eXPC' 113:7El_.) . -“ < y < w . The density function fw(w) can be found immediately by fY(y) - “ITO applying equation [2.2.5] f ( - f f + w w) xgk x(X) YCW X) X'l X'k k - 1- 2 (k-l) ( P) P 2 exp(;L!§5%El—J [3.5.1] 0 Applying equation [2.4.1], the density function of u is obtained as O “ 2 8(“) ' J 2 (i j)(1 p)x k pk p-—l—-exp('LE§Z%El—J dw, u=0 no a x-k “ 2 ' Z (k-1)(1' -p)x k pk p/zi exp('L!:z%El—-) , if u>0 x-k no 20 - O , otherwise [3.5.2] 53 with m n 2 E[U] ' I u 2 (k:1)(1'p)x k Pk 1 OXPC “fit-u ) du O x-O no 20 ' xix (k- 1)(1- M)”k 1‘ I-f; exp(- “2:” 2) du = x21: (1,"1)(1--"p)"‘k pkfpf—o: expCQ—zj—ZEJ-i) + (u-xm'i-g-‘i )} [3.5.3] and E[UZ] - 3. u2 at; (k-1)(1-‘p)" 1‘ pk p-—:;- cm" “2"" 2) du 0 °° x- -k pk -(x - 11 x21: (k_ 1) (1 p) { mm a an 20 2) + [oz + u - x)21F('-"—*—E )1 [3.5.41 0 54 3.6 Non-productive Time Distribution of Negative Binomal Input Times and Poisson Processing Times Consider the two statistically independent discrete random variables X and Y with probability density functions respectively equal to fx(x) - (§:})(1 - p)"k pk . x = k.k+1. and fY(y) I AY e-x/y! , Y = 0.1.2, The density function of W can be found by applying equation [2.2.5] f = 0, f 0 I f f or w W( ) xgk XCX) Y(X) I l, f (l I f ( f ( + 1 w W ) xgk X x) Y x ) w - 2, fw(2) - xzk fx(x)fY(x + 2) thus, in general for w I 0,1,2,....n,.... fw(w) I xgk fx(x)fY(x + n) on X+W 'A = 2k (iii)(1 - p)“k pk L?_:E§— [3.6.11 xI x w I 55 The density function of u immediately follows from equation [2.4.1] 0 as + -), 8(u) . I [k (’1: _i)(1 - 13))”k pk 5:23— dw, if u = 0 -. ka (x+w)! oo 4. ..A ' 2 (11:3)(1 - p)"'k pk —-)‘x we ’ if u > 0 ka (x+w)! I 0 , elsewhere [3.6.2] to on X'Hl 'A 'thEU - 1- x-pkkL——:— 3.6.3] w1 [ ] ugo u x21: (k_1)( p) (flu): [ I I x+u -A E[UZJ - X uz 2k (i j)(1 - p)"“‘pk l———9—— [3.6.41 uIO ka (x+u)! 4. APPLICATIONS The applications of the analytical solutions developed in the previous chapters involve selecting the appropriate probability distributions for input times and processing times. In general, when a machine designer offers a new logging concept, he may not have changed the whole existing concept. A change in component design, such as a new input device or a faster processing device, may be the objective of the improvement. If these devices are already in existence and tested, then historical data may be available. When a probability density function represents information of this kind, we shall term such a probability density function as a 'data based' probability density function. If the component is going to be newly designed, the information regarding the input and processing time distributions is not available. The probability density function must be developed or chosen based on introspection and theoretical considerations. We shall term such probability density functions as judgement based. S6 .57 4.1 Application of Analytical Solutions-«Data Based Suppose a design engineer is interested in developing a new harvestingmmachine concept. He may use available historical data obtained from similar existing devices to hypothesize certain convenient probability density functions to represent the input-time and processing-time distributions. One way to hypothesize a density function for the time elements is to depict the historical data as a histogram and look for a reasonable, convenient distribution. If, by this method, no convenient distribution can be assumed, the suggestion here is to compare the cumulative historical data with the cumulative distribution function of some common distributions. In practice, both the plotting and comparison of cumulative curves can be simplified by scale changes, that is, by using special probability graph paper. This probability paper provides prOperly scaled ordinates, such that the cumulative distribution function of the probability law plots as a straight line. With such paper, comparison between the hypothesized model and data is reduced to a comparison between the cumulative historical data and a straight line. The use of these papers will not be illustrated here; it can be found elsewhere (For example, Hald,1952). If the input or processing times are not related to stand factors measured, the methods suggested are quite 58 adequate. However, in cases that stand factors are related to the time element under study, the density functions of the factors involved must be taken into consideration. This point will be demonstrated in Illustration II of this section. Once both the processing«time and inputstime distributions are assumed, proper equations in Chapter 3 can be used to derive the nonfproductive time distributions and to find their expected means and variances. Illustration I and II are provided to demonstrate the proper procedures. 59 Illustration I Field-trail data supplied by the Pulp and Paper Research Institute of Canada for the Koehring short wood harvester have been tabulated in Table 4.1.1 and 4.1.2 and presented as histograms in Figures 4.1.1. It seems quite reasonable to assume that both the felling and processing- times per tree are normally distributed. Therefore, equations [3.3.3] and [3.3.4] can be used to calculate the expected values by letting f’I 54,}? I 62, S1 = 7.5 and 82 = 16 cmin. as estimates of the unknown parameters pl, “2’ 01 and 02 respectively. All the calculations involved are simple and straight-forward once the value of F((-u1 + u2)//¢lz+ 022) I F(-0.446) is found from any standard cumulative normal table.1 The mean and variance of idle times are found to be 12 and 13 cmin. respectively and, using Theorem 3, the mean and variance of the waiting times are 3.9 and 7.7 cmin. 1F (-.446) means that the probability of z < -.446 which can be read from the table of the standard cumulative normal distribution tabulated as 2 I 7%: exp(-uZ/2) du. For example, F(-0.446) = 0.33 .00 Tr approximately. 60 Table 4.1.1: Felling Times for the Koehring Harvester* ‘ V V '1 ‘Tfi V w—V—YV V V f Class Interval Midpoint Frequency Density (cmin.) (cmin./tree) ‘ . . 20 « 24 22 3 .008 25 - 29 27 5 .014 30 - 34 32 9 .025 35 - 39 _ 37 19 .052 40 - 44 42 11 .030 45 ~ 49 47 42 .115 50 - 54 52 19 .052 55 - 59 57 61 .167 60 - 64 62 16 .044 65 - 69 67 61 .167 70 - 74 72 20 .055 75 - 79 77 52 .142 80 - 84 82 19 .052 85 - 89 87 18 .049 90 - 94 92 7 .019 95 - 99 97 2 .005 140 - 144 142 1 .002 Mean = 62.2 cmin./tree Standard deviation = 16.3 cmin. Number of observations I 365 * Tabulated from prototype Koehring harvester field-trial data supplied by the u1p and Paper Research Institute of Canada. 61 Table 4.1.2: Processing Times for the Koehring Harvester* w V V v . V f fi , fi' v v w Class Interval Midpoint - Frequency Density (cmin./tree) (cmin./tree) 35 - 39 37 2 .004 40 - 44 42 25 .052 45 - 49 47 97 .200 50 - 54 52 178 .368 55 - 59 S7 97 .200 60 - 64 62 t 45 .093 65 - 69 67 22 .045 70 - 74 72 13 .026 75 - 79 77 3 .006 90 - 94 92 1 .002 105 - 109 107 1 .002 Mean = 53.9 cmin./tree Standard deviation = 7.5 cmin. Number of observations I 484 * Tabulated from prototype Koehring harvester field-trial data supplied by the Pulp and Paper Research Institute of Canada. 62 Af (y) .'6‘ r! _I e|2fi .04‘ ‘v‘ - . * - , . , sN-—II=:—e- v 20 3O 4O 50 60 7O 30 90 '00 I40 Felling Tune per Tree in Cmin. .24“ 20‘ r- ‘L—| .l6 - .l2 ‘ L... - .—. 1:, ,x V 35 45 55 as 7'5 9'0 30 9'5 [35:70 Processing Tine per Tree in Cain. Figure 4.1.1 Histograms of Felling and Processing Times. 63 Illustration II Assume that a design engineer wants to know what will be the waiting times if a Koehring harvester is to be operated in a stand other than the type of stand in which the machine had been tested. Assume, also from the. inventory data, that the numbers of bolts per tree follow a Poisson distribution with a parameter of 6.2 bolts per tree. To find the waiting time per tree, the design engineer must figure out both the felling and processing time distributions. According to the report 'Evaluation of Logging Machines: Koehring Short Wood Harvester' (Bredberg, 1970), the felling time is a random variable which is not related to tree size, number of unmerchantable trees per acre, etc. However, the processing time per tree is dependent upon the number of bolts per tree. Using these findings and his professional judgement, the design engineer feels he can assume that the felling-time distribution shown in Figure 4.1.1 can be used as a felling- time distribution for the problem at hand. The processing- time distribution can be found by using the regression equation1 PT I 15.11 + 10.6 (NB) 1 This regression equation was constructed from the data supplied by Pulp and Paper Research Institute of Canada. To conform to the definition of waiting time, delay times were removed from the processing time used in the report of Bredberg (1970). Consequently, the regression coefficients are different from those in his report. 64 where PT denotes the processing time and NB denotes the number of bolts. By substituting NB = 6.2 bolts, the mean processing time is found to be 81 cmin. Since NB has a Poisson distribution, the processing time is thus Poisson distributed with a parameter of 81 cmin. This parameter must be transformed by subtracting 37 and dividing by 5, since the X's in equations [3.4.4] and [3.4.5] are 0, l, 2, 3 . . .. Whereas the data at hand are 37, 42, . . . the result of transformation will make the processing times Poisson distributed with a parameter A = 8.8 in units of 5 cmin. Similarly, the felling times will be normally distributed with mean u = 3.4 and standard deviation 0 = 1.5. Substituting these parameters into equation [3.4.4] and [3.4.5], we have E(U) = .05 and E(Uz) = .099. From theorem 3 in Section 2.4, we have E(V) = E(X) - E(Y) + ECU) = 8.8 + .05 ~ 3.4 = 5.45 (S cmin.) or 27.25 cmin. and Var (V) = Var(X) + Var(Y) + (E(Y-X))2 - E(U2) - (Em)2 = 8.82 + 1.52 + (8.8 - 3.4)2 - .099 - 5.452 = 79.05. The standard deviation = 8.9 x 5 cmin. or 44.45 cmin. ECV) and Var(V) are inter- preted as mean and variance of waiting times (see Section 2.5). 65 4.2 Applications of Analytical Solution--Judgement Based In situations where no historical data is available, probability density functions must be determined from experience or based on theoretical considerations. For example, processing times can be derived from tree height distributions if the processing time is highly correlated with tree height. However, there are cases where such a type of relation may not exist. The suggestion is that if we know nothing about the input time or processing time we should treat all possible input or processing time-intervals as "equally likely”. More precisely, all time-intervals of equal length in (a,b) should be treated as "equally likely", thus implying the time should be uniformly distributed (Raiffa and Schlaifer, 1961). The specific values of the distribution's parameters (a,b) can be estimated based on introspection and experience. For the purpose of illustration, the following example is provided. Illustration Assume a design engineer is develOping a new processor concept as described in Section 1.2. In order to carry out the feasibility study, he is interested in knowing the non-productive times. Assume that the design engineer has no historical data to decide how the input times and processing times will r+ 66 be distributed. However, the design engineer may be able to design a pick-up device capable of picking up a tree from 20 - 40 cmin. and a processing device processing at a speed of 125 ft. per minute. Based on the reasoning at the beginning of this section, he can assume that the pick- up times are uniformly distributed. Based on his judge- ment, the processing time should be linearly related to the length of tree. If he knows the distribution of tree height in the region where the machine will be operated, he can then derive the distribution of processing times. For example, the machine will be able to process a tree of 62.5 feet high in 50 cmin., since the processing device can be designed to process a tree at a speed of 125 feet per minute. However, he figures that a certain amount of time should be allowed as fixed-time to account for the times of cutting the tree to log length and random delays. Therefore, he adds 5 cmin. as fixed-time for every processing time estimated. Assume that the height of trees are exponentially distributed with a mean of 40 feet and the minimum merchantable tree height is 25 feet. Using the notation in Section 3.1, we can assign values to the parameters: c = S cmin., r = 100/125 = .8 cmin./ft., a = 20 cmin., b I 40 cmin., h I 25 ft. and E(Z) = 40 ft. Parameter B = .06667 is estimated on the basis of equation (3.1.3). Since c + rh - a = S + .8 (25) - 20 > O and c + rR - b = 5 + .8 (25) - 40 < 0, equations (3.1.12) and (3.1.13) are used to calculate the expected values 67 E(U) and E(Uz). Consequently, we have the mean waiting time = E(U) = 8.77 cmin. and Var(D) I Variance of waiting times = E(Uz) - (E(U))Z 'a 28.12. Similarly, using equations [3.1.18] and [3.1.19], we have the idle time I E(V) = 1.8 cmin. and variance of idle times = 10.47. At this juncture, the engineer may ask "What will the non-productive time be if the processing speed is designed at 200 ft. per minute?" To answer this question, calculations must be started over again by changing r = 100 cmin./200 ft. = .S cmin./ft. For the purpose of comparison, the engineer may tabulate his results in Table 4.2.1. Table 4.2.1: Time per Tree at Different Processing Speeds Processing_Speed Time Elements 125 ft./min. 200 ft./min. (cmin./tree) Mean Std. Mean Std. Picking up 30 5.8 30 5.8 Waiting 8.8 11.6 1.9 5.0 Processing 37 12 25 7.5 Idle 1.8 3.2 6.9 6.3 Std. = standard deviation 68 4.3 Method of Enumeration The analytical selutions proposed in Chapter 2 involve the procedures of hypothesizing suitable input- times and processing-time distributions and deriving the analytical solutions if they are not available. However, there are cases where no suitable distributions can be assumed, or even if they can be assumed, the calculus needed for evaluating the resulting integrals is sometimes not tractable. Under such circumstances, the following computational methods are proposed: (1) Approximating continuous distributions by discrete ones and solving by enumeration. (2) Applying simulation technique to approximate results. The method of enumeration is first to find fw(w) by applying equation (2.2.5) or (2.2.6) and then to find g(u) and h(v) by applying equations (2.4.2) and (2.4.4). The mean and variances can be computed by using the ordinary formulas. These procedures can best be presented by the illustrative examples provided in this section. 69 Illustration 1 Detailed time study data for an Arbomatik processor (Bredberg, 1970) were supplied by the Pulp and Paper Research Institute of Canada. The density functions of processing cycle time and loading cycle time are respectively tabulated in Table 4.3.1 and Table 4.3.2. The joint density function of the difference of the processing and loading times, W’I‘Y -IX, can be found by applying equation [2.2.5], as shown in Table 4.3.3. The procedures can be illustrated by calculating the -660 and w = -620. From Table 4.3.1 probabilities of w -660 can only be obtained by and Table 4.3.2, w 13.5 - 673.5; thus its probability is (.279) (.002) = .0005. There are two ways to obtain w = -620. These are: 53.5 - 673.5 and 13.5 - 633.5. The probability of the former is (.0018) (.062) and the probability of the latter is (.0018)(.278). Consequently, the probability of obtaining w = -620 is (.002)(.062) + (.002)(.278) = .0006, as shown in Table 4.3.3. By applying equations [2.4.2] and [2.4.4] respect- ively, we are able to obtain the density functions of waiting times and idle times, as shown in Tables 4.3.4 and 4.3.5. The distribution g(u) is identically distributed as fw(w) when W = Y - X > 0. The value g(o) is obtained by adding up all the fw(w) from w = -660 to w = 0 in Table 4.3.3, 70 Table 4.3.1: Processing Times for the Arbomatik Processor* ................. v v. fi v f w Class No. of Processing Times Density _Ccminx) Observati9ns. _ ..chpflin'/55991.““¢“13£Y(Y15 4 - 23 90 13.5 .279 24 - 43 196 33.5 .607 44 - 63 20 53.5 .062 64 - 83 11 73.5 .034 84 - 103 3 93.5 .009 104 - 123 1 113.5 .003 124 - 143 2 133.5 .006 Mean = 32 cmin./tree Standard Deviation I 17 cmin. Number of Observations I 323 * Tabulated from Arbomatik prototype study data supplied by Pulp and Paper Research Institute of Canada. 71 Table 4.3.2: Pick«up Times for the Arbomatik Processor* 7‘ 'v 'v —v 'v fl ‘v wVV a v a Class No. of Pickw p Time Density (cmin.) . Observations .Xp(cmin./tree)i i‘.£xcx)7 4 - 23 97 13.5 .174 24 - 43 267 33.5 .478 44 - 63 118 53.5 .211 64 - 83 47 73.5 .084 84 - 103 10 93.5 .018 104 - 123 6 113.5 .011 124 - 143 4 133.5 .007 144 - 163 1 153.5 .002 184 - 203 3 193.5 .005 204 - 223 3 213.5 .005 244 - 263 1 253.5 .002 624 - 643 1 633.5 .002 664 - 683 1 673.5 .002 Mean I 45 cmin./tree Standard Deviation I 46 cmin. Number of Observations I 559 * Tabulated from Arbomatik prototype study data supplied by Pulp and Paper Research Institute of Canada. 72 Table 4.3.3: Density.Function of W I Y_- X* ..... W i fw(w) -660 .0005 -640 _ .0011 -620 .0006 -600 .0011 -580 .0001 -560 .0001 -240 .0005 -220 .0011 -200 .0016 -180 .0048 -l60 .0036 -l40 .0010 -120 .0033 -100 .0075 - 80 .0121 - 60 .0352 - 40 .1114 - 20 .2671 0 .3544 20 .1430 40 .0293 60 .0115 80 .0044 100 .0035 120 .0011 * The difference between waiting and idle times. 73 Table 4.3.4: Arbomatik Pick-up Device Waiting Times* Waiting time 8 Density Function (cmin./tree) [g(u) 0 .8072 20 , .1430 40 .0293 60 .0115 80 .0044 100 .0035 120 .0011 Mean = 5.6 cmin./tree Standard deviation I 13.9 cmin./tree * Values of g(u) which represent the probabilities of the pick-up device idle times are enumerated from density functions of felling times and processing times for the Arbomatik processor. 74 Table 4.3.5: Arbomatik Processing-device Idle Times* Idle time V Density Function (cmin./tree). ' h(v) ' 0 .5472 20 .2671 40 .1114 60 .0352 80 .0121 100 .0075 120 .0033 140 .0010 160 .0036 180 .0048 200 .0016 220 .0011 240 .0005 560 .0001 580 .0011 600 .0011 620 .0006 640 .0010 660 .0005 Mean I 18.5 cmin./tree Standard deviation I 45.5 cmin./tree * Values h(v) which represent the probabilities of processing- device idle times are enumerated from density functions of .felling times and processing times for the Arbomatik processor. 75 that is, g(o) I .3544 + ... + .0005 I .8072. Similarly, h(v) is identically distributed as fw(w), when W I Y - X < 0 and h(o) - .3544 + ... + .0011 - .5472. The calculations are straight-forward. When W takes a few number of values, hand calculations are practicable. However, if it is not the case, computer calculations may prove more efficient. A computer program for calculating fw(w), h(u) and g(v) is provided in Appendix II. Means and variances are also computed by this program. To illustrate how fw(w), g(u) and h(v) are related, Figure 4.3.1 is plotted based on Tables 4.3.3, 4.3.4 and 4.3.5. The method of enumeration requires grouping raw data of input times and processing times into class intervals. The fewer the number of class intervals, the less is the effort required in enumerating the probability density functions of waiting times and idle times. Table 4.3.6 was constructed by methods of enumeration to show whether the means and standard deviations of waiting times and idle times are affected by the different widths of time intervals. It can be seen from this table that a time interval of 10 cmin. or 20 cmin. gives, for practical purposes, the same estimates of means and standard deviations as those estimates from a one cmin. time interval. 76 {M «5472 m a m J U IMO IIZO IIOO “80 I60 I40 IZO 0 Processing Device Idle Times in Cmin. A n l _A__ A I - w -I4O IIZO IIOO IOO IOO I40 -20 0 20 4O 60 80 IOO Differences of Pick-op Times and Processing Times in Cmin. l J a -v 0 20 40 60 80 IOO Pick-up Device Woiiing Time in Cmin. Figure 4.3.1 Density Functions of Non-productive Times. 77 uconommfip mo .mcowumw>op puepmeum.pce modes :0 .uuommo .mosfiu mcflmmoooun use wanna ecu Mom mAupwz He>uou:fi-oafip esp mzonm manna mane e as ”H .e«. HN as ea ma ma ma aH can» use“ . ouw>op mcwmmououm ma 4 ma m a” a ma 0 ma a can» «enemas oUw>op-uma:H as me as me as me 64 me as me can» as-xunm as aN ma an 5” mm as ~m ca Nn mane unannouoam .eum as»: .eom ado: .eum ado: .eam ado: .eam can: as on cu OH n.dflsu. moeoaoam cans fi.aaauv Hasaaeem case no new“: emnn>houma oawh acoHoMMHQ How mcowumfi>on whevcnum use mane: no.m.v OHDMH 78 IIIUStratigg'II Assume that the processing times of a harvester concept have a gamma distribution with a mean of 62 cmin. and standard deViation l6, and the felling times are normally distributed with a mean of 54 Cmin. and standard deviation of 7.5. The problem.is to find the probability density function of waiting times for the design engineer. The gamma distribution has two parameters: k and 1. Using the relationships, mean I k/A and the standard deviation I v’Y/A (Hood, 1952) k I 15 and A I .24 'were obtained. The cumulative distribution of a gamma random variable can be approximated by using a tabulated cumulative chi-square distribution (Johnson and Kotz, 1970). If we want to find the probability that the processing time is less than a certain number of cmin., we use the relationship Pr{f §.y} I Pr {21f §_21y} I Pr {Y §_21y} and by assigning the number of degrees of freedom I 2k (truncated to the nearest integer). For example, if y I 30 cmin., then 21y I 2(.24)30 I 15. From Table 2.6.7 of the Handbook of Mathematical Functions CAbramowitz and Segun, 1965) we find Pr{Y _<_ 15} q ,1-.99 - ..01. The column of cumulative probability in Table 4.3.7 was found in this manne r e 70 Table 4.3.7: Processing Times.with Gamma Density.Functient .................................... v—v—vi —‘ v V v Via VwV—wvfifi j—r v—vvfiVVTV V—V “ 'I—‘firv ‘ Class . Interval .Midpoint Cumulative Density _ (resin-r). 5 . -. . (9.1.11. .3 .......... .- ............. 20.~ 30 25 .01 .01 30 I 40 35 ' .05 .04 40 - 50. 45 ‘ .23 .18 50 - 60 55 .48 .25 60 - 70 65 .72 .24 70 - 80 75 .85 .13 80 - 90 85 .94 .09 90 - 100 95 .98 .04 100 a 110 105 1.00 .02 Mean I 62.3 cmin. Standard deviation I 16.1 cmin. * Approximated from the assumption that processing time follows a Gamma distribution with I .24 and k I 15. 80 The probability. of X'. 11: can be found by simply using a tabulated cumulative normal distribution as shown in Table 4.3.8. ’ It is worthwhile pointing out here that computer subroutines‘for oValuating cumulative probabilities for many common distributions such as normal and chi-square are available in many programming packages. (e.g. IBM scientific subroutine package, 1968). However, if there are only a few calculations involved, their use is hardly justified. 4 Using equations(2.4.2) and (2.4.4) the density function of waiting times and idle times are computed and tabulated in Table 4.3.9. 81 Table 4.3.8: Felling Times.with Normal Density Function* fi v f i Vf Vi f Class .Midn Interval Point Cumulative' Density _ .Cci‘ii.“.'.) _ .“_C°9".i.n.'.)j a. . . . 20 - 30 25 .0006 .0006 30 - 40 35 .0310 .03041 40 - so 45 .2969 .26585 50 - 60 55 .7884 .4915 60 - 70 6S .9838 .1954 70 - 30 75 .9998 .0160 80 - 90 85 1.0000 .0002 Mean I 54 cmin. Standard deviation I 8 cmin. * Approximated from the assumption of normally distributed felling times. 82 Table 4.3.9: Density.Functions of Non-productive Times* fv Idle time 41 u,v Waitin. time‘ ..ficmiw ..... 49:15. . . . .-.-;th):-: - 0 .453 .767 10 .217 .152 20 .154 .062 30 .094 .016 40 .051 .003 50 .023 60 .007 70 .001 Mean waiting time I 12 cmin. mean idle time I 3 cmin. Std. I 14 cmin. Std. I 7 cmin. * Enumerated from normall distributed felling times with u I 54 cmin. and o I 7. cmin. and processing times having a Gamma distribution with A I .24 and k I 15. 83 4.4 Sensitivity Analysis and the Choice of Distributions The input device or processing device of a logging machine concept can either be: (1) same type of device already in existence on another type of logging machines, (2) an improved existing device, and (3) newly designed devices. Historical data will most likely dictate the choice of an input-time or processing-time distribution under the first case.; However, the second case requires professional judgement as well as historical data. Under the third case where there are no data available, he may have to choose the distributions based solely on judgement. In situations where judgement is involved--the second and third cases--the design engineer may have a very difficult time deciding on the exact probability density functions. However, one may try a few most probable distributions to see whether the results are the same for practical purposes as the results of any other distributions one may wish to consider. If the choices of probability density functions of input times and processing times prove to be critical in estimating the waiting-time or idle-time distributions, the design engineer would seek more data from a similar type of device. However, if there are no data available, the engineer might develop primary designs to assess the sensitivity of the choice of distributions. Finally, the design engineer must simply make 84 what appears to him at the design stage of a harvesting machine concept, with limited evidence available, to be the best choice. 85 4.5 Verification of Unproductive-time Distributions Bothdistributiens of waiting times and idle times, and their expeCted.values, are derived from theorems in Section 2. This implies that if the mathematical operations are correct, then the analytical solutions in Section 3 will be exactly correct. The mathematical derivations have been carefully validated by finding the waiting times by different approaches to see if the approximate same answers can be obtained for the same input- time and the same processing-time distributions. For example, for a given set of parameters, the analytical formula [3.3.2] can be evaluated and plotted to see if the form of g(u) derived from this formula conforms to that enumerated by the methods in Section 4.3. All the equations in Section 3 are validated in this manner. One may raise the question of how the waiting times found by the analytical methods compare to those actually observed. The answer to this question is dependent upon the closeness of the data and the assumed common distributions. For example, data from Tables 4.1.1 and 4.1.2 are both approximately normally distributed, using the analytical formulas, equations [3.3.3] and [3.3.4], the mean waiting time was found to be 11.9 cmin. and the variance 13.1 compared to 11.3 and 12.7 from the data actually observed. If the method of enumeration is used, 86 then the results derived from equations [2.4.2] and [2.4.4] should be exactly.the Same as observed, assuming no observing errors occurred, since these equations are proved theorems.. The verification of the waiting times and idle times obtained from the analytical formula thus involves only the verification of the hypothesized distributions of input times and processing times. The chi-square test is by far the most popular method in verifying the hypothesized distributions. It is necessary only to divide the region of a defined distribution into a finite number of intervals and compute the probability of the random variables being in each of the intervals. The chi-square test procedures can be found from many references (for example, Lindgren, 1968). The second method of verifying the hypothesized density functions of input or processing times is based on the Kolmogorov-Smirnov test. It concentrates on the deviatons between the hypothesized cumulative distribution Fn(z) and the observed cumulative histogram (Lindgren, 1968). This test requires the calculation of the statistic n Dn I TE§{FR(Z) - F(z)} where Fn(z) I i/n. In words, Dn is the largest of the absolute values of the n differences between the hypothesized cumulative density function and the observed cumulative histogram. Massey (1951), Birnbaum (1952) and Fisz (1963) 87 found that the statisticDn has a distribution, which is independent of the hypothesized distribution of Z. The critical value greater than a constant is used to test F(z) against the alternative, that the cumulative density function is not F(z). The Kolmogorov-Smirnov test is a competitor of the chi-square goodness of fit whenever the hypothesized form of the distribution is completely Specified. The former test has an advantage over the latter test in that it does not lump data into frequencies and compare discrete categories, but rather compares all the data in an unaltered form. The value Dn is usually computed more easily than the value chi-square. 88 4.6 Comparison of Methads When choosing probability density functions to represent the input times and processing times of a logging-machine concept, the question which may arise is whether to seek the theoretical probability density functions rather than using the frequency distributions of historical data. The former alternative is usually preferable, since it would seem to come closer to predicting expected future performance rather than reproducing the past (Hillier and Lieberman, 1967). Theoretical distributions may also be preferred for reasons of convenience or professional judgement. On the other hand, if no convenient probability density functions for the input times and processing times can be suggested, then frequency distribu- tions can be used to represent the density functions as illustrated in Section 4.3. If the former-~theoretica1 distribution--are suggested, non-productive time can be predicted by the method of either analysis, enumeration, or simulation. If the latter--frequencies of historical data--are suggested, then only the method of enumeration or simulation can be used. For the purpose of comparsion, the problem of Illustration I in Section 4.3 is reapproached by simulation and the problem of Illustration I in Section 4.1 is reapproached by enumeration and simulation. The results 89 are tabulated in Tablesh4-6.l tq 4.6.4. These tables indicate that non-productive times obtained by these methods-- analysis, enumeration and Simulation-«for practical purposes are the same. [A The question as to which method should be used merits discussion. In situations where theoretical distributions for input times and processing times can be assumed, analytical solutions provide exact answers for the probability density functions, means and variances of non-productive times. Methods of enumeration and simulation can only provide approximate results by the fact that the former may involve approximating the theoretical distributions by discrete ones and the latter involves sampling from the assumed distributions. Analytical solutions are general whereas the methods of enumeration and simulation can only provide results from a particular set of data. Furthermore, computations involved in the analytical formula such as equation [3.3.2] require less effort than those involved in methods of enumeration and simulation. It must be emphasized that in comparing numerical (either enumeration or simulation) and analytical procedures, Ackoff g£.'gl. (1962) pointed out that the latter are generally preferred when they can be used. Hillier and Lieberman (1967) also pointed out that the analytical approach is usually superior to simulation. 1 However, as pointed out in Section 4.3, there are 90 Table 4.6.1: Simulated Idleftime’DiStribution* ' Density Function h (v) V (cmin.) Enumeration _nyimulation o ‘ .5472 .5326 20 .2671 .2779 40 .1114 .1153 60 .0352 .0326 80 .0121 .0119 100 .0075 .0135 120 .0033 .0019 140 .0010 .0000 160 .0036 .0053 180 .0048 .0019 200 .0016 .0013 220 .0011 .0013 240 .0005 .0006 560 .0001 .0000 580 .0011 .0000 600 .0011 .0006 620 .0006 ' .0006 ' 640 .0010 .0006 660 .0005 .0013 Mean (cmin./tree) 18.5 18.7 Std. (cmin./tree) 45.5 44.8 a The simulated frequency is tabulated from 1,500 observations. The density function h(v) in Table 4.3.5 is listed here for the convenience of comparison. 91 Table 4.6.2: ‘Simulated'Waiting;time Distribution* 'DOHSitY'FUnCtiOn‘EQU) u (cmin.) Enumeration ~Simulation O .8072 .8106 20 .1430 .1366 40 .0293 .0326 60 ‘ .0115 .0126 80 .0044 .0019 100 .0035 .0033 120 .0010 .0019 Mean (cmin./tree) 5.6 5.5 Std. (cmin./tree) 13.5 14.0 * The simulated frequency is tabulated from 1,500 observations. The density function g(u) in Table 4.3.4 is listed here for the convenience of comparison. 92 Table 4.6.3: .Comparison,of.Methods in Deriving Waiting Times ................... V W V i f a V Time Analysis“ Enumerationb Simulationc .3780 .3799 .3259 .1108 .1090 .1111 10 . .1118 .1098 .1193 15 .1041 .1025 .1132 20 .0892 .0886 .0897 25 .0712 .0709 .0765 30 .0520 .0526 .0520 35 .0354 .0361 .0316 40 . .0220 .0230 .0285 45 .0127 .0136 .0133 50 .0669 .0074 .0071 55 .0032 .0038 .0010 60 .0014 .0018 .0000 65 .0010 .0008 . .0000 70 .0003 .0003 .0000 Mean (cmin./tree) 3.9 3.8 3.3 Std. (cmin./tree) 7.7 7.8 6.7 ‘ Calculated from equation [3.3.2] by letting X I Felling times and Y.- processing times. b Enumerated from equation [2.4.2] ° Tabulated from 1,000 simulated observations. 93 Table 4.6.4: Comparison of Methods in Deriving Idle Times V f w v V — Time‘ Analysis" Enumerationb Simulationc (cmin.) ‘ -. ' ~ - 0 .7250 .7203 .6950 s .0857 .0853 .0928 10 .0672 .0673 .0775 15 .0485 .0491 .0469 20 - .0323 .0332 .0265 25 .0200 .0208 .0183 30 .0112 .0120 .0051 35 .0060 .0064 .0041 40 .0027 .0032 .0010 45 .0014 .0014 .0010 Mean (cmin./tree) 11.9 11.9 12.0 Std. (cmin./tree) 13.1 13.1 12.6 Calculated from equation [3.3.2] by letting X processing times and Y I input times. b Enumerated from equation [2.4.4] ° Tabulated from 1,000 simulated observations. 94 cases where no convenient distributions can be assumed, or even if they can be assumed, the integrations involved in the analytical approach may prove difficult and sometimes not feasible. The results can only be approximated by methods of enumeration or simulation. The remaining question is whether the method of enumeration or simulation should be used in deriving the non-productive-time distributions. Although simulation can give approximate values which can only be obtained by simulating hundreds, even thousands, of simulated observations, the amount of calculation involved in the methods of enumeration is limited. Without the aid of a computer, simulation is impractical, whereas the method of enumeration can be used with hand calculations. Even in cases where computer calculations are necessary, the method of enumeration would undoubtedly require less computer time than simulation to achieve the same, or approximately the same, results. Therefore, it can be concluded that the use of analytical solutions is recommended whenever feasible; otherwise, the method of enumeration should be used. Simulation is not necessary.1 1 Simulation is very useful in studying logging systems involving complex operations such as felling, s idding, loading, hauling, etc; analytical solutions may not be obtainable. The method of enumeration may become cumbersome. In these cases, simulation is apprOpriate. i M0 104 studie and p: npro: [hat basic for a aerci wcodl the 1 ways 1969 arit simL thi: dis pot are car S. MODELS FOR PREDICTING OR ESTIMATING THE POTENTIAL MACHINE PRODUCTIVITIES During the stages of feasibility and prototype studies, many questions arise, such as 1) How can the input and processing devices best be coordinated? 2) How can the unproductive waiting or idle times be reduced? and, 3) What are the input and processing times? The methods and basic theories presented in previous chapters are the basis for answering these questions. However, whether a com- mercial type of machine can be produced, and whether woodlands managers will place advance orders, depend upon the potential productivities of the machine concept. Logging research has contributed in a number of ways to the understanding of productivity (Cottell and Winer, 1969). However, as pointed out in Section 1.6, the arithmetic method and regression are not suitable, and simulation is not necessary for the problems presented. In this chapter, a model for predicting or "estimating" the potential productivity and the associated errors will be discussed. "Estimate" here means to approximate the true potential productivity based upon the model whose parameters are either assigned by guess-work or as the outcome of careful consideration or expert knowledge. 95 96 5.1 Models Philosophical discussions of 'models' which can be found in many textbooks of Operations Research (for example, Hillier and Lieberman, 1967), systems analysis and some other fields will not be discussed here. A model can be defined in many ways. However, for the problems involved in this research, a deterministic model can be best defined as a description in which precisely determined inputs yield precisely determined outputs; a stochastic model is one which can not yield exact outputs even if the exact inputs were given. Arithmetic, regression and simulation models, described in Section 1.6, can be used as either deterministic or stochastic models, depending upon whether uncertainties can be safely assumed away. For example, if processing time has been very well established as a linear function of tree height, and if the uncertainty involved is, for practical purposes very small, the model could be considered deterministic. It is a stochastic model if, for some reason, uncertainties can not be ignored. Predicting the performance of a machine which is expected to operate in an unchanging environment has been a relatively simple problem. A small number of experiments will usually lead to the optimization of design performance. A decision maker with sufficient experience will be able to make an outright decision on the basis of a deterministic 97 model. Such has been true in the case of railroad and highway vehicles. However, the machine concepts deacribed in Section 1.3 and the environment where the machines will be operated are complex. Random changes of topographic and terrain conditions, and of forest stand factors, enormously complicate the problem which then requires a suitable stochastic model. not only to predict the machine productivities but also to take the uncertainties into consideration. Thus, formulas to approximate the variance of output time and productivity will be presented in this section, Machine performance can be measured through productivity studies from which models may be constructed to allow forest industries to choose the machine that best suits their needs. However, the choice of a machine for a particular job is often complicated by the fact that the field study has been carried out in a type of forest stand or a region not like the one in which it is to be used. For example, a woodlands manager might have to decide whether a prototype machine favourably tested in a black spruce stand in the province of Ontario will operate equally successfully in a black spruce stand in New Brunswick. Ideally, the only way to predict the potential performance of a newly developed machine is to test the machine in each type of stand and in each region in which it might be operated. This, however, will be both expensive 98 and time consuming. Considerable advantage will be gained if a model is developed that will allow reasonable extrapolation of field study results from one region to another. The output time per tree is simply the summation of all the time elements predicted or "estimated". To be precise, the output time for a processor can be written as T - t + t + t + t + t [5.1.1] or T - t + tL + tU + tM + tD [5.1.2] where tI - input time per tree t - waiting time per tree - unloading time per tree - moving time per tree - tran5porting time per tree - delay time per tree - processing time per tree - idle time per tree Since a harvester involves the function of transporting a load to the points designated, the time element of travelling per tree, tT, must be added. T - tI + tw + tU + tM + tT + tD [5.1.3] or T - tP + tL + tU + tM + tT + tD [5.1.4] Equations such as these seem deceptively simple. 99 They are useless if the time elements involved can not be predicted or "estimated". On an abstract level, ordinary mathematics is concerned with independent variables, the values of which are measurements of the stand factors, and dependent variables which are determined by values assigned to the former. Mathematically, each time element, ti’ can be written as t. - bio + b. + 0.. + b 1 11x11 inxin [5.1.5] where ti, the dependent variable, denotes any one of the relevant time elements, and xil' xiz ..., the independent variables, denote the measured values of stand factors. The b's are estimated values of parameters which can be predicted by regression analysis or "estimated" by theoretical considerations or professional judgement. Research aims at the discovery of laws interrelating natural phenomena. A value of a particular variable can be determined when certain conditions are prescribed, as, for example, when processing time is related to the stand factor of tree height, their relationship is ’established, and the value of processing time can be predicted given only the tree height. Nevertheless, there exist enormous areas of objective reality characterized by changes which do not seem to follow any definitive pattern or have any simple connection with recognizable quantifiable factors. 100 Delay time has been a random event according to many of the logging research findings (For example, Bredberg, 1970; Bennett 33. 51., 1965). This means that the values of the dependent variables may not be predicted. Once the output time is predicted or "estimated", the predicted or estimated potential productivity can be computed by dividing the volume per tree by the output time per tree, or p - V/T [5.1.6] where p denotes the predicted or "estimated" machine potential productivity, V denotes volume per tree and T denoted output time per tree. For V expressed in cunits and T in hours, p is given in cunits. 101 5.2 Estimating the Parameters and Associated Errors 5.2.1 Error of "Estimation" A machine manufacturer, who is thinking about deve10ping a new logging-machine concept, will want to obtain some information to guide his thinking in some general problem area. At the point of choosing or developing the machine devices or components to fit the logging-machine concept under consideration, he will need such information as moving time per tree, input time per tree, etc. He will make rough calculations in which he "estimates" his unknowns based upon his experience, expert knowledge and insight of the problem. He will use these "estimates" as if they were true values of the parameters, even though he knows that they are certainly in error by some amount. This type of error, which we shall term error of "estimate" also occurs during the earlier stage of prototype studies during which many components or devices may have to be redesigned or modified to achieve an acceptable or Optimal over-all machine design. The design engineer may want to assign a value to this error in order to allow a suitable margin in his estimates of potential productivity for the logging machine he is designing. When a design engineer "estimates" the value of a parameter, he has a real feeling for the "meaning" of the quantity in question and, therefore, his judgements about the possible value of this quantity will 102 have real meaning. The same thing may be true when he expresses judgements about the variance of "estimate". Although he may find it hard to think about variance as such, he probably can give us his quantified judgement about the range. Raiffa and Schlaifer [1961) have suggested a technique for doing this, provided the decision maker has some evidence or reasonable judgement of the type of density function to be expected. If, for example, a normal distribution is anticipated, the design engineer first estimates the expected parameter, b, with or without data (of the unknown parameters), then he picks up a number, e, for which he would consider that b ranges from b - e to b + e, and it would be extremely unlikely that b would lie outside this range. The designer may use the relationship e - to to find 0, because t can be found from a normal table once "extremely unlikely" is interpreted in the probability sense. For example, if the designer is willing to take a risk of 5%, then t - 1.96, which can be found from a normal table. Thus a - e/1.96. We now ask, "What is the justification of assuming the b's to be normally distributed?" If we add a strong assumption that ti-are normal, and b is a linear combination of ti,it follows that b will also be normal. However, even without assuming that the t1 are normal, as sample size increases, the distribution of b will usually approach normality. This can be justified by a generalized form of the Central Limit Theorem. 103 5.2.2 Error of Prediction The dependent variables ti are random variables with normal distributions haVing means bi0 + bilxil and variances 0.2. The predicted values have in in 1 two sources of error: in the first place bi0 + bilxil + + ... b x b x are merely estimates of the means, and the actual in in values of ti may, of course, deviate from their means; in the second place, the estimated mean is subject to the random sampling errors inherent in the b's. Many text books (Draper and Smith, 1966; Wonnacott and Wonnacott, 1970) have shown how these errors can be estimated. 104 5.3 Combined Errors In order to estimate the variance of output time and the variance of production, we must introduce the - following formula from Davis (1961). If X - f(x1,x2,...,xn), where f is any function, then approximately 2 2 Var(X) - [2§—] Var(xl) + [2£—] Var(xz) + ..... 3x1. 3x2 + 2 駗» EE— cov(x1x2) + 2 35—» EX— cov(x1x3) + 3x1 3x2 3x1 3x3 From this formula, the variance of output time, T, can be approximated by Var(T) = Var(t1)'+ Var(tz) + ..... + Var(tn) + 2cov(t1tz) + 2cov(t1t3) + ..... [5.3.1] and the variance of production, P, can be approximated by Var[P) - Var[X] T 2Var(T)/T4 - 2cov(v,T)/T3 [5.3.2] - Var(V)/T2 + v where V denotes volume per tree and T denotes output time per tree. 105 5.4 Applications A forest situated in the province of New Brunswick consists of stands mainly balsam fir and black spruce. Field measurement data show that the tree height of fir in these stands is distributed normally with a mean of 3.5 bolts and standard deviation of .92, and the tree heights of spruce form a Poisson distribution with a parameter of 6.2 bolts. To harvest these stands, a pulpwood producer wants to decide whether he should purchase a newly developed machine-- Koehring short wood harvester, which has been successfully tested in a forest stand in the province of Ontario. The only source of information available to him is the prototype- study report published by the Pulp and Paper Research Institute of Canada (Bredberg, 1970). Using the information in this report and the method demonstrated by Illustration I in Section 4.1 of this research, the wood producer found that the mean waiting times and standard deviation are respectively 3.9 and 7.7 cmin. for the fir stands and from Illustration I in Section 4.1 that the mean waiting time and standard deviation is 27.8 and 5.0 cmin. for the spruce stands. He understands that many of the figures presented in the report, such as moving times, felling and delays, may not be suitable for his operating conditions. Nevertheless, he feels justified to use them, since the values of these time elements are not affected by stand conditions such as branchiness, number of unmerchantable trees per acre and DBH 106 (Bredberg, 1970). The mean travelling distance is 300 feet and ranges from 200 to 400 feet, or D a 300 1 100 feet, according to the woOd producer's judgement. Using the regression equation 1462 + .93D in the report, the mean travelling time is 1741 cmin. both for the fir stands and spruce stands. The number of trees per load can be computed by dividing the load capacity of the machine by mean volume per tree. The former is 600 cu. ft. according to the manufacturer's specification, and the latter is 3.6 cu. ft. for the fir stands and 11.1 cu. ft. for the spruce stands, according to the wood producer's inventory data. The number of trees per load is thus 166 trees for the fir stands and 54 trees for the spruce stands. Consequently, the average travelling times are 9 cmin. and 32 cmin. respectively. Since the travelling distance where the machine being tested is quite different from the travelling distance in his stands, he is reluctant to use the variance of travelling time based on the data from the Pulp and Paper Research Institute of Canada as an estimate of variance of travelling time per load for his stands. Using the method suggested in Section 5.2.1, the standard deviation of travelling time per load is loo/1.96 or 50 feet, where 100 feet is the range from the mean distance and 1.96 is found from a normal table if 'most likely' has been interpreted as 95% of the chance that the range of distance is between 200 and 400 feet. Thus the standard deviation of travelling time per load is .93(50) = 107 46.5 ft. Consequently, the standard deviation of travelling time per tree is 46.5/ /166’-_3.6 cmin. for the fir stands and 46.5/ l5? = 6.3 cmin. for the spruce stands. To estimate the error of "estimate", he uses the variance formula equation [5.3.1]. This equation involves covariance terms which must be taken into consideration if any ti are related to another ti. However, time elements such as felling, moving and delays are independent events and assigning zero covariance between any two of them is thus justified. The remaining question concerns the covariance between felling and waiting times. The answer to this question is that waiting time is related to the processing time of the previous tree and has little relationship to the felling time. For this reason, the decision maker can assume that this covariance is zero. Consequently, in equation [5.3.2], the variance of output time is thus reduced to the form Var[T] = Var[Moving] + Var[Felling] + Var[Waiting] + Var[Delays] + Var[Travelling] All the computations are summarized into the following table (Table 5.2.1) 108 Table 5.2.1: Estimated Time per Tree for a Koehring Harvester Time Elements Fir Stands Spruce Stands (cmin./tree) Mean Std. Mean Std. Felling 54.0 7.5 54.0 7.5 Waiting 3.7 8.0 27.8 5.0 Delays 11.0 8.3 11.0 8.3 Moving 4.0 3.0 4.0 3.0 Travelling 9.0 3.6 32.0 6.3 Output 81.7 14.6 128.8 14.1 The production of the machine is simply computed by equation [5.1.6]. The wood producer would be able to produce at a rate of 3.6 cu. ft./81.7 cmin. = 2.63 cunits per hour in the fir stands and 11.1 cu. ft./128.8 - 5.17 cunits per hour in the spruce stands. Equation [5.3.2] can be used to approximate the variance of production. The difficulty is that the covariance between the volume per tree and output time per tree will be very difficult to estimate without testing the machine in the stands in which it will be used. However, it is reasonable to assume that the volume per tree and output time per tree are positively related, and this implies that the covariance is positive. By ignoring this 109 covariance, we can have a conservative estimate of the variance. Thus equation [5.3.2] becomes 1 2 lo Var[P] -'-7'Var(V) + V Var[T]/T . T Assuming that the standard deviation of volume per tree is 1.5 cu. ft. for the fir stand and 3 cu. ft. for the spruce stand, then Var[P] can be evaluated at the mean values of V and T according to Davis (1961). The wood producer obtained the variance of production rate for the fir stand as 2 2 2 u Var[P] - 1.5 /8l.7 + 3.6 (14.6)/81.7 . The standard deviation of P in the fir stand is found to be /VE?TPT = .018 cu. ft./cmin. or 1.1 cunits/hour. Similarly, the standard deviation of P in the spruce stands is 1.42 cunits/hour. It must be pointed out that an investment decision - whether or not to purchase a processor or a harvester - depends mainly on the cost effectiveness of the machine. The "estimated" or predicted means and standard deviations of machine productivity can only provide part of the information for the cost-effectiveness analysis. Other information such as maintainability, must also be considered before a decision can be made. 6. SUMMARY, CONCLUDING REMARKS AND FUTURE RESEARCH Machine concepts for harvesters and processors involve an input device (a pick-up device in the case of a processor, or a felling device in the case of a harvester) and a processing device, both of which can be operated simultaneously. The conceptual processor can pick up a tree while processing another and similarly the harvester can fell a tree while processing another. If these conceptual machines are developed, one of the following three cases may occur during their operations: (1) Both input and processing devices are busy. [2) The input device is idle, waiting for the processing device to complete processing of the previous tree. (3) The processing device is idle, waiting for the input device to complete its function. The principal objective of the research is to develop analytical solutions to the probablity density functions, means and variances of waiting times and/or idle times. A method of enumeration is also proposed when ever analytical solutions can not be obtained or can be obtained only with much difficulty. With the time elements, waiting times and idle times derived, the input device and processing device can be chosen or improved or newly designed to the 110 111 best advantage in terms of monetary values per unit of wood produced. The secondary objective of this research is to propose a model for estimating or predicting the potential productivities of either a conceptual machine or an existing machine (prototype or commercial type). Based on this model, time elements suCh as processing, waiting or idling, moving, etc. can be "estimated" or predicted individually so that the design engineer can decide which time elements can be reduced and thus reduce the output time per tree. In addition, the woodlands manager can use the model to choose the type of machine best suited to the logging conditions prevailing. It must be emphasized that this research is not concerned with the study of any particular machine. Rather, the concern is with methods of potential value to the more "applied" workers in the developmental field, with the ultimate clients being the decision-makers of the forest industries. To achieve the first objective, basic theorems in deriving the analytical solutions and the method of enumeration are developed. The analytical solutions are presented as formulas which serve two purposes: to conveniently compute the waiting times and/or the idle times and to demonstrate how analytical solutions can be derived from the basic theorems. 112 The application of analytical solutions involves assuming probability densityfunctions for the input and processing times. One way of assuming a density function is to depict historical data as a histOgram. Another way is to compare the cumulative frequency of historical data with that of the assumed probability density functions. In either case, the intent is to determine whether a convenient distribution can be applied. In situations where no historical data are available, the engineer may hypothesize a distribution based on theoretical considerations and experience. If the choice of probability density functions of input—times and processing times prove to be critical in estimating the waiting-time or idle-time distributions, the design engineer must seek more data from a similar type of device. However, if there are no data available, the engineer may develop primary designs to assess the sensitivity of the choice of distributions. Illustrative examples are given to demonstrate the method of enumeration and the application of analytical formulas. The Chi-square test or the Komogorov test is suggested for verifying the assumed distributions. In comparing the methods of deriving non-productive times of the input device and/or processing device of a harvester or a processor, it is recommended that the analytical solutions should be used whenever possible; 113 otherwise the method of enumeration should be used. Simulation is not necessary for estimating the waiting and/or idle times. It must be noted that for predicting the productivity of logging systems involving complex operations, simulation may be required. Computing advantages will be gained if formulas for analysis and enumeration can be incorporated into simulation programs. To achieve the second objective, a model is proposed, and formulas for calculating the variance are also provided. An example is given to demonstrate how field-trial data obtained from one region can be used for another region in "estimating" or predicting the potential machine productivity. Management decisions whether to purchase a newly develOped machine depend on the cost-effective analysis which requires not only the prediction of machine productivity, but also the maintainability and reliability of the machine. No reports of research could be found in literature reviewed on methods of evaluating the maintainability and reliability of a logging machine at eitherthe conceptual stage or the prototype stage. Future research on these aspects is needed, without which decisions such as the following can only be made with considerable uncertainty: (1) Whether manufacturers should put prototype machines into production, and [2) Whether advance orders should be placed by woodlands managers. APPENDICES APPENDIX I PROOF OF EQUATIONS APPENDIX I PROOF OF EQUATIONS Equation [3.1.8] 0 g(u) - I fWCw) dw, ' u = 0 = fw(u), u > 0 = 0, otherwise. Since c + rh - a < 0, therefore 0 c+rh-a 1 -BW/r f (w) dw - J {F-7__ - k e } dw _1 W c+rh-b a 2 0 + I kle'BW/T dw c+rh~a c+rh-a c+rh-a z 1 w - k2(-r/B)e-Bw/r b - a c+rh-b c+rh-b 0 c+rh-a + k1(-r/e)e'5"/r - 1 - klr/B Equation [3.1.9] E[U] ' I kle-gu/ru du 0 oo = kle‘Bu/r/(-e/r)2c-au/r - 1) 0 2 2 klr /B 114 115 Equation [3.1.19] E[UZ] - I kluze'Bu/r du o ‘ k1e'8u/rtzu2/oe/r) - zu/(-s/r)2 + z/(-s/r)310 - Zklrs/B3 Equation [3.1.11] Since c + rh - a > 0 and c + rh - b < 0, therefore, 0 h(u) = J fw(w) dw , u = 0 a fw(u) , u > 0 = 0 otherwise 0 0 1 -Bw/r where I f (w) dw - J (B—:——-- k e ) dw _w W c+rh-b a 2 0 0 - 1 - k2(-r/f3)e'8w/r b ‘ a c+rh-b c+rh-b - b ' rh " C + k (1 _ e'B(C+rh-b)/T)r/8 b - a 2 Using these results and applying equation [3.1.7], equation [3.1.11] is thus proven. 116 Equation[3.l.12] E[U] - I“ ug(u) du 0 c+rh-a ' p - I n(E—é—E-- kze‘Bu/r) du 0 + ukle'Bu/rdu c+rh-a c+rh-a c+rh-a = 1 2.2— - k2 i?£./_r_2.[-Bu/r .. 1] b ' a 2 o (-B/r) 0 e‘Bu/r/(-e/r)2t-eu/r - 11 + k 1 c+rh-a This can be simplified to equation [3.1.12]. Equation [3.1.13] E[Uz] - l” ungu) du c+rh-a . l u2(5—%—5-- kze‘Bu/r) du + uzkle‘eu/r du c+rh-a 117 c+rh~a 3, - kze‘au/r[-ru2/B . Zrzu/B2 u 3(b - a) 0 c+rh~a -2r3/83] + kle'Bu/r[-ru2/B - Zrzu/B 0 *2r3/83] c+rh—a This can be simplified to equation [3.1.13]. Equation [3.1.14] Applying Theorem 2, we have h(v) = in fw(w) dw , ' v = 0 - fw(v) , v < 0 Since this equation is under the case c + rh - a < O, we have I fw(w) dw - i kle'BW/r dw 0 ' RIC-r/B)e'Bw/r 0 - klr/B The proof follows immediately. Im 118 Equation [3.145] c+rh-a E[V] . v(s—%—a - kze'BV/r) dv c+rh-b 0 + I vkle Bv/r dv c+rh-a 2 c+rh-a c+rh-a . ___X____ - kzrz/eze‘BV/rl-Bv/r - 1] 2(b ' a) c+rh-b c+rh-b 0 + k1r2/82e- V/r(-Bv/r - l) c+rh-a - c + rh - (a+b)/2 - ker/BZ + r/B The same results can be obtained by using the relationship E[V] - E[W] - E[U] . E[Y] - E[X] - E[U] Where E[Y] - c + rh + r/B E[X] " (b "‘ 8)/2 Eil’id E[U] can be found from equation [3.1.9] . EiguationL3.1.l6] E[VZ] can be found either directly (similar to the Iproof of equation [3.1.10]) or indirectly using the relationship E[V2] - 13th] - E[Uzl - mm + (E[wnz - E[Uzl 119 . rZ/B2 + (b - a)2/12 + (c + rh - E—i—3)2 - ZklrS/B3 2 Equation [3. l . 17] Using the relationship -Q £m fw(w) dw - 1 - J fw(w) dw if' 0, we have, < ll h(v) 1 - [(b - rh - c)/(b - a) + (1 - e'5(c * rh ' b)/r)]k2r/B 1 - (1 - e'BCC * rh ‘ b)/r)k2r/B + (c + rh - b)/(b - a) Ezlgation [3.1.18] The proof is similar to the proof of equation [13.1.15]. .Ehayation[s.1.1gl The proof is similar to the proof of equation [.3.1.16]. Equation [3. 2. 2] 1 t E[U] - u I exp[-——7] dtdu l “_u 20 80V 211’ 120 a-u t+u 2 = 1 J ' J u exp[«lfl dudt 20 t+u °° 2 _._.. t + I u exp[-—-2-} dudt I 20 a-u t-a+p 2 2 = 1 I .02—Li)— exp(-._t_2.) dt 2 20 oo 2 2 2 + 1 1 MM] ‘(t'a+“)lexpE--£7)dt aoffi' 8'11 2 2 20, 2 let F(z) = I 7;: exp(-t2/2) dt 1: -oo then a-u a-u 1 I m e w 1 up. we Zao/Ti' _u ;2 am - 272— Z7 2 a. Z 2 +L[p§_;_ll 42-11.] __0_I exp-t d-t 2a ( o) (o) mam (2:2) (3:2) + (u - -)(1 - F(a “)) o 3'“ Since I t2 exp(-—-E2-] dt 20 -u 2 2 3 ‘OzeXPC‘La-g—ua—J (8'11) - ozueXp(-Ei‘7) + os/Zflnifli ‘7 o O + Fog-)1 121 “NAB—gig) + _.£’.J:‘_(exp(fi_1%i) .. exp.(_u2)) Zo 2a/2E' ~20 237- 2 2 2 + L(p(2.1‘.) - 13(3)) - JL[exp(La—1%—) - exp(lz)] ”1? 20 20 E U - '° [ 1 2/2? 2a 8 o a 2 2 + Lima—1) - PCP-)1 + exp((a‘“ ) 2a 0 o 11 20 +11 -?-+(?--u)1=(9—‘1‘- 2 2 0 After some simplification, equation [3.2.2] follows. Eguation [3. 2 . 3] z 0°2 a+u u t2 E[U ] = in J exp(———7) dtdu ao n -20 u- a'“ t‘“ u2 t2 = exp(-——7) dudt A ao/YF 20 -u 0 ” tI“ u2 ( t2 + eXp -——7) dt a‘u t-a+u 80/2? 20 a-“ 3 = I it + E) 1 exp(;£;) dt _u 3 ao/IF' 20 m 3 3 2 St+u2 ‘ [fi-afyj 1 -t + J I 3 I eXPIng) dt a-“ 3 30/2" + many—:1 - (L392. + Hamil]: + 1)} 3ao/2'1? 122 .. J 2 . I 2 _ -Eg—-(a-u) exp(‘L2:E%—J + HEXPCLEZ) /_— 20 20 a 2n 3"].1 2 - /2F oz] 1 exPclgg) dt no 20 ‘11 1120 -(a-u 2 ' 2 --———- exp( ) - exp(—E70 a n 20 20 3 3’“ 2 u I 1 -t + —— exp(——Z) dt 3 ao/YF 20 + _2_.(a-u) exP('L2;%lfs . yixszw __l—exp('t2)dt f7? 20 a- J2Fo 2;? 2 + SZU‘a! exp(- 3'“ ) n 20 2 2 m -t2 + (u -au+a /3)J eXp(——7) dt ao n 20 3‘“ This can be simplified to equation [3.2.3]. Equation [3.3.3] Let t = U - “1 - “2 E[U] [m ( ) -t2 ) ‘ t'“1*“2 2 2 exp( 2 2 dt u1_u2 /2?/ol +02 2(o1 +02 ) .1" —————-}-——-—-exp(. 'tz )dth/z) _ «it/012mg 2(alz+ozz) 123 . A.” + (u2~u1)[1 - F( 12‘2 211 0‘1 +02 -/o 2+0 2 2 m g -u +p. 3 1 2 6X ‘ it 2 ) + (UZ‘U1)F( ——;——1—2-) [2? 2(0‘1 +02 ) “1'“2 u/ol +02 V0 *0 -(u -u )2 u -u ' —_—1 2 exP( 12 2 2 ) + (U2'H1)F(_—“——§-——l-Z) l2? 2(o1 +02 ) lol +02 Equation [3.3.4] Let t . u + “1 - “2’ then on 2 21 1 2 -t E[U 1 ‘ -----—J (t-u +u ) exp( )dt (TE/o §+o 2 u u 1 2 2(o 2+0 2) 1 2 1- 2 l 2 = —--——""_—12“—2 [‘(“12*"22)t ”I“ 2t2 2 ) °° [271/01 +02 2(01 +02 ) “1.112 2 2 3/2 ” 1 --t2 + (01 +02 ) rm] (T 2 2 €XPCZ 2 2)dt] + + u1_u2 n(o1 o2 ) (01 02:) 2(u -u ) - 2 °° 4' -E_—7=-.2_-2-E-1-=.—_2- ['(0‘12+O‘22)6Xp( 2t 2 ] ul'uZ 4' (‘u1+u2)[1 " F(-—__—-—-__.__——)] /olz+oz2 This can be simplified to equation [3.3.4]. APPENDIX II COMPUTER PROGRAM FOR ENUMERATING DENSITY FUNCTIONS OF NON-PRODUCTIVE TIMES APPENDIX II COMPUTER PROGRAM; FOR ENUMERATING DENSITY FUNCTION OF NON-PRODUCTIVE TIMES w- I THAN SS THAN ’3 \— QCPAFILITY DENSITY FUNC (ZQCAT. TF5 v—x IF =Y-X IF Y-X L‘ IATF :Y‘X n e V (j . "_ I) TAU”? u 3CU3L3 Vo '7OUQL-S THbN C7 TC U: h. 7 <\. 1' :- <‘. D 2'” q 4 r: ; l ‘ A . Q (VIN H 1" 0- ' ‘ :z' ‘V‘ . ' '1 n- l_ \‘ . ' r; ‘\—’ \4 V L3 er x 21 U3 TH. “. J G? QWGQAM WILL p (1.7 INT? GS ' V“. i". f Y \ I 'T::TL-J N hU”%* iT (LA Ins Tar : “I" .‘ .. v I 5; 1 . TH];~ kg" r. .-. I szIcIraTIdws F‘ l - p. U (i ,‘\ IA- \ L. If“. ] «I ..J C. 4 > ’T sf X [- ,_: I.“ J r~ '2‘ J Q J‘ 4 U )r. I F. .f‘, U i U‘ 1’: 00—1 7)- ' ’1 L'_'(,' 71.. C ’1 x” n a. b .i‘ Y \I‘VL'Q '3, 9LI“(7C‘39’€) oKS‘?) 9CD( )ofjg(A,QIV( 9?).SUM(?) Y(7(‘,_i)’cX(71\,), x(7l.!l‘)g. ’L(7C097)91‘Lk(3» C- I [TM’KCIV 1" d ‘r I. A _I ‘v p f .. O H 1' I ~ ' ‘. ‘ In ,\ " \ h~ [I 3-9 f ' f" . 2x. 3 '1. 124 Z i 5 (K.L C Y o A '1! v > 0.. }_ o \f A [I .I.. 4 v o > 'i "0 H )( A xv \’ \l .I. .‘ o q . 'f r- 1" ' 4". O I? ._a 2:21. P l.’ AA :‘ 3.11L ‘ vv Y >. " 1 l ' ’. J .‘. {KJ (- r-» (F. 4*- A ' - O r. o f. . \v u 'o L !. .v' o —7 v—~ I. I" A \‘t I... _‘ 01'. f. 5.4 \ VV f »v ' .' I- ._l K.‘ ' _ O ‘ J I" 3 O y. 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I IIII «manmJuII I wfiww9.nu.n Ac uuunumwnoc “I "I n [\ \L A K.) \- O '3 r‘ ‘L .I I II I II I I II I II I Iv II II IHI II III II I MI N :\ .‘l “I ...wo . \ IF ?0Wwouu900 ww .-€|!III|i: I --lII::. I I..III>ILJm<fiWdH|WQIHflAH3umohm”a 3% II —VII I II I II III II; I III II ,. I. I lllll ly.l.1;} II-.. til. r..pl.>1:III \)\.J\r J:.-."‘ o VUHPW 442nm noJ+ mmn.mxwficqunz AHSMH Oo$fl mmn. tudfi~< I gCF unuu < 134 -L I ST OF REFERENCES 135 LIST OF REFERENCES Abramowitz, M. and I.A. Segun. 1965. Handbook of mathematical functions. Dover Publications, Inc., New York. Ackoff, R.L. 1962. Scientific method: Optimizing applies research decisions. John Wiley 6 Sons, New York. Anderson, R.L. and T.A. Bancroft. 1952. Statistical theory in research. McGraw-Hill, New York. Axelsson, S.A. 1971. Evaluation of logging machine prototype Ambomatik processor. Woodlands Report WR/SS, Pulp and Paper Res. Inst. of Canada. Bazovsky, I. 1961. Reliability theory and practice. 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