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A simulation model for planning and control of forest harvesting operations Bonita, Manuel Libres 1972

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A SIMULATION MODEL FOR PLANNING AND CONTROL OF FOREST HARVESTING OPERATIONS by MANUEL LIBRES BONITA B.S.F. cum laude University of the Philippines, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of FORESTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1972 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e at t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f F o r e s t r y The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date May 2 5 , 1972 ABSTRACT Chairman: Professor Antal Kozak This thesis describes a methodology for examining problems associated with the management and control of forest harvesting operations. The methodology developed i s one of a systems simula-tion with general applicability that permits experimentation with a wide class of logging configurations. A model, capable of simulating multi-source, single-sink configurations with variable internode distances, with -various equipment types and combinations, and with various parameters and functional relationships, i s described. Written in FORTRAN IV, the model allows independent users to make modifications in the routine to adapt them to the particular operating rules and policies of their operations. The "va l i d i t y " of the model i s tested and demonstrated for an actual West Coast logging division used as a vehicle for model formulation. The verification procedure involves the examination of the assumptions and rules of operation of the model subsystems, and the hi s t o r i c a l confirmation that for a particular situation the sub-systems together make up a system which displays the behavior and characteristics associated with the real system. Some design and tactical considerations i n the execution of the model runs are described. Some experimental design problems, together with possible ways of overcoming them, are discussed. In particular, i t is shown that the control variate technique can be effectively used with the model to reduce the variance of the difference between two means under comparison. Simulation experiments with various logging configurations indicated the nature of the interrelationships among the responses of the "logging system". These interrelationships are described with respect to a principal factor - the number of trucks in the hauling fleet. Some practical applications of the logging simulation model are discussed and illustrated. The model can be used to evaluate and compare existing operating policies or to formulate new policies. This application i s il l u s t r a t e d with reference to the comparison of two operation shutdown modes. The model can also be beneficial i n the determination of the equipment requirements of an operation under different operating conditions. Another benefit from the model can be derived from i t s capability of increasing our understanding of the "logging system" - through learning how the parts of the system behave and interact and through learning how the system responds to changes in i t s factors. This capability can be beneficial not only in the design of better policies but also in the exercise of better control of the system. TABLE OF CONTENTS Page LIST OF TABLES ix LIST OF FIGURES . . ' x i ACKNOWLEDGMENT xy Chap ter I. INTRODUCTION 1 PAST STUDIES ON THE ANALYSIS OF FOREST HARVESTING PROBLEMS . . . . . . . 4 THE OBJECTIVES OF THIS STUDY 8 METHOD OF PRESENTATION AND CHAPTER DESCRIPTION 9 II. THE DESCRIPTION OF THE PHYSICAL SYSTEM . . . . . . . 11 THE BOUNDARIES OF THE PHYSICAL SYSTEM CONSIDERED IN THIS STUDY 11 DESCRIPTION OF THE LOGGING SYSTEM 15 Log extraction 18 Log transport ac t i v i t i e s 21 Subsidiary ac t i v i t i e s 26 Chapter Page III. THE LOGGING SYSTEM SIMULATION MODEL 29 THE SIMULATION FRAMEWORK 30 THE EXECUTIVE PROGRAM 35 THE EVENT SCHEDULER . . 36 THE EVENT ROUTINES 37 The representation of the yarding process . . . 40 The generation of the yarding rate for each simulated day . 44 The loading-time-volume-loaded relationship 46 Travel time 47 Camp delay 49 Truck dispatching 50 Breakdown of equipment . 52 Repair times and resumption of duties after repair 53 Overtime 53 THE SIMULTANEOUS OCCURRENCE OF SEVERAL EVENTS . . . 54 THE PROBABILITY TRANSFORM THEOREM AND THE CONSTRUCTION OF AN INVERSE CUMULATIVE DISTRIBUTION FUNCTION 58 FLEXIBILITY OF THE MODEL FOR REPRESENTING VARIOUS CONFIGURATIONS 59 PROGRAM INPUT REQUIREMENTS AND PROGRAM OUTPUT 60 MODEL PROGRAMMING AND COMPUTING REQUIREMENTS 62 Chapter Page IV. MODEL VALIDATION 65 SOME CONCEPTS ON THE VALIDATION PROCESS 66 MULTI-STAGE VALIDATION 67 A COMPARISON OF INPUT-OUTPUT TRANSFORMATIONS 70 V. SOME DESIGN, TACTICAL, AND STATISTICAL CONSIDERATIONS IN THE EXECUTION OF THE SIMULATION RUNS 79 SOME CONSIDERATIONS IN THE DESIGN OF SIMULATIONS EXPERIMENTS 80 The problem of motive 80 The problem of stochastic convergence 81 The problem of size 85 The multiple response problem 87 TACTICAL CONSIDERATIONS IN THE EXECUTION OF SIMULATION RUNS 88 Choice of starting conditions 89 Deletion of the early part of the run 89 The measurement mode 90 VI. SOME APPLICATIONS OF THE LOGGING SIMULATION MODEL . 92 INTERRELATIONSHIP AMONG THE LOGGING SYSTEM RESPONSES 92 Yarding production vs. trucking pro-duction vs. production at the dump 93 Production vs. truck per cent u t i l i z a t i o n 93 Chapter Page Production vs. number of truck round trips 98 Production vs. unit cost 100 Production vs. yarder per cent u t i l i z a t i o n 104 COMPARISON OF DIFFERENT OPERATING POLICIES . . . 105 Sample problem 106 DETERMINATION OF THE EQUIPMENT REQUIREMENT FOR VARIOUS CONFIGURATIONS 115 The truck requirement for different number of High Lead production sides and setting distances 116 Truck requirement for a 5-High. Lead, 1-Grapple yarding configuration 124 Truck requirement for the basic confi-guration using different combi-nations of "small" and "large" trucks 125 SENSITIVITY ANALYSIS: THE RELATIVE EFFECT OF SOME EQUIPMENT AVAILABILITY PARAMETERS ON PRODUCTION 130 VII. SUMMARY AND CONCLUSIONS 135 LITERATURE CITED 142 APPENDICES 1 4 6 A. START-UP AND SHUTDOWN MODES 146 START-UP SEQUENCE 146 SHUTDOWN MODE I 146 SHUTDOWN MODE II 147 Appendix Page B. THE EVENT SCHEDULER 148 INITIALIZATION 1*8 EVENT INSERTION 148 EVENT DELETION 149 C. INPUTS TO THE LOGGING MODEL 153 FUNCTIONAL RELATIONSHIPS 153 INITIAL CONDITIONS . . . . 155 THE YARDING SEQUENCE AND SETTING STATISTICS . . 156 D. COSTS ASSUMED IN THE MODEL 159 E. SAMPLE OUTPUT FROM THE SIMULATION MODEL 160 OUTPUT TYPE 1 160 OUTPUT TYPE 2 160 F. STATISTICAL PROCEDURES FOR THE ANALYSIS OF TIME SERIES 164 DEFINITION AND ASSUMPTIONS 165 THE MEAN, VARIANCE, AUTOCORRELATION, AND SPECTRUM OF STATIONARY TIME SERIES 166 CALCULATION OF VARIANCES AND COVARIANCES . . . . 170 DETERMINATION OF THE LENGTH OF THE SIMULATION RUN 171 TEST FOR THE EQUIVALENCE OF TWO SPECTRA . . . . 172 IDENTIFICATION AND FITTING OF A STOCHASTIC TIME SERIES MODEL 175 G. PER CENT UTILIZATION GRAPHS 187 H. FLOW CHART OF THE VARIOUS ROUTINES IN THE PROGRAM. . 194 ix LIST OF TABLES Table Page 2.1 The elements, attributes, and act i v i t i e s of the yarding subsystems 20 2.2 The elements, attributes, and act i v i t i e s of the loading subsystems 23 2.3 The elements, attributes, and act i v i t i e s of the unloading subsystem 25 3.1 The means of the indicated dependent variable for the specified loading-time - volume-loaded relationship and the number of trucks i n the hauling fleet 48 4.1 The mean and variance for each of the four experiments ^3 4.2 The mean and variance for the yarding production and number of loads time series 77 6.1 Production and unit cost for the two shutdown modes for various configurations 114 6.2 Estimates of the purchase price, operating cost, capacity, and average loading time for s "small" and "large" trucks 127 6.3 The levels used for each of the four equipment av a i l a b i l i t y parameters 133 C.l The stochastic yarding models and the inverse cumulative density functions used in the simulation runs 153 C.2 The i n i t i a l values assumed for the landing inventory at each production side and the i n i t i a l location of each truck 155 X Table Page C.3 The yarding sequence and setting s t a t i s t i c s used during the validation runs . . . . 156 C. 4 The yarding sequence and setting s t a t i s t i c s used during the various simulation runs 157 D. l Summary of the costs assumed i n the model. . . . . . 159 E. l The trucking, setting, and dump summaries for a day's operation 161 E.2 The yarding, loading, trucking, and unloading cost summaries for a day's operation . 162 E. 3 Sample output: Daily series of values for several responses 163 F. l Observed and expected frequencies of the daily yarding production and the resulting chi-square for each of four yarders 178 F.2 Behavior of the autocorrelation functions for the dth difference of various ARIMA models 182 x i LIST OF FIGURES Figure Page 1.1 An i l l u s t r a t i o n of the product flow in a forest industrial system 2 2.1 Block diagram of a hypothetical integrated forest products company and the interactions among i t s parts and the environment 12 2.2 Block diagram of the wood production system . . . 14 2.3 Schematic diagram of a logging system configuration with n sources, m sinks, and an intermediate node linked by a network of roads 17 2.4 Hypothetical volume-yarded - vs. - time and setting - volume - vs. - time step functions showing the turn times and the turn volumes resulting from the turn a c t i v i t i e s . . 21 2.5 Schematic diagram of a truck round trip 22 3.1 Summary of truck events and landing volumes for a simulated day 31 3.2 A conceptual flow diagram of the forest harvesting simulation program 34 3.3 A flow diagram of the execution of a typical "within-day" event 36 3.4 A schematic diagram of the logging system 38 3.5 Hypothetical case: total trucking capacity less than total yarding capability resulting in plugged landings 41 x i i Figure Page 3.6 Hypothetical case: total trucking capacity greater than or equal to the total yarding capability 43 3.7 The computing time requirement as a function of the number of days of simulation and the number of round trips/day 64 4.1 Spectrum for each of the four experiments. . . . . . 73 4.2 Yarding production time series of CANFOR data and two simulation runs 76 4.3 Spectra for the yarding production and the number of loads time series 77 6.1 The truck per cent u t i l i z a t i o n as a function of the number of trucks i n the hauling fleet . . . . 94 6.2 Divisional daily production as a function of the number of trucks 98 6.3 The average load volume as a function by the number of trucks i n the fleet 99 6.4 The average number of round trips per day as a function of the number of trucks i n the fleet 100 6.5 The unit cost and i t s components as functions of the number of trucks in the hauling fleet . . . . 103 6.6 The unit cost curves for the basic configuration using two shutdown modes HO 6.7 The production curves for the basic configuration using two shutdown modes HO 6.8 The unit yarding, loading, trucking, and unloading costs for the basic configuration using different shutdown modes H I x i i i Figure Page 6.9 The unit cost and production curves for the 4- and the 8- yarder configurations using two different shutdown modes 112 6.10 The unit cost and production curves for the basic configuration using two different shutdown modes when the settings are 5 miles farther and when the settings are 5 miles closer to the camp 113 6.11 The unit cost response surface over different values of the number of trucks and the mean setting distances for the 4-, 5-, 6-, and 7- yarder configurations 120 6.12 The truck requirement for various combinations of estimated daily production and mean setting distance . 123 6.13 The production response surface for the basic configuration . . . . . . . . . . . . 123 6.14 The unit cost and production curves for a 5-H.L yarder configuration, a 5-H.L. 1- Grapple yarder configuration, and a 6-H.L. yarder configuration 126 6.15 The unit cost and production curves for the basic configuration using different fleet of trucks 129 6.16 Iso - unit cost curves over different combinations of "large" and "small" trucks 1 3 0 6.17 Production response surface for the basic configuration over different values of u, and u 134 t r t B.l Flow chart of the "event scheduler" i n i t i a l i z a t i o n routine 150 xiv Figure Page B.2 Flow chart of the event deletion routine 150 B.3 Flow chart of the event insertion routine 151 B.4 Examples of the i n i t i a l i z a t i o n , event insertion, and event deletion procedures 152 F.l Graphs of "smoothed" spectra i l l u s t r a t i n g the "window-closing" procedure 169 * * F.2 g c (X)/ gfc (X) - graphs comparing the daily yarding production spectrum and the daily number of loads spectrum of the CANFOR data with the corresponding spectra of the two simulation t r i a l s 174 F.3 The autocorrelation functions for each of the four yarders 181 F.4 The 12-lag spectrum for the adjusted yarding production data and the theoretical spectrum for the f i t t e d model for yarders 0905 and 0911 185 F. 5 The 12- lag spectrum for the adjusted yarding production data and the theoretical spectrum for the fit t e d model for yarders 0909 and 0904 186 G. l U - graphs for various configurations 188 G.2 Expected daily production per truck as a function of the truck per cent u t i l i z a t i o n for various configurations 191 XV ACKNOWLEDGMENTS The author wishes to acknowledge the help and guidance given to him during his entire stay at the University of British Columbia by his thesis supervisor, Dr. C.W. Boyd. His constructive criticism and encouragement given during the development of the thesis are greatly appreciated. The author also wishes to acknowledge the c r i t i c a l reviews by the other Committee members: Dr. L.G. Mitten of the Faculty of Commerce; Mr. H.A. Leach of the H.A. Leach and Company Ltd.; and Mr. G.G. Young, Dr. D.D. Munro, and Dr. A. Kozak of the Faculty of Forestry. The kind assistance extended to the author by Mr. G.G. Young and the author's chairman, Dr. A. Kozak, is especially appreciated. The author also wishes to acknowledge the help and coope-ration of the Canadian Forest Products Co., Ltd.. Thanks are due to Mr. J. McPhalen and to the personnel of the Harrison Mills Logging Division of the Canadian Forest Products Co., Ltd., for the collection of the data used in the thesis and for the technical assistance given. The financial assistance from the University of Br i t i s h Columbia and the educational leave from the University of the P h i l i p -pines are also appreciated. Finally, to many friends and colleagues in Canada and in the Philippines, sincere thanks are given for making the pleasant stay of the author i n Canada possible. 1 CHAPTER I INTRODUCTION An industrial activity generally involves a main flow of products or services. Industrial systems deal with this flow, the continuing transfer and transformation of materials as they pass through several intermediate nodes enroute to the consumer. A recognition of this basic nature puts many of the problems inherent in such systems into perspective. These problems generally revolve around improving the efficiency of the individual nodes of the flow and smoothing the flow through these nodes. In forestry the materials are trees which are harvested upon "maturity", transported to the m i l l , and manufactured into products for subsequent consumption. The different major phases of this product flow are illustrated in Figure 1.1. 2 FOREST LOGGING MANAGEMENT a. Cutting a. Stand b. Wood extraction establishment "mature" c. Transport to Long logs, b. Stand trees an intermediate short wood, management destination chips, etc. CONSUMP- lumber, Pulp, WOOD TRANSPORT TION UTILIZATION TO THE MILL paper Figure 1.1 An i l l u s t r a t i o n of the product flow in a forest industrial system Figure 1.1 also illustrates the relationship among forest management, forest harvesting or logging, and wood u t i l i z a t i o n . They form links in the chain of acti v i t i e s necessary to bring the finished product to the consumer. This study focuses on the logging aspect of this product flow. A study on this aspect i s necessary, considering that the forest industry, faced with a public with growing concern for the improvement of i t s environment,and machine and labour costs which are high and d i f f i c u l t to stabilize, must review i t s system and policies for procuring and transporting raw materials for i t s mills. The forest industry must institute better policies, design better systems, and intensify the use of i t s present equipment to reduce and stabilize costs, i f i t i s to survive competition from outside sources and i f wood substitutes are to be prevented from taking over a large share of the wood market. 3 Additional relevant studies on logging are needed. While studies on forest management featuring the quantitative systems analytic approach are becoming extensive, the treatment of the logging aspect as found in the present literature has generally been limited to i t s representation as a small module or as a cost function of a larger forest management model. Although this treatment of the logging aspect may have been adequate for the purpose of these studies, a more intensive treatment i s necessary for developing tools and policies which could aid the logging manager in making decisions. In many of the studies dealing primarily with logging, the emphasis has been generally on designing machines or on component activ i t i e s while the development of a total logging system for using these machines lagged. Several studies which involved the simulation of machine concepts have been designed to increase the efficiency of the individual steps in the log flow. There remains a need for a , comprehensive study of the over-all logging system encompassing not only the subsystems represented by these machines but the interactions among these subsystems as well. A study designed to f i l l this need faces the requirements of providing answers to the problems of total systems design while portraying the important characteristics of logging systems. Some of the characteristics of logging systems are: 1. They are dynamic, their states change with time. 2. They are diffused throughout with uncertainties. Their subsystems represent processes that are stochastic. For example, the truck travel times 4 the loading and unloading rates, and the yarding rates are a l l random variables. Since the various subsystems generally interact with each other, i t i s essential for an intensive logging study that their stochastic nature is maintained in their representation. 3 . Their responses to change in their independent variables are, i n general nonlinear. For example since each additional logging truck results i n longer queues, longer waits for logs, and heavier t r a f f i c density, the daily production increases at a decreasing rate with an increase in the number of trucks. 4. They are self-regulating. They return to their i n i t i a l state after being disturbed. For instance, barring any great calamity, any system disturbance is corrected - strikes are settled, broken down equipment is repaired. The extent to which past logging studies meet the above requirements i s investigated in the following section. 1.1 PAST STUDIES ON THE ANALYSIS OF FOREST HARVESTING PROBLEMS. Early studies directly involved with logging were limited to the investigation of individual logging a c t i v i t i e s . The emphasis was on improving production and reducing cost for these individual a c t i v i t i e s . While some of the problems investigated involved two-machine a c t i v i t i e s , e.g. skidding and hauling, these investigations were fragmentary and did not look at the over-all system. 5 Matthews (1942) was among the f i r s t to use mathematical techniques for solving certain types of logging problems involving production and cost control. A tool he repeatedly used when comparing two or more alternatives is the break-even point chart. He also used diff e r e n t i a l calculus to find the minimum point of the cost function derived for some design problems such as the determination of the optimum road spacing for sloping ground, the determination of the maximum skidding distance for economic direct skidding under various conditions, etc. The studies of Lussier (1959) also dealt with production and cost control. Statistics gathered from f i e l d studies on the performance of machines and their crews were used to set up production standards and control charts which then served as a basis for comparing the future performance of the machines and their crews. In the last decade, several studies applied operations research tools to logging problems. One of the f i r s t operations research techniques used i n logging and forest management was linear programming. Although this technique has been applied to a variety of forestry problems, the problems investigated involving forest harvesting generally dealt with forest production scheduling. These applications revolved around determining how much to cut and where to cut. For example, the work of Theiler (1959) and Curtis (1962) involved scheduling various compartments for cutting and subsequent replanting. Logging entered the calculations only as costs. Studies on logging production allocation and scheduling by Lonner (1968) and Carlsson (1968) used linear programming and some heuristic models which 6 incorporated some formalized decision rules. The planning of logging was divided into three separate plans - five-year plan, one-year plan, and one-month plans - each geared towards allocating logging units to each corresponding time period and determining the labor and machine resources needed for the various logging units. In the investigation of actual logging problems, the use of linear programming was generally limited to problems involving the determination of the optimum combination of elements (e.g. cords per acre, truck hours per cord, skidder hour per cord, ect.) subject to a certain set of scarce resources (e.g. forest acreage of various species compositions, quality, and geographical distribution; harvesting equipment; operating roads; and labour supply) and to the requirement of producing a certain amount of wood for a given period. The work of Donnelly (1963) is an example. While this class of linear programming application gives an indication of the equipment requirement for a given unit of production in terms of machine hours per unit volume, no indications can be given of the desirable equipment composition for a logging operation that would account for the various machine interactions. Also, where the comparison of system policies and designs are concerned, programming formulation and solution becomes extremely d i f f i c u l t unless drastic assumptions on the characteristics of the logging system are made. Another operations research tool used in logging studies is simulation. The scope of the system investigated in most logging simulation studies, however, were either too narrow or too broad for the purpose of examining total systems problems. For instance, several simulation studies have been made to investigate the f e a s i b i l i t y and productivity of particular machines and machine designs for a given 7 physiography of the logging site and spatial arrangement of trees. Newnham (1968) used simulation to test the effect of varying the minimum merchantable d.b.h. on the productivity of some feller-buncher machines. The underlying purpose of the study was to design an ef f i c i e n t harvesting machine to harvest pulpwood. Where the simulation studies considered the total system, the logging system was included only as a cost phenomenon or as a subsystem of a forest management model. For instance, the simulation model of a large industrial forestry enterprise designed by Clutter and Bamping (1965) to obtain management planning projections such as acres cut, volume cut, income, expenditure, present net worth, etc., and a similar study by O'Regan, e_t al_. (1965) incorporated harvesting and hauling of wood only as cost phenomena. The management game developed by Bare (1969) simulates the operation of an industrial forest property. Its purpose i s to provide an environment for students of forest management to observe and experience how the various biological and economic factors associated with operational forest management interact to effect the behaviour of the forest system. Harvesting i s included among the several basic management a c t i v i t i e s , but again only as a subsystem for inventory and cost calculation. Other simulation studies include the trucking game developed by McPhalen (1970) which functions primarily as a tool for giving the player insight into the truck dispatching process, and a river drive simulation model reported by Gillam (1968) which simulates the movement of wood through a lake and i t s river systems. 8 1.2 THE OBJECTIVES OF THIS STUDY The main task faced by a logging manager i s the production of a sufficient quantity of wood to f u l f i l l a prescribed demand at a low and competitive cost while meeting the constraints dictated by some necessary forest management practices, some environmental considerations, and the budget. The direction of the effort expended to meet this task has been towards the improvement of the production of each activity through mechanization. Very l i t t l e has been done about improving the system for planning and managing the use of these machines. Regarding this problem, W.I.M. Turner, the President of Consolidated-Bathurst, Ltd., in his keynote address to a seminar on wood costs remarked: "I am well aware that computer simulations have been applied to a limited extent to wood handling, delivery, river drives, inventory measurement, log sorting, etc.. Could we expand on a technique like this to help decide on a complete logging system? I am aware that differences exist in logging conditions from company to company, and even within the same operating divisions. But let us not allow the differences and variations to obscure the si m i l a r i t i e s . The variations are not i n f i n i t e . We might do well, therefore, to consider an industry-wide logging simulation that a l l companies in Canada could share. This could be aimed at discovering the best total system of wood handling for each company's specific requirements." While the objective of this study i s not to provide an ultimate answer to this c a l l , this study is certainly intended to be a step towards that direction. This study aims 1. To provide a conceptual framework for analyzing complex logging problems. 2. To develop a methodology for exploring this framework. 9 3. To develop the capability to model and experiment with logging operations with the ultimate aim of examining possible ways of improving the system. In meeting these objectives, i t i s essential that the important characteristics of logging systems are taken into consideration. Logging systems are characterized by uncertainty, non-linearity, and the presence of interdependence among their variables. Problems imbedded in these types of systems are generally too intricate and too big to handle with known "analytic" models unless drastic simplifications are made. Frequently, the decisions arising from these problems involve the spending of large sums of money. Thus i t i s imperative that these decisions are not based solely on intuition. The inadequancy of intuition and experience and the infeasi-b i l i t y of analytic models point to experimentation as a possible approach. Unfortunately, the risk of failure i s inherent i n each decision made and gambling on a decision could lead to ruin. This, plus the often prohibitive cost involved, puts the f e a s i b i l i t y of conducting real-world experiments in question. Thus, experimentation in an a r t i f i c i a l medium such as in a simulation model becomes a last resort. Since the model mimics the r e a l - l i f e situation, i t allows the manager to observe the outcome of his decisions without having to actually endure the consequences of his decisions. 1.3 METHOD OF PRESENTATION AND CHAPTER DESCRIPTION The subsequent parts of this thesis are developed to follow the logical steps involved i n the construction of a simulation model and i t s application to logging problems. In Chapter II the boundaries of the 10 physical system considered in this study are given. This is followed by a description of the different components of the physical system. The representation of a logging operation by a simulation model is discussed in Chapter III. The discussions include a description of the simulation framework, the routines which represent the physical process, and the model input and output. Following this description, the validation of the model is given in Chapter IV. Chapter V follows with a discussion on the design, tact i c a l , and s t a t i s t i c a l considerations in the execution of the simulation runs. Specific problems, which i l l u s t r a t e the classes of problems that can be handled with the model, are given in Chapter VI. Finally, the conclusions - the ramifications and general implications of the study -are presented i n Chapter VII. 11 CHAPTER II THE DESCRIPTION OF THE PHYSICAL SYSTEM 2.1 THE BOUNDARIES OF THE PHYSICAL SYSTEM CONSIDERED IN THIS STUDY In the real world, systems are embedded in larger systems. Any given system exists as a part of a larger system. The limits of a specific system i n a particular study extend only as far as is relevant to the objectives of that study. An improper resolution of the boundaries of a system under consideration not only needlessly complicates the problem but also obscures the significance of the results. There is consequently a need to carefully delineate the boundaries of the system relevant to the objectives of the investigation. The objectives of this study were outlined i n Chapter I. In this section, the boundaries of the physical system relevant to these objectives are defined in relation to an integrated forest products company which represents the entire organizational framework. Figure 2.1 illu s t r a t e s a hypothetical integrated forest products company. The wood production division i s shown i n context with the other parts of the company. Goals to attain for the f i s c a l year are transmitted to the wood production division by the head office of the company. These goals, i n the form of production quota, are based on projected m i l l and sales demand. An estimate of the proportion of the total wood requirements that the division can produce and the corresponding cost of production are Goals INTEGRATED FOREST PRODUCTS COMPANY Policy Formulating Sys HEAD OFFICE p o l i c y flow correlated. information policy information (productior/ goal) feedback information (estimated budget) WOOD PRODUCTION. DIVISION Intermediate Control System local decision making mobil zation perfc m info ance ifmation Production System 1. "Non-critical" 2. " c r i t i c a l " a c t i v i t i e s OTHER DIVISIONS 1. Other wood production divisions 2. Manufacturing divisions 3. Etc. marketing advertisement sales etc. envi ronmen tal influences orders other inputs ate. ENVIRONMENT Figure 2.1 Block diagram of a hypothetical integrated forest products company and the interactions among i t s parts and the environment 13 transmitted back to the head office. The demand, the production, and the cost of production are evaluated quarterly and corresponding changes are made to accomodate any developments of the previous quarter. While the planning and control responsibilities of the entire company rest with the head office, planning and control i n the wood production division are shouldered by the logging manager. Guided by the goals he has to attain, he mobilizes and manipulates the resources of the production system. It is this planning aspect of the production system that this study deals with. The details of the production system are shown i n Figure 2.2. The production system act i v i t i e s are ar b i t r a r i l y classified into two categories: 1. " C r i t i c a l " a c t i v i t i e s - the basic day-to-day operations connected with the main function of the production system which i s the production of logs at different locations i n the forest and the transport of these logs to an intermediate destination. The " c r i t i c a l " a c t i v i t i e s include log extraction, loading, hauling, unloading, and truck dispatching , together with their subsidiary activities such as equipment management and transfer of equipment from one production location to another. 2. "Non-critical" a c t i v i t i e s - the various a c t i v i t i e s which are subsidiary to the " c r i t i c a l " a c t i v i t i e s . The "non-critical" a c t i v i t i e s include surveying DIVISIONAL OPERATING POLICIES PRODUCTION SYSTEM SUBSIDIARY SYSTEM Surveys & Inventories Felling & Burking General Planning (Allocation & Scheduling) Road network Planning & Landing survey Si layout Moving of equipment Extraction of logs Landing inventory Loading Transportation — t o dump LOGGING SYSTEM Equipment Management a. repair b. service c. maintenance TRUCKING CYCLE Transportation tn CIMP Truck servicing & dispatching Unloading r dump inventory A A ENVIRONMENTAL INFLUENCES A. climate influences B. change in demand for wood C. change i n labour agreement D. etc. Figure 2.2 Block diagram of the wood production system 15 and timber c r u i s i n g , a l l o c a t i o n and sc h e d u l i n g of l o g g i n g areas, road c o n s t r u c t i o n , l a n d i n g survey and l a y o u t , and f e l l i n g and bucking. A common c h a r a c t e r i s t i c of these a c t i v i t i e s i s that they are done a few weeks to a year ahead of l o g e x t r a c t i o n . The l i m i t s of the system considered i n t h i s study i n c l u d e the " c r i t i c a l " a c t i v i t i e s . .These a c t i v i t i e s comprise what i s r e f e r r e d to i n t h i s study as the " l o g g i n g system". The l o g g i n g system boundaries encompass the e x t r a c t i o n and t r a n s p o r t of logs to an intermediate d e s t i n a t i o n . The " n o n - c r i t i c a l " a c t i v i t i e s comprising the " s u b s i d i a r y system" are considered as exogenous a c t i v i t i e s of the lo g g i n g system. They represent c o n s t r a i n t s and assumptions which are f i x e d f o r any p a r t i c u l a r examination of the l o g g i n g system. However, t h i s r e l a t i o n s h i p works both ways. Feedback from the study of a l o g g i n g system i s u s e f u l i n developing p o l i c i e s which govern the design and planning of the " n o n - c r i t i c a l " a c t i v i t i e s . 2.2 DESCRIPTION OF THE LOGGING SYSTEM The c h a r a c t e r i s t i c s of l o g g i n g problems were discussed i n Chapter I . Among the s i g n i f i c a n t c h a r a c t e r i s t i c s i s the e x i s t e n c e of many, g e n e r a l l y s t o c h a s t i c , v a r i a b l e s which are r e l a t e d to the system responses i n a n o n - l i n e a r manner, and which are i n t e r r e l a t e d by some form of l o g i c s t r u c t u r e . In t h i s s e c t i o n , the d i f f e r e n t subsystems of the l o g g i n g system, t h e i r elements, and t h e i r a c t i v i t i e s are presented 16 to give an appreciation of the nature of the logging system and i t s variables. Logging systems in general are multi-source, multi-sink situations whose nodes are linked by a network of roads which may pass through some intermediate nodes. This arrangement i s il l u s t r a t e d in Figure 2.3. The sources are the production locations which w i l l be referred to as "sides". At each of the sides is a harvesting unit and in most cases, a loader. In certain cases, a loader may serve two adjacent production locations. The sinks are the immediate destination of the logs and they are referred to as "dumps". The intermediate nodes are the repair shop and the truck service f a c i l i t i e s such as the fuel, o i l , water, and tir e shops. The camp usually houses these f a c i l i t i e s . The responsibility of the logging manager is the production of logs at the different production locations in the forest and the transport of these logs to the dumps from where they are eventually shipped to their f i n a l destination, e.g. the m i l l or the market. The ac t i v i t i e s connected with this responsibility may be classified into: (1) log extraction, (2) log transport, and (3) subsidiary a c t i v i t i e s . Each of the different "logging subsystems" that correspond to the activities under these categories is described in the subsequent subsections. The description given in this chapter is necessarily a description of the static aspects of the different logging subsystems, e.g. their elements and functions. The dynamic aspects of the different logging subsystems, e.g. their events and the interaction of events, are given in the next chapter. Where applicable, the description given in this chapter is fa c i l i t a t e d by the use of tables. These tables ^ 17 Figure 2.3 Schematic diagram of a logging system configuration with n sources, m sinks, and an intermediate node linked by a network of roads 18 include the elements of the logging subsystem, their attributes, their processing states, and their a c t i v i t i e s . At this stage i t is necessary to go from a generalized situation to a specific situation. In the subsequent development of this thesis, reference w i l l be made to a specific West Coast logging division, namely that of the Canadian Forest Products Co., Ltd. logging division at Harrison M i l l s , B.C. The terminology, methods, and the level of technology are consequently peculiar to West Coast operations. Nevertheless, this logging division merely serves as a vehicle for model formulation and the approach may be applied to other specific situations. This operation is a multi-source, single sink configuration with an intermediate node. Unless otherwise stated, each of the sources (sides) contains a harvesting unit (yarder) and a loader. The intermediate node is the camp where the repair shop and the several truck service f a c i l i t i e s are located. The sink (dump) contains a single unloading f a c i l i t y . Adjacent to the dump is the marshalling yard where the different crews start and terminate their working day. The "long-log" extraction and transport method, rather than the short-wood or the full-tree methods, i s employed in this particular operation. That i s , the trees are felled and limbed at the setting and the logs are yarded to a landing and transported to the dump unbucked when less than 60 feet. Otherwise, they are bucked once or twice to a length of 16 to 60 feet before they are yarded. A. LOG EXTRACTION The harvesting or yarding units may be classified into three types according to the nature of their interaction with the loader. The 19 f i r s t type, exemplified by a Grapple yarder, works independently of the loader; thus the yarding process continues regardless of whether a truck is being loaded or not. On the other hand, the type exemplified by a "Trakloader" assumes both the yarding and loading functions; thus i t can only do one function or the other at any given instant in time. The third type, exemplified by a High Lead yarder, f a l l s into an intermediate category. Although the yarder performs only a yarding function, at particular times i t interacts with the loader. Depending upon the position from which i t is yarding and the location of the loader, safety considerations dictate whether or not yarding may proceed while a truck is being loaded. Table 2.1 summarizes the components and ac t i v i t i e s of the yarding subsystem. The attributes l i s t e d directly or indirectly affect the rate at which the logs are yarded in and the amount of available yarding time. The sequence of ac t i v i t i e s l i s t e d indicates that the yarding process consists of a series of yarding cycles or "turns". Each turn results in the transfer of logs from the setting to the landing. This process is illustrated in Figure 2.4 in terms of volume-over-time step functions. The step functions consist of a series of alternating turn times and turn volumes whose magnitudes are stochastic as a result of the v a r i a b i l i t y of conditions. The slope and topography can be expected to be different for each side. When located on relatively steep h i l l s i d e s , the landing size -and thus capacity - is usually small. Consequently, space becomes a major consideration since the yarding process is stopped when the landing becomes f i l l e d to capacity. This usually results from having too few 20 Table 2.1. The elements, attributes, and acti v i t i e s of the yarding subsystems Elements Attributes Processing State Sequence of Activities Yarder Yarder type Yarding 1. Yarder hauls back Power and line speed (with choker) the line Yarding road change 2. Crews sets choker time around logs Rig-down and.rig-up 3. Yarder hauls in the lo time 4. Chaser releases the Grappling time (for choker from the logs Grapple yarders) 5. Back to 1 Yarding Hooking time Yarding 1. Yarder hauls back crews Unhooking time (with grapple) the grapple Efficiency 2. Logs are "grappled" Experience 3. Yarder hauls in Number of working the logs hours 4. Logs are released at the landing 5. Back to 1 Logs Sizes and size Not yarding distribution (in case of: Density (no./acre) Yarder breakdown Spatial distribution Landing f i l l e d to capacity Pre-set (or not) Lunch break Yarder interacting with loader Yarder moving to another setting) Landing Size and capacity Setting Degree of slope and topography Amount of slash or vegetation Other ground con-ditions Weather conditions 21 trucks dispatched to the landing. Yarding is also stopped in the case of: 1. yarder breakdown, 2. lunch break, 3. the transfer of the yarder to another landing, 4. the re-positioning of the yarder i n the same landing, and 5. other delays. SETTING VOLUME VOLUME YARDED WHERE A B C D E F turn | - p •• time iDAECljr DA. FFFFFFDABEC| DABC f tDABC DABEEC turn volume " ~ ~ * f ~ Start of the day TIME choker setting haul-in chasing haul-back delay plugged landing Figure 2.4 Hypothetical volume-yarded-vs.-time and setting-volume-vs. time step functions showing the turn times and the turn volumes resulting from the turn a c t i v i t i e s . B. LOG TRANSPORT ACTIVITIES The transport of logs from the different production sides to the dump is done with logging trucks. The trucking fleet is composed of "small" trucks and "large" trucks. The attribute "small" refers to off-highway trucks with a design payload of 75,000 lbs. and "large" to off-22 highway trucks with 100,000 lbs. design playload. Each of the trucks may f a i l or may be assigned non-hauling duties ("bull-cooking") such as pulling a lowbed; thus the number of trucks available for hauling logs at any one time varies. DUMP Truck i s Truck travels to camp empty Truck travels to side empty Truck j oins queue s Figure 2.5 Schematic diagram of a truck round trip 23 Figure 2.5 illustrates a normal truck log-hauling route. Five main trucking ac t i v i t i e s can be identified, namely that the truck i s : 1. being loaded, 2. travelling (empty or with a load), 3. being serviced at the camp (truck empty or with a load), 4. being unloaded, and 5. being dispatched. Loading Table 2.2. The elements, attributes, and acti v i t i e s of the loading subsystems Elements Attributes Processing State Sequence of Activities Loader Loader type The loader i s busy 1. Truck arrives at the Loading rate landing 2. Truck joins the queue Truck and Capacity The loader i s idle 1. Truck arrives at the Trailer Number and type landing of t r a i l e r 2. Truck positions for Number in the loading queue 3. Trailer i s set in place 4. Loader loads logs 5. Truck leaves the landing Logs Sizes Size distribut-ion Arrangement at The loader i s not Truck originally the landing in a working state dispatched to this State (whether in case of: landing i s re-pre-loaded or 1. Loader breakdown dispatched to another not) 2. Loader moving landing to another landing 3. lack of logs due to yarder break-down 24 The components and acti v i t i e s involved in the loading of a truck are given i n Table 2.2. To summarize, some of the characteristics of the loading activity are: 1. The loading of trucks is on a f i r s t - i n - f i r s t - o u t basis. If.more than one truck i s at the landing at any one time, a queue developes. In certain cases when a particular truck is urgently needed for other duties, e.g. lowbed duty, the. truck has a high priority in the queue. 2. As a result of the v a r i a b i l i t y from time to time of several factors (e.g. loader rate, log size and size distribution, landing conditions, queue length), the time to load a truck and the volume of the load are random variables. 3. The loading time i s dictated by the time necessary to yard more logs whenever the landing inventory i s depleted as a result of having too many trucks dispatched to the landing within a short time. 4. When there are trucks waiting i n the queue and the landing i s short of logs, the truck being loaded departs with less than f u l l load, provided "sufficient" volume has been loaded. 5. A breakdown of the loader closes the "side" temporarily. Yarding proceeds only u n t i l the landing i s f i l l e d to i t s maximum capacity. Trucks in the queue are re-dispatched to other landings. 25 Travel Times The travel time between two points of known distance for a particular vehicle type and vehicle state (e.g. loaded or empty) i s also a random variable. It i s a function of both the driver characteristics and the quality of the road system measured in terms of the t r a f f i c density, the number of turnouts, and the road standards of each section (i.e. the alignment, grade, width, surface, degrees of curvature, and sight distances of the road sections). Camp delays The truck service f a c i l i t i e s such as the machine shop and the diesel o i l , water, and tire shops are located in the camp. Trucks, whether i n their loaded or empty state, arriving at the camp may stop to make use of the camp f a c i l i t i e s . Since there are several f a c i l i t i e s available at the camp, queues rarely form. If two trucks arrive at the camp within a short interval of each other, the truck with the shortest camp delay usually departs f i r s t . Unloading Table 2.3 The elements, attributes, and acti v i t i e s of the unloading subsystem Element Attributes Processing States Sequence of Activities Dump Size, shape and The unloading 1 . Truck arrives at the length of approach f a c i l i t y i s busy dump 2. Truck joins the queue Unloading Type, efficiency The unloading 1. Truck arrives at the f a c i l i t y unloading speed f a c i l i t y i s idle dump number of 2. Truck positions for f a c i l i t i e s unloading 3. Truck i s unloaded Truck Capacity 4. Trailer i s set in tr a i l e r type place number of trailers 5. Truck leaves the dump Load Size 26 Table 2.3 shows the components and acti v i t i e s involved in the unloading of a truck. Some of the characteristics of the unloading activity are: 1. Only one unloading f a c i l i t y handles the unloading of trucks in this particular operation. A queue forms i f there is more than one truck at the dump at any one time. 2. Unloading i s done on a f i r s t - i n - f i r s t - o u t basis except that i n cases where a particular truck i s needed for lowbed duty, the truck is given a high priority. 3. The unloading time i s a random variable. It is influenced by the unloading speed, as well as by the load characteristics. Truck dispatching The dispatching of empty trucks to the landings i s carried out by the woods foreman. He i s constantly informed through radio of the situation at each of the landings such as whether or not the loader is in working condition, what the landing volume status i s , and what the length of the queue i s . On this basis and knowing the location of a l l the other trucks, he dispatches a l l empty trucks that are either i n or approaching the camp. C. SUBSIDIARY ACTIVITIES Moving of yarders and loaders to another landing Two main types of act i v i t i e s connected with the "re-location" of the yarders can be recognized: 1. the re-positioning of the yarder within the same landing, and 27 2. the transfer of the yarder to another landing. If one "face" of the setting is depleted of logs, i t may be necessary for the yarder to be turned around before the opposite side can be accessed. In this case, the spar is "rigged down" before the yarder is turned around and "rigged up" before yarding can resume. When the setting i s completely depleted of logs, the yarder i s transferred to the next landing i n the schedule. The transfer i s done with the use of a lowbed i f the moving distance i s over a mile and i f the yarder has a track-type undercarriage. If the moving distance i s less than a mile or i f the yarder has a tire-type of undercarriage, the yarder moves under i t s own tractive power to the next landing anchored where necessary by a logging truck. In both cases, i t is necessary to " r i g down" the spar and the lines and, as soon as the re-location i s effected, to " r i g up" the spar and the lines. Equipment maintenance, breakdown, and repair Since each yarder, loader, or truck has less than 100 per cent r e l i a b i l i t y , each of these items of equipment may f a i l . Upon failure, the machine stops the activity i t i s engaged i n . In cases where a piece of equipment, at the moment of failure, i s interacting with another equipment, the failure also affects the a c t i v i t i e s of the latter. Repair of a yarder or a loader which has failed i s normally undertaken in the landing by a mobile crew. When a truck f a i l s , i t i s towed to the machine shop for repairs at the later part of the day by another logging truck. The machine shop generally maintains several repair crews on a double-shift basis. However, since the number of repair crews is limited, repair i s done a priority basis. A yarder or a loader 28 has a higher priority than a logging truck; thus when a yarder f a i l s , a repair job on a truck i s stopped i f there i s no other available crew, whenever two machines of the same type are waiting for repair, repair i s done on a first-come-first-served basis. Each item of equipment i s usually given maintenance on a regular basis. For instance, trucks are checked-up every third night. The fact that a truck is scheduled for a check-up usually affects the activ i t i e s of the truck at the day's end. For instance, since the truck is required to be at the camp at the end of the day, i t i s not dispatched to a landing unless i t is certain that i t has time for a complete round tr i p . If the dump i s already closed, a truck leaving a landing and due for a check-up i s shutdown at the camp i n i t s loaded state. Start-up and shutdown modes The start-up sequence of acti v i t i e s dictates the time of the start of each of the acti v i t i e s of the operation at the beginning of each day or s h i f t . On the other hand, the shutdown mode determines the manner by which each of the machines and their crews are shutdown at the end of each regular s h i f t . The start-up sequence of ac t i v i t i e s and the two currently practiced shutdown modes used i n this study are outlined in Appendix A. The shutdown modes are further studied and compared i n Chapter VT. 29 CHAPTER III THE LOGGING SYSTEM SIMULATION MODEL A simulation model i s merely a laboratory where the manager can test empirically each alternative generated in the study of a problem. While the limitations of "analytic" models do not apply to simulation, the requirements for feasible experimentation must be satisfied when using simulation, i f the technique is not to be restricted by the same limitations inherent in real-world experimentation. The j u s t i f i c a t i o n for the use of simulation rests on the inadequacy of intuition, the i n a b i l i t y of known "analytic" models to cope with the complexity of the problem, and the i n f e a s i b i l i t y of real-world experimentation. Thus i t should be shown that the reasons that prevent the use of these other methods are eliminated by the use of simulation. It i s imperative that the simulation model be capable of: 1. providing for the important characteristics of logging problems such as the existence of interacting events of stochastic nature, which affect the manner in which the independent variables of the system are related with the various system responses, 2. f l e x i b i l i t y to enable the examination of a wide class of alternative system configurations, and 3. fast execution to make feasible the use of experimentation. The subsequent development of this chapter and of this thesis i s directed towards presenting how these requirements are met by the use 30 of a logging system simulation model in the analysis of a wide class of logging problems. 3.1 THE SIMULATION FRAMEWORK The model developed in this study i s a discrete-event simulation model- which mimics the behaviour of the real system by the examination of the system model and the updating of the variables indicated by the system operating rules at the event times. To i l l u s t r a t e how these are carried out, as well as to give an indication of the degree of resolution of the details incorporated i n the model, Figure 3.1 i s given. The figure illustrates the various events that could occur i n a typical day's operation for a 6-yarder, 13-truck configuration. In using the figure, the reader i s referred to the accompanying legend for the identification of the events that correspond to the given event symbols. The state of each of the various elements of the system i s described by one or more state variables. For instance, the number of round trips made, the volume hauled, the number of miles traveled, and the number of hours worked are some of the state variables that describe the performance of a logging truck. Collectively, the different state variables compose a state vector which describes the over-all state of the system at any one time. For instance, by drawing a ve r t i c a l line across the individual graphs i n Figure 3.1 at any given point on the time axis, and by reading or calculating the values of the different state variables, e.g. number of trucks at each location, volume at each landing, number of loads, etc., one can describe the logging system i n terms of the act i v i t i e s and performance of i t s various subsystems. 31 —• o U3 a tO <n o a s a (M a " g (M > a a aa 2 a s - g to *•> tn D aa a o EE tn a tn tn a o a s a m CD o s a tn a as i cn S a a CT) Q CO cr tn a a g aa > oo a s ^ a tn _i 8 a§ & a m m a SB cc a a a s r- 3 B s a in _j m O SB m a 8 a s U3 a s cn a tn tn O as a o a in 8 to CX en 3 aa & tn a in a a T a a a tn O CD CD ui) c/i tn co cr o o m a a a m a a § a a s tn in a a aa <N a a a ic • s -a a tn _j a aa a tn O ao CO mm as in tn o a r i i i ' i i i i i i i i ' i O J 50.0 100.0 150.0 3D0.D 250.0 300.0 £ 0 . 0 « 0 . 0 «0 .D 500.0 550.0 600.0 630.0 300.0 —i 1 1 1 i 1 1 1 1 1 i : i i 50 0 100.0 . 150.0 200.0 350.0 300.0 350.0 OO.O «O.0 500.0 550.0 600.3 650.0 300.0 100.0 150.0 0 350.0 300.0 350.0 « 0 . 0 « 0 . 0 500.0 550.0 600.0 650.0 100.0 »1 3* 50.0 100.0 150.0 200.0 350.0 300.0 350.0 400.0 « 0 . 0 500.D S50.0 600.0 650.0 XO.O ,0.0 SO.O 100.0 1S0.0 200.3 250.0 300.0 350.0 « 0 . 0 0 500.0 SO.0 600.0 65C.0 700.0 0.0 SO.O 100.0 1 50 3 300.0 350.0 300.0 35a.O « O 0 «O.0 530.3 550.0 600.3 650.0 W0.3 PLOT FOR DRY 2 WITH 13 SMALL TRUCKS ure 3.1 Summary of truck events and landing volumes for a simulated day 32 CODED TERMS USED FOR THE TRUCK EVENTS Code name Definition AD = arrival at the dump DD . = departure from the dump AC = ar r i v a l at the camp DC = departure from the camp ASn = arrival at side n (start of loading i f the loader i s idle LSn = starting of loading at side n (implies truck was waiting in the queue) DSn = departure from side n BRK = instant of breakdown MOV = departure from camp to support the moving of a yarder to another landing Note: The time interval between DSn and AD includes camp delay. Figure 3.1 - cont. 33 Each of the letter symbols given in Figure 3.1 represents an event performed by the corresponding logging truck. The event is either an i n i t i a l point or a terminal point of an activity whose duration i s indicated by the time interval between the adjacent events. For instance, the activity "incurring a camp delay" i s preceded by the event "a r r i v a l at camp" and superceded by the event "departure from camp". By pooling a l l the event times in one time axis, one can observe from Figure 2.1 the sequence of events for the system for the given day. In simulating the system, each event i n the schedule was "executed" one after another. The execution of each event involved (1) the updating of the state variables indicated by the operating rules of the corresponding event routine and (2) the scheduling of the next event to occur in the corresponding subsystem. A discrete-event simulation model therefore requires an event scheduler that maintains and updates the l i s t of events of the system and different event routines that execute the corresponding event chosen from the events l i s t as the next event to occur. These items, together with a set of executive routines that perform the functions of i n i t i a l i z -ing, coordinating, outputting, extending, or terminating a program run, constitute a simulation model of the system. A simulation model of the system can be viewed as the union of two highly interrelated logical structures: (1) technological logic that represents the technology of the system, i t s description and i t s rules of operation, and (2) structural logic that controls the operation of the model in simulated time and performs the executive and event-selection tasks associated with discrete-event simulation models. (Pritsker and Kiviat, 1969). 34 (start) Executive program I n i t i a l i z e a l l state variables I Executive program Schedule breakdowns and i n i t i a l events for day 1 INITIALIZATION SYSTEM OPERATING RULES Event scheduler Find the next event to occur "Execute" the event, i.e. Event a. update the relevant routine state variables b. schedule the next event of the system SHUTDOWN POLICY & j s j OVERTIME// POLICY Executive program WITHIN-DAY LOOP End of the day? Yes Executive program No DAY-TO-DAY LOOP Output the relevant variables Executive program End of the run? START-UP ROUTINE No Yes Executive program a. r e - i n i t i a l i z e the relevant system variables b. check "non-daily a c t i v i t i e s " , e.g. moving of yarder, breakdowns, end of repair; schedule i f due for next day c. schedule the i n i t i a l events for next day Figure 3.2 A conceptual flow diagram of the forest harvesting simulation program 35 Since the policies, operating rules, and level of technology of logging operations vary, each logging operation is unique and consequently each logging system simulation model is unique. However, as indicated previously, most processes involved i n harvesting logs are common to a l l logging operations. These common features and the common structural logic inherent i n a l l discrete-event simulation models allow the development of a general framework from which individual operations may be modeled. This framework is i l l u s t r a t e d i n a conceptual flow diagram shown in Figure 3.2. The figure shows three distinct parts of a discrete-event simulation model namely: (1) an executive program , (2) an events scheduler, and (3) a set of event routines. The succeeding sections w i l l contain discussions of each of these items. 3.2 THE EXECUTIVE PROGRAM Among the different functions of the executive program indicated i n Figure 3.2, the executive program effects and coordinates the examination of the system at the event times. Figure 3.3 il l u s t r a t e s how the executive program performs this function for an event inside the "within-day activity loop", f i r s t by directing the event chosen from the events l i s t to the proper event routine, and next by checking for end-of-day conditions before "asking" the event scheduler to select the next event. Each of the different a c t i v i t i e s included i n the model may be class i f i e d as a "daily" or a "non-daily" activity. An important function of the executive program is the checking before the start of the next simulated day for whatever "non-daily" a c t i v i t i e s , e.g. moving of yarder, 36 equipment breakdown, end of repair, etc., that may be due to occur. For instance, i f a yarder i s calculated to have yarded a l l logs in the setting within the next day, the transfer of the yarder to another setting i s i n i t i a l l y scheduled for the next day. Events scheduler Find the next event to occur , cancel i t - from the event l i s t Executive program Direct the event to the proper event routine Event routine Update the variables indicated by the operating rules of the subsystem Executive program Check for end-of-day condition Events Scheduler Store the event scheduled by the subsystem Figure 3.3 A flow diagram of the execution of a typical "within-day" event 3.3 THE EVENT SCHEDULER There are three basic operations connected with the event scheduler, namely: i n i t i a l i z a t i o n , event insertion, and event deletion. The i n i t i a l i z a t i o n operation, which i s performed only once during each model "run", sets up a l l the arrays needed by the scheduler for main-taining the event l i s t . The event l i s t stores the time of occurrence and other attributes of a l l scheduled events. In an event deletion operation, the event with the smallest time i s deleted from the l i s t and 37 passed on to the event routine for execution. During execution, the event routine schedules the next event to occur for the subsystems. The event insertion operation handles the inclusion of this event into the events l i s t . A detailed flow chart of each of these operations, as well as a table i l l u s t r a t i n g examples of the i n i t i a l i z a t i o n , event insertion, and event deletion procedures, i s provided in Appendix B. It i s evident from the above description of the event schedul-ing functions to be performed, that the event scheduler i s an important factor in the development of an ef f i c i e n t simulation program. This consideration j u s t i f i e d the development of an event scheduler with a "singly-linked-list" structure. The items in a "singly-linked-list" are linked by a "pointer array" which indicates, for each item in the l i s t , the address i n the computer memory of the next item to follow. Thus the deletion (or insertion) of items from the l i s t does not require the relocation of the other items i n the l i s t to f i l l the space vacated by the deleted item, since this only requires the updating of two items in the "pointer array". For a more detailed description of "singly-l i n k e d - l i s t " data structure, the reader i s referred to Knuth (1968). The routines i n GASP II (Pritsker and Kiviat, 1969) a FORTRAN-based simulation language, similarly employs a linked-list structure. However, GASP II uses a "doubly-linked-list" structure which has other features not found in a "singly-linked-list" structure, but which i s relatively less e f f i c i e n t , since two, rather than one, "pointer arrays" require updating each time an event i s inserted or deleted. 3.4 THE EVENT ROUTINES Figure 3.4 shows a schematic representation of the logging 38 Figure 3.4 A schematic diagram of the logging system 39 system. The different physical subsystems are depicted together with their stochastic components. The policies and operating rules which govern the operation of the logging system are also shown. The different physical subsystems are represented in one or more event routines. The various routines i n the model include: 1. Yarding (This routine appears in several places in the program where the updating of the landing inventory i s required.) 2. Loading 3. Hauling (This routine includes truck travel to the camp with a load, camp delay, and truck travel to the dump.) 4. Unloading 5. Truck travel to the camp (empty) 6. Camp delay (empty) 7. Truck dispatching and travelling to the side 8. Travel time generation 9. Time check 10. Start-up (This routine i s incorporated i n the executive program.) 11. Start of moving of yarder 12. Termination of moving and setting-up of yarder 13. Termination of moving of loader (The moving of a loader to another landing is initia t e d either i n the loading routine after the last load has been loaded or i n routine 12 as soon as the loader i s ready to move.) 14. Arrival of support truck for moving of yarder 15. Arrival of lowbed 16. Yarding crew lunch break 17. Equipment breakdown 18. End of equipment repair and resumption of duties 19. Overtime 20. Towing of broken down truck The flow diagrams for each of these routines are included i n Appendix H. 40 Chapter II provided the description of the components, a c t i v i t i e s , and operating rules of each of the physical subsystems. In the succeeding subsections, additional presentation of the highlights, assumptions, and justifications not included previously in the subsystem description are given. These include: A. The representation of the yarding process B. The generation of the yarding rate for each simulated day C. The loading-time - volume-loaded relationship D. Travel time E. Camp delay F. Truck dispatching G. Equipment breakdown H. Repair times and resumption of duties after repair I. Overtime . A_: THE REPRESENTATION OF THE YARDING PROCESS In Chapter II, the yarding process was described as a step function consisting of a series of alternating turn times and turn volumes whose magnitudes are stochastic (Figure 2.5). This suggests a r e a l i s t i c representation of the yarding process by a step function, where the turn times and the turn volumes are generated from separate distribution functions. These distributions should, e x p l i c i t l y or implicitly, account for the effect or influence of the attributes such as log size and size distribution, number and distribution of logs per acre, yarder type, crew composition, etc. Clearly, a big disadvantage of this representation i s i t s very fine resolution of detail which makes the entire model unnecessarily complicated. 41 Unless the particular problem on hand requires a r e a l i s t i c representation of the yarding process, a linear function, where the slope i s the mean yarding rate for the day, should suffice to represent the yarding process. In this case, the mean yarding rate for each day is generated from a stochastic yarding model. This "ramp" model and the more r e a l i s t i c "step" model are compared in Figures 3.5 and 3.6 from a theoretical standpoint. Both models are assumed to have the same yarding rate; thus the volume-yarded-plus-initial-volume (referred to as V) curve for the step model (V g-curve) i s shown to fluctuate about the V-curve for the ramp model (V"r-curve) . Figure 3.5 illu s t r a t e s the case where the total trucking capacity i s less than the total yarding capability. In this case, the presence of sufficient quantity of logs in the landing prevents any VOLUME YARDED OR LOADED u n i t i a l landing Figure I n i t i a l volume + volume yarded (ramp model) Landing plugged^ end-of-day difference maximum landing capacity z j-volu :: r I J*^C->Initial volume + volume yarded (step model) /I me in 1 truck load total volume loaded inventory T I M E 3.5 Hypothetical case: total trucking capacity less than total yarding capability resulting in plugged landings 42 direct interaction between yarding and loading. For a l l practical purposes, the step model is equivalent to the ramp model i n this particular case. In the case where the total trucking capacity i s equal to or greater than the total yarding capability, yarding directly interacts with loading whenever the landing inventory i s not sufficient to com-plete the load of a truck. In this particular case, a difference between the two models exists in both time and volume. When the landing inventory i s insufficient for a load and when V is less than V (lower r s part of Figure 3.6), the time of departure of the truck for the ramp model is a few minutes later than the departure time of the truck for the step model. This is because the truck has to wait for the yarder to supply more logs. Also since policy dictates i n this case that the truck has to leave with less than a f u l l load, the volume loaded for the ramp model is less than the volume loaded for the step model. When V r is greater than V and the landing inventory is insufficient for a s load (upper part of Figure 3.6), the reverse i s true. These time and volume differences between the two models are inconsistent; thus i t may be safe to assume that these differences tend to cancel each other. In the absence of sufficient data at the moment to build a step model, i t is d i f f i c u l t to assess the effect of using the linear approximation. For purely intuitive reasons, these differences are deemed small especially when the entire fleet of yarders and trucks 1 The discussion and figures in this section are representative of a Grapple or High Lead type of yarder. However, with a slight modification pertaining to the direct yarding-loading interaction, the discussion and figures should also hold for a "Trakloader" type of yarder. VOLUME I n i t i a l landing inventory Start of the day T I M E Figure 3 . 6 Hypothetical case: Total trucking capacity greater than or equal to the total yarding capability 44 are considered. Also, these differences appear only in those con-figurations where the number of trucks i n the fleet i s higher than the "optimum" number. This tendency to amplify these differences in the "non-optimal" configurations should be an advantage rather than a disadvantage. Therefore, unless the problem requires a detailed execution of each yard-ing turn, the ramp or linear model is deemed sufficient to simulate the yarding process, and i n the subsequent development of the model, the ramp model i s used.*" B. THE GENERATION OF THE YARDING RATE FOR EACH SIMULATED DAY Daily production figures for 4 High Lead yarders were collected from the 1968 operation of the CANFOR logging division at Harrison M i l l s , B.C. with the number of observations ranging from 100 to 131. Treating the series of daily yarding production values as a stationary time series, a stochastic time series model was fi t t e d for each of the four 2 yarders. The results show that the yarding production time series for each of the four yarders can be adequately represented by either an autoregressive process of a certain order or by a mixed autoregressive-moving average process. Of the four yarding production time series examined, two are third order autoregressive processes, one a f i r s t order autoregressive process, and the other a mixed f i r s t order autoregressive-f i r s t order moving average process. To simulate the yarding process, the ramp model i s employed using a yarding rate generated from a stochastic time series model fit t e d for the particular yarder. A mean yarding rate and a yarding T The arguments presented i n this subsection have not e x p l i c i t l y taken into account the dependence of the turn times on the yarding distance. Nevertheless, these arguments should hold regardless of this dependence. 2 Appendix F contains a discussion on time series analysis including the process of identifying, f i t t i n g , and checking a stochastic time series model. 45 rate variance are needed as parameters for the time series model. For problems involving the forecasting of logging production, the mean and the variances may be treated as endogenous variables which should reflect the stand, topographic, ground, and weather conditions for each setting. To i l l u s t r a t e the generation of the yarding rate using a stochastic time series model, a set of equations for a second order autoregressive model is provided below. al = ° 2 ( 1 - P 1 C 1 " P 2 C 2 > X t = y + C l ( X t - l " y ) + ° 2 ( X t - 2 " y ) + A t R - Xfc/475 where: 2 a = variance of the daily yarding 2 production time series = variance of the "white noise" process {Afc} A t = the white noise value at t generated from a normal d i s t r i b u ^ tion with mean 0 and variance a. A = the autocorrelation value for lag i C^,i=l,2 = the autoregressive parameters u = the mean daily yarding production (in cunits) Xfc ^ = the yarding production value for day t - i (in cunits) R = the yarding rate i n cunits/minute (There are 475 minutes available for yarding in each standard shift.) 46 C. THE LOADING-TIME-VOLUME-LOADED RELATIONSHIP The a c t i v i t y of loading a truck i s represented i n the model by the generation of an aggregate loading time and an aggregate volume loaded. Aggregation i s done to avoid c l u t t e r i n g the model with un-necessary d e t a i l s . Unaggregated, the loading a c t i v i t y includes: (1) the p o s i t i o n i n g of the truck f o r loading, (2) the s e t t i n g up of the t r a i l e r , (3) the loading of each log onto the t r a i l e r , and (4) the binding of the load. These four a c t i v i t i e s can be aggregated as the loading a c t i v i t y . With aggregation, the p h y s i c a l subsystem can be modeled with more c l a r i t y than can be achieved without aggregation. Inverse cumulative d i s t r i b u t i o n functions are a v a i l a b l e f o r generating both the loading time and the volume loaded.^ In the absence of previous studies r e l a t i n g these two v a r i a b l e s , the following hypothe-t i c a l r e l a t i o n s h i p s were tested: 1 . That the volume loaded i s independent of the loading time. i . e . volume loaded = f(X) where X~Uniform [o,lj loading time = g(Y) where Y~Uniform [0,l] X and Y are generated separately, and where f and g are inverse c.d.f.'s. 2. That the r e l a t i o n s h i p can be represented by the use of common random v a r i a t e s , i . e . volume loaded = f(X) Loading time = g(X) where X~Uniform [0, lj (Note that a d i r e c t r e l a t i o n : volume loaded = h(loading time) reduces to: volume loaded = h( g (X)) which further reduces to f(X) a f t e r s e t t i n g f=h.g .) 1 The data c o l l e c t i o n and f i t t i n g of inverse cumulative d i s t r i b u t i o n functions are discussed i n Section 3.6. 47 3. That the relationship can be represented by the use of "antithetic" random variates, i.e. volume loaded = f(l-X) loading time = g(X) where X~ Uniform lO,l] . Results of the several runs tabulated in Table 3.1 show no significant differences among the three relationships for each of the five system responses. This shows that the model is insensitive to change in the structure of the relationship between loading time and the volume loaded, at least in those forms considered. The model, in i t s present form, uses the third relationship. This relationship is preferred over the other two given since i t i s intuitively more appealing. Using this relationship, an average loading time results i n an average volume loaded. A low loading time results i n a high loaded volume as would happen when large logs are loaded. On the other hand, a high loading time results i n a low volume loaded as would happen when small logs are loaded. D. TRAVEL TIME It i s assumed i n the model that the road network is divided into distinct road sections, each having a different set of road standards. For a given direction of travel, generating a travel time between two points requires (1) the generation of an average velocity for each section included between the two given points, (2) the calculat-ion of the travel time for each section as a function of the section distance and the generated average velocity, and (3) the summing up of the individual travel times. Assuming that there are n sections and m truck types, the model w i l l require mn different average velocity generating functions for each of the two truck states (empty and loaded) or a total 48 Table 3.1 The means of the indicated dependent variable for the specified loading-time-volume-loaded relationship and the number of trucks in the hauling fleet. Dependent variable Relationship between the loading time and the volume loaded Number 9 of trucks 14 20 (Total) $/cun. Antithetic variates Common variates Independent variates 19.34 19.31 19.53 15.11 15.10 15.14 15.89 15.89 15.92 Cunits hauled Anthithetic variates Common variates Independent variates 261.54 259.54 256.31 378.49 378.81 375.78 383.48 383.03 382.90 % Utilization (Trucks) Antithetic variates Common variates Independent variated 96.42 96.46 96.66 92.90 92.75 92.82 83.17 83.58 83.32 % Ut i l i z a t i o n (Yarders) Antithetic variates Common variates Independent variates 69.88 69.92 68.83 97.86 97.76 97.54 98.96 98.62 99.30 Number of loads hauled Antithetic variates Common variates Independent variates 19.64 19.51 19.23 29.73 29.79 29.29 32.17 32.15 32.21 Length of the runs = 75 days Number of yarders = 6 The per standard error of the mean cent of the mean. varies from 0.30 to 4. 83 49 of 2mn generating functions. The above representation assumes that (1) t r a f f i c density differences do not significantly affect travel time over the entire set of logging configurations examined in the investigation, and that (2) the travel times can be generated independently. The f i r s t assumption arises because, although t r a f f i c density is implicitly incorporated i n the empirically based generating functions for the basic configuration where the data were collected, the model uses the same set of generating functions for configurations different from the basic configuration. Nevertheless, this assumption i s j u s t i f i e d in the case of loaded trucks since they always have the right-of-way. In the case of empty trucks, the assumption is j u s t i f i e d for road sections wide enough to accomodate two-way t r a f f i c . For narrow road sections with an adequate number of turnouts, the effect of this assump-tion should be negligible. As a result of the second assumption, i t i s not ensured in the model that when two trucks are dispatched to the same destination, the f i r s t truck dispatched is the f i r s t to arrive. This situation i s possible since minor breakdowns, e.g. f l a t t i r e , are incorporated i n the travel time distributions. In any case, the improvement resulting from a more r e a l i s t i c representation i s deemed insignificant and insufficient to ju s t i f y the added complexity i t imposes. E. CAMP DELAY The f a c i l i t i e s for minor truck repairs and for truck mainte-nance and service are located at the camp. While these are separate 50 f a c i l i t i e s , in the model the frequency distribution of the length of time spent by the trucks i n these f a c i l i t i e s are pooled into one frequency distribution. This representation is j u s t i f i e d by the relatively insignificant magnitude of these "camp delays" compared to the entire round trip time. F. TRUCK DISPATCHING The dispatching of empty trucks to the settings i s handled by a dispatching routine. The dispatching routine requires a set of information such as the setting distances, the inventory at each landing, the number of trucks previously assigned to each setting, the yarder productivity, and the length' of the different queues. This i s analogous to the real-world dispatcher possessing the most recent information and being kept informed through radio of the situation at each landing. The underlying objective followed for the dispatching routine is the maximization of production subject to the available resources. For the dispatching routine to meet this objective, both the yarding and trucking considerations must be appraised. Enough trucks must be assigned to each landing to prevent the stoppage of yarding resulting from lack of space in the landing. At the same time, truck delays due to queueing or waiting for more logs should be minimized. At any given instant that a dispatching decision is required, the underlying objective of maximizing production cannot be clearly l a i d out. Instead i t is translated to the above secondary objectives, i.e to balance the landing volumes and to minimize the trucking delays due to queueing or waiting for logs. A set of rules geared towards meeting these objectives is followed by the dispatching routine. The dispatching 51 routine goes through the following steps: 1. Check a l l landings for f e a s i b i l i t y of assignment: a. If the loader i s "down", cancel the ar r i v a l of a l l previously dispatched trucks and re-dispatch these trucks. b. If the yarder i s "down", check the number of loads the landing inventory i s equivalent to. This number should not be exceeded. c. If the yarder i s moving to another landing, check i f enough trucks have been dispatched to clear the old landing. d. Note the landing inventory, the yarding rate, the length of the queue, and the number of scheduled arrivals. 2. Classify the landings into two groups: a. Landings which can give a ready load, i.e landings without any queue and whose landing inventory i s at least a load. b. Landings which can give a truck load after only a few minutes wait (due to queueing or lack of logs). 3. Rank the landings according to the following set of rules: a. To enable the early transfer of the loader to the next landing, give the highest p r i o r i t y to landings whose yarder i s moving to the next landing provided that the remaining landing inventory can support another load and that there i s no queue at the landing. b. Rank landings in group 2a according to volume i f the time i s earlier than 1:00 P.M. or according to distance from the dump i f the time i s after 1:00 P.M.. c. Rank landings in group 2b according to waiting time. 4. Rank the trucks to be dispatched f i r s t according to capacity (large trucks f i r s t ) and next according to time of departure from camp. 5. Match the trucks to the landings according to rank, i.e. let T. denote truck with rank i and L. to landing 52 with rank i . Assuming that the number of trucks is greater than the number of landings by m, matching proceeds as follows: Match T to T to L n n T n +^ to L. (provided the landing is capable of loading the truck within the T Q + m to same day. Otherwise, the truck(s) stays at the camp.) G. EQUIPMENT BREAKDOWN Studies by Drinkwater and Hastings (1967) on army vehicles and by Lambe (1970) on Ford passenger vehicles show that the frequency of breakdown follows a Poisson distribution. Their studies also show that the cost of repair can be represented as a random sample from an exponential distribution. Vandenboom (1971) found for CANFOR logging truck components that the frequency of failure i s characterized by a time-dependent Poisson process. This implies that the inter-failure time for logging trucks may be represented as a random sample from an exponential distribution whose parameter i s dependent upon the age of the vehicle. In this model, i t is assumed that each logging truck, yarder, and loader has an inter-failure and repair time generated from separate exponential distributions. It i s also assumed that a rigorous and care-ful maintenance policy is followed for the unloading f a c i l i t y and thus i t s probability of breakdown can be considered negligible. As used here, inter-failure time is not measured in terms of clock time but in terms of the time the machine actually spent on i t s function. For instance, 53 the number of minutes that yarding stops, due to too much wood accumulating in the landing resulting from loader breakdown or too few trucks dispatched, i s excluded when calculating the "time to next breakdown" for yarders. At the start of each run, a "time to next breakdown" i s randomly assigned to each yarder, loader, and logging truck. This operation i s included in the breakdown i n i t i a l i z a t i o n routine. After each simulated day, "the time spent on the job" for each machine is sub-tracted from the "time to next breakdown". If the updated "time to next breakdown" is small enough, the breakdown of the equipment is scheduled for the following day. H. REPAIR TIMES AND RESUMPTION OF DUTIES AFTER REPAIR As previously mentioned, i t i s assumed that each yarder, loader, and truck has a repair time generated from an exponential distribution. After repairs, the resumption of duties for the yarder and the loader follows immediately. For the truck each breakdown is considered a major breakdown and since i t s repair does not start u n t i l after i t is towed to the shop, the resumption of ac t i v i t i e s after repair does not proceed unt i l the following day. I_. OVERTIME An interface for combining the simulation model with an over-time policy submodel is provided. The decision to go into overtime is made before 2:00 P.M. each day in accordance with labour agreements. The overtime routine relays information on the av a i l a b i l i t y , productivity, and landing inventory of each side; the production of the whole opera-tion u n t i l that time; and the expected production i f the operation 54 does not work overtime. Assuming a decision to work overtime has been made, the overtime routine requires as input the number of extra trips to be made and the identification of the sides (yarding and loading crews) to work overtime. The simulation of the day's operation proceeds u n t i l the rquired number of extra round trips i s completed. Since only a certain number of sides may be available for overtime, an overtime dispatching routine i s provided as an alternate to the main dispatching routine. The overtime dispatching routine con-sists of assigning s e r i a l l y each side on overtime to the trucks to be dispatched. 3.5 THE SIMULTANEOUS OCCURRENCE OF SEVERAL EVENTS So far the presentation has dealt only with events occuring singly or i n series, for instance truck travel followed by ar r i v a l at the side and loading or queueing. The real world, however, behaves in an unpredictable manner and several events may occur simultaneously or within short intervals of each other causing ac t i v i t i e s or conditions which overlap and interact. Assuming that there are n simple events known for the system, theoretically there are ( n)=2 n - 1 possible events and event combinations of order 5 2 that could occur. The model of the system should be capable of accounting for each combination of events that may occur. Of the total possible number of combination of events, the majority is composed of mutually exclusive or non-interacting events. The effect of third and higher order interactions are usually confounded with the effect of second order interactions; thus "pure" interactions of order greater than two are vir t u a l l y non-existent. Also, a consider-able number of event combinations do not produce significant effect on 55 the system performance. Nevertheless, regardless of whether or not the effects of a combination of events on the system responses are significant, the model should cope with such occurrences. Since in the real situation each simultaneous occurrence of several events is adequately handled, the representation should reflect this s t a b i l i t y . Earlier experiences with the logging model have shown that models show grossly unrealistic results i f no appropriate branches in the logic flow account for possible combinations of events, even when such combinations do not contribute significantly to the overall performance of the system. A significant problem i n the simulation of a system i s the immediate recognition of non-obvious combination of events which may affect the performance of the system. After recognizing these event combinations, the remaining modelling task consists of determining which event combinations to consider significant, providing extra routines to handle the more significant event combinations, and provid-ing default branches to handle the less significant event combinations. Whenever a possible event combination has been identified, the c r i t e r i a used to decide what courses of action to follow in i t s representation are: 1. The likelihood of i t s occurrence. 2. The magnitude of i t s effect on the system responses. ~3. The ease of i t s representation. 4. The computer time and storage requirements for i t s inclusion. In modelling a system, the f i r s t p r i o r ity i s placed on the 56 "adequate" representation of the main events occurring either singly or in sequence. When accounting for the second-order interactions, past experience with the logging model has shown that, although there is a need to represent these interactions in the model, simplifications can be made without significantly affecting the system responses. That i s , the representation of these interactions is more important than the degree of realism that i s introduced (assuming that the representation is reasonable). If the object of the study i s to compare two or more alternatives, i t i s not necessary to provide a r e a l i s t i c representation to interactions that have no obvious bearing on the comparison. There i s a certain danger that the. inclusion of provisions, which enhance the realism of the model but are nevertheless minor, may obscure the significance of the results. Some of the more significant second-order interactions are lis t e d below, together with their effects. Almost a l l of the inter-actions l i s t e d are results of equipment failure. 1. Loading on yarding - If the yarder i s of the type similar to a "Trakloader" or a "Mobile Logger" which performs both the loading and the yarding functions, yarding i s stopped so that the arriving truck may be loaded. Grapple yarders and, in general, High Lead yarders do not have this type of interaction. 2. Loader breakdown on yarding - The effect of this interaction i s delayed or indirect. Yarding continues u n t i l the landing is f i l l e d to capacity and then yarding stops. 3. Loader breakdown on landing - Loading stops. The truck being loaded proceeds to the dump i f the volume already loaded i s greater than 5 cunits. Otherwise, i t i s dispatched to the other sides. The trucks waiting in the queue are dispatched as well. 57 4. Loader breakdown on the scheduled moving of the yarder - This has no effect i f the landing has adequate space to accomodate the rest of the unyarded logs. Otherwise, the moving of the yarder i s postponed. 5. Loader breakdown on the moving of the loader -The moving is postponed u n t i l the loader is repaired. 6. Loader breakdown on truck dispatching - The side i s momentarily closed down. Trucks already dispatched are recalled and re-dispatched to the other sides. 7. Loader breakdown on overtime act i v i t i e s - If the side has been scheduled to work overtime and i t i s too late to get a replacement, the required number of extra round trips may have to be readjusted to avoid excessive queueing at the other sides. 8. Yarder breakdown on loading - Loading continues i f the landing inventory i s enough for a load (greater than 5 cunits). Otherwise, the truck being loaded, as well as the trucks in the queue ( i f any), are dispatched to the other sides. 9. Yarder breakdown on the moving of the yarder -The moving i s postponed. The trucks already scheduled to support the moving resume their log-hauling duties. 10. Yarder breakdown on overtime act i v i t i e s - The loader works overtime u n t i l the landing is depleted of logs. 11. Truck breakdown on the moving of a yarder or a loader - Since the probability of the event (that the truck supporting the mov-ing of a yarder breaks down) is small, i t is assumed i n the model that this event can never occur. 12. Truck breakdown on dispatching - If the truck dispatched to a side which urgently needs a logging truck (when the landing i s f i l l e d to capacity) breaks down, ideally another truck (which may have been already dispatched to another side) should be 58 dispatched to the (fi r s t ) side. In the dispatching routine of the model, this Is not taken into account owing to the dis-proportionate complexity of the respresent-ation involved relative to the improvement i t effects. 3.6 THE PROBABILITY TRANSFORM THEOREM AND THE CONSTRUCTION OF AN EMPIRICAL INVERSE CUMULATIVE DISTRIBUTION FUNCTION The previous discussion made mention of the provisions for the uncertainties in the physical system through the use of variates generated from inverse cumulative distribution function (c.d.f.'s). This is made possible by the use of a uniformly distributed pseudo-random number generator and the following "probability transform theorem . A variate with cumulative distribution function F(X) i s transformed into a variate P with a uniform cumulative distribution function by the transformation P = F(X). Thus to generate a variate X with c.d.f. F(X), a uniform pseudo-random number P i s f i r s t obtained from the generator, and then X is obtained from X = F 1 (P). From a set of samples collected through observations, the inverse c.d.f. may be obtained through: 1. a sequence of hypothesis tests to determine i f the distribution of the set of samples conform with a known function, or through 2. the construction of an empirical inverse c.d.f.. 1 Proof i s provided i n Evans, et. a l . , 1967 p. 186. 59 For instance, using a set of data on inter-failure times for CANFOR logging truck components, Vandenboom (1971) showed that the inter-failure time can be presented as a random sample from an exponential distribution whose parameter i s dependent upon the age of the vehicle. An empirical c.d.f. was constructed for each of the sets of data on travel times, loading times, volume loaded, camp delays, and unloading times, after a series of chi-squared tests failed to show at the 5% level of significance that the sets of data followed the normal, exponential, gamma, or log normal density functions. The various inverse c.d.f.'s, based on data gathered at the Harrison M i l l s Logging Division of CANFOR were constructed as follows:* 1. Assuming that n observations of, say, travel time t have been recorded, order the observations i n a monotonic sequence. fc2 * * C i * * V S l A X . 2. Form a set of vectors (r.,t.) , where r. = i/n . X 1 1 — 1 y • • t )tl 1 3. Using the method of least squares, f i t a low order polynomial to the set of vectors (r. t.)i=l,...,n , i . e . , t = P (r) = a + a.r + a_r 2 + ... + a r n = F _ l ( r ) n o 1 2 n for O^r^l such that t^O. 3.7 FLEXIBILITY OF THE MODEL FOR REPRESENTING VARIOUS CONFIGURATIONS The model i n i t s present form i s capable of representing various multiple-source, single-sink logging configurations. For instance, 1 The author is indebted to James McPhalen for the collection of data. 60 the following may be changed in the model: 1. The number of yarders. (Up to ten yarders may be used.) 2. The combination of the type of yarders. (Any of the yarders may be a Grapple yarder, a High Lead yarder or a "Trakloader".) 3. The number of trucks. (Up to 30 trucks may be used.) 4. The truck size. (A choice between 75,000 lbs. or 125,000 lbs. payload is provided.) 5. The distance between nodes. 6. The breakdown and repair parameters. 7. The yarding schedule, the setting volumes, and the landing capacities. 8. The functional relationships, e.g. the various inverse c.d.f.'s and the cost functions. The above items may be changed by merely manipulating the set of inputs to the model or, in the case of item 8, by substituting the particular relationship for the one provided i n the model. In addition, two different shutdown models are available. With some modifications, the model can be made capable of representing situations where one loader may serve two adjacent landings, where there i s more than one unloading f a c i l i t y , or where there is more than one s h i f t . 3.8 PROGRAM INPUT REQUIREMENTS AND PROGRAM OUTPUT The use of simulation requires complete information on the characteristics, behavior, and operating rules of each part of the system. It is based on the premise that much is known about the be-havior of each part of the system but not how the parts interact to produce the overall system behavior. Thus the entire system i s broken 61 down into parts for which operating rules can be given. Although adequate knowledge of the different parts of the system may be available to represent the "backbone" of the system, a meaningful use of the model requires a detailed knowledge of the functional relationships and numerical value of the parameters of the system. The set of inputs used in the simulation model developed in this study are li s t e d in Appendix C. Appendix D shows the cost values assumed for a l l the runs made with the model. The empirical derivation of each of the various inverse c.d.f.'s requires a sufficiently long time studies to obtain a reliable estimate of the required distributions. In this particular study, about a month of time studies was made to obtain the various inverse c.d.f.'s used as inputs to the program. To develop the yarding models for "predictive" purposes, i t is necessary to obtain an estimate of the mean and the variance of the performance of each yarder on each of i t s settings. These estimates may be obtained through data on the past performances of the given yarder on settings under similar conditions or through a person who is familiar with the given yarder's performance. Also, extensive data on the past consecutive daily performance of each yarder are needed to obtain a measure of the time-dependent properties of the yarding process corresponding to the given yarder. The model yields two types of output. 1. A detailed breakdown of the times spent by each of the trucks, yarders, and loaders in their corresponding ac t i v i t i e s , together with the cost and production 62 summaries for trucking, yarding, loading, and unloading. A sample output of this type is given i n Appendix E. 2. The values of the different dependent variables or responses of the system. These include the unit cost (broken down into yarding, trucking, loading, and unloading $/cunit); the production in cunits for each of the yarding, trucking and dumping a c t i v i t i e s ; per cent av a i l a b i l i t y and per cent u t i l i z a t i o n of the machines; and the total number of loads hauled by the trucking fleet. This information may be read into a f i l e or punched into cards for subsequent s t a t i s t i c a l analysis. Appendix E (part 2) shows a sample l i s t of the daily values of the dependent variables of the system. Both types of output are generated at the end of each simulated day. In addition, after the inclusion of extra output statements i n the program, the model i s capable of yielding a set of output from which a plot similar to Figure 3.1 may be produced. 3.9 MODEL PROGRAMMING AND COMPUTING REQUIREMENTS The programming and computing requirements for the imple-mentation of the logging simulation model are highly dependent on the experience of the modeller with simulation models and on his famil-i a r i t y with logging systems. For instance, the logging simulation 63 model described in this study was the f i r s t large-scale simulation model developed by the modeller and over half a year of program-ming and about five thousand "computer dollars" were spent to develop the model for the f i r s t shutdown mode. However, to develop the model for the second.shutdown mode, only two weeks and five hundred "computer dollars" were spent despite the extensive revisions on the f i r s t model that needed to be done. The computer time requirement for each simulation run varies depending upon the number of days and the logging configuration simulated. Using the number of truck round trips as an index for the different logging configurations, the computing time requirement for a simulation run of the logging model in an IBM 360/67 system i s shown in Figure 3.7 as a function of the number of days of simulation. The table included in Figure 3.7 shows that the "time compression" ratio ranges from 0.40 to 0.73 seconds per day. 64 Number of truck round trips/ 20 d a y No. of round trips/day Program loading + i n i t i a l i z a t i o n time (seconds) Time compression ratio (sees/day) 20 10.10 0.399 25 10.12 0.482 30 10.13 0.566 35 10.25 0.649 40 10.25 0.734 TO 20 ~30 40 50 60 Number of days of operation simulated 70 80 Figure 3.7 The computing time requirement as a function of the number of days of simulation and the number of round trips/day 65 CHAPTER IV MODEL VALIDATION Learning though a model of a real-world system involves: f i r s t , the representation or modelling of those aspects of re a l i t y which are to be investigated; second, the getting of relevant insights from the model or representation; and third, the conversion of these insights from the model into conclusions on the aspects of the real system under study. It i s clear that making valid con-clusions about the real system through insights from the model requires that there exists a "true" correspondence between the behavior of the real system and the behavior of the model of that system. There i s , therefore, a need to "validate" the model. Otherwise, insights derived from i t w i l l contribute nothing to the understanding of the system being studied. This need for model validation applies to a l l types of models - linear programming models, waiting line models, inventory models, or any other programming model. Why then the preoccupation on validation in the context of simulation? Richard Van Horn (1969) has this to say: "Simulations tend to become far more complex than other Management Science models. Most analytic models either deal with small problems - for example queuing models - or deal with common 66 parts of large problems - input-output models. Simulators allow the modeller to include many different parts and processes in one model and allow the parts to interact in non-linear, non-stationary modes. "In addition, simulators conceal their assumptions and processes, certainly from the casual observer, and often from their designer. The simple statement that model x is a linear programming model conveys a great deal of information about i t s structure, assumptions, and limitations. The statement that model y i s a simulation conveys virtu a l l y no information. Finally, simulators, either e x p l i c i t l y or implicitly often claim to represent " r e a l i t y " . " 4.1 SOME CONCEPTS ON THE VALIDATION PROCESS The problem of validating simulation models i s one of the most d i f f i c u l t of a l l unresolved methodological problems associated with computer simulation techniques. Much of the d i f f i c u l t y arises since.... "to prove that a model is "true" implies (1) that we have established a set of c r i t e r i a for differentiating between those models which are "true" and those which are not "true", and (2) that we have the a b i l i t y to apply the c r i t e r i a to any given model". (Naylor and Finger, 1967) Clearly i t i s d i f f i c u l t to agree upon a set of c r i t e r i a for differentiating when a model i s proven "true". Naylor and Finger (1967) reported three major methodological positions on the problem of verification i n economics. These are: (1) rationalism which... "holds that a model i s simply a system of logical deductions from a series of synthetic premises of unquestion-able truth not themselves open to empirical verification or general appeal to objective experience," (2) empiricism which rejects any postulate or assumption that cannot be independently verified through observation, and (3) positive economics which contends that a model should be validated on the basis of the accuracy of i t s predictions 67 on the behavior of the dependent variable of the model, rather than on the basis of the validity of i t s assumptions. Naylor and Finger (1967) suggested yet a fourth - multi-stage verification which incorporates the methodology of rationalism, empiricism, and positive economics. It contends that each of the aforementioned i s a necessary but not sufficient procedure for validating simulation experiments. Van Horn (1969) maintains that seldom, i f ever, w i l l validation result in a "proof" that the simulator i s a correct or "true" model of the real process. Thus i s seems more r e a l i s t i c to view validation as the process of building an acceptable level of  confidence that the inference about a simulated process is a valid  inference for the real process (Van Horn 1969). That i s , instead of focusing on whether or not a model is "true", the emphasis should be on the degree of confirmation of the model. The model is subjected to a series of tests and confidence in the model increases when no negative results are found. 4.2 MULTI-STAGE VALIDATION For building confidence in the model, i t appears that the Naylor and Finger approach to verification i s appropriate. This three-stage approach includes: 1. A formulation of a set of hypotheses or postulates for the process using a l l available information -observations, general knowledge, relevant theory, and intuition. 2. An attempt to "verify" the postulates on which the 68 model is based subject to the limitations of existing s t a t i s t i c a l tests. 3. A test of the model's a b i l i t y to predict the behavior of the system being studied. In short, this approach consists of checking the relation-ships, structure, and policy of each component of the model and con-firming that these components, when combined, display the overall characteristics and behavior associated with the real system. In an attempt to perform the steps indicated by the f i r s t two stages, the previous discussions on the important aspects of the components of the model were presented to consider one or more of the following actions: 1. Associating a high degree of confidence on the representation of processes which are easy to observe and measure. For instance, some physical processes of machines may be represented by production functions with a substantial degree of confidence. 2. Associating a high degree of confidence on the representation of processes which had undergone previous validation in similar experiments or studies. 3. Associating a high degree of confidence on the representation of processes backed up by the existence of an extensive body of research. 69 4. The empirical testing of those representations which are amenable to some form of s t a t i s t i c a l tests or the performing of sensitivity analyses as a substitute to the more costly empirical testing. Through sensitivity analyses, a result may be found to hold for a class of distributions or even for a general distribution. The third stage of the multi-stage verification procedure involves subjecting both the real system and the model to the same set of input, and comparing the input-output transformations generated by the model to those generated by the real system. This stage is of great importance since obviously a great deal of confidence on the model rests upon the confirmation of i t s a b i l i t y to transform input to output i n a similar manner to the real system. Several ways of making this comparison have been suggested. An appropriate way is by time series analysis. The real system, as well as the model, produces some stochastic processes - a set of time series of some relevant system responses. Assuming two processes are stationary, testing their equivalence involves the testing of the three elements that completely describe a stationary process; namely: the mean, the variance, and the auto-correlation structure or equivalently the spectrum of the process.'' Studies by Fishman and Kiviat (1967) on spectral analysis resulted i n a test for comparing two spectra. The next section w i l l be devoted to the input-output transformation in the 1 For a more detailed discussion on time series analysis, refer to Appendix F. 70 model and the application of the test to compare the time series produced by the model to the corresponding observed time series. 4.3 A COMPARISON OF INPUT-OUTPUT TRANSFORMATIONS Several arrays of daily values of some system dependent variables are generated both by the real system and the model. Among these arrays or time series of importance are unit cost, total yarding production, total trucking production, total number of loads, per cent u t i l i z a t i o n of trucks, per cent u t i l i z a t i o n of yarders etc. Of the several mentioned, the total yarding production time series i s particularly interesting for two reasons: (1) as w i l l be discussed later, the relationship between yarding production and the other dependent variables can be traced, and (2) the transformation of the total yarding production time series as- i t evolves from the input time series can also be traced. As mentioned earlier, the potential daily production of any given side i s generated from that side's parametric model. The sum of these generated values for a given day represents the ceiling of the total yarding production for that day. The series of these sums can be considered an input series. A model, unlike the real system, can be manipulated so that only certain particular aspects of i t can be observed while others are suppressed. In so doing, the effect of the suppressed aspect can be separated and the resultant behavior can be attributed only to the observable aspects. For instance, i n several consecutive experiments, the following aspects were investigated: 71 Experiment 1. The input series. (This corresponds to the time series of the sum of the individual yarding production. A l l the other aspects were suppressed.) Experiment 2. The input series plus the yarding-loading-trucking-unloading interactions. (This corresponds to a hypothetical case where a l l equipment has 100 per cent a v a i l a b i l i t y and a l l settings have inexhaustible supplies of logs. Experiment 3. The input series with the various subsystem interactions plus the breakdown of equip-ment. (In this case, only the moving of yarders and loaders was suppressed from the model.) Experiment 4. The input series with the various subsystem interactions and equipment breakdown plus the moving of yarders and loaders. (In this case, the model is complete.) For a l l four experiments, four high lead yarders and nine logging trucks were used. The input for these experiments is included as the entries in Appendix C corresponding to the f i r s t four yarders. The mean and variance of the total yarding production time series for each of the above cases are shown in Table 4.1. Figure 4.1 shows the spectrum of each of the four time series. Experiment 1 vs. Experiment 2 - Since series 1 i s the input 72 series, each daily element of the series represents the maximum total yarding production for that day. The mean yarding production should therefore decrease as more aspects that interact with the yarding subsystem are introduced into the model. For instance, the mean of series 2 (series 1 + yarding-trucking-dumping interactions) is less than the mean of series 1. The loss in production can be interpreted as the reduction due to the interaction among the processes, e.g. yarding-trucking interaction. The decrease in variance i s caused by the stabilizing effect of the different interactions. As w i l l be shown later, the optimum number of trucks for this configuration i s greater than 9. This implies that, on the average, the total trucking capacity i s less than the total yarding capability thus resulting in the loss of yarding production. This loss i s disproportionately higher in the "high daily production" elements of the series than in the "low daily production" elements. Consequently, the range of these values, as well as the variance of the series, decreases. Experiment 2 vs. Experiment 3 vs. Experiment 4 - A further decrease in the mean yarding production i s observed with the incor-poration of: (1) breakdown of equipment and (2) moving of yarders and loaders. The breakdown of a loader or a truck affects the yarding production as a consequence of the yarding-loading-trucking interaction, and clearly, the breakdown or moving of a yarder interrupts the yarding process. Each single or concurrent occurrence of the aforementioned events reduces the yarding production for that day. As a result, the range of the values of the series and the variance of the series increase. Table 4.1 The r.ean and variance for each of the four ey.Derinents EXPERIMENT >2AN (Mfbm) VARIANCE (Mfbm2) 1 183.45 354.47 2 180.64 291.64 3 174.34 392.08 4 165.41 588.84 S P E C T R U M F O R Y A R D I N G P R O D U C T I O N I 1 1 1 — i r 0 0.1 0.2 0.3 0.4 0.5 F R E Q U E N C Y (Cycles per day) Figure 4.1 Spectrurr. for each of the four experiments 74 Although the s h i f t in the spectrum as the observation proceeds from experiments 1 through 4 i s interesting from a time series transformation standpoint, the physical interpretation of the individual spectrum i s d i f f i c u l t . Normally, the interpretation of the spectrum i s done by looking for true and well resolved peaks i n the spectrum, noting the frequency at which the peak occurs, and looking back to the physical process to see i f the observed frequency relates to a regular physical' phenomenon. This interpretation i s highly sub-jective and d i f f i c u l t i e s arise when several sources of variation occur in a given frequency range, and when a given disturbance occurs at an irregular frequency. Series 1, the input series, i s the sum of four "potential" yarding production time series, two of which are generated from third-order autoregressive processes whose spectra are dominated by low frequencies.'' The big s h i f t in the spectrum from series 1 to series 2 indicates the effect of the various machine interactions on the input series. This i s contrasted by the insignificant change in the low -and mid-frequency bands, but a relatively greater change in the higher frequencies as the breakdown of equipment i s introduced into the model (Exp. 3 vs. Exp. 2 in figure 4.1). A pronounced peak forms at frequency 0.42 (=1/2.4) cycles per day in the spectrum of series 3. The pooled breakdown frequency of a l l trucks, yarders, and loaders at around 1/2.2 explains the difference in this part of the spectrum between series 2 and 3. The big s h i f t in the mean, variance, and 1 The individual spectrum for each of the four yarding processes i s given in Appendix F. 75 spectrum from series 3 to series 4 indicates the effect of the inclusion of the moving of equipment into the model. Not only does a yarder and a loader stop i t s main function to move to a new location, but one or two trucks are taken from the hauling fleet to support this transfer. Consequently a significant effect on production i s expected. It may be concluded that the input-output transformation i n the model is plausible and added confidence on the model may result from this fact. But how does the output series generated by the model compare with the time series generated by the real system i f both the real system and the model are subjected to as similar a set of inputs as possible? The answer to this question requires a test of equivalence between the time series generated by the real system and the time series generated by the model. Each of these time series i s a "record" or "realization" from a stochastic process. Testing their equivalence is determining whether these separate realizations come from the same ensemble; thus i t requires a comparison of their respective means, variances, and spectra. Two 69-day time series - the total yarding production and the total number of loads, time series - were compiled from the 1968 records from the Harrison Mills Division of CANFOR. In the 1968 con-figuration, your yarders and nine trucks were used. The parametric yarding model for each of the your yarders i s shown as the f i r s t four entries of Table C.l in Appendix C. The yarding schedule, setting distances, landing capacities, and setting volumes for the 69-day period are shown as the entries for the f i r s t four yarders of table C.3 Figure 4 . 2 Yarding production time series of CANFOR data and two simulation runs 100 + 0 10" 20" 3$ 40 5S 60 70 77 Table 4.2 The mean and v a r i a n c e f o r the yar d i n g p r o d u c t i o n and number of loads time s e r i e s Mean ya r d i n g production (Mfbm) Variance Mean number of loads Variance CANFOR 164.87 570.03 18.39 5.77 TRIAL 1 168.88 640.07 17.99 5.06 TRIAL 2 165.41 588.84 18.85 4.74 SPECTRUM FOR Mfbm YARDED / \ 400 -4- 4-0.0 0.1 0.2 0.3 0.4 F R E Q U E N C Y ( c y c l e s per day) 0.5 30 20 " 15 • 10 •• CANFOR data TRIAL 1 TRIAL 2 SPECTRUM FOR NUMBER OF LOADS 0.0 0.1 0.2 0.3 F R E Q U E N C Y ( c y c l e s per day) Figure 4.3 Spectra f o r the yar d i n g p r o d u c t i o n and the number of loads time s e r i e s 78 in Appendix C. These values were used in the model as part of the input. Two separate t r i a l s were performed using different series of random numbers. Figure 4.2 shows the graph of the yarding production time series for the CANFOR data and for the results of the two t r i a l s . The respective means and variances of the three time series are summarized in Table 4.2 while their respective spectra are shown i n Figure 4.3. The s t a t i s t i c a l comparison of the yarding production and number of loads mean and variance, between the CANFOR data and the model output for both t r i a l s , showed no significant difference at P = .95. Using a s t a t i s t i c a l test reported in Fishman and Kiviat (1967), no significant difference (at P = .95) was observed, for a l l frequencies, between the spectrum of the CANFOR data and the spectrum of the model output (for both t r i a l s ) for both the yarding production and the number of loads. These s t a t i s t i c a l comparisons are included i n Appendix F.5. Following the verification of the various components of the model and the a b i l i t y of these components to display the overall characteristics and behavior associated with the real system, the model is used to explore situations for which no empirical data exists. This i s essentially an extrapolation procedure, but i t is expected that the model w i l l yield acceptable results. The results from the various "runs" of the model are, themselves, basis for building confidence in the model. If the model continuously yields r e a l i s t i c and acceptable results, confidence i n the model increases. 79 CHAPTER V SOME.DESIGN, TACTICAL, AND STATISTICAL CONSIDERATIONS IN THE EXECUTION OF THE SIMULATION RUNS Following the construction and validation of the model, the next phases in an investigation of a problem by simulation involve: 1. The design of experiments - the selection of the factors and factor levels to use in the simula-tion runs, the determination of the sampling tech-niques to apply, and the determination of the number of replications. 2. The tactical planning of the simulation runs to ensure that the experimental design i s carried out properly. 3. The s t a t i s t i c a l analysis of results and the deri-vation of conclusions .about the problem. Depending upon the purpose and the nature of the problem to be investigated, a particular design i s chosen for the problem. For this reason, the experimental design for each of the problems consi-dered in this study is given i n the next chapter. However, some general considerations in the design of computer simulation experiments are given i n this chapter. Other considerations, tactical as well as s t a t i s t i c a l , are also presented in this chapter. 80 5.1 SOME CONSIDERATIONS IN THE DESIGN OF SIMULATION EXPERIMENTS Computer simulation experiments are considered synthetic experiments. They are, nevertheless, real experiments and their differences from real-world experiments can be directly attributed only to the difference i n experimental medium. Consequently, the same problems inherent in real-world experimentation are present in computer simulation experimentation. These problems are intensified i n the case of computer simulation experimentation by the fact that computer time is not a free resource. Naylor, et. a l . (1969) pointed out that i n the design of simulation experiments, careful attention should be given to; the problem of motive, the problem of stochastic convergence, the problem of size, and the multiple response problem. A. THE PROBLEM OF MOTIVE Let f denote a functional relationship or "response surface" which relates a given dependent variable or response X to the indepen-dent variables or factors Y ,i=0.,. »«,n, of the system; that i s , X = f(Y Y ) . i n The underlying objective i n a study of a system i s to learn more about the system. In particular, the objective may be: (1) to describe and explore the response surface over some region of interest i n the factor space, or (2) to optimize the response over some feasible region in the factor space. Associated with the objective of an experimental investiga-tion i s an appropriate experimental design. The design may only involve the planning for a simple t-test i f the objective i s to com-pare two levels of a particular factor while leaving the levels of the other factors fixed. The design may involve a more complicated response 81 surface design i f the objective of the experiment is to see how the response surface behaves with changes in the levels of the different factors. If the determination of the optimum combination of factor levels i s the objective, the design may involve the use of a random sampling method or a systematic sampling method such as the method of steepest ascent. B. THE PROBLEM OF STOCHASTIC CONVERGENCE An estimate.of the population average of one or more responses is generally required in a simulation experiment. Accompanying each estimate or sample average is a measure of i t s deviation from the population average, namely the standard error of the mean. As the number of samples used to calculate the sample average i s increased, the standard error of the mean is expected to decrease and converge to zero. The problem associated with stochastic convergence i s keep-ing the length of the run and the number of replications down to an economical level while meeting the desired standard error of the estimate. Several variance-reduction techniques have been introduced for simulation experiments (see Hammersley and Handscomb, 1964), two of which are found appropriate for this study, namely the use of control variates and the use of antithetic variates. When the outcomes of two similar processes differing only i n some minor respects are to be compared, the variance of the differences of their respective means can be reduced i f a positive correlation be-tween the outcomes of the two processes is induced. For instance, l e t 82 X^ and X 2 be some series of n values of a response X observed respectively from the f i r s t and second processes. That is X, = X,.,X,_,...,X, with mean X, 1 11 12 In 1 and X 2 = X2i» X22'* *" ,3^2n w i t n m e a n X2* The variance of the difference between the two means is expressed as . Var(X 1 - X 2) = Var (X ) + Var (X) - 2 CovCX^Xj). Clearly, Var (Xj-X 2) i s reduced i f Cov (X^,X2) can be made positive. Positive correlation between the outcome of two processes i s generally induced by the use of the same sequence of random numbers i n the simulation run for each process. Since i t i s not certain for most simulation models that the i t h random number w i l l generate the same events for the two processes i f the same sequence i s used in the simu-lation run for both processes, this technique of inducing positive correlation i s made more effective i n the model by the use of four sequences of random numbers, namely: 1. A sequence used i n the generation of interfailure and repair times for logging trucks. 2. A sequence used in the generation of interfailure and repair times for yarders and loaders. 3. A sequence used in the generation of the yarding rates. 4. A sequence used i n the generation of the rest of the variates. If the calculation within a specified precision of the mean 83 of a response i s desired, the variance of the mean may be reduced by increasing the length and hence the number of observations n of the run. However, because of autocorrelation, i t is not certain that the variance w i l l decrease in the order of 1/n i f n i s increased.* If the variance i s s t i l l large for a reasonable length of the run, a second replication may be made. A variance smaller than half the average of the variances of the two replications can be achieved i f a negative correlation between the outcomes of the two replications i s induced. Let the outcomes of the f i r s t replication be X^^, X^ ,,,..., with mean X^ and for the second replication X 2^,X 2 2, • • w ^ t n X l+ X2 mean X,,. The overall mean i s X = — 2 — with variance Var (X) = Var (^(X^+X^ ) = Jz;Var (X^ + h Var (X 2) + h Cov Q4,X^ Thus i f Cov (X^,X2) i s made less than 0, Var (X) i s reduced. Negative correlation between the outcomes of the two replications may be induced by the use of "antithetic" sequences of random numbers. That i s , i f the sequence r ^ , r ^ , . . . r n i s used i n the f i r s t replication, the sequence ( 1 - r ^ ) , ( l - r 2 ) , . . . , ( l - r Q ) i s used in the second replication. Again for large-scale models i t i s not certain that this technique w i l l cause Cov (X^,X2> to be negative. In the log-ging model, for instance, this technique, except for some rare instances, does not cause Cov (X^,X2) to be negative. 1 The determination within a specified standard error of the length of the run for single experiments or for comparing two experiments i s given in Appendix F. 84 Noting that a large part of the variation i s caused by the stochastic yarding rate, the method of using antithetic variates was •extended to induce a negative correlation between the respective yarding rates of two replications. To recapitulate, the yarding parametric model which generates the yarding rate i s of the form n m X = u + Z 4> (X - y) - Ze.a„ . + a #. t . . = 1 x t-x . = 1x t-x t where y = E(X) <{K = i t h autoregressive weight or parameter 6_£ = i t h moving average weight a = white noise generated from N \p,a 3 n a n = order of the autoregressive term (£0) m = order of the moving average terms (£0). 2 a t i n the model i s generated normally with mean 0 and variance aa» Since the variate generated i n the computer i s normally distributed with mean 0 and variance 1, the transformation (1) a = pa where p ~ N[o,l] is used. A second transformation (2) a t = -pa a is used for the second replication to induce a negative correlation between the yarding rates of the two replications. Since E(a) = E(-pa ) = -a E(p) =0 and a a Var (a) = Var (-pa ) = (-1)2 a * Var (p) = a 2, a a a the second transformation gives the same mean and variance for a f c as the f i r s t transformation. However, when at>0 using (1), at<0 using (2) and vice versa. 85 When the outcomes of two similar processes are compared and a second replication i s necessary to get the desired precision, a com-bination of the control variate and antithetic variate techniques may be applied. However, this technique may be inferior to either the control variate or the antithetic variate technique used singly, since i f X^ _. denotes the mean of jth replication of the i t h process, then for a two-process situation Var ( X r X 2 ) = Var h(%n+^12) ~ ^ (J^+X^) = *&Var ( X u ) +h Var (X^) + k Var (X 2 1) + h Var (X 2 2> + h cov ( x u , x 1 2 ) + h cov (x 2 1 ,x 2 2 ) - h cov ( x U jX 2 1) - h Cov ( x u , x 1 2 ) - h (x 1 2 ,x 2 1 ) - h cov (x 1 2 ,x 2 2 ) and when a positive correlation i s induced between the outcomes of the two processes while a negative correlation is induced between the out-comes of the replications, not only w i l l Cov (Xj^,X^ 2) and Cov (X 2j,X 2 2)be < 0 but also w i l l Cov (^n^2Z^ a n d C ° V ^ 1 2 ' X 2 p * I f t h e S U m ° f t h e last two covariances i s greater than the sum of the f i r s t two covar-iances, clearly the combination of the two techniques is inferior to the simple use of the control variate technique. C. THE PROBLEM OF SIZE The problem of size arises when too many factors and too many factor levels are thought to influence the response under consideration. The several factors - independent variables, functional relationships, and policies - of the logging system include the number and capacity of trucks, breakdown and repair parameters for each class 86 equipment, setting distances, number and composition of yarders, mean travel time between any two points in the road system, yarding rates, shutdown modes, and various functional relationships. Since some of these factors are non-quantitative (e.g. shutdown modes, functional relationships), they are held fixed for any particular investigation. The remaining factors, some integer-valued but nevertheless quanti-tative, may be grouped into classes according to the nature of the investigation. For instance, for a study on repair and maintenance policies of equipment, the breakdown and repair parameters for each class of equipment may be assigned different levels while the rest of the factors are held fixed. Thus for three classes of equipment, only six factors (the various breakdown and repair parameters) need be used. By noting that the range of the levels of some factors i s naturally narrow, further reductions in the number of observable factors can be made. An example is the repair time of a yarder. Since i t is l i k e l y that yarder repair times are already near-optimal, very l i t t l e can be done with the repair policy to significantly reduce the repair time. Hence, only one level for the yarder repair parameter need be used. Thus for some investigations, the number of factors and factor levels may in effect be only a few and the number of simulation runs required becomes manageable. In this case, a f u l l factorial design may be used, or where less than f u l l information from the design i s tolerable, designs which require fewer " c e l l s " than f u l l factorial may be used. 87 D. THE MULTIPLE RESPONSE PROBLEM The multiple response problem is the problem of selecting an appropriate index of performance for a system. For problems requiring the comparison of different combinations of the factors y^,y2»...ym> there may be several appropriate responses, ,X2»...,Xn, that may be' used. Which response should be examined more intensively? Among several responses of the logging system are: 1. unit cost 2. total daily cost 3. daily yarding production, daily trucking production, daily production at the dump 4. % u t i l i z a t i o n of the trucks 5. % u t i l i z a t i o n of the yarders 6. daily number of truck round trips. Logging managers invariably plan towards minimizing the unit cost of attaining the required level of production; that i s , the unit cost response is generally used as the index of performance of the logging system with production as the constraining response. The logging manager l i s t s the available alternatives that satisfy the production constraint and he selects the one offering the least unit cost. In this thesis, when optimization is the object of the problem under investigation, the unit cost is used as the index of performance. Indeed, business problems are easier resolved and understood when expressed in concrete terms, e.g. in terms of dollars. 88 As w i l l be seen later, the minimization of the unit cost does not proceed at the expense of production. It is compatible with the objective of maximizing production, but the reverse does not hold. Because of the close relationship among the given system responses, observations on the behavior of unit cost and production can usually be explained through the other responses. The f i r s t section of the next chapter w i l l be devoted solely to the exploration of the interrelationships among these responses. 5.2 TACTICAL CONSIDERATIONS IN THE EXECUTION OF THE SIMULATION RUN Two important problems arise i n the generation of data for a simulation experiment. These are the determination of the point where data gathering starts and the selection of the measurement mode. Since a simulated system i s started abruptly, the conditions existing during the i n i t i a l part of the run may be considered transient. The data collected from the i n i t i a l part of the run should be excluded since they tend to bias the results. The gathering of data should only start when the conditions or states of the system are deemed to be independent of the starting conditions and close to the equilibrium conditions. With respect to this consideration, the comment of Conway (1963) i s relevant: "It i s important to recognize that equilibrium i s a limiting condition which may be approached but actually never attained. This means that there i s no single point in the execution of the simulation experiment, beyond which the system is i n equilibrium. The difference between the temporal and limiting distributions presumably decreases with time and one seeks a point beyond which he is willin g to neglect the error that i s made by considering the system to be in equilbrium." 89 A. CHOICE OF STARTING CONDITIONS The e a r l y p a r t of the run, which i s considered as a t r a n s -i e n t phase, i s dependent upon the s t a r t i n g c o n d i t i o n s . Making t h i s phase as s h o r t as p o s s i b l e r e q u i r e s t h a t a reasonable s e t of s t a r t i n g c o n d i t i o n s i s chosen f o r each run. However, to e l i m i n a t e the p o s s i b i -l i t y of b i a s i n g the r e s u l t s , each of the program runs should use an i d e n t i c a l s e t of s t a r t i n g c o n d i t i o n s . The s e t of s t a r t i n g c o n d i t i o n s e s t a b l i s h e d f o r the s i m u l a t i o n model i n c l u d e s : 1. The i n i t i a l l o c a t i o n of yarders and l o a d e r s . 2. The i n i t i a l l o c a t i o n and a c t i v i t y of the truck s ( i m p l y i n g a l s o the i n i t i a l l e n g t h of the queues). 3. The i n i t i a l volume a t each l a n d i n g . 4. The i n i t i a l volume a t each s e t t i n g . 5. The i n i t i a l schedule of the breakdown of each equipment. 6. The i n i t i a l y a r d i n g r a t e s ( f o r the a u t o r e g r e s s i v e y a r d i n g p r o c e s s e s ) . The s e l e c t i o n of the values of the s e t of s t a r t i n g con-d i t i o n s was made a r b i t r a r i l y u s i n g means when a v a i l a b l e , e.g. mean ya r d i n g rates, and other values deemed " r e a l i s t i c " . The values f o r items 1 to 4 are i n c l u d e d w i t h the inp u t to the model given i n Appendix C. These same s e t of s t a r t i n g values are used i n a l l the runs made w i t h the model. B. DELETION OF THE EARLY PART OF THE RUN To determine how much of the e a r l y p a r t of the run to d e l e t e , 90 the "rough guide" suggested in Conway (1963) was used. This involves the truncation of the i n i t i a l series of measurements u n t i l the f i r s t of the series i s neither the maximum nor the minimum of the remaining set. This is not done for each run. A fixed-length stabilization period i s chosen from p i l o t runs and the same length i s deleted there-after from the results of each run. For. this study, the deletion of the f i r s t five observations is deemed sufficient to render the state of the system independent of the starting conditions. As used here, an observation corresponds to a day's performance. In over a hundred runs made with the model the sixth observation was found to satisfy Conway's criterion; that i s , i t is neither the maximum nor the minimum of the remaining set. C. THE MEASUREMENT MODE Two possible measurement modes may be used. The f i r s t one -periodic sampling - requires the collection of samples separated by an intervening period of time which should be long enough to safely assume the samples to be uncorrelated. The second measurement mode involves a continuous, rather than intermittent, measurement. Continuous measurement i s a better choice for economic reasons. Since continuous measurements are made, there are no inter-vening time periods between samples and, therefore, no measurements are discarded. The use of continuous measurements, however, has a major disadvantage. The series of values of the various system responses resulting from continuous measurements form a time series. The adjacent elements in a time series are correlated; thus the implementation of continuous measurements entails the use of more 91 complicated s t a t i s t i c a l methods. S t a t i s t i c a l techniques that require the assumption of independence of sample errors can not be used unless the presence of autocorrelation i s accounted for. Ignoring auto-correlation i s unacceptable since the r e l i a b i l i t y of the means and the variance are thereby overestimated (Fishman and Kiviat, 1967); thus the v a l i d i t y of t-tests becomes questionable. Continuous measurement is used in the generation of data from the model. The subsequent calculation of variances and covariances accounts for the presence of autocorrelation. The appropriate formulae used are given in Appendix F (parts 2 and 3). 92 CHAPTER VI SOME APPLICATIONS OF THE LOGGING SIMULATION MODEL The task of modelling is the creation of a medium for the investigation of real problems without having to deal directly with the real system. The usefulness of the model rests on i t s capacity to be manipulated and experimented upon. Assuming that confidence on the one-to-one correspondence between the behavior of a given system and the behavior of the model of that system has been established, operational plans, designs, or strategies are applied to the real system. Accordingly, the u t i l i t y of the logging simulation model rests on i t s capacity to aid i n the evaluation of the various alter-native configurations that are presented as possible means for s a t i s -factorily meeting the system objectives. The f l e x i b i l i t y of the model allows the investigation of a wide class of logging problems and i n this chapter, various examples are presented to demonstrate this capa-b i l i t y . These examples are in no way exhaustive; they are merely used to i l l u s t r a t e some of the problems that can be examined. 6.1 INTERRELATIONSHIP AMONG THE LOGGING SYSTEM RESPONSES Before the main examples on the applications of the logging simulation model are presented, the multiple-response problem intro-duced earlier w i l l be discussed further. In this study, the unit cost response i s used as the index of performance of the logging system with 93 production as the constraining response. In this section some j u s t i -fications are given for this choice. Using production as a basis, the interrelationship among the various responses of the logging system w i l l be explored. Several simulation runs for various logging configurations provided insights and observations about the nature of these interrelationships. These insights and observations provided a basis for the derivation of the relationships presented. A. YARDING PRODUCTION vs. TRUCKING PRODUCTION vs. PRODUCTION AT THE  DUMP Because of the existing limit on the capacity of the different f a c i l i t i e s , e.g. on the landing capacity, on the trucking capacity, etc., a "bottleneck" occurs whenever the amount of output from one sybsystem exceeds the amount of input that the next sub-system in the production sequence can handle. For instance, the land-ings become frequently f i l l e d to capacity whenever the total yarding capacity exceeds the total trucking capacity. The production of the system is limited by i t s lowest producing subsystem; consequently, in the long run the production of each of the main subsystems becomes identical. For this reason, production i s subsequently regarded as an output of the logging system without further cla s s i f i c a t i o n into yard-ing production, trucking production or production at the dump. B. PRODUCTION vs. TRUCK PER CENT UTILIZATION An important response of the logging system is the per cent u t i l i z a t i o n of each class of equipment. While equipment avai l a b i l i t y reflects the divisional repair and maintenance policies, u t i l i z a t i o n is an index of the management and operating policies. Since u t i l i z a -tion i s a measure of the degree of system interactions, i t i s 94 extremely d i f f i c u l t and perhaps impossible to evaluate the degree of equipment u t i l i z a t i o n in a complex and dynamic stochastic system of the type being considered without following a simulation approach. In this study, truck per cent u t i l i z a t i o n i s defined as follows: Let U = truck per cent u t i l i z a t i o n = truck per cent a v a i l a b i l i t y > = mean- truck interbreakdown time y = mean truck repair time. Then queueing time at the waiting time U = ( 1 - dump and at the side + for more logs) X 1 0 0 total available time where the total available time = Afc x total time 1 0 0 x total time . 100r I I 0 N N N. a c b Number of trucks ( N ) Figure 6.1 The truck per cent u t i l i z a t i o n as a function of the number of trucks in the fleet 9 5 Figure 6.1 shows the nature of the truck per cent u t i l i z a -tion as a function of the number of trucks in the hauling fleet. This characteristic behavior i s common in the various configurations examined. Sample U t graphs for several configurations are provided in Appendix G. In Figure 6.1, N & is the number of trucks when queueing time starts to become significant; that i s , when N>N , increasing 3. queueing time contributes to the decrease in the truck per cent u t i l i z a t i o n . For N less than or equal to the number of production sides (MX queueing time at the sides can be assumed to be zero. Since the unloading time is small relative to the total time, queue-ing time at the dump i s expected to be small (found to be near 1% of the total time). Thus N is expected to be approximately equal to 3i M . In Figure 6.1 a change in slope is seen at N=Nc> At this point the effect of waiting time for more logs starts to become significant. Clearly this condition occurs just before the total trucking capacity starts to exceed the total yarding capability ( i . e . at N<N^)• In fact, for a deterministic system with a "sound" dis-patching policy, N £ and coincide. Let N = the average daily production per truck given that there are N trucks i n the fleet; that i s P = P../N where P = the total daily production . N N N It i s seen in figure 6.2 that P.>AP =P -P.. , for N>N . Furthermore, b ' N N N N-l a regression results from the model runs given in Appendix G indicated that i s linearly proportional to the truck per cent u t i l i z a t i o n for 96 N trucks in the range examined (77 <U^ ^ 98), that i s P =k,* U. „ + k2 where k, and k„ are parameters peculiar y N 1 t,N 1 2 to the specific configuration, U „ = truck per cent u t i l i z a t i o n t,N for N trucks, and 77 * U ^ N $ 98. As w i l l be seen in the following discussions, this relationship, plus the linear behavior of the U - curve as a function of N, results in a quadratic and concave production curve at a given domain. This implies that for the particular domain, the "increase in marginal production per unit increase in number of trucks" i s a decreasing function. The relationship between production and truck per cent u t i l i z a t i o n may be summarized as follows: 1. 0<N^ N In Figure 6.1 i t i s seen that for 0<N$N , U = 100%. This implies that for this domain is a constant; thus the average daily production for N trucks (P^ ) i s a linear function of N since P = N/? . N N N < N $ N a c In this case, P = N p = N (K.U X T + K 0) and AP„ = P T -N ' N 1 t,N / N N pN _ i = N ( K i u t , N + V " ( N _ 1 ) ^IVN-I'^V- B y l i n e a r i t y of the U -curve, U" „ ,-U\ „ = some positive constant S. t t,N-l t,N Thus APN= N(K 1U t > N + K 2) - (N-1) ^  ( U ^ +S) + K 2 = K L (U f c > N - NS + S) + K 2. 97 The second difference A 2 ? N = K l Ut,N - N S + s " (U t ) N_ r(N-l)S+S) = - 2K^S, a negative constant. This implies that for N a<N£N c, may be expressed as 2 P J J =aN + bN + c where a,b, and c are parameters peculiar to the specific configuration with 2a = -2KjS<0 implying the concavity of the production curve. Similarly, at the "narrow" interval N < N < N, , P . . C D N may be expressed as P., = a'N2 + b'N + C where 2a' = - 2K,S? N 1 with S' = U „ . > S. t,N-l 3. For N £ N^, the yarding production becomes limiting so that production becomes independent of the truck per cent u t i l i z a t i o n . In this case the production curve levels off at P=Pmax (Figure 6.2). The relationship between truck per cent u t i l i z a t i o n and production, therefore, indicated that the production curve i s linear ( P „ = N A T ) for 0<N*N and quadratic (P =aN2 + bN + C ) for N <N*N N •' N ^ a ^ N a ^ c and for N <N<N, . A hypothetical production curve i s shown i n figure 98 max Daily production ( P ) -0 N i-1 i N c b Number of trucks ( N ) Figure 6.2 Divisional daily production as a function of the number of trucks C. PRODUCTION vs. NUMBER OF TRUCKS ROUND TRIPS For N number of trucks in the fleet, the expected daily production may be expressed as P^ = LjjT^ where = the average load volume given that there are N trucks i n the fleet. T^ = the average number of round trips per day given that there are N trucks i n the fle e t . This expression i s only an approximation for N>N£ since, as w i l l be discussed, T^ and are not independent at this range. In an earlier discussion on the loading policies, i t was stated that where there are trucks waiting in the queue and the land-ing i s short of logs, the truck being loaded departs with less than f u l l load, provided that a minimum load volume requirement (L m^ n) has been met. Consequently, the average load volume is a function of the 99 number of trucks in the hauling fleet. Figure 6.3 shows that at N=Nc, at which the waiting time for more logs starts to become significant, the average load volume begins to decrease. This decrease continues u n t i l the average load volume i s equal to L min Average load volume ( L ) 0 N Number of trucks ( N ) c ' Figure 6.3 The average load volume as a function of the number of trucks i n the fleet. Figure 6.4 shows the average number of round trips per day as a function of the number of trucks. This function reaches a ceiling at N>N^ , i.e. later than does the production curve, since the average load volume decreases after N>N^ . The ceiling represents the maximum number of loads that the given configuration can support. It i s given by T = P /L . where P = maximum daily prodution max max min max potential of the config-uration, and L . = minimum average load mm volume. 100 T max Average number of round trips-per day ( T ) 0 b Number of trucks ( N ) Figure 6.4 The average number of round trips per day as a function of the number of trucks in the fleet. D. PRODUCTION vs. UNIT COST The total unit cost is defined as C = daily yarding, loading, unloading, and trucking costs for N trucks # daily production for N trucks For any given configuration, the total daily yarding, loading, and unloading cost i s independent of the number of trucks in the hauling fleet, assuming that the time of shutdown of the yarders, the loaders, and the unloading f a c i l i t y i s not influenced by the number of trucks.* Thus the unit daily yarding, loading, and unloading cost (C^ ^ ) i s given by B l C. = where B. = the daily total yarding, loading N 1 This assumption holds for the second shutdown mode presented earlier. This assumption s t r i c t l y does not hold for the f i r s t shutdown mode, since in this case, the time of shutdown of the dump is affected by the number of trucks in the fleet. However, the variation in the unloading cost resulting from the difference in number of trucks i s , for a l l practical purposes, negligible. 101 and unloading cost for M sources and one sink = the daily production for N trucks. Since i s constant for a given configuration, while P^ i s an increasing function of N for N<Nb, N is a decreasing function of N. P.T levels off after N=N , so that C, „ is constant for N > N, N b 1,N x b. The daily trucking cost may be partitioned into a cost component fixed with "respect to mileage and the number of operating hours, a cost component varying with mileage, and a cost component varying with the number of operating hours. Although each standard sh i f t has a fixed number of hours, the number of truck operating hours exceeds this fixed number since trucks in their loaded state are generally shutdown at the dump and extra time i s necessary to bring the trucks to the dump. The number of overtime hours per day is influenced by the number of trucks in the fleet. For instance, i n a situation where there are too many trucks, the number of overtime hours i s less than when there are too few trucks. Let B 2 = the fixed cost per truck'' B^ = the cost per mile B^ = the cost per overtime hour G-j = the expected total truck mileage per day 1 B 2 is assumed to include the operator's wages for the standard sh i f t but not the overtime wages. 102 = the expected number of overtime hours per day* The average unit trucking cost (C 9 ) is given by '2,N and AC 2 ) N = C 2 ^ - C 2 > N - 1 is given by A C2.N " P N + for N>Nb. G N ~ G N _ 1 > 0 while H^-R^^O. As N increases, both G^-G^ and H J J - ^ decrease i n magnitude; thus the second term in AC 2 ^  tends to zero. The simulation results for various configurations indicated that for N>N^ , AC2 ^  may be assumed constant; i.e. the effect of the second term i n AC 2 ^  i s small. This implies that, for a l l practical purposes, may be assumed linear for N > N^ • Let C . = the minimum value of the total unit cost, min Snin o c c u r s a t ^ o n i y i f AC^ ^  > AC^ < AC^^^ where AC , <0 and AC„,.>0. For a l l N j N, , AC, „ = 0 while AC„ „>0; N-1 N+1 b 1 ,N I, N thus C . can not occur at N>N, . mxn b Let C . occur at some N . . For any specified production mm min curve the location of N . is dependent on the values of B. and B„. min 1 2 Figure 6.5 shows the values of N . in situations where B 0 is & min 2 fixed while B^ is varied, and where B^ i s fixed while B 2 is varied. 1 is the sum of the individual overtime hours after these are rounded off to the next half-hour as per labour agreement. 103 UNIT COST UNIT COST NUMBER OF TRUCKS ( N ) Figure 6.5 The unit cost and i t s components as functions of the number of trucks in the hauling fleet E. PRODUCTION vs. YARDER PER CENT UTILIZATION The yarder per cent u t i l i z a t i o n U^ i s defined as U = total time spent on the yarding function x 100 ^ total available time For a particular side the expected production P g i s given by P g = total time spent yarding x mean yarding rate = U x A x total time x mean yarding rate y»s y,s 3 100 100 (where A = per cent av a i l a b i l i t y of yarder s) y> s = U a where a = A x total time x mean yarding y,s s s __y 100 , 0° . rate of side s. Thus the total expected production for M sides i s given by M P = E P s=l M = E U a s , i y » s s S 1 100 U a where U = mean per cent u t i l i z a t i o n for the - jr y 100 yarders M a = E a . s=l 3 Assuming that the number of trucks dispatched to any given side i s proportional to the side's productivity, U^ g = U^ for a l l S. Thus production and yarder per cent unilization have a simple linear relationship. For N>N^ , U^ reaches a maximum which i s approximately 98% for the configuration examined with the model. The 2% accounts for the non-productive time spent i n moving and rigging the yarder. 105 In the i l l u s t r a t i v e application of the simulation model which w i l l be covered in the following sections, the unit cost response w i l l be emphasized. The unit cost is chosen as the index of performance in standard practice and in this study because i t provides a meaningful measure of the system performance. From the schematic production curve (Figure 6.2), i t is seen that the number of trucks at which production i s maximum is not unique, and that a rela-tively high production does not necessarily imply a low unit cost. For this reason, production i s an inferior index of performance of the system. For a similar reason, the yarder per cent u t i l i z a t i o n and the number of truck round trips are inferior indices of system perfor-mance. Since a high truck per cent u t i l i z a t i o n invariably implies that the system can s t i l l accomodate more trucks, the truck per cent u t i l i -zation i s also not a good index. These responses are nevertheless useful in explaining various unit cost phenomena. 6.2 SOME ILLUSTRATIVE APPLICATIONS OF THE SIMULATION MODEL In the sections to follow, some classes of problems which can be handled using a simulation approach w i l l be presented in a general context. Specific problems w i l l be given to i l l u s t r a t e the application of the model to these classes of problems. A. COMPARISON OF DIFFERENT OPERATING POLICIES For any given degree of mechanization, logging systems can be better designed through the examination of the different system operating rules and policies. In some specific situations, the cost of production can be reduced without an additional outlay of equipment 106 through the use of better operating policies. A comparison of different operating policies is illustrated in the following example. Sample Problem Two notable policies with regard to the shutting down of equipment at the end of the standard s h i f t are currently practiced in the West Coast. These shutdown modes were described earlier and are l a i d out in detail i n Appendix A. The major differences between the two shutdown modes may be summarized as follows; Shutdown Mode 1 Shutdown Mode 2 Dump shuts down as soon as a specified number of empty trucks are parked for the night Loaders shut down after 8 hours unless a truck i s waiting to be loaded or i s approaching the side for a load Yarders shut down after 8 hours unless a truck being loaded requires more logs for a reasonable load Trucks shutdown either at the camp or at the dump shuts down at a given time shut down after 8 hours of work shut down after 8 hours of work shutdown either at the camp or at the dump i f loaded, may be parked overnight, at at the landings or along roadsides i f empty The questions to be examined are: 1 . Under the conditions existing in a basic logging configuration, which shutdown mode gives a lower unit cost? 107 2. If the conditions are varied from those of the basic configurations, w i l l this shutdown mode consistently give a lower unit cost? As used in this thesis, the basic configuration i s defined as a 6-source, single-sink configuration whose sources are High Lead production sides and whose functional relationships, parameters, cost and yarding schedule are given in Appendices C and D. Logging trucks with 75,000 lbs. payload are used for this configuration. The other configurations examined i n this sample problem include: 1. A four-High Lead yarder configuration. 2. An eight-High lead yarder configuration. 3. An six-High lead yarder configuration whose settings are five miles closer to the camp then those of the basic configuration. 4. A six-High lead yarder configuration whose settings are five miles further from the camp than those of the basic configuration. For each of the five configurations examined, the number of trucks providing the least unit cost was determined f i r s t . A t-test was then made to test the significance of the difference in unit cost between the two shutdown modes. To find the number of trucks which provides the least unit cost for the configuration, a simulation run is done for each choice of number of trucks. Each simulation run consists of 75 autocorrelat-ed observations (simulated days). An additional run is made i f the 1 0 8 f i r s t run has failed to provide the desired standard error of under 2% of the mean unit cost. However, the more variable extreme points selected to complete the shape of the unit cost curve are exceptions. In these cases, a standard error of up to 5% of the mean i s allowed. The procedure used for making the runs i s as follows: 1. Run the model of the particular configuration 3 times i n i t i a l l y using a different number of trucks each time. Previous runs for the other configura-tions may be helpful in providing a good guess on what number of trucks to use. 2 . The number of trucks providing the least unit cost is N i f C ^ _ ^ > C N < C J J + ^ . If the runs made have not included the minimum, make additional runs. The unit cost derived from previous runs should be helpful i n locat-ing the minimum point. Discussion of results The unit cost curves and the production curves for the basic configuration are shown respectively in Figures 6 . 6 and 6 . 7 . Figure 6 . 6 shows that for the number of trucks ( N ) $ 1 0 , the second shutdown mode yields a lower unit cost. The t-test made at N = 1 4 shows that the difference of 0 . 5 4 $/cunit between the two modes i s highly significant. A breakdown of the unit cost into i t s component unit costs (Figure 6 . 8 ) shows that the difference in the unit trucking costs accounts largely for the difference in the total unit cost. Since, for the f i r s t mode, the trucks have to be shutdown either at the camp or at the dump, a 109 greater number of overtime hours results. This made the difference in the unit trucking cost although extra travel time costs for the truck drivers were accounted for in the model for the second mode. The f i r s t mode, however, tends to produce a higher number of round trips and i t i s probably for this reason that i t is being employed. Because of the relatively higher number of round trips for the f i r s t mode, i t i s seen in Figure 6.7 that for N < ^ the f i r s t mode yields a higher production. Consequently, the unit yarding and loading costs for the f i r s t mode are i n i t i a l l y lower than for the second mode (Figure 6.8). Nevertheless, i f the trucking production can be improved, then this implies that the productivity of the trucks, rather than of the yarders, is the limiting factor. An increase i n the number of trucks, in this case, w i l l not only improve production but w i l l also reduce the unit cost. Indeed at N > 14, the production for both shutdown modes becomes identical; yet the second mode yields a lower unit cost. It i s seen from Figures 6.9 and 6.10 that the same observa-tions apply when: 1. The number of yarders is varied, and 2. the distances of the settings from the camp is reduced or increased by five miles. Table 6.1 shows that, in a l l the five configurations examined, the second shutdown mode consistently yields a significantly lower unit cost than does the f i r s t shutdown mode in the given region of interest (about N = N . ). mm 110 7 8 9 10 11 12 13 14 15 16 17 18 19 20 N U M B E R O F T R U C K S I l l Figure 6.8 The unit yarding, loading, trucking, and unloading costs for the basic configuration using different shutdown modes 112 N U M B E R O F T R U C K S Figure 6.9 The unit cost and production curves for the 4- and the 8-yarder configurations using two different shutdown modes 113 22 • -21 • " 20 19 18 -• 17 -16 .. 15 14 •• 13 350 300 •• 250 •• 200 shutdown mode 1 shutdown mode 2 Basic configuration sX ssettings 5 miles farther + 4- 4- 4- 4- 4-10 11 12 13 14 15 16 17 N U M B E R O F T R U C K S 18 19 20 Figure 6.10 The unit cost and production curves for the basic confi-guration using two different shutdown modes when the settings are 5 miles farther and when the settings are 5 miles closer to the camp 114 Table 6.1 Production and unit cost for the two shutdown modes for various configurations Configuration/ No. of trucks Production (Cunits) Mode 1 Mode 2 Unit Cost ($/Cunit) Mode 1 Mode 2 U.Cost Diff. < w %-c2 Basic configuration 13 14 15 365.80 378.49 381.42 360.38 379.32 381.56 15.28 15.11 15.19 14.96 14.57 14.70 0.32 0.54** 0.49** 0.17 0.07 0.06 4 High Lead Yarders 10 11 251.43 254.59 251.88 254.30 15.49 15.55 14.88 15.06 0.61** 0.49** 0.15 0.07 8 High Lead Yarders 18 19 20 504.77 526.91 530.03 495.82 510.28 525.56 14.75 14.56 14.64 14.39 14.33 14.14 0.36** 0.23* 0.50** 0.08 0.12 0.02 6 High Lead Yarders, 5 miles closer 10 11 12 353.58 371.08 382.21 347.68 374.43 380.82 14.47 14.03 13.91 14.13 13.40 13.48 0.34 0.63** 0.43** 0.29 0.21 0.05 6 High Lead Yarders, 5 miles farther 16 17 379.13 381.60 372.44 380.83 16.24 16.35 15.62 15.60 0.62** 0.75** 0.07 0.06 * -significant ** -significant at 95% at 99% 115 B. DETERMINATION OF THE EQUIPMENT REQUIREMENT FOR VARIOUS  CONFIGURATIONS An important area of concern f o r l o g g i n g manager i s the determination of the equipment requirements f o r t h e i r o p e r a t i o n to a t t a i n a s p e c i f i e d l e v e l of pr o d u c t i o n . The c l a s s of problems d e a l i n g w i t h equipment requirement not only deals w i t h the s e l e c t i o n o f the d e s i r e d combination o f equipment, but a l s o the mode f o r u s i n g them. For i n s t a n c e , to in c r e a s e p r o d u c t i v i t y a l o g g i n g manager may have s e v e r a l a l t e r n a t i v e s a t h i s d i s p o s a l , e.g. 1. purchase or le a s e more equipment, 2. h i r e c o n t r a c t o r s , 3. operate the e x i s t i n g equipment overtime or on double s h i f t s . On the other hand, to reduce p r o d u c t i v i t y a l o g g i n g manager needs to l a y o f f some equipment. Which pi e c e of equipment should he l a y o f f ? As examples of t h i s c l a s s of problem, the f o l l o w i n g are considered: 1. The determination of the truck requirements f o r d i f f e r e n t numbers of High Lead production s i d e s as the d i s t a n c e s of the s e t t i n g s get p r o g r e s s i v e l y f u r t h e r from the camp. 2. The determination o f the tr u c k requirement f o r a 5-High Lead, 1-Grapple yarder c o n f i g u r a t i o n . 3. The determination of the truck requirement f o r the b a s i c c o n f i g u r a t i o n u s i n g d i f f e r e n t combinations of " s m a l l " and " l a r g e " t r u c k s . 116 The truck requirement for different number of High Lead production sides and setting distances In a newly established multi-source, single-sink logging operation, the sources or production sides are usually close to the camp. In the course of several years of operation, the closer settings are logged over; hence the production sides get progressively farther from the camp. relatively fewer production sides and additional sides are opened as soon as a more favorable weather condition permits. Also, an increase in demand for wood may have the same effect of causing an increase in the number of production sides. In this example, the combined effect of these two factors - the distances of the production sides from the camp and the number of production sides - on the truck requirement of the logging operation, as well as on unit cost and production, w i l l be explored. Experimental design -and since each yarder moves from one location to another, i t i s neces-sary to define a single measure of distance for each configuration to be used as one of the independent variable. This measure - the mean setting distance ( D ) for any specified configuration - i s defined as In the winter, a logging operation generally starts with Since a l l yarders have different locations at any one time, / where n = the number of yarders m. = the number of locations scheduled for yarder i 117 D. . = the distance from the camp X^ of the jth location of yarder i V.. = the volume of the jth setting of yarder i Three levels of D for each number of yarders are used, designated by )S -5, * , and +5 where & is the mean setting n n n n distance for n yarders whose individual setting distances are given in Appendix C. Four levels of the number of yarders are used: 4, 5, 6, and 7 yarders. For each of the configurations, or c e l l s , resulting from the combination of different levels of D and the number of yarders, the truck requirement was determined using the procedure described i n the previous section. partition the response surface into two sections, since as previously described, the unit cost as a function of the number of trucks has linear and non-linear components. The model i n i t i a l l y chosen for the f i r s t section of the unit cost response surface i s a second order model of the form To f i t a unit cost response surface, i t i s necessary to where N = the number of trucks M = the number of yarders D = the mean setting distance from the camp C = the unit cost i=0 , . • • , 13 • the regression coefficients for the region defined by a l l the specified levels of 118 D and M, and N. <N + 1, i=4,...,7, 1 = * -5, Since N i s different for each configuration i j , the values of OP i j • N „ are staggered; thus the design i s not orthogonal. However, this design i s necessary to ensure that the N Qp t's derived from the response surface are close to the observed N 's. Three levels of N are used opt for the four- and five- yarder configurations. For the six- and seven-yarder configurations, four levels of N are used since the region i s wider for these two cases. Thus a total of 2x3x3 + 2x3x4 = 42 simulation runs were used to f i t the f i r s t section of the unit cost response surface assuming that only one simulation run i s made for each combination of factor levels. To obtain a "smooth" surface when the two sections of the unit cost response surface are joined, the slope of the linear sections are f i r s t determined using a model of the form S(M,D) =qQ + qjM + q2D + q^ MD + q^M2 + q D 2 + qgM2D + e where S(M,D) = the slope of the linear section of the unit cost function for the configura-tion defined by M and D q^ = the regression coefficients. The unit cost for k trucks, i yarders, and mean setting distance j is then determined using K 1 J opt. . + 1 To f i t S(M,D) a total of 2x3x4 = 24 simulation runs were made. Discussion of results -119 After a stepwise elimination of the non-significant variables, the best regression equation for the second order section of the unit cost response surface i s C = 12.587 - 0.1005ND + 0.2570MD - 014471NM - 010085M2D +• 0.1472N2 + 0.0145D2 where M = the number of yarders D = the mean setting distance from camp N = the number of trucks and N ^ N +1. "I'm A l l the regression coefficients are highly significant in the above regression equation which accounts for 97.455% of the total variation. Similarly, the best equation obtained for S(M,D) i s S(M,D) = -0.9972 + 0.1201D - 0.0478MD -0.0509M2 = 0.0046M2D . The regression equation for S(M,D) accounts for 98.258% of the total variation. A l l of i t s regression coefficients are highly significant. For N >N _ +1. the unit cost response surface i s given by °P tMD C = C(N fc + 1, M, D) + N«S(M,D) where the f i r s t term is obtained through the model for the second order section. Figure 6.11 shows the unit cost response surface as a func-tion of the number of trucks (N) and the mean setting distance (D) for each of the 4-, 5-, 6-, and 7-yarder configurations. The follow-ing can be observed from Figure 6.11: 120 7-yarder configuration 6-yarder configuration Figure 6.11 The unit cost response surface over different values of the number of trucks and the mean setting distance for the 4- , 5- , 6- , and 7-yarder configurations 121 1. For any M-yarder configuration, when there are excess trucks, changes i n the mean setting distance have less effect on unit cost than when there are too few trucks. 2. Let A C J J ^ = the change in unit cost per unit increase in the number of trucks (N) for a given M-yarder con-figuration. In section 6.1 i t was shown that for any given M-yarder configuration and for N>N, * A C J J where TS^ = t n e added cost of an ' max,M extra truck P ,, = the maximum daily max,M , J production for the M-yarder configuration. Since B 2 is a constant and < ? m a x ^ < P m a x > 6 < P m a X ) 7 , i t follows that > C N > 5 > C N > 6 > C N J ? for any fixed D and for N > N^. This implies that the slope of the linear section of the unit cost curve for the confi-guration MD i s relatively less for greater number of yarders M. 3. At the "second order section" of the unit cost surface . > | A C N J for i > j . This implies that for this section, the addition of a truck reduces the unit cost more when there are more production sides. 4. For any given D, the 7-yarder configuration gives the least unit cost at N The 6-yarder configuration opt gives the next least unit cost. This i s brought about (i.e. N<N ), opt 1 N i s assumed equal to N ^ + 1 for these response surfaces, b opt 122 by two factors: a. Decreasing unit unloading cost. Since only one unloading f a c i l i t y i s used in a l l configurations, more production implies less unit unloading cost. b. Higher average productivity assigned to the sixth and seventh yarders. In the concluding paragraph of Chapter IV i t was stressed that results from the various model runs are, themselves, bases for build-ing confidence i n the model. The above observations have shown what can be regarded as an in t u i t i v e l y acceptable behavior of a system response. However, perhaps the greatest benefit from the results can be derived from their indication of the level and the rate of the response for various combinations of the given factors. Without making additional simulation runs, the truck require-ment for various mean setting distances of an M-yarder configuration, M=4,5,6,7, whose individual yarding potential i s not the same as those already simulated, may be determined. This requires a graph (Figure 6.12) showing N ^ as a function of the production at N _ and the mean ° opt r opt setting distances. A graph such as this can be constructed from the unit cost and production response surfaces.* The points in the graph The production response surface used to construct Figure 6.12 is given by _ = "141.097 + 1.1985ND + 9.7064NM -0.2020M2N - 2.0185N2 for N $ N _ o p t « 5 max, - for N > N A sample plot of this response surface for M = 6 yarders i s shown in Figure 6.13 123 H 500-r 1 • I " I 1 ! 1 r 1 1 1 1 ' • | 8 9 10 11 12 13 14 15 16 17 18 19 20 MEAN SETTING DISTANCE FROM THE CAMP (MILES) Figure 6.12 The truck requirement for various combinations of estimated d a i l y production and mean setting distance Figure 6.13 The production response surface for the basic configuration 124 are taken from these response surfaces. The points corresponding to the same number of trucks, say N, determine the band of optimum applicability for N trucks. A graph such as Figure 6.12 can be used to find the addition-a l truck requirement to match an increased yarding productivity, or to find the number of trucks to lay off as a result of a reduced yarding productivity. However, the graph applies only for the costs assumed in the model. Also, the effectiveness of the graph i s not known for configurations whose variance of their setting distances i s much different from that of the model. The use of Figure 6.12 i s illustrated by an example in the section to follow. Truck requirement for a 5-High Lead, 1-Grapple yarder configuration The logging equipment manufacturer's answer to the problem of reducing logging costs is the design of new machinery which is not only more effici e n t , but which requires less manpower as well. One of such machines produced i s the Grapple yarder. A description of some early developments of the Grapple yarder is given by Sommer (1969). The purpose of this example i s to i l l u s t r a t e the change in the truck requirement resulting from the introduction of a highly productive machine such as the Grapple yarder. In this hypothetical example, i t i s assumed that a Grapple yarder.is added to five High Lead yarders already In operation. The total daily yarding production for the five High Lead yarders and the Grapple yarder is given by 125 P = P(for the 5 yarders) + U A x (the potential daily y y productivity of the Grapple yarder) where A^ = the yarder % ava i l a b i l i t y assumed at 99% U = the yarder % u t i l i z a t i o n assumed at 98% ^ at N trucks, opt Using the mean of the Grapple yarder autoregressive model (given in Appendix C) as the potential Grapple yarder daily productivity, P was calculated as 423 cunits per day. The mean setting distance from the camp (D)for the settings scheduled for the six yarders is 14.6 miles. From Figure 6.12, i t i s seen that 15 trucks w i l l be required for this 5-High Lead, 1-Grapple yarder configuration. A series of simulation runs for this configuration arrived at 15 trucks as having the least unit cost. The unit cost and production as functions of the number of trucks for this configuration, for the original 5-yarder configura-tion, and for a 6-High Lead yarder configuration are shown in Figure 6.14. It i s seen that unless three or four additional trucks are provided to balance the added production of the Grapple yarder, the f u l l benefit of the introduction of this yarder w i l l not be realized. At 12 trucks, the 5-High Lead, 1-Grapple yarder operation produces only an extra 38 cunits and at a higher cost. At 16 trucks the same operat-ion produces around 417 cunits at 14.45 $/cunit. In comparison, a similar 6-High Lead yarder operation with 14 trucks produces 378 cunits at 15.11 $/cunit. Truck requirement for the basic configuration using different  combinations of "small" and "large" trucks 126 i J V> 1 f 1 1 1 1 1 1 1 1 1 1 } 8 9 10 11 12 13 14 15 16 17 18 19 20 450 " * N U M B E R O F T R U C K S Figure 6.14 The unit cost and production curves for a 5-High Lead yarder configuration, a 5-H.L.-l-Grapple yarder confi-guration, and a 6-High Lead yarder configuration 127 The presentation, so far, involved the use of only one type of truck referred to in this study as "small" truck with a 75,000 lbs. design payload. A second type of truck considered i n the model has a payload of 100,000 lbs. While the purchase price and operating costs of these "large" trucks are higher, these are more than offset by their increased capacity. In this section, the effect of using d i f f e r -ent combinations of "small" and "large" trucks on unit cost and production w i l l be explored. Table 6.2 shows an estimate of the purchase price, operating cost, capacity, and average loading time for the two types of trucks. Although separate loading time and load volume distributions are used in the model for these two truck types, the same travel time d i s t r i b -utions are used. The available data on the travel times for the "large" trucks indicated no significant difference between the travel times of these two truck types. "small" trucks "large" trucks Purchase price ($) 47,000 69,000 Depreciation and insurance ($/shift) 27 40 Operating cost (fuel, lube, tires, supplies, etc.) ($/mile) 0.77 0.89 Average load (cunits) 13.16 17.88 Average loading time (min) 38.93 40.83 Table 6.2 Estimates of the purchase price, operating cost, capacity, and average loading time for "small" and "large" trucks 128 Figure 6.15 summarizes the unit cost and production results of the simulation runs made with the basic (6-yarder) configuration using various combinations of "small" and "large" trucks. For the costs and functional relationships assumed in this study, Figure 6.15 indicates that' a lower unit cost can be realized when a fleet of a l l "large" trucks i s used. For any number of trucks N, the values for any system response for a fleet of a l l "small" trucks and for a flee t of a l l "large" trucks respectively represent the extreme values. That i s , for any N, either R <: R < R, or R > R > R, holds, where s ** c v 1 s c •* 1 R = the vaule of the response R when a fleet of a l l "small" trucks i s used R^  = the corresponding value when a fleet of a l l "large" trucks i s used R = the corresponding value when a combination of "small" and "large" trucks i s used. For instance, i n Figure 6.15 at N = 10, C > C > C. and P < P < P,, s c i s c i but at N = 15 C < C < C. and P = P = P.. s c i s c i The unit cost curve using "small" trucks and the unit cost curve using "large" trucks intersect at some point i on the N-axis. It can be shown that at N=i the unit cost i s the same for a l l combinations of "small" and "large" trucks; that i s , C = C =C, . At N=i, while 0 s c i the total fixed cost for a fleet of "small" trucks i s lower than for a fleet of "large" trucks, i t takes more round trips for the "small" trucks to yield the same production as the "large" trucks. Hence, the 129 fleet of a l l "small" trucks fleet of 4 "large" and the rest "small 1 trucks 7 8 10 11 12 13 14 15 16 17 18 19 20 400-H c_> 2 O =3 O 8 PL. M < 350 300--fleet of a l l "small" trucks fleet of 4 "large" and the rest "small" trucks fleet of a l l large trucks 25Q> i 1 1 r- I I I I H r 7 8 9 10 11 12 13 14 15 16 17 18 19 20 N U M B E R O F T R U C K S Figure 6.15 The unit cost and production curves for the basic configuration using different fleet of trucks 130 lower f i x e d cost for the "small" trucks i s n e u t r a l i z e d by a higher v a r i a b l e cost. Figure 6.16 shows the i s o - u n i t cost curves over d i f f e r e n t Figure 6.16 Iso-unit cost curves over d i f f e r e n t combinations of " l a r g e " and "small" trucks C. SENSITIVITY ANALYSIS: THE RELATIVE EFFECT OF SOME EQUIPMENT  AVAILABILITY PARAMETERS ON PRODUCTION The sample a p p l i c a t i o n s presented thus f a r have examined the e f f e c t on u n i t cost and production of se v e r a l f a c t o r s , namely: 1. number of trucks 2. truck type 3. number of yarders and yarder p r o d u c t i v i t y 4. mean s e t t i n g distance 131 5. shutdown mode. While these applications are, in effect, analyses of the sensitivity of the system to the l i s t e d factors, the approach has been to examine a class of factors while keeping the rest of the factors fixed at their respective specified levels. This approach, of course, has a major disadvantage in that i t cannot provide a measure of the inter-actions between the various levels of the "fixed" factors and the levels of the "varied" factors. This approach, however, has been necessitated because of the problem of too many factors. As Conway, et.al. (1959) have pointed out, while in principle simulation can be used to investigate the effect of any conceivable factor, relation-ship, or policy, in practice this results i n f a c t o r i a l experiments whose dimensions dwarf the most powerful computer or the most lavish budget. Because of this problem of size, a complete factorial experiment becomes a luxury. The grouping of factors into classes according to the nature of the investigation becomes a necessity. In the discussions to follow, the relative effect of some equipment a v a i l a b i l i t y parameters on production w i l l be examined. While the model may be used to examine particular problems*, the purpose of this investigation is to rank the various a v a i l a b i l i t y parameters according to their effect on the production of the basic configuration. 1 An example i s the following: Is a machine with relatively low productivity but with high a v a i l a b i l i t y more economical than a potentially more productive machine which produces disruptive surges of wood through the system? 132 Equipment a v a i l a b i l i t y can be measured i n terms of the equipment breakdown and repair parameters, and respectively. In the following example, the effect on production of four parameters w i l l be examined. These parameters are: 1. ^ t = the mean truck inter-breakdown time 2. P r t = the mean truck repair time 3. V = the mean yarder inter-breakdown time 4. ^ 1 = the mean loader inter-breakdown time. Noting that the aim of this example is to rank the above parameters according to their effect on production, a significant reduction in the number of simulation runs can be realized i f only the main effects are estimated while the interaction effects of the various levels of the parameters are sacrificed. Hence, a one-ninth 4 replicate of a 3 design i s used. Nine runs are required for this design, namely: 0000 1011 2022 0121 1102 2110 0212 1220 2201 where in each run the four numbers indicate the levels of the factors ^by» ^ t ' and y r t f respectively. The values of these levels are shown in Table 6.3. 133 Table 6.3 The levels used for each of the four equipment ava i l a b i l i t y parameters. " h i ^ y ^bt y r t Level (Thousand minutes) 0 24 15 10 0.7 1 . 30 20 16 1.0 2 36 25 22 1.3 The i n i t i a l regression model used was P = ao + a l y b l + a 2 V + a3**bt + a 4 V r t + ^ l 2 + ^^by 2 + a 7 U b t 2 + a8vrt + e« After a stepwise elimination of the non-significant variables, the regression equation obtained i s P = 374.223 + 0.4672ubt - 5.1579yrfc2. This equation accounts for 73.96% of the total variation. The regression analysis, therefore, indicated that for the basic configuration and for the given range of the levels of the different factors, the mean truck repair time has the greatest effect on production among the factors examined. The mean truck inter-breakdown time also significantly affects production, but the mean yarder and loader inter-breakdown times are indicated to contribute no significant effect on production. Noting that the levels of and y^ are quite high compared to the levels of Ujjj.* i t can be concluded that the av a i l a b i l i t y of the yarders and the loaders is sufficiently high so that changes in y ^ and y^ do not result in significant changes in production. Further examination of y, _ and y _ was undertaken to invest-bt r t igate their interaction effect. Using a f u l l factorial design, six 134 simulation runs were made to collect sufficient data for the model:* P = ao + V b t + a 2 V r t + a 3 V V r t + C " The regression equation obtained i s P= 433.523 - 2.4730ubt - 59.8124prt + 2.7395ubt r f c. A l l regression coefficients i n the above equation are significant. The equation accounted for 96.58% of the total variation. Figure 6.17 shows the daily production response surface over the values of the mean truck repair time and the mean truck inter-breakdown time. It i s seen that the higher the value of the mean truck inter-breakdown time i s , the lesser is the effect on production of the mean truck repair time. On the other hand, the higher the value of the mean truck repair time i s , the greater i s the effect on production of the mean truck inter-breakdown time. 1.0 Figure 6.17 Production response surface for the basic configuration over different values of and u 1 Two extra runs were made to enable the examination of a wide range of p b t . The levels used were 1.0 and 1.3 thousand minutes for p and 10, 16, and 22 thousand minutes for u^ . r t 135 CHAPTER VII SUMMARY AND CONCLUSIONS This thesis has described a methodology for examining problems associated with the management and control of forest harvesting operations. The methodology developed accounts for the relevant characteristics inherent i n forest harvesting systems in that: 1. i t preserves the probabilistic nature of these systems, which dictates the interplay of the variables that govern the system behavior; 2. i t preserves the dynamic character of these systems thus allowing the interplay of the system variables} 3. i t preserves the inherent response-factor relationships which account for the inter-actions of the system variables; and 4. i t preserves the self-regulating behavior of these systems. The methodology developed i s one of a systems simulation with general applicability that permits experimentation with a wide class of logging configurations. Of modular structure, the model developed is capable of simulating multi-source, single-sink 136 configurations with variable internode distances, with various equipment types and combinations, and with various parameters and functional relationships. Written in FORTRAN IV, the model allows independent users to readily make modifications in the routines to adapt them to the particular operating rules and policies of their operations. Furthermore, the model is capable of fast execution; thus i t permits extensive experimentation at a manageable cost. The validity of the model has been tested and demonstrated for an actual West Coast logging division used as a vehicle for model formulation. The verification procedure involved the independ-ent examination of the assumptions and rules of operation of the model subsystems, and the h i s t o r i c a l confirmation that for a particular situation the subsystems together made up a system which displays the behavior and characteristics associated with the real system. The hi s t o r i c a l verification involving two outputs of the system, namely daily yarding production and number of daily round trips, showed that the model can "process" the production "input series" to produce a resultant series which i s compatible with the series observed for a real situation. While no model can be said to be valid in the absolute sense, confidence i n the model rests largely on the confirm-ation of i t s a b i l i t y to display the behavior and characteristics expected of the real system. A simulation model i s constructed i n the hope that i t w i l l successfully mimic a particular real world system so that inferences on the real system can be made through the model. As Hunter and 137 Naylor (1969) pointed out, one's understanding of a complicated and involved system through a complex model such as a simulation proceeds only when one is able to synthesize the system in terms of simple explanations. This synthesis requires the identification of the major factors affecting the responses and the evaluation of the empirical relationships associating the factors with the responses. In other words, simple empirical models are super-imposed on the larger detailed simulation models. Conclusions on the real system can then be derived once an empirical understanding has been acquired. The investigation of the various system responses has established some simple relationships among the responses in associa-tion with a principal factor - the number of trucks. The comparison of the responses led to the choice of unit cost as the measure of system performance with production as a constraining response. The execution of the model runs has been discussed. Some experimental design problems and some tactical considerations have been presented. Some ways for overcoming such design problems have been suggested. In particular, i t has been shown in the text and presented i n more detail in Appendix F.3 that the control variate technique can be effectively used with the model to reduce the variance of the difference between two means. In addition, i t has been shown that the method of antithetic variates can be used to further reduce the variance of the mean of a response when a second replication i s necessary. Some practical applications of the logging simulation model have been discussed and illu s t r a t e d . The model can be used to 138 evaluate and compare existing operating policies or to formulate new policies. This application has been illustrated with reference to the comparison of two operation shutdown modes. The results showed that while the f i r s t shutdown mode results in more production than the second shutdown mode, this i s true only when the number of trucks is less than the optimum number required. The higher unit cost of the f i r s t mode resulting from a larger number of truck overtime hours makes i t inferior to the second mode. The model has also been shown to be useful i n the deter-mination of equipment requirements under different operating conditions, using a simulation approach, the equipment requirements for a particular operation are given not merely in terms of a required prod-uctivity per unit time, but i n terms of an explicit statement of the type and combination of equipment. The model can also be used to compare the costs and pro-ductivity of an operation using trucks of different capacities. This capability can be u t i l i z e d in the development of trucking policies and in truck design, addressed to the question of what truck specifications best suit a given operation. Perhaps the greatest benefit from a simulation model can be derived from i t s capability of increasing our understanding of the system - through learning how the parts of the system behave and interact and through learning how the system responds to changes in i t s factors. These capabilities can be beneficial not only in the design of better policies but also in the exercise of better control of the system. 139 Simulation has been regarded by many as a last resort technique, perhaps rightly so, because of the many d i f f i c u l t i e s inherent in a simulation approach. Simulation requires extensive data and complete information about the processes of the system. As Mi l l e r and Starr (1969) have pointed out, simulation is not a substitute for knowledge. The misuse of simulation has largely been due to the forgetting of this crucial fact. Simulation i s based on the premise that much is known about the parts of the system, but not how they interact to display the over-all system behavior. D i f f i c u l t i e s also arise i n the formulation of the model. Logging problems are diverse in nature, and no general model i s like l y to be developed that can handle a l l classes of logging problems. Since the degree of resolution of detail in a representation i s problem-dependent, the task of building a general model w i l l be set back by the problem of determining the proper balance between faith-fulness of reproduction and simplicity of representation while attain-ing the f l e x i b i l i t y to describe any specific problem. In the absence of an all-inclusive model, there may well be different models for different classes of problems. This study has been based on the real-ization that some classes of problems, for instance those involving the evaluation of different system designs, policies, and system input combinations, require adequate realism provided by a descrip-tive type of model. D i f f i c u l t i e s also stem from the nature of simulation. A simulation model can be equated to a manager's experimental laboratory. Hence, i t i s subject to some methodological problems concerning: 140 1. Validity. The desription of a system by a model invariably carries with i t some assumptions on the properties and behavior of the system. Confidence and use of the model should be based on the aware-ness of these assumptions. 2. Motive. Simulation is more suitable to problems involving relationships between a cause and i t s effects. Simulation i s not inherently optimizing, optimization must be superimposed on the model by varying the level of the pertinent factors one at a time. In essence, this i s translating the problem into a cause-and-effeet structure. 3. Var i a b i l i t y and size. There is a need to carefully consider the design of the experimental runs. As in real world experimentation, simulation is beset by the same experimental design problems of keep-ing the amount of experimental effort down to a manageable level in obtaining the desired precision of the results. Further developments of the model should be made in two areas: 1. Revision. Revisions based on extensive sensitivity analysis can be made on the model. Through s e n s i t i -vity analysis, the factors which cause more profound changes in the system behavior can be identified. On this basis, more aggregation and simplification can be made on the routines of the model. Extension. The model presented in this thesis describes a system as i t i s , rather than what i t should be or could be. A crucial need, however, l i e s in the design of a better system of handling the product flow. Further developments should be directed towards satisfy-ing this need. 142 BIBLIOGRAPHY Bare, B.B. • User's Manual for the Purdue University Forest Management Game, Seattle, Washington: University of Washington Centre for Quantitative Science i n Forestry, Fisheries and Wildlife, March, 1971. Box, G.E.P. and Jenkins, G.M. TIME SERIES ANALYSIS forecasting  and control. San Francisco: Holden-Day Inc., 1970. Carlsson, B. "Routines for Short-range Planning of Logging Operations." Skogsarbeten, Bulletin No 5. (1968). Clutter, J.L. and Bamping J.H. "Computer Simulation of an Industrial Forestry Enterprise." Proceedings, Soc. of Amer. For.»(1965). Conway, R.W. "Some Tactical Problems in Digital Simulation." Mgt. Science, 10 (1) (October, 1963), pp.47-61. Conway, R.W., Johnson, B.M., and Maxwell, W.L. "Some Problems of Digital Systems Simulation." Mgt Science, 6(1) (October, 1959), pp. 92-110. Curtis, F.H. "Linear Programming the Management of Forest Property." 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Fishman, G.S. and Kiviat. P.J. "The Analysis of Simulation-generated Time Series." Mgt. Science, 13(7) (March, 1967), pp. 525-557. Forrester, J.W. Industrial Dynamics. Cambridge, Mass: The M.I.T. Press, 1961 .. Gillam, D.S. "The Application of Computers in Woodland Operations!' Pulp and Paper Magazine of Canada, June 7, 1968, pp. 102-103. Hammersley, J.M. and Handscomb, D.C. Monte Carlo Methods. London Methuen & Co. Ltd., 1964. Hare, V.C. Jr. Systems Analysis: A diagnostic approach. N.Y. : Harcourt, Brace and World, Inc. 1967. Hunter, J.S. and Naylor, T.H. "Experimental Design." The Design  of Computer Simulation Experiments. Edited by Naylor, T.H.,Durham, N.C.: Duke University Press 1969. Jenkins, G.M. and Watts, D.G. Spectral Analysis and Its Applications. San Francisco: Holden-Day, Inc. 1968. Knuth, D. The Art of Computer Programming. Vol. 1. Reading, Mass.: Addison-Wesley Publishing Company Inc., 1968. Kleijnen, R.H. "Monte Carlo Techniques: A comment." 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The Design of Computer Simulation Experiments. Edited by Naylor, T.H., Durham N.C.: Duke University Press, 1969. Myers, R.H. Response Surface Methodology. Boston, Mass.: Allyn and Bacon, Inc., 1971. Naylor, T.H. et a l . Computer Simulation Techniques. N.Y. John Wiley & Sons, Inc., 1968. Naylor T.H., Burdick, D.S., and Sasser, W.E. Jr. "The Design of Computer Simulation Experiments." The Design of  Computer Simulation Experiments. Edited by Naylor, T.H., Durham, N.C.: Duke University Press, 1969. Naylor, T.H. and Finger, J.M. "Verification of Computer Simulation Models." Mgt. Science, 14(2) (October, 1971), pp. B-92 - B-101. Newnham, R.M. "Minimum Merchantable Tree Size and Machine Productivity -a simulation study.** Pulp and Paper Magazine of Canada, Convention issue (1968), pp 227-229. O'Regan, W.G., Arvanitis, L., and Gould E.M. Jr. "Systems, Simulation and Forest Management." Proceedings Soc. of Amer. For., (1965). Pritsker, A.A.B. and Kiviat, P.J. Simulation with GASP II a FORTRAN  based simulation language. Englewood C l i f f s , N.J.: Prentice-Hall, Inc., 1969. Sommer, H. "Grapple yarding and Thinning with a Running Skyline: The Shamley System." Skyline Logging Symposium Proceedings. O'Leary, J.E. ed.,Corvallis,Oregon:O.S.U. School of Forestry, 1969. 145 Teichroew, D . , and Lubin, J . , "computer Simulation - discussion of the technique and comparison of languages." Comm. of the A. C M . , IX (10) (October, 1966), pp.723-741. Theiler , T . C . "Linear Programming and Optimal Cutting Pract ices . " Paper Industry, 41(6) (1959), pp. 384-386, 391. Turner, W.I.M. "In Pursuit of Lower Costs." Pulp and Paper Mag. of Canada, February, 1972, pp. 77-79. United Sates Department of the Interior , Bureau of Land Management, Portland Service Center. Logging Costs. May 15, 1967. Vandenboom, H.W. A Technique for Analysing the Effect of Absolute  Vehicle and Component Age on the Failure Behavior  and Repair Costs of Logging Truck Components. Unpublished Bachelor's thesis, U . B . C . , 1971. Van Horn, R. " V a l i d a t i o n . " The Design of Computer Simulation  Experiments. Edited by Naylor, T . H . , Durham, N . C . : Duke University Press, 1969. Wagner, H.M. Principles of Operations Research. Englewood C l i f f , N . J . : Prentice-Hall Inc., 1969. 146 APPENDIX A START-UP AND SHUTDOWN MODES A.l START-UP SEQUENCE The start-up sequence of acti v i t i e s of the operation at the beginning of each day i s as follows: At 6:30 the shop crews leave the marshalling yard for the camp. At 7:00 the empty trucks at the camp and at the dump start up. At 7:20 the dump starts unloading the trucks parked over-night at the dump i n their loaded state. At 7:45 the trucks parked overnight at the side of the road start up. At 8:00 yarding and loading start. A. 2 SHUTDOWN MODE I The shutdown sequence for the f i r s t shutdown mode near the end of the regular s h i f t i s as follows: After 2:45 P.M any truck arriving at the dump is shut down. Assuming that there are n production sides, the f i r s t n trucks arriving after this time are un-loaded before they are shut down. A l l other arrivals are shut down in their loaded state. 147 After 3:00 P.M. no trucks are dispatched to the side except when required to support the moving of a yarder or a loader. At 4:25 P.M. yarding stops unless a truck being loaded requires more logs to complete the load. At 4:30 P.M. the yarding and loading crews leave for the marshalling yard. If there are trucks wait-ing to be loaded at the side or approaching the side for a load, the loading crew stays u n t i l these trucks are loaded. Thus no trucks stay overnight at the side. A.3 SHUTDOWN MODE II The regular s h i f t shutdown sequence for the second shutdown mode is as follows: After 3:00 P.M. no trucks are dispatched to the side except when required to support the moving of a yarder or a loader. At 3:20 P.M. the dump closes. Unloading goes beyond this time only i f the truck i s scheduled for check-up or for lowbed duty. At 4:25 P.M yarding and loading stops. If a truck i s being loaded, loading stops and i s completed the next day. A l l trucks queueing at the side stay overnight at the side. At. 4:45 P.M. trucks travelling to the side are parked by the roadside. All.loaded trucks, however, continue to the dump (or to the camp i f i t i s scheduled for check-up). 148 APPENDIX B THE EVENT SCHEDULER The event scheduler maintains a l i s t of a l l events scheduled to occur. The event l i s t stores the time of occurrence and other attributes of a l l scheduled events. The items in the l i s t form a "singly-linked l i s t " , the link being the array which chrono-logically links the items in the l i s t by indicating, for each event, the address of the next chronological event. There are three basic operations connected with the event scheduler namely: i n i t i a l i z a t i o n , event insertion, and event deletion. B.l INITIALIZATION Figure B.l shows the flow chart of the i n i t i a l i z a t i o n opera-tion. This operation sets up a l l the arrays needed by the scheduler. It i s performed only once and i t i s initia t e d by the calling sequence CALL ISCHED (1). The arrays set up include: 1. TMLIST, which contains the time the scheduled events are to occur. 2. LISTTP, three arrays which contain the attributes of the scheduled events. 3. KPOINT, the array which links the events chronologically. 149 4. IACVEC, which contains the address of the storage spaces available for event insertion. B.2 EVENT INSERTION An event insertion i s the inclusion of an event into the events l i s t s . The calling sequence CALL SCHED (2,TIME,NTYPE) causes the scheduler to search through TMLIST to determine the chronolo-gical order of the event described by the variables TIME and NTYPE. The array KPOINT i s then updated to accomodate the event, and the contents of TIME and NTYPE are transferred to their corresponding places in the array TMLIST and LISTTP. This operation i s shown in Figure B.2. B.3 EVENT DELETION The variable NEXT contains the address of the f i r s t item in the singly-linked l i s t . The calling sequence CALL SCHED (3,TIME, NTYPE) causes the transfer of the contents of TMLIST AND LISTTP at the address NEXT to TIME and NTYPE respectively. Figure B.3 shows the flow chart of this operation, - Examples of the i n i t i a l i z a t i o n , event insertion, and event deletion procedures are given i n Figure B.4. The figure shows the contents of the several events l i s t s when a series of calling sequences i s transmitted to the events scheduler. 150 INITIALIZATION NUMAVL — 50 1 1 ^ NUMAVL = NEXT 9999 3_ TMLIST(I) *- 999999 I I A C V E C ( I ) - — I | number of a v a i l a b l e spaces f o r event i n s e r -t i o n Figure B . l Flow chart of the "event scheduler" i n i t i a l i z a t i o n r o u t i n e DELETION TIME NTYFE(l) TMLIST(NEXT) NTYPE(2) I LISTTP(l.NEXT) NTYPEp) I LISTTP(2,NEXT) I LISTTP(3,NEXT) NUMAVL NUMAVL + 1 T IACVEC(NUMAVL) I NEXT NEXT KPOINT(NEXT) RETURN NTYPE(l) 8888 | Fig u r e B.2 Flow cha r t of the event d e l e t i o n r o u t i n e 151 INSERTION X J •*— IACVEC (NUMAVL) TMLIST(J) I LISTTP(1,J) I NTYPE(1) LISTTP(2,J) - — NTYPE(2) |  LISTTP(3,J) •*— NTYPE(3) 1 RETURN| Figure B.3 Flow chart of the event insertion routine 152 CALLING SEQUENCE IACVEC ADDRESS TMLIST LISTTP LISTTP KPOINT (J) J (l.J) (3,J) (J) CALL ISCHED(l) 1 1 co _ 2 2 00 _ (Initialization) • • • • • • « • • • • • • *rp I— -+~ 50 50 • CO • * • • - NEXT - 9999 CALL SCHED(2,15, NTIPE) (Insertion) • • • • • • • • • • • • • • • • • • • • • [ P r - ^ - 49 49 CO - - — -50 50 15 X X X 9999 NEXT - 50 CALL SCHED(2,35, NTIPE) (Insertion) • • • • • • • • 0 • • • • • • • • • • • • • IP 1—1 »- 48 48 CO — — — — 49 49 35 X X X 9999 50 50 15 X X X 49 NEXT - 50 CALL SCHED(2,20, NTYPE) (Insertion) • • • • • • • • 0 • • • • • • • • • • • • |P ! — — 47 47 CO - — - -48 48 20 X X X 49 49 49 35 X X X 9999 50 50 15 X X X 48 NEXT = 50 CALL SCHED(3,TIME,NTYPE) (Deletion) • • • • • • • • • • • • • • • • • • • • • 47 47 CO — — — — U-J— 50 48 20 X X X . 49 NEXT - 48 49 49 35 X X X 9999 50 50 contents transfered to TIME, NTYPE CALL SCHED(2,55, NTYPE) (Insertion) C O -• O • • • • « • • • • • • • • • • • • • -*- 47 47 CO — — — — 50 48 20 X X X 49 NEXT « 48 49 49 35 X X X 50 50 50 55 X X X 9999 CALL SCHED(3,TEE,NTYPE) (Deletion) • • • • • • • • • • • • • • • • • • • • • 47 47 CD — — — — 1 P 1— 48 48 contents transferee! to TIME. NTYPE 49 49 35 X X X 50 NEXT - 49 50 50 55 X X X 9999 NOTE: P = IACVEC(NUMAVL) — gives the immediately available space for event insertion when there are NUMAVL spaces left Figure F3.4- Examples of the initialization, event insertion, and event deletion procedures 153 APPENDIX C INPUTS TO THE LOGGING MODEL C.l FUNCTIONAL RELATIONSHIPS Table C.l The stochastic yarding models and the inverse cumulative density functions used in the simulation runs Stochastic yarding models Yarder 1 Yarder 2 Yarder 3 Yarder 4 X = 10.4937 + 0.7586X x + Afc - 0.4861A ^ where Xfc = the potential production for the 8-hour shift in Mfbm. A t~N(0,8.014) X^ = 36.9817 + 0.17855X t-1 + A. •N(0,9.76) Xfc = 26.6979 + 0.43390X„ . + 0.133LX _ - 0.1823Xt _ + Afc t t -1 t-2 t - j t A .— N(0,9.05) Xfc = 37.2363 + 0.3928X^ . - 0.0507X -0.046X,. _ + Afc t t-1 t-2 t-3 t A t^N(0,9.20) Yarder 5 X t = 14.35 + 0.5X x + 0.15Xt_2 4 A t » V —N(0,7.15) Yarder 6 X t = 30.0 + -0.4X + A_ , A ~ t-1 t t • N(0,8.93) Yarder 7 X t = 27.0 + 0.45X L + 0.1X 2 + A t , A t ~ -N(0,9.03) Yarder 8 X t " 31.5 + 0.3X x + A t , A t — ~N(0,9.05) Grapple yarder X t = 45.0 + 0.4Xt_1 + A t , A t ^ xN(0,10.0) Loading time for small trucks (minutes) T = 12.56295 + 178.05948X - 941.79077X + 2652.5993X - 3274.1838X + 1467.09253X" X^Uniform(0,l) 154 Table C.l - cont. Loading time for large trucks (minutes) T = 19.4 + 50.17746X - 139.01392X2 + 594.2793X3 - 965.54053X4 + 549.05298X5 , X~Uniform(0,1) Camp delay for loaded trucks (minutes) T = -0.52135 + 35.35252X - 268.17603X2 + 833.32178X3 - 1101.88208X4 + 521.74536X5 , X—•Uniform(O.l) and T £ 0 Camp delay for empty trucks (minutes) T = -0.57894 + 23.72041X - 205.78593X2 + 729.8051X3 - 993.3894X4 + 466.75806X5 , X—Uniform(0,1) and T £ 0 Travel time from camp to dump (7.8-mile distance) (loaded trucks) (min.) T = 10.0744 + 166.36107X - 965.48706X2 + 2490.88794X3 - 2845.25537X4 5 + 1184.40723X , X ~Uniform(0,l) Travel time from dump to camp (empty trucks) (minutes) T = 16.19785 + 23.13339X - 107.70125X2 + 368.198X3 - 511.14917X4 + 246.9175X5 , X—Uniform(0,l) Unloading time (minutes) T = 3.32663 + 34.88907X - 210.3049X2 + 573.31982X3 - 680.90576X4 + 294.97192X5 , X — Uniform(O.l) Travel velocity for empty trucks to the sides (miles per hour) Section 1 V - 6.41803 + 34.42258X - 50.94272X2 + 27.7094X3 Section 2 V = 21.5242 - 136.59778X + 688.72266X2 - 1341.01147X3 X ~Uniform(0,l) Travel velocity for loaded trucks to the camp (miles per hour) Section 1 V = 6.41803 + 34.42258X - 50.94272X2 + 27.7094X3 2 3 Section 2 V = 21.5242 - 136.59778X + 688.72266X - 1341.01147X + 1124.04687X - 326.03052X , X^Uniform(0,1) 155 C.2 INITIAL CONDITIONS Table C.2 The i n t i a l values assumed for the landing inventory at each production sides and the i n i t i a l location of each truck I n i t i a l landing inventory (Mfbm.) Side 1 : 20.0 Side 6 : 12.0 Side 2 : 16.0 Side 7 : 24.0 Side 3 : 10.0 Side 8 : 17.0 Side 4 : 6.3 Side 9 : 21.2 Side 5 : 5.0 Side 10: 9.7 I n i t i a l location of the logging trucks Truck No. Location Status Load volume (Mfbm.) 1 camp empty 2 dump empty 3 dump loaded 9.0 4 dump loaded 6.9 5 dump loaded 7.0 6 camp empty 7 dump empty 8 dump loaded 8.0 9 dump loaded 8.1 10 dump loaded 8.5 11 dump loaded 9.3 12 dump loaded 9.7 13 dump empty 14 dump loaded 7.6 15 camp empty 16 dump empty 17 dump empty 18 dump loaded 7.9 19 dump loaded 8.8 20 dump loaded 9.9 C.3 THE YARDING SEQUENCE AND SETTING STATISTICS Table C.3 The yarding sequence and setting s t a t i s t i c s used during the validation runs Yarder Setting Distance Distance Landing Setting Number Number from camp from prev. capacity volume (miles) (miles) (Mfbm.) (Mfbm.) 0904 9363 25.6 36 58 9365 25.7 0.1 36 285 9363 25.6 0.1 36 354 9355 26.0 2.0 18 740 7643 20.5 18.5 18 888 7644 20.6 0.1 27 120 0905 5462 17.0 24 312 5462 17.0 18 450 5557 16.8 2.0 45 839 5464 16.4 2.75 36 1304 6841 18.1 6.0 45 343 0909 9350 25.8 36 479 9343 25.7 0.1 36 504 9343 25.7 36 381 9345 25.9 0.2 36 214 9155 24.7 3.8 18 572 9154 24.5 0.2 45 139 9171 24.3 0.7 36 449 0911 8359 19.0 9 13 8358 18.9 0.1 45 299 7854 18.3 11.0 45 286 7896 20.8 2.5 36 387 7955 22.0 1.2 27 1274 7998 22.4 0.4 18 305 157 Table C.A The yarding sequence and setting s t a t i s t i c s used during the various simulation runs Moving Distance distance from Setting Landing Yarder Yarding of yarder camp volume capacity number order (miles) (miles) (cunits) (cunits) I . 1 0.10 17 .90 407.52 53.28 1 2 0.10 18.00 211 .64 53.28 1 3 2.00 17 .90 210.16 53 .28 1 4 18.50^ 18.30 1095.20 26.64 1 5 -1 .00 12.80 281 .20 26.64 1 6 -1 .00 12 .80 230.88 26.64 1 7 -1 .00 12.80 250.12 26.64 1 8 -1 .00 12 .80 398.12 26.64 1 9 0.10 12 .80 153.92 26.64 1 10 0.0 12.90 1200.00 39.96 1 1 1 0.0 12.90 500.00 39 .96 2 1 -1 .00 18.10 448.59 53 .28 2 2 -1 .00 18.10 279 .72 53 .28 2 3 0 .10 18.10 313.76 53 .28 2 0.0 18 .00 745.92 53.28 2 5 - 1 .00 18 .00 230.88 53 .28 2 6 0 .20 18.00 333.00 53.28 2 7 3 .80 18 .20 316.72 53 .28 2 8 -1 .00 17.00 390.72 26.64 2 9 0.20 17.00 455 .84 26 .64 2 10 0 .70 16 .80 205.72 66.60 2 1 1 -1 .00 16.60 540 .20 53.28 2 12 0.0 16.60 1200.00 53.28 2 13 0.0 16 .60 500.00 53.28 3 1 0.10 1 1 .30 227.70 14.80 3 2 1 1 .00 1 1 .20 442.52 66.60 3 3 2 .50 10.60 423.28 66 .60 3 4 1 .20 13.10 572.76 53 .28 3 5 -1 .00 14.30 136 .16 39 .96 3 6 - 1 .00 14.30 651.20 39.96 3 7 -1 .00 14.30 125.80 39.96 3 8 0 .40 14.30 972.36 39 .96 3 9 -1 .00 14 .70 267 .88 26.64 3 10 0.0 14.70 1200 .00 26.64 3 1 1 o.o 14.70 500.00 26 .64 * Moving distance = -1.00 means that the yarder i s only turning around, i.e. same setting. = 0.0 means that the moving distance i s negligible 158 Table C.4 - cont. Moving Distance distance from Setting Landing Yarder Yarding of yarder camp volume capacity number order (miles) (miles) (cunits) (cunits) 4 1 0 .0 9 .30 853.22 35 .52 A 2 2 .00 9 .30 666.00 26.64 4 3 -1 .00 9.10 566.84 66.60 A A 2 .75 9.10 674 .88 66.60 A 5 -1 .00 8 .70 740.00 53.28 A 6 -1 .00 8.70 759 .24 53 .28 A 7 6 .00 8 .70 324. 12 53.28 A 8 0.0 10.40 1200.00 66.60 A 9 0.0 8.70 500.00 53.28 5 1 0.10 15.00 1000.00 40.00 5 2 0.25 15.10 650.00 40.00 5 3 1 .05 15.35 425 .00 40.00 5 0 .40 16 .40 890 .00 40.00 5 5 0.80 16.00 1200.00 40.00 5 6 0 .0 16.80 900.00 40.00 5 7 0.0 16 .80 750 .00 40.00 6 1 0.20 15.40 890 .00 40.00 6 2 -1 .00 15 .60 550.00 27.00 6 3 1.10 15.60 625 .00 27.00 6 0.30 16 .70 720 .00 40.00 6 5 0.10 17 .00 357.00 53.00 6 6 0.30 16.90 502.00 53.00 6 7 0 .90 16 .60 1250.00 40.00 6 8 0.0 17 .50 900 .00 40.00 6 9 0 .0 17.50 700 .00 40.00 7 1 0 .60 17.10 760.00 40.00 7 2 1.10 17 .70 420.00 40.00 7 3 -1 .00 18.80 650.00 53.00 7 A 0.30 18 .80 1 100.00 53 .00 7 5 -1 .00 18 .50 420.00 53 .00 7 6 1 .05 18 .50 610.00 40.00 7 7 -1 .00 19 .55 352.00 40.00 7 8 0.25 19 .55 711.00 40.00 7 9 0 .0 19 .80 1500 .00 40 .00 7 10 0.0 19 .80 580.00 40.00 8 1 0.10 11.10 1100.00 40.00 8 2 -1 .00 1 1 .20 510*00 53.00 8 3 0.40 1 1 .20 920.00 53 .00 8 A 0.50 1 1 .60 450.00 40.00 8 5 -1 .00 12. 10 790.00 40.00 8 6 0.40 12.10 940.00 40.00 8 7 0.0 12.50 1400.00 40.00 159 APPENDIX D COSTS ASSUMED IN THE MODEL Table D.l Summary of the costs assumed i n the model Equipment Cost per 8 ( $ ) -hr. day Overtime ( $/hr.) No. in of men the crew High Lead yarder 358.48 47.16 5 Grapple yarder 390.52 43.56 3 Trakloader 293.50 39.00 4 Loader 167.00 19.92 2 Unloading f a c i l i t y 130.00 7.08 1 Logging truck Fixed cost Operating ($/8-hr.day) ($/mile) cost Overtime ($/hr.) "small" 27.00 0. 77 7.62 "large" 40.00 0. 89 7.62 The yarder, loader, and dump costs include (where applicable): Cost per 8-hr. day: machine fixed cost - depreciation ("straight-line" depreciation over 16,000 hours assuming a 15% salvage value) and insurance machine operating cost - wire rope (depreciated over 25,700 hours with no salvage value, chokers, fuel, o i l , t i r e s , repairs (63% of the depreciation) labour - the basic wage for the crew plus 20% induced overhead (to cover the workmen's compensation dues, unemployment insurance, paid holidays, paid vacations, pensions); one hour travel time pay; extra H hour for the yarding engineer for the maintenance (oiling) of the yarder Overtime: machine operating cost and labour cost j(basic wage + 20%) x l .sj The truck operating cost includes fuel and o i l , repair and main-tenance labour and supplies, and tires on a per mile basis. 160 APPENDIX E SAMPLE OUTPUT FROM THE SIMULATION PROGRAM E . l OUTPUT TYPE 1 This type of output consists of a detailed breakdown of the act i v i t i e s of each of the yarders, loaders, and trucks, and a summary of the cost and production i n each of the yarding, loading, trucking, and unloading a c t i v i t i e s . An example i s shown in Tables E . l and E.2. In Table E . l , a l l times are given in minutes, while a l l volumes are given in cunits. In the trucking summary, BCKNG stands for the amount of time spent i n "bullcooking" duties, e.g. moving of yarder, towing of truck. In the setting summary, the symbol (M) WAIT under yarding overtime denotes moving to another setting. L O O denotes the total number of minutes the logging trucks waited for the yarding of more logs i n the designated side. E.2 OUTPUT TYPE 2 This type of output consists of the daily series of values for several responses. Output type 1 and output type 2 are obtained from separate computer logical units; hence, a run may be allowed to yield one type of output while suppressing the other type. END OF DAY 2 Table E.l The trucking, setting, and dump summaries for a day's operation TRUCKING SUMMARY TRK PRODUCTIVE TIMES DELAY TIMES TOTAL C/T LOADS NO. SIDE DUMP NO. NO. VOL Ml LES LDNG UNLDG HLNG BCKNG CAMP QUEU QUEU MAID DMPD HALO TPVLD SU 1 11*4. 15. 417. 0. 10. 5k. 0. 6U0. 160. 3 2 34.9 118. 91. 6 100.0 2 0. 7. 44. 0. 0. 0. 11.. 63. 0. 0 1 0.0 18. 82. 0 13.2 3 106. 2k. 434. 0. 17. 0. 7. 588. 108. 3 3 k0.5 118. 98. 8 100.0 <t 88. 20. 331. 0. 7. 0. 19. 465. 0. 2 3 25.4 80. 9G. 0 100.0 5 03. 8. 469. 0. 23. 0. 3. 597. 117. 2 1 2G.6 110. 99. l* lon.o 6 107. 17. 1*10. 0. 33. 0. 33. 600. 120. 2 2 26.6 90. 91*. 4 10.0.0 7 101. 23. 379. 0. 3. 67. 15, 588. 108. 2 3 28. 2 88. 86. 0 100.0 8 116. 3k. 337. 0. 23. i»6. 2R. 582. 102. 3 k 39.8 106. 87. 6 100.0 9 101. 16. 243. 324. 29. 0. 5. 717. 237. 2 2 30.0 110. 98. 8 100.0 10 161. 22. 300. 0. k. 0. U l . 529. »*0. 2 3 26.6 82. 92. 3 100.0 11 115. 13. 509. 0. 10. 0. 0. 647. 167. 3 2 40.7 121*. 100. 0 100.0 12 71. 15. 382. Ilk. 5. 0. 0. 587. 107. 2 2 26.6 120. 100. 0 100.0 13 1*43. 12. 343. 0. 7. 0. 6. 511. 31. 2 2 26. R 90. 98. 7 100.0 SETTING SUMMARY DOWN TIMES V C 1 L U M E S SIDE YRDNG LDNG YOR LDR PLUG QUEU WAIT TOTL TOTL VOL REM HUM OF 0/T 0/T BKDN BKDfl LDNG TIME LOG YRDD TRKD NOW VOL LOADS YDR * l l Y »A L U 1 192. (M) 30. 0. 0. 0. 0. 56. 5k. 3 53.3 7.7 1500.0 1* 100.0 100.0 100.0 2 0. 0. 83. 0. 0. 0. ll». 37.8 1*0.0 3.3 811.3 3 100.0 82.7 100.0 3 0. 30. 0. 0. 0. 0. 0. 61*.8 54.9 12.0 102.8 I* 100.0 100.0 100.0 i* 0. 30. 0. 0. 0. 90. 0.101*.3 95.7 26.2 104. 1 7 100.0 100.0 100.0 5 0. 30. 0. 0. -0. 10. 57. 54.9 63.0 0.0 3l»5.F 5 100.0 100.0 100.0 6 0. 30. 0. 0. 0. 67. 0. 60.7 66.0 9.6 589.7 5 100.0 100.0 10P.0 DUMP SUMMARY TOTAL VOLUME DUMPED - 1*05.6 TIME SPENT DUMPING - 227. DUMP OVERTIME » 108. Table E.2 The yarding, loading, trucking, and unloading cost summaries for a day's operation END OF DAY 2 COST SUMMARY YARDING COST HIGH LEAD SPARS TOTAL TOTAL $/CUN COST YRDD 2294.50 376.3 G.09 GRAPPLE YARDERS TOTAL TOTAL $/CUN COST YRDD 0.0 0.0 0.0 T R A K L O A D E R S TOTAL TOTAL $/CUM COST YRDD 0.0 0.0 0.0 LOADING COST TOTAL TOTAL $/CUN COST LOADD 1051.30 372.7 2.32 TRUCKING COST TOTAL TOTAL $/CUN COST HAULD 2013.09 372.7 5.40 DUMPING COST TOTAL TOTAL $/CUN COST DUMPD 142.75 405.6 0.35 TOTAL COST FOR 2 DAYS ENDING DAY 2 =$ 1098 2.Gl S TOTAL VOLUME DUMPED DURING SAME PERIOD = 84 8.2 CUNITS 163 Table E.3 Sample output: Daily series of values for several responses DAY YAROED YAPDED YARDED CUNITS C U N I T S % U S A I u J A % A S / C N S / C N $ / C N J / C N t/CN HI LEAD GRPL TRKL HAULED DJHPED TRK TRK YDR YDR L D * YDRS LORS TRKS DUMP TOTAL LOADS 1 3 4 1 . 3 0 . 0 0 . 0 3 7 0 . 7 3 8 2 . 5 9 1 . 1 1 C C . P 1 0 0 . 0 1 0 0 . 0 1 0 0 . 0 6 . 7 2 2 .96 5 . 7 2 0 . 3 5 1 5 . 6 5 30 2 4 1 4 . 6 0 . 0 0 . 0 4 1 9 . 9 4 3 4 . 3 9 6 . Z 9 4 . 1 9 8 . 7 1 O C . 0 1 0 0 . 0 5 . 5 3 2 . 55 5 . 5 8 0 . 3 2 1 3 . 9 8 34 3 3 8 9 . 8 0 . 0 C O 3 8 6 . 9 3 8 3 . 7 94 .1 9 3 . 3 9 9 . 1 9 3 . 1 9 3 . 8 5 . 8 3 2 . 7 0 5 . 7 2 0 . 3 6 14 .61 31 4 4 0 5 . 0 0 . 0 0 . 0 3 9 3 . 8 3 9 3 . 1 9 5 . 6 9 1 . 7 9 9 . 6 1 0 0 . C 100 .0 6 . 1 3 2 . 7 0 5 . 6 6 0 . 3 5 1 4 . 8 3 31 5 4 3 2 . 3 0 . 0 0 . 0 4 3 0 . 9 4 0 0 . 6 8 9 . 7 9 3 . 3 1 0 C . 0 1 0 C . 0 1 P P . 0 5 . 3 0 2 . 4 9 5 . 4 8 0 . 3 5 13 .62 33 6 3 6 0 . 1 0 . 0 0 . 0 3 8 3 . 0 3 9 3 . 1 93 .1 1 0 0 . 0 9 9 . 8 1 0 C . 0 100 .0 6 . 6 9 2 . 7 7 5 . 9 2 0 . 3 4 1 5 . 7 3 31 7 3 5 4 . 0 0 . 0 0 . 0 3 4 0 . 3 3 4 2 . 1 8 9 . 7 9 B . 6 1 0 0 . 0 1 0 0 . 0 1 0 0 . 0 6 . 4 8 3 . 1 2 6 . 1 1 0 . 3 9 16 .10 29 8 3 2 1 . 1 0 . 0 C O 327.1 3 2 8 . 6 8 5 . 6 I C O . O 1 C C . 0 1 0 0 . 0 100 .0 7 . 1 4 3 . 2 5 6 . 1 6 0 . 4 1 1 6 . 9 5 27 9 3 4 7 . 3 0 . 0 0 . 0 3 3 6 . 3 3 3 4 . 5 9 1 . 1 1 C 0 . 0 1 0 0 . 0 1 0 0 . 0 1 C 0 . 0 6 . 6 0 3 . 1 6 6 . 0 2 0 . 4 0 1 6 . 1 7 27 10 3 6 1 . 3 0 . 0 0 . 0 3 6 7 . 3 3 8 3 . 7 9 4 . 5 8 9 . 8 1 0 0 . 0 1 0 0 . 0 100 .0 6 . 3 4 2 . 8 9 6 . 0 2 0 . 3 7 1 5 . 6 2 3 0 . . 11 4 3 6 . 3 0 . 0 0 . 0 3 8 4 . 3 3 8 0 . 9 9 5 . 9 8 6 . 7 1 C 0 . 0 1 0 C . 0 100 .0 5 . 2 5 2 . 7 9 5 . 5 3 0 . 3 7 1 3 . 9 4 30 12 3 0 3 . 9 0 . 0 0 . 0 3 5 8 . 8 3 7 7 . 9 9 5 . 6 8 6 . 7 9 9 . 7 1 0 0 . 0 1 0 0 . 0 7 . 7 0 3 . 0 4 5 . 8 5 0 . 3 8 1 6 . 9 8 27 13 4 1 1 . 7 0 . 0 0 . 0 3 3 7 . 6 3 2 0 . B 9 4 . 3 8 0 . 2 9 9 . 8 9 8 . 9 100 .0 6 . 0 2 3 . 1 5 5 . 8 8 0 . 4 3 1 5 . 4 7 27 14 2 8 8 . 8 0 . 0 0 . 0 3 3 8 . 3 3 8 0 . 2 89 .1 8 6 . 7 8 9 . 3 9 4 . 4 1 0 0 . C B . 2 9 3 . 1 4 6 . 1 2 0 . 3 7 1 7 . 9 2 26 15 4 1 6 . 5 0 . 0 0 . 0 3 8 2 . 6 - 3 3 0 . 0 9 4 . 3 8 8 . 6 9 9 . 3 1 0 0 . 0 100 .0 5 . 5 0 2 . 8 0 5 . 6 3 0 . 4 2 14 .35 31 16 3 7 1 . 4 0 . 0 0 . 0 395 .3 4 0 0 . 7 8 3 . 6 8 6 . 7 1 0 0 . 0 9 3 . 5 9 8 . 3 6 . 1 3 2 . 7 0 5 . 2 4 0 . 3 4 14 .41 30 17 3 9 2 . 2 0 . 0 0 . 0 4 1 0 . 9 4 0 4 . 1 8 7 . 8 9 3 . 3 1 0 0 . P 1 0 0 . 0 1 0 0 . 0 S . 8 5 2 . 6 3 5 . 6 5 0 . 3 4 1 4 . 4 6 34 18 4 0 5 . 3 O . C 0 . 0 3 8 6 . 1 3 9 6 . 3 9 2 . 6 100 .0 9 8 . 4 1 0 0 . 0 1 0 0 . 0 6 . 1 2 2 . 7 5 5 . 8 9 0 . 3 5 1 5 . 1 1 30 19 3 8 6 . 4 0 . 0 0 . 0 3 9 0 . 7 3 7 3 . 6 9 3 . 9 1 C 0 . 0 1 C C . 0 1 0 0 . 0 100 .C 5 . 9 3 2 . 7 2 5 . 7 3 0 . 3 6 1 4 . 7 4 32 20 3 8 4 . 6 0 . 0 0 . 0 3 8 9 . 6 3 9 2 . 1 9 4 . 7 100 .0 9 9 . 6 1 0 0 . 0 1 0 0 . 0 5 . 9 6 2 .75 5 . 7 9 0 . 3 5 14 .85 30 21 3 4 2 . 4 0 . 0 0 . 0 3 3 3 . 9 3 1 7 . 7 9 5 . 0 100 .0 9 9 . 3 1 0 0 . 0 9 6 . 7 7 .11 3. 15 6 . 5 5 0 .42 1 7 . 2 3 27 22 4 1 7 . 7 0 . 0 0 . 0 4 3 0 . 4 4 4 2 . 1 9 5 . 4 9 5 . 5 9 8 . 7 1 0 0 . 0 103 .P 5 . 4 9 2 . 5 1 5 . 4 2 0 . 3 1 1 3 . 7 3 34 23 4 1 3 . 7 0 . 0 0 . 0 4 0 0 . 1 4 0 7 . 1 9 3 . 3 9 7 . 4 1 C C . 0 1 0 C . 0 1 0 0 . 0 5 . 5 4 2 . 6 5 5 . 8 5 0 . 3 4 14 .38 32 24 3 7 0 . 1 0 . 0 0 . 0 3 6 7 . 7 3 7 7 . 4 9 3 . 2 9 3 . 3 1 0 0 . 0 1 0 C . 0 100 .0 6 . 1 9 2 . 9 9 5 . 7 9 u.3£ 1 5 . 2 3 30 25 3 6 0 . 0 0 . 0 0 . 0 3 6 9 . 7 3 7 2 . 0 8 8 . 4 9 3 . 3 1 O C . 0 1 0 0 . 0 lPO.tr 6 . 3 7 2 . 3 7 5 . 9 1 3 .36 15 .51 30 26 3 9 3 . 5 0 . 0 0 . 0 376 .1 3 6 6 . 0 9 3 . ^ 93.3 9 ^ . 5 1 0 0 . Q 100. P 5.83 2 . a 2 5.7-J C .37 1 4 . 3 2 30 27 4 2 7 . 4 C O 0 . 0 405 . 7 4 0 7 . 2 8 8 . 7 8 9 . 6 9 7 . 0 1 00 . 0 1 00 . 0 5 . 36 2 . 64 5 . 5 6 0 . 35 1 3 . 91 32 28 3 7 2 . 2 0 . 0 0 . 0 3 7 7 . 9 3 9 3 . 1 9 5 . 0 8 4 . 7 9 5 . 5 1 O C 0 1 0 0 . 0 6 . 1 6 2 . 8 1 5".45 0 . 3 6 1 4 . 7 8 28 29 3 6 4 . 0 0 . 0 0 . 0 317 . 3 3 4 1 . 1 B 9 . 2 7 3 . 5 9 C . 1 1 0 C . 0 1 0 0 . 0 6 . 4 9 3 . 3 5 5 . 9 9 0 .4C 1 6 . 2 3 24 30 3 1 2 . 8 0 . 0 0 . 0 3 3 6 . 5 3 6 2 . 4 8 0 . 4 e o . 0 83 .1 1 0 0 . 0 9 4 . 7 8 . 3 7 3 . 1 1 5 . 9 9 0 .36 17. 85 26 31 3 4 1 . 5 0 . 0 0 . 0 3 9 6 . 4 3 5 4 . 9 8 7 . 4 9 3 . 3 9 9 . 5 1 C 0 . 0 1 0 0 . 0 6 . 8 5 2 . 6 8 5 . 6 1 G .41 15 .54 29 32 2 8 2 . 0 0 . 0 0 . 0 282 .1 2 7 4 . 1 9 4 . 4 1 C 0 . 0 1 0 0 . 0 9 6 . 1 1 C 0 . P 8 . 4 3 3 . 7 6 7 . 2 7 0 .49 19 .94 23 33 3 6 3 . 9 0 . 0 0 . 0 3 6 8 . 9 3 6 0 . 8 9 2 . 4 94 .2 9 9 . 8 1 0 0 . 0 100 .0 6 . 6 2 2 . 8 3 5 . 8 5 0 . 3 7 15 .72 30 34 4 1 3 . 1 0 . 0 0 . 0 3 8 4 . 3 3 8 2 . 4 8 4 . 0 9 3 . 3 9 9 . 6 1 0 0 . 0 100 .0 5 . 5 5 2 . 7 6 5 . 6 0 C .36 14 .27 31 35 3 9 8 . 3 C O 0 . 0 3 9 6 . 6 3 7 2 . 5 9 2 . 7 1 0 0 . 0 1 0 P . C 1 0 0 . 0 100 .0 5 . 7 6 2 . 6 8 5 . 7 5 C i 3 6 14 .54 33 36 3 7 0 . 0 0 . 0 0 . 0 4 0 6 . 4 4 1 7 . 8 3 9 . 4 1 0 C . C 99 .8 1 3 0 . 0 1 0 0 . C 6 . 2 6 2 . 6 1 5 . 7 6 C . 3 3 1 4 . 9 6 33 37 4 0 3 . 5 O . C 0 . 0 3 8 4 . 3 4 0 2 . 0 9 1 . 5 1 C 0 . 0 100 .0 1 P 0 . 0 I P O . O 5 . 8 0 2 . 76 5 . 7 5 0 .33 14 .64 30 38 3 8 0 . 3 C O C O 3 3 3 . 9 3 4 2 . 5 9 0 . 0 9 3 . 3 100 .0 100 .0 9 7 . 7 6 . 0 3 3 . 1 6 6 . 2 5 0 . 3 9 1 5 . 8 3 26 39 2 8 3 . 8 0 . 0 0 . 0 2 8 7 . 0 2 9 3 . 3 8 9 . 7 1 C 0 . C 8 ? . 2 1 0 0 . 0 7 8 . 5 7 .94 3 . 5 0 6 . 6 3 0 . 4 6 18 .52 24 40 3 9 8 . 6 0 . 0 0 . 0 4 3 4 . 7 3 9 8 . 5 9 6 . 2 9 3 . 3 1 C 0 . 0 1 0 0 . 0 I C O . O 5 . 7 5 2 . 4 4 5 . 2 6 C . 3 4 1 3 . 8 0 34 41 4 3 1 . 7 0 . 0 0 . 0 4 1 6 . 6 4 0 1 . 8 9 2 . 6 1 0 0 . 0 1 0 0 . 0 1 0 0 . 0 9 9 . 2 5 .31 2 . 5 4 5 . 4 8 0 . 3 3 1 3 . 6 7 33 42 4 5 2 . 5 0 . 0 C O 4 4 8 . 3 4 7 3 . 7 9 3 . 5 1 P 0 . 0 9 9 . 8 1 0 0 . 0 1 0 0 . C 5 . 2 2 2 . 3 9 5 . 3 3 C . 29 1 3 . 2 3 34 43 4 2 5 . 4 0 . 0 0 . 0 4 0 4 . 0 3 9 0 . 8 9 3 . 5 9 6 . 5 100 .0 1 0 0 . 0 100 .P 5 . 3 9 2 . 6 3 5 . 5 9 L . 34 13 .95 32 4 4 4 0 6 . 3 0 . 0 0 . 0 4 1 4 . 8 4 1 9 . 7 9 5 . 0 9 3 . 3 1 0 0 . 0 1 0 0 . 0 ICO .C 5 . 6 4 2 . 5 6 5 . 2 0 0 . 3 2 13.72 31 45 3 8 7 . 8 ' 0 . 0 0 . 0 372 .9 4 0 3 . 0 8 9 . 7 8 8 . 2 9 8 . 6 1 0 0 . 0 100 .P 5 . 9 7 2 . 6 5 5 . 7 3 0 . 3 4 1 4 . 8 9 28 46 3 6 2 . 5 0 . 0 0 . 0 4 4 3 . 6 4 0 9 . 9 89 .1 9 3 . 3 97 .1 1 0 0 . 0 103 .0 6 . 7 8 2 . 4 4 5 . 2 3 0 . 3 4 14 .79 34 47 4 1 7 . 6 0 . 0 0 . 0 3 6 0 . 6 3 7 5 . 1 « 0 . 1 I C O . O 9 9 . 8 1 0 0 . 0 1 0 0 . P 5 . 7 2 2 . 9 4 6 . 1 6 C . 3 7 15 .18 30 48 3 7 4 . 2 0 . 0 0 . 0 3 9 8 . 1 3 8 1 . 6 9 1 . 8 100 .0 9P .8 1 3 0 . 0 100 .0 6 . 1 3 2 . =7 5 . 8 1 0 . 3 6 14 .96 33 49 3 4 5 . 5 0 . 0 0 . 0 3 2 4 . 7 3 4 4 . 6 87 .1 I C O . O 9 S . 8 1 3 0 . 0 1 C 3 . C 6 . 8 4 3 . 2 7 6 . 1 9 0 . 3 9 16 .68 27 50 4 0 5 . 7 0 . 0 0 . 0 4 3 6 . 4 4 1 3 . 9 9 3 . 9 100 .0 5 5 . 5 1 0 C . 0 9 4 . 6 5 .62 2 . 4 0 5 . 4 7 0 . 3 3 13 .82 36 51 4 1 8 . 8 0 . 0 0 . 0 4 1 9 . 4 4 1 0 . 0 8 9 . 9 1 0 0 . 0 9 9 . 4 100 .0 100 .0 5 . 4 7 2 . 5 3 5 . 5 7 0 . 3 3 13 .91 34 52 3 4 4 . 6 0 . 0 0 . 0 3 4 4 . 9 362 . 7 8 8 . 8 100 .0 5 9 . 8 1 0 C . 0 1 0 0 . 0 6 .93 3 . 0 8 6 . 5 3 0 . 3 8 1 6 . 8 8 2 9 53 4 5 3 . 8 0 . 0 0 . 0 4 3 3 . 5 4 3 2 . 5 9 3 . 2 1 0 0 . 0 9 8 . 8 9 5 . 3 9 9 . 8 5 .02 2 . 4 9 5 . 5 1 0 . 3 2 13 .35 35 54 3 8 6 . 0 0 . 0 0 . 0 3 9 2 . 9 3 8 4 . 3 9 4 . 2 I C O . O 1 0 0 . 0 1 0 0 . 0 1 0 3 . 0 5 . 9 4 3 . 1 1 6 . 0 0 0 . 3 5 1 5 . 3 9 32 55 3 7 3 . 3 0 . 0 0 . 0 3 4 5 . 3 3 5 7 . 1 9 2 . 2 9 9 . 2 I C O . O 1 0 0 . 0 1 0 0 . 0 6 .14 3 . 0 8 6 . 0 2 0 . 3 7 15 .61 29 5 6 3 92 . 9 0 . 0 0 . 0 4 1 9 . 7 4 0 0 . 8 9 3 . 5 1 0 0 . 0 1 00 . 0 1 00 . 0 1 00 . 0 5 . 8 3 2 . 5 5 5 . 5 0 0 . 33 1 4 . ? 2 33 57 3 6 0 . 2 0 . 0 0 . 0 3 4 1 . 1 3 5 8 . 1 9 1 . 6 9 9 . 9 1 0 0 . 0 1 0 0 . 0 103 .0 6 . 3 6 3 .11 6 . 2 7 0 . 3 7 16.12 29 58 3 8 9 . 1 0 . 0 0 . 0 3 9 1 . 5 3 7 4 . 9 8 9 . 1 9 3 . 3 1 0 C . 0 I C O . O 100 . 0 5 . 8 9 2 . 74 5 . 5 1 0 . 36 1 4 . 5 3 31 59 3 8 3 . 0 0 . 0 0 . 0 3 9 5 . 9 3 8 4 . 3 9 4 . 7 1 0 0 . 0 I C O . O 1 0 0 . 0 1 0 0 . 0 5 . 9 9 3 . 1 1 6 . 0 1 0 . 3 5 1 5 . A S 32" 60 3 9 1 . 1 0 . 0 0 . 0 3 8 9 . 5 3 9 3 . 5 9 1 . 1 1 0 0 . 0 1 0 0 . 0 1 0 0 . 0 100 .0 5 .86 2 . 7 3 5 . 7 4 0 . 3 4 14 .67 32 61 4 3 2 . 8 0 . 0 0 . 0 4 2 7 . 3 4 3 6 . 4 9 2 . 0 9 7 . 9 9 9 . 8 I C O . O 100 .0 5 . 8 4 2 . 5 3 5 . 5 2 0 . 3 1 14.23 33 62 3 3 8 . 3 0 . 0 0 . 0 333 .1 3 5 1 . 4 9 3 . 5 I C O . O 99^8 100 .0 1P0 .0 7 . 5 4 3 . 1 9 6 . 7 0 0 .38 17.81 27 63 4 2 3 . 5 0 . 0 0 . 0 4 0 4 . 5 "592.6 9 1 . 9 100 .C I C O . O I C O . O 1 C C . 0 5 .41 2 .65 5 . 6 7 0 . 3 4 14. ?T 33 64 3 9 4 . 7 0 . 0 0 . 0 4 0 5 . 3 4 3 0 . <- 9 3 . 7 9 9 . 9 9 9 . 6 1 ^ 0 . 0 I C " . 0 5 .31 2 . 6 2 5 . 6 7 0 . 33 I4 .<-4 32 65 4 1 9 . 9 0 . 0 0 . 3 438 . 1 4 4 1 . 1 1 5 . 8 1 0 0 . 0 1 0 0 . 0 l ^ C ? 1 P 1 . C 5 . 4 6 2 . « 2 5 . 62 0 . 32 13.82 35 66 3 8 1 . 5 0 . 0 0 . 0 3 7 8 . 7 3 8 0 . 9 ^ 3 . 3 1 0 T . 0 9 9 . 8 1 0 0 . 0 1 0 0 . 0 6 . 2 5 2 . 8 0 5 . 8 0 C . 3 5 15.21 31 67 4 3 9 . 4 0 . 0 0 . 0 3 8 9 . 5 3 a 3 . 7 9^..1 9 7 . 0 1 C C . 0 i r ; O . C 1 C C . C 5 . 2 2 2 . 73 5. 87 C . 36 14 .17 32 68 4 7 4 . 6 0 . 0 0 . 0 3 8 7 . 9 397 . C a 7 . 3 9 1 . 7 9 6 . 0 I SO.O 103 .0 5 . 4 0 2 . 7 6 5 . 5 8 0 . 3 4 14 .PS 30 69 4 C 4 . 4 0 . 0 C O 4 3 0 . 9 4 1 0 . 0 9 5 . 2 9 3 . 3 9 8 . 9 100 .0 l C ' . C 6 . 0 8 2 . 5 3 5 . 1 8 C . 3 3 14 .12 32 70 4 0 3 . 2 0 . 0 0 . 0 3 9 9 . 5 4 3 5 . 6 6 9 . C 9 0 . 2 9 9 . 0 1 0 0 . C 100 .0 5 . 6 9 2 . 6 3 5 . 5 8 0 . 3 2 14 .27 31 .71 3 0 8 . 3 0 . 0 0 . 0 3 2 8 . 3 3 4 5 . 3 8 6 . 4 8 8 . 4 9 C 2 1 0 0 . 0 1 P 0 . 0 8 . 2 0 3 .23 6 . 1 8 3 . 3 9 18 .CO 26 72 3 8 5 . 9 0 . 0 0 . 3 396 .7 3 6 2 . 2 8 6 . 3 9 3 . 3 8 8 . 3 100 .0 8 9 . 9 5 . 8 9 2 .63 5 . 6 0 0 . 3 9 14.51 32 73 3 4 8 . 8 0 . 0 C . O 3 5 3 . 8 3 6 4 . 3 9 5 . 3 9 7 . 2 99 .1 l ' O . O 1 C . 0 6 . 5 7 3 . C P t > . l l C . 3 7 16. P5 29 74 3 7 5 . 6 0 . 0 C-.O 4 2 3 . 4 4 0 2 . 7 9 3 . 8 9 ? . 3 I C C C 1 T . C 1C O.C 6 . 1 0 2 . 5 1 5 . 4 0 0 . 3 4 14 .35 33 75 3 5 4 . 9 0 . 0 0 . 0 343 .4 3 4 7 . 6 6 1 . 7 9 3 . 3 I C O . ? i : 0 . ° l C ' . O 6 . 4 6 3 . 1 2 6 . 0 6 0 . 3 8 16 .03 28 76 4 4 4 . 7 0 . 0 0 . 0 4 0 8 . 4 4 1 P . 9 9 3 . 1 9 ^ 8 1 0 0 . 0 U P . O 10">.3 5 . 1 3 2 . 6 5 5 . 8 1 C . 34 1 3 . 9 2 34 77 4 0 9 . 8 O . P 0 . 0 4 0 7 . 5 3 9 6 . 1 9 3 . 2 9 3 . 3 1 0 C . 0 1 C C C 1 0 P . C 5 . 5 9 ? . 61 5 . 3 0 0 . 3 4 13 .94 31 ' 78 3 8 8 . 3 0 . 0 0 . 0 391 .7 4 1 5 . 0 8 3 . 5 9 3 . 3 9 9 . 8 100 .C 100.0 6 . 4 5 2 . 7 1 5 . 5 4 0 . 3 2 15 .C2 30 79 3 9 0 . 8 0 . 0 0 . 3 419 . 3 3 9 7 . 4 9 3 . 3 1 0 0 . 0 1 0 C . O 9 6 . 1 10r..0 5 . 8 4 2 . 5 3 5 . 5 9 0 . 3 4 14.31 34 80 3 7 ? . 3 0 . 0 0 . 0 327..? ? ? 9 . 7 P 9 . 8 9 V . . 9 9 9 . =>_1 r T ^ n jr-v.__o_ j _ 3 . J 6 _ _ 6 6 0 . . ^ L - 4 J _ l _ 7 i S 2 _ 28 164 APPENDIX F STATISTICAL PROCEDURES FOR THE ANALYSIS OF TIME SERIES Sets of data generated s e r i a l l y from real-world or simulated processes are generally time-dependent. The elements in these sets of data are autocorrelated. In dealing with this type of data, the s t a t i s t i c a l techniques commonly applied to sets of independent observations cannot be applied since the r e l i a b i l i t y of the sample means and variances are overestimated when autocorrelation is ignored (Fishman and Kiviat, 1967). Thus the analysis requires that: (1) uncorrelated data are collected, (2) correlated data are transformed to remove the time dependences, or (3) s t a t i s t i c a l techniques that account for the time dependence are used. The third alternative i s used throughout this thesis. The s t a t i s t i c a l proced-ures used i n the analysis of autocorrelated data are described i n the following sections: 1. Definitions and assumptions 2. Calculation of the mean, variance, autocorrelation, and spectrum of a time series 3. Calculation of the variance to use i n defining the confidence limits for a mean of a response or i n comparing the means of two processes differing only in some minor respects 4. The determination of the length of the simulation runs 5. Test for equivalence of two spectra 6. Fitting of stochastic time series models 165 These sections contain some results from the references given below. These sections also contain some examples pertaining to logging processes. References 1. Box, G.E.P. and Jenkins, G.M. TIME SERIES ANALYSIS forecasting and control. San Francisco: Holden-Day, Inc. 1970 2. Fishman, G.S. and Kiviat, P.J. "The Analysis of Simulation-Generated Time Series'.1 Mgt. Science,13(7) (March, 1967), pp. 525-557. 3. Jenkins, G.M. and Watts, D.G. Spectral Analysis and Its Applications. San Francisco:Holden-Day, Inc. 1968 4. Fishman, G.S. "The Allocation of Computer Time i n Comparing Simulation Experiments." Operations  Research,V. 16 (1968), pp. 280-295. 5. Fishman, G.S. "Problems in the S t a t i s t i c a l Analysis of Simulation Experiments: the comparison of means and the length of sample records!' Comm. of the ACM, 10 (2) (February, 1967), pp. 94-99. In the following sections, the numbers in parenthesis indicate the number of the references from where the results were taken. F.l DEFINITIONS AND ASSUMPTIONS (1,2) Stochastic processes concern sequences of events occurring in time according to probabilistic laws. Symbolically, a stochastic process (Xfc , t e T) i s a sequence of random variables indexed on a continuous parameter t which takes on a l l values in the set T. The index t may correspond to discrete units in time T = (0,1,2,3,...). A time series i s a set of observations generated sequentially in time. It is one particular realization, produced by the under-lying probability mechanism, of the stochastic process. 166 The subsequent sections deal with a class of stochastic processes, called stationary processes. A stochastic process i s said to be s t r i c t l y stationary i f i t is invariant under translation along the axis; i.e. i f P ( V V V X 2 " " ' X t < V = P ( X t 1 + k « V " X t +k* Xn> 1 2 n 1 n for arbitrary real values of s and for a l l n. A stochastic process i s said to be weakly stationary of order m i f the moments (of i t s probab-i l i t y distribution) up to some order m are f i n i t e and depend only on time differences. If the probability distribution associated with the process is a multivariate Normal distribution, the process is called a Normal or Gaussian process. Since the multivariate distribution i s f u l l y characterized by i t s f i r s t and second moments, second order stationary plus Normality would be sufficient to produce s t r i c t stationarity. F.2 THE MEAN. VARIANCE. AUTOCORRELATION. AND SPECTRUM OF STATIONARY  TIME SERIES (1,2,3) A stationary process has a constant mean y =E(Xfc) = JxP(X) dx and a constant variance 0 2 = E (X t- y ) 2 = j(X- y) 2P(X)dX. The estimators of the mean and the variance are N N X = IE X and 6 = 1_ E (X Nt=l C X N t=l respectively. By the stationary assumption, the joint probability distribution P(X ,Xt+^) i s the same for a l l times t,t+k which are k time lag apart. The covariance between X and X ^ called the 167 autocovariance at lag k is defined by Y k = Cov(X t,X t + k) = E((X t - y ) ( X t + k - y)). The autocorrelation at lag k i s defined by E((X, - y ) ( X ^ - y)) Jjc_ pk E((X t - y ) 2 ) E ( ( X t + k - y) 2) * o J In (1) i t is indicated that the most satisfactory estimate of y k is C, given by N-k ^ = N t * / X t " ( X t + k -and the estimate of p, i s r, given by C, / d 2 . To test the hypothesis that the P^'s are essentially zero beyond k > q, the variance of r k given by 1 q 2 Var(r k)=^( 1 + 2 E r^ ), k > q , v=l i s used. The variance of a process is made up of the individual variations caused by some disturbances. The spectrum of a time series describes how the variance of a time series i s distributed with the frequency of occurrence of these disturbances. As shown in (1) pp. 45-46, the spectrum is the Fourier cosine transform of the autocovariance function; that i s 00 g( X ) = 2{y + 2 E y cos2TrXk) , 0 $ X $ h ° k=l * where g( X ) i s the spectrum X is the frequency in cycles per unit time. The sample spectrum is shown in (3) to fluctuate about the theoretical spectrum. A "smoothed" estimate of the spectrum is given by M g*( X ) = 2(C + 2 E (IW (k)cos2iTXk) , 0 < X $ h ° k=l * M 168 where W^(k) = suitably chosen weight called a lag window M = largest lag chosen after a "window-closing" procedure. Several "lag windows" have been designed. These are given, together with their advantages and disadvantages in (3). The "lag windows" used i n this study are: 1. Parzen lag window given by W (k) = 1 - k V M i r k 2. Tukey lag window given by W^(k) = 1 + c o s — ) . The "window-closing" procedure is f u l l y described i n (3). Essentially i t consists of observing the change in the spectral estimates as M is increased. Ideally, the spectrum w i l l change markedly as M increases and then w i l l settle down. The best M is chosen at this point since as M is further increased, sampling v a r i a b i l i t y w i l l intro-duce spurious detail in the spectrum. Figure F . l ill u s t r a t e s this "window-closing" procedure. As given in (3), the interval between vg*(X) vg*(X) V i - f ) xvHf) i s a 100(l-a)% confidence interval for g*(X). Here, v is the degrees of N freedom given by 3.71^ , where N is the length of the time series, N for the Parzen window and 2.667— for the Tukey window. The values M V V for and are given as functions of v and a in (3) x ^ i - f ) X^ f ) Figure 3.10 p. 82. On an ordinary scale, the confidence interval i s not a constant, since the limits depend on \ . When the spectrum i s plotted on a log scale, the confidence interval i s represented by ;constant inter-val about the spectral estimate (as shown in Figure F . l ) . 169 4000 2000 - -SPECTRUM for CANFOR data: daily yarding production N = 69 observations Parzen lag window 1000 •-s 400 200 0.0 Bandwidth 12-lag 18-lag 24-lag Confidence interval (95%) 0.1 0.2 0.3 0.4 0.5 F R E Q U E N C Y (Cycles per day) Figure F.l Graphs of "smoothed" spectra i l l u s t r a t i n g the "window-closing" procedure 170 F.3 CALCULATION OF VARIANCES AND COVARIANCES (4) To calculate the confidence interval of a mean u, the s t a t i s t i c t - AzJi Var (x) is regarded as Normally distributed with mean 0 and unit variance. For autocorrelated data, Var (X) i s calculated from M Var(X) = 1 ( C + 2 E (l-k/M)C ). N(l-M/N) ° k=l As in the "window-closing" procedure, M is chosen such that a good resolution i s obtained with acceptable r e a l i a b i l i t y . The factor • (1-M/N) compensates for the bias resulting from the formula used to calculate the autocovariances C, . k When two runs of equal length N, with means X^ and respectively, are made, the variance of the overall mean h(X^ + X^) i s given by Var teQ^ + X 2)) = kVar&J + %Var(X2> + J s C o v C ^ , ^ ) . To calculate Cov (X^,X2) when the method of antithetic variates i s used, the covariances between the f i r s t and second runs are calculated using and C12,k = N l=1 ( X l , k " V ( X 2 , i + k - V ' k " °» ±l.-.«< M Cov (X..X,) = E ( 1 -lk| /M)C 1 1 Z k—M ' N(l-1 -M/N) The method of antithetic variates as described in Chapter V was used to make two runs for the basic configuration with 14 trucks. For the 171 production and truck per cent u t i l i z a t i o n responses, the calculated CovCXj^xp was approximately h of (VarCX^) + Var(X 2)). Thus two antithetic replications of length N i s equivalent to 2.67N observa-tions on 1 replication. The increased efficiency as the result of using the method of antithetic variates was negligible in the case of the unit cost response for this particular configuration. To compare the means of two similar processes differing only in some minor respects, the control variate technique may be used. Assuming X^ and are the respective means of the two processes, the variance of their difference i s given by Var (Xj -Xp = Var (X^) + Var(X 2) - 2Cov(X 1,X 2). Cov(X^,X 2)is calculated as in the method of antithetic variates, with N = max(N^,N2) i f the respective length and N 2 of the two runs are not equal. While the method of antithetic variates was not always effective, the control variate technique was observed to be consistently effective* F.4 THE DETERMINATION OF THE LENGTH OF THE SIMULATION RUNS To determine the length of the simulation run(s) to obtain a given variance, the following formulae are used: 1. For the mean of a given response N = m/Var(X) 1 M where m = . ( C + 2 E (l-k/M)C, ) L-M/L) O K. and where the autocovariances are i n i t i a l l y calculated from a pil o t run of length Q. Q should be sufficiently long so that well-resolved estimate of m is obtained. If N is calculated to be too 172 long to be conveniently included in a single run, two antithetic replications may be made. 2. For the difference between two means When the p i l o t runs made to compare the respective means of two processes have indicated a difference, but the length of the runs was not sufficient to establish the s t a t i s t i c a l significance of the difference, the length N of each run to obtain a given Var(X^-X2) is calculated using N = m l + m2 - m3 where and m2 are calculated Var(X. - X.) . 1 2 as m m and m. 3 U--M n This assumes that the control variate technique is used and that the respective runs for the two processes have equal length. To save computer time, i t is advisable to design the simulation program so that a run can be continued after i t has been stopped. F.5 TEST FOR EQUIVALENCE OF TWO SPECTRA To test whether a sample spectrum g^ (X) is significantly different from another sample spectrum g^*(X), the test reported in (2) is applied. In this test, the s t a t i s t i c t = {ln(g* (X)) - ln(g*(X))} - ' { l n ( 8 l ( X ) ) - ln(g 2(X))} where g^ (X) = the theoretical spectrum at frequency X of series i 173 M. = the number of lags chosen in the calculation of the sample spectrum N^ = the length of series i ¥ (M.,N.) = .539Mi/N. for the Parzen window 1 M 1 = .750 i/N. for the Tukey window = the variance of ln(g^ (X)) is regarded- as Normally distributed with mean 0 and variance 1. The nu l l hypothesis g^(X) = g 2(X) is tested against the alternative hypothesis g^(X) > g 2(X). The nu l l hypothesis i s rejected i f t > t Q which, after some simplifications, i s equivalent to g* (X) /g*(X) > exptta/nMpT^ + ^(M2, T^ ) = & ^ This test i s applied to each frequency X. Thus i t i s convenient to * * graph g^ (X) /g 2 (X) versus X to determine i f each point along the graph f a l l s under the horizontal line of height a. In many simulation experiments, the time series of interest may not be Gaussian. However, as indi cated i n (2), the assumptions * about ln (g (X) ) are f a i r l y insensitive to the stochastic process {X t j being Gaussian and, consequently, the logarithmic test of the n u l l hypothesis i s reasonably valid for non-Gaussian processes. As examples, the spectrum for the observed data from CANFOR is compared with the corresponding spectra for two simulation t r i a l s for both the yarding production and the number of loads responses. Their respective g^ (X) /g 2 (X) - graphs are shown in figure F.2. Since each of the curves f a l l inside their corresponding acceptance level a_, i t can be concluded that for a l l X's, no significant difference is observed between the spectrum of the CANFOR data and those of the t r i a l s for both responses. 174 0.0 J. CANFOR vs. t r i a l 1 *' ^ CANFOR vs. t r i a l 2 0.0 Figure F.2 0.1 0.2 0.3 0.4 F R E Q U E N C Y (Cycles per day) * * g c(X)/g t(X)- graphs comparing the daily yarding production spectrum and the daily number of loads spectrum of the CANFOR data with the corresponding spectra of the two simulation t r i a l s 175 F. 6 FITTING OF STOCHASTIC TIME SERIES MODELS This section describes the f i t t i n g of stochastic time series models, specifically the f i t t i n g of the stochastic yarding models described in Chapter III. This section covers the following: 1. Data preparation 2. Verification of the stationarity assumption 3. Model identification 4. Estimation of parameters 5. Diagnostic checking. Data preparation From the hi s t o r i c a l records of CANFOR logging division at Harrison Mi l l s , B.C. was obtained the raw data of the daily product-ion in thousand board feet for each of four yarders numbered 0904, 0905, 0909, and 0911. These raw data correspond to the net yarding production of the yarder, i.e. i t includes: 1. reduction in production due to the moving and rigging of the yarder, 2. reduction in production due to the turning around of the yarder, 3. reduction in production due to "plugged landing" caused by loader breakdown, 4. reduction in production due to "plugged landing" caused by an insufficient number of trucks dispatched to the side, 5. reduction i n production due to yarder breakdown, and 6. reduction in production due to delays caused by accidents. Adjustments by linear extrapolation were made on the raw data to account for these reductions in production. For example, i f the records indicated that yarding for yarder 0904 was delayed for two hours due to "plugged landing", the per-hour production is calculated by 176 dividing the recorded production by 6. An extra production for 2 hours is then added to the recorded production to adjust the production to a value corresponding to 8 hours. The adjusted data i s then used i n the subsequent analyses. Verification of the stationarity assumption From section F.l i t was mentioned that second order station-arity plus Normality i s sufficient to produce s t r i c t stationarity. A second order stationarity implies the existence of a time-invariant mean, and autocovariances which are dependent only on time differences. The verification of the stationarity assumption for the time series of adjusted daily production for each of the four yarders proceeded with the test that the mean and autocovariances for the f i r s t half of each time series are not different from the corresponding values for the second half of the time series. Table F . l shows the frequency distribution of daily yarding production of each of the four yarders. To test the n u l l hypothesis that the daily yarding production is normally distributed, the expected frequency distribution and the resulting chi-square were computed for each of the four yarders. Table F.2 shows that the computed chi-squared for a l l four cases i s less than the tabular value. Thus there i s no reason to reject the n u l l hypothesis and i t i s con-cluded that the Normal distribution provides a good f i t for the dist-ribution of the daily production for a l l four yarders. After inspecting the mean production for the two halves of the daily production time series for each of the four yarders, i t was readily apparent that no differences exist and i t was concluded as such Class boundaries 0i E i ) 2 A i Yarder 0911 8.5 - 32.5 32.5 - 40.5 40.5 - 48.5 48.5 - 56.5 56.5 - 72.5 15 23 42 27 8 16.58 28.18 34.71 23.75 11.45 0.151 0.952 1.531 0.445 1.040 totals 115 114.67 4.119 Chi-square Yarder 0909 16.5 - 32.5 32.5 - 40.5 40.5 - 48.5 48.5 - 56.5 56.5 - 72.5 11 25 44 24 13 11.91 25.62 36.74 2S.10 U . 3 6 0.070 0.015 1.435 0.598 0.129 totals 117 116.73 2.247 Chi-square Yarder 0904 16.5 - 32.5 32.5 - 40.5 40.5 - 48.5 48.5 - 56.5 56.5 - 72.5 10 31 46 23 11 12.54 31.84 42.60 25.92 8.04 0.515 0.022 0.271 0.329 1.090 totals 121 120.94 2.227 Chi-square Yarder 0905 24.5 - 40.5 40.5 - 48.5 48.5 - 56.5 56.5 - 64.5 64.5 - 80.5 9 25 26 28 12 10.35 22.44 31.06 23.84 11.83 0.176 0.292 0.824 0.726 0.002 totals 100 99.52 2.020 Chi-square \ 0 5 9 2 " 5 * 9 9 1 Table F . l Observed and expected frequencies of the daily yarding production and the resulting chi-square for each of the four yarders 178 without any further t-test. Using the test for equivalence of two spectra described in the previous section, the hypothesis that the spectrum for the f i r s t half of the daily production time series i s not different from that of the second half, for a l l four yarders, was tested at the 90% level of significance. The results showed no significant difference for a l l frequencies for the yarders 0904 and 0905. A significant difference at the frequencies, 0.28 < X <0.38, was observed for yarder 0909. However, these frequencies are non-dominant since the peak in the spectrum occurs in the lower frequencies. For yarder 0911, a significant difference was observed at the domin-ant frequencies, 0.02 < X < 0.12. This fact i s temporarily ignored on the condition that i f i t results in poor f i t in the model, some data transformations are made. From the foregoing i t is seen that stationary yarding prod-uction time series do exist, at least for certain time periods. This resulted from the fact that the settings yarded were close to each other and are, therefore, reasonably homogeneous. For certain cases where the settings to be yarded are not homogeneous, i t may not be valid to assume stationarity in the mean and the variance of the production time series. However, i t may be assumed that the corre-lation of daily production on past days remains the same. Therefore, a procedure that could be followed is to f i t a model over a period where the settings are homogeneous and to use the same model for settings which are different from the previously observed settings, but only after substituting the estimated mean and variance into the model. 179 Model identification The model fi t t e d to each of the four given time series belong either to a class of autoregressive models or to a class of mixed autoregressive-moving average models (ARMA) models of the form m n X" - Z ty. X„ . + Z e.a„ . + a t i-1 1 ^ j=l J ^  fc where X = the deviation Xfc - E(X) ty^ = i t h autoregressive parameter 0 = jth moving average parameter afc . = white noise generated from N m = order of the autoregressive terms n = order of the moving average terms. The calculated autocorrelation and parti a l autcorrelation functions were used to i n i t i a l l y identify which ARMA model to use. The parti a l autocorrelations, ^y^* a r e calculated using k = 1 k-1 fkk ^ r k " . f A - l , j r k - j k = 2,3,...,L k-1 1 " E * k - l i r i where *kj = * k - l , j " *kk*k-l,k-j 3 = l»2,...,k-l r^ = the autocorrelation for lag k. The ARMA model i s i n i t i a l l y identified with the aid of Table F.2 given below. In the tr i p l e t (a,b,c), a refers to the order of the auto-regressive terms, c the order of the moving average terms, and b the 0 1 AUTOCORRELATION FUNCTION YARDER 0905 T YARDER 0911 YARDER 0909 J L YARDER 0904 180 PARTIAL AUTOCORRELATION FUNCTION 2a - limits 1 I 1 I ' I , 2a - limits L_'± 2a - limits - i — r 2a - limits L A G I N D A Y S L A G Figure F.3 The autocorrelation functions for each of the four yarders 181 degree of differencing made on the time series. Figure F.3 shows the autocorrelation functions for each of the four yarders. Table F.2 and Figure F.3 indicate i n i t i a l l y a (2,0,0) - model for yarders 0905 and 0911, a (1,0,0)- model for yarder 0909, and a (1,0,1) - model for yarder 0904. Table F.2 Behavior of the autocorrelation functions for the dth difference of various ARIMA models (source: (1) p.176) (l,d,0) (0,d,l) Behavior of r, k Behavior of <J>^  decays exponentially only <j>^  nonzero only r^ nonzero decays exp onen t i a l l y (2,d,0) . (0,d,2) Behavior of r, k Behavior of tyy^ mixture of exponentials or damped sine wave only <j>u and <f>22 only r^ and r 2 nonzero dominated by mixture of exponentials or damped sine waves (l,d,l) Behavior of r. k Behavior of decays exponentially from f i r s t lag dominated by exponential decay from f i r s t lag Calculation of parameters Preliminary estimates of the parameters were obtained using: 182 for (1,0,0) model °1 = r l » - 1 < < f > l < : l 2 2 f-i i 2. a a = a x (1 - + 1 ) 2 theoretical spectrum: g (X) = a o<X^s 1 + <j> 2 -2<j>^cos2TrX for 2,0,0) model 4>i = r l ( 1 " r 2 } i 2  1 _ r l 0>2 = r 2 - r l 2 - 1 < * 2 * 1 l - r i 2 ' *2 + V 1 <f>2 ~ hK 1 ° a 2 = CTx2 ( 1 ~ * l r l " *2 r2 ) 2a 2 theoretical spectrum: g (X) = ai 2 2 1 + <J>^  +cj>2 — 24>^(1—cfs )^ COS2TTX-2(J)2COS4 ITX for (1,0,1) model * x and 6 obtained by solving r± = ( 1 ~ < i > l 6 l ) ^ 1 " V a 2 = °x 2 (1 -cfr, 2) 1 + - 2$ 10 1 2 i + e x - 2 * 1 e 1 r i * i ' -i«l>1<i , - K c b ^ l 2 ( W - 26cos 2TTX) ,0^X^% d+* x 2 - 2<j>cos 2TTX) A more accurate estimates of the parameters can be obtained through a constrained least squares technique given in (1). For instance, for a (2,0,0) model, X £ = ^ i ^ t j_ + ^2 Xt-2 + a t ' t**e P a r a m e t e r s < f ' i a n c* $2 can be estimated through a regression model constrained so that the 183 intercept i s zero. a i s considered the residual and i s merely the residual variance. Diagnostic checking To determine the adequacy of the fi t t e d model, several tech-niques are given in (1). For this study i t is deemed sufficient to check i f the theoretical spectrum of the f i t t e d data i s not s i g n i f i c -antly different from the spectrum of the original data. Using this procedure, the (2,0,0) model was found to be inadequate for yarders 0905 and 0911. However, the (3,0,0) model subsequently f i t t e d was found adequate for yarders 0905 and 0911. The resulting f i t t e d models for the four yarders are given below. Their theoretical spectrum are plotted in Figure F.4 and F.5. 0905 X_ = 37.2363 + 0.3928Xk . - 0.0507X „ - 0.0460X _ + a t t — 1 t—z t—3 t a 2 = 84.6359 a 0911 X = 26.6979 + 0.4339X + 0.1331X - 0.1823X + a t t —X t — 2m t"~ %3 t a 2 = 81.9031 a 0909 X„ = 36.9817 + 0.1785Xt . + a_ , a 2 = 95.2049 t t-1 t a 0904 X„ = 10.4937 + 0.7586X , + a - 0.4861a , , a 2 = 64.2259 t t-1 t t-1 a 184 50 0.0 0.1 0.2 0.3 0.4 0.5 F R E Q U E N C Y (cycles per day) Figure F.4 The 12-lag spectrum for the adjusted yarding production data and the theoretical spectrum for the fit t e d model for yarders 0905 and 0911 185 X. 100-• Figure F.5 The 12-lag spectrum for the adjusted yarding production data and the theoretical spectrum for the fitted model for yarders 0909 and 0904 APPENDIX G PER CENT UTILIZATION GRAPHS 187 7 8 9 10 11 12 13 14 15 16 17 18 19 20 N U M B E R O F T R U C K S Figure G.l U -graphs for various configurations (Source: different replications from those used in the text) 188 N U M B E R S O F T R U C K S 7 8 9 10 11 12 13 14 15 16 17 18 19 20 N U M B E R S O F T R U C K S Figure G.l - cont. 189 7 8 9 10 11 12 13 14 15 16 17 18 19 20 N U M B E R O F T R U C K S 8 High Lead Yarders 100 T 7 8 9 10 11 12 13 14 15 16 17 18 19 20 N U M B E R O F T R U C K S Figure G.l - cont. 190 U (TRUCK PER CENT UTILIZATION) 85 90 95 U (TRUCK PER CENT UTILIZATION) Figure G.2 Expected daily production per truck as a function of the truck per cent u t i l i z a t i o n for various configurations 191 Figure G.2 - cont. ;ure G. 2 - cont. 193 APPENDIX H FLOW CHART OF THE VARIOUS ROUTINES IN THE PROGRAM In the following pages, the flow chart of each of the event routines l i s t e d on page 39, with the exception of four routines, are given. The four routines are: 1. Yarding 2. Travel time generation 3. Overtime 4. Start-up. The f i r s t three of the four routines l i s t e d above are adequately described in the text. The "Start-up routine", on the other hand, i s incorporated in the executive program (see Figure 3.2). The computer l i s t i n g of the logging simulation program (approxi-mately 55 "print-out" pages) i s available at the University of British Columbia Faculty of Forestry and may be obtained on request. 19* SU3R0rri.N1 LOADNG ( a r r i v a l a t s p a r ) l o a d e r down? s e c o n d a r y d i s p a t c h i n g r o u t i n e to send t r u c k ( s ) to a n o t h e r l a n d i n g r e t u r n l o a d e r b us v' Yes g e n e r a t e l o a d i n g t i me and v o l u n e t o be l o a d e d Yes truck, j o i n s the queue X return t r u c k - - a i t s as l o g s a r e y a r d e d s c h e d u l e :he c e p a r -t u r e o f the truc.< f r o - the r e t u r n 195 generate t r a v e l tine to camp, caz^ > delay, and travel time to dump, schedule a r r i v a l of the truck at the dunu ( \ \ re turn J trucks i n the queue advance forward, the f i r s t truck i s to be loaded, generate the loading time and the volume to be loaded  update landing volune generate the moving tine for the loader, schedule the time the loader i s set-up I re turn P secondary dispatching routine to send the truck(s to the other sides reduce o r i g i n a l loading tine proportionally truck v a i t s as logs are varded schedule departure of truck frcm spar I return 1 196 SUBROUTIN'E (generation c CDELAY f caxp delay) Yes schedule departure of the truck from the canp return ^ LN # 11 i a p l i e s this subroutine Is c a l l e d fron subroutine TRAVSD to generate can? delay for trucks approaching the cacp to be dispatched. I t i s assumed that two trucks are needed to p u l l the lowbed (as i n the Harrison K i l l s logging d i v i s i o n ) . GD 197 SU33DCTIKE TRAVSD ( t r u c k d i s p a t c h i n g and t r a v e l l i n g t o the s i d e ) u p d a t e volume a t a l l l a n d i n g s Yes o r e r t i n e , d i s p a t c h i n g r o u t i n e to send t r u c k s to the s i d e s — M re turn J i n c l u d e f o r d i s p a t c h i n g a l l e n p t y t r u c k s a p p r o a c h i n g the camp, gene-r a t e c a n ? d e l a y i f r e G u i r e d i n c l u d e t h e s e t r u c k s f o r d i s p a t c h i n g , updatt t h e i r " d e p a r t u r e " the ca.-p  c l a s s i f y as c l a s s 2 l a n d i n g r e ~ o v e i r o n tne e v e n t l i s t the o l d s c h e d u l e o f a r r i v a l o f the t r u c k s i f lowbed ( o r s u p p o r t t r u c k ) i s r e q u i r e d , e n t e r l a n d i n g i n t o l i s t o f l a n d i n g s n e e d i n g a lowbed ( s u p p o r t t r u c k ) s o r t l a n d i n g c l a s s 1 by v o l u n e ( o r by d i s t a n c e i f a f t e r 1 : 0 0 P.M.) and l a n d i n g c l a s s 2 by " w a i t i n g " time r e - c h e c k e a c h l a n d i n g f o r c a p a b i l i t y o f s u p p o r t i n g .r.ore l o a d s ( c l a s s 3 l a n d i n g s s o r t t r u c k s f i r s t by c a p a c i t y and n e x t bv time o f d e p a r t u r e I \ f o r e a c h t r u c k "to be dispatched!"") d i s p a t c h s u p p o r t t r u c k ( o r the lowbed i f r e a d y ) t o the s c h e d u l e d s i d e d i s p a t c h t r u c k t o the g i v e n l a n d i n g s c h e d u l e a r r i v a l o f t r u c k a t s p a r return GD 198 SUBROUTIST: U S T O A D ( a r r i v a l at the dvuap) du^p Yes closing tine,. generate unloading time, schedule the departure of the truck Yes truck j o i n s . / the queue ( truck i s p arked i n i t s loaded state at the dump 1 T return 1 I return I 199 C D SUBROUTINE (departure travel t TRAVCP fron ducp, o canp) truck needed for overtime ha u l i a * truck needed for lovbed dutv truck i s parked e~pty overnight at the dump generate t r a v e l time to camp, schedule the a r r i v a l of the truck at the camp Ye3 trucks i n the queue advance forward, f i r s t truck i s unloaded, schedule i t s departure from the dump return 200 update volume at the landing and at the setting yarder prepares to move CD 201 S U B R O U T I N E Y R S E T (termination of setting-up of yarder) yarder is set-up and yarding resunei >, update the setting volume and the other setting status variables CD SUBROUTINE LRSET (termination of moving and setting-up of loader) loader is set-up and loading can resume, update a l l intemode distances, landing volume, and other pertinent variables lowbed is noved to the appropriate side lowbed is roved back to the carrp, lowbed trucks resune hauling i f there is enough time for a trip CD 202 SUBROUTINE ANCHOR ( a r r i v a l of support truck to anchor the moving of the yarder or the loader) calculate the moving and set-up times for the yarder, schedule the time the yarder i s set up CD SUBRDUTIXE CARRY ( a r r i v a l of lowbed for moving the yarder or loader) calculate the coving and set-up times, schedule the time the yarder i s set-up 203 CD SUBROUTINE LUSCHN (noontime stoppage of yarding) for all landings: give the loading crew extra half hour overtime update the volume at the landing to include volume yarded until noontime (T = 300) yarding stops set the last t updated to T -for half hour, .me volume was 330 return 204 CD SUBROUTINE FAILED (equipment breakdown) re-schedule the day of breakdown generate repair time remove from the event l i s t the next scheduled event for the given truck repair i s done by an outside shop G E D repair job l i s t e i n the queue determine the day the repair w i l l se finished cancel the overtime for this side update the volume at the landing up to the time of breakdown Yes No 1 lav aside a low p r i o r i t y job r schedule the termination of the varder reoair detemine the day the repair w i l l be finished schedule the termination of the truck reoair > return • / repair job l i s t e d i n the queue * It i s assumed that truck f a i l u r e never occurs when the truck i s on lowbed duty. ** The routine for loader breakdown is s i m i l a r to the routine for yarder breakdown except that the landing volume i s not updated i n the case of a loader breakdown. 205 CD SUBROUTINE RESUME (resumption of duties after repair) type of 1 truck set a v a i l a b i l i t y of truck for next day tine aft e r std. tine aft e r std. ^ s h i f t ? lYes c yarding resumes next truck i n the queue i s repaired determine the day repair w i l l be finis h e d return schedule the termi-nation of the repair set a v a i l a b i l i t y of equip, for next day schedule the termina-tion of yarding and the s t a r t of moving loader duties resume next yarder i n the queue is repaired next loader i n the queue i s repaired determine the day repair of yarder ( 0 1 loader) w i l l be finished return ^ « schedule the termination of the repair of the yarder (or loader) CD 206 SUBROUTINE TOW (adjustment at day's end of hours worked and mileage for the tow truck) time and mileage adjustment for towing waits u n t i l a tow truck i s available return 207 CD SUBROUTINE YMOVED (Cernination of noving of yarder) indicate that the loader i s set-up and ready »f^^ return calculate the moving time, schedule the time the loader finishes moving support truck waits support truck resumes hauling duties support truck i s shut down at the camp return lowbed i s moved to the old landing to prepare the moving of the loader lowbed i s moved to the appropriate side lowbed i s moved back to the camp, the lowbed trucks resusie hauling i f there i s enough tine for a t r i p lowbed waits u n t i l the loader i i ready to move return Note: It i s assumed that a support truck Is needed to move any yarder with either a tire-type undercarriage regardless of moving distance or with a track-type of undercarriage for noving distances less than a mile. For longer distances, a lowbed i s required. 208 

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