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Marine log supply : a transport engineering analysis Kahkeshan, Siavoche 1986

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MARINE LOG SUPPLY: A TRANSPORT ENGINEERING  ANALYSIS  by SIAVOCHE  KAHKESHAN  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F REQUIREMENTS FOR THE D E G R E E OF D O C T O R OF PHILOSOPHY  in T H E F A C U L T Y O F G R A D U A T E STUDIES C i v i l Engineering  We accept this thesis as conforming to the required standard  T H E U N I V E R S I T Y O F BRITISH C O L U M B I A  June 1986  O  Siavoche Kahkeshan, 1986  THE  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study.  I further agree that  permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives.  It is  understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  C i v i l Engineering  The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date:  3une 1986  ABSTRACT  This thesis investigates the marine transport-inventory system of Coastal British Columbia. The intrinsic characteristics of marine log transportation in this region are the vulnerability of marine transportation to adverse weather conditions and the presence of time-related economic costs. The system is confined to the Powell River operation. Three origins, one storage area and three pulp log types are considered. The formulated problem is classified as a sequential decision-making process. A deterministic model using the network flow theory and a simulation model using GPSS are developed. Due to considerable uncertainty in the system operation, the computer simulation model is selected. The  model includes all of the important  interactions and assesses  system variables and their  alternative operational doctrines by calculating  variation in a key aspect of system performance, total logistic cost. It is found that: 1) the use of barges as the transportation mode leads to the least logistic cost, 2) the second best transportation alternative is the direct shipment of logs from origins to the mill and 3) if higher value saw log is considered, the log-taxi alternative may become attractive. However, to improve the capability of the developed model, more information  on the salt contamination  and teredo  damage and accurate  estimation of cost consequences of a mill shutdown are required. Future works should focus on these areas.  ii  T A B L E OF CONTENTS  Abstract  ii  L i s t of Figures  vii  L i s t of Tables  ix  Acknowledgement  I.  II.  x  INTRODUCTION  1  1.  Problem Identification  3  2.  C o m p l e x i n g Issues  5  3.  Objective Definition  6  4.  Outline of the Thesis  9  11  L I T E R A T U R E REVIEW 1.  General L i t e r a t u r e Review  11  2.  Log-Industry L i t e r a t u r e Review  13  3.  Summary  15  P A R T 1; THEORY  ID.  IV.  18  SYSTEM F O R M U L A T I O N  19  1.  System F o r m u l a t i o n  19  2.  L o g i s t i c Cost Components  20  3.  Assumptions  21  4.  Summary  22  SIMULATION M O D E L  2*  1.  Model Development  24  2.  Summary  25  iii  V.  NETWORK FLOW MODEL: FORMULATION  30  1.  Selection of Decision Variables and Parameters  30  1.1.  Decision Variables  30  1.2.  Parameters  31  2.  VI.  Network Programming Model  33  2.1.  Formulation 1  33  2.2.  Formulation 2  35  3.  Technical Calculation Aspects  36  4.  The Matrix Structure  36  5.  Summary  39  NETWORK FLOW MODEL:  ALGORITHMS  45  1.  Literature Survey  45  2.  Decomposition Theory  46  2.1.  Benders Decomposition: The Master Problem  46  2.2.  Benders Decomposition: The Subproblem and Procedure  49  Dantzig-Wolfe Decomposition: The Master Problem  53  Dantzig-Wolfe Decomposition: The Subproblems Procedure  54  2.3. 2.4. 3.  Specialization  55  3.1.  Dantzig-Wolfe Master Problem Translation  56  3.2.  Dantzig-Wolfe Subproblems Translation  56  3.3.  Benders Master Problem Translation  58  4.  Computational Aspect  60  5.  Summary  62  SUMMARY OF PART 1  iv  63  PART 2; APPLICATION VH.  ANALYSIS OF AVAILABLE DATA  65  1.  Production and Consumption  65  2.  Transit Time  69  3.  Transportation Cost  75  4.  Inventory Charges  78  4.1. Financial Cost  79  4.2. Quality Deterioration Cost  79  4.3.  79  5. Vin.  64  Shortage Cost  Summary  80  TACTICAL PLANNING  81  1.  Equilibrium  81  2.  Validation  96  2.1. Extreme-Case Analysis  98  2.2.  98  Review by Experts  2.3. Observation of Intermediate Results  100  2.4. Comparison of Model Behaviour with Real-System Comportment  IX.  100  3.  Variance of Observed Performances  101  4.  Determination of Sample Size  102  5.  Summary  104  ANALYSIS  105  1.  Transportation System  105  1.1. Log Taxi System  106  1.2. Barge-Dryland Storage System  106  1.3.  106  2.  3.  Direct Shipment  Inventory System  107  2.1. Minimum Inventory Level  107  2.2. Inventory Depletion Policies  107  Salt Contamination Function  108  v  IX.  ANALYSIS (Cont'd) 4.  Analysis of Variable Effects  108  4.1.  The Effect of the Inventory Depletion Modes  110  4.2.  TRANS-INV-FORM Interaction Effects  110  4.3.  Multiple Comparison Procedure  113  4.3.1. 4.3.2. 5. 6.  X.  Model Sensitivity to the Salt Functional Form Comparison of Transportation and  118  Inventory Policies  119  The Effect of the Vessel Capacity  123  5.1.  124  Nonlinear Effect of the Link Capacity  Probability Distribution Function of Inventory Levels  130  CONCLUSION  137  1.  Evaluation of the System-Analysis Models  137  1.1.  Deterministic Model  137  1.2.  Simulation Model  137  2.  Performance of Transportation Policies with Respect to Minimum Mill Inventory Levels  138  3.  Sensitivity Analysis  140  4.  Contribution of the Study  140  5.  Future Research  141  Appendix 1: Simulation Model Listing  142  Appendix 2: Transit Time Data  151  References  152  vi  LIST OF FIGURES 1.1. MAPI.l. 1.2. II. 1.  Logging and Log Transport Process Flow  2  Geographical Boundaries of the System  7  System Definition  8  Process Control and the Forest-harvesting System  16  III. l .  Salt Contamination Functional Forms  23  IV. 1.  Model Flow Chart  26  V.l.  Structure of the Technological Matrix  40  V. 1.1.  Structure of the Technological Matrix (Window 1)  41  Structure of the Technological Matrix (Window 2)  42  Structure of the Technological Matrix (Window 3)  43  V.1.2.  V.1.3.  V.1.4.  Structure of the Technological Matrix (Window 4)  44  VII. 1.1.  Production Rates at Port M c N e i l l and Kelsey Bay  67  VII. 1.2.  Production and Consumption Rates at Eve River and Powell River  68  Weibull F i t for the Transit Time Models (PMN-TA, KB-TA)  71  VII.2.1.  VII.2.2.  Weibull F i t for the Transit Time Models (ER-TA, TA-PR)  72  VII.2.3.  Weibull F i t for the Delay Models  73  VII. 2.4.  Poisson F i t for the Transit Time Model (TA-PR)  74  Daily Variation of the Large Pulp Inventory L e v e l at the M i l l  87  Daily Variation of the Large Pulp Inventory L e v e l at the Storage A r e a  88  VIII. 1.1.  VIII. 1.2.  vii  VIII. 1.3.  VIII. 1.4.  VIII. 1.5.  VIII. 1.6.  VIII. 1.7.  VIII. 1.8.  VIII. 1.9.  Daily Variation of the Large Pulp Inventory L e v e l at the Origins  89  Daily Variation of the Small Pulp Inventory L e v e l at the Mill  90  Daily Variation of the Small Pulp Inventory L e v e l at the Storage A r e a  91  Daily Variation of the Small Pulp Inventory L e v e l at the Origins  92  Daily Variation of the Camprun Pulp Inventory L e v e l at the Mill  93  Daily Variation of the Camprun Pulp Inventory L e v e l at the Storage A r e a  94  Daily Variation of the Camprun Pulp Inventory Level at the Origins  95  VIII.2.  U t i l i t y vis-a-vis Validity of Models  96  VIII. 3.  Age Distribution of Logs  99  IX. 1.  Linear and Constant Salt Contamination Functions  109  IX. 2.1.  The Transportation and Inventory E f f e c t s for the Theoretical Salt Function  115  IX.2.2.  The Transportation and Inventory E f f e c t s for the Time-Independent Salt Function  116  IX.2.3.  IX.3. IX.4.1. IX.4.2.  IX.4.3.  The Transportation and Inventory E f f e c t s for the Linear Salt Function  117  The C a p a c i t y and Inventory E f f e c t s  126  Weibull F i t for the Opening Inventory Models (January, April)  131  Weibull F i t for the Opening Inventory Models (Duly, September)  132  Weibull F i t for the Opening Inventory Models (December)  133  viii  LIST O F  II. 1.  TABLES  Example of Computer Processing Time  14  VII. 1.  Y e a r l y Production and Consumption Rates  66  VII.2.  Transit T i m e Statistics  70  VII.3.  Goodness of F i t Evaluation  76  VII. 4.  Transit T i m e Probability Distribution Parameters  77  VIII. 1.  Analysis of Variance Table for the Determination of the Equilibrium State  84  Fleet Size — Initial Condition Interaction E f f e c t s  86  VIII. 2.  IX. 1.  IX.2.  Four-Factor Analysis of Variance Table (TRANS-INV-FORM-IMODE)  111  Three-Factor Analysis of Variance Table (TRANS-INV-FORM)  112  IX.3.  Response C e l l Means for the Three Levels of F O R M  114  IX.4.  T R A N S - F O R M Response C e l l Means  120  IX.5.  Scheffe Multiple Comparison Test (TRANS-FORM)  121  IX.6.  Scheffe Multiple Comparison Test (TRANS-INV)  122  IX.7.  CAP-INV Analysis of Variance Table 1  125  IX.8.  CAP-INV Analysis of Variance Table 2  128  IX.9.  Linear, Quadratic, Cubic and Quartic E f f e c t s of C A P A C I T Y Goodness of F i t Evaluation for the Opening Inventory Model  IX. 10.  IX.ll.  Opening Inventory Probability Distribution Parameters  ix  129 134  136  ACKNOWLEDGEMENT  I wish to express  my immense gratitude  to my research  supervisor,  Professor F . P . D . N a v i n , for his invaluable advice and guidance throughout this research.  I would also like to thank Professors G . R . Brown, W. F . C a s e l t o n and S. O . Russell (Department of C i v i l Engineering); and Doctors P . L . C o t t e l l and T. D . Hinthorne ( M a c M i l l a n Bloedel Ltd.) for their valuable comments.  A p p r e c i a t i o n is given to M r . R . L . Greenough, M r . W. Judge ( M a c M i l l a n Bloedel Ltd.) and M r . M . J . MacNabb (Department of C i v i l Engineering) for their cooperation and t i m e given during this study.  I am grateful to M r s . M . Navin and M r . R . A . Y a w o r s k y , whose help has improved the style of this thesis.  F i n a l l y , I wish to thank my parents for their support and encouragement, without which this work would not have been possible.  x  1  I.  INTRODUCTION  The transportation of logs in coastal British Columbia is dominated by the topography of the British Columbia coast. Most of the log supply of coastal mills is provided by the Queen Charlotte Islands (QCI), Vancouver Island and the 1000 kilometer long coastal forest bounded by the Coast Range Mountains and the Pacific Ocean. Only 2% of the coastal timber has access to the mills by ground transportation. The space-time diagram in Figure 1.1 illustrates the general process involved in the marine movement of logs from camps to mills. Phase I consists of falling and bucking logs at the camp. Logs are then transported by trucks to the water where they are sorted. In phase II, logs are towed to the storage areas, scaled and then towed to the mill's storage area. In some cases, logs are sorted and scaled at the points of origin and then towed directly to the mill. Finally, in Phase III, logs are transported from the mill's storage area to the mill's pond and then to the mill. Two general methods of water transport have evolved: - Barging, either towed or self-propelled, is used for direct hauls across open waters, for example from QCI to Vancouver mills. - Booming, usually of bundled but sometimes loose logs, in protected waters with intermediate storage.  2  FIGURE 1.1: Logging and Log Transport Process Flow  3  1.  Problem Identification  The intrinsic characteristic of marine log transportation is the presence of a high interdependence between the transportation and inventory systems. Consideration of the time aspect of marine log transportation increases the system complexity.  Due to this fact, until recently, the time-related  economic costs of the transportation of logs have been neglected. In his report, Betz (1982) stated that:  ". . . unlike many industries, the  transportation and inventory phases are combined making optimization of either aspect difficult.  The operating philosophy apparently has been to optimize  transportation.  Large  volumes  of  wood  are  moved  to  minimize unit  transportation costs. This results in delays while accumulation of economic units is occurring." Time related economic costs include: - Financial Cost: Inventory carrying charge as working capital is tied up. - Quality Deterioration Cost: o Salt Contamination; o Damage caused by parasites such as teredos and ambrosia beetles; o Sinkage. The total cost can, therefore, be seen as the sum of the transportation cost plus the time-related economic costs. A recent study by MacMillan Bloedel (8) shows millions of dollars are lost annually solely from the inventory carrying charges.  The magnitude of these  losses is even greater if costs due to the deterioration or loss of logs from teredos, salt uptake and sinkage are considered.  4  One way of minimizing the total cost is to bring the inventory level to near zero. However, due to the following practical factors, zero-level inventory is undesirable: - The presence  of the time lag between log production and mill  consumption wood production may be accelerated in some seasons due to unforeseen favorable weather conditions whereas mill consumption remains unchanged. - Adverse weather may delay the log delivery. - Mills may change planned cut orders to meet a variable market. - Labor disputes can affect production and delivery. - Inventory may be deliberately increased to mitigate the risk of interrupting mill production. Reduced inventory levels dictate that the operation components will have stronger inter-relationships.  Therefore, the need for careful  management  planning and control becomes vital. MacMillan Bloedel has recently (1982-1983) implemented a computerized log movement control system.  The new system displays much of the same  information as previously used for the manual coding of boom identification, such as volume, species (e.g. Balsam), type (e.g. Pulp) and sort (e.g. camprun) for all locations. However, inventory levels at each location are still determined on the basis of past practices. "Min and Max levels of each species and assortment are set for each location. These estimates are based on past experiences and are related to mill demand, stand type and the harvesting system employed." (8)  5  2.  Complexing Issues  Three main issues contribute to the complexity of the log transportinventory problem: - Time Horizon: The time horizon of the study should be long enough (510 years) if the seasonal factors, which play a major role in the design of the inventory profiles, are to be considered. - Multi-Item Operation: A typical forest product company on the west coast produces 7 to 10 species (Fir, Hemlock, Balsam, etc.), which are classified into 10 to 15 log types (Peelable, Sawlog, Pulp, etc.) and furthermore subclassified into 20 to 30 log sorts (Standard, Large, Small, Camprun, etc.). - Multi-Echelon Inventory System: The activity locations of the system are divided into three groups:  production, storage and mill areas.  Inventories are carried in 30 to 40 production sites, 100 to 120 storage areas and 15 to 20 mills. Consideration of the aforementioned issues renders the formulation of the entire system unrealizable due to the fact that their incorporation into a model increases prohibitively the required computation storage and processing time. Therefore, a sub-system of the log transport-inventory operation should be considered.  6  3.  Objective Definition  This research is to model the marine log transportation and inventory system present on the coast of British Columbia for the purpose of: i. formulating a system control model for the logistic system, i i . measuring the performance of two transportation modes; barge and towvessel, and two transportation doctrines; direct shipment and shipment via a consolidated point, with respect to various minimum inventory levels at the mill. iii.  determining the sensitivity of the system under study to different salt contamination functions. The present study will particularly emphasize the stochastic nature of the  transportation time and the perishable aspect of the logs. To  reduce the problem to manageable  issues, the MacMillan  Bloedel  Powell  River  proportions, due to the conversion  supplying log production divisions are considered.  plant  and its  This plant processes  large, small and camprun Hembal (Hemlock-Balsam)  pulp logs.  It  above major mainly  is supplied  from the Teakern A r m storage area. Most of the plant's consumption is produced at Port M c N e i l l , Eve River and Kelsey Bay.  The geographical map of the region  under study is presented in Map 1.1. The log transport system in this study is defined as that time from the placement of the log in the water at an origin camp to the delivery at the mill's holding areas (see Figure 1.2).  MAP  1.1:  Geographical Boundaries of the System  Source:  MacMillan Bloedel Limited  Phase I \ Falling • Buclnng  LogTrajectOPj  Time  FIGURE 1.2: System Definition  9  *•  Outline of the Thesis  The analysis of the transportation-inventory system is composed of two parts. Part 1 develops the theoretical framework of the system models and Part 2 analyzes the system under study by means of the 'best' model obtained in the first part. The plan of the thesis is as follows:  Chapter 2  reviews the existing literature with respect to transportation and inventory systems of perishable commodities and forest products.  PART 1  considers the theoretical aspect of the system formulation. In this part, the implementation  of the simulation  and network  flow  models is evaluated and the most applicable model is selected. The following chapters constitute the structure of this part. Chapter 3  develops  the general  formulation  of  the transportation and  inventory system. The assumptions of the study and the magnitude of the logistic cost components are discussed.  Chapter *  presents the framework of the simulation model for the marine log transportation problem.  Chapter 5  formulates  the problem  as  a  fixed  charge  multicommodity  capacitated transhipment flow model. Chapter 6  develops  the  theoretical  framework  for  a  three-level  decomposition algorithm and investigates its implementation.  10  PART  2  applies the selected model in Part 1 to the system under study.  Chapter 7  discusses the collection and analysis of the available data.  Chapter 8  investigates the pre-simulation phase of the simulation process. It determines the starting and steady-state conditions, validates the model and proposes the variance reduction measures to adopt in order to obtain reliable simulation data.  Chapter 9  considers the post-simulation phase of the study.  It assesses the  relationship between important system variables and evaluates different options. Chapter 10  evaluates the success of the implementation and applicability of the techniques used. A summary of obtained results and practical implications of the developed discussed.  model for forest  industry are  11  II.  LITERATURE  REVIEW  This chapter reviews the available literature on the general inventory theory and the forest industry.  A summary of the obtained information is  presented.  1.  General Literature  Review  A review was undertaken of the transportation and inventory systems of other perishable commodities. One problem closely related to log inventory is that of blood bank inventory, where blood is conserved in eight species and its useful lifetime is limited (21 days). Jennings (1973) developed a simulation model to study a regional blood inventory control system. He considered one type of blood (B ) and examined a +  variety of inventory policies with the aid of computer simulation.  When the  inventory level at any bank drops below a predetermined level, transfers are made from other banks selected randomly between those most recently known to have had the largest inventory.  However, in the context of our problem, the  extension of this model should include the more complex distribution system pertaining  to  the  logging  industry,  accounting  for  the  variability of  transportation time and the consideration of several types of logs. Nahmias (1975, 1977) developed a model describing the optimal ordering policies for a perishable single product with a lifetime of m periods. The model is based on an FIFO assumption and can be seen as a dynamic program with a  12  state variable of dimension (m-1) age level.  representing the total initial inventory at each  His model has a multi-echelon structure where the p th echelon  corresponds to the amount of product that will perish in p periods. with  the  exception  of  small  values  of  m,  the  model  is  However,  computationally  impossible. In a later paper, Nahmias (1976) developed approximational methods to calculate the inventory cost function. received instantaneously;  demands  His assumptions include:  all orders are  in successive periods are independent and  identically distributed random variables; unsatisfied demand is backlogged; F I F O policy is adopted; if the product is m  age old and still inventoried then it  deteriorates and is discarded at a specific per-unit cost, holding and shortage costs are convex, and ordering and deterioration costs are linear.  He used the  simulation method to verify the results of his approximational models. However, a difference between the log inventory problem and that of blood bank inventory exists. Blood has a constant value during its useful l i f e t i m e and a zero value after then.  Whereas, the value of log decreases continuously  from the first two weeks of watering until the time that the log is processed. Since the blood deterioration function is convex, F I F O policy is optimal. In the log inventory problem, if the sapwood deterioration is considered, L I F O policy is optimal. In  the  general  inventory  theory, the  log  inventory  problem  classified as a multi-echelon, multi-product inventory problem with demand, random lead-time and lost sale.  can  be  constant  The perishability aspect of log is  translated into an increasing holding cost. Peterson and Silver (1979), and Schwartz (1981) developed mathematical models for the above problem type and gave numerous references.  13  2.  Log-Industry Literature Review  In addition to the previously noted more general literature search, a survey of forestry periodicals and a review of many internal reports of several forest product companies was undertaken. Newnham Newnham  (1973) adopted  encouraged  the  a system  application  analysis of  approach.  simulation  and  In  his  paper,  mathematical  programming methods to the planning of inventory process controls and logging operations.  Figure  II. 1 is  Newnham's  diagrammatic  representation  of  the  relationship between process control and the forest-harvesting system. He states that:  " . . . When developing a control system, management  has  to consider the intensity of control that is desirable. Generally, tight control of a process results in lower inventories and less risk of a failure to meet mill demand.  However, intensive control systems are more costly and there is a  break-even point, often difficult to calculate . . .". Sellers (1971), Kirby (1968) and Thibaut (1962) were primarily concerned with inventory monitoring.  They explained the methods of controlling inventory  as practiced by different companies.  However, no formal inventory levels are  obtained from their models. Sellers (1971) presents a good example of how improving the planning of the  road  system  and cutting  areas  reduces the peaks  and  valleys  of  both  deliveries and mill inventories. He reports that: "Plans are now being made to place the mechanics of the woods program on electronic data processing.  Such a sophisticated tool will  enable us, through linear programming, to investigate and analyse many of the  14  factors which affect a woodlands production program."  However, no further  recommendations are proposed. Lonner (1968) pioneered a mathematical approach to this problem. formulated a linear programming assortments and 4 time periods.  model limited to  1 mill, 23 origins,  He  5 log  Since the transportation was done by truck, no  variability in transit time was considered. He then solved the problem by means of the Simplex method.  An example of the computer processing time is provided  in Table II. 1.  . .. * * Density  Number of rovs in matrix  Number of columns in matrix  (incl. slacks)  154  264  1.54  3.41  IBM 7090  519  760  0.61  32.75  IBM 7044  561  2700  0.60  103.17  IBM 360/50  71?  952  0.35  19.48  IBM 360/50  717  952  0.35  11.89*  IBM 360/50  *  %  Processi ng time in minutes  Computer type  Utilizing an earlier solution Percentage of non-zero elements  TABLE II.1:  Example of Computer Processing T i m e Source: Lonner (1968)  15  Holemo (1971) conceptualized the problem of inventory management in lumber production. He showed how to determine the optimum inventory level by means of the balancing costs method (EOQ method). "When inventory costs have been identified and quantified, they must be balanced to determine the optimal level of inventory.  Mathematically we can write a cost equation including all  relevant costs and, utilizing calculus, solve to find the level of inventory that will minimize cost." He stressed the employment of analytical tools and firm management in the decision-making process. He states: "Policies based on intuition are not always easy to administer, easy to keep track of, or easy to keep up to date.  Inventory management is a  logical area in which to begin to use analytical methods." Erdle, Arp and Meng (1980) illustrated how yearly pulpwood inventory levels at the roadside and millyard can be optimized by way of computer simulation. They considered 1 roadside storage area, 1 mill and 5 2 time periods. The delivery of logs to the storage area and to the mill were assumed to be instantaneous.  3.  Summary  The blood bank inventory problem is considered to be closely related to that of log inventory since blood is conserved in different species and perishes over the time.  However, blood has a constant value during its useful lifetime  and a zero value after then. Whereas, the value of log decreases continuously from the first two weeks of "watering" until the time that the log is processed.  16  Preventive Plointenonce  Repair  Replacement —1  r  EQUIPMENT CONTROL "HI  DLL DE" DEMAND Forest Stand  Felling  Stumparea Inventory ,  Primary transportation (skidding, forwarding, etc.)  Roadside Roadside inventorg processing (gnpro(delimbing, cessed slashing, vood) etc.) | +  Roadside Inventory (semiprocessed wood)  Secondary transportation (truck, river, etc)  Mill Inventory  KEY  Action Information  FIGURE n.l: Process Control and the Forest-harvesting System Source: Newnham (1973)  17  In  the general  classified  inventory  as a multi-echelon,  theory,  the log inventory  multi-product  inventory  problem  can be  problem with  constant  demand and random lead-time, where backlogging is not allowed. In this class of problem, the perishability of logs can be considered as an increasing holding cost. The accounting  available literature on the forest industry  contains  few studies  for the transportation attributes of the road distribution network.  However, in order to reduce the complexity of the problem, deterministic transit time and direct shipments are assumed. The  marine log transportation along the coast of British Columbia is a  special case of the log transportation and inventory systems described in the literature.  This system is unique in its genre due to the vulnerability of marine  transportation to adverse weather conditions and the presence of time-related economic costs. In the past, due to the relatively large inventories, transport mistakes made were not serious. more  precise  analysis  implementation costs.  and  engineering  However, due to reduced inventory  controls  are necessary,  resulting  in  levels, higher  The break-even point between the implementation of  sophisticated mathematical models and the prohibitive inventory and shortage cost of "rules-of-thumb" policies is still generally difficult to evaluate. Based  in part  on  the computational  difficulties  literature review, it is proposed that simple, short-term developed.  discovered  in the  models should first be  Then, by increasing the model's degree of complexity step-by-step,  the analyst will be in a better position to estimate when a further increase in the model sophistication will not bring a corresponding increase in benefits to the organization.  18  P A R T 1: T H E O R Y  This part develops the theoretical  framework necessary to  the  formulation of the operation of the system under study. The simulation and network flow models are considered as candidates for modelling of the system operation. The selection of the final approach is based on the success of the implementation of the derived models, their evolutionary aspect and their degree of simplicity.  19  ffl. SYSTEM FORMULATION  In this chapter, first, the general formulation of the optimal control of the dynamic system under study is discussed. Then, the order of magnitude of the different components of the logistic cost is presented and finally, the assumptions used in the study are defined.  1.  System Formulation  The log transportation and inventory system can be formulated as a discrete-time dynamic system: s  n+l = *n ( n» n» £>n) s  n = 0, 1,  d  where the state S e 2 n »  n=u  n  >  N-l  •••» N, is the inventory balance at the beginning  of period n; the decision d £ D (S ) is the volume of logs ordered at the beginning n  n  n  of period n; and the random transit time S . n  Given an initial state, S A= (do, di,  OJ  the problem is to find an admissible control law  dN_i) which minimizes the estimated cost of the above  multistage decision process.  K (S ) = E I g (S ) + ^ g (S , d , 5 ) j N  A  Q  N  N  n  n  n  n  n n = 0, 1,  N-l  where: g (S , d , & ) = holding cost + transportation cost + shortage cost n  n  n  n  = the logistic cost  20  §N (Sjsj)  = the salvage value of the final stock which can be set equal to zero as a first order of approximation.  In addition, it is required that:  D (S ) = [0,00) n  n  2.  Logistic Cost Components  2.1  Holding Cost The holding cost is composed of the financial and quality deterioration  cost. The log inventory averages about 9 million m^/year. This represents an average direct inventory cost of about $365 million per year. As for the quality loss, a cost of $1.77/m^ for every 0.1% salt present in chips is estimated. This results in a loss of about $5 million per year.  2.2  Shortage Cost Experts at MacMillan Bloedel estimate that log shortages at the Powell  River mill would incur a representative cost of $350,000 per day.  2.3  Transportation Cost Towing from origins to the storage area of Teakern Arm costs $1.5/m3,  whereas towing from the storage area to the mill costs about $.3/mA  21  3.  Assumptions  To simplify  the system  structure, several assumptions  are made.  However, attempts have been made to maintain a reasonable degree of realism. The assumptions are: - The Port McNeill, Eve River and Kelsey Bay camps are the sole suppliers of the Powell River mill, which in turn, is the only consumer. - The Powell River mill uses Hembal (Hemlock-Balsam) large, small and camp run pulp logs. - The sapwood/heartwood ratio used in the production of chips is 1/9. - Similar to the real system, inventories are depleted according to the FIFO (First-In-First-Out) rule. - Due to the confidentiality of data, representative figures are used as input to the model. - The maximum inventory levels at the mill for large, small and camprun pulp logs are 30,000 - 30,000 and 15,000 m , respectively. 3  - Logging camps operate 5 days per week, whereas mill and towing operations are scheduled for 24 hours per day, 7 days a week. - Top production months are May, June, September and October. Production camps are closed during the fire season (July, August) and the month of December. The January startup is slow. - The Port McNeill and Kelsey Bay camps have a fixed tug-visiting frequency (every 10 days), whereas a tug is sent to Eve River when the production at this site forms a tow.  22  - If a tug is required when none is available, the requisition is held according to the FIFO (First-In-First-Out) rule, until a tug becomes free. - The tug's return trip is less than 24 hours. - No shipping is allowed between any two origins. - The log decay function due to the salt is illustrated in Figure III. 1. - The rate of (simple) interest is 10 per cent.  4.  Summary  In this chapter the log transportation and inventory problem is formalized. The problem is classified as a stochastic multistage decision process where multiple products are stored at numerous locations. Different log types carried in inventory should be considered distinct the one from the other since their market values, the rate at which they are produced and consumed and their shipped volumes are different. dynamic programming approach is not  suitable  for the  Therefore, the solution of the  aforementioned problem, since the state dimension grows with the number of log types carried in inventory. In the following two chapters, the development of a simulation model and the deterministic formulation of the problem are discussed.  23  SALT GCNTOT -MJUfl  &UK  WMXD >t_m_»  —  •  SAP woo o  N:--.^.  •v  / / *  • •  f  1  r"  i  I  l  l  l  1  1  1  1  OATS  l°°  toys  i  0  °  FIGURE m.l: Salt Contamination Functional Forms for Balsam and Hemlock Source: Betz(1982)  2k  SIMULATION MODEL  IV.  This chapter describes the framework of a simuiation approach.  The  simulation model objectives and description, the model structure are discussed.  1.  Model Development  The  purpose  of this  model  is to determine  the c r i t i c a l  parameters  expressing the system and establish a means of comparing different alternatives. Since i t is necessary and desirable for the model to be easily modified and become more complex only in conjunction with the need, the model is adaptive and evolutionary. The General Purpose Simulation System (GPSS) (35,61) was selected as the simulation language due to its block structure along with its use of transactions. The model consists of 253 blocks, 4 queues, 1 storage, 20 functions, 8 variables, 23 savevalues, 22 matrix savevalues, 5 user chains, 7 logical switches, 1 table and k random number generators. The model listing is presented in Appendix 1. Each transaction presents a production lot and carries the watering time, the  production  location  Furthermore, in order  and  volume,  to generate  experiences, four random-number  and  identical  generators  the log sort  as  parameters.  configurations for replicating are considered.  The model is  composed of 6 segments: - The initialization segment provides initial inventory at a l l locations in order to reduce the simulation transient time.  25  - The operation control segment shuts down the operation at production camps every weekend. - The inventory control segment generates the daily production and records the daily inventory level at all camps, the storage area and the mill. - The towing segment simulates the existing towing operation between every pair of locations. - The mill operation segment generates the mill's daily consumption and records the shortage volume if applicable, and finally, - The bookkeeping segment tabulates the weekly inventory levels at all locations and calculates the disaggregated operation cost figures. Figure IV.I illustrates the model flow chart.  2.  Summary  The structure of a GPSS simulation model is discussed in this chapter. The system under study, consisting of three origins, one storage area and one mill, is modelled. This system produces and consumes hembal large, small and camprun pulp logs. Inventories at all locations are depleted according to the FIFO (First-In-First-Out) rule.  START  SET INITIAL CONDITIONS  ® DAY TJ-  WEEK \ N .DAYS »> •  <D  PRODUCE LOGS  MOVE THE TUG BETWEEN ORIGIN T AND THE STORAGE AREA  RELEASE THE TUG  INCREASE ORIGIN AND SYSTEM INVENTORIES  F I G U R E I V . 1:  Model Flow Chart (Cont'd)  DAY "d  RECORD LOG TYPE  QUEUE FOR A TUG  MOVE THE TUG BETWEEN THE STORAGE AREA AND THE MILL RELEASE THE TUG  FIGURE IV. 1: Model Flow Chart (Cont'd)  28  ® DAY "d  -  DEPLETE ORIGINS INVENTORIES  CONSUME LOGS  ^ 1  RECORD THE SHORTAGE  •<  DEPLETE STORAGE AREA INVENTORIES  INCREASE STORAGE AREA INVENTORIES  *  DEPLET!• MILL INVENT ORIES 1  CALClJLATE INCREASE MILL  TOTAL DAILY  INVENTORIES  INVENTC)RY COST  FIGURE IV.l: Model Flow Chart (Cont'd)  FIGURE IV. 1: Model Flow Chart  30  NETWORK FLOW MODEL: FORMULATION  V.  In this chapter, a deterministic model based on the network flow theory is considered. The list of variables and parameters necessary to the development of the model is first presented. Then, the mathematical model of the system  under  study is formulated.  1.  Selection of Decision Variables and Parameters  Prior to presenting a mathematical formulation of the model, it is useful to  define  the decision  variables  and the parameters  of the model.  The  measurement units are lO^m^.  1.1  Decision Variables  Y  t jsp  X  sdp  V  j0p  :  °-  u a n t l t  :  Q  n  :  u  a  t  y  ^ i' t  Quantity in  period  °f 1 3 ' s h i p p e d Q<  °f  ± 0 <  3  1  shipped  of l o g/ shipped  j  from  from  s  to s  i n period  to d i n period  directly  from j  to d  p.  t t t I. , I I , , III : Inventory ]P dp SP during  a t j.d.s,  period  respectively, '  p.  tr  2 1  p.  p.  31  t Q^pi  Quantity of l o g type t bought a t d e s t i n a t i o n d i n period  : 1 i f route is  5.  *j^p . ^ :  6  1.2  p.  route jd  i s taken  i s taken  : i f route sd i s taken  , sdp  in period  p: 0 otherwise  i n p e r i o d p; 0 otherwise.  i n p e r i o d p; 0 otherwise, y  Parameters The parameters of the model represent the productivity, demand, haulage  capacity, number of vessel limitation and cost.  t S..p : Quantity of l o g type / produced  at j  i n p e r i o d p,  : Quantity of l o g type / consumed at d i n p e r i o d p. dp Haulage c a p a c i t y between j and s i n p e r i o d  p.  y., : Haulage c a p a c i t y between j and d i n p e r i o d _dp  p.  7j p S  :  32  Haulage c a p a c i t y  between  Number of v e s s e l s Transportation Unit  available  fixed cost  transportation  link  in period  Average c o s t  s and d i n p e r i o d  in period  per p e r i o d  v a r i a b l e cost  p.  p. (= wage + f o o d )  f o r the a p p r o p r i a t e  p.  of Phase I o p e r a t i o n  per 1u m 3  3  of l o g  produced. Market v a l u e Realisation  for log type-i. o f t h e random t r a n s i t  shipment on (..) l i n k q  : Interest  started  r a t e , decay  i n t e r e s t and d e c a y  in period  p.  r a t e and i n v e n t o r y  charges per p e r i o d . Daily  time f o r the  rate.  carrying  2.  Network Programming Model  2.1  Formulation 1 The mathematical formulation of the model is as follows: Min  I  p  f  Z r fs • Z r fs JJ JJP jsp sd 3dp sdp  •  I V Y ' tjs J ' P J » P  1  •  • Zrfi  jd Jdp jdp  v x • Zv v tsd sdp sdp tjd J P J l  S  d  *kw <'>i,  •  1  d  •  P  ]i  •  l/2[  *  t i d W ^ p > * ;  Z l U i ^ r l C l J ^ , )  •  *  p  fdW^'mP> ; v  t  ZnO^irHllWllJ^,)  Z Md^^rKlII^IIi;^,)  •  d  ] ] • Zli,Q  dp  subject to :  1)  2)  3  )  S t D t i Zv • 2 V •! »  jsp  d  J<*P  J.P*«  -I  t JP  . s  -  J£j;t6T;pcP  1  JP  Z x • Z V * * " * -II* *Q* = D* 3 sdp j jdp dP d,p*1 dp dp  d60;t€T;p P  i V *ZX*  S€S;UT; €P  !  j  d  JSP  •HI  3dp  -HI*  1  3P  = 0  ,  €  P  3.P*1  4)  ZY* - j , p * j , p  ^0  J€J;S6S; 6P  5)  ZX* - *, p*,dp  * 0  S€S;d€D; € P  sO  jcJ;d«D;p£P  8  t J9P  d  t  S  P  «P  6)  ZV* - I S t jdp j«*P J«lp  . ''  I J js J » P jd J * P  7  P  »d  Y.V.X.I.II.III » 0  a  d  P  ;  £ NV_  5,6.6 6(0,1)  j = j - { i / shipment from j has a delay ) S = s " { I shipment from s has a delay ) s  p £ P  p  S = (l'/.'.'.is) D =  [  ,  D  )  p = (1 ,...,P)  d p  34  The resulting model is a mixed (0-1) linear programming model. The objective function minimizes the fixed and variable transportation costs.  The seventh, eighth and ninth cost components penalize the transit  inventory due to the capital committed and insurance charges.  Next, the  average in situ inventory cost is accounted for by computing the cost of capital committed, the insurance cost and the inventory carrying charges. Finally, the last cost component considers the alternative of purchasing logs at the mills. Constraint (1) requires that the volume of logs shipped from origin j , either directly or via a storage area, should be equal to the change of the inventory level between two consecutive periods plus the volume supplied by logging operations. Constraint (2) states that the volume of logs shipped to mill d from an origin, either directly or via a storage area, plus the volume of logs bought at the mill should be the same as the volume of logs consumed plus the change in the inventory level. Constraint (3) requires that every log sent to storage area s should be either shipped to a destination or put in inventory. Constraints (4-6) determine the maximum volume of logs which can be hauled by a vessel. Constraint (7) limits the maximum number of vessels that can be operated at the same time. The presence of fixed transportation cost, different log types, capacitated links and intermediate points classifies the marine log transportation as a fixed charge multicommodity capacitated transhipment flow problem.  35  2.2  Formulation 2 In order to elucidate the mathematical notation of Formulation 1, the  following vectors are defined: o  _  [ 5. ., 5. ., 5. . ]  A. . =  X. . = [Y. ., X. ., V. .]  i]  3 13 I. = [ I . , I I . , Let  1  i] III.]  (N, GJ denote the simulated network, with N representing the set of  vertices and G the set of directed arcs. Then, the aforementioned formulation can be stated as follows:  Min  if  Z .  [  • Z(y  1/2  [  x* • <n.*ii p>x . ) ] • t  l  ( H i - q ^ . r  )(••„.!;„.,)  •  1  }  subject to :  •Q = 2x* - I x • i - I< i.P*« ip j £ N ijp k£N kip ip 1  ZZx*.  tj<« 119  Z  -  I  J  A »jP UP  j<H  A  <  0.j).0 'JP  U N ; teT; peP  1  l  NV  ip  U N ; peP  0  p  X.I SO ; A € ( 0 , 1 )  where  '  R*  tp  =  s  i€j  t IP  for  0  for i€S  0*  for i € D  0  for i € J  0  for  u  ICS  for i C 0  36  3.  Technical Calculation Aspects  The standard Simplex algorithm can theoretically be employed to solve the logistic model. The simplex tableau w i l l have: JTP+DTP+STP+JSP+3DP+SDP+P constraints and T3SP+T3DP+(T3(P+1))+TSPD+(TD(P+1))+TDP+(TS(P+1))+3SP+SDP+JDP variables. With 3 (total number of origins), S (total number of storage areas), D (total number of destinations), P (total number of periods) and T (total number of log types) equal to 3, 2, 1, 52 and 3 respectively, the tableau w i l l have 1560 rows and 3450 columns. However, i f the Revised Simplex method is used, the tableau can be reduced to a 1560 x 1560 real type m a t r i x occupying 20 Mbytes. Therefore, the implementation of the model requires some other methods which can convert the large problem into several s m a l l problems of manageable size. for  Fortunately, the decomposition principle provides a systematic solving large-scale  linear programs  which contain constraints  procedure of special  structure.  4.  The Matrix Structure  The technological m a t r i x of the log transportation and inventory system defined earlier is presented in Figure V . 2 . L e t ej be the unit vector in E ^ , where P is the total number of periods, w i t h the one in the i th position. i = l , . . . , P we have:  Then for  37  t A  r  •  __ = " [ e ^ ] s  with  produced  •  / indicating  the p e r i o d  j . It is clear  at origin  f o r which  that  ° t A ., = jd  p  is  t . =  i  [e.] with I  indicating  the p e r i o d  1  t  type  (1)  shipments  =  B  i n t h e i-1  i s produced  inventory. , = [e.] with sd I  t ^  t  arrive  ,  /  t is  th position  and  i n the  at j  a t d.  f o r which  a n d t h e r e f o r e c a n be s t o r e d i n  to the period  I f type  i  corresponding  arrive  a t d.  /  when  i s n o t consumed  consumed  [ e ' .] where l  a t d,  elsewhere.  B  t  t ., = jd  e! i s t h e v e c t o r l  (-1)  position,  then  to the period  I f type 0  type  the p e r i o d  (0)  at  d,  at  j ,  =0.  shipments  t , = d  (-1)  in E  indicates  / corresponding  [e^] with  not  with  i s the vector  Position  fc  then  type  1  elsewhere.  ~ B  f o r which  P e!  i th position,  B  then  j .  at origin  [ e ' .] w h e r e  3  B  period,  p  1  produced ~ A  X  t is  i f the  aforementioned c o n d i t i o n i s true f o r each A i s the i d e n t i t y matrix I . fc  type  i s not produced 0.  in E  P  with  i n t h e /-/ t h p o s i t i o n  Position  * i s consumed  when  / indicates  (1)  and  i n t h e J'th —  (0)  the period  f o r which  a t d a n d t h e r e f o r e c a n be s t o r e d i n  inventory. at  B  t , = a  [e.] with I  consumes C ^  ]S  =  i  type  [e. ] with  1  corresponding  t o t h e p e r i o d when  t and t h e r e f o r e type /' c o r r e s p o n d i n g  t c a n be  to the period  d  bought.  when  38  shipments then ° C  t , = sd  C  arrive  [e.] x  s  with  arrive  at  s-d  not  s.  If  type  /  s.  indicating If  type  t  the  ~~ t C s  =  [e'.]  connected,  where  1  H  t  , = sd  e!  I  (-1)  position, (0)  [e.] l  with  Without possible two  is  not  then  C  js,  jd  nodes  is  =  N . . =  M. js  =  i [e.]  3d  1  the  3d  produced  the  /-/  th  when  consumed t , = sd  vector  i [e.] I  at  j,  in  shipments  at  d  and  route  0. E  P  (1)  with  position  in  the  / th  —  and  of  I  log  period type  generality, sd  not  the  combinations  connected,  by  it  then  for  the  assuming  a  during  which  t. i s assumed are  that  a l l  technically  example  allowed.  because  route  connecting  large  cost.  of  the  these  Therefore:  _  N . , = I _d . i P w h e r e e.e E with I  indicating  where  where  position  i e.e I  E  P  with the  operation, i e.e I  E  P  indicating  shipping  the  operation,  indicating  shipping  i [e.]  the  consumes  and  shipping  the , = sd  the  limitation,  position  M  d  are  position  =  in  prohibited  o  N. Ds  ° M.,  is  /' i n d i c a t i n g  loss  geographical nodes  not  elsewhere.  destination  If  is  period  0  is  /  =0.  fc  ]  at  with the  operation,  ones  in  starting  the  / th —  and  and  ending  /'th  —  periods  of  respectively. ones  in  starting  the  /th —  and  and  ending  j t h  —  periods  of  respectively. ones  in  starting  the  i th  and  respectively.  —  and  ending  /th  —  periods  of  39  By assuming (i) 3 origins, 2 storage areas, 1 mill, 3 log types and 52 periods and (ii) each type of log is produced and consumed at all origins and the destination during all 52 periods; the total number of non-zero elements in the matrix is 4940.  The constraint matrix has a dimension of (1560, 3450) and,  therefore, has a density of 0.00092.  5.  Summary  A deterministic model based on the network flow theory is considered in this chapter. It is demonstrated that the optimization of the network, if approached by classical linear programming methods, requires 20 Mbytes of memory for the storage of its highly sparse technological matrix. Consequently, the use of large scale mathematical programming methods is recommended to accomplish the objectives of the present study.  FIGURE V.l:  Structure of the Technological Matrix  A T„.. M  .1 _ i  ;i  ,J J I V - ....A^T;, . A'J.V;, V ) A  (4*  •  A„T « U  •A T U  U  •A V » U  U  AM  . AJI)  1  Cj,!^  •A V, . A l | ^ t B  s  t  FIGURE V . l . l : Structure of the Technological Matrix Window 1  A  1  T  1  JI»JI  'A' J» JD V  V  X  FIGURE V.1.2: Structure of the Technological Matrix Window 2  1  A  Wi. A  W«  FIGURE V.1.3: Structure of the Technological Matrix Window 3  M  •>II"U II. A  •Sj»N A JB  JB  FIGURE V.1.4: Structure of the Technological Matrix Window 4  45  VI. NETWORK FLOW MODEL: ALGORITHMS  Since the late 1950s, numerous approaches to solving large-scale problems have appeared in the literature. Most of the techniques have been inspired by an early suggestion of Ford and Fulkerson (1958) and its generalization by Dantzig and Wolfe (1961). The suggestion is called the decomposition principle. These techniques take advantage of the sparsity and the patterns of non-zero elements in the technological matrix, in computational schemes.  1.  Literature Survey  Plant location models are basically single commodity transportation problems whereby fixed cost is incurred whenever a source is used. Location models are, therefore, a very special case of the multi-commodity distribution models. Davis and Ray (1969), Gray (1970), Zoutendijk (1970) and Khumawala (1973) considered the capacitated plant location problems by applying the Branch-and-Bound technique. In his paper, Tomlin (1965) described the theoretical formulation of the multicommodity network flow problems including only continuous variables, but gave no computational experiences. Geoffrion and Graves (1974) included binary variables in their formulation and used Benders' decomposition to solve the multi-commodity problem. They assumed a single-period, uncapacitated problem in which no final destination is  46  allowed to deal with more than one intermediate point. Sullivan and Koenigsberg (1970) applied a mixed integer programming model to a ship allocation problem. They dealt with 500 variables, of which 25 were integer, and 200 constraints.  They considered a planning horizon of one  year which was divided into four quarters. The computation was carried out by using the mixed integer (MILP) code (a general branch-and-bound procedure) in CEIR's LP 90/94 program. Computation times ranged from about 15 minutes to several hours for one quarter.  2.  Decomposition Theory  2.1  Benders Decomposition: The Master Problem It is possible to take advantage of the special structure of the matrix  under study, as shown in Figure V.2, in such a way that the multi-commodity aspect of the problem becomes less complex. Let us define vectors d, Y, X , Cf, bf and e, and matrices At, B-j, D andA, t  such that our problem can be stated as follows: T  PI :  Min  dY + I C, X  t  A X t  -ADY  +  t  IB X  eY X^O  t  1  1  =  b  t  s  0  <  NV  , Yc(OJ)  t-l  T  k7  where X{ is the vector of flows of log t in the 52-period network and At is the simulated node-arc incidence matrix of the graph for type t. This problem is computationally difficult because the objective function contains a minimization over the set ( Y / Y . (0,1)} U j x/x^o}. However, if the discrete optimization over JY/Y € (0,1)| can be isolated, the problem will be linear with a much smaller 0-1 integer programming component. Benders (7) suggests the following decomposition:  P2 :  Min dY + h (Y) Y6(0,l) eY  s NV  where h ( Y ) i s d e f i n e d a s : T  P3  h (Y) =  M'n x >o  I 1  t  V t  =  IB.X. t  1  s  1  b  t  A  D  t=1,...J Y  Now consider the dual problem for P3. Let us associate ut and  to the first and  second set of constraints of P3 and let v=-v. Therefore, the dual of P3 is:  P4:  h(Y)  =  Max 1 u. b. t  v (ADY )  t t  u A t  t  -  vB  t  5C  t  v a 0 , u unrestricted  t=l,...,T  48  By assuming, without loss of generality that Z S [ > Z ZJ ' jp jp dp dp  (t=l  T)  (see also section V.2.2) it can be concluded that P3 is feasible and hence P4 is Therefore, by defining U = [u^/v] and  bounded (see also section VI.2.4).  denoting the extreme points of the dual feasible region by  |u^,...,U' |, <  we can  conclude that P4 reaches its optimal value at one of the extreme points of the dual feasible region, say k th point. Thus the value of h(Y) is determined by:  h(Y) = Max Zujb, - v*(ADY) k  *  isksK  k  U, unrestricted . v > 0 Therefore, PI becomes:  Min { dY + [ Max 2 u b - v (ADY) k  k  isksK ] }  t  eY < NV Y€(0,1)  ;  ujunrestncted  ; v> o k  Now, given the fact that a maximum is really a least upper bound, we can write PI as:  BM: (Brl-l)  (Bn-2)  Min  dY + y  o  v (ADY) + k  Y  0  eY Ye (0,1)  >  2ujb  < NV ;  Y unrestricted 0  t  k=i  K  49  This is the Benders master problem. It is noticed that BM still belongs to the class of mixed (0-1) linear problems with the only continuous variable (y ). 0  Furthermore, we remark that in addition to the original integer constraints, one additional constraint for every extreme point of the dual feasible region is added. Since the number of these constraints is astronomical, generating them is practically  impossible.  Benders  decomposition  relaxes  these additional  constraints and generates them only when they are violated. Generally, any constrained optimization problem (P) can be relaxed by loosening its constraints, resulting in a new problem (P ) (30). r  The only  requirement for (P ) to be a valid relaxation of (P) is: F(P) ^ F(P ), where F(P) r  r  denotes the set of feasible solutions of (P) and c the inclusion sign. Let (P) be a minimization problem. Then, the property of relaxation implies the following: - F(P ) = * implies F(P) = <p r  - Min (P) > Min (P ) r  - If an optimal solution of (P ) is feasible in (P), then it is an optimal r  solution of (P).  2.2  Benders Decomposition: The Subproblem and Procedure The Benders subproblem is defined by either P3 or its dual P4. It is to our  advantage to consider P3 since it is a collection of T uncapacitated singlecommodity minimum-cost network problem (see Figure V.l) linked together by capacity constraintsIIB^X^ ^ DY. This structure, in turn, suggests the use of the Dantzig-Wolfe (D-W) decomposition which will be discussed later.  50  The following Benders procedure is based on a £ -optimal criterion:  Step 0:  Select a convergence tolerance parameter i £ 0 Set UB =«o, LB = -<*> and iteration counter C = 0 Relax BM  Step 1:  Solve BM Replace LB by the optimal value of BM.  Denote this value by  BM* Store the optimal solution Y  +  If UB$LB+£ then Stop, otherwise Step 2:  Solve P3 with new  Y  C + 1  C 1 Store the optimal solution X Let the optimal solution of h(Y  +  ) by h*  Set UB = BM* + h* If UB$ LB +C then Stop, otherwise Step 3:  From the dual variables of P3 generate the violated constraint of BM Return to Step 1  51  The above procedure involves iteration between BM and P3. The first is BM to which unsatisfied constraints are successively added. The second is P3 which tests the optimality of a solution to BM and, if necessary, generates new constraints to bring about feasibility. However, it is not necessary to solve BM to optimality, but rather to stop as soon as a feasible solution to it is produced which has value less than UB- . Therefore, this variant master problem is feasibility-seeking only (30):  Find Y e (0,1) and y to satisfy o  (BM-1), (BM-2) and dY + y $ UB- £ o  By eliminating y we will have: o  Find Y c (0,1) to satisfy (BM-2) and dY +^u^b - v (ADY) £ UB-1 C  t  c=l  C  Therefore, we may introduce any "convenient" objective function, say 0(Y), and form the following modified Benders master problem after C iterations:  MBM .  Min  9(Y)  Yt(O.l)  c=1  C  52  Geoffrion and Graves (1974) found the last C th function: dY - v ( DY) C  to be a good choice forO(Y). MBM  is a pure 0-1 integer programming as opposed to the mixed  programming problem (BM), and is more convenient to work with.  However,  MBM will no longer produce a lower bound on the optimal value of (PI). Therefore, LB should be deleted from Step 1 and the termination criterion in Step 2 should be modified. The procedure to implement will then be:  Step 0:  Select £. ^ 0 Set UB = oo and C = 0  Step 1:  Solve MBM Store the optimal solution Y  +  If MBM* ^ UB-£ then Stop. otherwise Step 2:  Solve P3 with new Y Store X  C  +  C  +  1  1  Set UB = MBM* + h* Step 3:  From the dual variables of P3 generate the violated constraint of MBM Return to Step 1  53  2.3  Dantzig-Wolfe Decomposition: The Master Problem The  subproblem  multicommodity  for  Benders  decomposition, problem  P3,  is a  minimum-cost flow problem where ADY=T represents the  vector of upper limits on the sum of all log species flowing in the arcs of the network. As Figure V.l illustrates, P3 possesses a block diagonal structure. Therefore, the application of Dantzig-Wolfe decomposition technique may be advantageous. L e t H = | X t  /A X  t  t  t  = b ; X ^ OJ andu) and<s"te its extreme t  t  te  points and directions respectively. Any X  t  can be expressed as a convex combination of the extreme points  and direction ofI2 : t  with  Z* X . e  <c  X w  =  1  >  o  e = t....,E  >  0  e = 1  <e  t  e  1  =  1  .-.  T  t  L,  where Et and Lt are the total number of extreme points and directions o f i i f It can easily be shown thatf2t is bounded. Therefore, by substituting Xt in P3, the following linear program inoJ will result: te  P5.  Min  21  (C » .) t  t  x  te  54  This is the D-W master problem. The strategy of D-W decomposition procedure is to operate on T+l separate linear programs consisting of the master problem P5 and T single-commodity flow subproblems. These subproblems are defined as: T  P6, :  Max  \>-0  2 (wB. - C, ) X + ox, t  t  t  V t  =  t  b  t  t  l  = •  T  where (w,o<^) is the vector of dual variables corresponding to the basic feasible solution of P5.  2A  Dantzig-Wolfe Decomposition: The Subproblems Procedure  Subproblem P6j, consists of T single-commodity minimum cost networks. P6j is equivalent to: T  P6  .  Min x >o  2 ( C - wB ) X t  t  +  t  «  t  x  t  \ \ =  b  t= i  f  T  This subproblem consists of finding the minimum cost flow in T uncapacitated single-commodity networks with arc cost equal to (CfwA ). Therefore, the best t  policy is to send all the required flow along the cheapest path. The shortest (cheapest) path can be obtained by one of the efficient algorithms available in the literature. The problem P6 is feasible if and only if the total supply of each type of log exceeds or equals the total demand for that type. That is, P6 is feasible if and only if:  55  The D-W decomposition procedure is as follows: Step 1:  Find an initial basic variable to P5 Store the master basis inverse Store the right hand side column and the vector of dual variables  Step 2:  Solve P6  Step 3:  If optimality not met then From the dual vector of P5 select the entering variable Update its column and pivot Return to Step 2 Otherwise Stop.  3.  Specialization  We shall now present the specialization of the proposed model in a form suitable for the aforementioned three-level decomposition.  To facilitate the  variables' definition, it is convenient to formulate the Benders subproblem first. We shall adopt the notation used in Formulation 2. At iteration C, we have:  56  3.1  Dantzig-Wolfe Master Problem Translation  Min  T E, P  i  n  II t  2  (y> lfi^p)  (i> X  ijp tc  e  x  l e  i.jeN , i * j ; p=1,...,P  ijp UP  t = 1.....T  te e=l  3.2  E ; t=1 t  T  Dantzig-Wolfe Subproblems Translation For t=l,...,T, we define the subproblem t:  Min  ( v . • lfiLp) l -  i ( i P  (i.j)€G •  'JP  ">  II'Q* • I i€H *P o , j )  Ix\  t  - I X* k6N kip k*i  € G  • I* - I* IP  m  i M  ijp w X. 'JP ' J 1  m  j€M IJP i*j  • 1/2 i (m*q)*i r)(-i ViJ  x  r  €  1  •Q*  I.P* 1  IP  p  H  m  1  '  v  ,)  IP 1.P*1  ] • « J t  = R* 'P  where (w,^*) is the dual vector of the D-W master problem.  ieN; teT; pcP  57  The class of least cost problems has received great attention for both practical and theoretical purposes. The shortest path problems with non-negative arc cost can be solved in a time proportional to n^.  When the weights are allowed to be negative, the  solution can be obtained in a time proportional to n by means of matricial 3  methods, such as the Floyd-Warshall (26, 73) and Cascade (21) algorithms. These techniques are convenient and efficient when the least cost paths are required between all pairs of vertices in a graph.  However, the Cascade algorithm has  demonstrated better efficiency in practice (21, 26, 38). In matricial techniques two basic operations are involved in finding shortest paths. The first is the selection of the minimum of two elements and the second is to take the sum of two elements.  The operation * is, then, defined  as follows: (A*B)ij =Min Jaiu + bkj] k  If A contains entries only for costs of arcs, then the (ij)th element of A(2)=A*A represents the least cost route from node i to node j formed with exactly 2 arcs and in general, the k squarings of A,  = A*...*A (k times), contains the least  cost paths composed of exactly k arcs. The Cascade algorithm obtains the least cost path between all pairs of nodes in a graph only in two squarings if: (21) 1.  Zero diagonal elements are first imposed  2.  Elements are replaced in the matrix as soon as they are calculated  3.  Elements are calculated first left to right in each row and downwards by rows (the forward process), and then right to left in each row and upwards by rows (the backward process).  58  3.3  Benders Master Problem Translation  Min I  A .  CU)€G  p=1  p  c=l  C  p  .. - v X. A.. C  UP  A..  >P U P  £  UP  (0,1)  Several methods are available for solving this group of problems: Gomory's cutting-plane algorithm (32) solves integer programs by working on a single linear problem to which new constraints are successively added in order to reduce the feasible region until an integer optimal solution is obtained. However in practice, the branch-and-bound procedures almost always outperform the cutting-plane algorithm (28, 31). The branch-and-bound is based on the  relaxation  approach.  This  procedure solves successively the linear programming problem derived from the integer problem by relaxing the integrality restrictions.  Therefore for a  minimization problem, an optimum non-integer solution is always a lower bound on a solution of the restricted problem. The algorithm partitions the feasible solution into subsets, then by further partitioning of the "best" subset (with the lowest bound) it searches for the optimum solution. Implicit enumeration procedures are combinatorial in nature. They can be considered as a special case of the branch-and-bound techniques.  Balas (1965,  1967) developed an additive algorithm which seems to perform very efficiently  59  for the class of 0-1 programming. In his procedure, the vast majority of all solutions (feasible and non-feasible) are enumerated implicitly and only a few are explicitly enumerated. For the purpose of the present study, a slightly modified additive algorithm developed by Plane and McMillan (1971) is considered. However, since a feasible solution to MBM is sought, a variant of the modified algorithm is implemented. Before discussing the algorithm, it may be useful to state some definitions: Let S be the set of all n-tuple solutions to the problem. S = {(6 ,...,6  )  1  ( 5, , . . . , 6 ) } n  n  Then, the partial solution S j=l is defined as the set S.  = {(8 ,...,6 ) 1  n  S  | 5. =1 S  }  and, the completion of the partial solution S[-l is some combination of other variables raised to 1 which will bring about feasibility in an unsatisfied constraint. If the partial solution S\=l cannot be completed in such a way as to avoid violating one or more of the constraints, we say that the partial solution has been fathomed and therefore, all S\=l completion are ignored. The following procedure searches for a feasible solution to an 0-1 integer problem: Step 0:  Set S = {*>] Set UB = 0(1): the objective function at all &'s=l  Step 1:  Find V, the set of constraints violated when partial solution S[ is completed by setting to zero all variables not in S If V is empty then, Stop (a feasible solution is obtained) Otherwise  60  Step 2:  Calculate  (S): the value of the objective function when S is  completed by setting to zero all variables not in S Store in T each variable not in S which has an objective coefficient less than UB and a positive coefficient in some constraint in V Step 3:  If T is not empty and all constraints in V can be made feasible by adding only variables in T, then Add to S the variable in T with the greatest coefficient sum Return to Step 1 Otherwise  Step 4:  If all elements in S are negative, then Stop (optimal solution is reached) Otherwise  Step 5:  Locate the rightmost positive element in S Replace it with its negative and drop any elements to the right Return to Step 1  4.  Computational Aspect  Since up to the present time, no computational statistics of the implementation of a three-level decomposition algorithm, using the DantzigWolfe and Benders principles, has been found in the literature, the determination of CPU time for the proposed model is almost impossible.  61  In this section, attempts are made to estimate the adequate CPU time for solving the network flow formulation of the system under study. The model utilizes a three-level decomposition procedure. In the first level, the modified Benders master problem (MBM) is optimized. In the second level, the subproblem (P3) is first decomposed into P4 and P6 problems and then optimized using the Dantzig-Wolfe decomposition principles. Swoveland (1976) studied a two-level decomposition algorithm for the multi-commodity flow problem including 3,438 constraints and 10,080 continuous variables.  He reports a total time of 6.38 minutes on IBM  360/67.  The  restricted master problem was optimized 433 times requiring a total of 1,073 pivot steps. The problem P3, with 1560 constraints and 3450 variables, has a structure similar to the problem solved by Swoveland. As a first order of approximation, the Swoveland's computation statistics can be regarded as being 2.5 times the CPU time required to solve the problem P3.  Assume that 300 iterations are  required to optimize the modified Benders master problem (MBM). This number of iterations represents almost half the number of binary variables in the problem (MBM). Therefore, the estimated CPU time necessary to optimize the three-level decomposition problem is (6.38/2.5) x 300 = 765 minutes or 11 hours on IBM 360/67. Since the scaling factor between IBM 360/67 and Amdahl 580 is about 8, the required CPU time to solve the network flow problem is in order of magnitude of one hour and 30 minutes. In recent studies based on the problem reformulation method, large-scale fixe-charge multi-item capacitated lot sizing problems have been successfully solved.  62  Van Roy and Wolsey (1984) solved to o p t i m a l i t y a (1461 x 2756) problem in 13 minutes on a D G M V 8000, and Eppen & M a r t i n (1985) reported a total C P U t i m e of 100 minutes on A m d a h l 470 for a problem w i t h 3210 rows and 9587 variables. Therefore, in the future works, the variable redefinition method should be considered as an alternative to D a n t z i g - W o l f e decomposition.  5.  Summary  First,  attempts  were  made  to  succinctly  illustrate  the  theoretical  background for Benders and D a n t z i g and Wolfe decomposition principles.  Then,  the modification and specialization of the existing theory necessary for p r a c t i c a l implementation of the proposed model were investigated in d e t a i l .  F i n a l l y , the  order of magnitude of the C P U t i m e necessary to o p t i m i z e the system under study is determined. Due to  the  number  of iterations  between  the  subproblems, the o p t i m i z a t i o n of the transport-inventory hour and 30 minutes on Amdahl 580 machine.  master problems  and  system requires  one  63  SUMMARY OF PART 1  In this part, the implementation of the stochastic and deterministic models is assessed. The deterministic model is based on the network flow theory and uses the Dantzig-Wolfe and Benders decomposition principles. Due to the large number of iterations between the subproblems and master problems, the implementation of the decomposition principles requires excessive amounts of CPU time.  Future  works should focus on the variable redefinition method to optimize the aforementioned network. However, since there is considerable uncertainty in the system operation, the simulation model is used to describe the system under study.  The model  conserves a high degree of simplicity and carries a satisfying evolutionary aspect. The following part is concerned with the application of the simulation model to the system under study.  64  P A R T 2: A P P L I C A T I O N  As the study of Part 1 indicates, simulation can be used to model the log transport-inventory system. In order to satisfy the objectives of the present dissertation, Part 2 investigates the application of the simulation model to the system under study. The collection of required data, the pre-simulation study and the analysis of results are discussed.  65  VII. ANALYSIS OF AVAILABLE DATA  The data gathering effort is oriented towards the collection of the data represented by the inputs required for the simulation programme. This chapter develops statistical models for the transit time on different links of the system's network.  Production, consumption and cost figures are also  estimated.  1.  Production and Consumption  Log production and consumption figures for different sorts of hembal pulp logs (large, small and camp run are available from the log supply department of MacMillan Bloedel L t d .  However, due to the confidentiality of this data, only  representative data are used in this study (see Table VII. 1). Figures VII. 1.1 and VII. 1.2 illustrate the annual log production schedule at Port M c N e i l l , Eve River and Kelsey Bay operations. Although in this study, the Powell River consumption rate is assumed to be constant throughout the year (see Figure VII. 1.2), the model allows for a variable m i l l consumption schedule.  Hembal Location  Production  Pulp  Large  Small  Camprun  Port McNeill  106  161  20  Eve River  176  170  40  Kelsey Bay  158  109  160  Total  440  440  220  Powell River  437  437  218  <IO m /Yr) S  3  Consumption (10 m /Vr) 3  3  T A B L E VTJ.1: Yearly Production and Consumption Rates  67  VOL. (m /day) 3  lOOOi  Port McNeill  900  .92P..  800  .MP..  8  - - --"..8?o, 4  Q  7004 600  510  500 400  340  340  310  •420"  300200100  03  33 Jon  JJQ_  93  Fev Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan  MONTH  VOL (nr?/day) Kelsey Bay 1000  -25JL  900  940  800  900  _25JL 900  380  600  840  ' 790  790  700  1  ....AQQ....  .....<JQ9  500 40CH 300  250 • 200  200  170  1004 Jan  Fev Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan  MONTH  Large  Pulp Log  Small  Pulp Log  Camprun Pulp Log  FIGURE Vn.1.1: Production Rates at Port McNeill and Kelsey Bay  68 VOL. (m /day) 5  10001 900' 800 700 600 500 400 300200 100-  Eve River 900 900  * 270 *  190  1000  1000  930  950  _25Q_  250  190  45  Jan  Fev Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan  MONTH  VOL. (m /day) s  Powell River  1500 1200 1200  1000 600  500  Jon  Fev Mor Apr May Jun Jul Aug Sep Oct Nov Dec Jan  MONTH  Large  Pulp Log  Small  Pulp Log  Camprun Pulp Log  FIGURE VII. 1.2: Production and Consumption Rates at Eve River and Powell River  69  2.  Transit Time  The collection of transit time and delay at the storage area covers the period from November 1984 through July 1985.  The data source is the Towing  History Report of the Log Flow Control System from MacMillan Bloedel (See Appendix 2 ) . The delay time is composed of the time necessary for the handling operations and the delay due to the adverse weather.  Table VII.2 displays the  data statistics. The Weibull distribution (Equation VII. 1) is selected to model the transit time: f (x;X,o, ) w  u  =  \/o  [(x-  )/a] ~ X  M  1  exp{-[(x- )/a] } X  M  VI1.1  where parameters A,(»and/* are the shape, scale and location of the distribution, respectively.  This selection is based on the fact that the Weibull distribution is extraordinarily flexible and capable of modelling a wide range of random phenomena (11).  Figures VII.2.1, 2.3 test the theoretical cumulative Weibull  distribution, obtained by means of maximum likelihood estimation method, vis-avis that of the observed data. Since the Weibull model performs poorly for the observed transit time between Teakern Arm and Powell River, a Poisson distribution (Equation VII.2) was fit to the data due to the positive coefficient of skewness of the underlined data set (see Figure VII.2.4).  70  Links  Port McNeill - Teakern Arm  Standard Coefficient of Deviation Skewness  Sample Size  Mean (days)  18  5.39  1.91  .29  Eve River  - Teakern Arm  38  3.36  1.31  1.45  Kelsey Bay  - Teakern Arm  21  3.08  1.55  1.73  Teakern Arm - Powell River  44  1.52  0.63  0.78  68  1 1.01  17.40  2.91  Delay  at  Teakern Arm  TABLE VII.2: Transit Time Statistics  71  16.2  FIGURE VII.2.1: Weibull Fit for the Transit Time Models (PMN-TA, KB-TA)  ia.o  72 o  -11  fEfiKERN ARM - POWELL RIVER  -DATA  /  COa  £=o a.  /  /  /  UJ  3  s:  o.o  0.4S  0.9  1.3S  1.8  2.2S TRANSIT TIME  2.7  3.!S  —I—  3.6  FIGURE VH.2.2: Weibull Fit for the Transit Time Models (ER-TA, TA-PR)  ~1 4.OS  I 4.S  73  FIGURE VH.2.3: Weibull Fit for the Delay Model  TEAKE3NE  FIRM - P O W E L L  RIVER  OfiTfl  1 3.3  3.4  1— 0.8  1 , 2  fR&NSJT rrflE (Oflrsj  4  2  8  FIGURE VII.2.4: Poisson Fit for the Transit Time Model (TA-PR)  3i  3.S  4.3  75  f (x;X) = [ X  U  M )  p  exp  (-X)]  / (x- )! M  VII . 2  where A and h are the shape and location parameter of the distribution.  Table VII.3 summarizes the result of testing the null hypothesis H at 5 % q  significance level.  It is observed that there is not a significant difference  between the shape of the origins-Teakern Arm transit time data and the Weibull models with parameters as tabulated in Table VII.4. Furthermore, the Teakern Arm-Powell River travel time does not deviate a statistically significant amount from the model, a Poisson distribution with parameter 0.52 days (see Table VII.4).  3.  Transportation Cost  In general, the total transportation cost is the sum of two components: fixed and variable cost. The transportation fixed cost is independent of the volume of logs hauled. This cost represents the expenses required to operate the vessel. Included are: food, operators' wages and maintenance. On the other hand, the transportation variable cost and the fuel cost vary with the volume transported.  76  Calculated  Degree of Freedom  Critical  2.33  3  7 81  Weibull  9.14  4  9.47  Weibull  Kelsey Bay - Teakem Arm  7 16  4  9.47  Weibull  Teakern Arm - Powell River  21.03  1  3.84  Weibull  Teakem Arm - Powell River  1.09  1  334  Poisson  22.42  8  13.51  Weibull  Links  X  Port McNeill - Teakern Arm  Eve River  - Teakem Arm  2  X  2  Model  i  Delay  at  Teakern Arm  TABLE VII.3:  Goodness of Fit Evaluation for the Transit Time Models  Links  Port McNeill - Teakern Arm  Eve River  - Teakern Arm  Kelsey Bay - Teakern Arm  Teakern Arm - Powell River  Delay  at  Teakern Arm  T A B L E VII.*:  Shape X  Scale  1.00  2.39  3.00  1.00  1.36  2.00  1 49  2.38  0.93  a  U  1.00  0.52  1.00  Location  11.29  1.00  Transit Time Probability Distribution Parameters  78  The fuel consumption F C is linearly proportional to the horsepower HP utilized: FC(HP) = ol HP Navin and MacNabb (1982) report a rate of 1.1 to 1.25 Imperial gallons per horsepower for tugs. They indicate that tugs operate at 80% of their effective maximum power because beyond this limit, there is a risk of breaking the bundles. The horsepower, in turn, is linearly proportional to the volume towed V. HP(V) = /3 V Therefore: FC(V) =o</3V = v V  O ^ V ^ link capacity  According to the same source (51), 13-15 horsepower is required to move 1 bundle section (20m x 20m) weighing 400-500 tonnes.  The maximum volume  towed weighs approximately 25000 tonnes. However, the cost figures used by the company are available in an aggregated form.  Towing from origins to the storage area costs $1.5/m3  whereas $.3/m3 is used for transportation between the storage area and the mill. These cost figures are only representative.  4.  Inventory Charges  Costs incurred from operating the inventory system are measured as follows:  79  Financial Cost The financial cost is composed of: (1) the real out-of-pocket costs such as the cost of insurance, and the costs of operating the holding areas; and (2) the opportunity cost of the funds committed to the inventories.  The latter is  calculated by applying the daily interest rate (10% per annum) to an estimated average coastal British Columbia unit production cost of logs ($45/m3) times the daily volume.  4.2  Quality Deterioration Cost The principal deteriorating agent for pulp logs is salt. "In pulp and paper  operation salt will increase the dead load in the liquor recovery system thereby reducing pulping capacity. This effect is quantifiable. Salt will also contribute to corrosion of process metallurgy" (8). The same reference reports that many factors influence the salt level in logs; among which the species, the salinity of water and the duration of residency in salt water are the most important. As for the penalty cost, the aforementioned report estimates a cost of $1.77/m^ for every 0.1% salt present in chips.  4.3  Shortage Cost The coastal logging industry's traditional inventory doctrine is: a mill will  not run out of the appropriate logs. In part because of this, little attempt has been made to determine the cost of going short. There are two main approaches to determining the shortage cost: direct and subjective methods.  80  The first approach assumes that in the case of a shortage, the mill is supplied from the outside market. This implies that the shortage cost equates to the spot market value for logs. Contrary to the first approach, the second method takes into account the intangible elements such as: convenience; cost of the mill idle time; and the subjective value that managers assign to the zero-shortage policy. The second approach reflects the true shortage cost in the logging industry. Experts at MacMillan Bloedel estimate that log shortages at the Powell River mill would incur a representative variable cost of $350,000 per day. Therefore, the unit shortage cost of $120/m3 is considered.  4.  Summary  The data collection effort is divided into two aspects: 1.  The  collection of available data such as the production and  consumption rate, the transit time along various links, the delay at the storage area, the transportation cost and all figures necessary in the calculation of the financial and deterioration cost of the inventory system. 2.  The estimation of unavailable data such as the shortage cost.  The transit time distributions are obtained by statistical means; the shortage cost is estimated by experts; and finally, the representative figures for the rest of the data are obtained from the MacMillan Bloedel Ltd.  81  Vin.  T A C T I C A L PLANNING  This chapter considers the pre-simulation phase of the study. The starting conditions and definition of equilibrium are first determined, followed by the process of validating the simulation model, and finally, the variance reduction measures and the determination of sample size are discussed.  1.  Equilibrium  Generally, simulation models are assumed to be stationary stochastic processes.^ They are based on the hypothesis that events are time-independently distributed random variables. Since initial values assigned to input variables of the simulation experiment greatly influence the observations made near the beginning of the experiment, the stationary process fundamental assumption is violated if early observations are included in the output analysis. To define the equilibrium, it is assumed that each system is characterized by the limiting probability distribution of the response variable. The equilibrium state is 'reached' when the response variable behaves according to the underlined limiting probability distribution. However, the mathematical derivation of the  1  |x(t),t € T }  is strictly stationary if  P[X(t,)*x, X(t )£x ,... X(t )*x ] f  2  a  f  n  n  =  P[X(t,+s)$x,,X(t +s)*x 2  for arbitrary real values of s and for all n (54).  2 /  ...,X(t +s)£x ] n  n  82  limiting probability distribution of the response variable of complex systems is very difficult if not impossible. This makes the determination of the equilibrium state one of the most difficult procedural questions in simulation modelling. No satisfactory method exists to define the steady-state conditions: Conway (1963) recognizes the equilibrium when the first of a series of measurements is neither the maximum or minimum of the remaining set. Gordon (1969) assumes that in the absence of initial bias, the standard deviation can be expected to be inversely proportional to (n=sample size).  n ^  He suggests plotting the variance of the sample  mean versus sample size on log-log paper. He assumes equilibrium is attained if the data lies on a straight line, sloping downward at the rate 1:2. Emshoff and Sisson (1970) assume steady-state when the moving average of the output no longer changes significantly over time. Fishman (1971) views the process of interest in the simulation as a covariance stationary * that has an autoregressive representation. The equilibrium is assumed at time k, after which the correlation between observations is considered to be low. In this study, in order to investigate the bias resulting from the starting state of the model, the interaction between the initial conditions and the important decision variable, the fleet size, is measured.  The sequence |X(t),teTJ is covariance stationary if E[x(t)X(t+s)] <<*>  83  For this purpose, a two-factor analysis of variance is used to determine that set of initial conditions which interacts the least with the fleet size variable. Table VIII. 1 tabulates the results of this analysis where: - The FLEET factor represents the fleet size and has three levels: 3-tug, 5-tug and 7-tug fleets. - The INIT factor represents the initialisation set: 1. INIT1 with the initial inventories of 1 day and 10 days of the mill's consumption at each origin and the storage area; and an initial mill's inventory corresponding to the minimum inventory level (14 days of consumption). 2.  INIT2 with the initial inventories of 3 and 20 days of the mill's consumption at each origin and the storage area; and an initial inventory of 1.2 times the minimum inventory level.  3.  INIT3 with the initial inventories of 5 and 30 days of the mill's consumption at each origin and the storage area; and an initial inventory of 1.3 times the minimum inventory level at the mill.  The F-statistic provided in Table VIII. 1 tests the null hypothesis that the main and interaction effects are zero. The inspection of this table suggests that the interaction effect between the decision variable, the fleet size, and the initialisation sets is not significant at 95% level of significance. However, in order to gain further insight into the relationship between the above factors, the interaction table for the effect on total logistic cost of fleet size i and initial condition j is computed from the knowledge that Yjjk represents the kt h observation of the total logistic cost when fleet size i and initial condition j are selected, the sum of squares due to  34  *  BY  » *  A N A L Y S I S  TOTCOST FLEET INIT  O F .  Total L o g i s t i c Cost FLEET SIZE I N I T I A L I S A T I O N SET  SOURCE OF VARIATION MAIN E F F E C T S FLEET INIT 2-WAY INTERACTIONS FLEET INIT EXPLAINED RESIDUAL TOTAL  18 CASES WERE PROCESSED. O CASES ( 0 . 0 P C T ) WERE  V A R I A N C E  « « •  (x$1000)  SUM OF SQUARES  DF  MEAN SQUARE  3 3 7 2 8 1 2 . 092 7 8 7 1 S . 607 3 2 9 4 0 9 5 . 485  4 2 2  8 4 3 2 0 3 ..023 39358. 304 1647047..743  8 .308 0 . 388 16. 228  0 004 0. 689 0. 001  119830. 884 119830. 884  4 4  29957 .721 29957 .721  0. 295 0. 295  0 .874 0 .874  3 4 9 2 S 4 2 . 976  8  4 3 6 5 8 0 .372  4 . 301  0 .022  9 1 3 4 5 5 . 767  9  101495,.085  4 4 0 6 0 9 8 . 743  17  259182..279  F  MISSING.  TABLE VTJI.1: Analysis of Variance Table for the Determination of the Equilibrium State  SIGNIF OF F  85  the FLEET-INIT interaction is: SSI  = m I..  i]  ( ! . . - ! .  13.  1..  -  ?  . + ?  •••  )  2  where m is the number of replicates (=2 in this case). Table VIII.2 tabulates the result of this investigation. Since INIT2 accounts for the least variation of the interaction effect (13%), the initial inventory levels at origins and the storage area are set at 3 days and 20 days of the mill's consumption, whereas the initial level of the mill's inventory is equal to 1.2 times the minimum level. The variation of the large, small and camprun pulp log inventory levels with respect to time for the selected initial inventory are presented in Figures VIII. 1.1 to VIII. 1.9. Although these illustrations suggest that the transient period is practically non-existent, a 35-day 'warm up' run is considered.  86  C E L L BY  TOTAL  TOTCOST FLEET INIT  M E A N S  Total L o g i s t i c Cost FLEET SIZE INITIALISATION SET  (x$1000)  POPULATION  6469.48 18)  (  FLEET 6562.83 ( 6)  INIT 6418.03 ( 6)  FLEET'  6427.57 ( 6)  INIT 6771.29 ( 6)  6772.64 ( 6)  5864.49 ( 6)  Initial  7027.16 ( 2)  6793.43 ( 2)  5867.89 ( 2)  6643.36 ( 2)  6761.88 ( 2)  5848.84 ( 2)  6643.36 ( 2)  6762.62 ( 2)  5876.75 ( 2)  .• i \|  Conditio^' I  1  2  3  1  26 412.75  5 264.95  8 091.00  2  5 849.19  1 655.68  1 281.64  3  7 399 4 4  1 016.97  2 934 39  79 322.76  15 875.20  24 614 06  13 25  20.55  Fleet Size  \  n >  \  2x(Totol) % of Sum of Squares due to Interaction  66.20  TABLE VIII.2: Fleet Size 'i' - Initial Condition "j Interaction Effects  87  FIGURE VIII.1.1: Daily Variation of the Large Pulp Inventory Level at the Mill (D.C. = Days of Mill Consumption)  88  o  0.0  «.0  80.0  120.0  FIGURE Vm.1.2:  160.0 200.0 TIME (DAYS)  240.0  200.0  Daily Variation of the Large Pulp  Inventory Level at the Storage Area (D.C. = Days of Mill Consumption)  320.0  380.0  89  FIGURE Vm.1.3:  Daily Variation of the Large Pulp Inventory Level at the Origins  (D.C. = Days of Mill Consumption)  90  FIGURE Vm.l.*: Daily Variation of the Small Pulp Inventory Level at the Mill (D.C. = Days of Mill Consumption)  91  FIGURE VIH.1.5:  Daily Variation of the Small Pulp  Inventory Level at the Storage Area (D.C. = Days of Mill Consumption)  92  FIGURE VIII. 1.6:  Daily Variation of the Small Pulp Inventory Level at the Origins  (D.C. = Days of Mill Consumption)  93  FIGURE Vni.1.7:  Daily Variation of the Camprun Pulp Inventory Level at the Mill  (D.C. = Days of Mill Consumption)  94  FIGURE Vffl.1.8: Daily Variation of the Camprun Pulp Inventory Level at the Storage Area (D.C. =Days of Mill Consumption)  95  FIGURE VIII. 1.9:  Daily Variation of the Camprun Pulp Inventory Level at the Origins  (D.C. =Days of Mill Consumption)  96  2.  Validation  Albeit that the validation stage is the most elusive of all the unresolved procedural problems associated with simulation techniques, the attainment of an acceptable level of confidence in the inferences drawn from the performance of the model is crucial. Anshoff and Hayes (1972) compare different degrees of validity of a model vis-a-vis the model's utility to the decision makers.  These curves are  reproduced in Figure VIII.2.  Cost of model Utiles Benefit Cost  0  0.5  1.0  Validity  FIGURE VIII.2: Utility vis-a-vis Validity of Models Source: Anshoff and Hayes (1972)  As the degree of validity of the model increases, the model development and  implementation cost become more expensive; and since the value of the  97  model to the decision makers increases with a decreasing rate, the benefit to cost ratio variation takes a concave shape, peaking around a validity score of less than 0.5. According to Shannon (1975), this score is "presumably less than the most valid model that money can buy." Four positions in validating a simulation model can be taken: - Rationalism, in which the problem of validation is reduced to the search of a set of basic assumptions underlying the behaviour of the system. If basic (but unproved) facts and the logic with which they are connected, are not rejected, the model is valid. - Empiricism, which suggests that the model should be built on verified facts and not assumptions. - Positive Economics or Absolute Pragmatism, in which the validity of the model is tested according to its ability to predict the real system behaviour. - Utilitarianism, which measures the validity of simulated models from rationalistic, empirical and absolute pragmatic perspectives. Fishman and Kiviat (1967) suggest a tree-step procedure in evaluating a simulation model: 1) Verification, to insure that the model behaves as expected, 2) Validation, to measure the forecasting ability of the model and 3) Output analysis which deals with the analysis and interpretation of generated data. Uyeno and Seeberg categories:  (1984) divide the process of evaluation into six  sensitivity analysis, individual component evaluation, review by  experts, observation of intermediate results, comparison of model results with actual data and experimental results. In this study, the evaluation process consists of four steps: extreme-case  98  analysis, review by experts, observation of intermediate results and comparison of the model behaviour with the real-system comportment.  2.1  Extreme-Case  Analysis  Transit time influences the length of the water residency of the logs. Hence, the model is used to simulate the two extreme cases where: i) the delivery of logs is instantaneous and ii) the delivery is severely delayed (say 20 days). Figure VIII.3 displays the fact that for long transit times, the percentage of old logs in the system increases. Therefore, it is expected that the cumulative age distribution of logs in the simulated system be bound by the cumulative age distribution curves obtained from the instantaneous and 20-day delivery models. Five different simulation runs are made. Figure VIII.3 illustrates that the cumulative age distribution of logs obtained from these five runs conforms to the expectation.  2.2  Review by Experts  Experts at MacMillan Bloedel Ltd. reviewed the model structure and assumptions.  They agreed on the model logic and suggested two additional  constraints which were incorporated into the model. These constraints are: i)  The tug's return trip is less than 24 hours.  ii)  The Powell River mill and the towing operation are scheduled for 7 days a week.  33  F I G U R E  VTJI.3:  Age Distribution of Logs  100  2.3  Observation of Intermediate Results Several deterministic test examples were run. The content of user chains,  transaction paths and queue statistics were checked rigorously. Outputs were compared with hand calculations. No discrepancy was detected.  2.4  Comparison of Model Behaviour with Real-System Comportment The system under study forms a subsystem of the MacMillan Bloedel  logging operation. That is: - The Powell River mill is supplied from numerous origins and storage areas. - The Teakern Arm storage area serves as an intermediate point for various mills, and - The Port McNeill, Eve River and Kelsey Bay operations supply several mills. This makes the comparison of the generated weekly inventory levels with the actual levels impossible since a disaggregation of the available information is not feasible. However, according to Section 6 of Chapter IX, the model simulates properly the behaviour of the real system. As will be seen later, in this section the probability frequency functions of the opening inventory at the storage area for the months of January, April, July, September and December are obtained by means of the simulation model. The cumulative probability distributions of these functions is low for the months of January and September, high for the months of July and December, and somewhere in between this range for the month of April.  101  Since this comportment conforms to the behaviour of the real system, it is concluded that the model is valid for the description and exploration of the real system.  3.  Variance of Observed Performances  Irrespective of the estimation confidence  method adopted, the width of the  interval is proportional to the sample standard  deviation and  inversely proportional to the square root of the sample size. Therefore, in order to have the random error, one must quadruple the sample size. Due to this slow stochastic convergence, obtaining accurate estimates by simulation models tends to be expensive. The subject of variance reduction has received considerable attention and several variance reduction methods have been developed (44, 48, 53, 63) among which the correlated sampling technique is of interest in this study. Correlated sampling is a method of reducing variance which is especially applicable when the performance of a system under two different operating policies is to be compared. Let jxjj,i=l,...,nj and jX2i,i=l,•••,"} represent two independent samples of n observations under policies 1 and 2. In order to compare these two alternatives, an estimate of the variance of (X1-X2) is desired.  Under the  assumption of independency between the two samples, the variance of (X1-X2) can be estimated by (Sf+S^/n.  102  Now if correlated sampling technique is used, an estimate of the variance of (X1-X2) will be (S?+S^-2Si2)/n- It is, therefore, clear that in order to reduce the variance, the covariance term, Si2> should be maximized.  Therefore,  common random number streams are used to correlate the experiments.  4.  Determination of Sample Size  The last problem of the tactical planning deals with the determination of the appropriate sample size. The objective of the planning of sample sizes is to provide the necessary protection against both Type I and Type II errors, and the sufficient precision which renders the estimates of the parameters of interest, useful. If different random number streams are used to generate the n replicates, one can argue that the model responses | X[,i=l,...,nj are independent. Moreover, due to the Central Limit theorem, one can assume that jXj,i=l,...,nj are normally distributed. Therefore, the assumption that the model responses Xj,i=l,...,n j are independent and normally distributed can be justified. Armed with this assumption, the planning of the sample size can be considered with the power of F test which permits controlling the risks of making Type I and II errors.  One procedure for implementing the power  approach would be the use of the power charts (22) which furnish the appropriate sample sizes. Entries for these charts are: - the level <* at which the risk of making a Type I error is to be  103  controlled, - the level fi at which the risk of making a Type II error is to be controlled for a specified value of  ,  - the value of dj> , the non-centrality parameter, which measures the unequality between the factor level means. One method of specifying d) is in terms of the difference between any pair of means (54): d> =A(l/2r)l/2/tf where: r = the number of factor levels A = the maximum difference between any pairs of factor level means that one wish to detect, and 6 = the model variability. By setting & =£, the non-centrality parameter will become: 4} = (l/2r)l/2 In the case where a four-factor analysis of variance with r i=5, r2=r3=r^=3 and  G"i)=.32 is performed, the power charts (22) give n  = 30.  Thus, the  number of replications, n, will be 30/(^3^=2. However, for three  and two-factor  analysis  of  variance, if  replications are desired, the level (i will drop to .7 and .5, respectively.  5.  Summary  The pre-simulation study is performed in this chapter.  two  104  It is found that the artificial nature of the model vanishes by providing 3 and 20 days of the mill's consumption of initial inventory for each sort of logs at origin camps and the storage area; and an initial inventory of 1.2 times the minimum inventory level at the mill. The transient period for this set of initial conditions is 5 weeks. The validation process consists of:  1) comparing the model response with  the extreme cases, 2) reviewing the model structure and assumptions with the aid of the forest industry experts, 3) testing the intermediate results with hand calculations, and 4) comparing the model's comportment with the real behaviour. In order to reduce the variance of the observed performances, the correlated sampling technique is used. Finally, the planning of the sample size is considered with the power of F test approach.  105 IX. ANALYSIS  The present chapter is concerned with the assessment of alternative policies available to decision-makers. Although there are infinite alternative scenarios, this study focuses on major transportation and inventory policies which  have been  subject to  considerable attention in practice. Post-simulation analyses are also performed to evaluate the magnitude of the bias resulting from simplifying assumptions on the salt contamination functional form. Finally, the simulation model is used to determine the probability frequency distribution functions of the opening inventory at the storage area for the months of January, April, July, September and December.  1.  Transportation System  The present transportation system is composed of a fleet of 3 tugs with 20,000  capacity each, towing logs between origins, the storage area and the  mill. It is recognized by experts that large inventories are a consequence of a slow transportation system. Three experiments are designed to assess the effects of transportation characteristics on the system. These experiments simulate the policies where log taxis, barges and direct shipments are used.  106  1.1  Log Taxi System The use of log taxis to tow logs consists of operating a greater number of  tugs with smaller capacity in the network. Two scenarios: i) 2 tugs with 30,000  each, and ii) 6 tugs with 10,000  each are considered.  1.2  Barge-Dryland Storage System The views concerning dryland storage combined with water transportation  of logs by towed barges are examined. As more emphasis is placed on recovering maximum value from the log, the industry is considering an increase in the number of dryland sorting and storage areas. The combination of dryland storage and transportation by barges has as its primary objective the preservation of the quality of logs. Moreover, the use of barges leads to shorter and more reliable transit time which in turn may reduce the minimum inventory levels at the mill. Since no historical data on barge travel time between the nodes of the network considered in this study exists, the estimates of 3, 2 and 2 days are selected as the transit time between the Port McNeill, Eve River and Kelsey Bay camps to the storage area, respectively; and an estimate of 1 day is chosen as the transit time between the storage area and the mill.  1.3  Direct Shipment The third transportation alternative is to send logs directly from the  107  origin camps to the mill. Therefore, the delay incurred at the storage area for the handling operations is eliminated.  2.  Inventory System  The main purpose of accumulating large inventories is to avoid production interruptions. In order to assist the decision-making process, the sensitivity of the system under study with respect to the minimum inventory levels, is measured.  2.1  Minimum Inventory Level  The present situation at the Powell River mill consists of carrying a minimum inventory level of 14 days of consumption. Two different scenarios are considered.  These scenarios depict the  situations where 4 and 24 days of consumption are used as the minimum inventory level.  2.2  Inventory Depletion Policies  The simulation model is used to evaluate the effectiveness of the two most common stock depletion policies: (Last-In-First-Out).  FIFO (First-in-First-Out) and LIFO  108  3.  Salt Contamination Function  The model sensitivity to the functional form of the salt contamination is investigated. Two functional forms are considered: (see Figure IX. 1) i)  Linear function, where a constant rate of deterioration is assumed and  ii)  Constant function, where the deterioration function is time-independent. It is, however, noticed that in both cases, any inventory depletion mode is  optimal.  This is due to the fact that the linear and constant deterioration  functions are convex and concave at the same time.  4.  Analysis of Variable Effects  The effects of five transportation alternatives, three inventory doctrines and three salt contamination functional forms on the total system cost are studied. Define: TRANS 1 = the alternative where barges are used as the transportation mode TRANS2 = the alternative where logs are transported from the origin camps directly to the mill TRANS3 = the alternative where 3 tugs with a capacity of 20,000 m3 each are used TRANS4 = the alternative where 6 tugs with a capacity of 10,000 m^ each are used TRANS5 = the alternative where 2 tugs with a capacity of 30,000 used.  each are  109  eomxT  iu.i  -WCM  i I  4  i  J  •  2 t  COKTIXT - w u n  tal  Constant Function  —unaao KWTVDCD  •  . •  / 3$  u, e&fcT^I l  •  M  l  l H  l W  IS  l  1  1  14  IN  IM  2M  04TS  0  FIGURE IX. 1:  i<3o  P^Y5  Linear and Constant Salt Contamination Functions  110  INV1  = the doctrine where 4 days of consumption is considered as the minimum level  INV2  = the doctrine where 14 days of consumption is considered as the minimum level  INV3  = the doctrine where 24 days of consumption is considered as the minimum level  FORM1 = the salt contamination function form obtained from reference (8) FORM2 = the constant form FORM3 = the linear form IMODE1 = the FIFO inventory depletion mode IMODE2 = the LIFO inventory depletion mode.  4.1  The Effect of the Inventory Depletion Modes From the analysis of variance table (Table IX. 1), it is concluded that at 5  percent level of significance: i)  The inventory depletion mode does not interact with the transportation and inventory policies and  ii)  The adoption of either FIFO or LIFO will not significantly alter the performance of the system measured by the total logistic cost. The overall F-statistic is equal to 206.92.  4.2  TRANS-INV-FORM Interaction Effects Table IX.2 presents the ANOVA table for the three factors: TRANS, INV  and FORM.  When the IMODE factor is deleted from the model, the overall F-  statistic increases to 291.23.  Therefore, the following analysis does not  Ill « * «  BY  A N A L Y S I S Total  TRANS INV IMOOE FORM  TRANSPORTATION POLICY INVENTORY POLICY INVENTORY DEPLETION MODE SALT FUNCTIONAL FORM  2388216962. 287 2299833063. 849 10830771. 246 177086. 665 77376040. 527  RESIDUAL TOTAL  « * •  (x$1000)  MEAN SQUARE  DF  9 265357440. 254 4 574958265. 962 2 5415385. 623 1 177086. 665 2 38688020. 264  F 2008 . 784 4352 . 494 40. 995 1 341 . 292 . 872  SIGNIF OF F 0. 0 0. 0 0. 000 0. 250 0. 0  1 1 . 100 16 .256 1 .338 19 .496 1 .500 4 . 101 0,,021  0. 000 0. 000 0. 262 0. 000 0., 229 0,.004 0..979  90034 .859 222045 .580 70551 .468 31972 .020 20072 .664  0,.682 1 .681 , 0,,534 0 . 242 0 . 152  0..901 0"..114 0 .922 0 .982 0 .962  16 16  8637 .364 8637 . 364  0 .065 0 .065  1 .000 1 .000  2432654170,.381  89  27333192 .926  206 .915  11888870 .913  90  132098 .566  2444543041 . 294  179  13656665 .035  41057755. 331 17179283. 127 706982. 309 20602730. 017 396402. 383 2166737. 720 5619. 774  28 a 4 8 2 4 2  1466348. 405 2147410. 39 1 176745..577 2575341 . 252 198201 .191 541684 .430 2809 .887  IMODE FORM FORM FORM  3241254. 938 1776364. 639 1128823.,480 255776., 161 80290,.658  36 8 16 a 4  IMODE  138197,.825 138197,.825  2-WAY INTERACTIONS TRANS INV TRANS IMODE TRANS FORM INV IMODE INV FORM IMODE FORM  4-WAY INTERACTIONS TRANS INV FORM  Cost  SUM OF SQUARES  MAIN EFFECTS TRANS INV IMOOE FORM  3-WAY INTERACTIONS TRANS INV TRANS INV TRANS I MODE INV IMODE  Logistic  V A R I A N C E  TOTCOST  SOURCE OF VARIATION  EXPLAINED  OF  180 CASES WERE PROCESSED. 0 CASES ( 0.0 PCT) WERE MISSING.  TABLE IX. 1: Four-Factor Analysis of Variance Table (TRANS-INV-FORM-IMODE)  0 .0  112  •  BY  SOURCE  * «  A N A L Y S I  TOTCOST TRANS INV FORM  SUM OF SQUARES  MAIN E F F E C T S TRANS INV FORM  1062941064 990048714 35170655 37721694  . 796 .625 .975 . 196  71858241 61114416 10078374 665450  . 156 .083 .088 .986  2-WAY INTERACTIONS TRANS INV TRANS FORM INV FORM  EXPLAINED  V A R I A N C E  *  *  *  MEAN SQUARE  OF  SIGNII OF F  1428. 948 2661 913 189. . 124 202 ..842  .058 .010 .761 .746  38. .641 82. . 158 13, .549 1 . 789  0. 0 0,.000 0,,000 0.. 148  0 .319 0 .319  0,.'992 0 .992  20 8 8 4  3592912 7639302 1259796 166362  16 16  29699 . 506 29699 .506  1135274498 .051  44  25801693 . 138  4184227 .251  45  92982 .828  1139458725 .302  89  12802907 .026  9 0 CASES WERE PROCESSED. 0 CASES ( 0 . 0 P C T ) WERE  F  8 132867633 .099 4 247512178 .656 17585327 .987 2 18860847 .098 2  475192 .099 475192 .099  FORM  RESIDUAL TOTAL  0 F  Total L o g i s t i c C o s t ( x$1000) TRANSPORTATION IPOLICY INVENTORY POLICY SALT FUNCTIONAL FORM  OF VARIATION  3-WAY INTERACTIONS TRANS INV  S  MISSING.  TABLE IX.2: Three-Factor Analysis of Variance Table (TRANS-INV-FORM)  277 . 489  0. 0. 0. 0.  000 000 0 0  0,,0  113  incorporate the effect of the inventory depletion mode. In order to gain further insight into the existing interaction between the above factors, the mean cell responses, using Table IX.3, are plotted against the levels of the transportation variable for each level of the inventory factor. This investigation produces three panels, Figures IX.2.1 - IX.2.3, which correspond to the three forms of the salt function. These plots indicate a strong interaction between the transportation and inventory factors as the degree of interaction is reflected by the lack of parallelism between the response curves within panels. Similarly, the TRANS-FORM interaction is measured by the degree of parallelism between the response curves between panels. Therefore, the TRANSFORM interaction cannot be disregarded. The significance of the aforementioned interactions are tested by the null hypothesis that the interaction between any two or three variables is equal to zero. Table IX.2 provides the required information. It is, therefore, concluded that at 5 percent level of significance, the effect of a proposed transportation policy should be measured with regard to the inventory doctrine used in the system. Furthermore, the bias incorporated into the model due to the simplifying assumption on the salt contamination function should be studied solely according to the operating transportation system.  4.3  Multiple Comparison Procedure The efficiency of the proposed transportation and inventory alternatives,  as well as the model sensitivity to the salt functional form are investigated in this section.  114  M E A N S  C E L L  TOTCOST TRANS INV FORM  BY  T o t a l L o g i s t i c C o s t (x$1000) TRANSPORTATION POLICY INVENTORY POLICY SALT FUNCTIONAL FORM  FORM INV 2  1  3  TRANS (  2837 ..08 2)  (  2833 . 12 2)  (  4 9 4 4 . 26 2)  (  5741 ,.99 2)  (  4791 .55 2)  (  6 2 5 0 . 37 2)  (  8488 .68 2)  (  5867 .89 2)  (  6 0 1 0 . 14 2)  (  9196 .20 2)  (  6055 . 73 2)  (  6434 . 48 2)  1  2  3  4  5  14006 .79 2) (  12130 .36 2) (  14895 .69 2) (  FORM  FORM INV  3  INV 1  1  3  2  2  3  TRANS  TRANS 2 8 3 7 ..02 2)  2 8 3 3 .,09 2)  (  4 9 4 4 . 29 2)  1  (  88 2)  (  3478, . 10 2)  (  4264, . 18 2)  2  (  7248 .94 2)  (  4659 .77 2)  (  4601 . 77 2)  3  (  7506 .87 2)  (  4666 .95 2)  (  4615 .50 2)  4  (  13144 .03 2) (  5  1 ( 2  3562  3  4  5  12349 . 19 2) (  11023 . 12 2) (  TABLE IX.3:  (  2837 ,,02 2)  (  2833 .07 2)  (  4944 ., 27 2)  (  5 7 4 2 , ,01 2)  (  5661 .82 2)  (  6 2 5 0 .. 37 2)  (  8939 .87 2)  (  6412 .33 2)  (  6428 ..59 2)  (  9 1 9 6 . . 23 2)  (  6426 . 16 2)  (  6434 , 48 2)  14006 .83 2) (  12779 .29 ( 2.)  Response Cell Means for the Three Levels of FORM  14895 . 7 1 2) (  FIGURE IX.2.1:  The Transportation and Inventory Effects for the Theoretical Salt Function  FIGURE IX.2.2: The Transportation and Inventory Effects for the Time-Independent Salt Function  FIGURE IX.2.3: The Transportation and Inventory Effects for the Linear Salt Function  118  Due to the existence of strong interactions between the variables, the analysis of factor effects must be based on the treatment means/*jjL;, which is estimated by Yijk, the mean total inventory cost where the TRANS, INV  and  FORM factors are at the level i, j and k. The Scheffe method is employed because the number of contrasts to be estimated is large. The Scheffe confidence interval is: L-Ss(L) £ L £ L+Ss(L) where: A  L is the estimate of the contrast L =/* ijk^i'j'k' s(L) = 2 MSE/n S = ( r - D F(l-« ;(rir )-l,(n-l)rir ) 2  ir2  4.3.1  2  2  Model Sensitivity to the Salt Functional Form Only the transportation factor interacts with the salt functional form.  Consequently, pairs of treatment means/*  (to simplify the notation,^  is used  instead of/* i.i<) are to be compared. To conclude that the constant function (k=2) or the linear function (k=3) can be used instead of the function obtained by previous studies (8) (see also Figure VII.4), the following contrast family (I) are estimated: (I):  Lu  =>« u - / i i 2  Li2 =/"i3-Al  for 1=1,2,3,4,5  The point estimators of the contrasts (I) are, using the data in Table IX.4, presented in Table IX.5. From the family confidence intervals in Table IX.5, it is concluded that  119  although the use of the linear salt function does not incorporate a bias into the measurement of the system response, the assumption that the salt contamination effect is time-independent, underestimates the total logistic cost.  4.3.2  Comparison of Transportation and Inventory Policies The procedure of comparing the proposed transportation and inventory  policies is based on the Scheffe multiple comparison test results presented in Table IX.6. From the 95 percent confidence intervals for the family of contrasts in this Table, the following conclusions may be drawn: 1.  The use of barges as the transportation mode, and dry-land storage as the storage technique, significantly outperforms other transportation and inventory policies if the minimum level of inventory at the mill does not exceed 14 days of consumption. As the mill inventory increases, the (salt) deterioration cost becomes important and hence, the superiority of the barge-dry land storage vanishes.  2.  The  second  best  transportation  alternative  is the  transportation of the logs from the origins to the mill.  direct  For this  alternative, the total inventory cost varies marginally with respect to the minimum level of inventory carried at the mill. However, if more than 14 days of consumption is selected as the minimum inventory level at the mill, then the direct shipment will not be significantly better than the present transportation system and the alternative system where two 30,000 m^-tugs are operating. This is  120  C E L L BY  TOTAL  (  TOTCOST  Total  TRANS  TRANSPORTATION  Logistic  INV  INVENTORY  FORM  SALT  M E A N S  Cost  (X$1000)  POLICY  POLICY  FUNCTIONAL  FORM  POPULATION  7022.41 90)  TRANS  3538.13 18)  (  (  5082.59 18)  (  6517.55 18)  (  6163.49 30)  (  7270.55 30)  (  6115.71 30)  (  7585.88 30)  (  6725.85 18)  13247.90 ( 18)  INV  (  7633.18 30)  FORM  (  7365.63 30)  FORM TRANS  (  3538.17 6)  (  3 5 3 8 . 11 6)  (  3538. 1 1 6)  (  5594.63 6)  (  3768.39 6)  (  5884.73 6)  (  6788.90 6)  (  5503.49 6)  (  7260.26 6)  (  7228.80 6)  (  5596.44 6)  (  7352.29 6)  13677.63 ( 6)  12172.11 ( 6)  TABLE IX.*:  13893.95 ( 6)  TRANS-FORM Response Cell Means  *  *  BY  CELL NO  *  SCHEFFE MULTIPLE COMPARISON TEST - * *  TOTCOST TRANS FORM  LOWER BOUND  T o t a l Log 1st i c Cost ( x$1000 TRANSPORTATION POLICY SALT FUNCTIONAL FORM  CONTRAST  UPPER BOUND  11-12 21-22 31-32 41-42 51-52  -684 936 395 742 616  205 1826 1285 1632 1506  1095 2716 2175 2522 2395  13-12 23-22 33-32 43-42 53-52  -491 1226 867 866 832  398 2116 1757 1756 1722  1288 3006 2647 2646 2612  13-11 23-21 33-31 43-41 53-51  -696 -599 -418 -765 -673  193 290 471 123 216  1083 1 180 1361 1013 1 106  TABLE IX.5:  SIGNIFK DIFFEREI  * * * *  * * * *  ** ** «* * *  «** * * *** * *** * ** *  Scheffe Multiple Comparison Test (TRANS-FORM)  * SCHEFFE MULTIPLE COMPARISON TEST «  BY  CELL NO  TOTCOST TRANS INV LOWER BOUND  TOTAL INVENTORY COST (10O0 TRANSPORTATION POLICY INVENTORY POLICY CONTRAST  UPPER BOUND  SIGNIFIC DI FFEREN  11-12 21-22 31-32 41-42 51-52  - 1772 -825 844 1364 100  4 950 2621 3 140 1876  1780 2727 4397 4917 3653  13-12 23-22 33-32 43-42 53-52  335 -317 - 1633 -1397 989  2111 1459 142 379 2765  3888 3235 1919 2 155 4542  ****  13-11 23-21 31-33 41-43 53-51  331 -1267 702 985 -887  2107 508 2479 2762 889  3884 2285 4255 4538 2665  ****  21-11 31-11 41-11 51-11 31-21 41-21 51-21 41-31 51-31 51-41  1 128 3875 4583 9393 970 1678 6488 -1068 3742 3034  2904 5651 6359 1 1 169 2746 3454 8264 707 5518 4810  4681 7428 8136 12946 4523 5231 1004 1 2484 7295 6587  **** ****  22-12 32-12 42-12 52-12 32-22 42-22 52-22 42-32 52-32 52-42  182 1258 1446 7521 -699 -511 5562 -1588 4486 4298  1958 3034 3222 9297 1076 1264 7338 187 6262 6074  3735 481 1 4999 1 1074 2853 3041 9115 1964 8039 7851  23-13 33-13 43-13 53-13 23-33 43-23 53-23 43-33 53-33 53-43  -469 -710 -285 8175 -1535 -1591 6869 -1351 7109 6685  1306 1065 1490 9951 240 184 8645 424 8885 8461  3083 2842 3267 1 1728 2017 1961 10422 2201 10662 10238  TABLE IX.6:  ***# **** ****  ****  **** ****  m * * m  * * * *  *** *** *** ***  **** **** * * * *  ** * *** *** ***  **** **** ****  ****  **** *** * ****  Scheffe Multiple Comparison Test (TRANS-INV)  123  mainly due to the fact that because of the large inventory at the mill, the mill operation becomes less sensitive to the reliable and fast transportation network. 3.  The present real-system transportation and inventory policy consists of operating 3 tugs with 20,000 m^ capacity each and carrying 14 days of consumption as the minimum inventory level. Although  allowing  more  inventory  at the mill  does not  significantly reduce the system performance, lowering the minimum inventory level will increase the total cost since shortages will occur at the mill. 4.  Irrespective of the level of the minimum inventory at the mill, operating two 30,000 m^-tugs is not significantly better than the present system. However, altering the fleet composition to 6 tugs with a capacity of 10,000 m^ each, will substantially increase the total system cost.  5.  In general, the sensitivity of the system to the transportation mode decreases as the inventory level at the mill increases.  5.  The Effect of the Vessel Capacity  The nature of the relationship between the quantitative variable, the vessel (link) capacity, and the response variable, the total logistic cost is studied.  124  Eight levels for the capacity variable are defined: CAP1 =  10,000  m  3  CAP2 =  12,500  m  3  CAP3 =  13,900  m  3  CAP4 =  15,000  m  3  CAP5 =  16,800  m  3  CAP6 = 20,000  m  3  CAP7 = 25,000  m  3  CAP8 = 30,000  m  3  Table IX.7 shows that the effect of the capacity and the inventory policy on the total inventory cost are significant. The variation of the total logistic cost with respect to the vessel capacity and the inventory policy is shown in Figure IX.3. This figure suggests that: i)  A nonlinear relationship exists between the vessel capcity and the total logistic cost.  ii)  Due to the presence of high shortage cost, the system becomes more sensitive to the vessel capacity if log volumes less than 14,000 m are 3  towed. iii)  The real system total cost is minimized with respect to the vessel capacity, if tow-vessels with a capacity within the range of 15,000 20,000 m  5.1  3  are operated.  Nonlinear Effect of the Vessel Capacity  According to Fisher and Yates (23), when the number of variable levels are selected at equal distances, nonlinear effects of the variable can be  • • •  C I L L  M E A N S  TOTCOST  Total  CAP INV  VESSEL CAPACITY ( M3I INVENTORY POLICY  Logistic  Cost  (x$1000)  TOTAL POPULATION 7983.31 ( 72) CAP 1  2  13138.69 ( 9)  8642.39 ( 9)  3 6291.66 ( 9)  4 6779.91 ( 9)  5 6743.79 ( 9)  6 6710.33 ( 9)  7  8  6720.86 ( 9)  6838.84 ( 9)  INV 1  2  9053.71 ( 24)  7299.95 ( 24)  3 7596.28 ( 24)  INV 1 2 3 4 5 6 7  8  1  2  13868 20 3) (  15022 89 ( 3)  <  9986 67 3) (  7918 54 3) (  I  8021 98 3)  6919 12 3) (  5916 43 3) (  <  6039 44 3)  8447 16 3) (  5873 21 3) (  <  6019 36 3)  8380 09 3)  5836 72 3) (  <  6014 56 3)  8275 39 3)  5847 65 ( 3)  <  6007 95 3)  8170.73 3) (  5964 05 ( 3)  8382 30 3)  6020 01 ( 3)  ( (  (  • • •  BY  3 16524 92 3)  (  6027 79 3)  6114 20 3) (  A N A L Y S I S  TOTCOST CAP INV  OF  V A R I A N C E  TOTAL INVENTORY COST ( 10OO t ) VESSEL CAPACITY (MS) INVENTORY POLICY  SOURCE OF VARIATION MAIN EFFECTS CAP INV 2-WAY INTERACTIONS CAP INV CXPLAINEO RESIOUAL TOTAL  SUM OF SQUARES  OF  MEAN SQUARE  600342724 644 558041669 017 42301055 627  9 7 2  66704747 183 79720238 .431 21150527 .814  34186168 34186168  151 151  14 14  2441869 . 154 2441869 154  634528892 795  23  27588212 730  7105825 018  48  148038 021  641634717 813  71  9037108 .702  F  SIGNIF OF F  450 592 538 512 142 872  0.000 0.000 0.0  16 495 16 495  0.000 0.000  186 359  0.0  81 CASES WERE PROCESSED. 9 CASES ( 11.1 PCT) WERE MISSING.  TABLE IX.7:  CAP-INV Analysis of Variance Table 1  126  INV1  0,^  INV3 INV2  0.0  1  1  50.0  100.0  1  150.0  1  VESSEL CAPACITY IM3) ( X 1 0  FIGURE IX.3:  1  200.0, 2  250.0  1  300.0  )  The Capacity and Inventory Effects  127  calculated by orthogonal polynomials. Therefore, five equidistant levels for the vessel capacity variable are defined: CAP1 =  10,000  CAP2 =  15,000 3  m  3  m  CAP3 = 20,000  m  3  CAP4 = 25,000 m3 CAP5 = 30,000 m3 Table IX.8 shows the corresponding table of analysis of variance. For a five-level factor, the coefficients of the polynomial are: Linear  C[ = { -2,-1,0,1,2 |  Quadratic c = (2,-1,-2,-1,2} 2  Cubic  C3 =  {-1,2,0,-2,1 \  Quartic  c^ =  [l,-4,6,-4,1 \  If SCAPj denotes the sum of the observations at level j; the linear (E[), quadratic ( E 2 ) , cubic ( E 3 ) and quartic ( E 4 ) component of the capacity effect can be calculated by the following orthogonal polynomials: 5 Ej = Z qjSCAPj and the sum of squares of the capacity variable can be decomposed according to:  SSj = (Ei) / n r ? 2  Ci  where n=9 in the present case. Table IX.9 presents the component effects, sums of squares and the Fstatistics. It is noticed that although the major effect of the capacity variable on the total cost is linear, the existence of nonlinear effects is significant.  128  « * • TOTCOST CAP INV  BY  TOTAL (  C E L L  M E A N S  *  T o t a l L o g i s t i c Cost (x$1000) VESSEL CAPACITY (M 3 ) INVENTORY POLICY  POPULATION  8437.73 45)  INV CAP  CAP  1  1  2  3  13868 24 3)  15022 91 3)  16524 94 ( 3)  (  15138.71 ( 9)  (  6779.91 9)  (  6710.33 9)  (  6720.86 9)  (  6838.84 9)  2 3  INV 9428.76 ( 15)  8447 (  (  7745.57 ( 15)  4  8138.85 ( 15)  (  5 (  * * *  A N A LY S I S  TOTCOST CAP INV  BY  8275 39 3)  <  <  8 1 7 0 73 3) 8382 30 3)  V A R I A N C E  *  (  5873 21 3)  (  5847 65 3)  (  6007 95 3)  5964 05 3)  (  6027 79 3)  6 0 2 0 01 3)  (  6114 20 3)  6019 36 3)  * +  T o t a l L o g i s t i c Cost (x$1000 ) VESSEL CAPACITY (M 3 ) INVENTORY POLICY SUM OF SQUARES  SOURCE OF VARIATION  5 2 8 5 1 5 4 1 9 .494 505257113 .395 23258306 .099  MAIN EFFECTS CAP INV 2-WAY INTERACTIONS CAP INV EXPLAINED RESIDUAL TOTAL 45 CASES WERE PROCESSED. 0 CASES ( 0 . 0 P C T ) WERE  TABLE IX.8:  0 F  16 3)  DF  MEAN SQUARE  F  SIGNI OF F  6 88085903 249 4 126314278 349 1 1629153 050 2  440 883 632 222 58 206  0.000 0.000 0.0  3 1 1 9 0 4 5 9 .594 3 1 1 9 0 4 5 9 .594  8 8  3898807 449 3898807 449  19 514 19 514  0.000 0.000  5 5 9 7 0 5 8 7 9 .088  14  39978991 . 3 6 3  200 101  0.000  5 9 9 3 8 2 9 .679  30  199794 . 3 2 3  5 6 5 6 9 9 7 0 8 .767  44  12856811 . 5 6 3  MISSING.  CAP-INV Analysis of Variance Table 2  129  component Erfect  Linear  Quadratic  Cubic  Quartic  Sum or squares SS,  F-5tatlsttc (u,=1,^2=18)  -149 929.1 1  249 763 755.90  1250.10  153 303.03  186 522 373.10  933.57  -73 635.93  60 247 224.30  301.55  74 128.05  8 722 171.10  43.66  TABLE IX.9:  Linear, Quadratic Cubic and Quartic Effects of CAPACITY  130  6.  Probability Distribution Function of Inventory Levels  The probability distribution function of the monthly opening inventory level for large pulp logs is estimated from a sample of 15 observations. The Weibull distribution (see Equation VII. 1) is selected. The shape (^), the scale (6 ) and the locaiton (/<) parameters of the function are estimated by the  maximum likelihood method.  Figures IX.4.1-IX.4.3 and Table IX. 10  investigate the goodness of fit of the models for different months of the year. The values of  and/< j for the probability distribution of the opening  inventory levelY: f  W.  =  V  i ny-M.J/o.lV  a  1  exp{-[(y-,i.)/a  ] i} X  are tabulated in Table IX. 11. Hence, the probability that at the starting of the month i, the inventory level be less than or equal to 1 is computed according to: F  = 1 - exp{-[(/-„.)/ .] i} X  W.  i  i  a  If the level 1 is equal to the mill one-month consumption (36,000 m3), then: P(Yj P(Y P(Y  an  ^36000)  =  1.00  A p r  ^ 36000)  =  .87  3ul  « ; 36000)  P(Y  S e p  P(Y  D e c  = 0.  ^ 36000)  =  1.00  ^36000)  =  0.05  131  FIGURE IX.4.1:  Weibull Fit for the Opening  Inventory Models (January, April)  FIGURE IX.4.2: Weibull Fit for the Opening Inventory Models (July, September)  FIGURE IX.4.3: Weibull Fit for the Opening Inventory Model (December)  134  Calculated  Degree of Freedom  Critical  2.71  1  3.84  Weibull  April  .14  1  3.84  Weibull  July  •14  1  3.84  Weibull  September  .08  1  3.84  Weibull  December  1.08  1  3.84  Weibull  Months  X  January  TABLE IX. 10:  2  X  2  Model  Goodness of Fit Evaluation for the opening Inventory Models  135  This simulated characteristic conforms to the real-system behaviour, id est it is certain that after the months of closure, either during the winter or summer season, the level of inventory at the storage area drops below the level corresponding to the mill one-month consumption.  After the top production  months, the probability that the inventory level at the storage area is less than or equal to the aforementioned volume, falls substantially.  Months  Shape X  Scale  Location U  January  3.2  9 133.2  0  April  2.2  8 442.3  24 307 0  July  3.3  11 327.6  54 200.9  September  2.2  8 211.4  1 277.4  December  3.4  12 431.5  30 877.6  TABLE IX.11: Opening Inventory Probability Distribution Parameters  137  X.  CONCLUSION  The marine log transportation along the coast of British Columbia has been considered.  This chapter  summarizes the  major results obtained.  Directions for future studies are presented.  1.  Evaluation of the System-Analysis Models  1.1  Deterministic Network Model the deterministic models belongs  to the class of the fixe charge  multiproduct capacitated lot sizing problems. The Dantzig-Wolfe  and Benders decomposition  optimize the formulated network.  Due  principles are used to  to the large number of iterations  between the subproblems and master problems, the implementation  of the  decomposition principles requires excessive amount of CPU time. Future works should focus on the variable redefinition method to optimize the above network.  1.2  Simulation Model the simulation model provides a more realistic image of the real system.  In addition to this property, the simulation model concserves a high degree of simplicity and carries a satisfying evolutionary aspect. The model utilizes a set of initial conditions in order to reduce the transient period and is validated in four steps: 1.  Extreme-case analysis  138  2.  Review by experts  3.  Observation of intermediate results, and  4. Comparison  of the model  behaviour  with  the  real-system  comportment.  2.  Performance of Transportation Policies With Respect to Minimum Mill Inventory Levels  The validated simulation model is used in the experimentation. The following results are obtained with 95 percent confidence interval: 1.  The effect of a proposed transportation policy should be measured with regard to the inventory doctrine used in the system, due to the interaction between transportation and inventory alternatives.  2.  The sensitivity of the system to the transportation mode decreases as the inventory level at the mill increases.  3.  A nonlinear relationship exists between the vessel capacity and the total logistic cost.  This relationship indicates that due to the  importance of the shortage cost, the sensitivity of the system to the vessel capacity increases, if volumes less than 14,000 m  3  are towed.  The real-system total cost is minimized with respect to the vessel capacity for tow-vessels with a capacity within the range of 15,000 20,000 m . 3  4. The use of barges as the transportation mode and dry-land storage as the storage technique significantly outperforms other transportation  139  and inventory policies, if the minimum level of inventory at the mill does not exceed 14 days of consumption. increase in the minimum  inventory  Beyond this limit, an  levels increases the salt  deterioration cost. Consequently, the logistic cost of this alternative increases. 5.  The  second  best  transportation  alternative  is the direct  transportation of logs from the origins to the mill. As the minimum inventory level increases, the mill operation becomes less sensitive to the fast transportation network and therefore, renders the direct shipment and the shipment via a consolidated point equal in performance. 6.  The log-taxi alternative (i.e. more tugs of smaller capacity) does not reduce the logistic cost if the system processes only pulp logs. However, the log-taxi alternative is expected to become attractive when more valuable logs are transported and kept in inventory.  7.  Given the present real-system transportation and inventory policy and a 10% simple interest rate, an increase in the minimum inventory volume at the mill does not reduce the system performance. However, if the interest is compounded and higher rates are used, it is expected that the system's sensitivity to large inventory volumes increases.  8.  Due to the high shortage cost used in this study, the reduction of the minimum inventory level (the safety stock) will increase the total logistic cost.  140  9.  There is no interaction between the inventory depletion mode (FIFO, LIFO) and  the transportation and  inventory  policies given  the  assumption that the system processes only pulp logs. Furthermore, the adoption of either FIFO or LIFO does not alter the performance of the system, since the salt contamination function is neither concave nor convex (See also Section III.3).  3.  Sensitivity Analysis  The simulation model is used to measure the system sensitivity to the linear and constant salt contamination functions. It is found that although the use of the linear salt function does not introduce a bias into the measurement of the system response, the assumption that the salt contamination effect is time-independent, underestimates the total annual inventory cost.  4.  Contribution of the Study  This study may  be among the first studies which formulated the marine  log transportation and inventory system existant in the west coast of British Columbia. A simulation model is developed to help decision-makers evaluate their strategic and operational decisions.  141  In developing this simulation model, considerable attention is given to preserving its adaptability and evolutionary aspects. The model allows for multiple origins, multiple log types, one storage area and multiple mill locations is solely limited by the maximum memory allocated by the host computer system. The model is easily extended to allow for more than one storage location. The model can be used to study the effect of other deterioration agents, such as teredos.  This attempt requires the inclusion of the deterioration  functions with respect to the water residency time. In the above cases, the probability of events, such as running out of logs of a given type prior to a particular date, can be assessed. These probability measurements could then be used by managers to evaluate the risk of certain situation and react accordingly.  5.  Future Research  The following is a list of future research to enhance the logistic model. i)  A model of teredo and ambrosia beetle damage should be developed.  ii)  The losses due to salt contamination, including increased maintenance costs of machinery and degradation of the quality of pulp log should be accurately evaluated.  iii)  The cost consequences of a mill shutdown should be assessed and finally,  iv)  The use of advanced technology in the transportation domain, such as the use of fast and powerful tug boats or helicopters, should be studied.  1  *  2  *  MULT.  3 4  ORIG  -  ONE  STORAGE  -  ONE  DESTINATION  -  MULT.  SIMULATE *  5  INTEGER  6  REALLOCATE  FUN,20,LOG,7,CHA,5.COM,200000,QUE,4  7  RMULT  230,11,319.556.195  &J  8  PMN  EOU  9  KLEt  EOU  10  EVR  EOU  3.C  1,C  4,C  11  STORA  EOU  MILL  EOU  5,C  13  ORDR  EOU  4,0  14  CUNSI  EOU  I.L  15  CUNS2  EQU  2,L  16  CUNS3  EOU  3,L  17  WKEND  EOU  4,L 11.L  18  T0W1  EOU  19  T0W2  EOU  12,L  20  T0W3  EOU  13,L  21  MILL1  EOU  1,2  22  MILL2  EOU  2,Z 3,Z  23  MILL3  EQU  24  PMN1  EQU  11.Z  25  PMN2  EQU  12,Z  26  PMN3  EQU  13,Z  27  KLB1  EQU  21.Z  28  KLB2  EQU  22,Z  29  KLB3  EQU  23,Z  30  EVR 1  EQU  31.Z  31  EVR2  EQU  32.2  32  EVR3  EQU  33,Z  33  PMNTA  EQU  101,Z  34  KBTA  EQU  102,2  35  ERTA  EQU  103,2  36  TAPR  EQU  104,Z  37  PSAPW  EQU  1,XH  38  PUF2  EQU  9.XH  39  SHCST  EQU  4,MH  40  MMIN  EQU  10,MH  41  *  42  *  43  * STORAGE *  46  *  47  *  Simulation Model Listing  STORAGE  44 45  APPENDIX 1  2.C  12  S$TUG,3  MATRICES  48  ORWI  MATRIX  X.3.52  ORIGINS  49  MWI  MATRIX  X,3,52  MILL'S  WEEKLY WEEKLY  50  STWI  MATRIX  X.3.52  STORAGE'S  51  SWI  MATRIX  X.3,52  SYSTEM'S  52  SUMIN  MATRIX  X.4,3  SUM  53  SHORT  MATRIX  H.3.52  WEEKLY  WIC  MATRIX  X,8,52  54  OF  *  R=1  56  *  R=3,4  57  *  R=6 INV  MATRIX  X.3,3  AVERAGE  WEEKLY  DAILY  AVERAGE  INV.  COST COST  DET.  &  SHORTAGE  CURRENT  INV.  (R=TYPE  INV.  AVERAGE  SHORTAGE  DET.  INV.  AVERAGE  WEEKLY  INVENTORY  55  58  SORT  INV. INV.  LEVELS  C=WEEK  R=1,2,3.4  )  FOR  MILL,STOR.,SYSTEM  AND  ORIGINS  (R=TYPE;C=WEEK) AT  THE  END  FOR  ALL  ORIGINS.STORAGE  FOR  LOGS  FIN. COST, LEV.  R=7  OF  C &  MILL;  CONSUMMEDDURING  TOTAL  (R = 1 , 2 , 3  WEEK  COST  &  R=8  R=2  THE  CUMULATIVE  MILL,STORAGE,SYSTEM;C  FINANCIAL  WEEK;  R=5  TOTAL  = TYPE)  COST;  TRANSPORTAION COST  COST  59 GO 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 1 16  OINV TOTIO OSHIP SHRTG DEMND SHCST ORTPN LOGV OUNIT SUNIT Mil MMIN  MATRIX MATRIX MATRIX MATRIX MATRIX MATRIX MATRIX MATRIX MATRIX MATRIX MATRIX MATRIX  X,1,3 X.1,3 X,1,3 H.1,3 H,1,3 H,1,3 H.1,3 H,1,3 H,3,3 H.1,3 H.1,3 H,1,3  SUM OF CURRENT ORIGINS I N V . (C=TYPE) CURRENT TOT. I N V . ( A L L TYPE) @ ORIG. C SHIPMENT VOLUME FROM ORIGIN C TOT. SHORTAGE OF LOG TYPE/SORT C DEMAND FOR LOG TYPE/SORT C UNIT SHORTAGE COST FOR LOG TYPE C IN $/M3 TOT. NO. OF LOG TYPE/SORT AT ORIG. C VALUE OF LOG TYPE C IN $/M3 TYPE C I N I T . I N V . AT ORIG. R (M3/DAY) TYPE C I N I T . I N V . AT THE STOR. (M3/DAY) TYPE C I N I T . I N V . AT THE MILL (M3) M I N . I N V . L E V . AT THE MILL FOR TYPE C  INITIAL INITIAL INITIAL INITIAL INITIAL INITIAL INITIAL  MH$ORTPN(1, 1-3) ,3 MH$LOGV(1,1-3),45 M H $ S H C S T ( 1 , 1 - 3 ) , 120 MH$OUNIT( 1 - 3 , 1 - 2 ) , 3 6 0 0 / M H $ 0 U N I T ( 1 - 3 , 3 ) , 1800 MH$SUNIT(1,1-2),12000/MH$SUNIT(1,3),6000 MH$MII(1,1-2),20000/MHSMII(1,3),10000 MH$MMIN(1,1-2),16800/MH$MMIN(1,3),8400  * *  * * *  SAVEVALUES INITIAL INITIAL INITIAL INITIAL INITIAL INITIAL INITIAL INITIAL INITIAL INITIAL  XH$PSAPW,10 XH$DRIGN,3 XH$T0TYP,3 XH$DIREC,0 XH$0DAYS,1 XH$SDAYS,2 XH$TOW,20000 XH$PUF1,10 XH$PUF2,10 X$CAP,20000  % OF SAPWOOD IN CHIPS TOTAL NUMBER OF CAMP SITES TOTAL LOG TYPE/SORT 0=VIA A STORAGE ; 1=DIRECT DAYS OF I N I T I A L I N V . AT ORIGIN DAYS OF I N I T I A L I N V . AT STORAGE CALL WHEN A TOW IS FORMED PICK UP FREQ. FOR 0RIG.1=PMN PICK UP FREO. FOR 0RIG.2=KB TUG CAPACITY IN M3  * * FUNCTIONS * *PRODUCTION * PMN1 FUNCTION P1,D8 PRODUCTION RATE FOR TYPE 31 , 4 2 0 / 1 2 0 , 5 1 0 / 1 8 1 , 5 4 0 / 2 1 2 , 0 / 2 4 3 , 1 4 5 / 3 0 4 , 5 4 0 / 3 3 4 , 5 1 0 / 3 6 4 , 0 PMN2 FUNCTION P1.D8 PRODUCTION RATE FOR TYPE 31,460/120,820/181,840/212,0/243,150/304,840/334,820/364,0 PMN3 FUNCTION P1.D7 PRODUCTION RATE FOR TYPE 31,35/120,85/181 , 130/243,0/304,130/334,85/364.0 KLB1 FUNCTION P1.D7 TYPE 31,170/120,840/181,900/243,0/304,900/334,840/364,0 KLB2 FUNCTION P1.D7 TYPE 31,200/120,580/181 ,600/243,0/304,600/334,580/364,0 K L B 3 FUNCTION P1.D7 TYPE 31,250/120,790/181,950/243,0/304,950/334,790/364,0 EVR1 FUNCTION P1.D7 TYPE 31,355/120,900/181,1000/243,0/304,1000/334,900/364,0 EVR2 FUNCTION P1.D7 TYPE 31,270/120,900/181,950/243,0/304,950/334,900/364,0 EVR3 FUNCTION P1.D7 TYPE 31,45/120,190/181,250/243,0/304,250/334,190/364.0 *  1 AT PMN 2 AT PMN 3 AT PMN 1 AT KB 2 AT KB 3 AT KB 1 AT ER 2 AT ER 3 AT ER  _ -P" ^  117 118 1 19 120 121 122 123 124 125 126 127 128 129 130 1.31 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169' 170 171 172 173 174  •CONSUMPTION *  MI LL 1 FUNCTION 364,1200 M I L L 2 FUNCTION 364,1200 M I L L S FUNCTION 364,600  PI,01  CONSUMPTION RATE FOR TYPE 1  P1.D1  CONSUMPTION RATE FOR TYPE 2  P1.D1  CONSUMPTION RATE FOR TYPE 3  *  •TRANSIT  TIME  *  RN1.C14 PMN - TA TRANSIT TIME PMNTA FUNCTION 0 , 3 . 0 / . 04100.3.10/ .24458,3.67/.41726,4.29/.54097,4.86/.64590,5.48 .75088,6.32/.83677 ,7.33/.89305,8.34/.93195,9.42/.95968,10.67 .98187, 1 2 . 5 8 / . 9 9 3 1 4 , 1 4 . 9 0 / 1 , 3 3 KBTA FUNCTION RN2.C13 KB-TA TRANSIT TIME 0 , . 9 3 / . 05378,1.27/ .17269,1.71/.49609,2.78/.68569,3.56/.78780,4.13 .87328,4.81/.92461 ,5.44/.96018,6.16/.98265,7.03/.99455,8.15 . 9 9 8 7 5 , 9 . 4 6 / 1 , 14 ERTA FUNCTION RN3.C12 ER-TA TRANSIT TIME 0,2./.07104,2.1/.30818,2.5/.52490,3.01/.68087,3.55/.78720,4.10 .84949,4.57/.90746 ,5.23/.94753,6.0/.97768,7.16/.99562,9.37/1,19 TAPR FUNCTION RN4.D7 TA-PR TRANSIT TIME .59452,1/.90367,2/ .98405,3/.99791,4/.99979,5/.99998,6/1,7 DELAY FUNCTION RN5.C12 0 . . 1 . / . 13610,2.65/ .35639,5.97/.55649,10.17/.70608,14.81/.80897.19.67 . 8 9 5 9 3 , 2 6 . 5 2 / . 9 4 6 5 7 , 3 4 . 0 4 / . 9 7 9 9 4 , 4 5 . 0 9 / . 9 9 5 3 0 , 6 1 . 4 5 / . 9 9 9 6 6 , 9 1 . 0 6 / 1 , 139 *  •GENERAL  *  MP1.C4 SALTS FUNCTION 0,0/20, .5/40,1.5/180,2.8 SALTH FUNCTION MP1.C5 0,0/20, .01/40,.03/80,.01/180,.05 XH$ABSIS,D4 1ST FUNCTION 1 , . 1 2 / 2 , 1 . 7 7 / 3 , 1 . 5 / 4 , .3 * * *  * * * *  SALT CONTAMINATION OF HEARTWOOD I N T . RATE, SALT D E T . 8. TRANS. UNIT COST  (0-  VARIABLE LOADO LOADS PHRTW DETER LOST FINAN TRTIM SUM SURP CHAIN  * *  SALT CONTAMINATION OF SAPWOOD  BVARIABLE BVARIABLE VARIABLE FVARIABLE FVARIABLE FVARIABLE FVARIABLE VARIABLE VARIABLE VARIABLE  P1'LE'XH$ARIVA P1'LE'XH$READY 100-XH$PSAPW ((FN$SALTS XH$PSAPW)+(FN$SALTH+V$PHRTW))*P4*FN$IST^10/100 MH$SHRTG(1,P2)+MH$SHCST(1,P2) P4*MH$L0GV(1,P3) MP1 FN$IST/364 (FN+3)+.5 MX$WIC(3,P1)+MX$WIC(4,P1)+MX$WIC(5,P1) P4-MH$DEMND(1,P3) 1+(CH$ST0RA)+(CH$MILL)+(CH1)+(CH2)+(CH3) t  +  +  TABLES RT I ME TABLE  SEGMENT 1  MP1,0,5.52  WATER R E S . TIME FOR SOLD LOGS  INITIALISATION  OF USER  CHAINS  175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 21 1 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 23 1 232  * * *  *  MILL GENE MARK IN I TL S P L I T TRANSFER PROCD ASSIGN ASSIGN MSAVEVALUE MSAVEVALUE PRIORITY LINK  1 , , , 1 ,7,5 1 XHSTOTYP,PROCD .STORE 3,(W$INITL+1) 4,MH$MII(1,P3) INV+,1,P3,P4 INV+,3,P3,P4 5 MILL,FIFO  BRANCHING BY TYPE  MILL  INITIALISATION  STORAGE AREA *  * * *  STORE TEST E ASSIGN DAYST PRIORITY INTRM S P L I T TRANSFER STO ASSIGN ASSIGN MSAVEVALUE MSAVEVALUE PRIORITY LINK SLOOP PRIORITY TEST E LOOP  ORIGINS DIR ASSIGN DAYOR PRIORITY SITE SPLIT TRANSFER SOURC ASSIGN SPLIT TERMINATE ORGIN TEST E SAVEVALUE SAVEVALUE SAME ASSIGN ASSIGN PRIORITY MSAVEVALUE MSAVEVALUE MSAVEVALUE TEST E TERMINATE LNKO LINK LOOP PRIORITY TEST E LOOP TERMINATE  *  XH$DIREC,0.DIR 2,XH$SDAYS 7 XH$TOTYP,STO ,SLOOP 3,(W$INTRM+1 ) 4,MH$SUNIT(1,P3) INV+.2.P3.P4 INV+,3,P3,P4 5 STORA,FIFO 6 W$INTRM.O 2,DAYST  2,XH$0DAYS 7 XH$T0TYP,SOURC .LOOP 3.(W$SITE+1) XH$ORIGN,ORGIN XH$COUNT,XH$ORIGN,SAME COUNT,0,H COUNT+, 1 , H 2,XH$C0UNT 4,MH$0UNIT(P2,P3) 4 OINV+,1,P3,P4 TOTIO+,1,P2,P4 INV+,3,P3,P4 P4.0.LNK0 P2,FIFO 5 W$SITE,0 2,DAYOR  BRANCHING BY TYPE BRANCHING BY  ORIGINE  NUMBER OF ORIGIN  -p-  233 234 235 236 237 238 239 240 24 1 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 26 1 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290  * * * * *  SEGMENT 2  GENE LOGIC S ADVANCE LOGIC R TERMINATE  * * * * * * * *  OPERATION CONTROL ***  7,,6,,6,1 WKEND 2 WKEND  WEEKEND = THE 6 & 7TH DAYS OF THE WEEK CLOSE OPERATION AT CAMPS WEEKEND IS OVER  SEGMENT 3  INVENTORY CONTROL * * * ORIGIN  DIVO DIVT FIN PRD  GENE GATE LR MARK SPLIT SPLIT ASSIGN ASSIGN TEST E TERMINATE MSAVEVALUE MSAVEVALUE MSAVEVALUE LINK  * * *  1 , , , ,5,5 WKEND,FIN 1 (XH$0RIGN-1),DIVO,2 (MH$ORTPN(1,P2)-1),DIVT,3 5,((P2*10)+P3) 4,FN*5 P4,0,PRD INV+,3,P3,P4 OINV+, 1 , P 3 , P 4 TOTIO+, 1 , P 2 , P 4 P2.FIF0  DO NOT WORK ON WEEKENDS RECORD WATERING TIME  PRODUCTION INCREASE SYSTEM INCREASE ORIG. ORIGIN  INV. INV.  INV. (FIFO)  STORAGE AREA ARIVS MSAVEVALUE TEST G LINK MSAVEVALUE J1 MSAVEVALUE SAVEVALUE SAVEVALUE TEST E MSAVEVALUE LINK  * * *  OSHIP+,1,P2,P4 MX$OSHIP(1,P2),X$CAP,J1 P2.LIF0 OINV-,1,P3,P4 TOTIO-,1,P2,P4 ABSIS,3,H TRCST+,(P4*FN$IST) XH$DIREC,0,J2 INV+,2,P3,P4 STORA.FIFO  TOW SHOULD BE < TUG C A P . EXTRA PRODUCTION STAYS @ ORIGIN DECREASE ORIG. I N V . TRANSPORTAION COST DIRECT SHIPMENT ? INCREASE STORAGE I N V . STORAGE I N V . ( F I F O )  MILL ARIVM ASSIGN GATE LS SAVEVALUE TEST G LOGIC R RELNK LINK SAVEVALUE J3 SAVEVALUE MSAVEVALUE  5,(10+P3) P5,RELNK SSHIP+.P4 X$SSHIP,X$CAP,J3 P5 STORA,LIFO ABSIS,4,H TRCST+,(P4*FN$IST) INV-.2,P3,P4  0\  TRANSP . COST DECREASE STORAGE I N V .  231 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 31 1 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 34 1 342 343 344 345 346 347 348  J2  MSAVEVALUE LINK CNSUM LOGIC S TEST GE MSAVEVALUE MSAVEVAULE ASSIGN ASSIGN MSAVEVALUE TABULATE SAVEVALUE SAVEVALUE SAVEVALUE SAVEVALUE ASSIGN TEST E TERMINATE BACK LINK MORE MSAVEVALUE MSAVEVALUE MSAVEVALUE TABULATE SAVEVALUE SAVEVALUE SAVEVALUE SAVEVALUE TERMINATE  INV+,1.P3.P4 MILL,FIFO P3 P4,MH$DEMND(1,P3) ,.MORE INV-,1,P3,MH$DEMND(1,P3) INV-,3,P3,MH$DEMND( 1 ,P3) 5,VSSURP 4,MH$DEMND(1,P3) DEMND,1,P3,0,H RTIME ABSIS.2,H DCC+,V$DETER ABSIS,1,H FCC+,V$FINAN 4,P5 0, P4,BACK MILL,LIFO DEMND-,1.P3.P4.H INV-,1,P3,P4 INV-,3,P3,P4 RTIME ABSIS,2,H DCC+,V$DETER ABSIS,1 ,H FCC+,V$FI NAN  INCREASE MILL I N V . MILL INVENTORY ( F I F O ) DEPLETE MILL INV. DEPLETE S Y S . INV. QUANTITY USED READ DET. COST UNIT DET. COST FOR CONSUMMED LOG DURING THE WEEK READ INTEREST RATE FINANCIAL COST FOR CONS. LOG DURING THE WEEK  TRANS. CARRYING THE SURPLUS GOES TO THE FRONT UNSATISFIED DEMAND MILL SYSTEM  * * ********************************************************************** * *  TOWING OPERATION  SEGMENT 4  * * * *  ***  ORIGIN -- STORAGE AREA GENE ASSIGN TRANSFER  XH$PUF1,,,,3,4 2, 1 , SENT  TUG IS SENT ACORDING TO THE DESIRED ORIGIN 1 = PMN  FREQ.  GENE ASSIGN TRANSFER  XH$PUF2,,,,3,4 2,2 .SENT  TUG IS SENT ACORDING TO THE DESIRED ORIGIN 2 = KB  FREQ.  GENE ASSIGN TEST L TERMINATE  1 ,, , ,3,4 TUG IS SENT WHEN A TOW IS FORMED 2,3 MX$TOTIO(1,P2),XH$TOW,SENT  •*  *  *  SENT  TEST E MARK ENRUT MSAVEVALUE TEST G QUEUE ENTER DEPART ASSIGN ASSIGN  Q*2,0,ABORT 1 OSHIP, 1 , P 2 , 0 CH*2,0,ABORT P2 TUG P2 1 , (P1+MP1) 3,(100+P2)  -pIF NO PRODUCTION,  ABORT THE OPERATION  SEND A TUG IF AVAILABLE DEPARTURE TIME FROM ORIGIN TRANS. TIME FUNCTION NO.  349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406  * * *  *  ADVANCE SAVEVALUE UNLINK TEST E LEAVE ABORT TERMINATE  V$TRTIM ARIVA.P1 ,H P2,ARIVS,ALL,BV$LOADO P4.0.MIL TUG  TRANSIT TIME TUG'S ARRIVAL TIME TUG ARIVES AT THE STORAGE AREA TUG IS  FREE  STORAGE AREA - MILL GENE MARK ASSIGN SAVEVALUE SPLIT ASIGN ASSIGN LOGIC R TEST G TEST LE LOGIC S TEST LE TEST E QUEUE TEST E ENTER ASSIGN ASSIGN ADVANCE SAVEVALUE UNLINK MIL DEPART LEAVE TERM TERMINATE  1 . . . .4,4 1 4, 1 SSHIP,0 (XH$T0TYP-1),ASIGN.2 3,(10+P2) P3 ( M X $ I N V ( 2 , P 2 ) + X H $ D I R E C ) , 0,TERM IS STORAGE EMPTY? M X $ I N V ( 1 , P 2 ) , ( M H $ M M I N ( 1 , P 2 ) * 1 . 5 ) , T E R M ORDER THOSE BELOW 1.5 TIMES MMIN P3 RECORD THE TYPE MX$INV(1,P2),MH$MMIN(1,P2).TERM ORDER IF M I N . LEVEL IS REACHED Q$ORDR,0,TERM IS AN ORDER ALREADY PLACED? ORDR XH$DIREC,0,DIRCT TUG 1 ,(P1+MP1) DEPARTURE TIME 2,(101+XH$0RIGN) TRANS. TIME FUNCTION NO. FN*2 READY,(FN$DELAY+P1),H BOOMS BECOME AVAILABLE STORA,ARIVM,ALL,BVSLOAOS ORDR TUG DIRECT  DIRCT  SHIPMENT  SELECT MAX 2 , 1 , 3 , , C H TRANSFER ,ENRUT  * * ****** **************************************************************** * * SEGMENT 5 MILL CONSUMPTION *** * * *  * *  GENE MARK SPLIT DIVM MSAVEVALUE NEXT TEST G UNLINK GATE LS TEST G LOGIC R TRANSFER TRUBL MSAVEVALUE END LOGIC R TERMINATE  1 2,3 1 (XH$T0TYP-1),DIVM,3 DEMND,1,P3.FN*3,H MX$INV(1,P3),0,TRUBL MILL,CNSUM,1,3 P3 MH$DEMND(1,P3),0.END P3 , NEXT SHRTG+, 1 ,P3,MH$DEMND( 1 , P 3 ) , H P3  DAILY  CONSUMPTION  IS THE DEMAND COMPLETLY S A T I S F I E D ? Co  407 408 409 410 41 1 412 413 414 415 416 4 17 4 18 419 420 42 1 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464  *********************************  * * * *  *  SEGMENT 6 GENE SPLIT SUITE MSAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE TERMINATE  LOGTP  SORCE OBOOK  SBOOK  MBOOK  GENE SAVEVALUE ASSIGN SPLIT MSAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE SAVEVALUE MSAVEVALUE SAVEVALUE SAVEVALUE SAVEVALUE UNLINK UNLINK SPLIT UNLINK TERMINATE SAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE SAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE LINK SAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE SAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE LINK SAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE SAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE LINK  t.  BOOK KEEPING  *************  +**  1, ... 1 ,1 (XH$T0TYP-1).SUITE,1 SUMIN+, 1 ,P1 , M X $ I N V ( 1 , P I ) SUMIN+.2.P1,MX$INV(2,P1) SUMIN+.3.P1,MX$INV(3,P1 ) SUMIN+.4.P1,MX$OINV(1 ,P1) 7,.,,0,2 WEEK+, 1 , H 1,XH$WEEK XH$TOTYP,BOOK WIC.3.P1,X$DCC WIC.4.P1,X$FCC WIC.5.P1,X$TRCST W I C 7 . P 1 ,V$SUM CUMUL+,V$SUM WIC.8.P1,X$CUMUL DCC . 0 FCC.O TRCST.O STORA,SBOOK,ALL MILL,MBOOK,ALL (XH$0RIGN-1),S0RCE,2 P2,OBOOK,ALL ABSIS,2,H WIC+,1.XHSWEEK,VSDETER WIC+,7,XH$WEEK,V$DETER WIC+,8,XH$WEEK,VSDETER ABSIS, 1 ,H WIC+,2,XH$WEEK,V$FINAN WIC+,7,XHSWEEK,V$FINAN WIC+,8,XHSWEEK,VSFINAN P2.FIF0 ABSIS,2,H WIC+,1,XH$WEEK,V$DETER WIC+,7,XHSWEEK,VSDETER WIC+,8,XHSWEEK,VSDETER ABSIS, 1 ,H WIC+,2,XHSWE E K , V $ F I N A N WIC+,7,XH$WEEK,V$FINAN WIC+,8,XHSWEEK,VSFINAN STORA,FIFO ABSIS,2,H WIC+,1,XHSWEEK,VSDETER WIC+.7,XHSWEEK,VSDETER WIC+,8,XHSWEEK,VSDETER ABSIS, 1 ,H WIC+,2,XHSWEEK,VSFINAN WIC+,7,XHSWEEK,VSFINAN WIC+,8,XHSWEEK,VSFINAN MILL,FIFO  -p-  465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 544 545  BOOK  ASSIGN MSAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE SAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE MSAVEVALUE TERMINATE  2,(W$LOGTP+1) MWI,P2,P1,(MX$SUMIN(1,P2)/7) STWI,P2.P1,(MX$SUMIN(2,P2)/7) SWI,P2,P1.(MX$SUMIN(3,P2)/7) ORWI,P2.P1,(MX$SUMIN(4,P2)/7) SHORT,P2,P1,MH$SHRTG(1,P2),H CUMUL+,V$LOST WIC+.6.P1,V$LOST WIC+,7,P1,V$LOST WIC+.8.P1,V$LOST SHRTG,1.P2.0.H SUMIN,1,P2.0 SUMIN.2.P2.0 SUMIN.3.P2.0 SUMIN,4,P2,0  GENE UNLINK UNLINK UNLINK UNLINK UNLINK TERMINATE TABULATE TERMINATE  364, , , , 1 , 1 STORA , A G E , A L L MILL,AGE,ALL 1,AGE,ALL 2,AGE,ALL 3,AGE,ALL  GENE TERMINATE  365,,,,9,1 1  START  1  * * ******************************************************** * * * AGE DISTRIBUTION OF LOGS IN THE SYSTEM * *  AGE  RTIME  * * ********************************************************************** * * * COUNTER * *  * * * *****  END  o  151  APPENDIX 2  Transit Time Data Days  PMN-TA  ER-TA  KB-TA  4 . 5. 4. 7. 4. 7. 6. 3. 10. 4. 6. 4. 6. 3. 5. 6.  3. 3.  2.  2. 3. 3. 2. 5.  2.  3. 8. 6. 3. 2. 4.  12 . 5.  PMN = Port McNeill ER =  Eve River  KB =  Kelsey Bay  PR =  Powell River  TA =  Teakern Arm  4. 4. 3. 6. 3. 3. 3.  2. 7.  2. 3. 4. 5. 3 3 2 4 3 3 6 3 2 4 2  1 ".  2.  2. 5. 3. 3. 3. 4. 2. 2. 2. 2. 2. 3.  TA-PR 1. 2. 1 . 1 . 1 . 2. 1 . 2.  1 .  3. 2. 2 . 2.  1 . 2. 1 . 2. 2. 3.  1 .  1 . 1 . 1 . 1 . 1 . 2. 1 . 2. 3. 1 . 1 . 2. 2. 2. 1 . 1 . 1 . 2. 1 . 1 . 1 . 1 . 2. 2.  Delay at T A 10. 2. 10. 5. 2. O. 1 . 4. 3. 0. 3. 2. 19 . 18 . 10. 4. 89 . 1 . 4. 13 . 5.  2.  88 . 8. 6. 3. 2. 1 . 10. 1 . 6. 15 . 4. 1 . 5. 39 . 34 . 30. 0. 1 . 7. 0. 1 . 2.  2. 3. 43 . 1 . 20. 27 . 40. 25 . 20. 14 . 3. 2. 0. 1 . 7. 0. O. 2. 3. 35 . 10. 4. 14 . 2.  152  REFERENCES 1.  Anshoff, H. I. and R. L. Hayes, 1972, "Role of Models in Corporate Decision Making", Proceedings of IFORS Sixth International Conference, Dublin, Ireland, August 1972.  2.  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