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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 SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF GRADUATE STUDIES Civil Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1986 O Siavoche Kahkeshan, 1986 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. Civi 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 system variables and their interactions and assesses 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 TABLE OF CONTENTS Abstract i i L i s t of Figures v i i L i s t of Tables ix Acknowledgement x I. INTRODUCTION 1 1. Problem Identification 3 2. Complexing Issues 5 3. Objective Defini t ion 6 4. Outl ine of the Thesis 9 II. LITERATURE REVIEW 11 1. General Li tera ture Review 11 2. Log-Industry Li terature Review 13 3. Summary 15 PART 1; THEORY 18 ID. SYSTEM FORMULATION 19 1. System Formulat ion 19 2. Logis t ic Cost Components 20 3. Assumptions 21 4. Summary 22 IV. SIMULATION MODEL 2* 1. Model Development 24 2. Summary 25 i i i V. NETWORK FLOW MODEL: FORMULATION 30 1. Selection of Decision Variables and Parameters 30 1.1. Decision Variables 30 1.2. Parameters 31 2. 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 VI. 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 2.3. Dantzig-Wolfe Decomposition: The Master Problem 53 2.4. Dantzig-Wolfe Decomposition: The Subproblems Procedure 54 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 63 iv PART 2; APPLICATION 64 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. Shortage Cost 79 5. Summary 80 Vin. TACTICAL PLANNING 81 1. Equilibrium 81 2. Validation 96 2.1. Extreme-Case Analysis 98 2.2. Review by Experts 98 2.3. Observation of Intermediate Results 100 2.4. Comparison of Model Behaviour with Real-System Comportment 100 3. Variance of Observed Performances 101 4. Determination of Sample Size 102 5. Summary 104 IX. ANALYSIS 105 1. Transportation System 105 1.1. Log Taxi System 106 1.2. Barge-Dryland Storage System 106 1.3. Direct Shipment 106 2. Inventory System 107 2.1. Minimum Inventory Level 107 2.2. Inventory Depletion Policies 107 3. 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. Model Sensitivity to the Salt Functional Form 118 4.3.2. Comparison of Transportation and Inventory Policies 119 5. The Effect of the Vessel Capacity 123 5.1. Nonlinear Effect of the Link Capacity 124 6. Probability Distribution Function of Inventory Levels 130 X. 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. Logging and Log Transport Process Flow 2 MAPI.l. Geographical Boundaries of the System 7 1.2. System Definition 8 II. 1. 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 V.1.2. Structure of the Technological Matrix (Window 2) 42 V.1.3. Structure of the Technological Matrix (Window 3) 43 V.1.4. Structure of the Technological Matrix (Window 4) 44 VII. 1.1. Production Rates at Port McNeill and Kelsey Bay 67 VII. 1.2. Production and Consumption Rates at Eve River and Powell River 68 VII.2.1. Weibull Fit for the Transit Time Models (PMN-TA, KB-TA) 71 VII.2.2. Weibull Fit for the Transit Time Models (ER-TA, TA-PR) 72 VII.2.3. Weibull Fit for the Delay Models 73 VII. 2.4. Poisson Fit for the Transit Time Model (TA-PR) 74 VIII. 1.1. Daily Variation of the Large Pulp Inventory Level at the Mill 87 VIII. 1.2. Daily Variation of the Large Pulp Inventory Level at the Storage Area 88 vii VIII. 1.3. Daily Variation of the Large Pulp Inventory Level at the Origins 89 VIII. 1.4. Daily Variation of the Small Pulp Inventory Level at the Mill 90 VIII. 1.5. Daily Variation of the Small Pulp Inventory Level at the Storage Area 91 VIII. 1.6. Daily Variation of the Small Pulp Inventory Level at the Origins 92 VIII. 1.7. Daily Variation of the Camprun Pulp Inventory Level at the Mill 93 VIII. 1.8. Daily Variation of the Camprun Pulp Inventory Level at the Storage Area 94 VIII. 1.9. Daily Variation of the Camprun Pulp Inventory Level at the Origins 95 VIII.2. Utility 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 Effects for the Theoretical Salt Function 115 IX.2.2. The Transportation and Inventory Effects for the Time-Independent Salt Function 116 IX.2.3. The Transportation and Inventory Effects for the Linear Salt Function 117 IX.3. The Capacity and Inventory Effects 126 IX.4.1. Weibull Fit for the Opening Inventory Models (January, April) 131 IX.4.2. Weibull Fit for the Opening Inventory Models (Duly, September) 132 IX.4.3. Weibull Fit for the Opening Inventory Models (December) 133 viii L I S T O F T A B L E S II. 1. Example of Computer Processing Time 14 VII. 1. Yearly Production and Consumption Rates 66 VII.2. Transit Time Statistics 70 VII.3. Goodness of Fit Evaluation 76 VII. 4. Transit Time Probability Distribution Parameters 77 VIII. 1. Analysis of Variance Table for the Determination of the Equilibrium State 84 VIII. 2. Fleet Size — Initial Condition Interaction Effects 86 IX. 1. Four-Factor Analysis of Variance Table (TRANS-INV-FORM-IMODE) 111 IX.2. Three-Factor Analysis of Variance Table (TRANS-INV-FORM) 112 IX.3. Response Cell Means for the Three Levels of FORM 114 IX.4. TRANS-FORM Response Cell 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 Effects of CAPACITY 129 IX. 10. Goodness of Fit Evaluation for the Opening Inventory Model 134 I X . l l . Opening Inventory Probability Distribution Parameters 136 ix ACKNOWLEDGEMENT I wish to express my immense gratitude to my research supervisor, Professor F . P. D . Navin , for his invaluable advice and guidance throughout this research. I would also l ike to thank Professors G . R. Brown, W. F . Caselton 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 (MacMil lan Bloedel Ltd.) for their valuable comments. Appreciat ion is given to M r . R . L . Greenough, M r . W. Judge (MacMil lan Bloedel Ltd.) and M r . M . J . MacNabb (Department of C i v i l Engineering) for their cooperation and t ime given during this study. I am grateful to Mrs . M . Navin and M r . R . A . Yaworsky, whose help has improved the style of this thesis. F ina l ly , 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 transport-inventory problem: - Time Horizon: The time horizon of the study should be long enough (5-10 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 tow-vessel, and two transportation doctrines; direct shipment and shipment via a consolidated point, with respect to various minimum inventory levels at the mill. i i i . 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 proportions, due to the above issues, the MacMillan Bloedel Powell River conversion plant and its major supplying log production divisions are considered. This plant processes mainly large, small and camprun Hembal (Hemlock-Balsam) pulp logs. It is supplied from the Teakern Arm storage area. Most of the plant's consumption is produced at Port McNeil 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 P A R T 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 model for forest industry are discussed. 11 II. L I T E R A T U R E R E V I E W This chapter reviews the available literature on the general inventory theory and the forest industry. A summary of the obtained information is presented. 1. G e n e r a l L i t e r a t u r e R e v i e w 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) representing the total initial inventory at each age level. His model has a multi-echelon structure where the p th echelon corresponds to the amount of product that will perish in p periods. However, with the exception of small values of m, the model is computationally impossible. In a later paper, Nahmias (1976) developed approximational methods to calculate the inventory cost function. His assumptions include: all orders are received instantaneously; demands in successive periods are independent and identically distributed random variables; unsatisfied demand is backlogged; FIFO 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 ifetime 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, FIFO policy is optimal. In the log inventory problem, if the sapwood deterioration is considered, LIFO policy is optimal. In the general inventory theory, the log inventory problem can be classified as a multi-echelon, multi-product inventory problem with constant demand, random lead-time and lost sale. 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 (1973) adopted a system analysis approach. In his paper, Newnham encouraged the application of simulation and 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. He formulated a linear programming model limited to 1 mill, 23 origins, 5 log assortments and 4 time periods. 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. Number of Number of . .. * * Density Processi ng Computer rovs in columns in % time in type matrix matrix (incl. slacks) minutes 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 * Utilizing an earlier solution Percentage of non-zero elements TABLE II.1: Example of Computer Processing Time Source: Lonner (1968) 1 5 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 Repair Replacement —1 Plointenonce r EQUIPMENT CONTROL "HI DE" Forest Felling Stump- Primary Roadside Roadside Roadside Second-Stand area trans- inventorg processing Inventory ary Inven- portation (gnpro- (delimbing, (semi- trans-tory (skidding, cessed slashing, processed portation , forwarding, vood) etc.) wood) (truck, etc.) | + river, etc) DLL MAND Mill Inventory KEY Action Information FIGURE n.l: Process Control and the Forest-harvesting System Source: Newnham (1973) 17 In the general inventory theory, the log inventory problem can be classified as a multi-echelon, multi-product inventory 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 available literature on the forest industry contains few studies accounting 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 engineering mistakes made were not serious. However, due to reduced inventory levels, more precise analysis and controls are necessary, resulting in higher implementation costs. 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 discovered in the literature review, it is proposed that simple, short-term models should first be developed. 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 PART 1: THEORY 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: sn+l = *n (sn» dn» £>n) n = 0, 1, N-l where the state S n e2n» n = u> •••» N, is the inventory balance at the beginning of period n; the decision d n £ Dn(Sn) is the volume of logs ordered at the beginning of period n; and the random transit time S n. Given an initial state, SOJ the problem is to find an admissible control law A= (do, di, dN_i) which minimizes the estimated cost of the above multistage decision process. K A (SQ) = E I g N (SN) + N ^ g n (Sn, d n ,5 n ) j n n = 0, 1, N-l where: g n (Sn, d n ,& n ) = holding cost + transportation cost + shortage cost = 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 n (Sn) = [0,00) 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 3 , respectively. - 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. Therefore, the dynamic programming approach is not suitable for 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 W M X D — >t_m_» •v N:--.^. SAP woo o • / / * • • 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 IV. SIMULATION MODEL 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 critical parameters expressing the system and establish a means of comparing different alternatives. Since it 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 and volume, and the log sort as parameters. Furthermore, in order to generate identical configurations for replicating experiences, four random-number generators are considered. The model is composed of 6 segments: - The initialization segment provides initial inventory at all 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. S T A R T SET INITIAL CONDITIONS ® DAY TJ-WEEK \ N .DAYS »> • < D PRODUCE LOGS INCREASE ORIGIN AND SYSTEM INVENTORIES MOVE THE TUG BETWEEN ORIGIN T AND THE STORAGE AREA RELEASE THE TUG F I G U R E IV. 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 -CONSUME LOGS DEPLETE ORIGINS INVENTORIES ^ 1 RECORD THE SHORTAGE •< * INCREASE STORAGE AREA INVENTORIES DEPLETE STORAGE AREA INVENTORIES INCREASE MILL INVENTORIES DEPLET! INVENT • MILL ORIES 1 CALCl TOTAL INVENTC JLATE DAILY )RY COST FIGURE IV.l: Model Flow Chart (Cont'd) FIGURE IV. 1: Model Flow Chart V. NETWORK FLOW MODEL: FORMULATION 30 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 t Y j s p : ° - u a n t l t y °f 1Q<3 ' shipped from j to s i n p e r i o d p. X s d p : Q u a n t ^ t i ' °f ± 0 < 3 1 shipped from s to d i n p e r i o d p. V j 0 p : Quantity of log / shipped d i r e c t l y from j to d in p e r i o d p. t t t I. , I I , , III : Inventory at j.d.s, r e s p e c t i v e l y , ]P dp SP ' tr 2 1 d u r i n g p e r i o d p. 31 t Q^pi Quantity of log type t bought at destination d in period p. 5. : 1 i f route is is taken in period p: 0 otherwise *j^p :. ^ route jd i s taken in period p; 0 otherwise. 6 , : i f route sd is taken in period p; 0 otherwise, sdp y 1.2 Parameters The parameters of the model represent the productivity, demand, haulage capacity, number of vessel limitation and cost. t S..p : Quantity of log type / produced at j in period p, : Quantity of log type / consumed at d in period p. dp 7 j S p : Haulage capacity between j and s in period p. y., : Haulage capacity between j and d in period p. _dp 32 Haulage c a p a c i t y between s and d i n p e r i o d p. Number of v e s s e l s a v a i l a b l e in p e r i o d p. T r a n s p o r t a t i o n f i x e d c o s t per p e r i o d (= wage + food) Un i t t r a n s p o r t a t i o n v a r i a b l e cost f o r the a p p r o p r i a t e l i n k in p e r i o d p. Average cost of Phase I o p e r a t i o n per 1u 3m 3 of l o g produced. Market value f o r l o g t y p e - i . R e a l i s a t i o n of the random t r a n s i t time f o r the shipment on (..) l i n k s t a r t e d i n p e r i o d p. q : I n t e r e s t r a t e , decay r a t e and inv e n t o r y c a r r y i n g charges per p e r i o d . D a i l y i n t e r e s t and decay r a t e . 2. Network Programming Model 2.1 Formulation 1 The mathematical formulation of the model is as follows: Min I f Zr fs • Zr fs • Zr f i • p 1 JJ JJP jsp sd 3dp sdp jd Jdp jdp • I V Y ' • S v xl • Z v v 1 • tjs J 'P J»P tsd sdp sdp tjd J d P J d P *kw]i<'>i, * t i d W ^ p > * ; d p * t f d W ^ ' m P > v ; d p • l / 2 [ Z l U i ^ r l C l J ^ , ) • Z n O ^ i r H l l W l l J ^ , ) • Z M d ^ ^ r K l I I ^ I I i ; ^ , ) ] ] • Z l i ,Q d p subject to : S t D t i t . 1) Z v • 2 V • ! -I - s 1 J £ j ; t 6 T ; p c P » jsp d J<*P J . P * « JP JP 2) Zx ! • Z V * * " * -II* *Q* = D* d 6 0 ; t € T ; p € P 3 sdp j jdp dP d,p*1 dp dp 3) i V * ZX * • H I 1 -HI* , = 0 S € S ; U T ; P € P j JSP d 3dp 3P 3.P*1 4) ZY* - 8 j , p * j , p ^ 0 J € J ; S 6 S ; P 6 P t J9P 5) ZX* - *,dp*,dp * 0 S€S;d€D; P € P t S « P 6) ZV* - I S s O j c J ; d « D ; p £ P t jdp j«*P J«lp 7 . I J £ NV_ p £ P ' ' js J»P jd J *P »d a d P p Y.V.X.I.II.III » 0 ; 5,6.6 6(0,1) S = (l'/.'.'.is) j = j - { i / shipment from j has a delay ) D = [ , D ) S = s " { s I shipment from s has a delay ) p = (1 ,...,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 _ A. . = [ 5. ., 5. ., 5. . ] X. . = [Y. ., X. ., V. .] i ] 13 13 i ] I. = [I., I I., III.] Let (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 i f Z [ • Z(y x* • <n.*iilp>xt. ) ] • . 1/2 [ ( H i - q ^ . r ) ( • • „ . ! ; „ . , ) • 1 } subject to : 2x* - I x 1 • i l -j£N ijp k£N kip ip I< • i.P*« Q1 = ip ip UN; teT; peP ZZx*. -tj<« 119 I J A j<H »jP UP 0 UN; peP Z A 0.j).0 'JP < NVp X.I SO ; A € (0,1) where ' s t for IP i € j 0 for i € J R* = tp 0 for i€S 0 for ICS 0* for i € D u for i C 0 36 3. Technical Calculation Aspects The standard Simplex algorithm can theoretically be employed to solve the logist ic 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 matrix occupying 20 Mbytes. Therefore, the implementation of the model requires some other methods which can convert the large problem into several small problems of manageable s ize . Fortunately, the decomposition principle provides a systematic procedure for solving large-scale linear programs which contain constraints of special structure. 4. The Matrix Structure The technological matrix of the log transportation and inventory system defined earlier is presented in Figure V.2 . Let ej be the unit vector in E ^ , where P is the total number of periods, with the one in the i th position. Then for i=l , . . . ,P we have: 37 t r • • A __s ="[e^] w i t h / i n d i c a t i n g the p e r i o d f o r w h i c h t y p e t i s p r o d u c e d a t o r i g i n j . I t i s c l e a r t h a t i f t h e a f o r e m e n t i o n e d c o n d i t i o n i s t r u e f o r e a c h p e r i o d , t h e n A fc i s t h e i d e n t i t y m a t r i x I p X p . i s 1 ° t A ., = [ e . ] w i t h i i n d i c a t i n g t h e p e r i o d f o r w h i c h t y p e t i s j d I p r o d u c e d a t o r i g i n j . ~ t P A . = [e' .] where e! i s t h e v e c t o r i n E w i t h ( -1 ) i n t h e 3 1 1 i t h p o s i t i o n , (1) i n t h e i-1 t h p o s i t i o n and (0) e l s e w h e r e . P o s i t i o n / i n d i c a t e s t h e p e r i o d f o r w h i c h t y p e t i s p r o d u c e d a t j and t h e r e f o r e c an be s t o r e d i n i n v e n t o r y . B fc, = [ e . ] w i t h / c o r r e s p o n d i n g t o t h e p e r i o d when sd I s h i p m e n t s a r r i v e a t d. I f t y p e / i s n o t consumed a t d, t h e n B t , =0. t B ^ = [ e ^ ] w i t h i c o r r e s p o n d i n g t o t h e p e r i o d when s h i p m e n t s a r r i v e a t d. I f t y p e t i s n o t p r o d u c e d a t j , 0 t not consumed a t d, t h e n B . , = 0. j d ~ t P B , = [ e ' . ] where e! i s t h e v e c t o r i n E w i t h ( 1 ) i n t h e J'th d l l — p o s i t i o n , ( - 1 ) i n t h e /-/ t h p o s i t i o n and ( 0 ) e l s e w h e r e . P o s i t i o n / i n d i c a t e s t h e p e r i o d f o r w h i c h t y p e * i s consumed a t d and t h e r e f o r e c a n be s t o r e d i n i n v e n t o r y . at t B , = [ e . ] w i t h i c o r r e s p o n d i n g t o t h e p e r i o d when d a I consumes t y p e t and t h e r e f o r e t y p e t c a n be b o u g h t . C ^ = [ e . ] w i t h /' c o r r e s p o n d i n g t o t h e p e r i o d when ]S 1 3 8 s h i p m e n t s a r r i v e a t s. I f t y p e / i s not p r o d u c e d a t j, then C fc = 0 . ] s ° t C , = [ e . ] w i t h / i n d i c a t i n g the p e r i o d when s h i p m e n t s sd x a r r i v e a t s. I f t y p e t i s not consumed a t d and r o u t e 0 t s-d i s not c o n n e c t e d , t h e n C , = 0. sd ~~ t P C = [ e ' . ] where e! i s t h e v e c t o r i n E w i t h ( 1 ) i n t h e / t h s 1 I — p o s i t i o n , ( - 1 ) i n the /-/ th p o s i t i o n and (0) e l s e w h e r e . H t , = [ e . ] w i t h /' i n d i c a t i n g t h e p e r i o d d u r i n g w h i c h sd l d e s t i n a t i o n d consumes l o g t y p e t. W i t h o u t l o s s o f g e n e r a l i t y , i t i s assumed t h a t a l l p o s s i b l e j s , jd and sd c o m b i n a t i o n s a r e t e c h n i c a l l y a l l o w e d . I f two nodes a r e n o t c o n n e c t e d , f o r example b e c a u s e of t h e g e o g r a p h i c a l l i m i t a t i o n , t h e n the r o u t e c o n n e c t i n g t h e s e nodes i s p r o h i b i t e d by a s s u m i n g a l a r g e c o s t . T h e r e f o r e : o _ N. = N . . = N . , = I Ds 3d _d . i i P M. = [ e . ] where e.e E w i t h ones i n t h e / t h and /'th j s 1 I — — p o s i t i o n i n d i c a t i n g t h e s t a r t i n g and e n d i n g p e r i o d s o f the s h i p p i n g o p e r a t i o n , r e s p e c t i v e l y . ° i i P M., = [ e . ] where e.e E w i t h ones i n t h e / t h and j t h 3d I I — — p o s i t i o n i n d i c a t i n g t h e s t a r t i n g and e n d i n g p e r i o d s o f t h e s h i p p i n g o p e r a t i o n , r e s p e c t i v e l y . i i P M , = [ e . ] where e.e E w i t h ones i n t h e i t h and / t h sd I I — — p o s i t i o n i n d i c a t i n g t h e s t a r t i n g and e n d i n g p e r i o d s o f the s h i p p i n g o p e r a t i o n , r e s p e c t i v e l y . 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 M T„ . . .1 _ i ; i ,J A J I V -(4* V 1) • C j , ! ^ ....A^T;, . A'J.V;, A M . AJI) A„T U « • A U T U • A U V U » • A t B V , s . A t l | ^ FIGURE V. l . l : Structure of the Technological Matrix Window 1 A 1 T 1 JI»JI 'A' J» VJD V X 1 FIGURE V.1.2: Structure of the Technological Matrix Window 2 A W i . AW« FIGURE V.1.3: Structure of the Technological Matrix Window 3 M • > I I " U A I I . •Sj»NJBAJB 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 t , Cf, bf and e, and matrices At, B-j, D andA, such that our problem can be stated as follows: T PI : Min dY + I C, Xt A t X t = bt t - l T -ADY + I B X s 0 t 1 1 eY < NV X^O , Yc(OJ) 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 as: T P3 h (Y) = M'n I x t>o 1 V t = b t t=1,...J I B . X . s A D Y t 1 1 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. - v (ADY ) t t t u t A t - v B t 5 C t t=l,...,T v a 0 , u unrestricted 48 By assuming, without loss of generality that Z S [ > Z ZJ ' (t=l T) j p j p dp dp (see also section V.2.2) it can be concluded that P3 is feasible and hence P4 is bounded (see also section VI.2.4). Therefore, by defining U = [u^/v] and 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) i s k s K * k k U, unrestricted . v >0 Therefore, PI becomes: Min { dY + [ Max 2 u k b t - vk(ADY) i s k s K ] } eY < NV Y€(0,1) ; ujunrestncted ; vk> o Now, given the fact that a maximum is really a least upper bound, we can write PI as: BM: Min dY + yo (Brl-l) vk(ADY) + Y0 > 2ujb t k = i K (Bn-2) eY < NV Ye (0,1) ; Y0 unrestricted 4 9 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 (y0). 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 (Pr) (30). The only requirement for (Pr) to be a valid relaxation of (P) is: F(P) ^  F(Pr), where F(P) 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(Pr) = * implies F(P) = <p - Min (P) > Min (Pr) - If an optimal solution of (Pr) is feasible in (P), then it is an optimal 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 single-commodity 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 o to satisfy (BM-1), (BM-2) and dY + y o $ UB- £ By eliminating y o we will have: Find Yc (0,1) to satisfy Therefore, we may introduce any "convenient" objective function, say 0(Y), and form the following modified Benders master problem after C iterations: (BM-2) and dY + t^u^b - vC (ADY) £ UB-1 c=l C MBM . Min Yt (O. l ) 9 ( Y ) c=1 C 52 Geoffrion and Graves (1974) found the last C th function: dY - vC( DY) 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 C + 1 Store X C + 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 for Benders decomposition, problem P3, is a multicommodity 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. LetH t = | X t / A tX t = bt; Xt ^ OJ andu)te and<s"te its extreme points and directions respectively. Any X t can be expressed as a convex combination of the extreme points and direction ofI2t: with Z* X . = 1 1 = 1 . - . T e <c X < e > o e = t....,Et  w t e > 0 e = 1 L, where Et and Lt are the total number of extreme points and directions of i i f It can easily be shown thatf2t is bounded. Therefore, by substituting Xt in P3, the following linear program inoJ t e will result: P 5 . Min 21 ( C t » t . ) x t e 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 2 (wB. - C, ) X + ox, \>-0 t t t t t V t = b 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 2 ( C t - wBt ) X t + « t x t>o x \ \ = b f t= i T This subproblem consists of finding the minimum cost flow in T uncapacitated single-commodity networks with arc cost equal to (CfwA t). Therefore, the best 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 T E, P Min i n 2 (y> l f i ^ p ) x l e I I (i> X t e ijp tc te ijp UP i.jeN , i * j ; p=1,...,P t = 1.....T e=l E t ; t=1 T 3.2 Dantzig-Wolfe Subproblems Translation For t=l,...,T, we define the subproblem t: M i n i ( i ( v . • lfiLp) x l - • 1/2 i (m*q)*itmr)(-i1ViJ ,) P (i .j)€G 'JP ">r i jp i €M H m ' v IP 1.P*1 • I I ' Q * • I w1 X1. ] • « i € H m *P o , j ) € G 'JP ' J p J t I x \ - I X* • I* - I* •Q* = R* j€M IJP k6N kip IP I.P* 1 IP 'P i* j k*i ieN; teT; pcP where (w,^ *) is the dual vector of the D-W master problem. 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 3 by means of matricial 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 =Mink Jaiu + bkj] 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 CU)€G A . p=1 p p .. - v C X. A.. U P >P UP U P c=l C A.. £ (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 1 , . . . , 6 ) ( 5, , . . . , 6 ) } n n Then, the partial solution S j=l is defined as the set S. = { ( 8 1 , . . . , 6 n ) S | 5.S=1 } 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 Dantzig-Wolfe 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 opt imali ty a (1461 x 2756) problem in 13 minutes on a D G M V 8000, and Eppen & Mart in (1985) reported a total C P U t ime of 100 minutes on Amdahl 470 for a problem with 3210 rows and 9587 variables. Therefore, in the future works, the variable redefinition method should be considered as an alternative to Dantzig-Wolfe decomposition. 5. Summary Firs t , attempts were made to succinctly i l lustrate the theoret ical background for Benders and Dantzig and Wolfe decomposition principles. Then, the modification and specialization of the existing theory necessary for pract ica l implementation of the proposed model were investigated in deta i l . F ina l ly , the order of magnitude of the C P U t ime necessary to optimize the system under study is determined. Due to the number of iterations between the master problems and subproblems, the opt imizat ion of the transport-inventory system requires one hour and 30 minutes on Amdahl 580 machine. 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 PART 2: APPLICATION 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 Ltd. 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 McNeil 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 mill consumption schedule. Location Hembal Pulp Large Small Camprun Production <IOSm3/Yr) Port McNeill 106 161 20 Eve River 176 170 40 Kelsey Bay 158 109 160 Total 440 440 220 Consumption (10 3m 3/Vr) Powell River 437 437 218 TABLE VTJ.1: Yearly Production and Consumption Rates 67 VOL. (m3/day) lOOOi 900 800 7004 600 500 400 300-200-100 Port McNeill •420" 33 .92P.. .MP.. 8-4-Q--"..8?o, 510 340 340 310 03 JJQ_ 93 Jon Fev Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan MONTH VOL (nr?/day) 1000 900 800 700 600 500 40CH 300 200 1004 Kelsey Bay 940 790 250 170 • 200 -25JL 900 380 ....AQQ.... _25JL 900 1 840 ' 790 .....<JQ9 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 VOL. (m5/day) 10001 900' 800 700 600 500 400 300-200 100-Eve River 900 900 * 270 * 190 45 1000 930 _25Q_ 68 1000 950 250 190 Jan Fev Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan MONTH VOL. (ms/day) Powell River 1500 1000 500 1200 1200 600 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 6 9 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 w ( x ; X , o , u ) = \/o [ ( x - M ) / a ] X ~ 1 e x p { - [ ( x - M ) / a ] X } V I 1 . 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-a-vis 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 Sample Size Mean (days) Standard Deviation Coefficient of Skewness Port McNeill - Teakern Arm 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 Delay at Teakern Arm 68 1 1.01 17.40 2.91 TABLE VII.2: Transit Time Statistics 71 16.2 ia.o FIGURE VII.2.1: Weibull Fit for the Transit Time Models (PMN-TA, KB-TA) 72 o fEfiKERN ARM - POWELL RIVER COa £=-o a. UJ 3 s: / / / / -11 -DATA o.o 0.4S 0.9 1.3S 1.8 2.2S 2.7 TRANSIT TIME 3.!S —I— 3.6 ~1 4.OS I 4.S FIGURE VH.2.2: Weibull Fit for the Transit Time Models (ER-TA, TA-PR) 73 FIGURE VH.2.3: Weibull Fit for the Delay Model T E A K E 3 N E FIRM - POWELL R I V E R O f i T f l 1 1— 3.3 3.4 0.8 1 , 2 fR&NSJT rrflE (Oflrsj4 2 8 3i 3.S 4.3 FIGURE VII.2.4: Poisson Fit for the Transit Time Model (TA-PR) 75 f p(x;X) = [ X U M ) 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 q at 5% 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 Links Calculated X 2 Degree of Freedom Critical X 2 Model Port McNeill - Teakern Arm 2.33 3 7 81 Weibull Eve River - Teakem Arm 9.14 4 9.47 Weibull Kelsey Bay - Teakem Arm 7 16 4 9.47 Weibull i Teakern Arm - Powell River 21.03 1 3.84 Weibull Teakem Arm - Powell River 1.09 1 334 Poisson Delay at Teakern Arm 22.42 8 13.51 Weibull TABLE VII.3: Goodness of Fit Evaluation for the Transit Time Models Links Shape X Scale a Location U Port McNeill - Teakern Arm 1.00 2.39 3.00 Eve River - Teakern Arm 1.00 1.36 2.00 Kelsey Bay - Teakern Arm 1 49 2.38 0.93 Teakern Arm - Powell River 0.52 1.00 Delay at Teakern Arm 1.00 11.29 1.00 T A B L E VII.*: Transit Time Probability Distribution Parameters 78 The fuel consumption FC 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 = vV 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. TACTICAL 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 , f X ( t 2 ) £ x a , . . . f X ( t n ) * x n ] = P [ X ( t , + s ) $ x , , X ( t 2 + s ) * x 2 / . . . , X ( t n + s ) £ x n ] for arbitrary real values of s and for all n (54). 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 ^ (n=sample size). 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 3 4 * » * A N A L Y S I S O F . V A R I A N C E « « • TOTCOST T o t a l L o g i s t i c Cos t (x$1000) BY FLEET FLEET SIZE INIT INITIALISATION SET SUM OF MEAN SIGNIF SOURCE OF VARIATION SQUARES DF SQUARE F OF F MAIN EFFECTS 3372812. 092 4 843203. .023 8 . 308 0 004 FLEET 7871S. 607 2 39358. 304 0. 388 0. 689 INIT 3294095. 485 2 1647047. .743 16. 228 0. 001 2-WAY INTERACTIONS 119830. 884 4 29957 .721 0. 295 0 .874 FLEET INIT 119830. 884 4 29957 .721 0. 295 0 .874 EXPLAINED 3492S42. 976 8 436580 .372 4 . 301 0 .022 RESIDUAL 913455. 767 9 101495, .085 TOTAL 4406098. 743 17 259182. .279 18 CASES WERE PROCESSED. O CASES ( 0 . 0 PCT) WERE MISSING. TABLE VTJI.1: Analysis of Variance Table for the Determination of the Equilibrium State 85 the FLEET-INIT interaction is: SSI = m I.. ( ! . . - ! . - ? . + ? ) 2 i ] 13. 1.. ••• 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 M E A N S TOTCOST T o t a l L o g i s t i c Cost (x$1000) BY FLEET FLEET SIZE INIT INITIALISATION SET TOTAL POPULATION 6469.48 ( 18) FLEET INIT 6562.83 6418.03 6427.57 ( 6) ( 6) ( 6) INIT 6771.29 6772.64 5864.49 ( 6) ( 6) ( 6) FLEET' 7027.16 ( 2) 6643.36 ( 2) 6643.36 ( 2) 6793.43 ( 2) 6761.88 ( 2) 6762.62 ( 2) 5867.89 ( 2) 5848.84 ( 2) 5876.75 ( 2) Initial .• i \| Conditio^ ' I Fleet Size \ n > \ 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 44 1 016.97 2 934 39 2x(Totol) 79 322.76 15 875.20 24 614 06 % of Sum of Squares due to Interaction 66.20 13 25 20.55 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) 8 8 o 0.0 « . 0 80.0 120.0 160.0 200.0 240.0 200.0 320.0 380.0 TIME (DAYS) FIGURE Vm.1.2: Daily Variation of the Large Pulp Inventory Level at the Storage Area (D.C. = Days of Mill Consumption) 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: (D.C. = Daily Variation of the Camprun Pulp Inventory Level at the Mill Days of Mill Consumption) 9 4 FIGURE Vffl.1.8: (D.C. = Daily Variation of the Camprun Pulp Inventory Level at the Storage Area Days of Mill Consumption) 95 FIGURE VIII. 1.9: (D.C. = Daily Variation of the Camprun Pulp Inventory Level at the Origins 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. Utiles Benefit Cost Cost of model 0 0.5 Validity 1.0 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 (1984) divide the process of evaluation into six categories: 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 E x t r e m e - C a s e A n a l y s i s 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 R e v i e w b y E x p e r t s 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 method adopted, the width of the confidence 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 two replications are desired, the level (i will drop to .7 and .5, respectively. 5. Summary The pre-simulation study is performed in this chapter. 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: FIFO (First-in-First-Out) and LIFO (Last-In-First-Out). 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 each are used. 109 iu.i eomxT - W C M i I 4 i J • 2 t tal COKTIXT - w u n Constant Function —unaao KWTVDCD • . / u, e&fcT^I l l l l l 1 1 • 3$ • M H W I S 14 I N IM 2M 04TS 0 i<3o P^Y5 FIGURE IX. 1: 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 I l l « * « A N A L Y S I S O F V A R I A N C E « * • TOTCOST T o t a l L o g i s t i c Cos t (x$1000) BY TRANS TRANSPORTATION POLICY INV INVENTORY POLICY IMOOE INVENTORY DEPLETION MODE FORM SALT FUNCTIONAL FORM SUM OF MEAN SIGNIF SOURCE OF VARIATION SQUARES DF SQUARE F OF F MAIN EFFECTS 2388216962. 287 9 265357440. 254 2008 . 784 0. 0 TRANS 2299833063. 849 4 574958265. 962 4352 . 494 0. 0 INV 10830771. 246 2 5415385. 623 40. 995 0. 000 IMOOE 177086. 665 1 177086. 665 1 . 341 0. 250 FORM 77376040. 527 2 38688020. 264 292 . 872 0. 0 2-WAY INTERACTIONS 41057755. 331 28 1466348. 405 1 1 . 100 0. 000 TRANS INV 17179283. 127 a 2147410. 39 1 16 . 256 0. 000 TRANS IMODE 706982. 309 4 176745. .577 1 . 338 0. 262 TRANS FORM 20602730. 017 8 2575341 . 252 19 . 496 0. 000 INV IMODE 396402. 383 2 198201 .191 1 . 500 0. , 229 INV FORM 2166737. 720 4 541684 .430 4 . 101 0, .004 IMODE FORM 5619. 774 2 2809 .887 0, ,021 0. .979 3-WAY INTERACTIONS 3241254. 938 36 90034 .859 0, .682 0. .901 TRANS INV IMODE 1776364. 639 8 222045 .580 1 , .681 0". .114 TRANS INV FORM 1128823. ,480 16 70551 .468 0, ,534 0 .922 TRANS I MODE FORM 255776. , 161 a 31972 .020 0 . 242 0 .982 INV IMODE FORM 80290, .658 4 20072 .664 0 . 152 0 .962 4-WAY INTERACTIONS 138197, .825 16 8637 .364 0 .065 1 .000 TRANS INV IMODE 138197, .825 16 8637 . 364 0 .065 1 .000 FORM EXPLAINED 2432654170, .381 89 27333192 .926 206 .915 0 .0 RESIDUAL 11888870 .913 90 132098 .566 TOTAL 2444543041 . 294 179 13656665 .035 180 CASES WERE PROCESSED. 0 CASES ( 0.0 PCT) WERE MISSING. TABLE IX. 1: Four-Factor Analysis of Variance Table (TRANS-INV-FORM-IMODE) 112 • * « A N A L Y S I S 0 F V A R I A N C E * * * TOTCOST T o t a l L o g i s t i c Cos t ( x$1000) BY TRANS TRANSPORTATION I POLICY INV INVENTORY POLICY FORM SALT FUNCTIONAL FORM SUM OF MEAN SIGNII SOURCE OF VARIATION SQUARES OF SQUARE F OF F MAIN EFFECTS 1062941064 . 796 8 132867633 .099 1428. 948 0. 000 TRANS 990048714 .625 4 247512178 .656 2661 913 0. 000 INV 35170655 .975 2 17585327 .987 189. . 124 0. 0 FORM 37721694 . 196 2 18860847 .098 202 . 842 0. 0 2-WAY INTERACTIONS 71858241 . 156 20 3592912 .058 38. .641 0. 0 TRANS INV 61114416 .083 8 7639302 .010 82. . 158 0, .000 TRANS FORM 10078374 .088 - 8 1259796 .761 13, .549 0, ,000 INV FORM 665450 .986 4 166362 .746 1 . 789 0. . 148 3-WAY INTERACTIONS 475192 .099 16 29699 . 506 0 .319 0, .'992 TRANS INV FORM 475192 .099 16 29699 .506 0 .319 0 .992 EXPLAINED 1135274498 .051 44 25801693 . 138 277 . 489 0, ,0 RESIDUAL 4184227 .251 45 92982 .828 TOTAL 1139458725 .302 89 12802907 .026 90 CASES WERE PROCESSED. 0 CASES ( 0 . 0 PCT) WERE MISSING. TABLE IX.2: Three-Factor Analysis of Variance Table (TRANS-INV-FORM) 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 TRANS-FORM 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 C E L L M E A N S TOTCOST T o t a l L o g i s t i c Cos t (x$1000) BY TRANS TRANSPORTATION POLICY INV INVENTORY POLICY FORM SALT FUNCTIONAL FORM FORM INV TRANS FORM 1 2 3 1 2837 . .08 2833 . 12 4944. 26 ( 2) ( 2) ( 2) 2 5741 , .99 4791 .55 6250. 37 ( 2) ( 2) ( 2) 3 8488 .68 5867 .89 6010. 14 ( 2) ( 2) ( 2) 4 9196 .20 6055 . 73 6434 . 48 ( 2) ( 2) ( 2) 5 14006 .79 12130 .36 14895 .69 ( 2) ( 2) ( 2) FORM 3 INV INV TRANS 1 2 3 1 2 3 TRANS 1 2837. .02 2833. ,09 4944. 29 1 2837 , ,02 2833 .07 4944 . , 27 ( 2) ( 2) ( 2) ( 2) ( 2) ( 2) 2 3562 88 3478, . 10 4264, . 18 2 5742, ,01 5661 .82 6250. . 37 ( 2) ( 2) ( 2) ( 2) ( 2) ( 2) 3 7248 .94 4659 .77 4601 . 77 3 8939 .87 6412 .33 6428 . .59 ( 2) ( 2) ( 2) ( 2) ( 2) ( 2) 4 7506 .87 4666 .95 4615 .50 4 9196. . 23 6426 . 16 6434 , 48 ( 2) ( 2) ( 2) ( 2) ( 2) ( 2) 5 12349 . 19 11023 . 12 13144 .03 5 14006 .83 12779 .29 14895 . 7 1 ( 2) ( 2) ( 2) ( 2) ( 2.) ( 2) TABLE IX.3: Response Cell Means for the Three Levels of FORM 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 2 = (r i r 2-D F(l-« ;(rir2)-l,(n-l)rir2) 4.3.1 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 direct transportation of the logs from the origins to the mill. 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 M E A N S T O T C O S T T o t a l L o g i s t i c Cost (X$1000) BY T R A N S T R A N S P O R T A T I O N P O L I C Y INV I N V E N T O R Y P O L I C Y FORM S A L T F U N C T I O N A L FORM T O T A L P O P U L A T I O N 7022 .41 ( 90) T R A N S 3 5 3 8 . 1 3 5 0 8 2 . 5 9 6 5 1 7 . 5 5 6 7 2 5 . 8 5 13247 .90 ( 18) ( 18) ( 18) ( 18) ( 18) INV 7 6 3 3 . 1 8 6 1 6 3 . 4 9 7270 .55 ( 30) ( 30) ( 30) FORM 7 3 6 5 . 6 3 ( 30) 6115 .71 ( 30) 7585 .88 ( 30) FORM T R A N S 3 5 3 8 . 1 7 ( 6 ) 3538 . 11 ( 6) 3538 . 1 1 ( 6) 5 5 9 4 . 6 3 ( 6 ) 3 7 6 8 . 3 9 ( 6) 5 8 8 4 . 7 3 ( 6 ) 6 7 8 8 . 9 0 ( 6 ) 5 5 0 3 . 4 9 ( 6) 7260 .26 ( 6 ) 7 2 2 8 . 8 0 ( 6 ) 5 5 9 6 . 4 4 ( 6) 7352 .29 ( 6) 13677 .63 ( 6 ) 12172.11 ( 6) 13893.95 ( 6) TABLE IX.*: TRANS-FORM Response Cell Means * * * SCHEFFE MULTIPLE COMPARISON TEST - * * TOTCOST T o t a l Log 1st i c Cost ( x$1000 BY TRANS TRANSPORTATION POLICY FORM SALT FUNCTIONAL FORM CELL LOWER CONTRAST UPPER S I G N I F K NO BOUND BOUND DIFFEREI 11-12 -684 205 1095 21-22 936 1826 2716 * * * * 31-32 395 1285 2175 * * * * 41-42 742 1632 2522 * * « * 51-52 616 1506 2395 * * * * 13-12 -491 398 1288 23 -22 1226 2116 3006 « * * * 33-32 867 1757 2647 * * * * 4 3 - 4 2 866 1756 2646 * * * * 53-52 832 1722 2612 * * * * 13-11 -696 193 1083 23-21 -599 290 1 180 33-31 -418 471 1361 43-41 -765 123 1013 53-51 -673 216 1 106 TABLE IX.5: Scheffe Multiple Comparison Test (TRANS-FORM) * SCHEFFE MULTIPLE COMPARISON TEST « TOTCOST TOTAL INVENTORY COST (10O0 BY TRANS TRANSPORTATION POLICY INV INVENTORY POLICY CELL LOWER CONTRAST UPPER SIGNIFIC NO BOUND BOUND DI FFEREN 11-12 - 1772 4 1780 21-22 -825 950 2727 31-32 844 2621 4397 * * * # 41-42 1364 3 140 4917 * * * * 51-52 100 1876 3653 * * * * 13-12 335 2111 3888 * * * * 23-22 -317 1459 3235 33-32 - 1633 142 1919 43-42 -1397 379 2 155 53-52 989 2765 4542 * * * * 13-11 331 2107 3884 * * * * 23-21 -1267 508 2285 31-33 702 2479 4255 * * * * 41-43 985 2762 4538 * * * * 53-51 -887 889 2665 21-11 1 128 2904 4681 * * * * 31-11 3875 5651 7428 * * * * 41-11 4583 6359 8136 m * * m 51-11 9393 1 1 169 12946 * * * * 31-21 970 2746 4523 * * * * 41-21 1678 3454 5231 * * * * 51-21 6488 8264 1004 1 * * * * 41-31 -1068 707 2484 51-31 3742 5518 7295 * * * * 51-41 3034 4810 6587 * * * * 22-12 182 1958 3735 * * * * 32-12 1258 3034 481 1 * * * * 42-12 1446 3222 4999 * * * * 52-12 7521 9297 1 1074 * * * * 32-22 -699 1076 2853 42-22 -511 1264 3041 52-22 5562 7338 9115 * * * * 42-32 -1588 187 1964 52 -32 4486 6262 8039 * * * * 52-42 4298 6074 7851 * * * * 23-13 -469 1306 3083 33-13 - 7 1 0 1065 2842 43 -13 -285 1490 3267 53-13 8175 9951 1 1728 * * * * 23-33 -1535 240 2017 43-23 -1591 184 1961 53-23 6869 8645 10422 * * * * 43-33 -1351 424 2201 53-33 7109 8885 10662 * * * * 53-43 6685 8461 10238 * * * * TABLE IX.6: 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 m3 CAP2 = 12,500 m3 CAP3 = 13,900 m3 CAP4 = 15,000 m3 CAP5 = 16,800 m3 CAP6 = 20,000 m3 CAP7 = 25,000 m3 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 m3 are 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 m3 are operated. 5.1 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 T o t a l L o g i s t i c Cost (x$1000) CAP VESSEL CAPACITY ( M3I INV INVENTORY POLICY TOTAL POPULATION 7983.31 ( 72) CAP 1 2 3 4 5 6 7 8 13138.69 8642.39 6291.66 6779.91 6743.79 6710.33 6720.86 6838.84 ( 9) ( 9) ( 9) ( 9) ( 9) ( 9) ( 9) ( 9) INV 1 2 3 9053.71 7299.95 7596.28 ( 24) ( 24) ( 24) INV 1 2 3 1 13868 20 15022 89 16524 92 ( 3) ( 3) < 3) 2 9986 67 7918 54 8021 98 ( 3) ( 3) I 3) 3 6919 12 5916 43 6039 44 ( 3) ( 3) < 3) 4 8447 16 5873 21 6019 36 ( 3) ( 3) < 3) 5 8380 09 5836 72 6014 56 ( 3) ( 3) < 3) 6 8275 39 5847 65 6007 95 ( 3) ( 3) < 3) 7 8170.73 5964 05 6027 79 ( 3) ( 3) ( 3) 8 8382 30 6020 01 6114 20 ( 3) ( 3) ( 3) • • • A N A L Y S I S O F V A R I A N C E TOTCOST TOTAL INVENTORY COST ( 10OO t) BY CAP VESSEL CAPACITY (MS) INV INVENTORY POLICY SUM OF MEAN SIGNIF SOURCE OF VARIATION SQUARES OF SQUARE F OF F MAIN EFFECTS 600342724 644 9 66704747 183 450 592 0.000 CAP 558041669 017 7 79720238 .431 538 512 0.000 INV 42301055 627 2 21150527 .814 142 872 0.0 2-WAY INTERACTIONS 34186168 151 14 2441869 . 154 16 495 0.000 CAP INV 34186168 151 14 2441869 154 16 495 0.000 CXPLAINEO 634528892 795 23 27588212 730 186 359 0.0 RESIOUAL 7105825 018 48 148038 021 TOTAL 641634717 813 71 9037108 .702 81 CASES WERE PROCESSED. 9 CASES ( 11.1 PCT) WERE MISSING. TABLE IX.7: CAP-INV Analysis of Variance Table 1 126 0,^ INV1 INV3 INV2 1 1 1 1 1 1 0.0 50.0 100.0 150.0 200.0, 250.0 300.0 VESSEL CAPACITY IM3) ( X 1 0 2 ) FIGURE IX.3: The Capacity and Inventory Effects 1 2 7 calculated by orthogonal polynomials. Therefore, five equidistant levels for the vessel capacity variable are defined: CAP1 = 10,000 m 3 CAP2 = 15,000 m 3 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 c2 = (2,-1,-2,-1,2} 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)2 / nr C i? where n=9 in the present case. Table IX.9 presents the component effects, sums of squares and the F-statistics. 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 « * • C E L L M E A N S * TOTCOST T o t a l L o g i s t i c Cost (x$1000) BY CAP VESSEL CAPACITY (M 3) INV INVENTORY POLICY TOTAL POPULATION 8 4 3 7 . 7 3 ( 45) CAP 15138.71 6779 .91 6710 .33 6720 .86 6 8 3 8 . 8 4 ( 9) ( 9) ( 9) ( 9) ( 9) INV 9428 .76 7745 .57 8138 .85 ( 15) ( 15) ( 15) CAP INV 1 2 3 1 13868 24 15022 91 16524 94 ( 3) 3) ( 3) 2 8447 16 5873 21 6019 36 ( 3) < 3) ( 3) 3 8275 39 5847 65 6007 95 ( 3) < 3) ( 3) 4 8170 73 5964 05 6027 79 ( 3) 3) ( 3) 5 8382 30 6020 01 6114 20 ( 3) ( 3) ( 3) * * * A N A L Y S I S 0 F V A R I A N C E * * + TOTCOST BY CAP INV T o t a l L o g i s t i c Cost (x$1000 VESSEL CAPACITY (M 3) INVENTORY POLICY ) SOURCE OF VARIATION SUM OF SQUARES DF MEAN SQUARE F SIGNI OF F MAIN EFFECTS CAP INV 528515419 505257113 23258306 .494 .395 .099 6 4 2 88085903 126314278 1 1629153 249 349 050 440 632 58 883 222 206 0 . 0 0 0 0 . 0 0 0 0 . 0 2-WAY INTERACTIONS CAP INV 31190459 31190459 .594 .594 8 8 3898807 3898807 449 449 19 19 514 514 0 . 0 0 0 0 . 0 0 0 EXPLAINED 559705879 .088 14 39978991 .363 200 101 0 . 0 0 0 RESIDUAL 5993829 .679 30 199794 .323 TOTAL 565699708 .767 44 12856811 .563 45 CASES WERE PROCESSED. 0 CASES ( 0 . 0 PCT) WERE MISSING. TABLE IX.8: CAP-INV Analysis of Variance Table 2 129 component Erfect Sum or squares SS, F-5tatlsttc (u,=1,^2=18) Linear -149 929.1 1 249 763 755.90 1250.10 Quadratic 153 303.03 186 522 373.10 933.57 Cubic -73 635.93 60 247 224.30 301.55 Quartic 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 a i n y - M . J / o . l V 1 exp{-[(y - , i.)/a ]Xi} 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{-[(/-„.)/ a. ] X i } W. i i If the level 1 is equal to the mill one-month consumption (36,000 m3), then: P ( Y j a n ^36000) = 1.00 P ( Y A p r ^ 36000) = .87 P ( Y 3 u l « ; 36000) = 0. P ( Y S e p ^ 36000) = 1.00 P ( Y D e c ^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 Months Calculated X 2 Degree of Freedom Critical X 2 Model January 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 TABLE IX. 10: 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 principles are used to optimize the formulated network. Due 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 m3 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 m3. 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. Beyond this limit, an increase in the minimum inventory 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 * M U L T . O R I G - O N E S T O R A G E - O N E D E S T I N A T I O N - M U L T . S O R T 2 * 3 S I M U L A T E 4 * 5 I N T E G E R & J 6 R E A L L O C A T E F U N , 2 0 , L O G , 7 , C H A , 5 . C O M , 2 0 0 0 0 0 , Q U E , 4 7 R M U L T 2 3 0 , 1 1 , 3 1 9 . 5 5 6 . 1 9 5 8 P M N E O U 1,C 9 KLEt E O U 2 . C APPENDIX 1 1 0 E V R E O U 3 . C 11 S T O R A E O U 4 , C 12 M I L L E O U 5 , C 13 ORDR E O U 4 , 0 14 C U N S I E O U I . L Simulation Model Listing 1 5 C U N S 2 E Q U 2 , L 1 6 C U N S 3 E O U 3 , L 17 WKEND E O U 4 , L 18 T0W1 E O U 1 1 . L 1 9 T 0 W 2 E O U 1 2 , L 2 0 T 0 W 3 E O U 1 3 , L 21 M I L L 1 E O U 1,2 2 2 M I L L 2 E O U 2 , Z 2 3 M I L L 3 E Q U 3 , Z 2 4 PMN1 E Q U 1 1 . Z 2 5 P M N 2 E Q U 1 2 , Z 2 6 P M N 3 E Q U 1 3 , Z 2 7 K L B 1 E Q U 2 1 . Z 2 8 K L B 2 E Q U 2 2 , Z 2 9 K L B 3 E Q U 2 3 , Z 3 0 E V R 1 E Q U 3 1 . Z 3 1 E V R 2 E Q U 3 2 . 2 3 2 E V R 3 E Q U 3 3 , Z 3 3 P M N T A E Q U 1 0 1 , Z 3 4 K B T A E Q U 1 0 2 , 2 3 5 E R T A E Q U 1 0 3 , 2 3 6 T A P R E Q U 1 0 4 , Z 3 7 P S A P W E Q U 1 ,XH 3 8 P U F 2 E Q U 9 . X H 3 9 S H C S T E Q U 4 , M H 4 0 M M I N E Q U 1 0 , M H 41 * 4 2 * S T O R A G E 4 3 * 4 4 S T O R A G E S $ T U G , 3 4 5 * 4 6 * M A T R I C E S 4 7 * 4 8 ORWI M A T R I X X . 3 . 5 2 O R I G I N S W E E K L Y A V E R A G E I N V . ( R = T Y P E 4 9 MWI M A T R I X X , 3 , 5 2 M I L L ' S W E E K L Y A V E R A G E I N V . 5 0 STW I M A T R I X X . 3 . 5 2 S T O R A G E ' S W E E K L Y A V E R A G E I N V . 51 SWI M A T R I X X . 3 , 5 2 S Y S T E M ' S W E E K L Y A V E R A G E I N V . C = W E E K ) 5 2 S U M I N M A T R I X X . 4 , 3 SUM OF D A I L Y I N V . L E V E L S R = 1 , 2 , 3 . 4 FOR M I L L , S T O R . , S Y S T E M A N D O R I G I N S 5 3 S H O R T M A T R I X H . 3 . 5 2 W E E K L Y S H O R T A G E ( R = T Y P E ; C = W E E K ) 5 4 W I C M A T R I X X , 8 , 5 2 I N V E N T O R Y C O S T A T T H E END O F WEEK C 5 5 * R=1 D E T . C O S T FOR A L L O R I G I N S . S T O R A G E & M I L L ; R=2 F I N A N C I A L C O S T ; 5 6 * R = 3 , 4 D E T . & F I N . FOR L O G S C O N S U M M E D D U R I N G T H E W E E K ; R=5 T R A N S P O R T A I O N C O S T 5 7 * R=6 S H O R T A G E C O S T , R = 7 T O T A L C O S T & R=8 C U M U L A T I V E T O T A L C O S T 5 8 I N V M A T R I X X . 3 , 3 C U R R E N T I N V . L E V . (R = 1 , 2 , 3 M I L L , S T O R A G E , S Y S T E M ; C = T Y P E ) 59 OINV MATRIX X , 1 , 3 SUM OF CURRENT ORIGINS INV. (C=TYPE) GO TOTIO MATRIX X . 1 , 3 CURRENT TOT. INV. (ALL TYPE) @ ORIG. C 61 OSHIP MATRIX X , 1 , 3 SHIPMENT VOLUME FROM ORIGIN C 62 SHRTG MATRIX H . 1 , 3 TOT. SHORTAGE OF LOG TYPE/SORT C 63 DEMND MATRIX H , 1 , 3 DEMAND FOR LOG TYPE/SORT C 64 SHCST MATRIX H , 1 , 3 UNIT SHORTAGE COST FOR LOG TYPE C IN $/M3 65 ORTPN MATRIX H . 1 , 3 TOT. NO. OF LOG TYPE/SORT AT ORIG. C 66 LOGV MATRIX H , 1 , 3 VALUE OF LOG TYPE C IN $/M3 67 OUNIT MATRIX H , 3 , 3 TYPE C I N I T . INV. AT ORIG. R (M3/DAY) 68 SUNIT MATRIX H . 1 , 3 TYPE C I N I T . INV. AT THE STOR. (M3/DAY) 69 M i l MATRIX H . 1 , 3 TYPE C I N I T . INV. AT THE MILL (M3) 70 MMIN MATRIX H , 1 , 3 MIN. INV. LEV. AT THE MILL FOR TYPE C 71 * 72 * 73 I N I T I A L MH$ORTPN(1, 1-3) ,3 74 I N I T I A L M H $ L O G V ( 1 , 1 - 3 ) , 4 5 75 I N I T I A L MH$SHCST(1 ,1-3) , 120 76 I N I T I A L MH$OUNIT( 1 -3 ,1 -2 ) , 3600 /MH$0UNIT( 1-3,3) , 1800 77 I N I T I A L M H $ S U N I T ( 1 , 1 - 2 ) , 1 2 0 0 0 / M H $ S U N I T ( 1 , 3 ) , 6 0 0 0 78 I N I T I A L M H $ M I I ( 1 , 1 - 2 ) , 2 0 0 0 0 / M H S M I I ( 1 , 3 ) , 1 0 0 0 0 79 I N I T I A L MH$MMIN(1 ,1 -2 ) , 16800 /MH$MMIN(1 ,3 ) , 8400 80 * 81 * SAVEVALUES 82 * 83 I N I T I A L XH$PSAPW,10 % OF SAPWOOD IN CHIPS 84 I N I T I A L XH$DRIGN,3 TOTAL NUMBER OF CAMP SITES 85 I N I T I A L XH$T0TYP,3 TOTAL LOG TYPE/SORT 86 I N I T I A L XH$DIREC,0 0=VIA A STORAGE ; 1=DIRECT 87 I N I T I A L XH$0DAYS,1 DAYS OF INITIAL INV. AT ORIGIN 88 I N I T I A L XH$SDAYS,2 DAYS OF I N I T I A L INV. AT STORAGE 89 I N I T I A L XH$TOW,20000 CALL WHEN A TOW IS FORMED 90 I N I T I A L XH$PUF1,10 PICK UP FREQ. FOR 0RIG.1=PMN 91 I N I T I A L XH$PUF2,10 PICK UP FREO. FOR 0RIG.2=KB 92 I N I T I A L X$CAP,20000 TUG CAPACITY IN M3 93 * 94 * FUNCTIONS 95 * 96 *PRODUCTION 97 * 98 PMN1 FUNCTION P1,D8 PRODUCTION RATE FOR TYPE 1 AT PMN 99 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 , 0 100 PMN2 FUNCTION P1.D8 PRODUCTION RATE FOR TYPE 2 AT PMN 101 3 1 , 4 6 0 / 1 2 0 , 8 2 0 / 1 8 1 , 8 4 0 / 2 1 2 , 0 / 2 4 3 , 1 5 0 / 3 0 4 , 8 4 0 / 3 3 4 , 8 2 0 / 3 6 4 , 0 102 PMN3 FUNCTION P1.D7 PRODUCTION RATE FOR TYPE 3 AT PMN 103 3 1 , 3 5 / 1 2 0 , 8 5 / 1 8 1 , 1 3 0 / 2 4 3 , 0 / 3 0 4 , 1 3 0 / 3 3 4 , 8 5 / 3 6 4 . 0 104 KLB1 FUNCTION P1.D7 TYPE 1 AT KB 105 3 1 , 1 7 0 / 1 2 0 , 8 4 0 / 1 8 1 , 9 0 0 / 2 4 3 , 0 / 3 0 4 , 9 0 0 / 3 3 4 , 8 4 0 / 3 6 4 , 0 106 KLB2 FUNCTION P1.D7 TYPE 2 AT KB 107 3 1 , 2 0 0 / 1 2 0 , 5 8 0 / 1 8 1 , 6 0 0 / 2 4 3 , 0 / 3 0 4 , 6 0 0 / 3 3 4 , 5 8 0 / 3 6 4 , 0 108 KLB3 FUNCTION P1.D7 TYPE 3 AT KB 109 3 1 , 2 5 0 / 1 2 0 , 7 9 0 / 1 8 1 , 9 5 0 / 2 4 3 , 0 / 3 0 4 , 9 5 0 / 3 3 4 , 7 9 0 / 3 6 4 , 0 _ 110 EVR1 FUNCTION P1.D7 TYPE 1 AT ER -P" 111 3 1 , 3 5 5 / 1 2 0 , 9 0 0 / 1 8 1 , 1 0 0 0 / 2 4 3 , 0 / 3 0 4 , 1 0 0 0 / 3 3 4 , 9 0 0 / 3 6 4 , 0 ^ 112 EVR2 FUNCTION P1.D7 TYPE 2 AT ER 113 3 1 , 2 7 0 / 1 2 0 , 9 0 0 / 1 8 1 , 9 5 0 / 2 4 3 , 0 / 3 0 4 , 9 5 0 / 3 3 4 , 9 0 0 / 3 6 4 , 0 114 EVR3 FUNCTION P1.D7 TYPE 3 AT ER 115 3 1 , 4 5 / 1 2 0 , 1 9 0 / 1 8 1 , 2 5 0 / 2 4 3 , 0 / 3 0 4 , 2 5 0 / 3 3 4 , 1 9 0 / 3 6 4 . 0 1 16 * 117 •CONSUMPTION 118 * 1 19 MI LL 1 FUNCTION P I , 0 1 CONSUMPTION RATE FOR TYPE 1 120 3 6 4 , 1 2 0 0 121 MILL2 FUNCTION P1.D1 CONSUMPTION RATE FOR TYPE 2 122 3 6 4 , 1 2 0 0 123 MILLS FUNCTION P1.D1 CONSUMPTION RATE FOR TYPE 3 124 3 6 4 , 6 0 0 125 * 126 •TRANSIT TIME 127 * 128 PMNTA FUNCTION RN1.C14 PMN - TA TRANSIT TIME 129 0 , 3 . 0 / . 0 4 1 0 0 . 3 . 1 0 / . 2 4 4 5 8 , 3 . 6 7 / . 4 1 7 2 6 , 4 . 2 9 / . 5 4 0 9 7 , 4 . 8 6 / . 6 4 5 9 0 , 5 . 4 8 130 . 7 5 0 8 8 , 6 . 3 2 / . 8 3 6 7 7 , 7 . 3 3 / . 8 9 3 0 5 , 8 . 3 4 / . 9 3 1 9 5 , 9 . 4 2 / . 9 5 9 6 8 , 1 0 . 6 7 1.31 . 9 8 1 8 7 , 1 2 . 5 8 / . 9 9 3 1 4 , 1 4 . 9 0 / 1 , 3 3 132 KBTA FUNCTION RN2.C13 KB-TA TRANSIT TIME 133 0 , . 9 3 / . 0 5 3 7 8 , 1 . 2 7 / . 1 7 2 6 9 , 1 . 7 1 / . 4 9 6 0 9 , 2 . 7 8 / . 6 8 5 6 9 , 3 . 5 6 / . 7 8 7 8 0 , 4 . 1 3 134 . 8 7 3 2 8 , 4 . 8 1 / . 9 2 4 6 1 , 5 . 4 4 / . 9 6 0 1 8 , 6 . 1 6 / . 9 8 2 6 5 , 7 . 0 3 / . 9 9 4 5 5 , 8 . 1 5 135 . 9 9 8 7 5 , 9 . 4 6 / 1 , 14 136 ERTA FUNCTION RN3.C12 ER-TA TRANSIT TIME 137 0 , 2 . / . 0 7 1 0 4 , 2 . 1 / . 3 0 8 1 8 , 2 . 5 / . 5 2 4 9 0 , 3 . 0 1 / . 6 8 0 8 7 , 3 . 5 5 / . 7 8 7 2 0 , 4 . 1 0 138 . 8 4 9 4 9 , 4 . 5 7 / . 9 0 7 4 6 , 5 . 2 3 / . 9 4 7 5 3 , 6 . 0 / . 9 7 7 6 8 , 7 . 1 6 / . 9 9 5 6 2 , 9 . 3 7 / 1 , 1 9 139 TAPR FUNCTION RN4.D7 TA-PR TRANSIT TIME 140 . 5 9 4 5 2 , 1 / . 9 0 3 6 7 , 2 / . 9 8 4 0 5 , 3 / . 9 9 7 9 1 , 4 / . 9 9 9 7 9 , 5 / . 9 9 9 9 8 , 6 / 1 , 7 141 DELAY FUNCTION RN5.C12 142 0 . . 1 . / . 1 3 6 1 0 , 2 . 6 5 / . 3 5 6 3 9 , 5 . 9 7 / . 5 5 6 4 9 , 1 0 . 1 7 / . 7 0 6 0 8 , 1 4 . 8 1 / . 8 0 8 9 7 . 1 9 . 6 7 143 . 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 , 139 144 * 145 •GENERAL 146 * 147 SALTS FUNCTION MP1.C4 SALT CONTAMINATION OF SAPWOOD 148 0 , 0 / 2 0 , . 5 / 4 0 , 1 . 5 / 1 8 0 , 2 . 8 149 SALTH FUNCTION MP1.C5 SALT CONTAMINATION OF HEARTWOOD 150 0 , 0 / 2 0 , . 0 1 / 4 0 , . 0 3 / 8 0 , . 0 1 / 1 8 0 , . 0 5 151 1ST FUNCTION XH$ABSIS,D4 INT. RATE, SALT DET. 8. TRANS. UNIT COST ( 0 -152 1 , . 1 2 / 2 , 1 . 7 7 / 3 , 1 . 5 / 4 , .3 153 * 154 * VARIABLE 155 * 156 LOADO BVARIABLE P 1 ' L E ' X H $ A R I V A 157 LOADS BVARIABLE P1 'LE 'XH$READY 158 PHRTW VARIABLE 100-XH$PSAPW 159 DETER FVARIABLE ((FN$SALTS t XH$PSAPW)+(FN$SALTH+V$PHRTW))*P4*FN$IST^10/100 160 LOST FVARIABLE MH$SHRTG(1,P2)+MH$SHCST(1,P2) 161 FINAN FVARIABLE P 4 * M H $ L 0 G V ( 1 , P 3 ) + M P 1 + F N $ I S T / 3 6 4 162 TRTIM FVARIABLE (FN+3)+.5 163 SUM VARIABLE MX$WIC(3,P1)+MX$WIC(4,P1)+MX$WIC(5,P1) 164 SURP VARIABLE P4-MH$DEMND(1,P3) 165 CHAIN VARIABLE 1+(CH$ST0RA)+(CH$MILL)+(CH1)+(CH2)+(CH3) 166 167 * TABLES 168 * 169' RT I ME TABLE M P 1 , 0 , 5 . 5 2 WATER RES. TIME FOR SOLD LOGS 170 * 171 * 172 173 * 174 * SEGMENT 1 I N I T I A L I S A T I O N OF USER CHAINS 175 * 176 * MILL 177 * 178 GENE 1 , , , 1 , 7 , 5 179 MARK 1 180 IN I TL S P L I T XHSTOTYP,PROCD BRANCHING BY TYPE 181 TRANSFER .STORE 182 PROCD ASSIGN 3, (W$INITL+1) 183 ASSIGN 4 , M H $ M I I ( 1 , P 3 ) 184 MSAVEVALUE I N V + , 1 , P 3 , P 4 185 MSAVEVALUE I N V + , 3 , P 3 , P 4 186 PRIORITY 5 187 LINK M I L L , F I F O MILL INITIALISATION 188 * 189 STORAGE AREA 190 * 191 STORE TEST E XH$DIREC,0 .DIR 192 ASSIGN 2,XH$SDAYS 193 DAYST PRIORITY 7 194 INTRM S P L I T XH$TOTYP,STO 195 TRANSFER ,SLOOP 196 STO ASSIGN 3,(W$INTRM+1 ) 197 ASSIGN 4 ,MH$SUNIT(1 ,P3) 198 MSAVEVALUE I N V + . 2 . P 3 . P 4 199 MSAVEVALUE I N V + , 3 , P 3 , P 4 200 PRIORITY 5 201 LINK STORA,FIFO 202 SLOOP PRIORITY 6 203 TEST E W$INTRM.O 204 LOOP 2,DAYST 205 * 206 * ORIGINS 207 * 208 DIR ASSIGN 2,XH$0DAYS 209 DAYOR PRIORITY 7 210 SITE SPLIT XH$T0TYP,SOURC BRANCHING BY TYPE 21 1 TRANSFER .LOOP 212 SOURC ASSIGN 3.(W$SITE+1) 213 SPLIT XH$ORIGN,ORGIN BRANCHING BY ORIGINE 214 TERMINATE 215 ORGIN TEST E XH$COUNT,XH$ORIGN,SAME NUMBER OF ORIGIN 216 SAVEVALUE COUNT,0,H 217 SAME SAVEVALUE COUNT+, 1 ,H 218 ASSIGN 2,XH$C0UNT 219 ASSIGN 4 ,MH$0UNIT(P2 ,P3) 220 PRIORITY 4 221 MSAVEVALUE O I N V + , 1 , P 3 , P 4 222 MSAVEVALUE TOTIO+,1 ,P2 ,P4 223 MSAVEVALUE I N V + , 3 , P 3 , P 4 224 TEST E P 4 . 0 . L N K 0 225 TERMINATE 226 LNKO LINK P 2 , F I F O 227 LOOP PRIORITY 5 228 TEST E W$SITE,0 229 LOOP 2,DAYOR 230 TERMINATE 23 1 232 * -p-233 234 * 235 * SEGMENT 2 OPERATION CONTROL *** 236 * 237 * 238 * 239 GENE 7 , , 6 , , 6 , 1 WEEKEND = THE 6 & 7TH DAYS OF THE WEEK 240 LOGIC S WKEND 24 1 ADVANCE 2 CLOSE OPERATION AT CAMPS 242 LOGIC R WKEND WEEKEND IS OVER 243 TERMINATE 244 * 245 * 246 247 * 248 * SEGMENT 3 INVENTORY CONTROL *** 249 * 250 * 251 * ORIGIN 252 * 253 GENE 1 , , , , 5 , 5 254 GATE LR WKEND,FIN DO NOT WORK ON WEEKENDS 255 MARK 1 RECORD WATERING TIME 256 SPLIT ( X H $ 0 R I G N - 1 ) , D I V O , 2 257 DIVO S P L I T ( M H $ O R T P N ( 1 , P 2 ) - 1 ) , D I V T , 3 258 DIVT ASSIGN 5 , ( ( P 2 * 1 0 ) + P 3 ) 259 ASSIGN 4 ,FN*5 PRODUCTION 260 TEST E P 4 , 0 , P R D 26 1 F I N TERMINATE 262 PRD MSAVEVALUE I N V + , 3 , P 3 , P 4 INCREASE SYSTEM INV. 263 MSAVEVALUE OINV+, 1 , P 3 , P 4 INCREASE ORIG. INV. 264 MSAVEVALUE TOTIO+, 1 , P 2 , P 4 265 LINK P 2 . F I F 0 ORIGIN INV. ( F I F O ) 266 * 267 * STORAGE AREA 268 * 269 ARIVS MSAVEVALUE O S H I P + , 1 , P 2 , P 4 270 TEST G M X $ O S H I P ( 1 , P 2 ) , X $ C A P , J 1 TOW SHOULD BE < TUG C A P . 271 LINK P 2 . L I F 0 EXTRA PRODUCTION STAYS @ ORIGIN 272 J1 MSAVEVALUE O I N V - , 1 , P 3 , P 4 DECREASE ORIG. INV. 273 MSAVEVALUE T O T I O - , 1 , P 2 , P 4 274 SAVEVALUE A B S I S , 3 , H 275 SAVEVALUE TRCST+,(P4*FN$IST) TRANSPORTAION COST 276 TEST E X H $ D I R E C , 0 , J 2 DIRECT SHIPMENT ? 277 MSAVEVALUE I N V + , 2 , P 3 , P 4 INCREASE STORAGE INV. 278 LINK STORA.FIFO STORAGE INV. (F IFO) 279 * 280 * MILL 281 * 282 ARIVM ASSIGN 5 , (10+P3) 283 GATE LS P5,RELNK 284 SAVEVALUE SSHIP+.P4 285 TEST G X $ S S H I P , X $ C A P , J 3 286 LOGIC R P5 287 RELNK LINK STORA,LIFO 288 J3 SAVEVALUE A B S I S , 4 , H 289 SAVEVALUE TRCST+,(P4*FN$IST) TRANSP . COST 290 MSAVEVALUE I N V - . 2 , P 3 , P 4 DECREASE STORAGE INV. 0\ 231 J2 MSAVEVALUE I N V + , 1 . P 3 . P 4 INCREASE MILL INV. 292 LINK M I L L , F I F O MILL INVENTORY (FIFO) 293 CNSUM LOGIC S P3 294 TEST GE P4,MH$DEMND(1,P3) , .MORE 295 MSAVEVALUE I N V - , 1 , P 3 , M H $ D E M N D ( 1 , P 3 ) DEPLETE MILL INV. 296 MSAVEVAULE INV- ,3 ,P3 ,MH$DEMND( 1 ,P3) DEPLETE SYS. INV. 297 ASSIGN 5,VSSURP 298 ASSIGN 4,MH$DEMND(1,P3) QUANTITY USED 299 MSAVEVALUE D E M N D , 1 , P 3 , 0 , H 300 TABULATE RTIME 301 SAVEVALUE A B S I S . 2 , H READ DET. COST UNIT 302 SAVEVALUE DCC+,V$DETER DET. COST FOR CONSUMMED LOG DURING THE WEEK 303 SAVEVALUE A B S I S , 1 , H READ INTEREST RATE 304 SAVEVALUE FCC+,V$FINAN FINANCIAL COST FOR CONS. LOG DURING THE WEEK 305 ASSIGN 4 , P 5 306 TEST E 0 , P4 ,BACK 307 TERMINATE 308 BACK LINK M I L L , L I F O TRANS. CARRYING THE SURPLUS GOES TO THE FRONT 309 MORE MSAVEVALUE D E M N D - , 1 . P 3 . P 4 . H UNSATISFIED DEMAND 310 MSAVEVALUE I N V - , 1 , P 3 , P 4 MILL 31 1 MSAVEVALUE I N V - , 3 , P 3 , P 4 SYSTEM 312 TABULATE RTIME 313 SAVEVALUE A B S I S , 2 , H 314 SAVEVALUE DCC+,V$DETER 315 SAVEVALUE A B S I S , 1 , H 316 SAVEVALUE FCC+,V$FI NAN 317 TERMINATE 318 * 319 * 320 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 321 * 322 * SEGMENT 4 TOWING OPERATION *** 323 * 324 * 325 * ORIGIN -- STORAGE AREA 326 * 327 GENE X H $ P U F 1 , , , , 3 , 4 TUG IS SENT ACORDING TO THE DESIRED FREQ. 328 ASSIGN 2, 1 ORIGIN 1 = PMN 329 TRANSFER , SENT 330 •* 331 GENE X H $ P U F 2 , , , , 3 , 4 TUG IS SENT ACORDING TO THE DESIRED FREQ. 332 ASSIGN 2 , 2 ORIGIN 2 = KB 333 TRANSFER .SENT 334 * 335 GENE 1 , , , , 3 , 4 TUG IS SENT WHEN A TOW IS FORMED 336 ASSIGN 2 ,3 337 TEST L MX$TOTIO(1,P2),XH$TOW,SENT 338 TERMINATE 339 * 340 SENT TEST E Q*2,0 ,ABORT 34 1 MARK 1 342 ENRUT MSAVEVALUE OSHIP, 1 , P 2 , 0 343 TEST G CH*2,0 ,ABORT IF NO PRODUCTION, ABORT THE OPERATION 344 QUEUE P2 345 ENTER TUG SEND A TUG IF AVAILABLE 346 DEPART P2 347 ASSIGN 1 , (P1+MP1) DEPARTURE TIME FROM ORIGIN 348 ASSIGN 3 , (100+P2) TRANS. TIME FUNCTION NO. -p-349 ADVANCE V$TRTIM TRANSIT TIME 350 SAVEVALUE A R I V A . P 1 ,H TUG'S ARRIVAL TIME 351 UNLINK P 2 , A R I V S , A L L , B V $ L O A D O TUG ARIVES AT THE STORAGE AREA 352 TEST E P 4 . 0 . M I L 353 LEAVE TUG TUG IS FREE 354 ABORT TERMINATE 355 * 356 * STORAGE AREA - MILL 357 * 358 GENE 1 . . . . 4 , 4 359 MARK 1 360 ASSIGN 4, 1 361 SAVEVALUE S S H I P , 0 362 SPLIT ( X H $ T 0 T Y P - 1 ) , A S I G N . 2 363 ASIGN ASSIGN 3 , (10+P2) 364 LOGIC R P3 365 TEST G (MX$INV(2,P2)+XH$DIREC) , 0 ,TERM IS STORAGE EMPTY? 366 TEST LE M X $ I N V ( 1 , P 2 ) , ( M H $ M M I N ( 1 , P 2 ) * 1 .5 ) ,TERM ORDER THOSE BELOW 1.5 TIMES MMIN 367 LOGIC S P3 RECORD THE TYPE 368 TEST LE MX$ INV(1 ,P2 ) ,MH$MMIN(1 ,P2 ) .TERM ORDER IF MIN. LEVEL IS REACHED 369 TEST E Q$ORDR,0,TERM IS AN ORDER ALREADY PLACED? 370 QUEUE ORDR 371 TEST E XH$DIREC,0 ,DIRCT 372 ENTER TUG 373 ASSIGN 1 ,(P1+MP1) DEPARTURE TIME 374 ASSIGN 2,(101+XH$0RIGN) TRANS. TIME FUNCTION NO. 375 ADVANCE FN*2 376 SAVEVALUE READY,(FN$DELAY+P1) ,H BOOMS BECOME AVAILABLE 377 UNLINK STORA,ARIVM,ALL,BVSLOAOS 378 MIL DEPART ORDR 379 LEAVE TUG 380 TERM TERMINATE 381 * DIRECT SHIPMENT 382 DIRCT SELECT MAX 2 , 1 , 3 , , C H 383 TRANSFER ,ENRUT 384 * 385 * 386 ****** **************************************************************** 387 * 388 * SEGMENT 5 MILL CONSUMPTION *** 389 * 390 * 391 * 392 GENE 1 2 , 3 393 MARK 1 394 SPLIT ( X H $ T 0 T Y P - 1 ) , D I V M , 3 395 DIVM MSAVEVALUE D E M N D , 1 , P 3 . F N * 3 , H 396 NEXT TEST G M X $ I N V ( 1 , P 3 ) , 0 , T R U B L 397 UNLINK M I L L , C N S U M , 1 , 3 DAILY CONSUMPTION 398 GATE LS P3 399 TEST G MH$DEMND(1,P3) ,0 .END IS THE DEMAND COMPLETLY SATISFIED? 400 LOGIC R P3 401 TRANSFER , NEXT 402 TRUBL MSAVEVALUE SHRTG+, 1 ,P3,MH$DEMND( 1 ,P3) ,H 403 END LOGIC R P3 404 TERMINATE 405 * 406 * Co 407 ********************************* t. ************* 408 * 409 * SEGMENT 6 BOOK KEEPING +** 410 * 41 1 * 412 GENE 1 , . . . 1 , 1 413 SPLIT ( X H $ T 0 T Y P - 1 ) . S U I T E , 1 414 SUITE MSAVEVALUE SUMIN+, 1 ,P1 , M X $ I N V ( 1 , P I ) 415 MSAVEVALUE S U M I N + . 2 . P 1 , M X $ I N V ( 2 , P 1 ) 416 MSAVEVALUE SUMIN+.3 .P1 ,MX$INV(3 ,P1 ) 4 17 MSAVEVALUE SUMIN+.4.P1,MX$OINV(1 ,P1 ) 4 18 TERMINATE 419 * 420 GENE 7 , . , , 0 , 2 42 1 SAVEVALUE WEEK+, 1 , H 422 ASSIGN 1,XH$WEEK 423 LOGTP SPLIT XH$TOTYP,BOOK 424 MSAVEVALUE W I C . 3 . P 1 , X $ D C C 425 MSAVEVALUE W I C . 4 . P 1 , X $ F C C 426 MSAVEVALUE WIC.5 .P1 ,X$TRCST 427 MSAVEVALUE W I C 7 . P 1 ,V$SUM 428 SAVEVALUE CUMUL+,V$SUM 429 MSAVEVALUE WIC.8 .P1 ,X$CUMUL 430 SAVEVALUE DCC .0 431 SAVEVALUE FCC.O 432 SAVEVALUE TRCST.O 433 UNLINK STORA,SBOOK,ALL 434 UNLINK MILL,MBOOK,ALL 435 SPLIT ( X H $ 0 R I G N - 1 ) , S 0 R C E , 2 436 SORCE UNLINK P2,OBOOK,ALL 437 TERMINATE 438 OBOOK SAVEVALUE A B S I S , 2 , H 439 MSAVEVALUE WIC+,1.XHSWEEK,VSDETER 440 MSAVEVALUE WIC+,7,XH$WEEK,V$DETER 441 MSAVEVALUE WIC+,8,XH$WEEK,VSDETER 442 SAVEVALUE A B S I S , 1 ,H 443 MSAVEVALUE WIC+,2,XH$WEEK,V$FINAN 444 MSAVEVALUE WIC+,7,XHSWEEK,V$FINAN 445 MSAVEVALUE WIC+,8,XHSWEEK,VSFINAN 446 LINK P 2 . F I F 0 447 SBOOK SAVEVALUE A B S I S , 2 , H 448 MSAVEVALUE WIC+,1,XH$WEEK,V$DETER 449 MSAVEVALUE WIC+,7,XHSWEEK,VSDETER 450 MSAVEVALUE WIC+,8,XHSWEEK,VSDETER 451 SAVEVALUE A B S I S , 1 , H 452 MSAVEVALUE WIC+,2,XHSWE EK,V$FINAN 453 MSAVEVALUE WIC+,7,XH$WEEK,V$FINAN 454 MSAVEVALUE WIC+,8,XHSWEEK,VSFINAN 455 LINK STORA,FIFO 456 MBOOK SAVEVALUE A B S I S , 2 , H 457 MSAVEVALUE WIC+,1,XHSWEEK,VSDETER 458 MSAVEVALUE WIC+.7,XHSWEEK,VSDETER 459 MSAVEVALUE WIC+,8,XHSWEEK,VSDETER 460 SAVEVALUE A B S I S , 1 , H 461 MSAVEVALUE WIC+,2,XHSWEEK,VSFINAN 462 MSAVEVALUE WIC+,7,XHSWEEK,VSFINAN 463 MSAVEVALUE WIC+,8,XHSWEEK,VSFINAN 464 LINK M I L L , F I F O -p-465 BOOK ASSIGN 2,(W$LOGTP+1) 466 MSAVEVALUE M W I , P 2 , P 1 , ( M X $ S U M I N ( 1 , P 2 ) / 7 ) 467 MSAVEVALUE S T W I , P 2 . P 1 , ( M X $ S U M I N ( 2 , P 2 ) / 7 ) 468 MSAVEVALUE S W I , P 2 , P 1 . ( M X $ S U M I N ( 3 , P 2 ) / 7 ) 469 MSAVEVALUE O R W I , P 2 . P 1 , ( M X $ S U M I N ( 4 , P 2 ) / 7 ) 470 MSAVEVALUE SHORT,P2 ,P1 ,MH$SHRTG(1 ,P2 ) ,H 471 SAVEVALUE CUMUL+,V$LOST 472 MSAVEVALUE WIC+.6 .P1 ,V$LOST 473 MSAVEVALUE WIC+,7 ,P1 ,V$LOST 474 MSAVEVALUE WIC+.8 .P1 ,V$LOST 475 MSAVEVALUE S H R T G , 1 . P 2 . 0 . H 476 MSAVEVALUE S U M I N , 1 , P 2 . 0 477 MSAVEVALUE S U M I N . 2 . P 2 . 0 478 MSAVEVALUE S U M I N . 3 . P 2 . 0 479 MSAVEVALUE S U M I N , 4 , P 2 , 0 480 TERMINATE 481 * 482 * 483 ******************************************************** 484 * 485 * 486 * AGE DISTRIBUTION OF LOGS IN THE SYSTEM 487 * 488 * 489 GENE 364, , , , 1 , 1 490 UNLINK STORA , A G E , A L L 491 UNLINK M I L L , A G E , A L L 492 UNLINK 1 ,AGE,ALL 493 UNLINK 2 , A G E , A L L 494 UNLINK 3 , A G E , A L L 495 TERMINATE 496 AGE TABULATE RTIME 497 TERMINATE 498 * 499 * 500 ********************************************************************** 501 * 502 * 503 * COUNTER 504 * 505 * 506 GENE 3 6 5 , , , , 9 , 1 507 TERMINATE 1 508 * 509 * 510 * 511 START 1 544 ***** 545 END o 151 APPENDIX 2 PMN-TA ER-TA 4 . 3 . 5 . 3 . 4 . 2 . 7 . 4 . 4 . 4 . 7 . 3 . 6 . 6 . 3 . 3 . 10. 2 . 4 . 3 . 6 . 3 . 4 . 2 . 6 . 7 . 3 . 2 . 5 . 3 . 6 . 4 . 12 . 5 . 5 . 3 3 2 4 3 3 6 3 2 4 2 Transit Time Data Days KB-TA 2 . 3 . 3 . 2 . 5 . 1 ". 3 . 8 . 6 . 3 . 2 . 4 . 2 . 2 . 5 . 3 . 3 . 3 . 4 . 2 . 2 . 2 . 2 . 2 . 3 . 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