Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Exploring alternative production strategies for a British Columbia coastal forest products supply chain Vergara, Francisco 2015

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2015_november_francisco_vergara.pdf [ 1.56MB ]
Metadata
JSON: 24-1.0166662.json
JSON-LD: 24-1.0166662-ld.json
RDF/XML (Pretty): 24-1.0166662-rdf.xml
RDF/JSON: 24-1.0166662-rdf.json
Turtle: 24-1.0166662-turtle.txt
N-Triples: 24-1.0166662-rdf-ntriples.txt
Original Record: 24-1.0166662-source.json
Full Text
24-1.0166662-fulltext.txt
Citation
24-1.0166662.ris

Full Text

    EXPLORING ALTERNATIVE PRODUCTION STRATEGIES FOR A BRITISH COLUMBIA COASTAL FOREST PRODUCTS SUPPLY CHAIN  by  Francisco Vergara   B.A., Universidad del Bio Bio, 1988 MSc., Universidad del Bio Bio, 2005  A THESIS SUBMITTED IN PARTIAL FULLFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF   DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES    (Forestry) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  August 2015 © Francisco Vergara, 2015 ii  Abstract  The BC Coastal forest industry has high production costs and the focus for improving efficiency has been to maximize capacity utilization. However, uncertainty in timber-grades and demand results in failure to meet client needs and excessive inventories. This research explores three major topics for improving forest supply chain performance: 1) how the error in timber grades negatively impacts customer demand fulfillment; 2) when lean and agile manufacturing environments, plus lumber demand patterns yield profit opportunities, and 3) how a planning-scheduling model can reduce order backlogs. The dissertation has three chapters that describe improvement to lumber planning through the application of firm-level case studies and mixed integer programming models. These supply chain models are guided by policies for over-under production control and policies for chase and level capacity usage control. Case studies provide the data for forest inventory yields, harvesting costs, log-to-lumber yields, manufacturing costs, lumber prices and sales orders. The first chapter shows that the level of error contained in timber volume and grades inventories underestimates profit with respect to perfect information and reduces the ability to fulfill customer orders. This drawback is not relevant for the current market, but it could compromise the industry’s ability to participate in other cut-to-order low value lumber markets. The second chapter identifies when lean and agile manufacturing should be adopted based on changes in lumber demand. The third chapter presents methodology to reduce backlogs and logistics costs by solving the lumber planning problem with a planning-scheduling model which imposes heuristics sequences to process orders. This planning tool can be advantageous for competing in cut-to-order lumber business scenarios. It would also provide lumber cost, log-lumber inventory levels, and backlogs levels as planning benchmarks for the current BC forest supply chain.  The improvements to lumber planning decision making in this dissertation identify the benefits of moving BC Coastal forest industry firms to lean, agile or cut-to-order manufacturing environments, especially in the context of participating in new lumber product portfolios. Forest inventory, manufacturing and economic data were extracted from publicly available reports, thus metrics calculated may be inaccurate and limit the general validity of some results.    iii  Preface  The work developed contains three research chapters that have been or will be submitted for publication in peer reviewed journals. The DSSs, gathering forest and case study data, results and analyses were developed by the author of this thesis.  In relation to co-authorship, the contribution of Dr. John Nelson should be recognized. He, as the thesis supervisor, helped on decision making formulations, the results analysis process, and editing and improving this manuscript. Dr. Cristian Palma, advised on decision making systems formulations, validations, and scenario analysis development. Dr. Peter Marshall advised on establishing research questions, statistics and correct research objectives. As a consequence of the development of this doctoral thesis, versions of the research chapters have been submitted or will be submitted for publication. A list of chapters follows: Chapter 2: Francisco Vergara, John Nelson, Cristian Palma, and Peter Marshall (submitted). Impact of timber volume and grade estimation uncertainty on BC coastal forest supply chain Chapter 3: Francisco Vergara, Cristian Palma, and John Nelson (to be submitted). Economic impact of implementation of lean and agile manufacturing systems on the BC coastal forest industry Chapter 4: Francisco Vergara, Cristian Palma, and John Nelson (to be submitted). The value of due date fulfillment for production orders when solving the operational lumber manufacturing problem Finally, this research recognizes the collaboration with FP Innovations. Dr. Darrell Wong, Dr. Bruce Lehmann, Dr. Catalin Ristea, and Joel Mortyn advised on real problems and features of the BC Coastal forest industry.   iv  Table of Contents  Abstract ..................................................................................................................................................................... ii Preface .................................................................................................................................................................... iii Table of Contents..................................................................................................................................................... iv List of Tables .......................................................................................................................................................... vii List of Figures .......................................................................................................................................................... ix Acknowledgements .................................................................................................................................................. x Dedication ................................................................................................................................................................ xi Chapter 1 Introduction ............................................................................................................................................. 1 1.1 The British Columbia coastal forest supply chain ..................................................................................... 1 1.1.1 SWOT analysis of BC coastal forest industry ......................................................................................... 2 1.2 Literature Review: Forest supply chain decision support systems ........................................................... 5 1.2.1 Supply chain background ....................................................................................................................... 5 1.2.2 Existing forest supply chain decision support systems ........................................................................... 8 1.2.3 Optimization and simulation modeling methods for supply chain management ................................... 11 1.2.4 Research gaps ..................................................................................................................................... 13 1.3 Research objectives .................................................................................................................................... 15 1.4 Thesis organization ..................................................................................................................................... 16 Chapter 2 Impact of timber volume and grade estimation uncertainty on BC Coastal supply chains ..................... 18 2.1 Summary ..................................................................................................................................................... 18 2.2 Introduction .................................................................................................................................................. 19 2.3 Literature review .......................................................................................................................................... 19 2.4 Methods ....................................................................................................................................................... 23 2.5 Results ........................................................................................................................................................ 27 2.5.1 Profit analysis ....................................................................................................................................... 27 2.5.2 Risk analysis ........................................................................................................................................ 29 2.5.3 Value of information ............................................................................................................................. 31 2.6 Discussion ................................................................................................................................................... 33 2.7 Conclusion ................................................................................................................................................... 35 Chapter 3 Exploring manufacturing environments for the BC coastal forest industry ............................................ 37 3.1 Summary ..................................................................................................................................................... 37 3.2 Introduction .................................................................................................................................................. 38 3.3 Literature review .......................................................................................................................................... 39 3.3.1 Conceptual supply chain manufacturing strategies .............................................................................. 39 v  3.3.2 Conceptual framework for Lean and Agile manufacturing strategies ................................................... 42 3.3.3 Aggregate production planning drivers ................................................................................................. 44 3.3.4 Modeling the production planning problem with lean and agile approaches......................................... 44 3.3.5 BC coastal forest supply chain as a case study ................................................................................... 48 3.4 Methodology ................................................................................................................................................ 50 3.4.1 Lumber demand scenarios ................................................................................................................... 51 3.4.2 Description and formulation of ME models ........................................................................................... 52 3.4.3 Mathematical formulations ................................................................................................................... 55 3.5 Results ........................................................................................................................................................ 64 3.5.1 Results for MEs .................................................................................................................................... 65 3.6 Discussion ................................................................................................................................................... 68 3.7 Conclusion ................................................................................................................................................... 71 Chapter 4 Assessing due date fulfillment for lumber manufacturing production orders ......................................... 72 4.1 Summary ..................................................................................................................................................... 72 4.2 Introduction .................................................................................................................................................. 73 4.3 Literature review .......................................................................................................................................... 74 4.4 Methodology ................................................................................................................................................ 78 4.4.1 Decision-making model formulations .................................................................................................... 79 4.4.2 Problem setting .................................................................................................................................... 81 4.4.3 Lumber manufacturing data ................................................................................................................. 81 4.5 Results ........................................................................................................................................................ 83 4.5.1 Feasibility analysis ............................................................................................................................... 83 4.5.2 Lumber manufacturing costs analysis, without backlog and due date relaxation ................................. 83 4.5.3 Economic and manufacturing effects of backlog and due date relaxation ............................................ 85 4.6 Discussion ................................................................................................................................................... 88 4.7 Conclusions ................................................................................................................................................. 91 Chapter 5 Conclusions ........................................................................................................................................... 93 5.1 General conclusions .................................................................................................................................... 93 5.2 Limitations, strengths and weakness of this research ................................................................................. 97 5.3 Future research and potential applications .................................................................................................. 99 References .......................................................................................................................................................... 101 Appendices .......................................................................................................................................................... 109 Appendix A  Model formulation for chapter 2 and logit regression analysis ............................................ 109 A1 Formulation of the BC coastal forest industry production planning problem .......................................... 109 A.2 Logit regression analysis ...................................................................................................................... 113 Appendix B ...................................................................................................................................................... 117 vi  B.1 Additional tables for Chapter 3 .............................................................................................................. 117 Appendix C Formulation of the lumber manufacturing planning and scheduling problem ............................... 118 A Plann (PL) model ..................................................................................................................................... 118 B Plann-sched (PS) model .......................................................................................................................... 121 C Plann model accepting backlog ............................................................................................................... 125 D Plann-sched model accepting overdue order .......................................................................................... 127 Appendix D DSS results for chapter 4 ............................................................................................................. 129 D.1 Non relaxed run results ......................................................................................................................... 129 D.2 Relaxed run results ............................................................................................................................... 132    vii  List of Tables  Table 2.1 Error scenarios ....................................................................................................................  .........  24 Table 2.2 Profit statistics by demand level, and error scenarios ...................................................................  29 Table 2.3 Case study lumber products targets, and expected target reduction given error scenarios ..........  30 Table 2.4 Profit variation statistics based on error scenarios for target 1,406,000 m3 ...................................................... 32 Table 3.1 Efficient versus Responsive SC attributes ....................................................................................  41 Table 3.2 Attributes of Lean and Agile manufacturing supply chain drivers ..................................................  45 Table 3.3 Summary of manufacturing environment model formulation .........................................................  52 Table 3.4 Over and under lumber demand satisfaction penalty framework ..................................................  53 Table 3.5 Over and under capacity usage penalty framework ......................................................................  54 Table 3.6 Economic performance by manufacturing environment and lumber demand scenario ($) ............  65 Table 3.7 Summary of profit variation between MEs and lumber demand scenarios .................................. .. 66 Table 4.1 Lumber products demand scenarios .............................................................................................  82 Table 4.2 Lumber manufacturing costs for planning and scheduling approaches .........................................  87 Table A.1 Logit regression analysis ..............................................................................................................  113 Table A.2 Logit regression analysis ..............................................................................................................  114 Table A.3 Logit regression analysis ..............................................................................................................  115 Table A.4 Logit regression analysis ..............................................................................................................  116 Table B.1 Manufacturing performance by ME and lumber demand scenario................................................  117 Table D.1 PL and Pl-S run results, no backlog and no overdue orders ........................................................  129 viii  Table D.2 PS-E with large, small, mixed, high variation, and low variation lumber products demand ..........  130 Table D.3 PS-L with large, small, mixed, high variation, and low variation lumber products demand ...........  131 Table D.4 PL with large, small, mixed, high variation, and low variations demand .......................................  132 Table D.5 Pl-S-E with large, small, mixed, high variation, and low variation lumber products demand.. ......  133 Table D.6 Pl-S-L with large, small, mixed, high variation, and low variation lumber products demand .........  134 Table D.7 Pl-S-S with large, small, mixed, high variation, and low variation lumber products demand.. ......  135   ix  List of Figures  Figure 1.1Coast forest products industry SWOT analysis .............................................................................  3 Figure 1.2 Coast Management and Production Strategies ...........................................................................  4 Figure 2.1 Profit variation for normal cut block error, and normal cut block nirvana error .............................  28 Figure 2.2 Profit variation for super cut block error, and super cut block nirvana error .................................  28 Figure 2.3 Predicted failure of lumber demand fulfilment ..............................................................................  30 Figure 2.4 Profit variations based on error scenarios Ne (left) and Nne (right) for target 1,406,000 m3 ............. 31 Figure 2.5 Profit variations based on error scenario Se and Sne for target 1,406,000 m3 ............................................. 32 Figure 3.1 Profits by ME and lumber demand scenario ................................................................................  64 Figure 3.2 Flexibility parameters by ME and demand scenario.....................................................................  65 Figure 4.1 PSAs feasibility percentage as a percentage of the runs .............................................................  83 Figure 4.2 Lumber manufacturing costs by size of the order ........................................................................  84 Figure 4.3 Lumber manufacturing costs by PSAs .........................................................................................  84 Figure 4.4 Lumber manufacturing costs by size of the order (relaxed PSAs) ...............................................  85 Figure 4.5 Lumber manufacturing costs by relaxed PSA ..............................................................................  86 Figure 4.6 Deviation of relaxed PSA costs relative to the corresponding non-relaxed PSA costs ................  86 Figure 4.7 Backlogged volumes by relaxed PSA ..........................................................................................  87 Figure A1 Nodes and variables of the decision support system....................................................................  109   x  Acknowledgements  This part of my dissertation should be the longest, because I have to recognize the economic, emotional, and spiritual support of many people during these years in Canada.  First of all, I would like to express my appreciation to Dr. John Nelson my second and definite scientific advisor. He were always available for my questions, discussions, and even to advise me on the hardest moment that I passed during this journey.  However, I would like to mention my first scientific advisor Dr. Tom Maness, who unfortunately moved to the United Stated when I just started this journey, although he found the way to maintain himself connected with what I was doing. He encouraged me to start my Doctoral degree at UBC, and I always counted on his economic and scientific support. I would to say thanks for your friendship as well.  Another key person was a post-doc fellow called Cristian Palma. Dr. Palma was the toughest fellow, but, he was the one which always put the highest scientific standard in our laboratory. The first day in Canada was chaotic, delayed flights, lack of money, lack of language, nothing was familiar, but a fellow called Catalin Ristea appears at Vancouver airport with a solution for everything. Sometimes, we asked our self if this person is supposed to be our guardian angel. Therefore, I would like to recognize his help and interested support.    xi  Dedication  To my beloved wife Jessica, Joaquin and Francisco Jr, they were the central support to keep this dream alive. I know that you made my dream yours, without considering the externalities. My highest appreciation for you family, now I know that the largest life dreams’ demand a large proportion of naivetés and love. I’m greatly indebted to you.   1  Chapter 1 Introduction  1.1 The British Columbia coastal forest supply chain  Over the last decade, the British Columbia (BC) Coastal forest industry has declined because of more restrictive harvesting policies, high timber costs, old sawmills, and market changes affecting BC lumber. These factors have resulted in a lack of competitiveness in the global economy. B.C.Coastal sawmilling costs are at the high end of the global spectrum due to the use of slower head rig type mills to process large logs, which restricts the mills from participating profitably in commodity markets. While this strategy achieves high revenue for major coastal tree species, including western red cedar (Thuja plicata), Douglas-fir (Pseudotsuga menziesii) and some minor species such as yellow-cedar (Chamaecyparis nootkatensis) and Sitka spruce (Picea sitchensis), it does not offset the high costs of logging, hauling and milling of less profitable species like western hemlock (Tsuga heterophyla) and true fir (Abies sp.) (Cohen 2007), which represent more than half of the BC coast wood supply. This results in poor operating margins relative to other global regions. At the same time, the coastal timber supply is declining in quality because of the transition from old-growth to second-growth forests. Second-growth wood lacks the unique qualities of old growth, and there are many competing suppliers of this lower quality softwood (Pearse, 2001).Currently, coastal mills process large, high-grade logs to produce a wide range of specialty products, while interior mills use smaller, lower quality logs to produce commodity products such as dimension lumber. Consequently, B.C. interior mills focus primarily on North American markets while coastal mills have more diversified markets. Given the current market conditions, the BC coast is milling below capacity, which is a major contributor to the industry’s uncompetitive condition. Accordingly, material flows are primarily directed towards push production of price-sensitive commodity products. The predominant focus for improving efficiencies involves increasing throughput and capacity utilization and reducing inventories. However, long lead times make it exceedingly difficult to control inventory. Also, uncertainty in supply and variation in supply quality trigger large amounts of by-products, raw material and end product in inventory (Haartveit et al. 2004).  2  1.1.1 SWOT analysis of BC coastal forest industry  The competitiveness of the BC Coastal forest industry is limited due to both internal and external changes made over the last decade (Pearse 2000). High production costs are offset somewhat by the relatively high value products manufactured, but not entirely. On the other hand, supply chain attributes such as long length and complexity (Maness 2008), unsuitable log allocation methods (FP Innovations 2008), and volume maximization focuses (Haartveit et al. 2004) fail to meet client needs and inventories can grow without control (Gaudrealts et al, 2009). Many variables play significant roles in the future of B.C Coast lumber industry. Therefore it is urgent to re-engineer the supply chain (SC) design and manufacturing alternatives, with a focus on unprofitable species. The potential for effective SC management has been proven in other countries and industries. For instance, production and storage exchanges between sawmills have increased operational revenues in Chile by 15% per month (Singer 2008). Also, analyses of uncertainty in forest inventory parameters help to develop alternative plans, which could generate 9% more profit than plans developed without uncertainty (Beaudoin et al. 2006). Inventory policies that minimize backorders and ordering costs at each SC stage can increase order fulfillment and reduce inventories (Mason-Jones et al. 1997). In terms of manufacturing systems, Goldsby et al. (2006) conclude that lean systems are the lowest cost alternative in comparison with agile systems, which were 15% higher than lean, and 21% higher than hybrid systems.  This research addresses redesign of the forest SC, where integrated forest companies are mostly managed in a centralized manner. Currently, forest management plans do not link to manufacturing plans; they do not have appropriate and consistent goals, and planning approaches are not suitable for new scenarios regarding forest supply and product markets. Therefore, in order to craft strategies that help the coastal lumber industry, I prepared a Strengths, Weaknesses, Opportunities and Threat analysis (SWOT). The goal was to identify appropriate strategies for the next decade, with a focus on the supply chain, manufacturing, old- and second-growth species, and the forest products market (Figure 1.1).    3  STRENGTHS  Extensive knowledge in the lumber industry  The BC Coast has some of the world’s most desirable softwood  timber  Location close to US market  Flexibility of smaller niche players  Fibre basket has 55% of first growth, which contains 35% High-value Douglass fir and Western red cedar species, and 65% low value Western hemlock and Balsam; and second growth harvest (45%) largely commodity (Pearse 2000)  Reasonably good working relationships between Government, First Nations, and lumber companies that will likely lead to developing cooperative relationships.     WEAKNESSES  Low profitability leaves no room for investment, which triggered the failure of industry to invest in new technology and products  High labour costs and low levels of productivity  Uncertainty about land use because of First Nations land claims  Uncertainty in wood supply and wood quality, which command products distribution  Increased in administrative duties and stumpage in the 90s  Limited use by managers of decision support systems  Head saw have old scanners, low productivity and high operational cost  Only 40% of board edgers at sawmill are optimized  Low productivity in manual bucking and sorting operations  Value of Western hemlock products has declined, making this species unprofitable THREATS  Increase in competition from other regions and non-wood substitutes means that customers have more choices than ever before  Changes in the BC Coast traditional export such as the U.S., Europe, Japan, Australia, North Africa, the Caribbean, South Korea and China  Weak demand for lumber commodity products in the U.S  Softwood Lumber Export Tax to the USA. OPPORTUNITIES  Global emerging markets, including China and India  Healthy market for specialty products made from Western red cedar and Douglas fir.   Figure 1.1 Coast forest products industry SWOT analysis As a result, a set of strategies were developed to overcome weaknesses and threats, as well as capturing the strengths and opportunities (Figure 1.2). Hence, supply chain management and production, manufacturing facilities, and forest products strategies can be seen as potential areas of research to improve the competitiveness of the BC Coastal forest industry.       4  Supply chain management and production strategies  Develop integrated decisions support systems for the forest supply chain to simulate scenarios and respond to market opportunities and value optimization  Develop and implement inventory control tools that integrate mill requirements, log quality and value  Develop integrated value-chains using  efficient sawmills cutting to order; small log inventories sorted into diameter and quality classes and intensively managed forest lands  Determine the effects and strategies to follow to deal with wood quality uncertainty in forest supply chain decision making process  Simulators, supply chain management and optimization tools already used in others industries need to be adopted to the lumber industry  Reinforce entrepreneurial initiatives in coastal communities by exploring opportunities for partnerships, collaboration, and outsourcing between forest companies and small community-based contractors. Manufacturing facilities strategies  Explore the feasibility of centralized log merchandizing lines, using x-rays scanners and precision bucking to maximize value  Develop specific log grade information for forest stands by image processing Forest product strategy  Explore a new commodity products mix, including low appearance products as pallets as alternatives for second growth Western hemlock and Balsam  Specialty and some semi-commodity products produced with Western red cedar; custom-cut or specialty products and niche markets with all the major species (Western red cedar, Douglass fir and Yellow cedar, Sitka spruce). Veneer and plywood focusing on Douglass fir with further opportunities possibly exist for Western Hemlock and Balsam  Processing small-diameter logs of second-growth Western hemlock and Douglas fir logs, in cost efficient mill is a logical fit for commodity markets Figure 1.2 BC Coast Management and Production Strategies Potential areas of research identified with the SWOT, plus the lack of research determined in the literature review helped in formulating the central research hypothesis, and goals of this research. The next part of this introduction provides a literature review on SC management to introduce related and relevant research, identifies research gaps, states the central research hypothesis, the specific research objectives, and gives the thesis structure.     5  1.2 Literature Review: Forest supply chain decision support systems  1.2.1 Supply chain background  A strategic fit of the SC requires that the firm achieve a balance between the responsiveness and efficiency that best meets its competitive needs. There are four essential SC drivers that require attention: 1) inventory, 2) transportation, 3) facilities and 4) information (Chopra and Meindl 2001). In addition to these factors, there are several other concepts such as planning strategies and demand behaviour to consider; thus, a brief review of these was included as they impact the forest industry. SC management deals with a total flow of materials from suppliers through end users. Jones and Riley (1984) pointed out that the key to efficient management is to plan and control inventories and operations as one single integrated activity. This insight offers relevant suggestions for the currently disjointed decision making process in the BC coastal region. Therefore, three factors must be considered: 1) recognition of customer service needs; 2) placement and quantity of inventories; and 3) development of the right policies for managing the SC as a single entity. Lee and Billington (1992) noted that the lack of common SC metrics is a common pitfall to appropriate inventory management. Strategic goals should involve key elements that consider consumed resources, output and flexibility (Beamon 1999). The use of resources (i.e. level of efficiency), the desired output (i.e., level of customer service) and flexibility (i.e. ability to react to uncertainty) are vital components of SC success. In the BC coastal industry, the traditional management model has been based on operational efficiency and economic performance, leaving room to include other key factors in the decision making process. Min and Zhou (2002) noted that SC decision variables set limits on the range of decision outcomes and they are functionally related to SC performance. Thus, performance measures should be expressed as a function of those variables. Particularly, the aggregate planner must make a trade-off among factors like capacity, inventory and backlog costs to satisfy SC goals. Arriving at the most profitable trade-off is the goal of aggregate planning and requires a dynamic and reflexive decision support system (DSS). Given that demand varies over time, the relative level of these three costs is essential to maximizing profits. There are three aggregate planning strategies for achieving balance among these costs (Chopra and Meindl 2001): 6  (1) chase strategy, in which the production rate is synchronized with demand rate by varying machine capacity or by hiring/reducing the workforce; (2) time flexible strategy, in which, if there is an excess of machine capacity, the workforce is kept stable but the number of worked hours is reduced; and (3) level strategy, in which stable machine capacity and workforce are maintained with a constant output rate. Mason-Jones et al. (1997) observe that SC continue to operate inventory policies on the outdated basis of sequential upstream transmission of orders from the market place. The result is inevitably demand distortion and amplification at every stage. This is particularly true in forest SCs where decision making is disjointed and uncoordinated. However, if inventory management policies are applied at every stage, demand and order noise reduction can be achieved by stock-holding and stock-out minimization.  Another important aspect of SC management is lean and agile supply paradigms (Bruce and Daly 2004). A lean SC can be defined as an operationally synchronized SC with focus on the elimination of all waste, including the time to enable a level production schedule (Naylor et al., 1999). The term “lean” is often used in connection with manufacturing to imply a “zero inventory” and a “ just in time” approach (i.e. when demand is predictable, product diversity is low and volume is high) (Christopher, 2000). An agile SC works with minimal lead times to be able to service volatile consumer demand with high levels of availability. Therefore, agility might be defined as the ability of the SC to respond rapidly to changes in demand, both in terms of volume and variety. Thus, there will be occasions when a strictly agile or lean strategy might be appropriate. However, Christopher (2000) pointed out the possibility of situations in which a combination of the two may be appropriate (e.g. hybrid SC’s). Hybrid SC strategies recognize that within a mixed portfolio, there will be products where demand is stable and predictable and products where the opposite is true.  Lean strategies involve advance production on a make-to-stock basis and on speculated demand. This planning method supports level scheduling of production and pull-based ordering upstream. While lean management focuses on efficiency, agile management focuses on flexible accommodation of unique customer demands. Instead of relying on speculative demand, agile strategies employs a “wait and see” approach. Thus, while lean strategies call for make-to-stock replenishment driven by short-term forecasts, agile strategies employ make-to-order provisions, producing only what has already been sold or committed to in the marketplace.  The key to agile response is its ability to produce large or small batches, minimizing the “pain” of setups, as well as flexible workforces and product design that provide for quick conversion of materials from raw to 7  completed states (Christopher 2000). Many companies are realizing that the cost and risk of holding speculative inventories is too high. For instance, once a long stem is bucked or a sawlog is sawn without a production order they are held in inventory with a high risk of value loss. Thus, companies are adopting agile strategies, maintaining inventories of components that be easily assembled for different end products. They can readily mix and match materials and produce products that meet customer needs. A limitation of agile strategies is the lack of advance assembly and, hence, every order becomes a backorder (Goldsby et al. 2006). Therefore, the necessity to adopt correct allocation and manufacturing systems is even more important in the forestry industry than in others, given the size of operations, volumes transported, demand-supply seasonality, and the properties of different forests. In an effort to re-engineer the SC process, answering questions of “when” and “where” to adopt lean and agile approaches should involve an analysis of decoupling points (Naylor, 1999). Decoupling separates the part of the SC that responds directly to customers from the part that uses forward planning and stock to buffer against changes in demand. Upstream, the SC is initially forecast-driven. The position of the decoupling point in the SC determines its ability to handle different factors. Thus, the lean paradigm can be applied upstream of the decoupling point as the demand is smooth and standard products flow through a number of value streams. The agile paradigm can then be applied downstream from the decoupling point as demand is variable and the product variety per value stream has increased. Forest harvesting operations are generally not in balance throughout the year. In BC harvesting is often carried out during winter to protect soils. In Chile it is the other way around and harvesting is concentrated to the summer when the forests are dry (Ronnqvist 2003). Rather than continuing the application of make-to inventory systems (e.g. massive production system), a critical analysis of the position of the decoupling point presents an opportunity to develop a SC management system that is responsive to particular demands.  Goldsby et al. (2006) studied how lean, agile and hybrid SCs behave in terms of three factors: 1) customer service (i.e. order-to-ship time), 2) inventory, and 3) total cost. The results show that order-to-ship time for a hybrid model was 8 times higher than a lean model and an agile model was 3 times higher than a lean model. However, lean models resulted in considerably more inventory than agile and hybrid models. In terms of total cost, the lean model represented the lowest cost alternative among the three approaches. Agile was 15% higher than lean, and hybrid was 21% higher than lean.  8  Although Goldsby et al. (2006) demonstrated appropriate aggregate planning and inventory policies for lean, agile, and hybrid systems under certain conditions, these assertions cannot be generalized to different fields of study or circumstances. As a result, the ability of these strategies and their combinations to improve a particular SC performance will be strongly influenced by specifics of each business (e.g., demand dynamics). In the BC coastal forest industry there are products and markets such as custom-cut lumber in which agile SC models appear to be an obvious approach.  Conversely, lean SC models seem appropriate for construction grade lumber. Thus, research to identify particular factors in the context of the BC forest industry is crucial to developing a competitive SC model that considers agile, lean, and hybrid approaches to improve the current forest SC.  1.2.2 Existing forest supply chain decision support systems   Operations research has helped to improve the SC performance of many forest companies. Ronnqvist (2003), and D’Amours et al. (2008) argue that challenges of integrating different planning problems remain relevant in SC development, demonstrating the need for increased integration between forests and industrial SCs to improve efficiency and efficacy. Analyses of research initiatives have shown solutions for specific problems encountered in softwood production planning on different levels. An early model that simultaneously determines the optimal bucking and sawing policies based on demand and final product price was developed by Maness and Adams (1991). Their model used linear programming and dynamic programming with Dantzig-Wolfe decomposition. The model included a log sawing algorithm and a price volume relationship used to control output. The DSS maximized sawmill revenue by determining optimal bucking and sawing policies for the operational sawmill planning problem. This model was later modified to handle several time periods (Maness & Norton 2002). However, the contribution of this research is limited to a given SC design and was developed using deterministic data.  Donald et al. (2001) addressed integration of decisions between primary and secondary lumber manufacturing. One model maximized economic benefits, penalizing under and over-production in a value added facility, and the second integrated processes from the sawmill log yard to the value added facility, 9  linked by market constraints. Their results highlighted how integrated decisions influence sort yard and sawmilling model solutions, increasing the revenues of the entire system under analysis. In addition, Maness and Norton (2002) developed a tool that introduced flexible marketing constraints based on a piece-wise linear objective function to simulate product demand curves of cut-to-order lumber. The tool had to adapt the production strategy in response to production and inventory shortages, a particularly relevant insight for the development of a responsive SC.  For sawmill planning production, Todoroki and Ronnqvist (2002) developed a secondary log breakdown tool to evaluate volume, value and a hybrid model which maximized both. They critiqued formulations that assume elastic demand for products. They controlled the grade-volume relationship by applying dynamic board values to reflect the changing levels of demand for various timber grades. Their results show the advantages of product optimization logic over volume and value optimization logic. Gunnarsson et al. (2007) determined procurement, distribution, and product recipe policies for the pulp SC in a Swedish company. A mixed integer revenue maximization model was formulated for this SC. They used binary variables to handle strategic decisions for new products. This large formulation and related case studies were useful to evaluate alternative products, suppliers, production recipes and distribution as well as to determine, when and where different transportation methods and terminals should be used.  Beaudion et al. (2006) developed scenarios for tactical wood procurement decisions, with randomly generated values for uncertain parameters. An optimal plan was determined for each scenario with a mixed integer deterministic model. Harvesting and transportation capacity feasibility was later analyzed by a human planner. The outcomes show how uncertainty impacted profit and the influence of forest inventory parameters and market conditions. This was a useful approach to handle uncertainty, but it did not support the selection of the probability distribution applied (usually a normal distribution) and this can impact decisions significantly.  For sugar, molasses and the ethanol distillery processes Paiva and Marabito (2008) built a decision model in which process selection was adopted as a binary variable and storage capacity constraints and holding inventory costs were applied. A decision variable of inventory by mill was considered to penalize revenues. However, they concluded that the number of binary variables, case studies and scenarios must be carefully considered because they drastically affect model solution times. 10  Singer and Donoso (2009) modeled a cluster of sawmills to probe whether or not collaboration, understood as the exchange of warehousing, production orders, and raw materials between sawmills can help to improve revenues. They showed outcome exhibits 15% higher revenues between collaborative mills compared to independent operations. Unfortunately, there was no mention of what kind of mechanism was applied to distribute the market-orders in the sawmill cluster. Schwab et al. (2009) calibrated an agent-based simulation system combined with optimization models to conduct strategic analysis in the forest sector. The system assessed the impact that demand and inventories can have on the structure and economic viability of the forest sector. This environment was modeled as a group of interacting autonomous economic agents, using a rule interaction between agents which reinforced learning. The model was an interesting mix of a multi-agent simulation and a linear program. While this research offers useful decision making techniques for decentralized systems, it is unable to account for centralized systems, like that of the vertically integrated BC Coastal forest industry case study. Beaudoin et al. (2010) adopted a similar method to solve the procurement planning problem on shared procurement lands. They developed decision and economic environments, which combined individual and collective decision models connected by one buy-sell wood auction negotiation mechanism. Their results show the benefits of sharing information between SC members. Particularly, favorable economic environments and centralized decisions resulted in the highest profits for SC members. However, in unfavorable economic environments collaborative and decentralized decisions garnered better results than centralized approaches. Despite the contributions of past research, there are many aspects of this field of study that have not been fully elucidated. The majority of research in this field has focused on revenue maximization. Comprehensive forest SC models require further research to specifically include inventory reduction, customer response and value maximization.  Haartveit et al. (2004) emphasized the significance of uncertainty in the planning process for the BC Coast forest industry. There is not enough knowledge about uncertainty in supply deliveries, inconsistencies in timber grades and their impacts on SC management.  The majority of analyses have been done for particular facilities, with no attention given to the ways that alternative aggregate facility design can affect outcomes. Timber supplies have been changing in the BC Coast forest industry, so product profiles should 11  be adaptive to face new market opportunities. The massive and make-to-stock manufacturing systems should be the subject to further research, including value output, customer response and SC flexibility.  1.2.3 Optimization and simulation modeling methods for supply chain management  For researchers like Beamon (1998), SC management models are multi-stage tools for design and analysis. These models can be divided into four categories, according to inputs and objectives: 1) deterministic analytical models; 2) stochastic analytical models; 3) economic models; and 4) discrete event or continuous simulation models (Lee et al. 2002). These categories provide a general framework for current SC methods. The following review analyzes the methods used to model SCs and identifies connections with problems in the BC coast forestry SC. Hillier (2005) comments on the strengths of wide-perspective operations research methods that adopt an organizational priority and attempt to resolve conflicts between the components of the organization as a whole. However, Buongiorno and Gilless (2003) state that only one objective function, most commonly economically focused, can be optimized in traditional linear programming models, and multiple management objectives may not be satisfied with one objective function. Other objectives, for example, maintaining an even flow of timber production or maintaining a specific number of large trees in a stand, are expressed only as constraints. This way of handling multiple management objectives may not be adequately reflective of actual conditions. Buongiorno and Gilless argue that representing some goals as constraints, in effect, gives them priority over the goal reflected in the objective function. Goal programming can provide a way of striving toward selected objectives simultaneously; however, determining their different weights remains challenging.  Weintraub et al. (2007) pointed out that heuristics and meta-heuristics, as well as operations research methods, have been proposed to solve forest planning problems. The first fit well when the number of integers or binary variables, non-linearities and stochasticities increase in mixed integer programming problems. However, Jones et al. (2002) pointed out that most meta-heuristics are naturally discrete, meaning this method can handle models with integers, discrete and-or logical (binary or zero or one) variables well. Another advantage is their flexibility, because the range of models capable of being solved 12  by meta-heuristics is far greater than conventional methods. However, meta-heuristic methods are not function optimizers; their purpose is to seek good solutions to particular problems. Another disadvantage is the large number of parameters required to be set by the user compared to exact solution methods. Thus, a number of executions with different parameter settings are needed before a good solution is produced. Distributed artificial intelligence is a subfield of artificial intelligence that has experienced rapid growth (Balaji and Srinivasan 2010). Multi-agent systems are an artificial intelligence approach recently applied to SC management. This system can be defined as a network of individual agents that share knowledge and communicate with each other in order to solve a problem (El Habib and Yacine 2006). Multi-agent simulation is the execution of a multi-agent model and this method can be used to create systems which mimic real scenarios. At every model iteration, each agent individually assesses its situation and makes a decision about what action to take based on a set of rules defined by the modeler. Multi-agent systems are easier to understand and implement when the problem is decentralized. However, Moyaux et al. (2006) argued that one major disadvantage over direct solution methods is that there are no solution algorithms to identify an optimal solution. Additionally, efficient and effective negotiation between SC partners to reflect and meet dynamic and volatile requirements for SCs is very hard to establish. Buongiorno and Gilles (2003) stated that the best forest system models are the simplest ones that reflect the key elements of the question to be answered. Similarly, Weintreaub et al. (2007) pointed out that modelers should resist adding complexity to the models. Good modeling is not a way of computing, but rather a way of thinking. More than finding a particular solution, good models should help forest resource managers reason though problems in a logical manner. Thus, although the quality of data underlying the model is important, it is not critical. Much useful understanding of a problem can be acquired by building a model with rough data when important decisions must often be made quickly. Thus, in order to create appropriate SC management systems in Coastal BC, appropriate research must account for current problems in the BC Coast SC, specific attributes and limitations of mixed integer programming methods, simulation methods, and artificial intelligence methods.   13  1.2.4 Research gaps  During the last 20 years many operations research applications have focused on the forest SC, mathematical programming mixed with heuristics, discrete event simulation (DES) or even multi-agent simulation systems (MAS) methods, depending on the problem. However, there are many aspects of this field of study that have not been fully explored. For instance, a survey of the literature shows that less attention has been focused on the forest to lumber SC. Most of the research has been carried out on harvesting models, transportation and road design models. Given the managerial attributes of integrated forest companies, where the decision making process is hierarchical and centralized, mixed integer programming (MIP) methods can provide the opportunity to analyze SC policies efficiently. The ability to model the SC by knowing its relevant features without introducing a large number of binary variables is the challenge. Therefore, mixed integer programming plus case study scenarios were chosen as modeling and analysis tools in this thesis. The majority of research in this field has focused on revenue maximization. This research has been done around the massive make-to-inventory and make-to-order manufacturing systems for commodities and custom cut lumber. Until now, this system has worked with sawmill facilities that are flexible and productive enough to efficiently and profitably handle the current combination of products, market and supply. However, given market and forest supply changes that are occurring, and considering the comments of researchers in this field (e.g. already mentioned on the SWOT analysis), the previous assumptions are not adequate for the near future and should be the subject of further research. Accordingly, this research is focused on testing manufacturing strategies for the forest-to-lumber SC. Specifically, this research will carry out a comprehensive analysis in three main areas: 1) uncertainty in supply deliveries and inconsistencies in timber grades; 2) lean, agile and hybrid manufacturing systems; and 3) planning and scheduling policies for cut-to-order lumber manufacturing. I will assess how these impact outputs, flexibility and SC value.  The first area of research will provide knowledge for understanding interactions between uncertainty in timber grades, manufacturing systems, and lumber manufacturing planning-scheduling procedures on SC outcomes. A DSS will be created that can be applied to different scenarios in order to test SC assumptions and conditions. The core methodology is an empirical case study approach, where a BC Coastal forest-to-14  lumber integrated company is the focal case study. Accordingly, the lumber planning case study will be modeled as a baseline, with alternative decision making policies and forest resources and lumber demand scenarios. The cut-to-order lumber planning environment is not realistic for the BC Coastal forest-to- lumber integrated industry yet, but it is important because it represents an opportunity to address emergent market scenarios. Therefore a Chilean lumber planning cut-to-order case will be analyzed for the last research chapter, highlighting implications which will be applicable to the BC Coastal forest-to-lumber Industry. Case study outcomes will be analyzed, and implications made for industry improvements by comparing manufacturing and economics supply chain metrics. The first study will evaluate the impacts of two levels of errors – timber grades and volumes – in the tactical decision-making processes. The first step will develop the necessary inventory data with its corresponding level of errors. Three errors will be calculated and used to hypothesize a fourth, which is currently unavailable. Once these four errors have been calculated, two inventory data sets containing volume and timber grades by stands will be developed; the first, with low error called “the hypothetical”, and the second with high error called the “current”. These data will then be used in a DSS to produce two SC plans. Two sets of decision outcomes will determine the impacts of levels of error on SC plans. The second study will provide knowledge to evaluate the economic and logistic convenience of adopting lean, agile or hybrid manufacturing principles when lumber product demand changes. Three manufacturing environments to simulate lean, agile, and hybrid manufacturing systems will be developed and compared. These manufacturing environments will highlight economic and logistics advantages and disadvantages from changes in lumber product demands. The third component of my thesis will evaluate benefits of solving the lumber planning problem with a planning-scheduling model that imposes heuristic sequences to process orders rather than the traditional lumber planning model. It will also evaluate the conditions of customer demands or orders under which cut-to-order lumber manufacturing approaches should be adopted. The traditional lumber manufacturing planning approach does not recognize the value of satisfying lumber customer orders on time and in volume and only penalizes over- and under-production in comparison with aggregate lumber product demands. This goal will be fulfilled by determining the value of due date fulfillment for production orders when solving the operational lumber manufacturing problem. Two lumber manufacturing planning 15  approaches and lumber demands and orders will test whether adding sequences of processing orders increases the efficiency of operational lumber manufacturing planning.   1.3 Research objectives  The general objectives of this study are to determine and understand the relationships between the chosen SC drivers and decision outcomes and to highlight their advantages and disadvantages. This goal will be achieved by the development of a comprehensive DSS that is appropriate to test multiple SC drivers and diverse economic conditions. The DSS will be a key tool to address the following specific objectives: 1. to determine the relative impact of forest inventory errors in volume and timber grades on the SC decision-making process;  2. to determine the best combination of agile, lean, and hybrid manufacturing systems for the BC coastal forest SC; 3. to determine the ability of a production planning and scheduling approach which uses sequences to process lumber production orders, in comparison with a cost minimization multi-period approach, to solves the operational lumber production planning problem. Thus, this study proposes the development of a DSS based on mixed integer programming, which will be able to determine an effective strategy under various case study assumptions. By using case studies to test different SC environments, the DSS will be able to calculate the value of timber grades in forest inventories for planning purposes, the appropriate use of agile and lean manufacturing systems for the forest SC, and the benefits of a cutting to order lumber manufacturing approach focused on meeting lumber orders due dates.   16  1.4 Thesis organization  The scientific contributions of this thesis are organized into three research chapters. This introduction, which includes the problem context, literature review, research gaps, and research objectives precedes the research chapters.  The research chapters are: 1. Chapter 2.  Impact of timber volume and grade estimation uncertainty on BC coastal SCs.  A multiple site, products and period linear program model is formulated to solve production planning for the BC coastal forest case study. Four error scenarios are analyzed based on the size of the sample used to estimates timber grade error, the size of the cut-block in use and grade-volume error assumptions. Solutions are evaluated based on four error scenarios and 12 lumber demand targets. A failing probability curve for fulfilling lumber demand is determined, as well as SC profit variations for error scenarios. 2. Chapter 3.  Exploring manufacturing systems for the BC coastal forest industry. In this chapter the ability of lean, and agile principles to improve SC performance in comparison to the current BC coastal forest SC is tested. Manufacturing drivers are translated into agile, lean, and BC-SC forest manufacturing environments (ME) based on MIP formulations to solve the forest-to-lumber SC production-planning problem. Additionally, ME formulations include the possibility of lumber interchanges between sawmills (Singer and Donoso 2007) and sawmilling manufacturing outsourcing (Babazadeh et al. 2012), which are common attributes of SCs.  I also explore capacity constraints for modelling a level manufacturing strategy within a lean ME, and a chase manufacturing strategy within the agile ME (Chopra and Meindl 2011). Two extra features are added to the ME formulations. The first applies penalties when lumber demand is exceeded or cannot be satisfied by sawmill production. Accordingly, over and under demand are heavily penalized by an Agile ME, moderately penalized by a BC-SC ME, and lightly penalized by a Lean ME. The second penalization is applied when capacity (i.e., time availability) is exceeded or is not completely used over a period by loggers, sort-yards, and sawmills. Accordingly, over- and under-production capacity is heavily penalized by a lean ME, moderately penalized by a BC-SC  ME, and lightly penalized by an agile ME. The objective is to determine how each ME performs on its own and when lumber demand changes.  17  3. Chapter 4.  The value of due date fulfilment for production orders when solving the operational lumber manufacturing problem. In this chapter I study the cost of scheduling sequences to process production orders when solving the lumber manufacturing planning problem. First, I built a multi-period MIP model where demands must be satisfied in a certain period of the planning horizon, without explicit orders and due dates. One version does not allow backlogs while a second version allows backlogs but does not penalize them. Second, I test manufacturing sequences to process orders that must meet their due dates.  Three manufacturing sequences are evaluated: 1) earliest due date; 2) shortest processing time; and 3) longest processing time. The periods of the first formulation were dropped and a “sequence” constraint was added that forced orders to be processed with a predetermined heuristic sequence. Again, one version was formulated without allowing overdue orders, and a second version allowed overdue orders, but without penalizing them. The multi-period models and planning-sequences models were fed with five sizes of demand, and five sizes of orders. Demand and orders were equivalent in volume and periods were equivalent with due dates. The objective was to test whether adding sequences of processing lumber production orders as Earliest Due Date, Shortest Processing Time or Longest Processing Time helped the efficiency of the MIP formulation in solving the operational lumber manufacturing planning problem.  In chapter 5, thesis conclusions are presented along with limitations and suggestion for future research. Finally, references and an appendix with model formulation details, and results are presented.   18  Chapter 2 Impact of timber volume and grade estimation uncertainty on BC Coastal supply chains  2.1 Summary  Timber supplies are particularly affected by volume and timber grade uncertainty in the current forest inventory. The performance of the BC Coastal timber industry can be improved by developing policies to manage the uncertainty in timber supplies. A multiple-site, multiple-product and multiple-period linear program was formulated to solve production planning for the BC coastal forest case study. Four error scenarios were analyzed based on the size of the sample used to estimate timber grade error, the size of the cut-block in use, and the volume/grade error assumptions. Solutions were evaluated based on four error scenarios and 12 lumber demand targets. A probability curve for fulfilling lumber demand was determined, as well as SC profit variations for the error scenarios. Profit variation was consistent with the magnitude of error applied. When using the normal cut block error scenario, the rate of change of average profit was 6.05% less than the perfect profit estimation. I also tested a 25% reduction in timber grade estimation, which reduced the profit variation to 1.43%. However, when using the super cut block (cut block >50ha) error scenario, the rate of change of average profit was 2.24% less than the profit determined by perfect estimation. On the other hand, I tested a 25% reduction on timber grade estimation, which reduced the profit variation to 0.48%. The ability to fulfill lumber demand was compromised by timber volume and grade error. For all error scenarios, the lumber demand target (i.e., low commodity product) should be reduced at least one quarter from the case study target to ensure that 80% of demand will be fulfilled. Naturally, the reductions were greater for higher error scenarios; however, if timber grade error can be reduced, the lumber demand targets could be as well (i.e., a less risky operation). This performance drawback is not extremely relevant for the U.S. housing market. However; in changing market conditions, it could compromise the industry’s ability to participate in other cut-to-order low value lumber markets with higher due dates and lumber demand fulfillments requirements (i.e. Japanese thin board market). While the proposed approach provides a straightforward tool to analyze uncertainty of model parameters, only timber volume and grade yields were analysed.    19  2.2 Introduction  The evolution to second-growth forests and changes in forest supply composition is triggering lower profitability for BC coastal forest operations. Considering this, the hypothesis of this study is that the poor performance of the BC coast forest SC is due, in part, to inefficient procedures for handling timber volume and grade uncertainties. There is a lack of accurate, long-term prediction models to estimate timber volumes and grades. The aim of this study is to determine the impact of forest inventory errors in timber volume and grade estimation, and to propose inventory policies to deal with this source of uncertainty. I used a hybrid approach that applied normally distributed random numbers to the historic bucking yields, and a combination of timber volume and grades estimation errors, which represent several levels of uncertainty, in the forest inventory data. I then formulated a DSS and ran the modified forest inventory databases to simulate how the timber and log grade supplies behave relative to the existing forest inventory. I first present a literature review, then the methods to estimate levels of error in forest inventory and create two forest databases with these errors. A linear programming DSS was formulated to model the forest BC coast SC, and feed data containing various levels of error for grades and volume. I then compare SC profit determined with different level of error impacts. Finally, I present the conclusions and limitations of this study, and suggest future lines of research.    2.3 Literature review  Long-term timber production analyses and uncertainty in timber inventory data rarely consider the impact of accuracy, and how accuracy improvement could lead to uncertainty reduction (Eid, 2000). Although decision makers are increasingly concerned with the effect of uncertainty in forest parameters on the quality of their decisions, forest planning problems have been solved mostly by deterministic formulations, where sensitivity analysis and scenario analysis are common. However, the ability to evaluate uncertainty is limited, given the lack of information about the probability of occurrences of each scenario (Beaudoin, 2006). As a consequence, this approach results in inferior planning decisions compared to methods that explicitly account for the uncertainty. A short analysis of the research carried out in this field follows, in 20  order to choose and support a suitable approach for the inherent uncertainty in the planning problem for the BC Coastal forest SC.  Stand variables are fed to growth and yield simulators to calculate forest volume. Inevitably, these variables are inaccurate because inventory is carried out using a probability sample design and is subject to sampling error. In practice, timber inventory data (TID) is used for short- and long-term decision making (Borders, 2008). As perfect estimation does not exist, there is a trade-off between improving decision making and the added cost of reducing error. Although all of these decisions need high quality information, the understanding of quality is not always clear. In general, the value of information (VOI) is defined as the difference between the project value with information and the project value without information (Kangas, 2010). The VOI has been evaluated by using cost-plus-loss analyses and recently by Bayesian decision theory. Methods of acquiring data for timber inventory are judged by accuracy and the cost of acquiring the data. Cost-plus-loss analysis calculates the expected losses due to suboptimal decisions made with inaccurate data and added to the cost of inventory. In particular, analytics and simulation methods are commonly applied in a cost-plus-loss analysis. The analytics approach is appropriate when losses can be expressed as a function of the accuracy. Uncertainty in TID could be estimated by expressing the value of information derived from the inventory in economic terms. Hamilton (1978) proposed cost-plus-loss analyses and suggests that the total cost should be minimized (cost of performing inventory and the expected loss resulting from future incorrect decisions due to uncertain timber inventory data). Similarly, Duveno and Lamas (2006) pointed out that the analytical approach is best used when optimal sampling and timing of inventory are the goals. The optimal inventory intensity is found by minimizing the expected value of the function. Since the analytical approach is exact, it becomes very hard to solve when more than one stand variable is considered in the analysis. Simulation studies are an efficient way to compare forest management plans made based on TID with error. Real forest data containing observed errors or simulated errors can be used. In the first case; the validity is limited by the accuracy of observed forest data. In the second case, the effects or errors can be analyzed in all inventory variables, but the probability distribution of error should be known in advance (usually assumed as normal). In both cases, errors in TID variables could be correlated or uncorrelated, and systematically or randomly applied. 21  In general, estimating loss given uncertainty has been dealt with by imitating random errors through simulations and Bayesian frameworks, where TID variables have been treated as a probability distribution (Eid 2000). In particular, TID uncertainty in forest planning can be analyzed by studying final harvest timing and net present value (NPV) changes as a consequence of incorrect timing. Results show different levels of impact produced by TID variables on the NPV losses, at different levels of errors. Jointly all these effects heavily influence decisions regarding the timing of final harvests, and NPV losses may become large. For instance, Borders et al. (2008) showed that expected NPV loss increased with sampling error, and as the discount rate increased, the expected NVP loss decreased. Management plans with stand-level data that had a sampling error from 5% to 25%, and a discount rate of 6%, experienced expected losses in NPV from 20 to 170 US$/ha. Also, Beaudoin, 2006 combined a MIP deterministic model and scenarios by randomly generating values for the uncertain parameter. For each scenario, the optimal plan was determined by solving a MIP problem, and analyzed in terms of operational feasibility. Using a case study, profitability increased 9% compared with an approach based on a deterministic MIP model using average forest parameter values. Furthermore, Peter and Nelson (2005) incorporated fire disturbance into a forest scheduling model in order to estimate variations in forest management indicators. The fire disturbance was modeled by considering probability functions and the historic occurrence of these phenomena, and then linked period by period with the forest scheduler simulator. Subsequently, the risk associated with these indicators was assessed by establishing the relationship between indicator rates (e.g. annual allowable cut) and the associated risk that those rates would not be satisfied.  Approaches to incorporate uncertainty depend on how to represent the uncertain parameter. Scenario-based and distribution-based methodologies can be used to represent uncertainty. In scenario-based approaches, the description is done by scenarios that capture how the uncertainty might play out in the future. Each scenario has a probability that represents the decision-makers expectation of the occurrence of this scenario. The drawback of the scenario-based approach is that it needs forecasts for all possible outcomes of the uncertain variable. The distribution-based method is appropriate when scenarios are not clear and only a continuous range of potential futures can be predicted. In this case, allocating a probability distribution to the potential range of outcomes makes sense. However, such formulations increase the problem size drastically (Gupta, 2003).  Once the representation of the uncertain parameter is decided, two-stage stochastic programming is widely used. Decisions and constraints are sorted into two sets. The first-stage variables are design variables (i.e., 22  efficient allocation of production capacity), which are determined before the resolution of the uncertainty. The second-stage variables (i.e., demand satisfaction and inventory management) are determined to optimize the results of the first-stage variables in the face of uncertainty (Dantzig, 1955). The presence of uncertainty is modeled by the fact that second-stage decisions are probabilistic in nature. Thus, a production-planning model under demand uncertainty can be solved using a two-stage stochastic formulation, assuming manufacturing variables as first-stage here-and-now decisions, and logistics as second-stage wait-and-see decisions. The objective function is divided into two terms. The first captures manufacturing cost and the second captures logistics cost, determined by using the average cost obtained from optimizations of the embedded problem (Gupta, 2003). However, the central question for decision making under uncertainty is how to assess the sensitivity of the optimal solution to variations in the uncertain parameter. It is impossible, under uncertainty, to find a solution that is ideal in all circumstances. In stochastic formulations, decisions should be balanced with the various scenarios. Thus, the value of stochastic programming over a similar deterministic model is based on the quality and availability of data from uncertain parameters prior to decision making. If the decision-maker has perfect information (a high quality forecast) for the uncertain parameter, he/she can always make the optimal decision in advance. When the decision-maker does not have perfect information (a poor forecast) for the uncertain parameter, he/she may solve the problem by using stochastic programming. In the long run, the difference between these solutions represents the value of decisions made with perfect information, which is called the expected value of perfect information (EVPI). If the decision-maker assumes the expected value for the uncertain parameter, and always allocates the optimal solution based on this value, then after evaluating the results, the value is called the expected value solution (EVS). This approach is common in optimization; however, it has unfavorable consequences, such as under-estimating the problem solution.  Finally, the loss caused by not considering the random or cyclical variation of the uncertain parameter must be quantified. The difference between the evaluated solution produced with expected uncertain parameters, and the value of the stochastic solution, is called the value of stochastic solution (VSS) (Birge and Louveaeux, 2011).  23  2.4 Methods   The first step was to develop a DSS to solve the production planning problem for the case study. The DSS was formulated keeping in mind an integrated BC Coastal forest SC, where harvesting, sort yard, and sawmilling operations do not always happen simultaneously (Haartveit et al. 2004). In this SC lumber manufacturing operations are planned individually, and sales are company centralized. I chose a planning horizon of 1 year, which was divided into 4 periods (4 quarters). The DSS took the form of a linear programming profit-maximization model, which is extensively described in Appendix A, section A1.  The timber volume and grade errors were calculated using a sample of stands from British Columbia Timber Sales (BCTS). Volume and grade estimates were taken from a standard and intensive inventory and compilation method called a timber cruise. Additionally, real volume and grade data were provided by scaled volumes and log grades from the same BCTS stands provided by the Ministry of Forests, Lands and Natural Resources Operations billing system. This sample provided estimates and true scales of volumes and grades to determine errors in those estimates. However, as the size of the sample area was approximately 500 ha, the errors determined by this sample were too limited to be useful. Thus, only the determined timber grade error was considered as an acceptable representation for the purposes of this study. Timber volume estimation error from cruising in BC coastal forests is set by standard to be within 15% of true volume1.                                                             1 British Columbia cruising manual, Ministry of Forest, Land and Natural Resources Operations   24  Table 2.1 Error scenarios Error scenario Description i. Ze: Normal size cut block non error scenario. Zero error in timber volume and grade is considered a “non-error scenario”:  ii. Ne:Normal size cut blocks errors.  A timber volume estimation error of 15%, and timber grade estimation error of 60% [Western Hemlock], 60% [Douglas fir] and 60% [Western red cedar]:  iii. Nne: Normal cut blocks nirvana errors.  A timber volume estimation error of 15%, and a 25% reduction in timber grade estimation error, which is 45% [Western Hemlock], 45% [Douglas fir] and 45% [Western red cedar] iv. Se: Super size cut blocks errors. A timber volume estimation error of 5% and timber grade estimation error of 20% [Western Hemlock], 20% [Douglas fir] and 20% [Western red cedar]   v. Sne Super cut blocks nirvana errors. A timber volume estimation error of 5%, and a 25% reduction in timber grade estimation error, which is 15% [Western Hemlock], 15% [Douglas fir] and 15% [Western red cedar]. A combination of errors was applied to the forest data in order to explore SC decision outcomes (Table 2.1). The error level scenarios were chosen based on the results of timber grade errors (e.i., ±60%), and the timber volume estimation error ( ±15%). Also, I assumed a common error for the three log grades families (i.e., timber, sawlogs, and utility), differentiated by species. SC decisions involve both errors, and the available forest database contains stands as big as 225 ha, which are three times bigger than a normal cut block at these harvesting operations (Work et al., 2003). Since BCTS data was based on normal size cut blocks, both errors were reduced by a factor of three to make them comparable (e.g., see Scenarios iv and v).  For error Scenarios ii and iii, the error reduction based on the size of the cut block was not considered in order to simulate a dramatic error situation. However, the third and fifth scenarios were designed keeping in mind the progress of non-destructive evaluation techniques (NDE), where visual grading of timber is perhaps the easiest technique (Wang, 2006). During visual grading, the ratio of clear fiber is established by a grader’s judgement. In these scenarios, I assumed that this ratio could be improved by 25% with the use of the new NDE techniques. Thus, Scenario iii considered a timber grade error of 45% (e.g., 60% reduced by 25%), and Scenario v considered a timber volume error of 15% (i.e. 20% reduced by 25%). Finally, Scenario i was a non-error scenario; the DSS was run with expected volume and grade yields and those 25  decisions were used as a benchmark. The benchmark decisions were not subject to any verification because they did not contain error.  These scenarios were created by simulating forest data with errors from a set of forest stands used as a case study. The case study forest database is a subset containing 30 stands of comprising 225 ha chosen from 4500 stands on Vancouver Island (in TFLs 6, 19, 37, 39, 44, 46 and 47). The forest database was developed by Vahid (2012), using a dispersal algorithm (Schwab and Maness, 2009) to rebuild spatial forest inventories from aggregated data. The volume, by stands, was projected with the stand growth model Variable Density Yield Prediction (VDYP 7), based on stand variables contained in the forest inventory. Historic timber grade distributions came from the Campbell River BCTS data and were assumed representative for this case study. Once the errors and the case study database were determined, the DSS was run for 12 lumber demand targets. Subsequently, I assumed that the uncertainty in grades could be applied at the log bucking stage because it is here that these errors are first realized. Thus, the uncertainty was applied in two steps: first on timber volume, and second on timber grade inventory through the log grades distribution during the bucking operations. The stand volumes in the database were modified by Equation [2.1] to simulate the effect of uncertainty when errors in timber volume inventory exist. The log grade distribution during the bucking operations was modified by Equation [2.2] to simulate the effect of uncertainty when errors in timber grades exist. Each new generated data point became part of a specific error scenario database where the original data were modified by adding a random number multiplied by its level of error.  Yield_Stand 𝑖 𝑘 e = 𝑌𝑖𝑒𝑙𝑑_stand  𝑖[1  + 𝑅𝑁𝑘 × 𝑒𝑡𝑣] ……… [2.1]  Yield_Log   𝑝 𝑛 𝑒 = Yield_log𝑝𝑛[1 + 𝑅𝑁𝑚 × 𝑒𝑙𝑔] ………… . [2.2]  where: Yield_stand i k e  : Modified yield of stand “i”, iteration k, by error “e” in m3/ha. Yield _stand  i 0 : Unmodified yield of stand “i” in m3/ha. (i.e. calculated with stand growth model VDYP 7) RNk and RN m : “k” and “m” normally distributed random numbers (generated between -1 and 1, using an   average of 0, and a standard deviation of 0.33)  26  e tv    : Timber volume inventory error of estimation in %  k : Number of generated stand volumes with error, with k=1 to 30.  Yield_Log p n e : Modified log yield in species “p”, log grade “n”, by error e, in %  Yield_log p n  : Historic log yield in specie “p”, grade “n” in % (taken from Vahid, 2012) M  : Number of generated log grades yield (bucking policies) with error, with m=1 to 30. e l g : Log grade error of estimation in % (calculated from BCTS sample) Solutions from the non-error optimal solution were evaluated against an Excel matrix that contained the modified inventory database. The database was modified for each error scenario with 30 cuts block (stands) modified by equations 2.1. and 2.2. Subsequently, 12 lumber demand targets were verified in the Excel matrix to explore SC behaviors against uncertainty, representing variations of the original lumber demand. Thus, the DSS was run 12 times, and the optimal solutions were evaluated 30 times per scenario. The outcomes of the DSS and the evaluated decisions determined a probability curve for fulfilling lumber demand at different levels. Next, the demand target was processed through the DSS and repeatedly verified in order to determine the magnitude of change based on error scenarios.  The whole model (objective function and constraints) were transferred to an Excel matrix (i.e. verification matrix), which re-assessed the solution using the error yield changes. The verification was made 30 times for each error scenario and lumber target. The failure probability was determined by dividing the number of times that the lumber demand was not fulfilled by the number of verifications performed. This exercise was performed on “low commodity value” lumber products (59% of the lumber demand), in all species, and annual quarters of the planning horizon. However, other decision variables or indicators can be used. Subsequently, observed probabilities versus lumber demand targets were treated with a logistic regression approach (see Appendix A, section A.2 for details).  This goal of this research was to establish a policy to deal with uncertainty produced by timber inventory errors. In order to perform this analysis, I considered that customers could tolerate at most 20% of unfulfilled demand for lumber commodity products. Using the previous function I determined the targets for 27  “low commodity value demand” by error scenario (which ensured at most 20% of un-fulfilled demand). The following results were produced: i. For the  Ne scenario, the target should be reduced to 595,000 m3 (Fig 2.3, 1st interception line) ii. For the Nne scenario, the target should be reduced to  591,500 m3 (Fig 2.3, 2nd interception line), iii. For the Se and Sne scenarios, the targets should be reduced to 615,500 m3 (Fig 2.3, 3rd interception line). Finally the EVS was determined only for one demand target (i.e., the case study lumber demand). For every error scenario, this value corresponds to the difference between the SC profits obtained by the verification method and the DSS solution based on the same timber volume and grade yields used for verifications.  2.5 Results  2.5.1 Profit analysis  The variation in SC profits corresponds to a proportional variation in the error scenarios for the different demand levels. For instance Figure 2.1 (left) shows profits with Ne error, and Figure 2.1 (right) shows profits with Nne error. Figure 2.2 (left) shows profits with Se error, and Figure 2.2 (right) shows profits with Sne error.  28     Figure 2.1 Profit variations for normal cut block error, and normal cut block nirvana error   Figure 2.2 Profit variation for super cut block error and super cut block nirvana error Lumber demand levels ranged from 928,000 m3 to 1,491,000 m3 per year. However, for simplification, only the average, maximum and minimum profits, and standard deviation values are shown in Table 2.2. It is important to note that the spread around the average becomes wider as the scenario errors increase.     $23,000$24,000$25,000$26,000$27,000$28,000$29,000$30,000$31,000$32,000$33,000$34,000$35,000$36,000$37,0001406 1322 1238 1153 1125 1097 1069 1041 1013 985 956 928ThousandsLumber demand *1000 m 3Profit  variation for Ne scenario[normal cut block error]Quartile 1 Min Average Max Quartile 3 Ze$23,000$24,000$25,000$26,000$27,000$28,000$29,000$30,000$31,000$32,000$33,000$34,000$35,000$36,000$37,0001406 1322 1238 1153 1125 1097 1069 1041 1013 985 956 928ThousandsLumber demand *1000 m 3Profit  variation for Nne scenario                                             [normal cut block nirvana error]Quartile 1 Min Average Max Quartile 3 Ze$23,000$24,000$25,000$26,000$27,000$28,000$29,000$30,000$31,000$32,000$33,000$34,000$35,000$36,000$37,0001406 1322 1238 1153 1125 1097 1069 1041 1013 985 956 928ThousandsLumber demand *1000 m 3Profit  variation for Se scenario   [super cut block error]Quartile 1 Min Average Max Quartile 3 Ze$23,000$24,000$25,000$26,000$ 7,000$28,000$29,000$30, 00$31, 00$32, 00$33,000$34,000$35,000$36,000$37,0001406 1322 1238 1153 1125 1097 1069 1041 1013 985 956 928ThousandsLu ber demand *1000 m 3Pro it variation for Sne scenario   [super cut block nirvana error]Quartile 1 in Averag Max Quartile 3 Ze29  Table 2.2 Profit statistics by demand level and error scenario  2.5.2 Risk analysis  In order to assess the effect of the uncertainty produced by estimation errors, a probability of failure curve for fulfilling lumber demand was built. The DSS was run for several lumber demand targets. Later, the decisions were verified by using an Excel worksheet developed to evaluate the optimal solutions to these changing yields. The failure probability was established when verifying the decision against yield changes, assuming that as targets are reduced, the frequencies of shortfall will decrease (Peter and Nelson, 2005).   Error Statistics Lumber demand *1000 m3                   Scenario   1,406  1,322  1,238  1,153  1,125  1,097  1,069  1,041  1,013  985  956  928  Ze  *1000$ 24,468  26,960  29,361  31,645  32,389  33,133  33,872  34,601  35,326  35,966  36,594  36,949    ave. 24,217 26,604 29,097 31,368 32,092 32,852 33,569 34,385 35,107 35,734 36,340 36,737 Sne min. 24,502 26,834 29,305 31,579 32,329 33,036 33,787 34,556 35,263 35,892 36,522 36,915   max 23,843 26,213 28,810 31,131 31,847 32,495 33,207 34,236 34,902 35,216 36,082 36,450   stdev 120 161 120 124 112 143 142 95 90 147 113 106   ave. 24,097 26,580 28,989 31,249 32,035 32,773 33,460 34,255 35,023 35,643 36,292 36,621 Se min. 23,711 23,711 28,667 30,869 31,742 32,361 32,703 33,950 34,279 35,285 36,044 36,226   max 24,371 24,371 29,301 31,582 32,308 33,046 33,795 34,506 35,296 35,882 36,515 36,853   stdev 195 195 147 148 148 197 225 150 206 167 123 162   ave. 23,240 26,024 28,431 30,351 31,343 31,845 32,761 33,478 34,364 34,961 35,455 35,921 Ne min. 22,242 24,947 27,303 29,256 30,401 30,874 31,661 31,712 33,564 33,555 34,405 34,190   max 23,914 26,798 29,148 31,289 32,101 32,806 33,776 34,479 34,942 35,682 36,235 36,558   stdev 475 378 401 543 435 490 555 625 370 571 449 531   ave. 23,568 25,799 28,224 30,882 31,625 32,184 33,007 33,817 34,605 35,179 35,931 36,142 Nne min. 22,311 24,714 27,142 30,214 30,803 31,019 32,001 33,205 33,822 34,034 35,248 34,974   max 24,497 26,351 29,123 31,377 32,188 32,926 33,801 34,329 35,162 35,677 36,397 36,777   stdev 440 422 483 277 359 467 421 350 326 395 284 471  30  Table 2.3 Case study lumber products targets and expected target reduction given error scenarios  Case study Suggested reduction on lumber demand to ensure at most 20% of failure in fulfillment Scenarios Lumber product Demand [m3]     [%] Ne Reduction [%] Nne Reduction [%] Sne Reduction [%] Se Reduction [%] Specialty lumber 191,717 m3          14% 138,608 m3      -27.7% 137,793 m3     -28.8%    143,384 m3 -25.2% 143,384 m3 -25.3% High value commodity 391,797 m3         28% 283,262  m3     -27.7% 281,596 m3       -28.8% 293,022 m3-25.2% 293,022 m3 -25.2% Low value commodity 822,980 m3         59% 595,000  m3    -27.7% 591,500 m3       -28.1 615,500 m3-25.2% 615,500 m3 -25.3% Total lumber demand 1,406,494 m3 1,016,870 m3 1,010,889 m3 1,051,905 m3 1,051.905 m3   Figure 2.3 Predicted failure of lumber demand fulfilment  The ability to fulfill lumber demand for low commodity value lumber products was compromised by timber volume and grade error. For a normal cut block error scenario, the lumber demand target should be reduced from 822,980 to 595,000 m3 (27.7% reduction). This ensures that the demand will be fulfilled 80% of the time (Table 2.3). On the other hand, if timber grade error can be reduced by 25% by improving NDE 0%5%10%15%20%25%30%35%40%45%50%55%60%65%70%75%80%85%90%95%100%500 550 600 650 700 750 800 850 900Probability of failure Lumber target *1000 m3 Ne:15%-60% Se:5%-20% Sne:5%-15% Nne:15%-45%31  techniques (i.e., the Nne error scenario), then the effects on target reduction did not contribute to reducing the risk; in fact risk was increased by 1%. For the Se Scenario the lumber demand target should be reduced from 822,982 to 615,500 m3 (25.2% reduction) to ensure that 80% of the time the demand will be fulfilled for low value commodity products. The Sne Scenario for the super cut block did not contribute to reducing the risk.  2.5.3 Value of information  Another impact of timber inventory estimation errors can be observed in profit variations. For this purpose, 30 verifications were done for a target of 1,406,000 m3, which is the case study lumber demand. Also, 30 DSS runs were completed with timber volume and grade yields affected by error scenarios to represent when the decision maker assumes the perfect estimation of timber volume and grades yield before they are realized. The profit averages of the verifications minus the profit averages determined by the optimal solutions helped determine the expected value solution (EVS), which was used as baseline. Thus, there are four options for the EVS value one for each error Scenario.  As was expected the verified profits were always lower than the profits based on the perfect estimation. The level of these differences was proportional to the level of error used (Figures 2.4, left and right).  Figure 2.4 Profit variations based on error scenarios Ne (left) and Nne (right) for target 1,406,000 m3 32    Figure 2.5 Profit variations based on error scenario Se and Sne for target 1,406,000 m3 Table 2.4 Profit variation statistics based on error scenarios for target 1,406,000 m3  The trends in profit variation were consistent with the magnitude of the error applied; however, the magnitudes of variation were not as great as expected. Profits determined by verification were smaller than the profits determined by the DSS with perfect information. This behavior can be explained by how the verification matrix works in comparison with the optimized solution offered by the DSS. The DSS determined an optimal production plan for every run (PI). However, the verification matrix determined a production plan when timber volume and grade yield increased or decreased randomly by the error Profit analysis for 1,406,000 m3 target Error scenarios: vol-grades Average [Av] Std. Dev.[Sd] Variation wrt PI Ne :15%-60% 23,239,624$               475,185$                      -6.05%PI: Ne 24,736,537$               164,733$                      EVPI:Ne 1,496,913$                 Nne :15%-45% 23,568,177$               439,627$                      -4.62%PI: Nne 24,709,593$               138,914$                      EVPI:Nne 1,141,416$                 :5%-20% 24,097,484$               194,639$                      -2.24%PI: Se 24,650,501$               42,568$                         EVPI:Sne 553,017$                     Sne:5%-15% 24,216,555$               120,220$                      -1.76%PI: Sne 24,650,238$               38,947$                         EVPI:Sne 433,683$                     33  scenario (based on DSS allocated volumes). Also, verification happens if the decision-maker assumes the expected value for the uncertain parameter and always allocates the optimal solution based on this value (i.e., evaluating these results after realization). Thus, the verification method represents an empirical solution, which should never be as good as the optimal solution. The value of information (VOI) in decision making can be established by comparing the value of decisions made with high and low quality information (Kangas, 2010). However, here, VOI was established by determining the how profits were affected by errors. Therefore, when using the Ne Scenario, the rate of change of average profit compared to perfect information profit was 6.05% less than the perfect estimation profit. On the other hand, I tested the assumption that NDE techniques could reduce timber grade estimation error by 25%, represented by the Nne scenario. The change of average profit was 4.62 % less than the perfect estimation profit, which represents a reduction of 1.43%. However, when using the Se Scenario, the average profit compared to perfect information profit was 2.24% less than the perfect estimation profit. Under the Sne Scenario, the average profit was 1.76 % less than the perfect estimation profit, which represents a reduction 0.48%.  2.6 Discussion  Errors in timber volume and grade volume estimation impact SC profit and the failure probability of fulfilling lumber demands. The ability to satisfy lumber demand during the planning horizon was compromised by errors in timber volume and grade estimation. Only the demand fulfilment of one of the lumber products was analyzed. Nevertheless, a proportional impact of the error scenarios on SC profit was observed when using the solution verification method and the DSS solution with perfect estimations. Furthermore, considering the limited size of the sample used to determine timber grade estimation error and the error reduction applied based on the cut block size, the impact on profits in reality could be higher than those reported in this research.  Because of its empirical nature, the approach to verify solutions given uncertainty could affect the magnitude of variation in my results. It is intended to simulate a realistic version of the decision making process, where decision makers do not have resources to run DSSs every time that unexpected changes 34  happen during the planning process. My results are in agreement with Birge and Louveaeux, (2011), who mentioned that evaluating the optimal solution after realization has always unfavorable consequences compared to decisions taken with perfect information for the uncertain parameter prior to realization. This situation is common when the decision-maker assumes an expected value for the uncertain parameter and allocates the optimal solution based on this value. Deterministic formulations do not consider uncertainty on any model parameters (Gupta 2003). Scenario-based analyses must explicitly identify the probability of occurrence of each scenario and a distribution-based method (DB) assigns a probability distribution to the range of potential outcomes. However, problems should be split and separately solved, when using DB, to make them treatable; thus, it turns into a high memory and resource-demanding approach. In my approach, the probabilistic nature of timber volume and grade yield was solved by increasing and reducing yields randomly, with a known magnitude. While solving a normal sized SC production planning LP problem, instead of running an extensive DSS with thousands of different yields, I evaluated the first DSS solution several times based on yield changes. This avoided the excessive use of resources, but with the limitation that is an empirical solution and cannot guarantee high quality solutions. Related studies compared decisions made with and without uncertainty and often determined the risk of operations with uncertainty (Gupta, 2003). In this research, risk analysis helped to establish lumber production targets for four error scenarios that ensured a high level of products services (i.e. 80%). Alternatively, Borders (2008) evaluated the effect of VOI on DSS as a net present value loss, with regard to long-term harvesting scheduling problems. However, as my planning horizon was shorter, my losses were based on profits with different levels of errors and without rates of interest, which may have made my results less dramatic.   One value for timber volume error was used, and one value for timber grade error by species was used for all cut blocks. However, this assumption is naive for old-growth forest cut blocks, and both error values should differ from cut block to cut block. Thus, a dynamic errors framework could be considered as further analysis in order to expand on my approach; however, it would make the solution method much more complex and time-consuming. Different targets for lumber production were used to explore demand fulfillment. However, an unexpected DSS behaviour was observed when the lumber target decreased, because profit increased. I formulated a 35  DSS to maximize profits. Sawmills produced as much as possible of high value specialties first (highest price), then high commodity lumber (medium price), and finally low commodity lumber (lowest price). However, as I used a market constraint that allowed the lumber production of sawmills to go over and under the market demand, sawmills over-produced high and medium price lumber, and fell short in low price lumber.  The profit was penalized with an overproduction fee (half of lumber products’ price), and an underproduction fee (lumber products’ price). Thus, when lumber targets decreased, the DSS reduced lands harvested, and logs transported, reducing total costs. Also, sawmills reduced lumber production, which reduced underproduction, and its penalization. Sawmills did not decrease production at the same rate as harvesting, which kept lumber sales high, and as consequence, pushed profits higher. Further research on this unexpected behavior is needed.  2.7 Conclusion   The approach developed in this research was used to determine the impact of timber volume-grade estimation errors on SC production planning, using the BC coastal integrated forest industry as a case study. Particularly, the risk of operating the SC was analysed with the probability of failing to fulfill lumber demand during the planning horizon, and exploring rates of change on profits.  The SC profit was negatively and proportionally affected by timber volume and grade estimation errors. Also, the ability to fulfill lumber demand was compromised by timber volume and grade error magnitudes. This fact suggests that decision makers should be aware that demand fulfillment will be compromised depending on the timber volume and grade estimation errors in use for low commodity lumber products, which for this case study represent 29% of the lumber demand. However, this SC performance drawback is not relevant for the current low commodity lumber products market orientation (i.e., the U.S. housing market). Nevertheless, it could compromise the BC coastal industry’s ability to efficiently participate in other cut-to-order low value lumber markets with tighter due dates and lumber demand fulfillment requirements (i.e., the Japanese thin board market).  36  Using normal cut block-size errors (Ne scenario), the SC annual profit was under-achieved by 6.02%. However, if the timber grade error is reduced by 25%, SC annual profit would be under-achieved by only 4.62%. Second, when using super cut block-size errors (Se scenario), the SC annual profit was under-achieved by 2.24%. If only the timber grade error was reduced by 25%, SC annual revenues were under-achieved by only 1.76%.  My results are limited by the size of the sample used to estimate timber grade error, the value of the timber volume estimation error, and using these error values for all cut blocks. Another limitation is the verification method (Excel matrix), given its empirical nature in comparison with stochastic programming. A dynamic error framework for different timber stands and natural combinations, mixed with a stochastic programming formulation of the case study, will help to evaluate changes on forest combinations and their economic impacts.   37  Chapter 3 Exploring manufacturing environments for the BC coastal forest industry  3.1 Summary  This research tested the ability of lean and agile manufacturing principles to improve SC performance in the BC coastal integrated forest industry. Mixed integer programming formulations of agile, lean and hybrid manufacturing environments were used to solve the forest-to-lumber SC production-planning problem. The objective was to determine how these manufacturing environments perform when lumber demand changes. My results show that the manufacturing environment that should be adopted depends on the attributes of lumber demand. First, when lumber demand is stable, with low variation and large batch volumes, agile or lean principles should be adopted, and when lumber demand is unstable with high variation and large batch volumes, hybrid or agile principles should be adopted. Second, when lumber demand is unstable, with high variation and small batch volumes, an agile environment result in higher profits than lean or hybrid environments. Third, for unstable lumber demand with low variation, an agile environment should be adopted, although the profit differences between manufacturing environments are not as large in the previous cases. Further analysis is required because British Columbia SC theoretically should be largely superior for this kind of lumber demand condition.  Opportunities for profit improvement were 11.1% for adopting agile principles when lumber demand is stable, with low variation and large volumes. However, profit improvement was non-existent when the same demand attributes apply, but with high variation. The opportunities for profit improvement were 12.1% when agile or lean principles are adopted when lumber demand is stable with low variation and small volumes. However, opportunities for profit improvements of 15.5% existed when lumber demand was unstable with high variation and small volumes. These profit increments were subject to the over-under production capacity, and over-under demand fulfillment penalties used, my lumber demand assumptions, and the exploratory nature of this work. Opportunities for future research include introducing statistical analysis, as well as adding mixed lumber product portfolios to the lumber demand scenarios. Additionally, the postponement concept can be included in the mixed integer programming formulation to explore lead time inventory reduction. In the lumber SC, postponement means postponing the last lumber manufacturing step (i.e., grading or final sizing) beyond 38  sawmills. For example, lumber matrices (i.e., double width, and/or double thickness lumber pieces) receive a light manufacturing to quickly produce end-products and satisfy changing lumber orders in a distribution center close to markets.  3.2 Introduction   The competitiveness of the BC coast forest industry is limited due to both internal and external changes made over the last decade (Pearse 2000). High production costs are offset somewhat but not entirely by the relatively high value of manufactured products. On the other hand, SC attributes, such as long length and complexity (Maness 2008), unsuitable log allocation methods (FP Innovations 2008), and volume maximization focuses (Haartveit et al. 2004), fail to meet client needs and inventories can grow without control (Gaudrealts et al, 2009). The majority of research in this field to date has focused on profit maximization, using the massive make-to-inventory and make-to-order manufacturing systems for commodities and custom cut lumber, respectively. Until now, these systems have worked well for sawmills that efficiently and profitably handle the current combination of products, markets and supplies. However, given current market and forest supply changes, previous assumptions are not adequate for the near future and should be the subject of further research. The decision to adopt a different manufacturing system is hard for both employees and management. It means changes to raw materials planning, plant layouts, flows and scheduling. Thus, it is difficult to predict the magnitude of the benefits gained by implementing lean-agile-hybrid principles (Detty and Jon 2000).  This research conducts an analysis of lean-agile-hybrid manufacturing systems, assesses their impacts, and recommends the best combination of them for a case study. A DSS is developed to model agile and lean manufacturing environments and to evaluate them against a hybrid environment assumed as a baseline scenario (i.e., the current BC coastal forest industry). The goal of this chapter is to explore the relationships between lean-agile-hybrid drivers and decision outcomes and to highlight their benefits and tradeoffs. A comprehensive DSS tests multiple SC drivers, lean-agile-hybrid drivers, and market scenarios for the BC Coastal integrated forest SC. Performance measurements are based on the use of resources 39  (i.e., efficiency), SC output (i.e., customer service), and flexibility (i.e., the ability to respond to a changing environment) (Beamon, 1999). This chapter first provides descriptions of the essential drivers of manufacturing procedures. Previous research on this problem and a detailed case-study are given. Then, a methodology to model manufacturing principles is explained, followed by the results of the assessment of the performance of the manufacturing environment. The final sections contain discussion and conclusions, where highlights and conclusions on selecting manufacturing principles are explained.  3.3 Literature review   The BC forest industry is very conservative in terms of technology innovation and typically implements technology previously proven in other fields. For instance: 1) regular creation, and rapid replenishment of small batches of new goods adopted by Spanish retailer Zara (Harvard Business Review 2004), and 2) the elimination of waste from the Toyota automaker manufacturing system (Goldsby et al. 2006) are SC manufacturing paradigms in use in other fields that have not been adopted in the forest industry. Certainly, the forest industry is divergent and uncertainty in raw material supplies does not make innovation easy. This section describes SC manufacturing strategies and what drives them.  3.3.1 Conceptual supply chain manufacturing strategies  Generally, the goal of SC management is to find the best balance between responsiveness and efficiency, given company product portfolios and customer behavior. Thus, is it essential to understand how SC drivers and their trade-offs impact SC performance. For instance, when managers increase inventory, making the SC more responsive, such actions come at a cost, which is reduced efficiency. In terms of transportation, the speed by which supplies and products are transported strongly impacts SC costs, responsiveness, and efficiency. Facilities are another key driver, as their locations and capacity decisions impact SC costs as well as manufacturing methods. A products-oriented factory can be efficient but rigid, and flexible factories can handle more variety of products with less efficiency. SC managers should decide 40  how flexible and/or dedicated they want capacity to match product portfolios and, thus, manufacturing lines. All SC driver decisions involve immediate changes in efficiency and responsiveness; however, all of them add SC costs, and hence should be carefully balanced (Chopra and Meindl 2011). A company’s competitive strategy defines the customer needs that it seeks to satisfy through its products and services; in particular the ways in which customers prioritize product cost or response time, product variety and quality. Hence, the SC strategy determines the nature of the supplies and their transportation, manufacturing, and distribution, which can only be achieved with high customer and SC understanding (Chopra and Meindl 2011). Understanding which market segment is being served determines the quantity of products available, the response time, variety, service level, price, and level of innovation in the product. Each segment tends to have similar needs; although there are many needs, they can be grouped under the demand uncertainty concept. The level of uncertainty in product demand is correlated with product margin, product life cycle, errors in demand forecasts, stock out rates, and markdown rate attributes. The higher the demand uncertainty is, the larger the attributes (Fisher 1997). SC understanding means understanding how to meet demand needs, which implies a balance between efficiency and responsiveness. A SC responding to a high variety of products with short lead times and innovative products can be called responsive. Nevertheless, most of these SC attributes come at a high cost and decrease efficiency (Chopra and Meindl 2011). Thus, strategic fit is achieved when the SC responsiveness is consistent with the implied demand uncertainty (Table 3.1).  41  Table 3.1 Efficient versus Responsive SC attributes Strategy Efficient SC Responsive SC Primary goal Supply demand at the lowest cost Respond quickly to demand Products design Maximize performance at minimum product cost Create modularity to allow postponement product differentiation Pricing  Lower margin because price is a prime customer driver Higher margin, as price is not a prime customer driver Manufacturing  Lower cost through high utilization Maintain capacity flexibility to meet unexpected demand Inventory Minimize inventory to lower cost Maintain buffer inventory to meet unexpected demand Lead times Reduce but not at expense of cost Aggressively reduce even if the cost is significant Transportation Based on cost and quality Based on speed, flexibility and quality  The SC should be modified to handle demand uncertainties; for instance, manufacturing or transportation strategies are customized to handle different product levels of responsiveness (Chopra and Meindl 2011). In practice, all drivers are traded at the aggregate planning level to determine levels of capacity, production, inventory, and backorders. Depending on how flexible and costly varying capacities are, inventories and backorders can be used as levelers. As a result, chase, time flexibility and level strategies are three strategies used to balance costs. The chase strategy, in which production rates are synchronized with demand rates by varying machine capacities or the workforce, is expensive and should be adopted only when the cost of carrying inventory is prohibitive. The time flexibility strategy is suitable when there is an excess of machine capacity; thus, the workforce is kept stable but varies over time. Time flexibility leads to low levels of inventory and capacity use. In the level strategy, stable machine capacities and workforces are maintained to produce at constant production rates. This leads to inventory and backorders growth, which is suitable when the costs of carrying inventories and backorders are low (Chopra and Meindl 2011).  42  3.3.2 Conceptual framework for Lean and Agile manufacturing strategies  Fisher (1997) noted that the patterns of demanded products determine whether products are functional or innovative. Functional products satisfy basic needs; they do not change much and they present a stable and predictable demand because they have long life cycles. Innovative products have short life cycles, great variety and unpredictable demand. For functional products, market operations are easy and cheap (i.e., there is a close match between supply and demand). Thus, the focus is on minimization of costs, and manufacturing schedules are developed in advance to minimize inventory and maximize efficiency. For innovative products, uncertain market reactions to innovations increase mismatched supplies and shortages. Thus, market operation costs dominate innovative products, so market signals and quick reactions determine inventory and capacity decisions. Here, the SC focus is on where in the chain to hold inventory and available production capacities (i.e., a responsive SC).  Lean production is accomplished by minimizing waste due to unnecessary, inefficient and excessive buffering in operations. Alternatively, agile production efficiently changes operation states in response to uncertain and changing demands placed upon it (Hallgren and Olhager 2008). Lean production depends on the type of market, dominant technology, and SC structure. A lean value approach develops a level schedule to eliminate all waste, involving a manufacturing process undisturbed by uncertainty and volatility and enabling high capacity utilization, cost minimization, and highly efficient operations. Agile production describes a system that operates profitably under unpredictable, changing customer demand. An agile system reacts quickly and effectively to changing markets driven by customer designed products, handles a variety of products and introduces new products quickly with minimal or zero investment (i.e., customized products). In terms of competitive strategies, operation capabilities and performance, empirically-based analyses show that cost leadership strategies and cost performance are aligned with lean manufacturing. Agile manufacturing is aligned with a product differentiation strategy, a strong impact on volume-product flexibility, and delivery performance (Hallgren and Olhager 2008). In intense industry competition, SCs engage in cost leadership or differentiation strategies. Thus, the cost leadership strategy is best supported by lean production, with high positive impacts on cost efficiency, quality, and delivery, based on a repetitive production schedule and flow-oriented layout. For a SC that is oriented by differentiation strategy, agile 43  production provides a better fit, with a significant improvement in flexibility and delivery performance over lean.  However, lean and agile paradigms tend to be seen as isolated solutions. Naylor et al. (1999) noted that agility and leanness depend on the SC strategy, market knowledge, and decoupling point positions. In terms of SC flow, the decoupling point (DP) is the point in the SC at which inventory should be held to buffer fluctuations between supplies and markets orders. Upstream from the DP, the SC is forecast driven, while downstream from the DP, production is pulled from the market. Consequently, on the downstream side of the DP, demand is highly variable with a large variety of products, and upstream it is smoothed with reduced variety (Naylor et al. 1999). Hence, upstream of the DP, with smooth demand and a low variety of products, a lean approach can be adopted, while downstream of the DP, where demand is variable and high product variety exists, an agile system is appropriate. The adoption of SC manufacturing systems should be based upon product market demand behaviors and systems attributes. Hence, the combination of volume and a variety of products that the SC should produce strongly determines the applicability of lean or agile manufacturing. On one hand, if a high variety of products should be produced in low volumes, the SC should adopt agile manufacturing. On the other hand, if a low variety of products should be produced in high volumes, the SC should adopt a lean manufacturing approach (Christopher 2000). The lean approach operates with minimum inventory, where any inventory reduction does not add value, while the agile approach operates with higher inventory levels to increase SC flexibility against sudden demand changes. Accordingly, the agile approach requires being able to conduct fast production reconfigurations to quickly respond to changes in demand. The same attribute is important, but not essential, for the lean approach. Despite the individual advantages of the lean and agile approaches, Christopher (2000) suggests that both can coexist successfully in a SC that manages hybrid product portfolios and markets, part of which has products with stable and predictable demand and part where the opposite is true.  44  3.3.3 Aggregate production planning drivers  As manufacturing initiatives guide the manner in which production plans should be executed, a brief explanation of the drivers of aggregate production planning follows. A production plan determines how much and when to make each product, so as product demand fluctuates, determining production levels is not obvious. A production plan should balance capacity, aggregate units and costs (Sipper and Bulfin 1997). A production plan specifies the quantities of each final product, subassemblies, and parts needed at various points in time. Thus, estimates of end product demand and a master production schedule (MPS) are required to determine the production plan. The MPS includes exact amounts and delivery timing for each product. It is derived from demand estimates, manufacturing constraints (i.e., capacity), and end product inventory. The MPS is broken into production schedules for each component of an end product (called material requirement planning) and requires detailed capacity planning. While not all drivers mentioned in the literature can be translated and used in a mathematical formulation, a summary of the key drivers is found in Table 3.2.  3.3.4 Modeling the production planning problem with lean and agile approaches    Linear and mixed integer programming techniques have been used to optimize production planning. The central goal is to efficiently minimize production and inventory costs. Nevertheless, these aggregate plans are determined with a rough capacity analysis and with much aggregation of production units and time. Unfortunately, they do not ensure the feasibility of the disaggregated production plan. Decisions such as how much of each product should be produced in a given period, and within what time period considering setup, batch size, and sequences, must be made (Sipper and Bulfin 1997). This is called the lot-sizing problem and it has a short-term production-planning horizon. The aggregate production-planning problem represents a suitable scenario for testing manufacturing approaches. Hence, I reviewed related literature in order to suggest a mathematical framework to address this chapter’s objective.   45  Table 3.2 Attributes of Lean, Agile and Hybrid manufacturing supply chain drivers  Driver Lean Agile Hybrid SC Strategy Costs leadership, Cost reduction, flexibility and incremental improvements for existing product production Differentiation-responsiveness. Provides customized products with short lead times by reducing the costs of variety Mass customization by postponing product differentiation until final assembly Product attributes Functional-commodity: Highly predictable, long life cycles (e.g., a staple)  Innovative: Uncertain demand, Short life cycles (e.g., a customized laptop)  Mixed portfolio: functional and innovative components. Long-Short life cycles (e.g., a car) Volume-variety Large volumes of low variety products Small volumes of high variety products Both Demand   Stable-Predictable  Unstable-Unpredictable  Both Emphasis Zero waste and economic efficiency Site of inventory-capacity: responsiveness Determined by the products Manuf. Focus Maintain high average utilization rate. Level strategy Deploys excess buffer capacity to ensure that raw materials/products are available to manufacture. Chase strategy Part Chase strategy and part level strategy  Back orders Allowed but penalized Not allowed or highly penalized Both mixed Inventory policy Generate high turnover and minimize inventory Deploys significant stocks of parts to tide over unpredictable market needs Minimize functional component inventories  Lead time focus Shorten lead times as long as it does not increase cost Invest aggressively in ways to reduce lead times Similar to lean at component level. However, at products level follows agile focus Decouple Point At warehouse site At manufacturing site At manufacturing site 46  The SC uses raw materials, resources with limited capacities to produce products that satisfy customer demand, optimizing the tradeoff between setup and inventory holding costs (Hege et al. 2010). Basically, the problem has been solved with MIP formulations. Furthermore, Billington et al. (1983) suggest that the problem begins when material requirement planning (MRP) systems assume no constraints for facilities; hence, any amount of production is presumed to be possible in each facility. However, lead times (setup and production time) can increase due to bottleneck operations, triggering unpredictable lead times. This problem has been called a capacity-constrained production-scheduling problem.  Depending on the manufacturing environment, different modeling approaches can be applied. A tradeoff analysis between setup and holding inventory costs should be conducted (Sipper and Bulfin 1997). Multiple products and capacity constraints increase the complexity of the problem. The mathematical formulations usually focus on minimizing setup and holding inventory costs, subject to material balance constraints, plus capacity and market demand constraints. However, in practice, the amount of binary and continuous variables (i.e., production quantities by product and setup cost) makes the problem intractable with exact algorithms. MRP calculates requirements for all items, including raw materials, parts, components, and subassemblies. MRP determination is based on MPS, capacity requirement planning, and lot sizing determinations. Lot sizing decisions may be made based on previous requirements and always consider materials, machines and labor constraints. If there are capacity constraints, this problem could become a single- or multiple-level capacitated lot-sizing problem (Hung and Chien 2000). There is another group of researchers who adopted discrete event simulation (DES) with value stream mapping technique (VSM) as a method to test manufacturing paradigms. In this context, the well-defined and rich literature on lean tools and methods is contradicted by a few documented quantitative implementations. Yang-Hua and Van Landeghem (2009) test the value of lean manufacturing using DES and VSM. They assessed a pull strategy to reduce work in process (WIP) and lead time (LT) in relation to a push strategy (i.e., large WIP and LT), but there are no details on how the pull future state was developed. Although there is evidence that lean manufacturing techniques (i.e., JIT, Total Preventive Maintenance, and cellular manufacturing) improve performances in discrete manufacturing (e.g., furniture assembly lines), evidence on continuous production (e.g., pulp digesters) is scarce. Aldulmalek and Rajgopal (2007) performed an analysis of a continuous manufacturing process based on DE, VSM and historical data. They 47  introduced buffers and scheduling around the bottleneck work station. Later, a DES was run and setup with two levels of TPM, two setup times, and push and hybrid pull manufacturing. Their results showed that pull hybrid manufacturing and TPM trigger significant lead-time reductions, as well as strong reductions in WIP.  Lean and agile manufacturing approaches easy to understand philosophies; however, their complexity appears during implementation. Adopting a different perspective, Golsby (2005) modeled a SC for lean, agile, and hybrid manufacturing to evaluate their benefits and tradeoffs. A DES model was used to simulate the SC. The results showed that the lean approach was the best in terms of lead times, followed by hybrid and agile approaches. The hybrid approach held fewer raw materials in inventory than the other approaches, due to the advantage of generic subassemblies and the hybrid and agile approaches held fewer end items in inventory than the lean approach. There was no mention of how product demand patterns should be assessed to better understand manufacturing philosophies. Another attempt to quantify lean manufacturing benefits was made by Al-Aomar (2011). The manufacturing environment was modeled with DES; a Tabu search (TS) was used to find the model parameter that optimized lean measures (e.g.,WIP, and LT). As lean measures with one objective could be contradictory, a cost (processing) function to rank solutions at each TS step was applied. Although this approach considered the effect of lean techniques on profits, is not clear how this method balanced lean measures as the author claimed because it appears to only determine the value of each combination and select the most valuable one. Despite efforts to measure the benefits of manufacturing methods with models, most research has been conducted for only one facility, and without considering product demand changes (Huang, et al. 2002). Until now, researchers have been applying DES, VSM, experimental design, and TS to find the setting that optimizes the balance between manufacturing measures. Those approaches have been successful when modeling floor variables, such as machine reliability, product failure rates, and the impact of machine failures rates on production, WIP and LT given their stochastic nature. Unfortunately, those efforts fail to explicitly measure the impact of the manufacturing principles on economic performance. MIP models have been used to solve the aggregate production-planning problem when testing agile principles (Babazadeh et al. 2012), but do not consider that manufacturing measures would be in conflict. The adoption of emergent manufacturing principles, as with any technological change, should be subject to accurate economic analyses. However, there is not substantive economic evidence at the bottom economic 48  forest company level to support their implementation. However, it should be recognized that VSM, and DES have been applied to evaluate them, and indirect SC metrics have been determined when shop floor lean and agile principles were used. Thus, the question of when and where those principles should be applied still remains without answer, at least at the aggregate forest company level. Therefore, there is a lack of research on determining explicitly and quantitatively the effects of lean, agile and hybrid manufacturing drivers on BC coastal forest-lumber SC performance.  To fill this gap, I suggest a MIP formulation to solve an aggregate production-planning problem, in which one primary manufacturing principle will be used as an objective function, with a set of secondary manufacturing drivers as a set of constraints. The impact of those manufacturing drivers will be assessed on SC use of resources, flexibility, and output. My formulations should avoid large uses of integer-binary variables, keeping in mind the intractability warning for such a problem as suggested by Hege et al. (2010).  3.3.5 BC coastal forest supply chain as a case study  The Canadian sawmilling industry is concentrated in the province of BC, producing 34 million m3 of lumber. Sawmills consume a wide range of large and high-grade logs from old- and second-growth forests to produce specialty products and commodities (Salehirad and Sowlatti 2005). Large, integrated companies have access to old growth forests containing harvestable mature timber. In this context, forest companies’ objectives are to create value for shareholders, maximizing the extraction of high quality timber to produce high value lumber (WFP, 2011). A year-based forest SC operation begins with logging, and then the logs for sawmilling are sorted by species, grade, and size and hauled by ground or water directly to sawmills. Head rig-based sawmills, kilns, and remanufacturing mills process large, high-quality logs into high-grade lumber, commodity-grade lumber and wood chips. On the coast of BC, the predominant focus for improving efficiencies involves increasing throughput and capacity utilization and reducing inventories. However, long lead times, large production batches, and uncertainty in fiber supply and deliveries make it exceedingly difficult to control inventory (Haartveit et al. 2004). As a result, sawmills try to extract the highest lumber grades possible from each log 49  to produce mainly commodity lumber products. However, there is a portion of lumber production that is “cut to order” (i.e., customized orders), which must be shipped with accurate lead times and quantities2. BC coastal forest companies sell their lumber products to distributors or major manufacturers in the USA. Sales to Japan are made directly to trading houses or to wholesale and manufacturing customers (WFP 2011). When commodity grades are traded, wholesalers are more tolerant with over- and under-fulfillment of lumber orders. However, when trading higher grades and lumber sizes, industrial customers and wholesalers tend to be less tolerant with not meeting order fulfillment and lead times. As a consequence, they apply high penalties for under-over fulfillment of orders and low tolerances for over-fulfillment and lead delivery times3. The coastal forests of BC are composed of five main coniferous species, which are tall and are, to a large extent, self-pruning. This results in a straight clean trunk that develops clear wood fiber with each annual ring of growth. Consequently, most of the large coast trees have distinct growth zones, each of which yields a particular type and quality of lumber product (Coast Forest Products Association 2014). Because my research focus is on the BC coastal forest, knowledge of stand level models is necessary to simulate forest inventory with lumber grade attribute relationships. However, given that this issue is outside of the scope of this research, I assume that wood quality attributes are stand type, age and growing site dependent (Zhang 2009). Hence, to model the forest inventory, I worked with a forest inventory that contains age, elevation, and site index descriptors for each stand and assumed that stand timber grades are positively correlated with age and with site index and negatively correlated with elevation. Furthermore, for lumber grades, I assumed that the older the timber was the larger logs would be. As a consequence, they produce a higher proportion of high-grade lumber (Coast Forest Products Association 2014). In summary, the focus in coastal BC is to create value while maintaining high volume and value recoveries and capacity utilization. Forest operations push log flows, where throughput and capacity maximization predominate. Downstream, sawmills extract the highest lumber grades possible from logs to produce commodity and customized lumber products. When construction lumber grades are traded, wholesalers tend to be tolerant with over/under fulfillment of orders. However, when trading high grade lumber, industrial and wholesaler customers are less tolerant (i.e., high penalizations for missing targets).                                                      2 Personal conversation with Joel Mortyn, 2013, Lumber Manufacturing Department Scientist at FP Innovations.  3 Personal conversation with Pablo Crespell, 2013, Markets and Economics - Business Analysis at FPInnovations. 50   3.4 Methodology  The theoretical literature about Lean, agile, and hybrid manufacturing environments is extensive, however, few quantitative implementations can be found, and almost no publications are available for forest industry applications. Therefore, for the purposes of this research the drivers of manufacturing principles were sorted hierarchically (see table 3.2). It was necessary to translate soft drivers into hard drivers to model the aggregate/tactical forest-to-lumber planning problem. Accordingly, the supply chain strategy driver of the manufacturing environment was considered as the central driver, and used as the objective function to model the manufacturing environment. Manufacturing focus and inventory policy were translated into capacity usage penalty policies. Backorder and lead time focus were translated into demand satisfaction penalty policies. Products attributes, demand and volume and variety were considered as lumber demand scenarios.  There are many drivers of each manufacturing environment, and it was not possible to include all of them in a mathematical model.  Therefore, I selected only those drivers that would best represent an efficient modelling abstraction of the real-world system for my case studies. I built mathematical formulations to solve the production-planning problem for a case study forest to lumber SC. Those formulations simulate agile, lean, and existing BC coastal forest SCs. As Haartveit et al. 2004 showed, I assumed the existing coastal British Columbia SC (BC-SC) to be a hybrid manufacturing environment. Furthermore, demand variation is a key factor in manufacturing principles selection (Agarwal et al. 2003). Consequently, I used publically available information to determine that the lumber sales variation from 2011 to 2013 was 6% for all families of lumber products. The US new privately owned housing units authorized for the same period was, on average, 62,091 units with rate of variation of 23%, which can be used as an indirect estimate of lumber sales variation. Although none of these variations represent the exact variation that lumber product sales experienced, I assumed that they could guide the rate of variation introduced to lumber demand.   51  3.4.1 Lumber demand scenarios  Only one period of demand was forecast. Demand variations were made to simulate demand changes through the time. Random variations were made to lumber product demand volume forecasts. As a consequence, four lumber demand scenarios (LDS) plus a base case were created representing combinations of production batch sizes and varieties of lumber products. The lumber demand scenarios are: 1. Base LD: Lumber demand by grades and species, based on BC companies’ public annual reports. 2. LB_LV: Large batch and low variety of lumber products demanded, obtained by multiplying the original lumber demand data by a random binary number. This arrangement ensures large values similar to Base_LD, but when multiplied by a random binary number the variation was reduced, because when the binary random number takes a zero value, it makes demand zero as well.  3. LB_HV: Large batch and high variety of lumber products demanded, obtained by multiplying the original lumber demand data by a continuous random number generated between 0 and 1 plus 0.25. This arrangement ensured large values similar to Base_LD, or even bigger, but all are non-zero values, because continuous random numbers were used. This means all values exist, thus ensuring high variation. 4. SB_LV: Small batch and low variety of lumber products demanded, obtained by multiplying the original lumber demand data by a random binary number, and by a continuous random number generated between 0 and 1.This arrangement ensured lower values in comparison with Base_LD, or at most the same, but the variation was reduced. 5. SB_HV: Small batch and high variety of lumber product demanded, obtained by multiplying the original lumber demand data by a continuous random number generated between 0 and 1. This arrangement ensured lower values in comparison with Base_LD, or at most the same, but all are non-zero values ensuring high variety  Lumber demand scenarios were processed in Excel producing 30 set of values per scenario, except for the Base_LD, which contained only one set. All scenarios were fed to optimization programs, which were run in 52  ILOG CPLEX. Then, 3 DSSs representing each ME (lean, agile and BC-SC) were run under these five lumber demand scenarios. A description of the manufacturing environment drivers used in my DSS formulations is given in Table 3.2.  3.4.2 Description and formulation of ME models  3.4.2.1 Primary drivers and penalty framework  The primary and secondary drivers identified from the literature were summarized and used in my mathematical formulation (Table 3.3). The agile, lean and BC-SC formulations included the possibility of lumber interchanges between sawmills (Singer and Donoso 2007) and sawmilling manufacturing outsourcing (Babazadeh et al. 2012), which are common attributes of current SCs. Table 3.3 summary of manufacturing environment model formulations ME Formulation Central manufacturing principle applied Demand satisfaction penalty policy Capacity usage penalty policy Objective  Function Agile Reduce the cost of variety to create the highest possible value Agile only allows a tiny percentage of lumber demand to be over or under demand The penalization emulates a time flexibility policy and CHASE manufacturing strategy Agile is focused on creating maximum value. Thus, I applied a profit max. objective function Hybrid:  BC-SC Profit max by harvesting the highest value forest resources and satisfying customer lumber demand Hybrid=BC SC assumed to be middle ground, because it produces a mixed portfolio of lumber  LEVEL for procurement, and CHASE for sawmilling DP at sort yards; thus, harvesting and sort yards minimizes cost, satisfying log demand. Sawmills max. profit, satisfying lumber demand Lean SC cost minimization reduces all form of waste (i.e., muda) through the SC Lean allows the highest percentage of lumber demand, which can be over or under demand  The penalization emulates a LEVEL strategy  Lean is focused on SC cost minimization; thus, a cost minimization objective function was applied    53  3.4.2.2 Secondary goals of the manufacturing environment and constraints  Capacity constraints were added to the models because operation working times are divided into shifts. The lean approach used a level manufacturing strategy, which was translated as a capacity constraint to ensure that at least a minimum level of capacity must be utilized. The agile approach used a chase manufacturing strategy, which was translated as a capacity constraint to ensure a higher flexibility of capacity usage (Chopra and Meindl 2011). The BC-SC approach used mixed capacity constraints, which was a level strategy for the timber supply problem, and a chase strategy for the lumber manufacturing problem.  Two additional features were added to the ME formulations. The first applied penalties when lumber demand was below or exceeded demand. A second penalty was applied when capacity (i.e., time availability) was exceeded or was not completely used over a period by loggers, sort-yards, and sawmills. The literature emphasizes that the MEs satisfy customer demand with different customer service levels and manage manufacturing capacities differently as a function of customer demand behaviors. Thus, in term of demand satisfaction, over- and under-demand were heavily penalized under the agile approach, moderately penalized under the BC-SC approach, and lightly penalized under the lean approach. Furthermore, in terms of capacity, over- and under-production capacity was heavily penalized under the lean approach, moderately penalized under the BC-SC approach, and lightly penalized under the agile approach (Tables 3.4, and 3.5). Table 3.4: Over and under lumber demand satisfaction penalty framework Criteria for over and under-production     Manufacturing environments  Lumber grade Agile BC-SC  Lean % of lumber price used to penalize under-production on cost Lumber grade Sawing grade Utility grade 90% 80% 70% 85% 75% 65% 80% 70% 60% Allowed under demand lumber production as a (%) of lumber demand 10%              15%            20% % of lumber price used to penalize over-production on incomes Lumber grade Sawing grade Utility grade 50% 40% 30% 55% 45% 35% 60% 50% 40% Allowed over demand lumber production as a (%) of lumber demand      10%             15%               20%  54   Table 3.5 Over and under capacity usage penalty framework Operation Penalization for operations with under-over period capacity Agile BCSC Lean Loggers Percent increased cost when over period capacity 105% 150% 200% Percent of over capacity allowed 50% 30% 5% Percent increased cost when under period capacity 90% 95% 110% Percent of under capacity allowed 50% 30% 5% Sort yards Percent increased cost when over period capacity 110% 200% 400% Percent of over capacity allowed 50% 30% 5% Percent increased cost when under period capacity 90% 190% 400% Percent of under capacity allowed 50% 30% 5% Sawmills Percent increased cost when over period capacity 105% 200% 400% Percent of over capacity allowed 50% 30% 5% Percent increased cost when under period capacity 95% 200% 400% Percent of under capacity allowed 50% 30% 5%   55  3.4.3 Mathematical formulations The BC coastal forest SC has 3 nodes, forest stands, sort-yards. and sawmills. I chose a planning horizon of 1 quarter, which was divided into 4 periods. The MIP formulation follows: 3.4.3.1 Model’s sub-index i: Cut-block    j: cut-block growth    k: Species    l: Sort yard  m: Bucking policy n: log grade o: Sawmill    p: Sawing policy q: Sawn-woods products r: Planning periods 3.4.3.2 Operational and inventory costs: C HI : Harvesting cost in $ by m3 harvested at stand i. C Stu k j : Stumpage cost in $/m3 when harvesting specie k, growth j CSCB   i : Cost of keep 1 m3 of stem at cut-block i. CiSYS   l : Cost of keep 1 m3 of stem at sort-yard l. CiSYL   l : Cost of keep 1 m3 of logs at sort-yard l. CiSAL  o : Cost of keep 1 m3 of logs at sawmill o. CiSAP o : Cost of keep 1 m3 of sawn-wood products at sawmill o. CT1    i l : Transportation cost in $/ m3 of stems from stand i, to sort yard l CT2  l o : Transportation cost in $/ m3 of logs from sort yard l to sawmill o CT3  o oo   : Transportation cost in $/ m3 of lumber from sawmill o to sawmill oo CSY  l    : Sort yard production cost in $/m3, in sort yard l.  CSetup kno     : Sawmills setup cost in $, charged when a sawmill change specie and log grade.  CSW  o          : Sawmilling production cost in $/m3, in sawmill o.  CSWOUT  o    : Outsourced sawmilling production cost in $/m3, for sawmill o.  Price k q o r  : Lumber price in $/m3 of species k, product q, sell by sawmill o, in period r.   Price k r  : Chips price in $/m3 of species k, in period r Log_price k n l r  : Log price in $/m3 of species k, grade n, sort yard l, in period r 3.4.3.3 Yields, capacities and productivities Yield_f    I j k        : Forest yield in m3 by hectares in stand i, to growth j, and species k. Yield_sy j k m n   : Sort yard yield in m3 of product n by m3 of stem growth j, species k, by applying bucking policy m. Yield_sw k n p q  : Sawmilling yield in m3 of product q by m3 of log grade n, species k, by applying sawing policy p. Cap_f     r   : Harvesting capacity (in hours) on period r. Cap_sy  r     : Sort yard capacity (in hours) on period r. Cap_sw r     : Sawmill capacity (in hours) on period r. Prod_f  j       : Harvest productivity (in hour/m3) in a stand predominantly growth j. Prod_sy j k l  : Sort yard productivity (in hour/m3) when bucking-sorting stems growth j, species k, at sort yard l. Prod_sw j k l   : Sawing productivity (in hour/m3) when sawing species k, log grade n, at sawmill o 56  io_sys    j k l  : Zero inventory of stems of growth j, species k at sort yard l. io_syl      k n l  : Zero inventory of log of species k, log grade n,  at sort yard l io_sal     k n o  : Zero inventory of log of species k, log grade n,  at sawmill o. io_sap    k q o  : Zero inventory of sawn-wood of species k, products q, at sawmill o. D k q o r  : Demand (in m3) of species k, product q, in sawmill o, for period r.   Log_D k n l r  : Logs demand (in m3) of species k, log grade n, in sort yard l, for period r.   3.4.3.4 Allowable quantities to penalize lumber production, capacity usage, and used economic penalties. %O_1 k q o r  : Max. % for over-production defined in function of the demand of lumber product specie k, product q, produced/sold by sawmill o, period r,   %B_1 k q o r  : Maximum % for below production defined in function of the demand of lumber product specie k, product q, produced/sold by sawmill o, period r,   %O_LOT1 r  : Maximum over hours usage for logging operation in period r, defined as percentage of the hours available in period r. %B_LOT1 r  : Maximum below hours usage for logging operation in period r, defined as percentage of the hours available in period r. %O_SYT1 l r  : Maximum over hours usage for sort-yarding operation in sort-yard l, period r, defined as percentage of the hours available in period r. %B_SYT1 l r  : Maximum below hours usage for sort-yarding operation in sort-yard l, period r, defined as percentage of the hours available in period r. %O_SWT1 o r  : Maximum over hours usage for sawmilling operation in sawmill o, period r, defined as percentage of the hours available in period r. %B_SWT1 o r  : Maximum below hours usage for sawmilling operation in sawmill o, period r, defined as percentage of the hours available in period r. PO_1 k q o r  : Percentage of the lumber price of lumber product specie k, product q, produced-sell by sawmill o, period r, to penalize 1 m3 produced over the lumber demand.  PB_1 k q o r  : Percentage of the lumber price of lumber product specie k, product q, produced-sell by sawmill o, period r, to penalize 1 m3 produced below the lumber demand  PB_2 k n l rr  : Percentage of the log price specie k, log grade n, produced-sell by sortyard l, period r, to penalize 1 m3 produced below the log demand  PO_LOT1 r  : Percentage of the logging cost charged for every over hour usage in logging operation in period r. PB_LOT1 r  : Percentage of the logging cost charged for every below hour usage in logging operation in period r. PO_SYT1 l  r  : Percentage of the sort-yarding cost charged for every over hour usage in sort-yarding operation in sort yard l, period r. PB_SYT1 l r  : Percentage of the sort-yarding cost charged for every below hour usage in sort-yarding operation in sort yard l, period r. PO_SWT1 o r  : Percentage of the sawmilling cost charged for every over hour usage in sawmilling operation in sawmill o, period r. PB_SWT1 o r  : Percentage of the sawmilling cost charged for every below hour usage in sawmilling operation in sawmill o, period r. 3.4.3.5 Decision variables:  H   i r         :  Land harvested (in ha.) at stand I, in period r 57  T   i j k  r     : Harvested volume (in m3) at stand “i”, growth type “j”, species “k” in period “r”  ISCB I j k r     : Inventory of stems (in m3) at cub-block I, of growth j, species k, in period r.”  U  i j k l r     : Volume of stems  (in m3) of growth “j”, specie “k” sent from stand “i” to sort yard l, in period r. V  j k l m r   : Sort yard input of stems (in m3) of growth j, species k, at sort yard l, bucked with policy m, period r.  ISYS j k l r     : Inventory of stems (in m3) of growth j, species k, at sort yard l, in period r.”  W    k n l r    : Volume of logs (in m3) of species “k”, log grade “n”, produced in sort yard l in period r Y  k n o p r    : Volume of logs (in m3) of species k, log grade n, sawn with sawing policy p, at sawmill o, in period r. YO k n o p r   : Volume of logs (m3) of species k, log grade n, sawn with sawing policy p, at the outsourcing sawmill of sawmill o, in period r. ISAL  k n o r   : Inventory of logs (in m3) of species k, log grade n at sawmill o, in period r.  ISYL k n l r   : Inventory of logs (in m3) of species k, log grade n, at sort yard l, in period r. X   k n l o r   : Volume of logs (in m3) of species k, log grade n sent from sort yard l to sawmill o, in period r Z   k q o r   : Volume of sawn-wood (in m3) of specie k, product q, produced-sell by sawmill “o”, in period r ZO k q o r   : Volume of sawn-wood (in m3) of specie k, product q, produced-sell by outsourced sawmill of sawmill “o”, in period r ZI O_OO k q o r  : Volume of sawn-wood (in m3) of specie k, product q, arrived in interchanges to sawmill o, in period r ZI OO_O k q o r  : Volume of sawn-wood (in m3) of specie k, product q, sent in interchanges to sawmill o, in period r DO_1 k q o r  : Volume in m3 of over demand lumber production for specie k, product q, produced-sell by sawmill “o”, in period r. DB_1 k q o r  : Volume in m3 of below demand lumber production for specie k, product q, produced-sell by sawmill “o”, in period r. DO_LOT1 r  : Quantity of over capacity usage (hours) in logging operation in period r. DB_LOT1 r  : Quantity of below capacity usage (hours) in logging operation in period r. DO_SYT1 l  r  : Quantity of over capacity usage (hours) in sort-yarding operation at sort yard l in period r. DB_SYT1 l r  : Quantity of below capacity usage (hours) in sort-yarding operation at sort yard l in period r. DO_SWT1 o r  : Quantity of over capacity usage (hours) in sawmilling operation at sawmill o in period r. DB_SWT1 o r  : Quantity of below capacity usage (hours) in sawmilling operation at sawmill o in period r. BA  k q o r   : Lumber balance for specie k, product q, produced-sell by sawmill “o”, in period r. C   k r         : Produced chips (in m3)  in species k in period r. ISAP k q o r   : Inventory of sawn-wood products (in m3) of species k, sawn-wood products q, in period r. LOT  r         : Logging time in hours for period r  SYT  l r        : Sort yard l processing time in hours for period r  SWT o r       : Sawmill o processing time in hours for period r  BY k n o r      :Binary variable, which is 1 if sawmill o, consume logs species k, grade n, in period r, zero otherwise.  3.4.3.6 Objective functions The objective function for the agile approach maximizes profit (3.1Agile). Its components are: lumber and chip incomes, penalized incomes for over-production, less all SC operational costs, and cost penalties for 58  below-demand lumber production, and cost penalties for over/under capacity usage. The objective function for the lean approach minimizes cost (3.1Lean) and contains the same components as the objective function for the agile approach. The BC-SC was divided into two problems; 1) timber supply and 2) lumber manufacturing. In order to compare this formulation with the other two, it was assumed that procurement areas transfer logs to the lumber production area with no profits. The timber supply problem satisfies a forecasted log demand; the log production solution was used to solve the lumber manufacturing problem. The objective function of the timber supply problem minimized cost (3.1Timber), its component were: all SC operational costs and cost penalties for over/under capacity usage in logging and sort-yard operations. The lumber manufacturing problem had a profit maximization objective function (3.1Lumber), and solved the lumber planning problem. The components of the objective function were: lumber and chip incomes, penalized incomes for over production, log costs, all operational costs, cost penalties for under-demand lumber production and cost penalties for over/under capacity in sawmilling operations.  𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑜𝑓𝑝𝑟𝑜𝑓𝑖𝑡= ∑ 𝑍𝑘𝑞𝑜𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑞𝑜𝑟 +∑𝐶𝑘𝑟𝑘𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑟𝑘𝑞𝑜𝑟+ ∑ 𝐷𝑂_1𝑘𝑞𝑜𝑟𝑃𝑂𝑘𝑞𝑜𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑞𝑜𝑟 −∑𝑇𝑖𝑗𝑘𝑟𝐶𝑗𝑘𝑆𝑇𝑈 −∑𝑇𝑖𝑗𝑘𝑟𝐶𝑖𝐻 −𝑖𝑗𝑘𝑟𝑖𝑗𝑘𝑟∑ 𝑈𝑖𝑗𝑘𝑙𝑟𝑇𝑖𝑙𝑇1𝑖𝑗𝑘𝑙𝑟𝑘𝑞𝑜𝑟−∑𝐼𝑖𝑗𝑘𝑟𝑆𝐶𝐵𝐶𝑖𝑆𝐶𝐵 −𝑖𝑗𝑘𝑟∑𝐼𝑗𝑘𝑙𝑟𝑆𝑌𝑆𝐶𝑙𝑆𝑌𝑆 −𝑗𝑘𝑙𝑟∑ 𝐼𝑘𝑛𝑙𝑟𝑆𝑌𝐿 𝐶𝑙𝑆𝑌𝐿 −𝑘𝑛𝑙𝑟∑𝑊𝑘𝑛𝑙𝑟𝐶𝑙𝑆𝑌 −𝑘𝑛𝑙𝑟∑ 𝑋𝑘𝑛𝑙𝑜𝑟𝐶𝑙𝑜𝑇2𝑘𝑛𝑙𝑜𝑟− ∑ 𝐼𝑘𝑛𝑜𝑟𝑆𝐴𝐿 𝐶𝑜𝑆𝐴𝐿 − ∑ 𝐼𝑘𝑞𝑜𝑟𝑆𝐴𝑃 𝐶𝑜𝑆𝐴𝑃 −𝑘𝑞𝑜𝑟𝑘𝑛𝑜𝑟∑ 𝑍𝑘𝑞𝑜𝑟𝐶𝑜𝑆𝑊 − ∑ 𝐷𝐵1𝑘𝑞𝑜𝑟𝑃𝐵𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑞𝑜𝑟−∑𝐷𝑂𝐿𝑂𝑇1𝑟[𝑃𝑂𝐿𝑂𝑇1𝑟]𝐶𝑖𝐻𝑖𝑟−∑𝐷𝐵𝐿𝑂𝑇1𝑟[𝑃𝐵𝐿𝑂𝑇1𝑟]𝐶𝑖𝐻 −∑𝐷𝑂𝑆𝑌𝑇1𝑙𝑟[𝑃𝑂𝑆𝑌𝑇1𝑙𝑟𝑙𝑟]𝐶𝑙𝑟𝑆𝑌 −𝑖𝑟∑𝐷𝐵𝑆𝑌𝑇1𝑙𝑟[𝑃𝐵𝑆𝑌𝑇1𝑟𝑙𝑟]𝐶𝑙𝑟𝑆𝑌−∑𝐷𝑂𝑆𝑊𝑇𝑜𝑟[𝑃𝑂𝑆𝑊𝑇1𝑜𝑟]𝐶𝑜𝑟𝑆𝑊 −𝑜𝑟∑𝐷𝐵𝑆𝑊𝑇𝑜𝑟[𝑃𝐵𝑆𝑊𝑇1𝑜𝑟]𝐶𝑜𝑟𝑆𝑊 −𝑜𝑟∑ 𝑍𝐼𝑘𝑞𝑜𝑜𝑜𝑟𝐶𝑜𝑜𝑜𝑟𝑇3𝑘𝑞𝑜𝑜𝑜𝑟− ∑ 𝐵𝑌𝑘𝑞𝑜𝑟𝐶𝑘𝑛𝑜𝑆𝐸𝑇𝑈𝑃 −𝑘𝑞𝑜𝑟∑ 𝑍𝑂𝑘𝑞𝑜𝑟𝐶𝑜𝑆𝑊𝑂𝑇𝑘𝑞𝑜𝑟                          [3.1𝐴𝑔𝑖𝑙𝑒]  59  𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑐𝑜𝑠𝑡𝑠= ∑𝑇𝑖𝑗𝑘𝑟𝐶𝑗𝑘𝑆𝑇𝑈 +∑𝑇𝑖𝑗𝑘𝑟𝐶𝑖𝐻 +𝑖𝑗𝑘𝑟𝑖𝑗𝑘𝑟∑ 𝑈𝑖𝑗𝑘𝑙𝑟𝑇𝑖𝑙𝑇1𝑖𝑗𝑘𝑙𝑟+∑𝐼𝑖𝑗𝑘𝑟𝑆𝐶𝐵𝐶𝑖𝑆𝐶𝐵 +𝑖𝑗𝑘𝑟∑𝐼𝑗𝑘𝑙𝑟𝑆𝑌𝑆𝐶𝑙𝑆𝑌𝑆 +𝑗𝑘𝑙𝑟∑ 𝐼𝑘𝑛𝑙𝑟𝑆𝑌𝐿 𝐶𝑙𝑆𝑌𝐿 +𝑘𝑛𝑙𝑟∑𝑊𝑘𝑛𝑙𝑟𝐶𝑙𝑆𝑌 +𝑘𝑛𝑙𝑟∑ 𝑋𝑘𝑛𝑙𝑜𝑟𝐶𝑙𝑜𝑇2𝑘𝑛𝑙𝑜𝑟+ ∑ 𝐼𝑘𝑛𝑜𝑟𝑆𝐴𝐿 𝐶𝑜𝑆𝐴𝐿 + ∑ 𝐼𝑘𝑞𝑜𝑟𝑆𝐴𝑃 𝐶𝑜𝑆𝐴𝑃 +𝑘𝑞𝑜𝑟𝑘𝑛𝑜𝑟∑ 𝑍𝑘𝑞𝑜𝑟𝐶𝑜𝑆𝑊 + ∑ 𝐷𝐵_1𝑘𝑞𝑜𝑟𝑃𝐵𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑞𝑜𝑟+∑𝐷𝑂_𝐿𝑂𝑇1𝑟[𝑃𝑂_𝐿𝑂𝑇1𝑟]𝐶𝑖𝐻𝑖𝑟+∑𝐷𝐵_𝐿𝑂𝑇1𝑟[𝑃𝐵_𝐿𝑂𝑇1𝑟]𝐶𝑖𝐻 +∑𝐷𝑂_𝑆𝑌𝑇1𝑙𝑟[𝑃𝑂_𝑆𝑌𝑇1𝑙𝑟𝑙𝑟]𝐶𝑙𝑟𝑆𝑌𝑖𝑟+∑𝐷𝐵_𝑆𝑌𝑇1𝑙𝑟[𝑃𝐵_𝑆𝑌𝑇1𝑟𝑙𝑟]𝐶𝑙𝑟𝑆𝑌+∑𝐷𝑂_𝑆𝑊𝑇𝑜𝑟[𝑃𝑂_𝑆𝑊𝑇1𝑜𝑟]𝐶𝑜𝑟𝑆𝑊 +𝑜𝑟∑𝐷𝐵_𝑆𝑊𝑇𝑜𝑟[𝑃𝐵_𝑆𝑊𝑇1𝑜𝑟]𝐶𝑜𝑟𝑆𝑊𝑜𝑟+ ∑ 𝑍𝐼𝑘𝑞𝑜_𝑜𝑜𝑟𝐶𝑜_𝑜𝑜𝑟𝑇3𝑘𝑞𝑜_𝑜𝑜𝑟+ ∑ 𝐵𝑌𝑘𝑞𝑜𝑟𝐶𝑘𝑛𝑜𝑆𝐸𝑇𝑈𝑃 +𝑘𝑞𝑜𝑟∑ 𝑍𝑂𝑘𝑞𝑜𝑟𝐶𝑜𝑆𝑊𝑂𝑇𝑘𝑞𝑜𝑟           [3. 1𝐿𝑒𝑎𝑛]  𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡𝑖𝑚𝑏𝑒𝑟 𝑝𝑟𝑜𝑐𝑢𝑟𝑒𝑚𝑒𝑛𝑡 𝑐𝑜𝑠𝑡= ∑𝑇𝑖𝑗𝑘𝑟𝐶𝑗𝑘𝑆𝑇𝑈 +∑𝑇𝑖𝑗𝑘𝑟𝐶𝑖𝐻 +𝑖𝑗𝑘𝑟𝑖𝑗𝑘𝑟∑ 𝑈𝑖𝑗𝑘𝑙𝑟𝑇𝑖𝑙𝑇1𝑖𝑗𝑘𝑙𝑟+∑𝐼𝑖𝑗𝑘𝑟𝑆𝐶𝐵𝐶𝑖𝑆𝐶𝐵 +𝑖𝑗𝑘𝑟∑𝐼𝑗𝑘𝑙𝑟𝑆𝑌𝑆𝐶𝑙𝑆𝑌𝑆 +𝑗𝑘𝑙𝑟∑ 𝐼𝑘𝑛𝑙𝑟𝑆𝑌𝐿 𝐶𝑙𝑆𝑌𝐿 +𝑘𝑛𝑙𝑟∑𝑊𝑘𝑛𝑙𝑟𝐶𝑙𝑆𝑌𝑘𝑛𝑙𝑟+∑ 𝐷𝐵_1𝑘𝑛𝑙𝑟𝑃𝐵_2𝑘𝑛𝑙𝑟𝐿𝑜𝑔_𝑝𝑟𝑖𝑐𝑒𝑘𝑛𝑙𝑟𝑘𝑛𝑙𝑟+∑𝐷𝑂_𝐿𝑂𝑇1𝑟[𝑃𝑂_𝐿𝑂𝑇𝑟]𝐶𝑖𝐻 +∑𝐷𝐵_𝐿𝑂𝑇1𝑟𝑖𝑟𝑖𝑟[𝑃𝐵_𝐿𝑂𝑇1𝑟]𝐶𝑖𝐻+∑𝐷𝑂_𝑆𝑌𝑇1𝑙𝑟[𝑃𝑂_𝑆𝑌𝑇1𝑙𝑟]𝐶𝑙𝑟𝑆𝑌 +𝑙𝑟∑𝐷𝐵_𝑆𝑌𝑇1𝑙𝑟[𝑃𝐵𝑆𝑌𝑇1𝑙𝑟]𝐶𝑙𝑟𝑆𝑌          [3. 1𝑇𝑖𝑚𝑏𝑒𝑟] 𝑙𝑟  60  𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑙𝑢𝑚𝑏𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑓𝑖𝑡= ∑ 𝑍𝑘𝑞𝑜𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑞𝑜𝑟 +∑𝐶𝑘𝑟𝑘𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑟𝑘𝑞𝑜𝑟+ ∑ 𝐷𝑂_1𝑘𝑞𝑜𝑟𝑃𝑂𝑘𝑞𝑜𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑞𝑜𝑟 −𝑘𝑞𝑜𝑟∑ 𝐼𝑘𝑛𝑙𝑟𝑆𝑌𝐿 𝐶𝑙𝑆𝑌𝐿 −𝑘𝑛𝑙𝑟∑𝑊𝑘𝑛𝑙𝑟𝐿𝑜𝑔𝑃𝑟𝑖𝑐𝑒𝑘𝑛𝑙𝑟 −𝑘𝑛𝑙𝑟∑ 𝑋𝑘𝑛𝑙𝑜𝑟𝐶𝑙𝑜𝑇2𝑘𝑛𝑙𝑜𝑟− ∑ 𝐼𝑘𝑛𝑜𝑟𝑆𝐴𝐿 𝐶𝑜𝑆𝐴𝐿 − ∑ 𝐼𝑘𝑞𝑜𝑟𝑆𝐴𝑃 𝐶𝑜𝑆𝐴𝑃 −𝑘𝑞𝑜𝑟𝑘𝑛𝑜𝑟∑ 𝑍𝑘𝑞𝑜𝑟𝐶𝑜𝑆𝑊 − ∑ 𝐷𝐵1𝑘𝑞𝑜𝑟𝑃𝐵𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑞𝑜𝑟−∑𝐷𝑂_𝑆𝑊𝑇𝑜𝑟[𝑃𝑂_𝑆𝑊𝑇𝑜𝑟]𝐶𝑜𝑟𝑆𝑊 −𝑜𝑟∑𝐷𝐵_𝑆𝑊𝑇𝑜𝑟[𝑃𝐵_𝑆𝑊𝑇𝑜𝑟]𝐶𝑜𝑟𝑆𝑊 −𝑜𝑟∑ 𝑍𝐼𝑘𝑞𝑜_𝑜𝑜𝑟𝐶𝑜_𝑜𝑜𝑇3𝑘𝑞𝑜_𝑜𝑜𝑟− ∑ 𝐵𝑌𝑘𝑞𝑜𝑟𝐶𝑘𝑛𝑜𝑆𝐸𝑇𝑈𝑃 − ∑ 𝑍𝑂𝑘𝑞𝑜𝑟𝐶𝑞𝑆𝑊𝑂𝑈𝑇𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟                      [3.1𝐿𝑢𝑚𝑏𝑒𝑟]  3.4.3.7 Constraints of the models Subject to: ∑ 𝐻𝑖𝑟𝑖𝑟 ≤ 𝐴𝑖     ∀𝑖     [3.1] 𝑇𝑖𝑗𝑘𝑟 = 𝐻𝑖𝑟Yield_f 𝑖𝑗𝑘     ∀𝑖 , 𝑗 , 𝑘, 𝑟        [3.2] ∑ 𝑇𝑖𝑗𝑘𝑟 Prod_f j    𝑖𝑗𝑘 ≤ Cap_f r    ∀ 𝑟          [3.3] 𝐿𝑂𝑇𝑟 = ∑ 𝑇𝑖𝑗𝑘𝑟 Prod_f j    𝑖𝑗𝑘 ∀𝑟       [3.4] 𝐿𝑂𝑇𝑟−1 = 𝐿𝑂𝑇𝑟 − 𝐷𝑂_𝐿𝑂𝑇𝑟 + 𝐷𝐵_𝐿𝑂𝑇𝑟  ∀ 𝑟 > 1     [3.5] 𝐷𝑂_𝐿𝑂𝑇1𝑟 ≤ [%𝑂_𝐿𝑂𝑇1𝑟] × Cap_f r    ∀ 𝑟         [3.6] 𝐷𝐵_𝐿𝑂𝑇1𝑟 ≤ [%𝐵_𝐿𝑂𝑇1𝑟] × Cap_f r    ∀ 𝑟 >   1     [3.7] 𝐷𝑂_𝑆𝑌𝑇1𝑟 ≤ [%𝑂_𝑆𝑌𝑇1𝑟] × Cap_sy lr    ∀ l𝑟     [3.8] 𝐷𝐵_𝑆𝑌𝑇1𝑟 ≤ [%𝐵_𝑆𝑌𝑇1𝑟] × Cap_sy lr    ∀lr     [3.9] 𝐷𝑂_𝑆𝑊𝑇1𝑟 ≤ [%𝑂_𝑆𝑊𝑇1𝑟] × Cap_swor    ∀or     [3.10] 𝐷𝐵_𝑆𝑊𝑇1𝑟 ≤ [%𝐵_𝑆𝑊𝑇1𝑟] × Cap_swor    ∀or     [3.11] 𝑇𝑖𝑗𝑘𝑟 + 𝐼𝑖𝑗𝑘𝑟−1𝑆𝐶𝐵 − 𝐼𝑖𝑗𝑘𝑟𝑆𝐶𝐵 =  ∑ 𝑈𝑖𝑗𝑘𝑙𝑟𝑙  ∀ijkr      [3.12] ∑ 𝑈𝑖𝑗𝑘𝑙𝑟𝑖  + 𝐼𝑗𝑘𝑙𝑟−1𝑆𝑌𝑆 − 𝐼𝑗𝑘𝑙𝑟𝑆𝑌𝑆 = ∑ 𝑉𝑗𝑘𝑙𝑚𝑟𝑚   ∀ j, k, l, r     [3.13] ∑ 𝑉𝑗𝑘𝑙𝑚𝑟 × 𝑌𝑖𝑒𝑙𝑑_𝑠𝑦𝑗𝑘𝑚𝑛𝑗𝑚 = 𝑊𝑘𝑛𝑙𝑟   ∀ k, n, l, r     [3.14] ∑ 𝑉𝑗𝑘𝑚 × 𝑃𝑟𝑜𝑑_𝑠𝑦𝑗𝑘𝑙𝑗𝑘𝑚 ≤ 𝑐𝑎𝑝_𝑠𝑦𝑙𝑟   ∀ l, r      [3.15] 𝑆𝑌𝑇𝑙𝑟 = ∑ 𝑉𝑗𝑘𝑙𝑚𝑟 × 𝑃𝑟𝑜𝑑_𝑠𝑦𝑗𝑘𝑙𝑗𝑘𝑚    ∀ l, r      [3.16] 𝑆𝑌𝑇𝑙𝑟−1 =  𝑆𝑌𝑇𝑙𝑟−1 − 𝐷𝑂_𝑆𝑌𝑇1𝑙𝑟 +  𝐷𝐵_𝑆𝑌𝑇1𝑙𝑟 ∀ 𝑙, 𝑟 > 1    [3.17] 61  𝑊𝑘𝑛𝑙𝑟 + 𝐼𝑘𝑛𝑙𝑟−1𝑆𝑌𝐿 − 𝐼𝑘𝑛𝑙𝑟𝑆𝑌𝐿 =   ∑ 𝑉𝑗𝑘𝑙𝑚𝑟 × 𝑌𝑖𝑒𝑙𝑑_𝑠𝑦𝑗𝑘𝑚𝑛𝑗𝑚 =    ∀ k, n, l, r   [3.18] ∑ 𝑋𝑘𝑛𝑙𝑜𝑟 + 𝐼𝑘𝑛𝑜𝑟−1𝑆𝐴𝐿 − 𝐼𝑘𝑛𝑜𝑟−1𝑆𝐴𝐿 = ∑ 𝑌𝑘𝑛𝑜𝑝𝑟 +𝑝𝑙 ∑ 𝑌𝑂𝑘𝑛𝑜𝑝𝑟  ∀ 𝑘, 𝑛, 𝑜, 𝑟 𝑝    [3.19] ∑ 𝑌𝑘𝑛𝑜𝑝𝑟 × 𝑌𝑖𝑒𝑙𝑑_𝑠𝑤𝑘𝑛𝑝𝑞𝑛𝑝 = 𝑍𝑘𝑞𝑜𝑟    ∀ k, q, o, r     [3.20] ∑ 𝑌𝑂𝑘𝑛𝑜𝑝𝑟 × 𝑌𝑖𝑒𝑙𝑑_𝑠𝑤𝑘𝑛𝑝𝑞𝑛𝑝 = 𝑍𝑂𝑘𝑞𝑜𝑟    ∀ k, q, o, r     [3.21] ∑ 𝑌𝑘𝑛𝑜𝑝𝑟𝑝 ≤ 𝐵𝑌𝑘𝑞𝑜𝑟×1,000,000   ∀ k, n, o, r     [3.22] ∑ 𝑍𝑘𝑞𝑜𝑟 × 𝑌𝑖𝑒𝑙𝑑_𝑐ℎ𝑘𝑞𝑜 = 𝐶𝑘𝑟   ∀ k, r      [3.23] ∑ 𝑍𝑘𝑞𝑜𝑟 × 𝑃𝑟𝑜𝑑_𝑠𝑤𝑜𝑘𝑞 ≤ 𝑐𝑎𝑝_𝑠𝑤𝑜𝑟   ∀ o, r      [3.24] ∑ 𝑍𝑂𝑘𝑞𝑜𝑟 × 𝑃𝑟𝑜𝑑_𝑠𝑤𝑜𝑘𝑞 ≤ 20% × 𝑐𝑎𝑝_𝑠𝑤𝑜𝑟   ∀ o, r    [3.25] 𝑆𝑊𝑇𝑜𝑟 = ∑ 𝑍𝑘𝑞𝑜𝑟 × 𝑃𝑟𝑜𝑑_𝑠𝑤𝑜𝑘𝑞    ∀ o, r      [3.26] ∑ 𝑍𝑂𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟 ≤ ∑ 𝑍𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟    ∀r       [3.27] 𝑆𝑊𝑇𝑜𝑟−1 = 𝑆𝑊𝑇𝑜𝑟 − 𝐷𝑂_𝑆𝑊𝑇1𝑜𝑟 + 𝐷𝐵_𝑆𝑊𝑇1𝑜𝑟  ∀ o, r > 1   [3.28] 𝑍𝑘𝑞𝑜𝑟 + 𝑍𝑂𝑘𝑞𝑜𝑟 + 𝐼𝑘𝑞𝑜𝑟−1𝑆𝐴𝑃 − 𝐼𝑘𝑞𝑜𝑟𝑆𝐴𝑃 − 𝐷𝑂1𝑘𝑞𝑜𝑟 + 𝐷𝐵1𝑘𝑞𝑜𝑟 −∑ 𝑍𝐼𝑘𝑞𝑜𝑜𝑜𝑟𝑜𝑜+∑ 𝑍𝐼𝑘𝑞𝑜𝑎𝑜𝑟𝑜𝑎= 𝐷𝑘𝑞𝑜𝑟   ∀ k, q, o, r          [3.29] 𝐵𝐴𝑘𝑞𝑜𝑟 = 𝑍𝑘𝑞𝑜𝑟+ 𝑍𝑂𝑘𝑞𝑜𝑟 + 𝐼𝑘𝑞𝑜𝑟−1𝑆𝐴𝑃 − 𝐼𝑘𝑞𝑜𝑟𝑆𝐴𝑃 − ∑ 𝑍𝐼𝑘𝑞𝑜_𝑜𝑜𝑟𝑜𝑜 + ∑ 𝑍𝐼𝑘𝑞𝑜𝑎_𝑜𝑟𝑜𝑎  ∀ k, q, o, r  [3.30] 𝑍𝐼_𝑂_𝑂𝑂𝑘𝑞𝑜𝑟 = ∑ 𝑍𝐼𝑘𝑞𝑜𝑜_𝑜𝑟𝑜𝑜  ∀ k, q, o, r      [3.31] 𝑍𝐼_𝑂𝑂_𝑂𝑘𝑞𝑜𝑟 = ∑ 𝑍𝐼𝑘𝑞𝑜_𝑜𝑜𝑟𝑜𝑜  ∀ k, q, o, r      [3.32] 𝐷𝑂_1𝑘𝑞𝑜𝑟 ≤ [%_𝑂1𝑘𝑞𝑜𝑟] × Dkqor    ∀ k , q, o, r     [3.33] 𝐷𝐵_1𝑘𝑞𝑜𝑟 ≤ [%_𝐵1𝑘𝑞𝑜𝑟] × Dkqor    ∀ k , q, o, r     [3.34]  𝐼𝑖𝑗𝑘𝑟𝑆𝐶𝐵  ≥ 0 ∀ i, j, k, r = 4        [3.35] 𝐼𝑗𝑘𝑙𝑟𝑆𝑌𝐿  ≥ 0 ∀ k, n, l, r = 4        [3.36] 𝐼𝑘𝑛𝑜𝑟𝑆𝐴𝐿  ≥ 0 ∀ k, n, o, r = 4        [3.37] 𝐼𝑘𝑞𝑜𝑟𝑆𝐴𝑃  ≥ 0 ∀ k, q, o, r = 4        [3.38] ∑ 𝐼𝑖𝑗𝑘𝑟𝑆𝐶𝐵 ≤𝑖𝑗𝑘 = ∑ 𝑇𝑖𝑗𝑘𝑟𝑖𝑗𝑘    ∀ r       [3.39] ∑ 𝐼𝑗𝑘𝑙𝑟𝑆𝑌𝑆𝑗𝑘𝑙 ≤ 20,000 ∀ r        [3.40] ∑ 𝐼𝑘𝑛𝑙𝑟𝑆𝑌𝐿𝑘𝑛𝑙 ≤ ∑ 𝑊𝑘𝑛𝑙𝑟𝑘𝑛𝑙  ∀ r       [3.41] ∑ 𝐼𝑘𝑛𝑜𝑟𝑆𝐴𝐿𝑘𝑛𝑜 ≤ 20,000 ∀ r        [3.42] ∑ 𝐼𝑘𝑞𝑜𝑟𝑆𝐴𝑃𝑘𝑞𝑜 ≤ ∑ 𝑍𝑘𝑞𝑜𝑟𝑘𝑞𝑜  ∀ r       [3.43]  62  Constraint (3.1) ensures that all period cut-block harvests do not exceed the cut-block area. Constraint (3.2) computes the cut–block volume harvested, and constraint (3.3) ensures that the time expended in logging operations will not exceed the available logging capacity. As I constrained the rate of capacity usage between periods, constraint (3.4) computes the time used for logging operations per period and constraint (3.5) adds slack variables to track logging time deviations between periods. The same principle was applied for sort-yards and sawmills; slack variables were defined as percentages of the available period capacity (i.e., constraints 3.5 to 3.11). The first flow balance constraint was on the harvesting side (3.12), which ensured that all volume harvested, plus the stems inventory, are going to the sort-yards. At the sort yard, constraint (3.13) ensured a balance between arrived stems and inventory with sort-yard inputs, and constraint (3.14) determined log production at the sort-yards. However, sort-yards should not exceed the period time capacity, and constraint (3.15) ensured this. Constraint (3.16) calculated the time used at sort-yards per period, and constraint (3.17) works in the same fashion as constraint (3.5). Constraint (3.18) ensured that sort-yard production plus log inventory was in balance with log flows going to the sawmills. Constraint (3.19) ensured a balance between arrived logs at sawmills, log inventory, and log inputs at internal and outsourced sawmills. Constraints (3.20 and 3.21) determined lumber production at internal and outsourced sawmills by multiplying log inputs by lumber recovery factors of sawing policies. A setup sawmilling cost was applied when sawmills changed log species and grades; thus, a binary variable, which can be 1 if the sawmill changes log species and grades and zero otherwise, was activated with constraint (3.22) and then multiplied by a setup cost. The chips produced are determined by constraint (3.23). Constraint (3.24) ensured that the time expended in sawmilling will not exceed the available period’s sawmilling capacity. Constraint (3.25) ensured that the time expended outsourcing sawmilling operations will not exceed 20% of the available period’s sawmilling capacity, and constraint (3.27) ensured that internal sawmill production will not be exceeded by outsourced sawmill production. As I constrained the deviations of capacity usage between periods, constraint (3.26) computes the time used on sawmilling operations per period. Constraint (3.28) limited the sawmilling time deviations between periods by adding slack variables.  A market constraint (3.29) ensured that lumber demand would be satisfied with all sources of lumber production, including internal and outsourced sawmills, lumber inventory, lumber sent in interchanges with other sawmills, and lumber from interchanges with other sawmills. Balance constraint (3.30) determined the volume of lumber to be considered for order fulfillment parameter calculations. Constraints (3.31 and 3.32) 63  determined the volume of lumber that arrived in interchanges from other sawmills to sawmill “o” and determined the lumber that is sent to other sawmills from sawmill “o.” Constraint (3.33) ensured that the amount of lumber produced over demand will not exceed a given percentage of the lumber demand. Constraint (3.34) ensured that the amount of lumber produced below the demand will not exceed a given percentage of the lumber demand. Constraints (3.35 to 3.39) ensured that all forms of final inventories are positive and the volume inventoried per period should not exceed the physical warehouse capacity. Finally, for the sake of simplicity, several parameters calculated by MIP formulations were not included in the formulations, although brief conceptual explanations follow. Process flow time is the ratio between the work of the process inventory and the process throughput. Over/under production is the difference between all sources of lumber production and lumber demand (i.e., positive or negative volumes). Order fulfillment is the ratio between all sources of lumber production divided by lumber demand. Procurement cost is the ratio between the summation of all costs prior to the sawmill log yard and the logs produced by sort yards. Finally, manufacturing cost is the ratio between the summation of all costs after logs are fed to the sawmills and the lumber produced by the sawmills. The agile objective function (3.1 agile) maximizes profit and the lean objective function (3.1 lean) minimizes costs.  Both objective functions operate on the same feasible region (convex hull defined by equations 3.2-3.39).  Therefore, 3.1.agile will always dominate 3.1 lean.  However, this dominance is minimized by the use of penalty coefficients for over/under production and order fulfillment. The penalty values are important components of the models and strongly influence how they behave. They will be discussed further when results are presented.  64  3.5 Results  The three formulations representing manufacturing environments were run with their specific penalties for over- and under-demand lumber production, over- and under-capacity usage, and lumber demand scenarios sets. A summary of profits is shown by ME and by lumber demand scenario in figure 3.1.    Figure 3.1 Profits by ME and lumber demand scenario Three key results were analyzed: 1) SC outputs, which involved profit, procurement, and manufacturing costs; 2) the use of resources, including harvested land, timber supply, and SC inventory; and 3) flexibility, which analyzes SC flow time and order fulfillment (see Appendix B, section B1). Unfortunately, I encountered frequent infeasibilities when the high variability lumber demand scenarios were used.  This was due to capacity constraints, which constrained the rate of change of capacity used between periods. Therefore, the ME formulations were run only 30 times with the corresponding lumber demand scenario data. 65    Figure 3.2 Flexibility parameters by ME and demand scenario  3.5.1 Results for MEs  Incomes for LDSs were similar (i.e., the same demand data); however, SC operation costs determined different profit and performances (see Table 3.6). A descriptive results analysis per ME and LDS follows.   Table 3.6 Economic performance by manufacturing environment and lumber demand scenario ($)   AGILE: LB_LVAGILE: LB_HVBC-SC: LB_LVLEAN: LB_LVBC-SC: LB_HVLEAN: LB_HVAGILE: SB_LVAGILE: SB_HVBC-SC: SB_HVBC-SC: SB_LVLEAN: SB_LVLEAN: SB_HVAGILE: BASE_LDBC-SC: BASE_LDLEAN: BASE_LD- 5 10 15 20 PERIODSFLOW TIME AGILE: LB_LVBC-SC: LB_LVLEAN: LB_LVAGILE: LB_HVBC-SC: LB_HVLEAN: LB_HVAGILE: SB_LVBC-SC: SB_LVLEAN: SB_LVAGILE: SB_HVBC-SC: SB_HVLEAN: SB_HVAGILE: BASE_LDBC-SC:BASE_LDLEAN: BASE_LD80% 90% 100% 110% 120%ORDERFULLFILLMENTAGILE: LB_LVLEAN: LB_LVLEAN: LB_HVBC-SC: LB_LVAGILE: SB_LVAGILE: SB_HVAGILE: LB_HVLEAN: SB_LVLEAN: SB_HVBC-SC: LB_HVBC-SC: SB_LVBC-SC: SB_HVAGILE: BASE_LDBC-SC: BASE_LDLEAN: BASE_LD(20,000) (10,000) - 10,000 20,000 M3OVER-UNDERLUMBERPRODUCTIONManufacturing Lumber SC_Profit Incomes Procurement Manufact. Over-below Penalized Cost penalty Total Environment Demand cost cost capacity incomes under operationalScenario cost over production production costAgile Large batch $21,083,812 $42,976,754 $14,878,728 $2,899,425 $27,677 $467,923 $4,555,034 $22,828,788hybrid low variation $18,736,347 $43,154,522 $18,992,888 $3,204,731 $21,309 $798,001 $2,903,414 $25,920,343Lean $20,920,577 $42,184,183 $15,290,763 $ ,9 2,356 $11,047 $915,114 $3,974,554 $23,093,834Agile Large batch $17,975,964 $29,763,280 $8,938,361 $2,035,174 $10,651 $310,916 $1,114,048 $12,409,149hybrid high variation $17,857,175 $29,949,019 $8,576,841 $2,357,296 $15,283 $1,188,978 $315,850 $12,454,247Lean $17,504,715 $29,012,748 $7,920,327 $1,825,886 $11,211 $69,998 $1,820,606 $11,648,029Agile Small batch $13,549,835 $18,367,724 $3,079,911 $1,299,064 $4,315 $209,722 $644,320 $5,237,333ybrid low variation $11,903,553 $18,635,191 $5,998,998 $1,717,081 $7,550 $1,146,727 $0 $8,870,355Lean $12,721,461 $18,437,293 $4,743,264 $1,340,208 $13,313 $897,123 $516,170 $7,510,078Agil Small batch $13,561,516 $17,842,480 $2,607,007 $1,180,764 $4,674 $182,110 $670,629 $4,645,184hybrid high variation $11,464,684 $18,073,694 $5,982,777 $1,572,740 $10,746 $1,185,514 $0 $8,751,777Lean $12,471,090 $17,960,676 $4,863,570 $1,322,901 $12,568 $1,089,765 $380,311 $7,669,116Agile Base LD $19,671,215 $34,263,929 $10,643,360 $2,225,832 $14,661 $121,198 $1,830,059 $14,835,110Hybrid $20,402,483 $34,442,541 $11,820,911 $2,573,995 $13,834 $1,068,479 $1,033,733 $16,510,952Lean $18,868,779 $33,453,660 $10,326,227 $2,186,995 $10,943 $0 $2,060,716 $14,584,88166  The agile approach was the most profitable, followed by the lean (3.9% below the agile) and the BC-SC (6.4% below agile). Regardless of the variation, for large batches, the agile approach was still the most profitable, followed by the lean (1.6% below the agile) and the BC-SC (6.3% below agile). For large batches, with low variation, the agile and lean approaches obtained almost equal profits, followed by the BC-SC (11.1% below agile and lean). For high variation, the agile and lean profits were almost equal, followed closely by the BC-SC (2.6% below lean and agile). No matter the variation, differences were larger for small batches than for large batches, where the agile approach remained the most profitable, followed by the lean (7.1% below) and the BC-SC (13.8% below). For small batches with low variation, the agile approach still produced the highest profits, followed by the lean (6.1% below) and then the BC-SC (12.1% below). For high variation, the differences were larger, with the agile approach being the most profitable, followed by the lean (8.0% below) and then the BC-SC (15.5% below).   Table 3.7 Summary of profit variation between MEs and lumber demand scenarios   Agile profit Variation w/r  agile  (%) BC-SC  profit Variation w/r  agile (%) Lean  profit Variation w/r  agile (%) Grand average  $ 17,168,468  0%  $ 16,072,849  -6%  $ 16,497,324  -4% Large batches regarless variation  $ 19,529,888  0%  $ 18,296,761  -6%  $ 19,212,646  -2% Small batches regarles variation  $ 13,555,675  0%  $ 11,684,119  -14%  $ 12,596,275  -7% High variation regarles batches  $ 15,768,740  0%  $ 14,660,930  -7%  $ 14,987,903  -5% Low variation regarless batches  $ 17,316,824  0%  $ 15,319,950  -12%  $ 16,821,019  -3% Large batch low variation (LB_LV)  $ 21,083,812  0%  $ 18,736,347  -11%  $ 20,920,577  -1% Large batch high variation (LB_HV)  $ 17,975,964  0%  $ 17,857,175  -1%  $ 17,504,715  -3% Small batch low variation (SB_LV)  $ 13,549,835  0%  $ 11,903,553  -12%  $ 12,721,461  -6% Small batch high variation (SB_HV)  $ 13,561,516  0%  $ 11,464,684  -15%  $ 12,471,090  -8%  The agile manufacturing environment led to lower harvests levels, followed by the lean and the BC-SC, with the exception of the large batch high variation LDS. The timber volume supplied to sawmills followed the same trend. As a consequence, procurement and manufacturing costs followed the same order, and because the lumber demand data were the same in each LDS, incomes were similar for the agile, lean, and 67  BC-SC approaches, although the BC-SC and lean approaches had higher costs. Over- and under-capacity costs were not significant due to the low allowable capacity deviation between periods. In my ME formulations, over-production was paid for with lower prices, while under-production was penalized by charging a cost for the under-production, determined by multiplying the volume of under-production by a fraction of the lumber price. This mechanism was efficient at controlling production and brought flexibility that was translated to a customer service emphasis in the agile, lean and BC-SC approaches. The agile approach had the smallest magnitude of over- and under-lumber production, with the exception of large batches with low variation. Lumber production was mostly below lumber demand (Table B.1 in appendix B, and Figure 3.2). The lean approach produced large volumes over and under the demand for most of the small and large batches. The BC-SC approach produced large over-demand volumes for most of the small and large batches, with the exception of large batches with low variation (Table B.1, in appendix B, and Figure 3.2). Another metric I used was order fulfillment, which measured how closely lumber production was to lumber demand. The agile approach led to production levels being as close as possible to lumber demand, with 89%, 99%, 99% and 98% of demand met by production for large batches of low and high variation and small batches of low and high variation, respectively. For the lean approach, order fulfillment was less stable, with 91% and 92% of demand met by production for large batches of low and high variation, respectively, and 107% and 111% of demand met by production for small batches of low and high variation, respectively. The BC-SC approach produced over the demand most of the time, with 96% and 108% of demand met by production for large batches of low and high variation, respectively, and 114% and 115% of demand met by production for small batches of low and high variation, respectively. Consequently, the BC-SC approach was the worst at controlling lumber production. SC flow time was defined as the ratio between average inventory and throughput; thus, lower flow time indicates higher responsiveness. On average, the agile approach showed 10.4 periods, the BC-SC approach shows 11.7 periods, and the lean approach shows 12.2 periods of flow time. These results agree with Narasimham et al. (2006), who found higher delivery speeds and reliability for the agile approach over the lean approach.   68  3.6 Discussion   My results can be compared with Goldsby et al. (2006), with respect to the economic performance of MEs, although they used discrete event simulation and I used MIP to model the SC. Nevertheless, both analyses determined operational costs. The lowest costs were produced by the agile approach, followed by the lean (5% over) and the BC-SC approaches (14% over). However, Goldsby et al. (2006) showed only a small difference between the lean and agile costs (e.g., only 2%) and a hybrid approach with 15% higher costs. Although the differences in percentages between my costs were larger, both results are close in terms of magnitudes and trends. Furthermore, I found that the operational costs agree with the results of Narasimham et al. (2006), in which lean costs were lowest, closely followed by the agile. Goldsby et al. (2006) analyzed the concept of “order-to-ship time” as a customer service parameter. In my study, I did not explicitly model delivery time, although order fulfillment or over/under production attempts to capture this. They stated that lean approaches exhibits three times lower order-to-ship times than hybrid approaches (i.e., BC-SC) and eight times slower order-to-ship times compared to agile approaches, without mentioning backorder numbers. I considered customer service parameters as order fulfillments as Babazadeh et al. 2012 did. Nevertheless, my results show that the lean approaches produced double the number of backorders of the agile approach, which agrees with the results of Narasimham et al. (2006), which showed higher delivery speeds and delivery reliability for agile approaches over the lean. The lean approach accumulates more inventory. Goldsby et al. (2006) found more inventory in agile than in hybrid approaches, but I found the opposite. The differences in inventory can be related to my over- and under-demand lumber production constraints, which cannot be stated in a DES model as explicitly as in a MIP model. There are also ship-to-order and order fulfillment conceptual differences between the studies. However, my results for order fulfillment, over- and under-production of lumber, and inventory are in agreement with the results of Hallgren and Olhager (2009), which showed that the lean approach had a significant impact on cost efficiency while the agile did not. However, the agile approach had a higher delivery performance (e.g., order fulfillment). In an effort to measure the impacts of demand variation and batch size, I tested the impact of lumber demand scenarios on the agile, lean and BC-SC approaches. Unfortunately, I found no other quantitative studies to compare with my results, so I can only benchmark my results with available theoretical research. My results show that the agile approach has the highest profit, closely followed by the lean and BC-SC 69  approaches. Goldsby et al. (2006) showed that the agile and lean approaches were equally cost efficient, while the hybrid approach was by far the most expensive (the most cost efficient ME was assumed to be the one with highest profits). My results showed that the agile and lean approaches continued to have higher profits the than the BC-SC approach for large batches without considering the effect of demand variation. When lumber demand showed low variation, no change was observed between the agile and lean profits, while the profits of the BC-SC approach decreased by half. These profits are comparable to those identified by Christhoper (2000). While the lean approach was expected to have higher profitability than the agile approach for this type of lumber demand, instead it showed the same costs as the agile approach. However, when lumber demand shows high variation, differences in profits were tiny, with the agile and BC-SC approaches leading profits, followed closely by the lean. These profits are in agreement with those of Christhoper (2000). The BC-SC approach had the highest profits for product demands characterized by high variation and high volumes. For small batches, the agile approach had higher profits than the lean and BC-SC approaches, but with larger profit differences. My results for high variation again agreed with those of Christhoper (2000), who stated that “agile profitably responds to high variety and low volumes.” However, for low variation my results did not show that the BC-SC approach had a higher profit as Christhoper (2000) suggested, probably because I did not explicitly work with mixed portfolios of demands, where the hybrid approach is supposed to perform better. Over and under demand with economic penalties were established to make the agile approach tight, the lean relaxed, and the BC-SC in the middle of the two. Capacity usage deviations between periods and economic penalties were relaxed for the agile approach, tight for lean, with again, BC-SC in the middle. As a consequence, the agile approach always showed lower procurement and manufacturing costs than the lean, which was able to use more capacity and deviate farther from demand, while compromising costs. The BC-SC approach showed the highest procurement and manufacturing costs because log demand was a forecasted average per period and was not exactly the required log demand. As a consequence, order fulfillment (%), and lumber production deviations were higher than they were in the lean and agile approach, where timber was pulled directly from sawmills every period.  The models determined the expected result patterns with an objective function (i.e., central driver) plus a set of constraints as second order drivers. Although Al-Aomar (2011) claimed that lean objectives are in 70  conflict and a multi-objective formulation is required, I showed that ME attributes, such as customer service, chase and level manufacturing strategies, can be modeled with MIP, as Babazadeh et al. (2012) did when modeling the design of a large agile SC. Alternatively, if the agile, lean and hybrid approaches are required to be measured in the short term, their stochastic shop floor techniques could be modeled with DES and VSM, and multi-objective techniques.   Lumber production outsourcing, interchanges of lumber between sawmills, over/under capacity usage, and over/under demand features increase the ability of my MIP formulations to represent MEs. Although over/under production capacity usage penalties did not play a large role in total costs, because its related constraints were too tight, these constraints actively helped to model chase and level manufacturing strategies. However, over/under demand lumber production constraints played a large role in relaxing and controlling order fulfillment and in controlling lumber production, another central feature of MEs (Fisher 1997).   71  3.7 Conclusion  My research shows that manufacturing environments for my case studies can be modeled with a reasonable level of detail with MIP and that these formulations react to the lumber demand scenarios as stated in my objectives. Indeed, my results highlight how manufacturing drivers and demand attributes influence the operational performance of lean, hybrid and agile approaches for the forest-to-lumber SC. When suggesting manufacturing approaches to be adopted for the particular case of integrated BC Coastal forest-to-lumber SCs, attributes of demand should be considered. When lumber demand is stable with low variation and large volumes, agile or lean principles should be adopted. However, when lumber demand is unstable with high variation and large volumes, hybrid, or agile, principles should be maintained. When lumber demand is unstable with high variation and small volumes, an agile approach clearly provides higher profits than any other ME. For the same conditions of lumber demand, but with low variation, agile principles are recommended; however, further analysis is required, because a hybrid (i.e., BC-SC) approach is in theory supposed to be the most appropriate under these circumstances. There is an opportunity to increase profits by 11.1% by adopting an agile strategy when lumber demand is stable with low variation and large volumes. However, the opportunity for increased profits is zero under the same demand attributes but with high variation. Contrarily, there is an opportunity to increase profits by 12.1% by adopting agile or lean strategies when lumber demand is stable with low variation and small volumes, and there is an opportunity to increase profits by 15.5% when lumber demand is unstable with high variation and small volumes. However, these profit increments are subject to over/under capacity and demand penalties, the lumber demand scenarios, and the exploratory nature of this work. These results are specific to my case studies and the over-under demand and production penalties I used. Opportunities for future research include adding statistical analyses and mixed lumber product portfolios to the lumber demand scenarios. Additionally, the postponement concept can be included in the MIP formulation to explore lead time and inventory reduction. In the lumber SC, postponement is delaying the final lumber manufacturing step (i.e., grading or final sizing) beyond sawmills, for instance, in a distribution center closer to lumber markets, where lumber matrices (i.e. double width, and/or double thickness lumber pieces) receive a light manufacturing to quickly produce end-products, and satisfy changing lumber orders demands.  72  Chapter 4 Assessing due date fulfillment for lumber manufacturing production orders  4.1 Summary  Lumber production planning can be classified as a general lot sizing and scheduling problem, which has been addressed with MIP formulations and myopic methods. These models allow backlogs, inventory, and sequence dependent set-ups based on due date of orders, but they do not analyze the effect on costs. The research conducted in this field until now focused on fulfillment of demand for lumber in a certain period without questioning the manufacturing cost of this policy. My research determines the opportunity cost of a multi-period LP approach in comparison with order-processing sequences. I conducted a study to determine the cost of scheduling sequences to process production orders. First, I built a multi-period MIP model (PL) where demands must be satisfied in a certain period of the planning horizon, without explicit orders and due dates. A relaxed version of the PL model allowing backlogs without penalties was also built. Second, I added a production sequence to test how sequences to process orders affected their due dates and called this model planning-scheduling (PS). The periods of the formulation were dropped and an “earliness” constraint that forced orders to be processed without violating their due dates was added. Earliest due date (E), longest processing time (L), and shortest processing time (S) heuristics sequences were used to sort orders which were then fed to the DSS. A relaxed version of this PS model allowing overdue orders (i.e., relaxing the earliness constraint) without penalization was also built. The PL model and its relaxed version PLr were solved with five sizes of demand, and the PS model and its relaxed PSr version were solved with five sizes of orders. The demand and sum of the orders were equivalent in volume, and order due dates occurred in the corresponding periods. The PL approach was the cheapest, although with a small cost gap in comparison to the PS with an E sequence. However, when relaxing overdue orders, this last approach was as efficient as PL, with lower backlogged volumes (high fulfillment of order due dates). Thus, the PS with an E sequence is proposed as the best approach to plan lumber production orders in sawmilling operations. Although the percentage differences in costs of the planning and scheduling approaches PSA-scenarios were not dramatic, the economic effect of the PS with an E sequence could be significant given the operational scale of this type of facility. Moreover, the PS with an E sequence showed the ability to reduce the backlog ratio close to zero, when the corresponding PL backlog ratio was 14%. I did not apply any penalties for backlogs 73  because of the subjective nature of these costs; however, when shortages occurred the demand was lost or backlogged with substantial costs. Therefore, if I adopt my planning approach using a conservative backlog cost, a sawmill which produces 300,000 m3 or 125,000 mbf per year would reduce backlogged orders by 42 thousand m3 (i.e., 14% backlog reduction). If the holding lumber cost is 2 US$/m3, annual savings would be $336,000.  4.2 Introduction  Production-planning in the lumber industry is based on the demand behavior of the customer, the nature of the timber, the attributes of the lumber, and processing technologies. Lumber production planning tools vary from elaborate electronic worksheets to MIP models; although discrete event simulations and pieces count worksheets are also used for sawmill design.  Lumber manufacturers develop aggregate production plans based on facility capacities, expected sales, and timber supplies. Linear programming profit maximization or cost minimization DSSs usually solve the problem, assuming log supplies, sales orders, and machine center parameters are known during the planning horizon. Mill profit goals are based on this analysis. However, in practice, during the day or shift, the problem parameters change, causing re-evaluations and analysis of the aggregate plan to guide short-term planning and scheduling. A short-term problem consists of determining the number of logs, by diameter and length, to be sawn and which sawing pattern should be used. Additionally, the production schedule for the various processes needs to satisfy order volumes and due dates must be determined. Unfortunately, economic deviation occurs when comparing the results of aggregate and short-term production plans, and the human resources used to solve the problem are extensive. The sawmilling industry needs better tools for combining planning and scheduling tasks. Accordingly, the goal of this research was to determine the effectiveness of the earliest due date scheduling policy, which is used extensively in lumber production planning. The relationship between the size of production batches and the lumber manufacturing costs also needs to be examined. To address these challenges, I propose the development of a MIP production planning-scheduler model that explicitly handles due dates, manufacturing sequences, machine center capacities, and sawmill set up costs. A multi-period production planning MIP formulation was built and used as benchmark model. Then, the periods were deleted, and 74  demand by periods were turn into orders with due dates. This planning-scheduling model forced orders to be processed without violating their due dates, but followed the heuristic scheduling rules to process orders. This chapter begins with a review of the subject, and after identifying areas which lack research, the research objectives are established. A detailed methodology to explain the approach follows, particularly, the mathematical model formulations, the data used, and the comparison strategy. After the results are presented, and benchmarked with previous research, further discussion and exploration of the limitations are discussed. Conclusions, analysis of the implications of the results, and the limitations of the approach are addressed at the end of the chapter.  4.3 Literature review  The operational lumber production planning problem (OLPPP) can be classified as a lot sizing and scheduling problem as to when sales orders should be satisfied, considering mill capacity to minimize backlog and inventory costs. However, because log-sawing patterns should be selected with certain criteria to transform logs into lumber, and because the number of processed logs should be scheduled in a sequence to rationally use capacity, this problem can also be categorized as a scheduling, and a cutting and stock problem.  In practice, mill schedulers must be a type of intuitive “optimizer” to balance conflicting goals, such as low inventories and efficient use of capacity. MIP DSSs help to minimize the costs of raw material, end-product inventory, and backlog orders, while keeping within available manufacturing capacity (Clark 2003). Unfortunately, operational mill schedules exhibit sequence-dependent set ups and multiple set ups within a planning period. As a consequence, the lot sizing and scheduling MIP formulation results in large numbers of binary variables, which cause computational intractability. Thus, exact formulations produce high quality solutions but with impractical solution times. A balance is needed between this approach and myopic methods that produce low quality solutions in a relatively short time. Relaxing the binary variables and constraints and breaking the large problem into a number of smaller problems to determine lot size and a sequence of lot setups overcomes the complexity. The poor quality of short-term solutions is partially overcome by considering future demand and capacity using an auxiliary MIP to increase unit production 75  times to factor in typical set ups (Clark 2003). Thus, my sawmill scheduler executed the same work as Clark (2003) noted. However, in addition to solving the cutting pattern optimization problem simultaneously, it also included short-term lumber orders and log inventories. The proposed model should minimize holding and backlog costs, and should handle heuristic scheduling rules to process lumber production orders. Analyzing the same challenge, Araujo et al. (2007) agreed with Clark (2003) on the intractability of the general lot sizing and scheduling problem (GLSS) and when machines with sequence dependent set up costs should be scheduled. There are large numbers of integer variables to model lot size, set up, and sequence-related changeover. Thus, Araujo et al. (2007) applied a rolling planning horizon, implementing only decisions for the next one or two periods in advance, and then the planning horizon was rolled, and the model was applied again with updated orders, demands and inventories. As a consequence, a few integer variables were relaxed to drastically reduce solution times. The ability to roll a planning horizon can be applied to the OLPPP when lumber processes must be scheduled, though the raw material and the nature of sawmills will determine if sequence-dependent set ups exist or are negligible. Maness and Adams (1990) and then Maness and Norton (2002) conducted early research in production planning and scheduling, which shared similarities with the GLSS problem. The linear programming (LP) model maximized revenues by generating a bucking-sawing pattern and by choosing the mix of log booms (i.e., log bundles) that satisfied sales demand. The LP model worked as a scare resources allocator, and the bucking-cutting pattern was generated by a dynamic programming sub-problem. The approach was particularly suitable when backorders and inventories were allowed, but not when there were lumber product orders that should be accurately satisfied. Furthermore, the B.C. coastal forest first-growth timber is supplied to sawmills without accurate diameter-length-grades, so the logs must be assessed for bucking and sawing policies. However, when logs are supplied to a sawmill with greater accuracy, and sorted by diameter classes and length (e.g., second-growth timber), generation of the bucking and sawing patterns do not play a significant role. Moreover, when a sawmill consumes batches of logs previously sorted by diameter-length-grade, a library of optimized sawing patterns is applied to the logs and cant breakdown machine centers. Thus, the efficient use of process capacity, inventory and volume control, and fulfillment of customer due dates become the focus. Currently, sawmills use optimization techniques that are volume or value focused. Unfortunately, in conditions of an open market, over-production and under-production may occur. However, if dynamic prices 76  of demanded lumber can be applied and updated regularly, while sawing is in progress, then optimized production plans result in lumber manufacturing plans that use fewer logs and that reduce over- and under-production (Todoroki and Ronqvist 2002). This use of dynamic prices in the objective function is an efficient way to control production. However, lumber orders are satisfied without a specific due date or manufacturing sequence. Thus, it is expected that these performance measures were optimized for order tardiness, without regard to holding costs, capacity usage, completion time, or process flow time. Adopting the lumber manufacturing problem for second growth forests or plantations, for which logs are accurately sorted by diameter-length and then processed in batches, the OLPPP has been solved in a similar way. In some cases, it was combined with timber transfer decisions among plants (i.e., sawmills) in the context of a SC (Singer and Donoso 2007). Additionally, when solving the OLPPP, a large source of concern was the effect of uncertain lumber recovery factors. For this reason, Zanjani et al. (2010) and Alvarez and Vera (2011) addressed the problem with stochastic and robust programming models to produce production plans that compared performance in controlling backlog orders, total costs and variability.  Alternatively, Maturana et al. (2010) modeled the OLPPP with a LP model and a heuristic approach that mimicked planner decisions. For this purpose, an ideal production scenario was created, and log supply and lumber demand perturbations were added. The LP application solved the OLPPP faster than the heuristic approach, for most of the scenario perturbations. However, the LP formulation did not exhibit explicit scheduling rules, in comparison with longest processing time (LPT) rule that was explicitly present in the heuristic method. Pradenas et al. (2004) also used an MIP formulation to solve the aggregate lumber production planning problem. They attempted to develop efficient solving methods by comparing LP results with a heuristic procedure based on a Tabu search algorithm. Consequently, the heuristic problem solution showed promising performance when compared with an exact problem solution.  Although mathematical programming has been used to solve the OLPPP problem, it has also helped to make other lumber manufacturing economic decisions. For example, Dutrow and Granskog (1986) and, Carino and Willis (2001) showed the ability of LP models to decide on log class purchases, to provide extra capacity to existing sawmills, and to manage rates of consumption and production. Yaghubian et al. (2001) developed a “dry or buy” cost model for a dry kiln scheduling problem. Customer due date orders, kiln availability and capacity, and lumber buying and drying costs were integrated into an MIP model. The dry or 77  buy scheduling problem is a particular version of scheduling “n” jobs on “m” parallel machines. The approach mixed planning and scheduling activities as demands are satisfied over a planning horizon. In particular, jobs were assigned to kilns to be processed in order of due date, which is a property of an optimal solution for a single machine schedule that minimizes maximal tardiness. Consequently, the optimization problem becomes one of allocating jobs to kilns. Thus, jobs are associated with demands (e.g., orders), and due date is satisfied by imposing a constraint that ensures that the last job on demand is fulfilled by the due date. Scheduling involves determining the timing of resource use to conduct activities to produce outputs on time and to satisfy time and relationship constraints among activities and resources. Jobs are activities, machines are resources, and a schedule determines the time that each job starts and finishes on each machine. A schedule accommodates jobs in a certain order and determines the start and completion times. However, it is difficult to determine the sequence that optimizes a performance measure, triggering a combinatorial explosion of possible sequences. The optimum schedule implies a measure of performance; however, costs and profits cannot be related to schedules. Therefore, indirect objectives are used. Thus, completion time, flow time, lateness, and tardiness are common goals, which can be pursued with exact and heuristic scheduling algorithms (Sipper and Bulfin 1997). An intuitive solution approach for scheduling problems is to prioritize jobs and schedule them based on a rule that recognizes the priorities of all the orders that are waiting to be processed on a machine. Prioritization rules can be based on orders or machines attributes. These rules are constructed based statics that prioritize jobs before building the schedule; thus, they are a function of the orders data (Possani, 2001). Static rules provide optimal solutions for certain problems. Shortest processing time (SPT) prioritizes jobs based on their processing times. This creates a sequence of orders with non-decreasing processing times. SPT minimizes the flow time (i.e., minimizes cost) in a single machine environments. Earliest due dates (EDD), prioritizes jobs based on their due dates. EDD yields optimal schedules to minimize tardiness of jobs in single machine environments. Longest processing times (LPT) prioritizes jobs in the same way as the SPT rule, but in a sequence with non-increasing processing times. LPT yields optimal schedules to minimize flow time in single and parallel machine environments. Unfortunately, neither SPT nor LPT consider due dates, because these rules have a flow time minimization focus (i.e., equivalent to cost minimization). Contrarily, if customer satisfaction is important, the earliest due date sequence (EDD), which minimizes tardiness, is the best sequence (Sipper and Bulfin 1997). Thus, if lumber demands by periods can be turn into lumber orders with due dates, the test of such scheduling rules can provide knowledge to assess the value of due date fulfillment. 78  The lumber manufacturing short-term planning problem and the sawmill operation order-jobs scheduling problem have been solved mostly in a hierarchic manner. However, complexity represented by a large number of binary variables in MIP formulations has caused these problems often to be difficult to solve. However, Gaudreaut et al. (2009) formulated MIP models to solve the lumber production planning and scheduling problem for sawmilling, drying and finishing operations.  The MIP formulation for scheduling batches of green lumber drying was relaxed for a simple greedy heuristic based on LPT sequences. This relaxation provided the opportunity to test several strategies of coordination on an agent-based simulation platform. Consequently, relaxing the scheduling problem by imposing a particular sequence to process orders (i.e., jobs) on machines (i.e., resources) rather than generating and testing a large number of combinations of orders schedules, significantly reduced complexity and made the problem tractable ( Yaghubian et al. (2001))  Although there has been effort to solve the GLSS problem with sequence dependent set ups using MIP and myopic methods, limited attention has been given to scheduling rules to efficiently organize production orders (i.e., jobs). When solving the OLPPP, most of research has considered the due date of orders as paramount, without analyzing their effect on manufacturing costs. Hence, my research proposes to explicitly test alternative due date policies, determine their production costs, and analyze the effect of the size of the order on these costs.  4.4 Methodology  The methodology was designed to test whether adding sequences of processing lumber production orders, such as EDD (E), SPT (S) or LPT (L), affected cost backlog and overdue orders when solving the OLPPP. The DSSs formulated to solve the problem are described along with how the demand for lumber products was used to create lumber demand scenarios. The procedure to run the DSSs and how the problem data were fed to relaxed and non-relaxed versions to explore the problem variation are explained below.  79  4.4.1 Decision-making model formulations  4.4.1.1 Formulation with no overdue orders or backlogs  I built a MIP model to solve the OLPPP, in which lumber demands must be satisfied in a certain period of the planning horizon (i.e., days of the week). The formulation did not have due dates, but lumber product demands were placed on certain periods. Thus, the model implicitly satisfied lumber demands in a certain period, which was the equivalent of satisfying an order containing a set of lumber demand products for a certain due date.  Consequently, to test the E, L and S sequences to satisfy lumber orders, the Yaghubian et. al (2000) approach was applied to add a sequence to process orders in the production planning problem. The first optimization MIP multi-period model (PL) was turned into a single period allocation of orders problem. The periods of the formulation were dropped off, then an “earliness” constraint was added that ensured that the processing time of order p, and processing times of preceding times orders must be less than or equal to the due date of order p (constraints 4.3 and 4.6, in model B; Appendix A). Accordingly, this constraint forced orders to be processed without violating their due dates. Consequently, as data orders are E, S, and L sorted, and then fed to the PS model, heuristic sequences can be tested. Mathematically, this constraint is the summation of the processing time of a certain order, plus the processing time of all preceding orders, which must be less than or equal to the order due date to make the order processed on time. Hence, two models were formulated to solve the same OLPPP problem from different perspectives. First, a MIP multi-period model satisfied lumber products demand per period, which implicitly satisfied the due dates of the set of lumber product orders that made up that demand. For every period, this model (PL) produced lumber products to satisfy demand until the production capacity was used, and at the same time, decided the volume of raw materials and products for inventory (see model A; in Appendix C). This approach satisfies customers because they always receive their product on time; however, the approach is expensive because it does not exploit the mill capacity (Sipper and Bulfin 1997). Second, a MIP model (PS) was formulated in which lumber product demands are packaged into orders with due dates, which must be processed in a sequence. Orders are applied to the problem in a certain sequence based on processing times or due dates as Yaghubian et. al (2000) suggested. For the E sequence, orders were scheduled in 80  increasing order of due date. For the S sequence, orders were by increasing processing times. For the L sequence, orders were scheduled by decreasing processing times (see model B; in Appendix C).  4.4.1.2 Formulation with allowed overdue orders and backlogs   Unfortunately, as the planning and scheduling approaches (PSA) with conflicting objectives were tested, not all the model runs were feasible for the lumber demand and order size scenarios that were used. As a result, both model formulations were relaxed to accept backlogs in the PL model and to accept delayed orders in the PS model. No economic penalties were applied either to backlogs or delayed orders because of the subjective nature of those values and the absence of any satisfactory criteria. However, these delays were used in determining backlog ratios. In the PL model, the total of backlogs divided by the lumber products demand determined a value called the backlog ratio ( A_BO1(%), and A_B02(%) in the model). The lower the ratio is, the better the solution is because it means lower backlog volumes in relation to demand.  In the PS model, metrics were based earliness and lateness of orders. Order earliness or lateness in hours was determined based on the time that an order was produced in relation with its due date (A1p, A2p, B1p, and B2p in the model). The hours of earliness or lateness of each order in relation with it due date determined the ratio of earliness or lateness (%) (Tfea1p,Tfea2p, Tfde1p,Tfde2p in the model). The ratio of earliness or lateness of each order multiplied by the order’s volume determined the early or late volume of the order (Ad_1_volp, Ad_2_volp, Ba_1_volp, and Ba_2_volp in the model). The summation of early and late volumes per order determined the total late or early volumes (AV1, AV2, BV1,and BV2 in the model). Finally, the summation of the earliness or lateness ratio of orders determined the total earliness or lateness ratio, which was equivalent to the backlog ratio of the PL model (A_A01, A_A02, A_B01, and A_B02 in the model). The relaxed version of the PL model called PLr accepted backlogs and only a few parts of the original formulation were changed (see constraints 4.2 and 4.9, in model C; Appendix C). Backlogs had to be produced immediately in the following planning period (see constraints 4.5 and 4.9, in model C; Appendix C).  81  The PS relaxed version was identified as PSr. It accepted overdue orders (constraints 4.3, and 4.6, in model D; Appendix C), which resulted in delayed orders relative to the due dates (i.e., hours of delay). Technically, these delays were transformed into volumes using sawing and anti-stain process productivity, which represented the equivalent backlogged volumes of the PL formulation. For PSr, an overdue order, and the equivalent backlog volumes, were not penalized and only a few parts of the original formulation were changed (see model D; in Appendix C).  4.4.2 Problem setting   The lumber production planning problem was set with 10 log diameter classes, regular and pruned log grades, a 100 sawing pattern library, sawmilling and anti-stain treatment processes, and 10 anti-stain lumber products. For the PL model, six periods, equivalent to days of the week were used. For the PS model, a single period of 138 hours was used (i.e., equivalent to 6 days of 23 hours). Each order contained a set of 10 lumber products and every order was equivalent in volume to the lumber product demand that should be satisfied in a certain period for the PL model. Additionally, each order had a due date (in hours), which was equivalent to the period of the PL model.   4.4.3 Lumber manufacturing data  The goal of this chapter is to explore how the cut-to-order lumber manufacturing approach might benefit the BC forest-to-lumber SC as it moves to a second growth log supply. However, as data for a cut-to-order lumber planning environment was not available for the BC Coastal forest to lumber integrated industry, I used a Chilean lumber planning cut to order case study.  Since the log supply and manufacturing methods are consistent with BC second-growth and industrial forecasts, this represents an opportunity to address emergent market scenarios for the BC forest-to-lumber SC. I assumed that this hypothetic-theoretic case study can be taken from the well documented Chilean lumber manufacturing case study (Singer and Donoso 2007, and Maturana et al. 2010).  82  Therefore, the data set used for this study came from a southern Chilean lumber manufacturing system, for which the productivity and capacity of processes and raw material productivity are similar to BC Coastal second growth. However, to protect company confidentiality, logs, labour and inventory costs were modified, but are close to reality. Log supply was assumed to be unlimited to avoid model infeasibilities because of log availability and to keep the focus on effects of sequences rather than other issues (Singer and Donoso 2007). The initial lumber products demand was the lumber products that the library of sawing patterns could produce on average when using all sawmill capacity. The demand was based on a sawmill which consumed up to 111 m3 of logs per hour, and produced up to 62 m3 per hour of lumber depending  on log class distributions, and the sawing pattern library. This assumption matches current company working conditions and is consistent with the Maturana et al. (2010) case study. To test the sequences on the OLPPP, I created five lumber demand scenarios based on the initial lumber products demand. Accordingly, scenarios for lumber products demand focused on rates of variation applied to initial lumber products demand between periods and between lumber products. Thirty values were generated for each scenario. A brief description of these values is given in Table 4.1. Table 4.1 Lumber products demand scenarios. Scenarios  Description Rate of variation Large (La) Strong market The values of the initial lumber products demand were changed by  3%, applied up or down randomly Small (Sm) Weak market The values of the initial lumber products demand were reduced by 20 % and then changed by 3%, applied up or down randomly. Mixed sizes (Mi) Mixed strong and weak market This demand can take a large or a small lumber products demand value. The data generation was made by a choice between a large or a small lumber product demand value generated previously. High rate of Variation (Hi) High volatility in market The values of the original lumber products demand were changed by 50%, applied up or down randomly Low rate of Variation (Lo) Low volatility in market The values of the original lumber products demand were changed by 20%, applied up or down randomly These five scenarios containing 30 sets of lumber demand by period, or order of lumber products, were input to the four PSA models (PL, PS-E, PS-L, PS-S) which resulted in a total of 600 runs. As expected, not all PSA runs were feasible. Therefore, the PL model was relaxed to accept backlogs, and the PS model was relaxed to accept due date delays. These relaxed PLr, and the PSr-E, PSr-L, and PSr-S models were then run with the five scenarios, 30 runs each, for a total of 600 runs for the relaxed cases.   83  4.5 Results  4.5.1 Feasibility analysis  Because I was testing PSAs with conflicting objectives to minimize the cost of imposing heuristic sequences to process orders, not all runs were feasible for all order sizes. My first results determined whether PSA runs were or were not feasible. The PL and PS-E models were the most often feasible (see Figure 4.1). Generally, the PL model satisfied lumber demand by period and could only carry inventories from one period to the following period. The PL model reinforced lumber production in the period in which the lumber demand was ordered, and the PS-E model used a sequence of production in which the order with the earliest due date was produced first. By contrast, the PS-L and PS-S models applied sequences to process orders that reinforced processing times instead of due dates; thus, the feasibility was poor.  Figure 4.1. PSAs feasibility percentage as a percentage of the runs 4.5.2 Lumber manufacturing costs analysis, without backlog and due date relaxation The lumber manufacturing costs were log cost, processing cost, and inventory cost, of which the costs for logs and processing were the most significant (Figure 4.2). 84    Figure 4.2. Lumber manufacturing costs by size of the order.  Figure 4.3. Lumber manufacturing costs by PSAs. Low lumber manufacturing costs were observed for small orders when using the PL and PS-E approaches. However, if I considered lumber manufacturing costs for all PSAs, the lowest cost was the PL, for all order sizes (Figure 4.3), except for two cases. First, the lumber manufacturing cost for PS-L when using small orders was lower. In this case, only one of 30 runs was feasible and produced a lumber manufacturing cost 0.24% lower than the average cost for the PL when using the same type of orders. The second case was with the PS-E model when using orders with high variation in volumes, for which the cost was 0.2% lower than the PL model (Table 4.2). Unfortunately, the PS-S model was never feasible nor was the PS-L model with large, mixed sizes, high volume variation, and low volume variation orders. 85  4.5.3 Economic and manufacturing effects of backlog and due date relaxation  4.5.3.1 Lumber manufacturing costs analysis  To explore the potential of each PSA, market constraints (i.e., constraints 4.3, 4.4, 4.6, and 4.7 in model A) and earliness constraints (i.e., constraints 4.3, and 4.6 in model B) were relaxed. Backlogs were accepted in the PL model and due date delays were accepted in the PS model. Although an overdue order penalty could have been applied, it was not invoked to reduce subjectivity. Consequently, after relaxing the market and earliness constraints, I determined overdue and backlogged volumes. These volumes, in relation with the ordered demand, determined how close every PSA-scenario was to being feasible. Thus, when no value was charged to late orders or backlogs, the average of lumber manufacturing costs by PSA-scenario did not change significantly with respect to the non-relaxed PSA-scenario costs. Accordingly, for large orders, the lumber manufacturing cost did not change more than 0.1%. For small orders, lumber cost did not change over 1.7%, and for mixed orders, cost did not change more than 0.5%. For orders with high variation, cost did not change over 0.1%, and for orders with low variation, cost did not change over 0.2% (see Table 4.2 and Figures 4.4 and 4.5).  Figure 4.4 Lumber manufacturing costs by size of the order (relaxed PSAs). 86   Figure 4.5 Lumber manufacturing costs by relaxed PSA.  Figure 4.6 Deviation of relaxed PSA costs relative to the corresponding non-relaxed PSA costs  87  Table 4.2 Lumber manufacturing costs for planning and scheduling approaches.   Lumber demand No backlogs or overdue orders  Allowing backlogs or overdue orders Backlog-Late  Backlog-Lateness PSA scenario Cost ($) Δ (%)  Cost ($) Δ (%) vol [m3] ratio PL Large 448,498  0% 448,498  0% 1,199  16%   Small 359,539  0% 359,539  0% 1,285  21%   Mixed orders  402,218  0% 402,317  0% 1,257  18%   High  451,397  0% 451,417  0% 1,371  18%   Low   450,796  0%   450,759  0% 1,149  15% PS-E Large 448,880  0.1% 448,880  0.1% 0  0%   Small 365,779  1.7% 365,731  1.7% 0  0%   Mixed orders 404,256  0.5% 404,045  0.4% 0  0%   High   450,401  -0.2% 450,787  0.1% 3  3%   Low   451,036  0.1% 451,036  0.1% 0  0% PS-L Large   -    n/a  449,018  0.1% 3,130  41%   Small  358,694  -0.24% 365,760  1.7% 1,980  31%   Mixed orders -    n/a 404,326  0.5% 2,555  38%   High  -    n/a 450,919  0.1% 3,207  44%   Low  -    n/a 451,552  0.2% 3,832  51% PS-S Large -    n/a  449,034  0.1% 3,891  50%   Small -    n/a 365,760  1.7% 3,353  42%   Mixed orders -    n/a 404,326  0.5% 2,661  38%   High  -    n/a 450,919  0.1% 3,973  47%   Low    -    n/a 451,552  0.2% 3,293  41%   Figure 4.7 Backlogged volumes by relaxed PSA 88  4.5.3.2 Due dates and backlog relaxation analysis  The effect of the relaxation on overdue volumes was larger than for lumber manufacturing costs. Larger averages of delayed orders (hours), and as a consequence, backlogged volumes, were found with the PS-S and PS-L relative to PS-E. However, the average of backlogged volumes for the PL approach was significant as well. For example, the PS-E average of backlogged volumes was 0.5% of the volumes of lumber demand ordered. For the PLr, the average of backlogged volumes was 17% of the volumes of lumber demand ordered. For PSr-L, the average of backlogged volumes was 41% of the volume of orders ordered, and for PSr-S, the average of backlogged volumes was 44% of the volume of orders ordered, independent of the order size (Table 4.2). Figure 4.7 shows the distribution of backlogged volumes by PSA-scenarios.   4.6 Discussion  This research determined the opportunity cost of the multi-period LP approach in comparison with order processing sequences. Most research conducted in this field has focused on fulfillment of demand for lumber in a certain period without questioning the manufacturing cost of this policy.  The PSA-scenarios were run without demand/order relaxation, which resulted in infeasible runs for the PS-L and PS-S approaches for most of the lumber demand order sizes. The reason for the infeasibility was due to the processing time of a certain order exceeding the due date of the order and violating the earliness constraints. This occurred because the PS-L approach began processing the order with the largest processing time. However, when orders were very small, the earliness constraints could still be satisfied because of the short processing time, explaining why one run with PS with the LPT sequence was feasible with small orders.  The PS-S and PS-L models, regardless of order due dates, imposed sequences to process orders that focused on processing time instead of due dates (i.e., orders with the longest or shortest processing times). Because of PS-L and PS-S, sequences were focused on cost reduction rather than customer satisfaction (Sipper and Bulfin 1997).  89  The distribution of backlogged volumes indicated how close a PSA-scenario was to being feasible (i.e., backlogged volumes close to zero). For example, for one run of PS with an LPT sequence, small orders were feasible because backlogged volumes were zero (Figure 4.7). Moreover, a few sets of orders with mixed sizes and high variation in sizes were close to a backlog volume of zero, which would make them feasible. Additionally, the PS with the SPT sequence and large, small, and mixed sizes orders were close to having backlog volumes of zero, which would make them feasible as well. Despite the infeasibilities for PS-L and PS-S models, the PL and PS-E models produced lumber manufacturing costs results that were very similar among order sizes. The PS-E sequence was similar to how the multi-period PL model processed lumber demands in the period when the demand was placed. As a consequence, the PL and the PS-E produced very similar costs. However, because of the heuristic nature of the EDD sequence, the PL model always produced lower lumber manufacturing costs. Those findings are in agreement with the results of Maturana and Pissani (2008), in which an LP multi-period model and a heuristic scheduling planning tool based on due dates were compared. However, they reported larger differences between (e.g., 52%) costs determined by the LP model and the heuristic approach for lumber demand scenarios, likely because they chose larger and specific data perturbations (e.g., increasing the demand of a low value product by 10%). In contrast, I applied random changes within certain rates to all lumber demand data. Therefore, my differences were smaller; for example, the PS-E cost was only 1.7% more than the PL cost when processing small orders.  As Gaudreaut et al. (2009) noted for industrial settings, overdue orders cannot be avoided. Because the earliness constraints (constraints 4.3 and 4.6, in model B; Appendix 1) were responsible for infeasibilities, I added surplus and slack variables to determine whether orders were early or delayed and their respective early or late (i.e., backlogged) volumes. Thus, the efficiency of planning approaches can be judged by late deliveries. Accordingly, the relaxed results indicated that the values of ratios of backlogged orders were on average 0.5% for the PSr-E, 17% for the PLr, 41% for the PSr-L, and 44% for PSr-S approaches, regardless of the size of the order. These values were in agreement with those of Gaudreaut et al. (2009), who found levels of ratios of backlogged volume of orders between 26% and 50% when solving the same of problem. However, they solved the problem using a planning-scheduling MIP and an LPT heuristic model with coordination protocols over an agent-based platform. 90  The PS-E approach illustrated some interesting abilities of PSAs. It saved lumber manufacturing costs when planning orders with high variation in volume of orders, with 0.2% lower cost than the PL for the same order sizes. However, the PS-E was as efficient as the PL when processing large and small variations orders as well (see Table 4.2). If I consider the results of the relaxation case, the PSr-E showed a greater ability to reduce due date delays and therefore overdue volumes in comparison with the PL. Additionally, the PS-L approach might be applied when small orders are planned (although only one in 30 runs was feasible) with a cost 0.24% lower than the average of the PL for the same order size. Generally, this PSA was the approach closest to being feasible for small and mixed size orders, and thus, the potential use would be extended to these order sizes. The PS-S approach was not as close as the PS-L approach to being feasible because the PS-S did not have any feasible runs when no due date relaxation was used (see Figure 4.2).  The LP multi-period formulation was highly efficient at solving the lumber manufacturing production planning problem as a general lot sizing and scheduling problem with cost minimization (Maness and Adams 1990, Maness and Norton 2002, Singer and Donoso, 2007, and Pradenas et al 2004, Maturana and Pisani, 2008). LP was a very effective solution method when changing the size of the orders and testing sequences to process orders. However, this research showed that this model formulation did not guarantee the cheapest production plan, or the lowest backlogged volumes in any circumstance. Although the percentage differences in costs of the PSA-scenarios were small, the economic effect of the PS-E approach could be significant given the operation scale of this type of facility. Moreover, the PS-E approach showed the ability to reduce the backlog ratio close to zero, when the PL backlog ratio was 14%. I did not apply a backlog penalty cost, so no economic impacts were observed; however, when shortages occurs the demand is lost or backlogged with substantial costs. In manufacturing, shortages costs are based on an estimate of lost revenue. Gupta and Martin (2015) suggested a cost ratio of 1:4 between holding inventory cost and backlog cost, but West (1988) assumed that a reasonable value for a backlogged unit must be the delivery price. Therefore, if I adopt my planning approach using a conservative backlogs cost (i.e., a ratio of 1:4 between inventory and backlog costs), with a holding lumber cost of 2 US$ per m3, a sawmill which produces 300,000 m3 or 127,120 Mbf per year would reduce backlogged orders by 42 thousand m3 (i.e., by 14%). As consequence, annual savings would be US$336,000. 91  The set-up cost was not added to lumber processing costs to avoid the use of binary variables to model the fixed cost of changing production plans, which meant an LP formulation. The main reason for this relaxation decision was to help to solve the problem within a reasonable time. Maturana and Pissani (2008) and Yaghubian et al. (2000) applied it to the lumber manufacturing production planning problem and to the dry kiln scheduling problem. Also, because high production sawmills today are under CNC control on log and cant breakdown machines, set up times are negligible4.  4.7 Conclusions  The PL approach produced the lowest manufacturing costs, albeit with only a small advantage in comparison with the PS-E. However, when relaxing overdue orders, this last approach was as efficient as PL with lower backlogged volumes. Therefore, the PS-E model is proposed as the best approach to plan lumber production orders in sawmilling operations.  Integrated forest companies have customers with different due date requirements, and a large proportion of their production is focused on the internal customer with high due date flexibility. Most of the lumber production planning research was conducted without consideration of any due date flexibility, and only backlogs in term of volume of lumber products were considered. Overdue orders can be weighted differently, depending on customer willingness to accept delays. An opportunity for new research would be to explore approaches to model the problem with flexible due dates for customers rather than excessively tightening the lumber manufacturing production planning problem. The timber supply change in BC from old to second growth provides an opportunity to craft an efficient future cut-to-order lumber industry. The homogeneity of the logs plus oriented sawmills processing accurate diameter-length log classes and smartly chosen timber supply areas may bring a business alternative for the BC coastal forest industry (e.g., as has been the case in Sweden, Chile and Brazil). In such a case, lumber planning must address accurate customer orders on volume and opportunity, but keeping healthy levels of inventory and backlogs. Thus, a lumber planning DSS such as PS-E may provide a useful tool to assess the ability to compete in a cut-to-order lumber business scenario. However, it would                                                      4 Personal conversation with Mr. Pedro Cardenas, mill chief planner 92  also provide lumber cost, log-lumber inventory levels, and backlogs levels and their cost to benchmarks current BCSC lumber planning practices.  The limitations in this research should be recognized. First, only 30 model runs were performed by order size. A larger number of data sets and model runs would improve the reliability of the results. Second, I created data sets based on the ideal lumber demand for this sawmill case study, and then randomly chose rates of variation. Thus, those rates of variation should be further analyzed. Finally, I formulated decision-making models for sawmilling and anti-stain operations only, excluding kiln drying operations. To include such operations could be a challenging task because of their use of parallel resources.  93  Chapter 5 Conclusions  5.1 General conclusions  The British Columbia (BC) Coastal forest industry has declined over the last few decades because of more restrictive harvesting policies, high timber costs, old sawmills, and market changes affecting BC lumber. These factors have resulted in a lack of competitiveness in the global economy. In particular, B.C. coastal sawmilling costs are at the high end of the global spectrum due to the use of slower head rig mills to process large logs to produce specialty lumber, which restricts the mills from participating profitably in commodity markets. While this strategy achieves high revenue for high value species it does not offset the high costs of less profitable species (Cohen 2007). At the same time, the coastal timber supply is declining in quality because of the transition from old-growth to second-growth forests (Pearse, 2001). The predominant focus for improving efficiencies for the BC coastal forest industry involves increasing throughput and capacity utilization and reducing inventories. However, long lead times make it exceedingly difficult to control inventory. Also, uncertainty in supply and variation in supply quality trigger large amounts of by-products, raw material and end products in inventory (Haartveit et al. 2004). I determined that there were research gaps in SC management and manufacturing strategies, and this research tested alternative manufacturing strategies for the forest-to-lumber SC. The objectives of this thesis set out to determine: 1) the value of timber grade information for lumber production planning purposes; 2) what parts of the SC should be lean, agile or hybrid, and 3) the value of planning and scheduling policies for cut-to-order lumber manufacturing. A comprehensive DSS using MIP to model the forest-lumber SC was formulated. Through the use of case studies and adding SC policy features to the DSS, impacts on SC performance were assessed. In particular, the value of timber grades in forest inventories for planning purposes was determined, the appropriate use of agile, lean, and hybrid manufacturing systems for the forest-lumber SC, and the benefits of cutting to order lumber manufacturing approach focused on meeting lumber orders due dates on manufacturing cost was determined. 94  The first objective was to determine how the performance of the BC coastal forest industry could be improved by developing policies to better manage the uncertainty in timber supplies. A hybrid approach combined distribution-based timber yield inventory data with a LP DSS and analytical decision verifications were applied. A multiple site-products-period model was formulated to solve the production planning problem for the BC coastal forest case study. The problem was formulated as profit maximization of three nodes containing logging, sort-yard and sawmill manufacturing units, connected by appropriate balance equations. Four error scenarios were analyzed based on the size of the sample used to estimate timber grade error, the size of the cut-block in use and the cut-block data used in this study. The four error scenarios were run with 12 lumber demand targets. A failure probability curve for fulfilling lumber demand was determined, as well as SC profit variations for the error scenarios.  The second objective was to determine the ability of lean and agile manufacturing principles to improve SC performance in comparison to current BC coastal forest-lumber production methods. Manufacturing drivers were translated into agile, lean and hybrid manufacturing models using MIP formulations. The objective was to determine how these manufacturing environments performed relative to each other when lumber demand changes. Demand variations were made to simulate the way that demand changes through the time. Four LDSs, plus a base case, were created representing combinations of production batch sizes and a variety of lumber products. The LDSs were analyzed by three DSSs, representing the three manufacturing environments to assess the performance of the manufacturing environment. The third objective was to determine whether adding sequences for processing lumber production orders, such as EDD, SPT and LPT helped the efficiency of the operational lumber production planning problem. First, a multi-period MIP model was built where demand must be satisfied in a certain period of the planning horizon, without explicit orders and due dates. A relaxed version of this first model that allowed backlogs without penalty was also built. Second, I added a manufacturing sequence to test how this method processed orders with respect to their due dates. The periods of the first formulation were dropped, and then a “sequence” constraint was added that forced orders to be processed with a predetermined heuristic sequence. A relaxed version of this model that allowed overdue orders without penalties was also built. The first model and its relaxed version were fed with five sizes of demand, and the second model and its relaxed version were fed with five sizes of orders to explore how decisions, SC metrics, and costs were impacted. 95  In Chapter 2, it was concluded that the SC revenue was negatively and proportionally affected by timber volume and grade estimation errors. Also, the ability to fulfill lumber demand was compromised by timber volume and grade error magnitudes. This suggests that decision makers should be aware that demand fulfillment for low commodity lumber products, which for this case study represent 29% of the lumber demand, will be compromised, depending on the timber volume and grade estimation errors in use. This SC performance drawback is not extremely relevant for the current low commodity lumber products market orientation (i.e. U.S. housing market). However, it could compromise the BC coastal industry’s ability to efficiently participate in other cut-to-order lumber markets with tighter due dates and lumber demand fulfillments requirements (i.e., Japanese thin board market). In relation to estimation errors and cut-block size, first, the SC profit will be under-achieved by 6.02% when using normal cut block-size errors. However, if the timber grade error is reduced by 25%, SC profit will be under-achieved by only 4.62%. Second, the SC profit will be under-achieved by 2.24% when using super cut block-size errors. If the timber grade error is reduced by 25%, SC annual profit will be under-achieved by only 1.76%.  In Chapter 3, it was concluded that lean, agile, and hybrid manufacturing principles can be modeled with a reasonable level of detail using MIP formulations for my case studies. These react to lumber demand behaviors as stated in the chapter’s objectives. When recommending manufacturing approaches for the BC-SC, attributes of demand play a large role and need to be considered. First, when lumber demand is stable, with low variety and large volumes, agile or lean principles should be adopted. When lumber demand is unstable with high variety and large volumes, hybrid or agile principles should be adopted. Second, when lumber demand is unstable, with high variety and small volumes, an agile approach clearly brings higher profit than any other approach. However, this needs to be qualified with respect to the objective functions (3.1 agile v. 3.1 lean). Agile should always dominate lean (maximization v. minimization), but the over-under demand and production penalties I used make the models comparable in terms of profit. For the same lumber demand conditions, but with low variability, an agile approach is also recommended. However, further analysis is required, because the hybrid approach should be the best approach under these conditions according to the relevant literature. If the results in this chapter hold in real situations, the opportunity for profit improvement is 11.1% for adopting an agile approach when lumber demand is stable, with low variety and large volume. However, the opportunity for profit improvement is nill with the same demand attributes, but with high variability. The opportunity for profit improvement is 12.1% for adopting agile or lean approaches, when lumber demand is stable with low variability and small 96  volumes. The opportunity of profit improvement is 15.5% when lumber demand is unstable with high variability and small volumes.  In Chapter 4, the opportunity cost was determined for a multi-period LP (Plann) approach in comparison with heuristic sequences (Plann-sched) to solve the lumber manufacturing planning problem. It was concluded that the Plann approach was the cheapest, although without a large cost gap in comparison with the Plann-sched with an EDD sequence. However, if overdue orders were relaxed, the latter approach was as efficient as Plann with lower backlogged volumes. Therefore, Plann-sched with an EDD sequence is proposed as the best approach to plan lumber production orders for the case study. Although the percentage differences in costs of the PSA-scenarios were not dramatic, the economic effect of the Plann-sched with an EDD sequence could be significant given the operation scale of this type of facility. Moreover, Plann-sched with an EDD sequence was able to reduce the backlog ratio close to zero, while the Plann backlog ratio was 14%. I did not apply any penalties, so the economic significance was not quantified; however when shortages occurred, the demand was lost or backlogged with substantial costs. If this planning approach is adopted using conservative backlogs cost (i.e., a ratio of 1:4 between inventory and backlog costs), with a holding lumber cost of 2 US$ per m3, a sawmill which produces 300,000 m3 or 127,120 Mbf per year would reduce backlogged orders by 42 thousand m3 (i.e., by 14%). As a consequence, annual savings would be US$336,000. Change in BC coastal timber supply from old- to second-growth provides an opportunity to craft an efficient future cut-to-order BC lumber industry. The homogeneity of second-growth logs plus oriented sawmills processing accurate diameter-length log classes and smartly chosen timber supply areas, may often a business alternative for BC coastal forest industry (e.g., as the case of: Sweden, Chile and Brazil). In such a case, lumber planning must address accurate customer orders on volume and opportunity, keeping healthy levels of inventory and reducing backlogs. Thus, a lumber planning DSS like Plann-sched may prove a useful tool to assess the ability to compete in a cut-to-order lumber business scenario. However, it would also provide lumber cost, log-lumber inventory levels, and backlogs levels and their cost to benchmarks current BC SC lumber planning practices.  SC management policies and production strategies can be addressed with a high level of precision using MIP. The ability of mathematical programming to model the forest to lumber SC was reasonable enough to test the improvement potential of manufacturing management policies. Some of the challenges were finding 97  appropriate BC coastal forest data, choosing appropriate values for economic penalties, and translating “soft” problem drivers to model objectives and constraints.  5.2 Limitations, strengths and weakness of this research  Several general limitations of this research must be noted. For instance, manufacturing and economic data from the case-study used in Chapters 2 and 3 were extracted from publicly available annual reports. Second, the forest database was developed by Vahid, 2012, using a dispersal algorithm (Schwab and Maness, 2009), to rebuild spatial forest inventories from aggregated data. Third, the forest roads and transportation costs were determined assuming Euclidian distances. Fourth, sawing pattern and bucking patterns came from a theoretically generated library of patterns. Five, since this research focused on the old-growth BC coastal forest, timber and lumber quality attributes were assumed age and site dependent, and lumber grades assumed that the older the timber, the proportionally larger and thicker then logs would be (Zhang 2009). Due the use of non-accurate forest data, costs/incomes, and SC data, SC metrics calculated may not accurately reflect reality. Six, no delays on time or seasonality were considered. As consequence of this assumption, the DSS decided to harvest logs mostly in the period when the customer demanded lumber. Otherwise, the DSS would be forced to begin harvesting higher volumes in previous periods to satisfy customer demands, leading to higher SC inventories and costs. It is also important to note that my results are specific to the case studies that model a single firm, so results do not necessarily reflect the industry as a whole. For example, competition between firms and how that influences the choice of manufacturing environments is not assessed in my research. The firm modelled is a vertically integrated firm, and it will behave differently than a non-integrated firm that has to compete in a competitive log supply market. In Chapter 2, the results are limited by the size of the sample used to estimate timber grade error, the value of the timber volume estimation error, and using these error values for all cut blocks. Another limitation could be the verification method (Excel matrix), given its empirical nature. A larger sample to estimate timber grade error could increase the reliability of the profit under estimate. The consequence of using a decision verification method rather than running the DSS again when timber grades were affected by uncertainty could affect the quality of the solution. It is important to keep the level of inventory and grade 98  error in perspective. Regardless of how much we spend trying to reduce these errors, error will always be present, and we must plan accordingly. For Chapter 3, it should be noted that the profit increments suggested by assuming manufacturing environments were subject to: 1) capacity and demand fulfillment penalties; and 2) the way soft and hard manufacturing drivers from lean, agile, and hybrid principles were translated into the MIP model. On one hand, if the size of over- and under- capacity penalties increase, the differences in profits between ME’s will be dramatic or weak depending on lumber demand shape. If it is increased, will help the agile environment to be more profitable, and if it is decreased will help the lean environment to be more profitable. Economic demand fulfillment penalties work in the same manner. For instance, if the value of the penalty is high for over-production, that policy will help the agile environment to be more profitable, and if this value is low for over-production that policy will help the lean approach to be more profitable. Lean, agile and hybrid manufacturing soft principles were translated into hard mathematical programming objective functions and constraints. Those model formulations provided an opportunity to explore the forest to lumber planning SC, however, my formulations did not capture the whole richness of the manufacturing environments. They do capture the essentials that provide economic and market attributes, but future research should be done to better capture this richness. Research methods should include simulation modeling techniques because manufacturing principles could be in conflict, or very hard to optimize, or even hard to sort based on their relative importance. For Chapter 4, the data sets were based on the ideal lumber demand for the sawmill, and random changes were made around chosen rates of variation. Thus, those rates of variation should be further analyzed. Larger rates could have affected the cost of lumber manufacturing plans, but as no setup cost was considered in my research, their impact was not determined.  A DSS for sawmilling and anti-stain operations was formulated, excluding the kiln drying operation. Adding the last operation would be a challenging task because of its use of parallel resources. In such a case, MIP and non-exact algorithm formulations should be considered as a solution method to avoid large solution times. The sawmilling setup costs were not included in either the Plann or Plann-sched formulations. This allowed avoiding the use of binary variables.  Maturana and Pisani, 2008, and Yaghubian et. al 2000 also applied 99  this relaxation when solving similar problems. In highly automated sawmills today, headrig machines are CNC controlled, so setup times no longer depend on production plans. Hence, setup costs can be added to the manufacturing costs. However, small differences in setup costs could trigger large differences in lumber manufacturing cost when processing small lumber orders. Despite of the ability of lumber planning DSS developed  to model the cut-to-order lumber manufacturing scenarios, and the similarities between the Chilean and BC Coastal cases studies, there are differences related to the nature of logs, sawmill configuration and performance that limit my results. The BC Coastal case study complexity could make the mathematical programming approach developed harder to solve, because of a higher variety of products, log sorts, heterogeneity of logs sort, and sawing patterns. The cut-to order lumber manufacturing scenario only holds for stable and homogeneous log grades supply, as well as for oriented sawmills feeds with log sort batches which make the production very efficient. Thus, these lumber manufacturing conditions could be harder to immediately implement in the BC Coast, involving large investment, radical changes in market orientation and timber supply policies.   5.3 Future research and potential applications  There are several new questions and future work that I identified in my research. For instance, in Chapter 2, a common value for timber volume error was used, and a common value by species for timber grade error was used for all cut blocks (these values were affected by uncertainty later). However, for old-growth forest cut blocks, this assumption is naive, and both error values should differ from cut block to cut block to capture the whole uncertainty of the problem. Also, in the DSS formulation, different targets for lumber production were used to explore demand fulfillments. However, an unexpected DSS behaviour was observed when the lumber target decreased (i.e., the profit increased). The DSS was formulated to maximize profits, so sawmills produced as much as possible of the high value products first, and fell short of low value products. This is because the market constraint allowed the lumber production of sawmills to go above and below the market demand. However, the profit penalties used for sawmill over-production (half of lumber product price), and under-production (lumber product price) were not as effective as was initially thought. Thus, further exploration of this unexpected behavior is suggested. 100  For assessing the performance of the manufacturing environments in Chapter 3, future research should focus on: 1) increasing model runs for MEs and LDS and adding statistical analysis; 2) adding a mixed lumber product portfolio to LDS; and 3) adding a postponement concept and a new framework to penalize over under SC capacity to the MIP formulation (i.e., postponing cant breakdown until final lumber sizes).  Finally, although integrated forest companies have customers with different due date requirements, a large proportion of their production is focused on internal customers who have a high due date flexibility. Lumber manufacturing planning research has been conducted on economic penalties applied to backlogged lumber products volumes, without consideration of due date flexibility. Thus, future research is needed to assess the lumber customer’s willingness to accept overdue orders in order to reduce costs and improve feasibility of OLPPP.  Research for short term lumber manufacturing planning should also examine adding more processes to the lumber manufacturing SC and exploring the impact of log supply and lumber demand uncertainty on lumber manufacturing costs. The modelling framework I developed has potential application for a number of larger policy issues, such as industry integration/concentration, forest management, emerging markets, lumber recovery factors, log exports and currency exchange rates. Integration and concentration versus a decoupled log supply would be an interesting case study to see how the value of forest land and forest management is affect by open log supply competition. Internal log transfer pricing and taxation rates are interesting components of this study.  Emerging markets for BC lumber products (e.g. China) increases the size and complexity of the supply chain, thus models such as mine can contribute to planning in this environment.  My models use a series of discrete lumber recovery factors for each size/grade/species of log.  An interesting field of research is to introduce some randomness to these discrete lumber recovery factors to better reflect the real system that does vary due to internal log defects, etc. One approach might to be to have lumber recovery factors modelled as continuous, random variables.  Finally, log exports and Canadian dollar exchange rates could be added to the models to explore government, industrial and firm impacts related to these major economic drivers.    101  References  Agarwal A., Shankar R., and Tiwari M.K. (2006). Modeling the metrics of lean, agile and leagile supply chain: An ANP-based approach. European Journal of Operation Research. 173(2006):211-225.  Al-Aomar R. (2011). Handling multi-lean measures with simulation and simulating annealing. Journal of the Franklin Institute. 348(2011): 1506-1522. Aldulmalek F., and  Rajgopal J. (2007).  Analyzing the benefits of lean manufacturing and value stream mapping via simulation: A process sector case study. Int. J. Production Economics. 107(2007): 223-236. Araujo S.A., Arenales M.N., and Clark A.R. (2007). Joint rolling-scheduling of materials processing and lot-sizing with sequence-dependent setups. J Heuristics. 13(2007): 337-358. Babazadeh R., Razni J., and Ghodsi R. (2012). Supply chain network design problem for a new market opportunity in an agile manufacturing system. Journal of industrial Engineering International. 8(19):1-8. Balaji P.G., and Srinivasan D. (2010). An Introduction to Multi-Agent Systems. Innovations in MASs and Applications. 310(1): 1–27.  Beamon B. (1999). Measuring supply chain Performance. International Journal of Operations & Production Management. 19(3): 275-292. Beaudoin D., Frayret J., and LeBel L. (2010). Negotiation-based distributed wood procurement planning within a multiform environment. Forest Policy and Economics. 12 (2010): 79-93.  Beaudoin D., LeBel L., and Frayret J.M. (2006). Tactical supply chain planning in the forest products industry through optimization and scenario based analysis. Canadian Journal of Forest Research. 37(1): 128-140. 102  Billington P.J., McClain J.O., and Thomas L.J. (1983). Mathematical Programming Approaches to Capacity-Constrained MRP Systems: Review, Formulation and Problem Reduction. Management Science. 29(10): 1126-1141. Birge John R., and Francois Louveaeux. (2011). Introduction to Stochastic programming. Second edition. Springer series in operational research. Springer.  Blakcwell P., and Walker J. (1984). “SAWMILLING”. Chapter 7. School of Forest and Ecosystem Science.  University of Melbourne, Australia. Borders B.E., Harrison W.M., Clutter M.L. Shiver B.D., and Souter R.A. (2008). The value of timber inventory information for management planning. Can. J. For. Res. 382009): 2287-2294.   Bruce M. and Daly L. (2004). Lean or agile A solution for SCM in the textiles and clothing industry?. International Journal of Operations & Production Management.  24(2): 151-70. Buongiorno J. and Gilles K. (2003). “Decision Methods for Forest Resources Management”. 1st edition. Academics Press. San Diego, California. Calderon J. and Lairo F. (2007). Simulación de cadenas de suministro: nuevas aplicaciones y áreas de desarrollo. Informacion Tecnologica. 18(1): 137-146. Carino HF, D Willis. (2001). Enhancing the profitability of a vertically integrated wood products production system: Part 2. A case Study. Forest Products Journal. 51(4): 45–53. Chopra S., and Meindl P. (2000). Supply chain management: Strategy, planning and operation. 1st edition. Prentice Hall College Div. United States. Christopher M. (2000). The Agile Supply Chain Competing in Volatile Markets. Industrial Marketing Management. 29: 37–44. Clark A. (2003). Optimization approximations for capacity constrained material requirement planning. International Journal Production economics. 84:115-131. Coast Forest Products Association. (2004). Products Directory. http://www.coastforest.org/products/species/western-hemlock/. Accessed on May, the 5th 2014. 103  Cohen D. (2008). B.C Coast Strategic Option: Current Business, Future Opportunities and Outlook to 2010. Draft paper. Coast Forest Products Association. Vancouver, March of 2007; 1-181.  BC Ministry of Forest, Lands and Natural Operations, Revenue Branch. (2009). Cruising manual. Retrieved 28 June 2011, from https://www.for.gov.bc.ca/hva/manuals/cruising.htm D’Amours S., Ronnqvist M., and Weintreaud A. (2008). Using Operational Research for Supply Chain Planning in the Forest Products Industry Infor: Information Systems and Operational research, University of Toronto Press. 46(4): 1-18 pp. Davis L.S., Johnson K.N., Bettinger P., and Howard T.E. (2005). Forest Management: To Sustain Ecological, Economic, And Social Values. Fourth edition. Waveland Press Incorporated. Illinois. Detty R., and Jon Y. C. (2000). Quantifying benefits of conversion to lean manufacturing with discrete event simulation: a case study. Int. J. Prod. Res. 38(2):429-445. Donald S., Maness T., and Marinescu M. (2001).“Production planning for integrated primary and secondary lumber manufacturing”. Wood and Fiber Science. 33(3): 334-344. Dutrow GF, JE Gransckog. (1973). A sawmill manager adapts to change with linear programming. U.S.D.A. Southern Forest Experiment Station. New Orleans, La. Forest Services research Paper, SO (88):1-11. Duvemo K., Lamas T. (2006). The influence of forest data quality on planning process in forestry. Scandinavian Journal of Forest Research. 2006(21): 327-339. Eid T. (2000). Use of uncertain inventory data in forestry scenario models and consequential incorrect harvest decisions. Silva Fennica 34(2): 89-100. El Habib N. and Yacine O. (2007). An Approach of Agent-Based distributed simulation for Supply chain: Negotiations protocols between collaborative agents. The 20th annual European Simulation and Modeling  Conference (2006), Toulouse, France. 1-6pp. Fisher M.L. 1997. What is the right supply chain for your products?. Harvard business review. 1997(97205):105-116. 104  FP Innovations. (2008). BC Coastal Forest Sector Hem-Fir Initiative. H.02. Identification of Key Barriers and Opportunities. FP Innovations. Vancouver, BC. FP Innovations. (2009). BC Coastal Forest Sector Hem-Fir Initiative. Bulletin of February of 2009. FP Innovations. Vancouver, BC.  Gass S. (1983). Decision-Aiding Models: Validation, Assessment, and Related Issues for Policy Analysis. Operations Research. 31(4): 603-631. Gaudreault J., Forget P.,Frayet J.,Rousseau A., and D’Amours S. (2009). Distributed Planning in the Lumber Supply Chain:Models and Corrdination. Interuniversity research Centre on enterprise Network, Logistics and Transportation CIRRELT. 9: 1-17. Gobakken T. (2000). Models for assessing timber grade distribution and economic value of standing birch trees. Scand. J. Res. 15:  570-578. Goldsby T., Griffis S., and Roath A. (2006). Modeling lean, agile and leagile supply chain strategies. Journal of Business Logistics. 27 (1): 57-80.  Gunnarsson H., Ronnqvist M. And Carlsson D.  (2007). “Integrated Production and Distribution Planning for Södra Cell AB”. Journal of Mathematical Modelling and Algorithm. 6(1): 25–45. Gupta A., and Maranas C. (2003). Managing demand uncertainty in supply chain planning. Computer and Chemical Engineering. 27(2003): 1219-1227. Gupta S., and Martin S. Production and Operation Management Systems. CPC Press Taylor and Francis Group. Boca Raton, Florida.  Haartveit E. Kozak R. and R. Maness T. (2004). Supply Chain Management Mapping for the Forest Products Industry: Three Cases from Western Canada. Journal of Forest Products Business Research. 1(5): 1-30. Hallgren M., and Olhager J. (2009). Lean and agile manufacturing: external and internal drivers and performance outcomes. International Journal of Operations & Production Management. 29 (10): 976-999. 105  Hege L., Gicquel C., and Minoux M. (2010). Lagrangian Relaxation for Multi-level Lotsizing problems. Paper presented at the French Society of Operation Research and Decision Support Cconference ROADEF, Toulouse, France.  Hiller F. and Lieberman G. (2005). Introduction to Operations Research. Mc Graw Hill. 8th edition. New York. Huang S., Uppal M., and Shi F. (2002). A products driven approach to manufacturing supply chain selection. Supply chain management: An international journal. 7(4):189-199. Hung Y.-F., and Chien K.-L. (2000).  A Multi-Class Multi-Level Capacitated Lot Sizing Model. The Journal of the Operational Research Society. 51(11):1309-1318 International Series in Operations Research and Management Sciences. (2007) by Springer Science Business Media. 1-609 pp. International Wood Markets Group Inc. (2007). B.C. Coast Strategic Options: Current Business, Future Opportunities and Outlook to 2020. Coast Forest Products Association. Vancouver, BC. Jones D.F., Mirrazavi S.K., and Tamiz M. (2002). Multi-objective meta-heuristics: An overview of the current state- of-the-art. European Journal of Operational Research. 137 (2002): 1-9. Jones T., and Riley D. (1984). Using Inventory for Competitive Advantage through Supply Chain Management. International Journal of Physical Distribution & Logistics Management. 17( 2): 94 – 104. Kangas, A. (2010). Value of forest information. European Journal of Forest Research. Eur. J. Forest Res. 129: 863-874. Lee H., and Billington C. (1992). Managing supply chain Inventory: pitfalls and opportunities. Sloan Management Review; Spring 1992; 33, 3; ABI/INFORM Global pg. 65-73. Maness T. (2008). Latin America Best Practices Report. Draft paper. The University of British Columbia. Vancouver, BC. 106  Maness T. and Farrell R. (2004). A multi-objective scenario evaluation model for sustainable forest management using criteria and indicators. Canadian Journal of Forest Research. Can. J. For. Res. 34: 2004–2017. Maness T. and Norton S. (2002). Multiple period combined optimization approach to forest production planning. Scandinavian Journal of Forest research, Scan. J. For. Res. 17: 460-471. Maness T., and Adam D. (1991). The combined optimization of log bucking and sawing strategies. Wood and fiber science. 23(2): 296-314. Maness T.C. (2008). An agent based integrated production model of the BC Coastal forest sector. Application for grant part 1, Form 101. The University of British Columbia. Vancouver, BC. Maness T.C. and Norton S.E. (2002). Multiple Period combined optimization approach to forest production planning. Scandinavian journal of forest research. 17: 460-471. Maness T.C., and Adams D.M. (1990). The combined optimization of log bucking and sawing strategies. Wood and Fiber Science. 23(2):296-314. Mason-Jones R., Naim M.,and Towill D. (1997). ”The Impact of Pipeline Control on Supply Chain Dynamics”. The International Journal of Logistics Management. 8(2): 47-62. Maturana S., Pizani E., and Vera J. (2010). Scheduling production for a sawmill; A comparison of mathematical model versus a heuristic. Computers & Engineering. 59(4):1-8.  Min H. and Zhou G. (2002). “Supply chain modeling: past, present, and future”. Computer & Industrial Engineering. 43: 231-249  Moyaux T., Chaib-draa B., and D’Amours S. (2006). Supply Chain Management and Multi-agents Systems: An Overview. Springer (SCI). 28: 1-27. Narasimhan R., Morgan Swink M., and Kim S.W. (2006). Disentangling leanness and agility: An empirical investigation. Journal of Operations Management. 24: 440–457. Naylor B., Nain M., and Berry D. (1999). Leagility: Integrating the lean and agile manufacturing paradigms in the total supply chain. Int. J. Production Economics. 62: 107-118. 107  Paiva R. and  Marabito R. (2008). ”An optimization model for the aggregate production planning of a Brazilian sugar and ethanol milling company”.  Ann. Oper. Res. 169: 117–130. Pearse P. (2001). Ready to change, Crisis and Opportunity in the coast forest industry. A Report to the Minister of Forest on British Columbia’s Coastal Industry. BC Ministry of Forests. Vancouver, BC. Peter, B., and Nelson J. (2005). “Estimating harvest schedules and profitability under the risk of fire disturbance”. Canadian Journal of Forest Resources. Can. For.Res. 35: 1378-1388.  Possani, E. (2001) Lot streaming and batch scheduling: splitting and grouping jobs to improve production efficiency (Ph.D. Thesis). University of Southampton, Southampton. Robson P.A. (1995). The Working Forest of British Columbia. Harbour Publishing for I.K. Barber. Bristish Columbia. Ronnqvist M. (2003). Optimization in Forestry. Mathematical Programming. 97 (1)1-18. Salehirad N., and Sowlati T. (2005). “Performance of primary wood producers in British Columbia using data envelopment analysis”. Canadian Journal of Forest Research. Can.J. For.Res. 34 :285-294.  Schmidt M., and Kandler G. (2009). An analysis of Norway spruce stems quality in Baden-Wurttemberg: results from the second German national forest inventory. European Journal of Forest Resources. 128: 515-529. Schwab O., Maness T., Bull G., and Roberts D. (2009). Modeling the effect of changing market conditions on mountain pine beetle salvage harvesting and structural changes in BC forest products industry. Canadian Journal of Forest Research. 39: 1806–1820. Singer M., and Donoso P. (2009). “Internal supply chain management in the Chilean sawmills industry”. Total Quality Management & Business Excellence. 20(9): 905-919. Sipper D., and Bulfin R. (1997). Production Planning, Control, and integration. The McGraw-Hill Companies. New York. Todoriki C., and Ronnqvist M. (2002). “Dynamic Control of Timber Production at a Sawmill with Log Sawing Optimization”. Scand. J. For. Res. 17: 79–89. 108  Vahid, S. (2011). An Agent-Based Supply Chain Model for Strategic Analysis in Forestry (PhD thesis). The University of British. Vancouver.  Wang, X. ( 2006). Sorting Red Maple Logs for Structural Use, Chapter 3, Forest Products Society, ISBN1-892529-32-7, Publication No. 7234. Madison, Wisconsin. Weintreaub A., Romero C., Bjorndal T., and Epstein R. (2007). Handbook of Operation Research in Natural Resources. Weintreaub A., Romero C., Bjorndal T., Epstein R., and Springer. United States West KD. (1988). Order backlogs and production smoothing. The economics of inventory management. National Bureau of Economic Research Cambridge. NBER w2385:305-317. Western Forest Products Inc. (2010) Annual Information Form: http://www.westernforest.com/. Accessed on September, the 5th of 2010. Western Forest Products Inc. (2010), (2011), and (2012). Annual Information Form: http://www.westernforest.com/. Accessed on October of 2013. Woodbridge P. (2009). Opportunity BC 2020 BC’s Forest Industry, Moving from a Volume focus to a Value Perspective. Woodbridge Associates, prepared for The Business Council of BC. Vancouver, BC. Work T., Spence J., Volney W., Morgantini L., and Innes J (2003). Integrating biodiversity and forest practices in western Canada. The forestry chronicle. 79(5): 906-916.  Yaghubian R.A., Hodson J.T., and Joines A.J. (2001).  Dry-or-buy decision support for dry kiln scheduling in furniture production. IIE transactions. 33(2):131-136.  Yang-Hua L., and Van Landeghem H. (2009). An application of simulations and value stream mapping in lean manufacturing. Paper presented at 14th European simulation symposium. Dresden, Germany. Zanjani K.M., Ait-Kadi D., and Nourefath M. (2010). Robust production planning in a manufacturing environment with random yield: A case in sawmill production planning. European Journal of Operational Research. 201::882-891.  109  Appendices  Appendix A Model formulation for chapter 2 and logit regression analysis  A1 Formulation of the BC coastal forest industry production planning problem  The BC Coastal forest supply chain can be seen as a 3 node problem (Figure A1). I assume these nodes are forest stands, sort-yards and sawmills. Thus, the problem can be formulated as a profit maximization of three manufacturing units, connected by appropriate balancing equations. A detailed linear programming formulation follows:  Figure A1 Nodes and variables of the DSS Sets  i :Stand index         j :Stand growth      k  :Species in the stand    l :Sort yard index   m: Bucking policy   n : log grade o :Sawmill index    p:Sawing policy      q :Sawn-woods products   r : Planning periods  Data declaration:  C H    i           : Harvesting cost in $ by m3 harvested at stand i. CSYS   l  : Cost of keep 1 m3 of stem at sort-yard l. CSYL   l  : Cost of keep 1 m3 of logs at sort-yard l. CSAL  o  : Cost of keep 1 m3 of logs at sawmill o. ISYSk l rStand iU i j k l rH i rSort yard lV j k l m rW  k n l rISYLk n l rISYSk l rSort yard lV j k l m rW  k n l rISYLk n l rISYSk l rSort yard lV j k l m rW  k n l rISYLk n l rISALk n o rSawmill oY k n o p rZ k q o rISAPk q o rX k n l o rX k n l o rX k n l o rU i j k l rU i j k l rISALk n o rSawmill oY k n o p rZ k q o rISAPk q o rISALk n o rSawmill oY k n o p rZ k q o rISAPk q o r110  CSAP o  : Cost of keep 1 m3 of sawn-wood products at sawmill o. CT1    i l     : Transportation cost in $ by m3 of stems from stand i, to sort yard l. CT2  l o   : Transportation cost in $ by m3 of logs from sort yard l to sawmill o. CSY  l  : Sort yard production cost in $/m3, in sort yard l.  CSW  o  : Sawmilling production cost in $/m3, in sawmill o.  Price_L k q o r : Price of lumber in $/m3 of species k, product q, sell by sawmill o, in period r.   Price_C k  r : Price of chip in $/m3 of species k, in period r.   OP_P k q o r : Economic penalty in $/m3 for producing over than the lumber demand on species k,   product q, sell by sawmill o, in period r.   UP_P k q o r : Economic penalty in $/m3 for producing under than the lumber demand on species k,   product q, sell by sawmill o, in period r.  Yields, capacities and productivities coefficients  Yield_f    I j k : Stand yield in m3 by hectares in stand i, to growth j, and species k. Yield_sy j k m n : Bucking yield in m3 of product n by m3 of stem growth j, species k, by applying bucking policy m. Yield_sw k n p q : Sawing yield in m3 of product q by m3 of log grade n, specie k, by applying sawing policy p. Cap_f     r : Harvesting capacity (in hours) on period r. Cap_sy  l r : Sort yard capacity of sort yard l (in hours) on period r. Cap_sw o r : Sawmill capacity of sawmill o (in hours) on period r. Prod_f    j  : Harvest productivity (hour/m3) in a stand predominantly growth j. Prod_sy j k l  : Sort yard productivity (hour/m3) when bucking stems growth j, species k, at sort yard l. Prod_sw o  : Sawing productivity (in hour/m3) at sawmill o IO_sys    j k l : Initial inventory of stems of growth j, species k at sort yard l. IO_syl      k n l : Initial inventory of log of species k, log grade n, at sort yard l. IO_sal     k n o : Initial inventory of log of species k, log grade n, at sawmill o. IO_sap    k q o : Initial inventory of sawn-wood of species k, products q, at sawmill o. A  i : Land available in hectares for stand i D         k q o r : Lumber demand (in m3) of species k, product q, in sawmill o, for period r Definition of decision variables: H    i r   Land harvested (in ha.) at stand i, in period r T    i j k r  : Harvested volume (in m3) at stand “i”, growth type “j”, species “k” in period “r” U  i j k l r    : Volume of stems  (in m3) of growth “j”, specie “k” sent from stand “i” to sort yard l, in period r. V  j k l m r   : Sort yard stems input (in m3) of growth j, species k, at sort yard l, bucked with policy m, period r.  ISYS j k l r    : Stem inventory (in m3) of growth j, species k, at sort yard l, in period r.”  W    k n l r   : Volume of logs (in m3) of species “k”, log grade “n”, produced in sort yard l in period r. ISYL k n l r   : Logs inventory (in m3) of species k, log grade n, at sort yard l, in period r. 111  X   k n l o r   : Volume of logs (in m3) of species k, log grade n sent from sort yard l to sawmill o, in period r. Y  k n o p r   : Volume of logs (in m3) of specie k, grade n, sawn with sawing policy p, at sawmill o, in period r. ISAL  k n o r   : Log inventory (in m3) of species k, log grade n at sawmill o, in period r.  ISAP k q o r   : Sawn-wood products inventory (in m3) of species k, sawn-wood products q, in period r. Z   k q o r   : Volume of sawn-wood’s (in m3) of specie k, product q, produced-sell by sawmill “o”, in period r C   k  r      : Volume of chips (in m3) of specie k, produced  in period r O_P   k q o r  : Sawn-wood’s over-produced (in m3) of specie k, product q, by sawmill “o”, in period r U_P   k q o r  : Sawn-wood’s under-produced (in m3) of specie k, product q, by sawmill “o”, in period r  Objective function  𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑜𝑓𝑝𝑟𝑜𝑓𝑖𝑡= ∑ 𝑍𝑘𝑞𝑜𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑞𝑜𝑟 +∑𝐶𝑘𝑟𝑘𝑟𝑃𝑟𝑖𝑐𝑒𝑘𝑟𝑘𝑞𝑜𝑟−∑ 𝑇𝑖𝑗𝑘𝑟𝐶𝑖𝐻− 𝑖𝑗𝑘𝑟∑ 𝑈𝑖𝑗𝑘𝑙𝑟𝐶𝑖𝑙𝑇1𝑖𝑗𝑘𝑙𝑟−∑𝐼𝑗𝑘𝑙𝑟𝑆𝑌𝑆𝑐𝑖𝑙𝑆𝑌𝑆 −𝑗𝑘𝑙𝑟∑ 𝐼𝑘𝑛𝑙𝑟𝑆𝑌𝐿 𝑐𝑖𝑙𝑆𝑌𝐿 −𝑘𝑛𝑙𝑟∑𝑊𝑘𝑛𝑙𝑟𝐶𝑙𝑆𝑌 −𝑘𝑛𝑙𝑟∑ 𝑋𝑘𝑛𝑙𝑜𝑟𝐶𝑙𝑜𝑇2 −𝑘𝑛𝑙𝑜𝑟∑ 𝐼𝑘𝑛𝑜𝑟𝑆𝐴𝐿 𝑐𝑖𝑜𝑆𝐴𝐿𝑘𝑛𝑜𝑟− ∑ 𝐼𝑘𝑞𝑜𝑟𝑆𝐴𝑃 𝑐𝑖𝑜𝑆𝐴𝑃 −𝑘𝑞𝑜𝑟∑ 𝑍𝑘𝑞𝑜𝑟𝐶𝑜𝑆𝑊 − ∑ 𝑂_𝑃𝑘𝑞𝑜𝑟𝑂𝑃_𝑃𝑘𝑞𝑜𝑟 − ∑ 𝑈_𝑃𝑘𝑞𝑜𝑟𝑈𝑃_𝑃𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟𝑘𝑞𝑜𝑟      [2.0] The objective function maximizes the supply chain profit, satisfying all constraints during the planning horizon. The components of the objective function are: lumber and chip sales incomes, harvesting cost, transportation cost from stand to sort yard, inventory cost of stems and logs at sort yard, sort yard cost, transportation cost from sort yard to sawmill, inventory cost of logs and sawn-wood products at sawmill, sawmilling cost, lumber over-production penalization, and lumber under-production penalization. Constraints    ∑ 𝐻𝑖𝑟  ≤ 𝐴𝑖𝑟   ∀ 𝑖                                                                                 [2.1] 𝑇𝑖𝑗𝑘𝑟 = 𝐻𝑖𝑟Yield_f 𝑖𝑗𝑘     ∀𝑖 , 𝑗 , 𝑘 𝑎𝑛𝑑 𝑟                                          [2.2] ∑ 𝑇𝑖𝑗𝑘𝑟  Prod_f j     𝑖𝑗𝑘 ≤ Capf r    ∀ 𝑟                                           [2.3] 𝑇𝑖𝑗𝑘𝑟 = ∑ 𝑈𝑖𝑗𝑘𝑙𝑟     ∀ 𝑖 , 𝑗, 𝑘 , 𝑎𝑛𝑑  𝑟𝑙                                               [2.4] ∑ 𝑈𝑖𝑗𝑘𝑙𝑟 + Ijklr−1SYS − IjklrSYS𝑖 = ∑ 𝑉𝑗𝑘𝑙𝑚𝑟    ∀ 𝑗, 𝑘, 𝑙,𝑚 𝑟                      [2.5] ∑ 𝑉𝑗𝑘𝑙𝑚𝑟𝑗𝑚 Yield_sy𝑗𝑘𝑚𝑛 = 𝑊𝑘𝑛𝑙𝑟  ∀𝑘, 𝑛, 𝑙, 𝑎𝑛𝑑 𝑟                      [2.6]  ∑ 𝑉𝑗𝑘𝑙𝑚𝑟𝑗𝑘𝑚 Prod_sy 𝑗𝑘𝑙 ≤ Cap_sy 𝑙𝑟 ∀ 𝑙 𝑎𝑛𝑑  𝑟                        [2.7]  𝑊𝑘𝑛𝑙𝑟 + 𝐼𝑘𝑛𝑙𝑟−1𝑆𝑌𝐿   −  𝐼𝑘𝑛𝑙𝑟𝑆𝑌𝐿 = ∑ 𝑋𝑘𝑛𝑙𝑜𝑟  ∀ 𝑘, 𝑛, 𝑙, 𝑟                  𝑜     [2.8]  ∑ 𝑋𝑘𝑛𝑙𝑜𝑟𝑙 + 𝐼𝑘𝑛𝑜𝑟−1𝑆𝐴𝐿 − 𝐼𝑘𝑛𝑜𝑟𝑆𝐴𝐿 = ∑ 𝑌𝑘𝑛𝑜𝑝𝑟𝑝  ∀ 𝑘, 𝑛, 𝑜, 𝑟                [2.9]  112  ∑ 𝑌𝑘𝑛𝑜𝑝𝑟  Yield_sw𝑘𝑛𝑝𝑞𝑛𝑝 = 𝑍𝑘𝑞𝑜𝑟 ∀ 𝑘, 𝑞, 𝑜, 𝑎𝑛𝑑 𝑟                   [2.10]  ∑ 𝑍𝑘𝑞𝑜𝑟  Yield_ch𝑘𝑛𝑝 = 𝑍 𝑘𝑟 ∀ 𝑘 𝑎𝑛𝑑 𝑟                                      [2.11]  ∑ 𝑍𝑘𝑞𝑜𝑟𝑘𝑞 Prod_sw 𝑜 ≤  Cap_sw 𝑜 𝑟    ∀ 𝑜 𝑎𝑛𝑑 𝑟                    [2.12]  𝑍𝑘𝑞𝑜𝑟 + 𝐼𝑘𝑞𝑜𝑟−1𝑆𝐴𝑃  − 𝐼𝑘𝑞𝑜𝑟𝑆𝐴𝑃 − 𝑂𝑃𝑘𝑞𝑜𝑟 + 𝑈𝑃𝑘𝑞𝑜𝑟 = D𝑘𝑞𝑜𝑟    ∀ 𝑘, 𝑞, 𝑜, 𝑟      [2.13]  Ijkl0SYS = IOjklr                           SYS ∀j, k, l, o and q                                        [2.14] Iknl0SYL = IOknl                           SYL ∀ k, l, n , o and q                                    [2.15] Ikno0SAL = IOkno                 SAL       ∀ k, l, n , o and q                                   [2.16] Ikqo0SAP = IOkqo                 SAP       ∀ k, q and o                                          [2.17] Ijkl4              SYS = 0                                                                                     [2.18] Iknl4     SYL = 0                                                                                       [2.19] Ikno4         SAL = 0                                                                                   [2.20] Ikqor4   SAP = 0                                                                                        [2.21]   The supply chain production planning problem was subject to several sets of constraints by operations, and balance constraints to connect flows of supplies. For harvesting operations: land availability constraint (2.1), stand stems production constraint (2.2), harvesting capacity constraint (2.3), and a balance constraint (2.4) to connect stand stems production with stems flows to sort yards. For sort yards: a balance constraint to connect stems flows from stands, and stem inventories with sort yards input (2.5).  Sort yard log production constraints (2.6), sort yard capacity constraint (2.7), and a balance constraint to connect sort yard log production and log inventories with log flows from sort yards to sawmills (2.8). For sawmills: a balance constraint to connect log flows from sort yards, log inventories with sawmills input (2.9), sawmill lumber production constraints (2.10), chip production constraints (2.11), sawmills production capacity constraints (2.12), a balance constraint that connect lumber production, lumber inventories, over-production, and under-production with lumber demand (2.13). Finally, initial and final inventory constraints are written in constraints 2.14 to 2.21.   113  A.2 Logit regression analysis  To analyze lumber demand targets, a regression curve was fitted. Accordingly, the relationship between lumber demand targets and the proportion of DSS runs that fulfill lumber demand targets was determined by using a logit regression analysis approach. The estimated risk curve for lumber demand target and data used in the analysis can be seen in Tables A1 to A4. Further statistical analysis of was done using Minitab 17.  Table A.1     Scenario Ne :15%-60%Lumber Observed Observed Logit Predicteddemand frequency of probability of transformation Slope & Probabilitytarget m3failure (%) failure f (%) f/(1-f) Ln(f/1-f) ATAN(f/1-f) y' EXP(y'') Intercept y=(y'/(1+y'))510,249 0 0% 0                                   -17.2 0.00 -8 0                                 0.0%518,479 0 0% 0                                   -17.2 0.00 -7 0                                 0.1%526,708 0 0% 0                                   -17.2 0.00 -6 0                                 0.2%543,168 2 7% 0                                   -2.6 0.07 -5 0                                 Slope 0.5%559,628 0 0% 0                                   -17.2 0.00 -4 0                                 7.E-05 1.8%567,858 19 63% 2                                   0.5 1.05 -3 0                                 Intercept 3.2%576,087 4 13% 0                                   -1.9 0.15 -3 0                                 -45.316 5.8%584,317 19 63% 2                                   0.5 1.05 -2 0                                 10.1%592,547 30 100% 299,999                      12.6 1.57 -2 0                                 17%600,777 30 100% 299,999                      12.6 1.57 -1 0                                 27%609,007 29 97% 29                                 3.4 1.54 0 1                                 41%625,446 24 80% 4                                   1.4 1.33 1 2                                 70%641,926 9 30% 0                                   -0.8 0.40 2 8                                 89%658,386 30 100% 299,999                      12.6 1.57 3 27                              96%674,845 28 93% 14                                 2.6 1.50 4 89                              99%691,305 30 100% 299,999                      12.6 1.57 6 302                            100%724,224 30 100% 299,999                      12.6 1.57 8 3,425                         100%740,684 30 100% 299,999                      12.6 1.57 9 11,542                      100%773,603 30 100% 299,999                      12.6 1.57 12 131,069                    100%822,982 30 100% 299,999                      12.6 1.57 15 5,015,967                 100%839,442 30 100% 299,999                      12.6 1.57 17 16,903,751              100%855,901 30 100% 299,999                      12.6 1.57 18 56,961,242              100%872,361 30 100% 299,999                      12.6 1.57 19 191,958,726            100%114  Table A.2    Scenario Nne :15%-45%Lumber Observed Observed Logit Predicteddemand frequency of probability of transformation Slope & Probabilitytarget m3failure (%) failure f (%) f/(1-f) Ln(f/1-f) ATAN(f/1-f) y' EXP(y'') Intercept y=(y'/(1+y'))510,249 0 0.0% 0                                   -17.2167 3.33333E-08 (8)      0                                 0%518,479 0 0.0% 0                                   -17.2167 3.33333E-08 (7)      0                                 0%526,708 0 0.0% 0                                   -17.2167 3.33333E-08 (6)      0                                 0%543,168 2 6.7% 0                                   -2.63906 0.071307465 (5)      0                                 1%559,628 0 0.0% 0                                   -17.2167 3.33333E-08 (4)      0                                 2%567,858 0 0.0% 0                                   -17.2167 3.33333E-08 (3)      0                                 4%576,087 6 20.0% 0                                   -1.38629 0.244978663 (3)      0                                 7%584,317 17 56.7% 1                                   0.26826 0.917949696 (2)      0                                 13%592,547 30 100.0% 299,999                      12.6115 1.570792993 (1)      0                                 21%600,777 30 100.0% 299,999                      12.6115 1.570792993 (1)      1                                 Slope 34%609,007 30 100.0% 299,999                      12.6115 1.570792993 (0)      1                                 7.7.E-05 49%625,446 30 100.0% 299,999                      12.6115 1.570792993 1       3                                 Intercept 77%641,926 11 36.7% 1                                   -0.54654 0.524795772 2       12                              -46.937 92%658,386 30 100.0% 299,999                      12.6115 1.570792993 4       43                              98%674,845 29 96.7% 29                                 3.3673 1.536327226 5       152                            99%691,305 30 100.0% 299,999                      12.6115 1.570792993 6       541                            100%724,224 30 100.0% 299,999                      12.6115 1.570792993 9       6,824                         100%740,684 30 100.0% 299,999                      12.6115 1.570792993 10     24,238                      100%773,603 30 100.0% 299,999                      12.6115 1.570792993 13     305,722                    100%822,982 30 100.0% 299,999                      12.6115 1.570792993 16     13,695,995              100%839,442 30 100.0% 299,999                      12.6115 1.570792993 18     48,643,755              100%855,901 30 100.0% 299,999                      12.6115 1.570792993 19     172,753,624            100%872,361 30 100.0% 299,999                      12.6115 1.570792993 20     613,565,144            100%115  Table A.3    Scenario Se :5%-20%Lumber Observed Observed Logit Predicteddemand frequency of probability of transformation Slope & Probabilitytarget m3failure (%) failure f (%) f/(1-f) Ln(f/1-f) ATAN(f/1-f) y' EXP(y'') Intercept y=(y'/(1+y'))510,249 0 0.0% 0                                   -17.2167 3.33333E-08 (11)   0                                 0%518,479 0 0.0% 0                                   -17.2167 3.33333E-08 (10)   0                                 0%526,708 0 0.0% 0                                   -17.2167 3.33333E-08 (9)      0                                 0%543,168 0 0.0% 0                                   -17.2167 3.33333E-08 (8)      0                                 0%559,628 0 0.0% 0                                   -17.2167 3.33333E-08 (6)      0                                 0%567,858 0 0.0% 0                                   -17.2167 3.33333E-08 (6)      0                                 0%576,087 0 0.0% 0                                   -17.2167 3.33333E-08 (5)      0                                 1%584,317 24 80.0% 4                                   1.38629 1.325817664 (4)      0                                 2%592,547 30 100.0% 299,999                      12.6115 1.570792993 (3)      0                                 3%600,777 30 100.0% 299,999                      12.6115 1.570792993 (3)      0                                 Slope 7%609,007 30 100.0% 299,999                      12.6115 1.570792993 (2)      0                                 9.E-05 13%625,446 30 100.0% 299,999                      12.6115 1.570792993 (1)      1                                 Intercept 37%641,926 3 10.0% 0                                   -2.19722 0.110657221 1       3                                 -54.927 72%658,386 30 100.0% 299,999                      12.6115 1.570792993 2       11                              91%674,845 30 100.0% 299,999                      12.6115 1.570792993 4       44                              98%691,305 30 100.0% 299,999                      12.6115 1.570792993 5       184                            99%724,224 30 100.0% 299,999                      12.6115 1.570792993 8       3,231                         100%740,684 30 100.0% 299,999                      12.6115 1.570792993 10     13,528                      100%773,603 30 100.0% 299,999                      12.6115 1.570792993 12     237,153                    100%822,982 30 100.0% 299,999                      12.6115 1.570792993 17     17,407,888              100%839,442 30 100.0% 299,999                      12.6115 1.570792993 18     72,889,416              100%855,901 30 100.0% 299,999                      12.6115 1.570792993 20     305,172,272            100%872,361 30 100.0% 299,999                      12.6115 1.570792993 21     1,277,801,679         100%116  Table A.4    Scenario Sne :5%-15%Lumber Observed Observed Logit Predicteddemand frequency of probability of transformation Slope & Probabilitytarget m3failure (%) failure f (%) f/(1-f) Ln(f/1-f) ATAN(f/1-f) y' EXP(y'') Intercept y=(y'/(1+y'))510,249 0 0.0% 0                                   -17.2167 3.33333E-08 (11)   0                                 0%518,479 0 0.0% 0                                   -17.2167 3.33333E-08 (10)   0                                 0%526,708 0 0.0% 0                                   -17.2167 3.33333E-08 (9)      0                                 0%543,168 0 0.0% 0                                   -17.2167 3.33333E-08 (8)      0                                 0%559,628 0 0.0% 0                                   -17.2167 3.33333E-08 (6)      0                                 0%567,858 0 0.0% 0                                   -17.2167 3.33333E-08 (6)      0                                 0%576,087 0 0.0% 0                                   -17.2167 3.33333E-08 (5)      0                                 1%584,317 21 70.0% 2                                   0.8473 1.165904541 (4)      0                                 2%592,547 30 100.0% 299,999                      12.6115 1.570792993 (3)      0                                 3%600,777 30 100.0% 299,999                      12.6115 1.570792993 (3)      0                                 Slope 6%609,007 30 100.0% 299,999                      12.6115 1.570792993 (2)      0                                 8.7.E-05 12%625,446 30 100.0% 299,999                      12.6115 1.570792993 (1)      1                                 Intercept 36%641,926 5 16.7% 0                                   -1.60944 0.19739556 1       2                                 -54.995 70%658,386 30 100.0% 299,999                      12.6115 1.570792993 2       10                              91%674,845 30 100.0% 299,999                      12.6115 1.570792993 4       41                              98%691,305 30 100.0% 299,999                      12.6115 1.570792993 5       172                            99%724,224 30 100.0% 299,999                      12.6115 1.570792993 8       3,018                         100%740,684 30 100.0% 299,999                      12.6115 1.570792993 9       12,639                      100%773,603 30 100.0% 299,999                      12.6115 1.570792993 12     221,563                    100%822,982 30 100.0% 299,999                      12.6115 1.570792993 17     16,263,502              100%839,442 30 100.0% 299,999                      12.6115 1.570792993 18     68,097,700              100%855,901 30 100.0% 299,999                      12.6115 1.570792993 19     285,110,391            100%872,361 30 100.0% 299,999                      12.6115 1.570792993 21     1,193,799,602         100%117  Appendix B  B.1 Additional tables for Chapter 3  Table B.1 Manufacturing performance by ME and lumber demand scenario Lumber demand scenario Manufacturing environment  Flow time [period] Orders fulfillment Land  Harvested [ha] Over-below lumber  production [m3] Timber supply [m3] Inventory  [m3] Large batch low variation (LB_LV) Agile 5.26  89% 457 (18,393) 414,360  111,492  BC-SC  7.19  96% 479 (7,184) 495,210  111,245  Lean  7.53  91% 459 (15,417) 416,062  123,239  Large batch high variation (LB_HV) Agile 6.98  99% 327 (1,253) 234,961  50,378  BC-SC  8.28  108% 237 9,100  213,422  71,047  Lean  8.98  92% 293 (9,073) 197,093  106,304  Small batch low variation (SB_LV) Agile 14.28  99% 81 (1,756) 52,224  103,020  BC-SC 15.75 114% 169 9,744   138,661  110,173  Lean 15.95  107% 144 3,515  93,218  132,123  Small batch high variation (SB_HV) Agile 15.14  98% 55 (1,534) 34,930  113,422  BC-SC  15.45  115% 156 10,258  134,688  103,660  Lean 16.28 111% 149  6,222  95,827  132,963 Base lumber demand (Base LD) Agile 5.73 94% 382 (7,409) 285,775 40,808 BC-SC 6.19 103% 328 3,993 308,219 70,162 Lean 8.14 91% 366 (11,170) 268,718 119,129   118  Appendix C Formulation of the lumber manufacturing planning and scheduling problem  The short term lumber production planning problem consists of determining the number of logs of certain diameter and grade to be sawn with a certain sawing pattern to satisfy lumber products demand in a certain period. To address this challenge, I proposed the development of two MIP production planning and planning-scheduler models. The first, a multi-period DSS which minimizes costs, and satisfies lumber product demands strictly within a certain period, with no backlog allowed (called “PL”). And the second, a mono-period DSS which minimizes costs and satisfies orders of lumber products on certain due dates (called PS). Accordingly, lumber product demands by periods were switched to orders of lumber products that must be satisfied on a certain due date. The due date in hours was equivalent to the period where the set of lumber products demanded must be produced. Hence, switching from a set of lumber products that must be produced in a certain period to orders (i.e. jobs) with due date`s is equivalent. This formulation change helps to explicitly handle scheduling heuristics sequences to process the orders. Both model formulations follow:   A Plann (PL) model  Sets i  :Diameter class                        j :Log grade class         k   :Sawing pattern p  :Lumber product demand      o :Lumber product       s :Planning periods Data declaration: Logcost  i j        :Log price in diameter class i, and log grade j in $ by m3   Cilog   s  :Cost to keep 1 m3 of log in inventory in period s. Crogr s  :Cost to keep 1 m3 of rough green lumber in inventory in period s. Croat s  :Cost to keep 1 m3 of anti-stain treated lumber in inventory in period s. Sawing_cost  :Cost of sawing in $ per hour At_cost :Cost of anti-stain treatment in $ per hour Yields, capacities, and productivities coefficients Yield_sw ijko  :Sawmilling yield in [%] of product o, recovered from a log of diameter class i, grade with sawing pattern k. Yield_at o  : Anti-stain yield in [%] of lumber product o  Cap_sw   s : Sawmilling capacity (in hours) on period s. Cap_at    s : Anti-stain capacity (in hours) on period s. 119  Prod_sw  i  : Sawmilling productivity (hour/m3) when sawing log diameter class i. Prod_at  o  : Anti-stain productivity (hour/m3) when treating lumber product o. IOLuSA  o  : Initial inventory of green rough lumber product o. IOLuAT o : Initial inventory of anti-stain lumber product o. D_roat o s : Lumber demand (in m3) of lumber product o, in period s Definition of decision variables: U i j k s  : Volume of logs  (in m3) of diameter class i, log grade class j, sawn with sawing pattern k, in period s. ISA o s    :Rough green lumber inventory (in m3) of lumber product o, in period s  V_sa o s   :Sawmill lumber production (in m3) of lumber product o, produced  in period s.  W_at o s  : Volume (in m3) of lumber product o, transferred to anti-strain treatment in period s V_at o s   :Anti-stain treatment production (in m3) of lumber product o, produced  in period s  Objective function 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒: 𝑍: 𝐿𝑢𝑚𝑏𝑒𝑟 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡𝑠   [4.0]  Subject to:  ∑ 𝑈𝑖𝑗𝑘𝑠𝑌𝑖𝑒𝑙𝑑_𝑠𝑤𝑖𝑗𝑘𝑜𝑖𝑗𝑘𝑠 = 𝑉_𝑠𝑎𝑜𝑠      ∀𝑜, 𝑠   [4.1] ∑ 𝑈𝑖𝑗𝑘𝑠𝑝𝑟_𝑠𝑤𝑖𝑗𝑘 ≤ 𝑐𝑎𝑝_𝑠𝑎𝑤𝑠     ∀𝑠     [4.2] 𝑉_𝑠𝑎𝑜𝑠 − 𝐼𝑆𝐴𝑜𝑠 −𝑊_𝑎𝑡𝑜𝑠 = 0 ∀𝑜, 𝑠 = 1     [4.3] 𝑉_𝑠𝑎𝑜𝑠 + 𝐼𝑆𝐴𝑜𝑠−1 − 𝐼𝑆𝐴𝑜𝑠 −𝑊_𝑎𝑡𝑜𝑠 = 0 ∀𝑜, 𝑠 > 1    [4.4] 𝑊_𝑎𝑡𝑜𝑠𝑌𝑖𝑒𝑙𝑑_𝑎𝑡𝑜 = 𝑉_𝑎𝑡𝑜𝑠           ∀ 𝑜, 𝑠     [4.5] 𝑉_𝑎𝑡𝑜𝑠 − 𝐼𝐴𝑇𝑜𝑠 = 𝐷_𝑎𝑡𝑜𝑠           ∀ 𝑜, 𝑠 = 1     [4.6] 𝑉_𝑎𝑡𝑜𝑠 + 𝐼𝐴𝑇𝑜𝑠−1 − 𝐼𝐴𝑇𝑜𝑠 = 𝐷_𝑎𝑡𝑜𝑠           ∀ 𝑜, 𝑠 > 1    [4.7] ∑ 𝑊_𝑎𝑡𝑜𝑠𝑝𝑟_𝑎𝑡𝑜𝑠 ≤ 𝑐𝑎𝑝_𝑎𝑡𝑠     ∀𝑠     [4.8] 𝐼𝑆𝐴𝑜𝑠 = 0                                        ∀ 𝑜, 𝑠 = 6     [4.9] 𝐼𝐴𝑇𝑜𝑠 = 0                                        ∀ 𝑜, 𝑠 = 6     [4.10] 𝑆𝑎𝑤𝑖𝑛𝑔_𝑡𝑖𝑚𝑒𝑖 = ∑ 𝑈𝑖𝑗𝑘𝑠𝑝r _𝑠𝑤𝑗𝑘𝑠       ∀𝑖     [4.11] 𝐴𝑡_𝑡𝑖𝑚𝑒𝑜 = ∑ 𝑊_𝑎𝑡𝑜𝑠𝑝r _𝑎𝑡𝑠             ∀𝑠     [4.12]  120  Where: 𝐿𝑜𝑔 𝑐𝑜𝑠𝑡: 𝐿𝑜𝑔_𝑐𝑜𝑠𝑡 = ∑ 𝑈𝑖𝑗𝑘𝑠𝐿𝑜𝑔𝑝𝑟𝑖𝑐𝑒𝑡𝑖𝑗𝑖𝑗𝑘𝑠     [4.13] 𝐿𝑜𝑔 𝑖𝑛𝑝𝑢𝑡: 𝐿𝑜𝑔_𝑖𝑛𝑝𝑢𝑡 = ∑ 𝑈𝑖𝑗𝑘𝑠𝑖𝑗𝑘𝑠      [4.14] 𝑆𝑎𝑤𝑖𝑛𝑔 𝑐𝑜𝑠𝑡: 𝑆𝑎𝑤𝑖𝑛𝑔_𝑐𝑜𝑠𝑡 = ∑ 𝑠𝑎𝑤𝑖𝑛𝑔_𝑡𝑖𝑚𝑒𝑖𝑠𝑎𝑤𝑖𝑛𝑔𝑐𝑜𝑠𝑡𝑖     [4.15] 𝐴𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑐𝑜𝑠𝑡: 𝐴𝑡_𝑐𝑜𝑠𝑡 = ∑ 𝐴𝑡_𝑡𝑖𝑚𝑒𝑜𝑎𝑡_𝑐𝑜𝑠𝑡𝑜    [4.16] 𝐶𝑜𝑠𝑡 𝑜𝑓 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑔𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦: 𝐿𝑢_𝑔𝑟_𝑖𝑐𝑜𝑠𝑡 = ∑ 𝐼𝑆𝐴𝑜𝑠𝑐𝑖𝑟𝑜𝑔𝑟𝑠𝑜𝑠    [4.17] 𝐶𝑜𝑠𝑡 𝑜𝑓 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑎𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦: 𝐿𝑢_𝑎𝑡_𝑖𝑐𝑜𝑠𝑡 = ∑ 𝐼𝐴𝑇𝑜𝑠𝑐𝑖𝑟𝑜𝑎𝑡𝑠𝑜𝑠   [4.18] 𝐺𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛: 𝐺𝑅𝑉𝐿𝑢𝑚𝑏𝑒𝑟 = ∑ 𝑉_𝑠𝑎𝑜𝑠𝑜𝑠     [4.19] 𝐴𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛: 𝐴𝑇𝐿𝑢𝑚𝑏𝑒𝑟 = ∑ 𝑉_𝑎𝑡𝑜𝑠𝑜𝑠     [4.20] 𝐺𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑑𝑒𝑚𝑎𝑛𝑑: 𝐴𝑇_𝐷𝑒𝑚𝑎𝑛𝑑 = ∑ 𝐷_𝑟𝑜𝑎𝑡𝑜𝑠𝑜𝑠     [4.21] 𝐿𝑢𝑚𝑏𝑒𝑟 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = 𝐿𝑜𝑔𝑐𝑜𝑠𝑡 + 𝐿𝑢𝑔𝑟𝑖𝑐𝑜𝑠𝑡 + 𝐿𝑢𝑎𝑡𝑖𝑐𝑜𝑠𝑡 +  𝑆𝑎𝑤𝑖𝑛𝑔𝑐𝑜𝑠𝑡 + 𝐴𝑡𝑐𝑜𝑠𝑡 [4.22]  The objective function minimizes lumber manufacturing costs, which are: log costs, green and anti-stain treated lumber inventory costs and sawing and anti-stain processing costs. No backlogs were allowed; thus, lumber demands should be strictly satisfied. I also considered a set of constraints to solve the problem, which are: sawn wood production constraint 4.1, where logs of a diameter class, and grade class, are processed with a certain sawing pattern to a produce certain lumber product demand. The sawmilling capacity constraint [4.2], is the summation of the time expended to saw logs, which cannot exceed sawing capacity per period. In flow balance constraint [4.3] for period 1, the green lumber production less the period inventory is transferred to the anti-stain process. The flow balance constraint [4.4] for periods >1, transfers the sawn wood production plus the previous inventory less the inventory of the period to the anti-stain process. Anti-stain treated production is handled by constraint [4.5], where rough green lumber products are processed with anti-stain yields to produce anti-stain lumber products. The market constraint [4.6] for period 1 ensures that the anti-stain lumber production less the inventory of the period must be equal to the anti-stain lumber products demand. The market constraint [4.7] for periods > 1 ensures that the anti-stain lumber production plus the previous period inventory less the period inventory must equal the anti-stain lumber products demand. The anti-stain process capacity constraint [4.8] ensures that the summation of time to process green lumber products orders does not exceed the anti-stain process capacity. Constraint [4.9] ensures that the final inventory of green lumber is zero. Constraint [4.10] ensures 121  that the final inventory of anti-stain lumber is zero. Constraint [4.11] determines the sawing processing time per log diameter class (in hours). Constraint [4.12] determines the anti-stain processing time per lumber product sort (in hours). Equations 4.13 to 4.22 are model metrics. Finally, all decision variables must be non-negative.  B Plann-sched (PS) model  The PS model does not work with lumber product demand by periods. Instead, it works with orders containing a set of lumber products, which should be satisfied on certain due dates. However, as the objective of this research was to test SPT, EDD, and LPT heuristic schedules, the DSS solves the problem with only these predetermined schedules. The model formulation is as follows: Sets i  :Diameter class                     j :Log grade class k   :Sawing pattern                   p:Orders of lumber products, 1…m o  :Lumber products, from 1…n  m, and n integer numbers Data declaration: Logpricet i j : Log price in diameter class i, and log grade j in $/m3   Cilog      : Cost in $ of keep 1 m3 of log in inventory. Cirogr    : Cost in $ of keep 1 m3 of rough green lumber in inventory. Ciroat    : Cost in $ of keep 1 m3 of anti-stain treated lumber in inventory Sawing_cost  : Cost of sawing in $/hour At_cost : Cost of anti-stain treatment in $/hour Yields, capacities, and productivities coefficients Yield_sw  i j k o  : Sawmilling yield in [%] of lumber product that can be recovered from a log of diameter class i, log grade j, and sawing pattern k Yield_at o : Anti-stain yield in [%] of lumber product o  Cap_sw    : Sawmilling capacity (in hours). Cap_at      : Anti-stain capacity (in hours). Prod_sw  i  : Sawmilling productivity (hour/m3) when sawing log diameter class i. Prod_at  o  : Anti-stain productivity (hour/m3) when treating lumber product o. IOLuSA  o  : Initial inventory of green rough lumber product o. 122  IOLuAT o  : Initial inventory of anti-stain lumber product o. D_roat p  o  : Lumber demand (in m3) for lumber product o contained in order p. d_roat p o  : Due date (in hours) for order p  Definition of decision variables: U  i j k p : Volume of logs (in m3) of diameter class i, log grade class j, sawn with sawing pattern k, to satisfied order p LogV p  : Volume of logs consumed to satisfy order p (in m3) V_sa p o   : Sawmill lumber production (in m3) of lumber product o, produced  for order p  GrV p  : Volume of green lumber produced to order p ISA  p o     : Rough green lumber inventory (in m3) of lumber product o, for order p W_at p o  : Volume of green lumber product o, transferred to anti-strain for order p V_at p o    : Anti-stain treatment production (in m3) of lumber product o, for order p  Sawing_time p  : Sawing time expend to process order p Ad_1 p : Ratio of earliness for green lumber order p in relation with its due date (in %) Ad_2 p : Ratio of earliness for anti-stain lumber order p in relation with its due date (in %) Ea1 p : Earliness of the green lumber order p (in hours) Ea2 p : Earliness of the anti-stain lumber order p (in hours) Ad_1_vol p : Volume produced in advance to the due date of the green lumber order p (in m3) Ad_2_vol p : Volume produced in advance to the due date of the anti-stain lumber order p (in m3) Tfea1 p   : Time fraction of earliness of green lumber order p (%) Tfea2p   : Time fraction of earliness of anti-stain lumber order p (%) OV p   : Volume of anti-stain lumber containing order p At_time p  : Time expend to process order p on the anti-stain process (hour)  Objective function 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒: 𝑍: 𝐿𝑢𝑚𝑏𝑒𝑟 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡𝑠   [4.0]  Subject to:  ∑ 𝑈𝑖𝑗𝑘𝑝𝑌𝑖𝑒𝑙𝑑_𝑠𝑤𝑖𝑗𝑘𝑜𝑖𝑗𝑘 = 𝑉_𝑠𝑎𝑝𝑜      ∀𝑝, 𝑜   [4.1] ∑ 𝑈𝑖𝑗𝑘𝑝𝑝𝑟_𝑠𝑤𝑖𝑗𝑘𝑝 ≤ 𝑐𝑎𝑝_𝑠𝑎𝑤     ∀𝑝     [4.2] ∑ 𝑈𝑖𝑗𝑘𝑙𝑝r _𝑠𝑤𝑖𝑗𝑘𝑙 − 𝑑_𝑟𝑜𝑎𝑡𝑝 ≤ 0      ∀𝑝, 𝑙 = 1…𝑝    [4.3] 𝑉_𝑠𝑎𝑝𝑜 − 𝐼𝑂𝐿𝑢𝑆𝐴𝑝𝑜 −𝑊_𝑎𝑡𝑝𝑜 = 0 ∀𝑝, 𝑜     [4.4] 𝑊_𝑎𝑡𝑝𝑜𝑌𝑖𝑒𝑙𝑑_𝑎𝑡𝑜 = 𝑉_𝑎𝑡𝑝𝑜           ∀ 𝑝, 𝑜     [4.5] 123  ∑ 𝑊_𝑎𝑡𝑙𝑜𝑝r _𝑎𝑡𝑙,𝑜 − 𝑑_𝑟𝑜𝑎𝑡𝑝 ≤ 0      ∀𝑝, 𝑙 = 1…𝑝    [4.6] 𝑉_𝑎𝑡𝑝𝑜 + 𝐼𝑂𝐿𝑢𝐴𝑇𝑝𝑜 = 𝐷_𝑎𝑡𝑝𝑜           ∀ 𝑝, 𝑜     [4.7] 𝑇𝑓𝑒𝑎1𝑝 = 𝐸𝑎1𝑝/𝑑_𝑎𝑡𝑝           ∀ 𝑝     [4.8] 𝑇𝑓𝑒𝑎2𝑝 = 𝐸𝑎2𝑝/𝑑_𝑎𝑡𝑝           ∀ 𝑝     [4.9] 𝑂𝑉𝑝 = ∑ 𝐷_𝑟𝑜𝑎𝑡𝑝𝑜𝑜             ∀ 𝑝     [4.10] ∑ 𝑊_𝑎𝑡𝑝𝑜𝑝𝑟_𝑎𝑡𝑝𝑜 ≤ 𝑐𝑎𝑝_𝑎𝑡     ∀𝑝     [4.11] 𝐿𝑜𝑔𝑉𝑝 = ∑ 𝑈𝑖𝑗𝑘𝑝𝑖𝑗𝑘𝑝   ∀𝑝      [4.12] 𝐺𝑟𝑉𝑝 = ∑ 𝐷_𝑟𝑜𝑎𝑡𝑝𝑜𝑜               ∀ 𝑝     [4.13] 𝑆𝑎𝑤𝑖𝑛𝑔_𝑡𝑖𝑚𝑒𝑝 = ∑ 𝑈𝑖𝑗𝑘𝑝𝑝r _𝑠𝑤𝑖𝑗𝑘       ∀𝑝     [4.14] 𝐴𝑡_𝑡𝑖𝑚𝑒𝑝 = ∑ 𝑊_𝑎𝑡𝑝𝑜𝑝r _𝑎𝑡𝑜             ∀𝑝     [4.15] 𝐸𝑎1𝑝 = 𝑑_𝑎𝑡𝑝 − ∑ 𝑈𝑖𝑗𝑘𝑙𝑝r _𝑠𝑤𝑖𝑗𝑘𝑙       ∀𝑝, 𝑙 = 1…𝑝    [4.16] 𝐸𝑎2𝑝 = 𝑑_𝑎𝑡𝑝 − ∑ 𝑊_𝑎𝑡𝑙𝑜𝑝r _𝑠𝑤𝑙𝑜       ∀𝑝, 𝑙 = 1…𝑝    [4.17] 𝐴𝑑_1𝑝 = 𝐸𝑎1𝑝/𝑑_𝑟𝑜𝑎𝑡𝑝                      ∀𝑝     [4.18] 𝐴𝑑_2𝑝 = 𝐸𝑎2𝑝/𝑑_𝑟𝑜𝑎𝑡𝑝                      ∀𝑝     [4.19] 𝐴𝑑_1_𝑣𝑜𝑙𝑝 = 𝐴𝑑_1𝑝 × 𝐺𝑟𝑉𝑝                 ∀𝑝     [4.20] 𝐴𝑑_2_𝑣𝑜𝑙𝑝 = 𝐴𝑑_2𝑝 × 𝐺𝑟𝑉𝑝                 ∀𝑝     [4.21] where: 𝐿𝑜𝑔 𝑐𝑜𝑠𝑡: 𝐿𝑜𝑔_𝑐𝑜𝑠𝑡 = ∑ 𝑈𝑖𝑗𝑘𝑝𝐿𝑜𝑔𝑝𝑟𝑖𝑐𝑒𝑡𝑖𝑗𝑖𝑗𝑘𝑝     [4.22] 𝐿𝑜𝑔 𝑖𝑛𝑝𝑢𝑡: 𝐿𝑜𝑔_𝑖𝑛𝑝𝑢𝑡 = ∑ 𝑈𝑖𝑗𝑘𝑝𝑖𝑗𝑘𝑝      [4.23] 𝑆𝑎𝑤𝑖𝑛𝑔 𝑐𝑜𝑠𝑡: 𝑆𝑎𝑤𝑖𝑛𝑔_𝑐𝑜𝑠𝑡 = ∑ 𝑠𝑎𝑤𝑖𝑛𝑔_𝑡𝑖𝑚𝑒𝑝𝑠𝑎𝑤𝑖𝑛𝑔𝑐𝑜𝑠𝑡𝑝    [4.24] 𝐴𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑐𝑜𝑠𝑡: 𝐴𝑡_𝑐𝑜𝑠𝑡 = ∑ 𝐴𝑡_𝑡𝑖𝑚𝑒𝑝𝑎𝑡_𝑐𝑜𝑠𝑡𝑝    [4.25] 𝐶𝑜𝑠𝑡 𝑜𝑓 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑔𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟𝑖𝑛 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦: 𝐿𝑢_𝑔𝑟_𝑖𝑐𝑜𝑠𝑡 = ∑ 𝐺𝑟𝑉𝑝𝑇𝑓𝑒𝑎1𝑝𝑐𝑖𝑟𝑜𝑔𝑟𝑝   [4.26] 𝐶𝑜𝑠𝑡 𝑜𝑓 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑎𝑛𝑡𝑖𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑖𝑛𝑣𝑒𝑛𝑡. : 𝐿𝑢_𝑎𝑡_𝑖𝑐𝑜𝑠𝑡 = ∑ 𝐺𝑟𝑉𝑝𝑇𝑓𝑒𝑎2𝑝𝑐𝑖𝑟𝑜𝑎𝑡𝑝   [4.27] 𝐺𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛: 𝐺𝑅𝑉𝐿𝑢𝑚𝑏𝑒𝑟 = ∑ 𝑉_𝑠𝑎𝑝𝑜𝑝𝑜     [4.28] 𝐴𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛: 𝐴𝑇𝐿𝑢𝑚𝑏𝑒𝑟 = ∑ 𝑉_𝑎𝑡𝑝𝑜𝑝𝑜    [4.29] 𝐺𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑑𝑒𝑚𝑎𝑛𝑑: 𝐴𝑇_𝐷𝑒𝑚𝑎𝑛𝑑 = ∑ 𝐷_𝑟𝑜𝑎𝑡𝑝𝑜𝑝𝑜     [4.30] 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑒𝑎𝑟𝑙𝑖𝑛𝑒𝑠 𝑓𝑜𝑟 𝑔𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟: 𝐴_𝐴𝑂1 =∑ 𝐴𝑑_1𝑝𝑝6⁄    [4.31] 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑒𝑎𝑟𝑙𝑖𝑛𝑒𝑠 𝑓𝑜𝑟 𝑎𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟: 𝐴_𝐴𝑂2 =∑ 𝐴𝑑_2𝑝𝑝6⁄    [4.32] 124  𝐺𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟  𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑖𝑛 𝑎𝑑𝑣𝑎𝑛𝑐𝑒: 𝐴𝑉1 = ∑ 𝐴𝑑_1_𝑣𝑜𝑙𝑝𝑝    [4.33] 𝐴𝑛𝑡𝑖 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟  𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑖𝑛 𝑎𝑑𝑣𝑎𝑛𝑐𝑒: 𝐴𝑉2 = ∑ 𝐴𝑑_2_𝑣𝑜𝑙𝑝𝑝    [4.34] 𝐿𝑢𝑚𝑏𝑒𝑟 𝑚𝑎𝑛𝑢𝑓. 𝑐𝑜𝑠𝑡 = 𝐿𝑜𝑔_𝑐𝑜𝑠𝑡 + 𝐿𝑢_𝑔𝑟_𝑖𝑐𝑜𝑠𝑡 + 𝐿𝑢_𝑎𝑡_𝑖𝑐𝑜𝑠𝑡 +  𝑆𝑎𝑤𝑖𝑛𝑔_𝑐𝑜𝑠𝑡 + 𝐴𝑡_𝑐𝑜𝑠𝑡      [4.35]  The objective function minimizes lumber manufacturing costs, which are log costs, green lumber and anti-stain inventory costs, plus sawing and anti-stain processing costs. No overdue orders were allowed. The main difference between DSS B (PS), and DSS A (PL) is that DSS B contains one period equivalent to one week of time while DSS A considers six periods, equivalent to days of the week. Also, lumber demands by period for DSS A are linked to orders containing a set of lumber products that must be satisfied on a due date. Accordingly, orders must be processed with a sequence, which were the EDD, LPT, and SPT heuristic schedules. Additionally, I applied a set of constraints which were the following: The sawn wood production constraint [4.1], where logs of certain diameter class, and grade class, are processed with certain sawing pattern to produce certain lumber product o, to satisfy order p; the sawmilling capacity constraint [4.2], the summation of time to process logs of all orders cannot exceed sawing capacity in hours. The earliness constraint [4.3] for green lumber orders, which ensures that the processing time of order p, and processing times of preceding orders must be ≤ the due date of order p. The flow balance equation [4.4], where the sawn wood production plus the green lumber products inventory are transferred to the anti-stain process. The anti-stain treated production constraint [4.5], where rough green lumber products are processed with certain anti-stain yields to produce certain anti-stain lumber products to satisfy order p. The earliness constraint [4.6] for anti-stain lumber orders, which ensures that the processing time of order p, and processing times of preceding orders must be ≤ to the due date of order p. The market constraint [4.7], which ensures that the anti-stain lumber production plus the anti-stain inventory must be equal to the anti-stain lumber demand for all orders of anti-stain product. Additionally, constraint [4.8] determines the time fraction of earliness of green lumber order p (in %) as the ratio between the earliness of order p (in hours), and it due date (in hours). Constraint [4.9] determines the time fraction of earliness of anti-stain lumber order p (in %) as the ratio between the earliness of order p (in hours), and it due date (in hours). Constraint 4.10 determines the volume (in m3) of each anti-stain lumber product order. Constraint [4.11] ensures that the summation of the time expended to process green lumber products orders does not exceed anti-stain process capacity. Constraint [4.12] determines the volume of logs consumed (m3) for order p. Constraint [4.13] determines the volume (m3) of order p of anti-stain lumber products. Constraint [4.14] determines the sawing processing time of order p (in hours). 125  Constraint [4.15] determines the anti-stain processing time of order p (in hours). Constraint [4.16] determines the earliness of order p of green lumber (in hours). Constraint [4.17] determines the earliness of order p of anti-stain lumber (in hours). Constraint [4.18] determines the ratio of earliness of order p of green lumber (in %) in relation with its due date. Constraint [4.19] determines the ratio of earliness of order p of anti-stain lumber (in %) in relation with it due date. Constraint [4.20] determines the volume of green lumber (m3) produced in advance of order p. Constraint [4.21] determines the volume of anti-stain lumber (m3) produced in advance of order p. Equations [4.22] to [4.35] are model metrics. Finally, all decision variables must be non-negative.  C Plann model accepting backlog  This version of the PL model accepts backlogs, only a few parts of the DSS A model formulation needed to be changed. Backlogs produced in one period must be produced immediately in the next planning period;, however, backlogs were not penalized. New decision variables B_gr os : Backlog of green lumber products o, in period s (in m3) B_at os : Backlog of anti-stain lumber products o, in period s (in m3) Model formulation  𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒: 𝑍: 𝐿𝑢𝑚𝑏𝑒𝑟 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡𝑠    [4.0] Subject to: ∑ 𝑈𝑖𝑗𝑘𝑠 × 𝑌𝑖𝑒𝑙𝑑_𝑠𝑤𝑖𝑗𝑘𝑜𝑖𝑗𝑘𝑠 = 𝑉_𝑠𝑎𝑜𝑠     ∀𝑜, 𝑠 = 1    [4.1] ∑ 𝑈𝑖𝑗𝑘𝑠 × 𝑌𝑖𝑒𝑙𝑑_𝑠𝑤𝑖𝑗𝑘𝑜𝑖𝑗𝑘𝑠 = 𝑉_𝑠𝑎𝑜𝑠 + 𝐵_𝑔𝑟𝑜𝑠−1     ∀𝑜, 𝑠 > 1    [4.2] ∑ 𝑈𝑖𝑗𝑘𝑠 × 𝑝𝑟_𝑠𝑤𝑖𝑗𝑘 ≤ 𝑐𝑎𝑝_𝑠𝑎𝑤𝑠      ∀𝑠     [4.3] 𝑉_𝑠𝑎𝑜𝑠 − 𝐼𝑆𝐴𝑜𝑠 −𝑊_𝑎𝑡𝑜𝑠 = 0 ∀𝑜, 𝑠 = 1     [4.4] 𝑉_𝑠𝑎𝑜𝑠 + 𝐼𝑆𝐴𝑜𝑠−1 − 𝐼𝑆𝐴𝑜𝑠 + 𝐵_𝑔𝑟𝑜𝑠 −𝑊_𝑎𝑡𝑜𝑠 = 0 ∀𝑜, 𝑠 > 1    [4.5] 𝑊_𝑎𝑡𝑜𝑠 × 𝑌𝑖𝑒𝑙𝑑_𝑎𝑡𝑜 = 𝑉_𝑎𝑡𝑜𝑠           ∀ 𝑜, 𝑠 = 1    [4.6] 𝑊_𝑎𝑡𝑜𝑠 × 𝑌𝑖𝑒𝑙𝑑_𝑎𝑡𝑜 = 𝑉_𝑎𝑡𝑜𝑠 + 𝐵_𝑎𝑡𝑜𝑠−1    ∀ 𝑜, 𝑠 > 1    [4.7] 126  𝑉_𝑎𝑡𝑜𝑠 + 𝐼𝐴𝑇𝑜𝑠−1 = 𝐷_𝑟𝑜𝑎𝑡𝑜𝑠           ∀ 𝑜, 𝑠 = 1    [4.8] 𝑉_𝑎𝑡𝑜𝑠 + 𝐼𝐴𝑇𝑜𝑠−1 − 𝐼𝐴𝑇𝑜𝑠 + 𝐵_𝑎𝑡𝑜𝑠 = 𝐷_𝑟𝑜𝑎𝑡𝑜𝑠           ∀ 𝑜, 𝑠 > 1   [4.9] ∑ 𝑊_𝑎𝑡𝑜𝑠 × 𝑝𝑟_𝑎𝑡𝑜𝑠 ≤ 𝑐𝑎𝑝_𝑎𝑡𝑠     ∀𝑠     [4.10] 𝐵_𝑔𝑟𝑜𝑠 = 0     ∀ 𝑜, 𝑠 = 6    𝑎𝑛𝑑        𝐵_𝑎𝑡𝑜𝑠 = 0     ∀ 𝑜, 𝑠 = 6        [4.11] 𝐼𝑆𝐴𝑜𝑠 = 0                                        ∀ 𝑜, 𝑠 = 6     [4.12] 𝐼𝐴𝑇𝑜𝑠 = 0                                        ∀ 𝑜, 𝑠 = 6     [4.13] 𝑆𝑎𝑤𝑖𝑛𝑔_𝑡𝑖𝑚𝑒𝑖 = ∑ 𝑈𝑖𝑗𝑘𝑠 × 𝑝r _𝑠𝑤𝑗𝑘𝑠       ∀𝑖     [4.14] 𝐴𝑡_𝑡𝑖𝑚𝑒𝑜 = ∑ 𝑊_𝑎𝑡𝑜𝑠 × 𝑝r _𝑎𝑡𝑠             ∀𝑠     [4.15] Where: 𝐿𝑜𝑔 𝑐𝑜𝑠𝑡: 𝐿𝑜𝑔_𝑐𝑜𝑠𝑡 = ∑ 𝑈𝑖𝑗𝑘𝑠 × 𝐿𝑜𝑔𝑝𝑟𝑖𝑐𝑒𝑡𝑖𝑗𝑖𝑗𝑘𝑠     [4.16] 𝐿𝑜𝑔 𝑖𝑛𝑝𝑢𝑡: 𝐿𝑜𝑔_𝑖𝑛𝑝𝑢𝑡 = ∑ 𝑈𝑖𝑗𝑘𝑠𝑖𝑗𝑘𝑠      [4.17] 𝑆𝑎𝑤𝑖𝑛𝑔 𝑐𝑜𝑠𝑡: 𝑆𝑎𝑤𝑖𝑛𝑔_𝑐𝑜𝑠𝑡 = ∑ 𝑠𝑎𝑤𝑖𝑛𝑔_𝑡𝑖𝑚𝑒𝑖 × 𝑠𝑎𝑤𝑖𝑛𝑔𝑐𝑜𝑠𝑡𝑖     [4.18] 𝐴𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑐𝑜𝑠𝑡: 𝐴𝑡_𝑐𝑜𝑠𝑡 = ∑ 𝐴𝑡_𝑡𝑖𝑚𝑒𝑜 × 𝑎𝑡_𝑐𝑜𝑠𝑡𝑜    [4.19] 𝐶𝑜𝑠𝑡 𝑜𝑓 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑔𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦: 𝐿𝑢_𝑔𝑟_𝑖𝑐𝑜𝑠𝑡 = ∑ 𝐼𝑆𝐴𝑜𝑠 × 𝑐𝑖𝑟𝑜𝑔𝑟𝑠𝑜𝑠   [4.20] 𝐶𝑜𝑠𝑡 𝑜𝑓 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑎𝑛𝑡𝑖𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦: 𝐿𝑢_𝑎𝑡_𝑖𝑐𝑜𝑠𝑡 = ∑ 𝐼𝐴𝑇𝑜𝑠 × 𝑐𝑖𝑟𝑜𝑎𝑡𝑠𝑜𝑠   [4.21] 𝐺𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛: 𝐺𝑅𝑉𝐿𝑢𝑚𝑏𝑒𝑟 = ∑ 𝑉_𝑠𝑎𝑜𝑠𝑜𝑠     [4.22] 𝐴𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛: 𝐴𝑇𝐿𝑢𝑚𝑏𝑒𝑟 = ∑ 𝑉_𝑎𝑡𝑜𝑠𝑜𝑠     [4.23] 𝐺𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑑𝑒𝑚𝑎𝑛𝑑: 𝐴𝑇_𝐷𝑒𝑚𝑎𝑛𝑑 = ∑ 𝐷_𝑟𝑜𝑎𝑡𝑜𝑠𝑜𝑠     [4.24] 𝐵𝑎𝑐𝑘𝑙𝑜𝑔𝑔𝑒𝑑 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑔𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 ∶ 𝐵𝑉1 = ∑ 𝐵_𝑔𝑟𝑜𝑠𝑜𝑠     [4.25] 𝐵𝑎𝑐𝑘𝑙𝑜𝑔𝑔𝑒𝑑 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 ∶ 𝐵𝑉2 = ∑ 𝐵_𝑎𝑡𝑟𝑜𝑠𝑜𝑠    [4.26] 𝐿𝑢𝑚𝑏𝑒𝑟 𝑚𝑎𝑛𝑢𝑓. 𝑐𝑜𝑠𝑡 = 𝐿𝑜𝑔_𝑐𝑜𝑠𝑡 + 𝐿𝑢_𝑔𝑟_𝑖𝑐𝑜𝑠𝑡 + 𝐿𝑢_𝑎𝑡_𝑖𝑐𝑜𝑠𝑡 +  𝑆𝑎𝑤𝑖𝑛𝑔_𝑐𝑜𝑠𝑡 + 𝐴𝑡_𝑐𝑜𝑠𝑡   [4.27]  I added new variables, constraints, and metrics to account for backlog volumes. The model constraints are: constraint [4.1] and [4.2], which ensure that the production of green lumber products for period 1, and for periods >1 includes backlogs from the previous period. The sawmilling capacity constraint [4.3], the summation of the time expended to saw logs, cannot exceed sawing capacity per period; the flow balance 127  constraint [4.4] for period 1, the green lumber production less the period inventory is transferred to the anti-stain process. The flow balance constraint [4.5] for periods >1, which ensures that the flow of green lumber products from sawmill to anti-stain processes include the backlogs. The anti-stain treated production for period 1 [4.6], where rough green lumber products are processed with anti-stain yields to produce anti-stain lumber products; constraint [4.7] ensures that the production of anti-stain lumber for periods >1 include backlogs from the previous period; constraint [4.8] is the market constraint for period 1; constraint [4.9] ensures that the demand of anti-stain lumber products for periods >1 includes inventory variation and backlogs; constraint [4.10] is anti-stain process capacity constraint; constraint [4.11], [4.12], and [4.13] ensures no backlogs, green lumber products inventory and anti-stain lumber products inventory for the final period; constraints [4.14] and [4.15] determine sawing time, and anti-stain processes time; constraints [4.16] to [4.24] are model metrics, and constraints [4.25] and [4.26], determine the total backlogged volumes of green lumber products, and the total backlogged volume of anti-stain lumber products, respectively, finally, all variables must be non-negative.   D Plann-sched model accepting overdue order  This version of the PS model accepts overdue orders, which means delays in hours of a certain order relative to its due date. I added new variables, constraints, and metrics to account for backlogged volumes due to orders delays. Delays were transformed in volumes by using sawing and anti-stain process productivity, which enabled me to compare backlogged volumes between relaxed formulation of models C and D. Similar to model C, in model D overdue orders and their equivalent backlogs volumes were not penalized. Thus, only a few parts of the B formulation changed. The new parts of this relaxed PS formulation follow: New decision variables A1 p  : Advanced time of green lumber order p B1 p  : Backlogged time of green lumber order p A2 p  : Advanced time of anti-stain lumber order p B2 p  : Backlogged time of anti-stain lumber order p 128  B_gr os  : Backlog of green lumber products o, in period s (in m3) B_at os  : Backlog of anti-stain lumber products o, in period s (in m3) BA_1_vol p : Delayed volumes produced of the green lumber order p (in m3) BA_2_vol p : Delayed volumes produced of the anti-stain lumber order p (in m3) Tfde1 p   : Time fraction of delay of green lumber order p (%) Tfde2 p   : Time fraction of delay of anti-stain lumber order p (%)  Objective function  𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒: 𝑍: 𝐿𝑢𝑚𝑏𝑒𝑟 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡𝑠   [4.0] Subject to: ∑ 𝑈𝑖𝑗𝑘𝑝 × 𝑌𝑖𝑒𝑙𝑑_𝑠𝑤𝑖𝑗𝑘𝑜𝑖𝑗𝑘 = 𝑉_𝑠𝑎𝑝𝑜     ∀𝑝, 𝑜  [4.1] ∑ 𝑈𝑖𝑗𝑘𝑝 × 𝑝𝑟_𝑠𝑤𝑖𝑗𝑘𝑝 ≤ 𝑐𝑎𝑝_𝑠𝑎𝑤     ∀𝑝   [4.2] ∑ 𝑈𝑖𝑗𝑘𝑙 × 𝑝r _𝑠𝑤𝑖𝑗𝑘𝑙 +𝐴1𝑝 − 𝐵1𝑝 − 𝑑_𝑟𝑜𝑎𝑡𝑝 ≤ 0      ∀𝑝, 𝑙 = 1…𝑝  [4.3] 𝑉_𝑠𝑎𝑝𝑜 − 𝐼𝑂𝐿𝑢𝑆𝐴𝑝𝑜 −𝑊_𝑎𝑡𝑝𝑜 = 0 ∀𝑝, 𝑜  [4.4] 𝑊_𝑎𝑡𝑝𝑜 × 𝑌𝑖𝑒𝑙𝑑_𝑎𝑡𝑜 = 𝑉_𝑎𝑡𝑝𝑜           ∀ 𝑝, 𝑜   [4.5] ∑ 𝑊_𝑎𝑡𝑙𝑜 × 𝑝r _𝑎𝑡𝑙,𝑜 + 𝐴2𝑝 − 𝐵2𝑝 − 𝑑_𝑟𝑜𝑎𝑡𝑝 ≤ 0      ∀ 𝑙 = 1…𝑝, 𝑜  [4.6] 𝑉_𝑎𝑡𝑝𝑜 + 𝐼𝑂𝐿𝑢𝐴𝑇𝑝𝑜 = 𝐷_𝑎𝑡𝑝𝑜           ∀ 𝑝, 𝑜  [4.7] 𝑇𝑓𝑒𝑎1𝑝 = 𝐴1𝑝/𝑑_𝑎𝑡𝑝           ∀ 𝑝  [4.8] 𝑇𝑓𝑒𝑎2𝑝 = 𝐴2𝑝/𝑑_𝑎𝑡𝑝           ∀ 𝑝  [4.9] 𝑇𝑓𝑑𝑒1𝑝 = 𝐵1𝑝/𝑑_𝑎𝑡𝑝           ∀ 𝑝  [4.10] 𝑇𝑓𝑑𝑒2𝑝 = 𝐵2𝑝/𝑑_𝑎𝑡𝑝           ∀ 𝑝  [4.11] 𝑂𝑉𝑝 = ∑ 𝐷_𝑟𝑜𝑎𝑡𝑝𝑜𝑜             ∀ 𝑝  [4.12] ∑ 𝑊_𝑎𝑡𝑝𝑜 × 𝑝𝑟_𝑎𝑡𝑝𝑜 ≤ 𝑐𝑎𝑝_𝑎𝑡     ∀𝑝  [4.13] 𝐿𝑜𝑔𝑉𝑝 = ∑ 𝑈𝑖𝑗𝑘𝑝𝑖𝑗𝑘             ∀𝑝  [4.14] 𝐺𝑟𝑉𝑝 = ∑ 𝐷_𝑟𝑜𝑎𝑡𝑝𝑜𝑜               ∀ 𝑝  [4.15] 𝑆𝑎𝑤𝑖𝑛𝑔_𝑡𝑖𝑚𝑒𝑝 = ∑ 𝑈𝑖𝑗𝑘𝑝 × 𝑝r _𝑠𝑤𝑖𝑗𝑘       ∀𝑝  [4.16] 129  𝐴𝑡_𝑡𝑖𝑚𝑒𝑝 = ∑ 𝑊_𝑎𝑡𝑝𝑜 × 𝑝r _𝑎𝑡𝑜             ∀𝑝 [4.17] 𝐴𝑑_1𝑝 = 𝐴1𝑝/𝑑_𝑟𝑜𝑎𝑡𝑝  × 100%                     ∀𝑝  [4.18] 𝐴𝑑_2𝑝 = 𝐴2𝑝/𝑑_𝑟𝑜𝑎𝑡𝑝  × 100%                     ∀𝑝  [4.19] 𝐵𝑎_1𝑝 = 𝐵1𝑝/𝑑_𝑟𝑜𝑎𝑡𝑝   × 100%                     ∀𝑝  [4.20] 𝐵𝑎_2𝑝 = 𝐵2𝑝/𝑑_𝑟𝑜𝑎𝑡𝑝    × 100%                    ∀𝑝 [4.21] 𝐴𝑑_1_𝑣𝑜𝑙𝑝 = 𝐴𝑑_1𝑝 × 𝐺𝑟𝑉𝑝/100%                ∀𝑝  [4.22] 𝐴𝑑_2_𝑣𝑜𝑙𝑝 = 𝐴𝑑_2𝑝 × 𝐺𝑟𝑉𝑝/100%                ∀𝑝  [4.23] 𝐵𝑎_1_𝑣𝑜𝑙𝑝 = 𝐵𝑎_1𝑝 × 𝐺𝑟𝑉𝑝/100%                ∀𝑝   [4.24] 𝐵𝑎_2_𝑣𝑜𝑙𝑝 = 𝐵𝑎_2𝑝 × 𝐺𝑟𝑉𝑝/100%                ∀𝑝   [4.25] Where: 𝐿𝑜𝑔 𝑐𝑜𝑠𝑡: 𝐿𝑜𝑔_𝑐𝑜𝑠𝑡 = ∑ 𝑈𝑖𝑗𝑘𝑝𝐿𝑜𝑔𝑝𝑟𝑖𝑐𝑒𝑡𝑖𝑗𝑖𝑗𝑘𝑝   [4.26] 𝐿𝑜𝑔 𝑖𝑛𝑝𝑢𝑡: 𝐿𝑜𝑔_𝑖𝑛𝑝𝑢𝑡 = ∑ 𝑈𝑖𝑗𝑘𝑝𝑖𝑗𝑘𝑝   [4.27] 𝑆𝑎𝑤𝑖𝑛𝑔 𝑐𝑜𝑠𝑡: 𝑆𝑎𝑤𝑖𝑛𝑔_𝑐𝑜𝑠𝑡 = ∑ 𝑠𝑎𝑤𝑖𝑛𝑔_𝑡𝑖𝑚𝑒𝑝𝑠𝑎𝑤𝑖𝑛𝑔𝑐𝑜𝑠𝑡𝑝   [4.28] 𝐴𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑐𝑜𝑠𝑡: 𝐴𝑡_𝑐𝑜𝑠𝑡 = ∑ 𝐴𝑡_𝑡𝑖𝑚𝑒𝑝𝑎𝑡_𝑐𝑜𝑠𝑡𝑝  [4.29] 𝐶𝑜𝑠𝑡 𝑜𝑓 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑔𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟𝑖𝑛 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦: 𝐿𝑢_𝑔𝑟_𝑖𝑐𝑜𝑠𝑡 = ∑ 𝐺𝑟𝑉𝑝𝑇𝑓𝑒𝑎1𝑝𝑐𝑖𝑟𝑜𝑔𝑟𝑝   [4.30] 𝐶𝑜𝑠𝑡 𝑜𝑓 ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑎𝑛𝑡𝑖𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑖𝑛𝑣𝑒𝑛𝑡. : 𝐿𝑢_𝑎𝑡_𝑖𝑐𝑜𝑠𝑡 = ∑ 𝐺𝑟𝑉𝑝𝑇𝑓𝑒𝑎2𝑝𝑐𝑖𝑟𝑜𝑔𝑟𝑝   [4.31] 𝐶𝑜𝑠𝑡 𝑜𝑓 𝑠𝑒𝑡𝑢𝑝 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑖𝑛𝑔 𝑝𝑙𝑎𝑛 𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛: 𝑆𝑒𝑡𝑢𝑝𝑐𝑜𝑠𝑡 = ∑ 𝐵𝑈𝑖𝑘𝑠𝑒𝑡𝑢𝑝𝑐𝑜𝑠𝑡𝑖𝑖𝑘   [4.32] 𝐺𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛: 𝐺𝑅𝐿𝑢𝑚𝑏𝑒𝑟 = ∑ 𝑉_𝑠𝑎𝑝𝑜𝑝𝑜   [4.33] 𝑇𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑏𝑎𝑐𝑘𝑙𝑜𝑔𝑔𝑒𝑑 𝑔𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑜𝑟𝑑𝑒𝑟𝑠: 𝐺𝑅𝐵𝑎𝑐𝑘𝑙𝑜𝑔 = ∑ 𝐵_𝑔𝑟𝑝𝑜𝑝𝑜   [4.34] 𝐴𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛: 𝐴𝑇𝐿𝑢𝑚𝑏𝑒𝑟 = ∑ 𝑉_𝑎𝑡𝑝𝑜𝑝𝑜   [4.35] 𝑇𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑏𝑎𝑐𝑘𝑙𝑜𝑔𝑔𝑒𝑑 𝑎𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑜𝑟𝑑𝑒𝑟𝑠: 𝐴𝑇𝐵𝑎𝑐𝑘𝑙𝑜𝑔 = ∑ 𝐵_𝑎𝑡𝑝𝑜𝑝𝑜   [4.36] 𝐴𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟 𝑑𝑒𝑚𝑎𝑛𝑑: 𝐴𝑇_𝐷𝑒𝑚𝑎𝑛𝑑 = ∑ 𝐷_𝑟𝑜𝑎𝑡𝑝𝑜𝑝𝑜   [4.37] 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑒𝑎𝑟𝑙𝑖𝑛𝑒𝑠 𝑓𝑜𝑟 𝑔𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟: 𝐴_𝐴01 =   ∑ 𝐴𝑑1𝑝/6𝑝   [4.38] 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑒𝑎𝑟𝑙𝑖𝑛𝑒𝑠 𝑓𝑜𝑟 𝑎𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟: 𝐴_𝐴𝑂2 = ∑ 𝐴𝑑1𝑝/6𝑝   [4.39] 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑑𝑒𝑙𝑎𝑦 𝑓𝑜𝑟 𝑔𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟: 𝐴_𝐵01 =   ∑ 𝐵𝑎1𝑝/6𝑝   [4.40] 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑑𝑒𝑙𝑎𝑦 𝑓𝑜𝑟 𝑎𝑛𝑡𝑖 − 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟: 𝐴_𝐵𝑂2 = ∑ 𝐵𝑎2𝑝/6𝑝    [4.41] 130  𝐺𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟  𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑖𝑛 𝑎𝑑𝑣𝑎𝑛𝑐𝑒: 𝐴𝑉1 = ∑ 𝐴𝑑_1_𝑣𝑜𝑙𝑝𝑝    [4.42] 𝐴𝑛𝑡𝑖 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟  𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑖𝑛 𝑎𝑑𝑣𝑎𝑛𝑐𝑒: 𝐴𝑉2 = ∑ 𝐴𝑑_2_𝑣𝑜𝑙𝑝𝑝   [4.43] 𝐺𝑟𝑒𝑒𝑛 𝑙𝑢𝑚𝑏𝑒𝑟  𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑖𝑛 𝑑𝑒𝑙𝑎𝑦: 𝐵𝑉1 = ∑ 𝐵𝑎_1_𝑣𝑜𝑙𝑝𝑝   [4.44] 𝐴𝑛𝑡𝑖 𝑠𝑡𝑎𝑖𝑛 𝑙𝑢𝑚𝑏𝑒𝑟  𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑖𝑛 𝑑𝑒𝑙𝑎𝑦: 𝐵𝑉2 = ∑ 𝐵𝑎_2_𝑣𝑜𝑙𝑝𝑝   [4.45] 𝐿𝑢𝑚𝑏𝑒𝑟 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 = 𝐿𝑜𝑔𝑐𝑜𝑠𝑡 + 𝐿𝑢𝑔𝑟𝑖𝑐𝑜𝑠𝑡 + 𝐿𝑢𝑎𝑡𝑖𝑐𝑜𝑠𝑡 +  𝑆𝑎𝑤𝑖𝑛𝑔_𝑐𝑜𝑠𝑡 + 𝐴𝑡_𝑐𝑜𝑠𝑡 [4.46]  In Model D, I added surplus and slack variables to the earliness constraint to make the solution feasible. The following new constraints were added to the model C formulation. Constraint [4.3], which ensures that the processing time of green lumber order p, and processing times of preceding orders can be lower or exceed the due date of order p. Thus, the surplus or slack variables captures the delay or earliness of each order. Constraint [4.6], which ensures that the processing time of anti-stain lumber order p, and processing times of preceding orders can be lower or exceed the due date of order p, thus, these surplus or slack variables capture the delay or earliness of each order. Constraints [4.8], and [4.9] help to determine the time fraction of earliness of green orders and anti-stain lumber orders in relation with its respective due dates. Constraints [4.10], and [4.11], help to determine the time fraction of delay of green lumber orders and anti-stain lumber orders in relation with its respective due dates.129   Appendix D DSS results for chapter 4 D.1 Non relaxed run results Table D.1 PL and Pl-S run results, no backlog and no overdue orders: Pl with large, small, mixed, high variation, and low variations demand PLANN La 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 446,398 448,783 448,905 449,480 448,931 449,245 449,015 447,587 446,953 449,223 448,134 448,443 446,930 448,982 446,577 448,418 449,976 448,764 449,673 448,203 449,538 451,167 448,051 449,033 447,353 449,801 447,160 446,945 450,113 447,168 Log_Cost 381,405 383,446 383,554 384,041 383,568 383,843 383,646 382,422 381,883 383,819 382,882 383,150 381,862 383,617 381,560 383,132 384,465 383,423 384,207 382,953 384,083 385,486 382,819 383,662 382,220 384,317 382,061 381,878 384,582 382,068 Lugr_icost 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Luat_icost 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Sawing_cost 63,441 63,775 63,790 63,875 63,802 63,839 63,808 63,608 63,515 63,841 63,734 63,515 63,803 63,464 63,726 63,945 63,781 63,901 63,691 63,892 64,111 63,674 63,810 63,578 63,919 63,544 63,513 63,965 63,544 At_cost 1,553 1,562 1,562 1,564 1,561 1,563 1,562 1,557 1,555 1,563 1,558 1,559 1,554 1,562 1,553 1,560 1,565 1,560 1,564 1,559 1,563 1,570 1,558 1,562 1,556 1,565 1,555 1,555 1,566 1,556Log_input 11,748 11,810 11,813 11,829 11,815 11,822 11,816 11,779 11,762 11,822 11,795 11,803 11,762 11,815 11,753 11,801 11,842 11,811 11,834 11,795 11,832 11,872 11,791 11,817 11,774 11,837 11,767 11,762 11,845 11,767GRLumber 7,667 7,713 7,712 7,722 7,709 7,719 7,713 7,687 7,678 7,717 7,692 7,701 7,675 7,714 7,671 7,701 7,731 7,706 7,725 7,701 7,719 7,752 7,695 7,713 7,683 7,729 7,681 7,677 7,732 7,684ATLumber 7,659 7,705 7,704 7,714 7,701 7,711 7,705 7,679 7,670 7,710 7,684 7,693 7,668 7,706 7,663 7,694 7,723 7,698 7,717 7,693 7,711 7,745 7,687 7,705 7,675 7,721 7,673 7,669 7,724 7,676ATDemand 7,659 7,705 7,704 7,714 7,701 7,711 7,705 7,679 7,670 7,710 7,684 7,693 7,668 7,706 7,663 7,694 7,723 7,698 7,717 7,693 7,711 7,745 7,687 7,705 7,675 7,721 7,673 7,669 7,724 7,676PLANN S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 359,927 359,905 360,529 360,529 357,747 356,286 359,986 359,199 359,472 357,725 359,480 359,555 358,936 358,976 359,098 361,208 362,267 361,185 359,269 359,036 360,583 360,638 358,342 361,036 359,904 357,759 359,342 360,392 359,373 358,496 Log_Cost 307,529 307,510 308,039 308,039 305,662 304,409 307,577 306,906 307,139 305,648 307,150 307,207 306,674 306,711 306,820 308,622 309,528 308,609 306,964 306,767 308,085 308,139 306,163 308,475 307,508 305,670 307,023 307,928 307,049 306,302 Lugr_icost 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Luat_icost 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Sawing_cost 51,146 51,143 51,236 51,236 50,841 50,638 51,157 51,043 51,082 50,833 51,079 51,098 51,015 51,017 51,028 51,329 51,478 51,319 51,055 51,020 51,244 51,244 50,933 51,305 51,144 50,845 51,070 51,209 51,074 50,947 At_cost 1,252 1,252 1,254 1,254 1,244 1,239 1,252 1,250 1,251 1,244 1,251 1,250 1,248 1,248 1,249 1,257 1,261 1,257 1,250 1,249 1,254 1,255 1,246 1,256 1,252 1,244 1,249 1,255 1,250 1,247Log_input 9,471 9,471 9,488 9,488 9,415 9,377 9,474 9,452 9,460 9,413 9,459 9,463 9,447 9,448 9,450 9,505 9,533 9,503 9,455 9,448 9,490 9,490 9,432 9,501 9,471 9,416 9,457 9,483 9,458 9,435GRLumber 6,184 6,184 6,192 6,192 6,144 6,118 6,183 6,171 6,176 6,145 6,179 6,175 6,163 6,164 6,170 6,206 6,225 6,208 6,172 6,169 6,194 6,198 6,151 6,202 6,183 6,144 6,170 6,195 6,172 6,159ATLumber 6,178 6,178 6,186 6,186 6,138 6,112 6,177 6,165 6,170 6,139 6,173 6,169 6,157 6,158 6,164 6,200 6,219 6,202 6,166 6,163 6,188 6,192 6,145 6,196 6,177 6,138 6,164 6,189 6,166 6,153ATDemand 6,178 6,178 6,186 6,186 6,138 6,112 6,177 6,165 6,170 6,139 6,173 6,169 6,157 6,158 6,164 6,200 6,219 6,202 6,166 6,163 6,188 6,192 6,145 6,196 6,177 6,138 6,164 6,189 6,166 6,153PLANN BS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 400,158 390,656 396,173 402,987 401,611 407,985 408,524 400,387 409,950 405,489 398,339 404,858 410,119 403,869 398,328 396,330 408,299 403,490 397,689 391,271 408,655 408,655 393,705 400,293 403,661 404,534 401,222 410,278 399,693 399,317 Log_Cost 341,878 333,770 338,491 344,330 343,165 348,551 349,085 342,104 350,246 346,452 340,338 345,896 350,403 345,062 340,320 338,620 348,842 344,739 339,796 334,287 349,121 349,121 336,339 342,018 344,895 345,655 342,789 350,522 341,489 341,183 Lugr_icost 2 0 0 0 0 0 0 0 0 12 0 0 11 9 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 11 Luat_icost 9 0 0 0 0 0 6 9 0 12 8 12 4 0 8 14 10 6 0 8 0 0 12 11 16 0 14 11 10 14Sawing_cost 56,879 55,528 56,303 57,256 57,046 58,017 58,010 56,880 58,282 57,600 56,607 57,544 58,274 57,395 56,616 56,317 58,027 57,335 56,509 55,617 58,119 58,119 55,988 56,870 57,344 57,469 57,024 58,319 56,803 56,718 At_cost 1,390 1,358 1,379 1,402 1,399 1,417 1,424 1,393 1,423 1,413 1,386 1,406 1,426 1,403 1,384 1,379 1,420 1,405 1,385 1,359 1,415 1,415 1,366 1,394 1,406 1,410 1,395 1,425 1,391 1,391Log_input 10,533 10,283 10,426 10,603 10,564 10,744 10,743 10,533 10,793 10,667 10,483 10,656 10,792 10,629 10,484 10,429 10,746 10,618 10,465 10,299 10,763 10,763 10,368 10,531 10,619 10,642 10,560 10,800 10,519 10,503GRLumber 6,864 6,705 6,810 6,922 6,910 6,997 7,030 6,881 7,025 6,976 6,846 6,942 7,044 6,928 6,835 6,809 7,012 6,939 6,838 6,712 6,990 6,990 6,744 6,884 6,943 6,962 6,888 7,039 6,868 6,869ATLumber 6,857 6,698 6,803 6,915 6,903 6,990 7,023 6,874 7,018 6,969 6,839 6,935 7,037 6,921 6,828 6,802 7,005 6,932 6,831 6,705 6,983 6,983 6,737 6,877 6,936 6,955 6,881 7,032 6,861 6,862ATDemand 6,857 6,698 6,803 6,915 6,903 6,990 7,023 6,874 7,018 6,969 6,839 6,935 7,037 6,921 6,828 6,802 7,005 6,932 6,831 6,705 6,983 6,983 6,737 6,877 6,936 6,955 6,881 7,032 6,861 6,862PLANN H 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 478,382 430,316 433,450 442,139 462,149 450,195 456,095 464,627 456,020 462,988 428,823 455,992 506,772 437,663 459,011 444,282 471,544 459,951 467,127 435,848 422,182 457,268 461,411 433,167 445,649 408,055 474,955 423,911 460,551 Log_Cost 408,604 367,600 370,271 377,513 394,834 384,582 389,604 396,751 389,736 395,835 366,204 389,625 429,665 373,757 392,178 379,625 402,708 393,098 399,131 372,241 433,786 360,514 390,532 393,907 370,054 380,841 348,326 405,745 361,894 393,420 Lugr_icost 4 20 32 256 95 121 170 87 5 0 65 31 1,046 16 41 0 108 99 65 2 1,087 144 86 148 0 0 11 0 230 0 Luat_icost 351 68 116 22 165 52 98 265 73 122 124 77 2,120 254 93 28 79 47 15 156 1,764 168 67 101 243 68 289 105 86 48Sawing_cost 67,748 61,135 61,524 62,820 65,443 63,876 64,633 65,905 64,615 65,407 60,949 64,667 72,333 62,119 65,097 63,088 67,015 65,094 66,287 61,934 73,127 59,884 64,998 65,660 61,362 63,188 58,024 67,454 60,231 65,489 At_cost 1,674 1,494 1,506 1,527 1,612 1,565 1,591 1,618 1,590 1,624 1,481 1,592 1,608 1,517 1,602 1,541 1,634 1,614 1,629 1,516 1,608 1,473 1,585 1,594 1,509 1,551 1,405 1,651 1,470 1,593Log_input 12,546 11,321 11,393 11,633 12,119 11,829 11,969 12,205 11,966 12,112 11,287 11,975 13,395 11,503 12,055 11,683 12,410 12,054 12,275 11,469 13,542 11,090 12,037 12,159 11,363 11,701 10,745 12,491 11,154 12,128GRLumber 8,266 7,376 7,436 7,542 7,962 7,728 7,855 7,988 7,854 8,018 7,315 7,860 7,941 7,491 7,910 7,610 8,070 7,969 8,043 7,484 7,939 7,272 7,827 7,874 7,449 7,661 6,938 8,155 7,259 7,869ATLumber 8,258 7,369 7,429 7,534 7,954 7,720 7,847 7,980 7,846 8,010 7,308 7,852 7,933 7,484 7,902 7,602 8,062 7,961 8,035 7,477 7,931 7,265 7,819 7,866 7,442 7,653 6,931 8,147 7,252 7,861ATDemand 8,258 7,369 7,429 7,534 7,954 7,720 7,847 7,980 7,846 8,010 7,308 7,852 7,933 7,484 7,902 7,602 8,062 7,961 8,035 7,477 7,931 7,265 7,819 7,866 7,442 7,653 6,931 8,147 7,252 7,861PLANN L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30     OF_Z 443,051 463,091 455,870 442,231 452,749 439,103 447,250 460,762 457,308 455,317 448,869 446,849 443,892 457,509 453,423 451,460 454,758 460,318 466,975 441,471 458,722 447,892 456,970 449,157 439,532 440,931 437,675 436,941 468,904 444,898Man_cost 443,051 463,091 455,870 442,231 452,749 439,103 447,250 460,762 457,308 455,317 448,869 446,849 443,892 457,509 453,423 451,460 454,758 460,318 466,975 441,471 458,722 447,892 456,970 449,157 439,532 440,931 437,675 436,941 468,904 444,898 Log_Cost 378,579 395,672 389,456 377,842 386,793 375,101 382,086 393,600 390,757 388,997 383,515 381,781 379,274 390,842 387,412 385,672 388,557 393,280 398,994 377,239 391,963 382,697 390,434 383,808 375,529 376,758 373,980 373,311 400,659 380,114 Lugr_icost 0 0 0 13 0 21 28 0 10 0 0 0 0 40 3 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 Luat_icost 12 0 31 8 48 60 0 25 0 22 6 40 0 0 20 56 17 19 0 0 4 23 35 0 30 0 1 6 20 4Sawing_cost 62,916 65,813 64,801 62,829 64,335 62,393 63,584 65,542 64,947 64,716 63,788 63,473 63,071 65,040 64,411 64,165 64,599 65,418 66,355 62,692 65,137 63,610 64,911 63,784 62,445 62,636 62,169 62,107 66,590 63,236 At_cost 1,545 1,607 1,583 1,540 1,573 1,528 1,552 1,594 1,594 1,582 1,560 1,555 1,546 1,587 1,579 1,566 1,585 1,601 1,627 1,541 1,600 1,563 1,591 1,565 1,528 1,538 1,525 1,517 1,635 1,545Log_input 11,651 12,188 12,000 11,635 11,914 11,554 11,775 12,137 12,027 11,984 11,813 11,754 11,680 12,044 11,928 11,882 11,963 12,114 12,288 11,610 12,062 11,780 12,021 11,812 11,564 11,599 11,513 11,501 12,331 11,710GRLumber 7,629 7,937 7,817 7,605 7,766 7,544 7,664 7,872 7,873 7,814 7,706 7,678 7,637 7,838 7,798 7,736 7,827 7,907 8,033 7,609 7,901 7,718 7,855 7,729 7,546 7,593 7,532 7,490 8,075 7,628ATLumber 7,621 7,929 7,809 7,597 7,758 7,536 7,656 7,864 7,865 7,806 7,698 7,670 7,629 7,830 7,790 7,728 7,819 7,899 8,025 7,601 7,893 7,710 7,847 7,721 7,538 7,585 7,524 7,483 8,067 7,620ATDemand 7,621 7,929 7,809 7,597 7,758 7,536 7,656 7,864 7,865 7,806 7,698 7,670 7,629 7,830 7,790 7,728 7,819 7,899 8,025 7,601 7,893 7,710 7,847 7,721 7,538 7,585 7,524 7,483 8,067 7,620130   Table D.2 PS-E with large, small, mixed, high variation, and low variation lumber products demand  EDD La 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 449,707 449,007 449,130 449,707 449,155 449,472 449,464 447,819 447,184 449,545 448,365 448,671 447,401 449,208 446,809 448,991 450,196 448,990 450,051 448,434 449,763 451,637 448,281 449,260 447,687 450,028 447,390 447,320 450,332 447,399Log_Cost 384,041 383,446 383,554 384,041 383,568 383,843 383,834 382,422 381,883 383,904 382,882 383,150 382,066 383,617 381,560 383,428 384,465 383,423 384,336 382,953 384,083 385,694 382,819 383,662 382,307 384,317 382,061 382,000 384,582 382,068Lugr_icost 100 99 99 100 98 100 100 102 102 97 101 100 101 100 102 99 97 99 99 102 99 99 101 100 102 100 102 102 96 102Luat_icost 127 125 126 127 126 127 127 130 129 124 129 128 129 127 130 127 124 127 127 129 126 126 129 127 129 127 129 129 123 129Sawing_cost 63,875 63,775 63,790 63,875 63,802 63,839 63,840 63,608 63,515 63,856 63,695 63,734 63,550 63,803 63,464 63,776 63,945 63,781 63,924 63,691 63,892 64,147 63,674 63,810 63,594 63,919 63,544 63,534 63,965 63,544   At_cost 1,564 1,562 1,562 1,564 1,561 1,563 1,562 1,557 1,555 1,563 1,558 1,559 1,555 1,562 1,553 1,561 1,565 1,560 1,565 1,559 1,563 1,571 1,558 1,562 1,556 1,565 1,555 1,555 1,566 1,556Log_input 11,829 11,810 11,813 11,829 11,815 11,822 11,822 11,779 11,762 11,825 11,795 11,803 11,768 11,815 11,753 11,810 11,842 11,811 11,838 11,795 11,832 11,879 11,791 11,817 11,777 11,837 11,767 11,766 11,845 11,767GRLumber 7,722 7,713 7,712 7,722 7,709 7,719 7,716 7,687 7,678 7,719 7,692 7,701 7,679 7,714 7,671 7,707 7,731 7,706 7,727 7,701 7,719 7,756 7,695 7,713 7,684 7,729 7,681 7,679 7,732 7,684ATLumber 7,714 7,705 7,704 7,714 7,701 7,711 7,708 7,679 7,670 7,711 7,684 7,693 7,671 7,706 7,663 7,699 7,723 7,698 7,719 7,693 7,711 7,748 7,687 7,705 7,676 7,721 7,673 7,671 7,724 7,676ATDemand 7,714 7,705 7,704 7,714 7,701 7,711 7,708 7,679 7,670 7,711 7,684 7,693 7,671 7,706 7,663 7,699 7,723 7,698 7,719 7,693 7,711 7,748 7,687 7,705 7,676 7,721 7,673 7,671 7,724 7,676   A_AO1 16 15 15 16 15 16 16 16 16 15 16 16 16 16 16 16 15 16 16 16 15 15 16 16 16 16 16 16 15 16   A_BO1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   A_AO2 17 17 17 17 17 17 17 18 18 17 18 18 18 17 18 17 17 17 17 18 17 17 18 17 18 17 18 18 17 18   A_BO2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV1 1200 1189 1191 1200 1184 1203 1203 1229 1226 1170 1222 1211 1217 1202 1231 1197 1166 1197 1197 1226 1194 1190 1220 1201 1223 1203 1223 1226 1157 1227     BV1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV2 1336 1321 1327 1336 1323 1340 1340 1364 1360 1306 1362 1349 1354 1336 1365 1334 1301 1335 1335 1359 1331 1324 1355 1338 1362 1335 1358 1360 1294 1359     BV2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0EDD S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 359794 360269 359484 360891 358114 356649 360348 359564 359837 358093 359845 448671 359921 359344 446809 361567 362628 361550 359632 359402 360947 361002 358708 361401 360270 358121 359707 360754 359737 358862Log_Cost 307101 307510 306839 308039 305662 304409 307577 306906 307139 305648 307150 383150 307207 306711 381560 308622 309528 308609 306964 306767 308085 308139 306163 308475 307508 305670 307023 307928 307049 306302Lugr_icost 166 166 167 165 167 166 165 166 166 168 166 100 167 167 102 164 164 167 166 167 166 166 167 166 167 165 166 165 166 167Luat_icost 198 198 199 197 200 198 197 199 199 200 198 128 199 200 130 196 196 199 198 199 198 198 199 199 199 197 198 197 198 199Sawing_cost 51079 51143 51029 51236 50841 50638 51157 51043 51082 50833 51079 63734 51098 51017 63464 51329 51478 51319 51055 51020 51244 51244 50933 51305 51144 50845 51070 51209 51074 50947   At_cost 1250 1252 1250 1254 1244 1239 1252 1250 1251 1244 1251 1559 1250 1248 1553 1257 1261 1257 1250 1249 1254 1255 1246 1256 1252 1244 1249 1255 1250 1247Log_input 9459 9471 9450 9488 9415 9377 9474 9452 9460 9413 9459 11803 9463 9448 11753 9505 9533 9503 9455 9448 9490 9490 9432 9501 9471 9416 9457 9483 9458 9435GRLumber 6175 6184 6171 6192 6144 6118 6183 6171 6176 6145 6179 7701 6175 6164 7671 6206 6225 6208 6172 6169 6194 6198 6151 6202 6183 6144 6170 6195 6172 6159ATLumber 6169 6178 6165 6186 6138 6112 6177 6165 6170 6139 6173 7693 6169 6158 7663 6200 6219 6202 6166 6163 6188 6192 6145 6196 6177 6138 6164 6189 6166 6153ATDemand 6169 6178 6165 6186 6138 6112 6177 6165 6170 6139 6173 7693 6169 6158 7663 6200 6219 6202 6166 6163 6188 6192 6145 6196 6177 6138 6164 6189 6166 6153   A_AO1 32 32 33 32 33 33 32 32 32 33 32 16 33 33 16 32 32 32 32 33 32 32 33 32 33 32 32 32 32 33   A_BO1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   A_AO2 34 34 34 33 34 34 34 34 34 34 34 18 34 34 18 33 33 34 34 34 34 34 34 34 34 34 34 34 34 34   A_BO2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV1 1995 1995 2010 1982 2017 1995 1988 2003 2005 2022 2004 1211 2008 2017 1231 1972 1981 2007 1997 2007 1998 2001 2010 2004 2009 1986 2000 1994 1998 2007     BV1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV2 2081 2081 2095 2069 2104 2083 2077 2092 2092 2109 2088 1349 2094 2106 1365 2059 2067 2092 2083 2093 2088 2086 2100 2092 2097 2073 2088 2077 2085 2095     BV2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0EDD BS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 400478 390987 396474 403318 401927 401927 408833 400710 410254 405809 398672 405193 410434 404171 446809 396675 408614 403815 398009 391599 408965 412378 394043 400610 403966 404853 401562 410582 400018 399668Log_Cost 341897 333770 338491 344330 343165 343165 349096 342120 350246 346493 340352 345917 350429 345077 381560 338645 348859 344758 339796 334301 349121 352088 336360 342038 344923 345655 342814 350542 341507 341227Lugr_icost 139 150 136 149 143 143 137 143 136 134 147 145 136 132 102 150 138 142 145 145 138 139 146 138 131 145 147 131 142 148Luat_icost 170 181 165 181 173 173 166 172 168 164 179 178 165 162 130 181 168 172 175 176 171 168 180 168 159 174 179 163 174 179Sawing_cost 56881 55528 56303 57256 57046 57046 58011 56882 58282 57605 56609 57547 58277 57397 63464 56320 58029 57338 56509 55619 58119 58549 55991 56872 57347 57469 57027 58321 56805 56723   At_cost 1390 1358 1379 1402 1399 1399 1424 1393 1423 1413 1386 1406 1426 1403 1553 1379 1420 1405 1385 1359 1415 1435 1366 1394 1406 1410 1395 1425 1391 1391Log_input 10534 10283 10426 10603 10564 10564 10743 10534 10793 10668 10483 10657 10792 10629 11753 10430 10746 10618 10465 10300 10763 10842 10369 10532 10620 10642 10561 10800 10519 10504GRLumber 6864 6705 6810 6922 6910 6910 7030 6881 7025 6976 6846 6942 7044 6928 7671 6809 7012 6939 6838 6712 6990 7085 6744 6884 6943 6962 6888 7039 6868 6869ATLumber 6857 6698 6803 6915 6903 6903 7023 6874 7018 6969 6839 6935 7037 6921 7663 6802 7005 6932 6831 6705 6983 7078 6737 6877 6936 6955 6881 7032 6861 6862ATDemand 6857 6698 6803 6915 6903 6903 7023 6874 7018 6969 6839 6935 7037 6921 7663 6802 7005 6932 6831 6705 6983 7078 6737 6877 6936 6955 6881 7032 6861 6862   A_AO1 24 27 24 26 25 25 24 25 23 23 26 25 23 23 16 27 24 25 26 26 24 24 26 24 23 25 26 23 25 26   A_BO1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   A_AO2 26 28 26 28 26 26 25 26 25 25 28 27 25 25 18 28 25 26 27 28 26 25 28 26 24 26 27 24 27 27   A_BO2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV1 1676 1804 1636 1793 1720 1720 1649 1718 1638 1620 1768 1744 1638 1587 1231 1811 1658 1715 1743 1744 1665 1670 1763 1663 1576 1744 1771 1582 1717 1778     BV1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV2 1789 1902 1738 1909 1824 1824 1749 1813 1764 1723 1881 1874 1740 1706 1365 1906 1765 1811 1844 1852 1804 1766 1899 1772 1673 1833 1889 1713 1827 1885     BV2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0EDD H 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 478662 430657 433749 442491 462645 451080 456431 465561 456350 463447 429232 456411 461999 438578 459270 444860 471815 460521 467374 436530 463132 423033 457803 462807 433973 446057 408722 475196 424418 460811Log_Cost 408998 367718 370457 377839 395282 385248 390006 397732 389905 396154 366546 389844 394639 374610 392371 379863 402987 393606 399284 372815 395606 361334 390941 395114 370704 381039 348877 405899 362306 393518Lugr_icost 66 131 116 106 82 98 100 65 96 78 103 112 69 78 75 136 78 82 61 69 58 112 87 89 125 106 147 73 158 90Luat_icost 89 162 146 140 106 126 125 86 119 91 129 140 87 105 96 173 102 101 86 91 84 136 117 126 152 130 181 97 197 119Sawing_cost 67836 61152 61523 62879 65562 64043 64610 66061 64640 65501 60974 64723 65596 62268 65127 63147 67014 65118 66314 62041 65776 59978 65073 65882 61484 63231 58112 67475 60287 65490   At_cost 1674 1494 1506 1527 1612 1565 1591 1618 1590 1624 1481 1592 1608 1517 1602 1541 1634 1614 1629 1516 1608 1473 1585 1594 1509 1551 1405 1651 1470 1593Log_input 12562 11324 11393 11644 12141 11860 11965 12234 11970 12130 11291 11986 12147 11531 12060 11694 12410 12059 12280 11489 12181 11107 12051 12200 11386 11709 10761 12495 11164 12128GRLumber 8266 7376 7436 7542 7962 7728 7855 7988 7854 8018 7315 7860 7941 7491 7910 7610 8070 7969 8043 7484 7939 7272 7827 7874 7449 7661 6938 8155 7259 7869ATLumber 8258 7369 7429 7534 7954 7720 7847 7980 7846 8010 7308 7852 7933 7484 7902 7602 8062 7961 8035 7477 7931 7265 7819 7866 7442 7653 6931 8147 7252 7861ATDemand 8258 7369 7429 7534 7954 7720 7847 7980 7846 8010 7308 7852 7933 7484 7902 7602 8062 7961 8035 7477 7931 7265 7819 7866 7442 7653 6931 8147 7252 7861   A_AO1 10 22 19 17 13 15 15 10 15 12 17 17 11 13 12 22 12 13 9 12 9 19 14 14 20 17 26 11 27 14   A_BO1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   A_AO2 11 23 21 20 14 17 17 11 16 12 19 19 12 15 13 25 13 13 11 13 11 20 16 17 22 18 28 13 29 16   A_BO2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV1 792 1580 1400 1276 988 1185 1206 786 1151 935 1240 1354 831 938 899 1633 941 992 740 832 705 1347 1049 1074 1505 1273 1772 883 1904 1088     BV1 0 0 0 0 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV2 932 1705 1535 1475 1114 1327 1311 900 1248 958 1355 1473 918 1101 1010 1826 1072 1063 904 954 887 1436 1230 1331 1598 1372 1902 1018 2077 1257     BV2 0 0 0 0 0 0 0 0 0 0 0 0 79 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0EDD L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 443300 463282 456094 442469 453019 439444 447505 460971 457535 455554 449125 447105 444106 457742 453643 451741 454990 460525 467173 441723 458933 448133 457226 449402 439806 441188 437942 437184 469094 445128Log_Cost 378599 395672 389508 377869 386876 375239 382115 393642 390774 389033 383525 381849 379274 390911 387448 385768 388585 393312 398994 377239 392001 382741 390494 383808 375580 376758 373982 373321 400685 380121Lugr_icost 105 82 84 100 99 118 108 79 96 94 109 96 95 85 90 100 95 83 87 112 83 96 98 110 108 114 118 104 74 99Luat_icost 133 109 111 126 126 148 141 107 122 122 141 123 120 112 114 129 121 107 110 139 107 121 125 136 137 142 147 133 99 127Sawing_cost 62918 65813 64808 62833 64346 62412 63589 65549 64949 64722 63789 63482 63071 65048 64412 64178 64604 65422 66355 62692 65142 63611 64918 63784 62452 62636 62169 62109 66601 63237   At_cost 1545 1607 1583 1540 1573 1528 1552 1594 1594 1582 1560 1555 1546 1587 1579 1566 1585 1601 1627 1541 1600 1563 1591 1565 1528 1538 1525 1517 1635 1545Log_input 11651 12188 12002 11636 11916 11558 11776 12139 12028 11986 11813 11756 11680 12046 11928 11885 11964 12115 12288 11610 12063 11780 12022 11812 11565 11599 11513 11502 12333 11711GRLumber 7629 7937 7817 7605 7766 7544 7664 7872 7873 7814 7706 7678 7637 7838 7798 7736 7827 7907 8033 7609 7901 7718 7855 7729 7546 7593 7532 7490 8075 7628ATLumber 7621 7929 7809 7597 7758 7536 7656 7864 7865 7806 7698 7670 7629 7830 7790 7728 7819 7899 8025 7601 7893 7710 7847 7721 7538 7585 7524 7483 8067 7620ATDemand 7621 7929 7809 7597 7758 7536 7656 7864 7865 7806 7698 7670 7629 7830 7790 7728 7819 7899 8025 7601 7893 7710 7847 7721 7538 7585 7524 7483 8067 7620   A_AO1 17 12 13 16 15 19 17 12 15 14 17 15 15 13 14 16 15 13 13 18 13 15 15 17 17 18 19 17 11 16   A_BO1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   A_AO2 18 15 15 18 17 21 19 14 16 17 19 17 17 15 15 18 16 14 15 19 14 17 17 19 19 20 21 19 13 18   A_BO2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV1 1270 986 1009 1208 1187 1425 1302 947 1159 1130 1314 1159 1139 1018 1080 1200 1147 998 1047 1354 1005 1158 1186 1320 1305 1377 1427 1253 897 1193     BV1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV2 1398 1148 1168 1330 1331 1555 1480 1123 1284 1289 1485 1297 1258 1177 1205 1362 1272 1124 1163 1467 1129 1279 1321 1429 1444 1496 1552 1397 1040 1333     BV2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0131   Table D.3 PS-L with large, small, mixed, high variation, and low variation lumber products demand    1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30LPT S 358,694Man_cost 358,694Log_Cost 306,163Lugr_icost 160Luat_icost 192Sawing_cost 50,933   At_cost 1,246Log_input 9,432GRLumber 6,151ATLumber 6,145ATDemand 6,145   A_AO1 31    A_AO2 33        AV1 1,931          AV2 2,023132   D.2 Relaxed run results Table D.4 PL with large, small, mixed, high variation, and low variations demand  PLANN B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30     OF_Z 446,398 448,783 448,905 449,480 448,931 449,245 449,015 447,587 446,953 449,223 448,134 448,443 446,930 448,982 446,577 448,418 449,976 448,764 449,673 448,203 449,538 451,167 448,051 449,033 447,353 449,801 447,160 446,945 450,113 447,168Man_cost 446,398 448,783 448,905 449,480 448,931 449,245 449,015 447,587 446,953 449,223 448,134 448,443 446,930 448,982 446,577 448,418 449,976 448,764 449,673 448,203 449,538 451,167 448,051 449,033 447,353 449,801 447,160 446,945 450,113 447,168 Log_Cost 381,405 383,446 383,554 384,041 383,568 383,843 383,646 382,422 381,883 383,819 382,882 383,150 381,862 383,617 381,560 383,132 384,465 383,423 384,207 382,953 384,083 385,486 382,819 383,662 382,220 384,317 382,061 381,878 384,582 382,068 Lugr_icost 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Luat_icost 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Sawing_cost 63,441 63,775 63,790 63,875 63,802 63,839 63,808 63,608 63,515 63,841 63,695 63,734 63,515 63,803 63,464 63,726 63,945 63,781 63,901 63,691 63,892 64,111 63,674 63,810 63,578 63,919 63,544 63,513 63,965 63,544 At_cost 1,553 1,562 1,562 1,564 1,561 1,563 1,562 1,557 1,555 1,563 1,558 1,559 1,554 1,562 1,553 1,560 1,565 1,560 1,564 1,559 1,563 1,570 1,558 1,562 1,556 1,565 1,555 1,555 1,566 1,556Log_input 11,748 11,810 11,813 11,829 11,815 11,822 11,816 11,779 11,762 11,822 11,795 11,803 11,762 11,815 11,753 11,801 11,842 11,811 11,834 11,795 11,832 11,872 11,791 11,817 11,774 11,837 11,767 11,762 11,845 11,767GRLumber 5,722 5,809 6,111 6,265 5,732 6,192 6,059 6,617 5,740 6,074 5,682 6,357 6,326 5,808 5,407 6,043 6,413 6,026 5,937 5,699 5,927 5,885 6,179 5,973 5,773 5,849 5,883 5,762 5,919 6,004BV1 1,944 1,903 1,600 1,457 1,977 1,527 1,653 1,070 1,937 1,644 2,009 1,344 1,349 1,906 2,264 1,658 1,317 1,680 1,788 2,001 1,792 1,867 1,515 1,740 1,909 1,880 1,798 1,915 1,813 1,680ATLumber 6,457 6,760 6,806 5,683 6,835 7,163 6,243 6,368 7,034 6,244 6,501 6,974 5,813 6,479 6,268 6,248 5,967 6,140 6,612 7,078 6,218 6,879 6,502 7,043 6,931 6,925 5,796 6,495 6,172 6,284BV2 1,202 945 898 2,031 866 548 1,462 1,311 636 1,466 1,183 719 1,855 1,227 1,395 1,445 1,756 1,558 1,105 615 1,493 865 1,185 662 744 796 1,877 1,175 1,552 1,392ATDemand 7,659 7,705 7,704 7,714 7,701 7,711 7,705 7,679 7,670 7,710 7,684 7,693 7,668 7,706 7,663 7,694 7,723 7,698 7,717 7,693 7,711 7,745 7,687 7,705 7,675 7,721 7,673 7,669 7,724 7,676A_BO2 % 16% 12% 12% 26% 11% 7% 19% 17% 8% 19% 15% 9% 24% 16% 18% 19% 23% 20% 14% 8% 19% 11% 15% 9% 10% 10% 24% 15% 20% 18%PLANN S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30   OF_Z 359,927 359,905 360,529 360,529 357,747 356,286 359,986 359,199 359,472 357,725 359,480 359,555 358,936 358,976 359,098 361,208 362,267 361,185 359,269 359,036 360,583 360,638 358,342 361,036 359,904 357,759 359,342 360,392 359,373 358,496Man_cost 359,927 359,905 360,529 360,529 357,747 356,286 359,986 359,199 359,472 357,725 359,480 359,555 358,936 358,976 359,098 361,208 362,267 361,185 359,269 359,036 360,583 360,638 358,342 361,036 359,904 357,759 359,342 360,392 359,373 358,496 Log_Cost 307,529 307,510 308,039 308,039 305,662 304,409 307,577 306,906 307,139 305,648 307,150 307,207 306,674 306,711 306,820 308,622 309,528 308,609 306,964 306,767 308,085 308,139 306,163 308,475 307,508 305,670 307,023 307,928 307,049 306,302 Lugr_icost 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Luat_icost 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Sawing_cost 51,146 51,143 51,236 51,236 50,841 50,638 51,157 51,043 51,082 50,833 51,079 51,098 51,015 51,017 51,028 51,329 51,478 51,319 51,055 51,020 51,244 51,244 50,933 51,305 51,144 50,845 51,070 51,209 51,074 50,947 At_cost 1,252 1,252 1,254 1,254 1,244 1,239 1,252 1,250 1,251 1,244 1,251 1,250 1,248 1,248 1,249 1,257 1,261 1,257 1,250 1,249 1,254 1,255 1,246 1,256 1,252 1,244 1,249 1,255 1,250 1,247Log_input 9,471 9,471 9,488 9,488 9,415 9,377 9,474 9,452 9,460 9,413 9,459 9,463 9,447 9,448 9,450 9,505 9,533 9,503 9,455 9,448 9,490 9,490 9,432 9,501 9,471 9,416 9,457 9,483 9,458 9,435GRLumber 4,279 4,590 4,107 4,107 4,502 4,000 4,143 4,794 4,884 4,538 4,881 4,908 4,538 4,451 4,764 4,144 4,125 4,415 4,462 4,875 4,409 4,743 4,336 4,640 4,313 4,060 4,658 3,955 4,429 4,701BV1 1,906 1,594 2,085 2,085 1,642 2,118 2,041 1,378 1,292 1,607 1,298 1,267 1,625 1,713 1,406 2,062 2,100 1,793 1,710 1,294 1,785 1,455 1,815 1,563 1,870 2,084 1,513 2,240 1,743 1,458ATLumber 5,026 5,334 4,610 4,610 4,546 4,426 4,958 4,494 5,222 4,639 4,758 4,740 5,032 4,820 5,301 4,724 5,338 5,187 5,245 4,861 5,077 4,971 4,778 4,306 4,792 4,528 5,229 4,210 5,462 5,323BV2 1,152 844 1,576 1,576 1,592 1,686 1,219 1,671 948 1,500 1,415 1,429 1,125 1,338 863 1,476 881 1,015 921 1,302 1,111 1,221 1,367 1,890 1,385 1,610 935 1,979 704 830ATDemand 6,178 6,178 6,186 6,186 6,138 6,112 6,177 6,165 6,170 6,139 6,173 6,169 6,157 6,158 6,164 6,200 6,219 6,202 6,166 6,163 6,188 6,192 6,145 6,196 6,177 6,138 6,164 6,189 6,166 6,153A_BO2 % 19% 14% 25% 25% 26% 28% 20% 27% 15% 24% 23% 23% 18% 22% 14% 24% 14% 16% 15% 21% 18% 20% 22% 30% 22% 26% 15% 32% 11% 13%PLANN BS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 OF_Z 400,149 390,656 396,173 402,970 401,556 407,985 408,505 400,378 409,946 405,477 398,331 404,846 410,115 403,869 398,320 396,303 408,268 403,484 397,669 391,263 408,655 412,025 393,676 400,249 403,645 404,534 401,208 410,266 399,683 399,303Man_cost 400,149 390,656 396,173 402,970 401,556 407,985 408,505 400,378 409,946 405,477 398,331 404,846 410,115 403,869 398,320 396,303 408,268 403,484 397,669 391,263 408,655 412,025 393,676 400,249 403,645 404,534 401,208 410,266 399,683 399,303 Log_Cost 341,878 333,770 338,491 344,314 343,117 348,551 349,073 342,104 350,243 346,452 340,338 345,896 350,403 345,062 340,320 338,595 348,823 344,739 339,778 334,287 349,121 352,039 336,324 341,989 344,895 345,655 342,789 350,522 341,489 341,183 Lugr_icost 2 0 0 0 0 0 0 0 0 12 0 0 11 9 0 14 0 5 0 0 0 8 0 0 0 0 0 0 0 11 Luat_icost 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Sawing_cost 56,879 55,528 56,303 57,254 57,040 58,017 58,008 56,880 58,281 57,600 56,607 57,544 58,274 57,395 56,616 56,314 58,025 57,335 56,506 55,617 58,119 58,543 55,986 56,866 57,344 57,469 57,024 58,319 56,803 56,718 At_cost 1,390 1,358 1,379 1,402 1,399 1,417 1,424 1,393 1,423 1,413 1,386 1,406 1,426 1,403 1,384 1,379 1,420 1,405 1,385 1,359 1,415 1,435 1,366 1,394 1,406 1,410 1,395 1,425 1,391 1,391Log_input 10,533 10,283 10,426 10,603 10,563 10,744 10,742 10,533 10,793 10,667 10,483 10,656 10,792 10,629 10,484 10,429 10,745 10,618 10,464 10,299 10,763 10,841 10,368 10,531 10,619 10,642 10,560 10,800 10,519 10,503GRLumber 4,623 4,059 5,246 5,373 5,429 5,173 5,004 5,507 4,904 4,953 4,642 5,657 5,093 4,907 5,180 4,598 4,698 4,484 5,623 5,113 4,866 4,969 4,606 5,311 5,606 5,408 5,712 4,395 4,786 5,221BV1 2,241 2,646 1,564 1,549 1,481 1,824 2,026 1,374 2,121 2,023 2,204 1,285 1,951 2,020 1,655 2,211 2,314 2,455 1,215 1,598 2,124 2,116 2,138 1,572 1,337 1,554 1,176 2,644 2,082 1,648ATLumber 5,890 4,939 5,315 5,899 5,162 5,364 5,689 5,718 6,080 6,108 5,379 6,036 5,953 5,779 5,158 5,300 5,989 5,727 5,446 4,855 5,830 5,964 5,413 5,493 5,635 5,444 6,487 5,552 5,791 5,970BV2 967 1,759 1,488 1,016 1,741 1,626 1,334 1,156 938 861 1,460 899 1,084 1,142 1,670 1,502 1,016 1,205 1,385 1,850 1,153 1,114 1,324 1,384 1,301 1,511 394 1,480 1,070 892ATDemand 6,857 6,698 6,803 6,915 6,903 6,990 7,023 6,874 7,018 6,969 6,839 6,935 7,037 6,921 6,828 6,802 7,005 6,932 6,831 6,705 6,983 7,078 6,737 6,877 6,936 6,955 6,881 7,032 6,861 6,862A_BO2 % 14% 26% 22% 15% 25% 23% 19% 17% 13% 12% 21% 13% 15% 16% 24% 22% 15% 17% 20% 28% 17% 16% 20% 20% 19% 22% 6% 21% 16% 13%PLANN H 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30   OF_Z 477,891 430,204 433,265 442,040 461,726 449,835 455,846 464,323 455,843 462,786 428,694 455,735 1 437,353 458,724 444,068 471,336 459,657 466,921 435,611 462,843 421,918 457,120 461,260 432,889 445,293 407,675 474,813 423,803 460,373Man_cost 477,891 430,204 433,265 442,040 461,726 449,835 455,846 464,323 455,843 462,786 428,694 455,735 502,663 437,353 458,724 444,068 471,336 459,657 466,921 435,611 462,843 421,918 457,120 461,260 432,889 445,293 407,675 474,813 423,803 460,373 Log_Cost 408,435 367,512 370,073 377,429 394,554 384,251 389,409 396,686 389,563 395,633 366,201 389,428 417,179 373,644 391,936 379,358 402,696 392,891 398,945 372,177 395,410 360,457 390,512 393,900 369,924 380,514 348,245 405,583 361,799 393,274 Lugr_icost 0 20 114 250 0 85 175 78 0 0 65 0 376 16 65 0 0 25 65 2 89 106 0 108 0 0 11 138 316 0 Luat_icost 0 0 0 0 0 0 0 0 0 0 0 0 13,523 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 0 0Sawing_cost 67,783 61,178 61,572 62,834 65,560 63,934 64,672 65,941 64,690 65,529 60,948 64,715 69,978 62,176 65,121 63,169 66,972 65,128 66,282 61,916 65,736 59,883 65,024 65,658 61,456 63,228 58,014 67,441 60,219 65,505 At_cost 1,674 1,494 1,506 1,527 1,612 1,565 1,591 1,618 1,590 1,624 1,481 1,592 1,608 1,517 1,602 1,541 1,634 1,614 1,629 1,516 1,608 1,473 1,585 1,594 1,509 1,551 1,405 1,651 1,470 1,593Log_input 12,552 11,329 11,402 11,636 12,141 11,840 11,976 12,211 11,980 12,135 11,287 11,984 12,959 11,514 12,059 11,698 12,402 12,061 12,275 11,466 12,173 11,089 12,041 12,159 11,381 11,709 10,743 12,489 11,152 12,131GRLumber 6,087 6,030 5,515 5,602 6,017 5,962 6,171 5,623 6,157 6,301 5,049 5,583 6,057 6,178 6,042 6,225 5,990 6,295 5,675 5,275 5,922 5,558 6,488 6,508 5,820 5,999 4,277 5,996 5,646 5,891BV1 2,179 1,346 1,921 1,940 1,945 1,766 1,684 2,365 1,697 1,717 2,266 2,277 1,883 1,314 1,868 1,385 2,080 1,674 2,368 2,210 2,017 1,714 1,338 1,366 1,629 1,662 2,661 2,159 1,614 1,978ATLumber 7,079 5,917 5,937 6,955 6,766 6,738 6,744 6,250 6,504 6,671 6,016 6,611 4,673 5,736 6,785 6,965 6,709 6,299 7,095 5,345 5,899 5,489 6,737 6,132 5,650 6,500 5,477 7,645 6,149 7,124BV2 1,179 1,452 1,492 579 1,188 982 1,103 1,730 1,342 1,339 1,292 1,241 3,260 1,748 1,117 637 1,353 1,662 940 2,132 2,032 1,776 1,082 1,734 1,792 1,153 1,454 502 1,103 737ATDemand 8,258 7,369 7,429 7,534 7,954 7,720 7,847 7,980 7,846 8,010 7,308 7,852 7,933 7,484 7,902 7,602 8,062 7,961 8,035 7,477 7,931 7,265 7,819 7,866 7,442 7,653 6,931 8,147 7,252 7,861A_BO2 % 14% 20% 20% 8% 15% 13% 14% 22% 17% 17% 18% 16% 41% 23% 14% 8% 17% 21% 12% 29% 26% 24% 14% 22% 24% 15% 21% 6% 15% 9%PLANN L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30    OF_Z 442,981 463,081 455,821 442,223 452,701 438,978 447,250 460,737 457,294 455,165 448,863 446,762 443,892 457,509 453,362 451,389 454,705 460,279 466,975 441,425 458,708 447,869 456,903 449,157 439,503 440,931 437,670 436,890 468,855 444,894Man_cost 442,981 463,081 455,821 442,223 452,701 438,978 447,250 460,737 457,294 455,165 448,863 446,762 443,892 457,509 453,362 451,389 454,705 460,279 466,975 441,425 458,708 447,869 456,903 449,157 439,503 440,931 437,670 436,890 468,855 444,894 Log_Cost 378,506 395,662 389,439 377,842 386,793 375,062 382,086 393,600 390,732 388,881 383,515 381,739 379,274 390,842 387,375 385,659 388,525 393,262 398,994 377,198 391,954 382,697 390,405 383,808 375,529 376,758 373,964 373,271 400,624 380,114 Lugr_icost 0 0 0 13 0 0 28 0 24 0 0 0 0 40 3 0 0 0 0 0 18 0 0 0 0 0 0 0 0 0 Luat_icost 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Sawing_cost 62,930 65,811 64,799 62,829 64,335 62,389 63,584 65,542 64,944 64,702 63,788 63,468 63,071 65,040 64,406 64,163 64,595 65,416 66,355 62,687 65,136 63,610 64,907 63,784 62,445 62,636 62,181 62,103 66,596 63,236 At_cost 1,545 1,607 1,583 1,540 1,573 1,528 1,552 1,594 1,594 1,582 1,560 1,555 1,546 1,587 1,579 1,566 1,585 1,601 1,627 1,541 1,600 1,563 1,591 1,565 1,528 1,538 1,525 1,517 1,635 1,545Log_input 11,654 12,187 12,000 11,635 11,914 11,553 11,775 12,137 12,027 11,982 11,813 11,753 11,680 12,044 11,927 11,882 11,962 12,114 12,288 11,609 12,062 11,780 12,020 11,812 11,564 11,599 11,515 11,500 12,333 11,710GRLumber 5,126 5,819 6,234 5,043 6,355 5,573 6,218 6,658 6,214 6,076 6,538 5,715 5,637 6,048 6,041 5,416 6,077 5,584 6,235 6,261 5,824 5,760 6,047 5,580 5,708 5,991 5,477 5,777 6,731 5,509BV1 2,503 2,118 1,583 2,561 1,411 1,970 1,446 1,214 1,659 1,738 1,168 1,963 1,999 1,790 1,757 2,320 1,750 2,323 1,798 1,348 2,077 1,958 1,808 2,148 1,837 1,602 2,055 1,714 1,344 2,118ATLumber 6,837 7,248 6,648 5,810 6,864 6,445 6,792 6,635 6,999 6,383 7,020 6,182 6,147 7,069 6,544 6,437 6,575 6,969 6,732 5,704 6,809 5,909 6,573 7,134 6,271 6,444 6,789 6,231 6,800 6,644BV2 784 681 1,161 1,787 894 1,091 864 1,229 866 1,423 678 1,488 1,482 761 1,246 1,291 1,244 930 1,293 1,897 1,084 1,801 1,274 587 1,267 1,141 735 1,252 1,267 976ATDemand 7621 7929 7809 7597 7758 7536 7656 7864 7865 7806 7698 7670 7629 7830 7790 7728 7819 7899 8025 7601 7893 7710 7847 7721 7538 7585 7524 7483 8067 7620A_BO2 % 10% 9% 15% 24% 12% 14% 11% 16% 11% 18% 9% 19% 19% 10% 16% 17% 16% 12% 16% 25% 14% 23% 16% 8% 17% 15% 10% 17% 16% 13%133   Table D.5 Pl-S-E with large, small, mixed, high variation, and low variation lumber products demand EDD B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 449,707 449,007 449,130 449,707 449,155 449,472 449,464 447,819 447,184 449,545 448,365 448,671 447,401 449,208 446,809 448,991 450,196 448,990 450,051 448,434 449,763 451,637 448,281 449,260 447,687 450,028 447,390 447,320 450,332 447,399Log_Cost 384,041 383,446 383,554 384,041 383,568 383,843 383,834 382,422 381,883 383,904 382,882 383,150 382,066 383,617 381,560 383,428 384,465 383,423 384,336 382,953 384,083 385,694 382,819 383,662 382,307 384,317 382,061 382,000 384,582 382,068Lugr_icost 100 99 99 100 98 100 100 102 102 97 101 100 101 100 102 99 97 99 99 102 99 99 101 100 102 100 102 102 96 102Luat_icost 127 125 126 127 126 127 127 130 129 124 129 128 129 127 130 127 124 127 127 129 126 126 129 127 129 127 129 129 123 129Sawing_cost 63,875 63,775 63,790 63,875 63,802 63,839 63,840 63,608 63,515 63,856 63,695 63,734 63,550 63,803 63,464 63,776 63,945 63,781 63,924 63,691 63,892 64,147 63,674 63,810 63,594 63,919 63,544 63,534 63,965 63,544   At_cost 1,564 1,562 1,562 1,564 1,561 1,563 1,562 1,557 1,555 1,563 1,558 1,559 1,555 1,562 1,553 1,561 1,565 1,560 1,565 1,559 1,563 1,571 1,558 1,562 1,556 1,565 1,555 1,555 1,566 1,556Log_input 11,829 11,810 11,813 11,829 11,815 11,822 11,822 11,779 11,762 11,825 11,795 11,803 11,768 11,815 11,753 11,810 11,842 11,811 11,838 11,795 11,832 11,879 11,791 11,817 11,777 11,837 11,767 11,766 11,845 11,767GRLumber 7,722 7,713 7,712 7,722 7,709 7,719 7,716 7,687 7,678 7,719 7,692 7,701 7,679 7,714 7,671 7,707 7,731 7,706 7,727 7,701 7,719 7,756 7,695 7,713 7,684 7,729 7,681 7,679 7,732 7,684ATLumber 7,714 7,705 7,704 7,714 7,701 7,711 7,708 7,679 7,670 7,711 7,684 7,693 7,671 7,706 7,663 7,699 7,723 7,698 7,719 7,693 7,711 7,748 7,687 7,705 7,676 7,721 7,673 7,671 7,724 7,676ATDemand 7,714 7,705 7,704 7,714 7,701 7,711 7,708 7,679 7,670 7,711 7,684 7,693 7,671 7,706 7,663 7,699 7,723 7,698 7,719 7,693 7,711 7,748 7,687 7,705 7,676 7,721 7,673 7,671 7,724 7,676   A_AO1 16 15 15 16 15 16 16 16 16 15 16 16 16 16 16 16 15 16 16 16 15 15 16 16 16 16 16 16 15 16   A_BO1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   A_AO2 17 17 17 17 17 17 17 18 18 17 18 18 18 17 18 17 17 17 17 18 17 17 18 17 18 17 18 18 17 18   A_BO2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV1 1200 1189 1191 1200 1184 1203 1203 1229 1226 1170 1222 1211 1217 1202 1231 1197 1166 1197 1197 1226 1194 1190 1220 1201 1223 1203 1223 1226 1157 1227     BV1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV2 1336 1321 1327 1336 1323 1340 1340 1364 1360 1306 1362 1349 1354 1336 1365 1334 1301 1335 1335 1359 1331 1324 1355 1338 1362 1335 1358 1360 1294 1359     BV2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0EDD S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 359794 360269 359484 360891 358114 356649 360348 359564 359837 358093 359845 448671 359921 359344 446809 361567 362628 361550 359632 359402 360947 361002 358708 361401 360270 358121 359707 360754 359737 358862Log_Cost 307101 307510 306839 308039 305662 304409 307577 306906 307139 305648 307150 383150 307207 306711 381560 308622 309528 308609 306964 306767 308085 308139 306163 308475 307508 305670 307023 307928 307049 306302Lugr_icost 166 166 167 165 167 166 165 166 166 168 166 100 167 167 102 164 164 167 166 167 166 166 167 166 167 165 166 165 166 167Luat_icost 198 198 199 197 200 198 197 199 199 200 198 128 199 200 130 196 196 199 198 199 198 198 199 199 199 197 198 197 198 199Sawing_cost 51079 51143 51029 51236 50841 50638 51157 51043 51082 50833 51079 63734 51098 51017 63464 51329 51478 51319 51055 51020 51244 51244 50933 51305 51144 50845 51070 51209 51074 50947   At_cost 1250 1252 1250 1254 1244 1239 1252 1250 1251 1244 1251 1559 1250 1248 1553 1257 1261 1257 1250 1249 1254 1255 1246 1256 1252 1244 1249 1255 1250 1247Log_input 9459 9471 9450 9488 9415 9377 9474 9452 9460 9413 9459 11803 9463 9448 11753 9505 9533 9503 9455 9448 9490 9490 9432 9501 9471 9416 9457 9483 9458 9435GRLumber 6175 6184 6171 6192 6144 6118 6183 6171 6176 6145 6179 7701 6175 6164 7671 6206 6225 6208 6172 6169 6194 6198 6151 6202 6183 6144 6170 6195 6172 6159ATLumber 6169 6178 6165 6186 6138 6112 6177 6165 6170 6139 6173 7693 6169 6158 7663 6200 6219 6202 6166 6163 6188 6192 6145 6196 6177 6138 6164 6189 6166 6153ATDemand 6169 6178 6165 6186 6138 6112 6177 6165 6170 6139 6173 7693 6169 6158 7663 6200 6219 6202 6166 6163 6188 6192 6145 6196 6177 6138 6164 6189 6166 6153   A_AO1 32 32 33 32 33 33 32 32 32 33 32 16 33 33 16 32 32 32 32 33 32 32 33 32 33 32 32 32 32 33   A_BO1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   A_AO2 34 34 34 33 34 34 34 34 34 34 34 18 34 34 18 33 33 34 34 34 34 34 34 34 34 34 34 34 34 34   A_BO2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV1 1995 1995 2010 1982 2017 1995 1988 2003 2005 2022 2004 1211 2008 2017 1231 1972 1981 2007 1997 2007 1998 2001 2010 2004 2009 1986 2000 1994 1998 2007     BV1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV2 2081 2081 2095 2069 2104 2083 2077 2092 2092 2109 2088 1349 2094 2106 1365 2059 2067 2092 2083 2093 2088 2086 2100 2092 2097 2073 2088 2077 2085 2095     BV2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0EDD BS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 400478 390987 396474 403318 401927 401927 408833 400710 410254 405809 398672 405193 410434 404171 446809 396675 408614 403815 398009 391599 408965 412378 394043 400610 403966 404853 401562 410582 400018 399668Log_Cost 341897 333770 338491 344330 343165 343165 349096 342120 350246 346493 340352 345917 350429 345077 381560 338645 348859 344758 339796 334301 349121 352088 336360 342038 344923 345655 342814 350542 341507 341227Lugr_icost 139 150 136 149 143 143 137 143 136 134 147 145 136 132 102 150 138 142 145 145 138 139 146 138 131 145 147 131 142 148Luat_icost 170 181 165 181 173 173 166 172 168 164 179 178 165 162 130 181 168 172 175 176 171 168 180 168 159 174 179 163 174 179Sawing_cost 56881 55528 56303 57256 57046 57046 58011 56882 58282 57605 56609 57547 58277 57397 63464 56320 58029 57338 56509 55619 58119 58549 55991 56872 57347 57469 57027 58321 56805 56723   At_cost 1390 1358 1379 1402 1399 1399 1424 1393 1423 1413 1386 1406 1426 1403 1553 1379 1420 1405 1385 1359 1415 1435 1366 1394 1406 1410 1395 1425 1391 1391Log_input 10534 10283 10426 10603 10564 10564 10743 10534 10793 10668 10483 10657 10792 10629 11753 10430 10746 10618 10465 10300 10763 10842 10369 10532 10620 10642 10561 10800 10519 10504GRLumber 6864 6705 6810 6922 6910 6910 7030 6881 7025 6976 6846 6942 7044 6928 7671 6809 7012 6939 6838 6712 6990 7085 6744 6884 6943 6962 6888 7039 6868 6869ATLumber 6857 6698 6803 6915 6903 6903 7023 6874 7018 6969 6839 6935 7037 6921 7663 6802 7005 6932 6831 6705 6983 7078 6737 6877 6936 6955 6881 7032 6861 6862ATDemand 6857 6698 6803 6915 6903 6903 7023 6874 7018 6969 6839 6935 7037 6921 7663 6802 7005 6932 6831 6705 6983 7078 6737 6877 6936 6955 6881 7032 6861 6862   A_AO1 24 27 24 26 25 25 24 25 23 23 26 25 23 23 16 27 24 25 26 26 24 24 26 24 23 25 26 23 25 26   A_BO1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   A_AO2 26 28 26 28 26 26 25 26 25 25 28 27 25 25 18 28 25 26 27 28 26 25 28 26 24 26 27 24 27 27   A_BO2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV1 1676 1804 1636 1793 1720 1720 1649 1718 1638 1620 1768 1744 1638 1587 1231 1811 1658 1715 1743 1744 1665 1670 1763 1663 1576 1744 1771 1582 1717 1778     BV1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV2 1789 1902 1738 1909 1824 1824 1749 1813 1764 1723 1881 1874 1740 1706 1365 1906 1765 1811 1844 1852 1804 1766 1899 1772 1673 1833 1889 1713 1827 1885     BV2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0EDD H 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 478662 430657 433749 442491 462645 451080 456431 465561 456350 463447 429232 456411 461999 438578 459270 444860 471815 460521 467374 436530 463132 423033 457803 462807 433973 446057 408722 475196 424418 460811Log_Cost 408998 367718 370457 377839 395282 385248 390006 397732 389905 396154 366546 389844 394639 374610 392371 379863 402987 393606 399284 372815 395606 361334 390941 395114 370704 381039 348877 405899 362306 393518Lugr_icost 66 131 116 106 82 98 100 65 96 78 103 112 69 78 75 136 78 82 61 69 58 112 87 89 125 106 147 73 158 90Luat_icost 89 162 146 140 106 126 125 86 119 91 129 140 87 105 96 173 102 101 86 91 84 136 117 126 152 130 181 97 197 119Sawing_cost 67836 61152 61523 62879 65562 64043 64610 66061 64640 65501 60974 64723 65596 62268 65127 63147 67014 65118 66314 62041 65776 59978 65073 65882 61484 63231 58112 67475 60287 65490   At_cost 1674 1494 1506 1527 1612 1565 1591 1618 1590 1624 1481 1592 1608 1517 1602 1541 1634 1614 1629 1516 1608 1473 1585 1594 1509 1551 1405 1651 1470 1593Log_input 12562 11324 11393 11644 12141 11860 11965 12234 11970 12130 11291 11986 12147 11531 12060 11694 12410 12059 12280 11489 12181 11107 12051 12200 11386 11709 10761 12495 11164 12128GRLumber 8266 7376 7436 7542 7962 7728 7855 7988 7854 8018 7315 7860 7941 7491 7910 7610 8070 7969 8043 7484 7939 7272 7827 7874 7449 7661 6938 8155 7259 7869ATLumber 8258 7369 7429 7534 7954 7720 7847 7980 7846 8010 7308 7852 7933 7484 7902 7602 8062 7961 8035 7477 7931 7265 7819 7866 7442 7653 6931 8147 7252 7861ATDemand 8258 7369 7429 7534 7954 7720 7847 7980 7846 8010 7308 7852 7933 7484 7902 7602 8062 7961 8035 7477 7931 7265 7819 7866 7442 7653 6931 8147 7252 7861   A_AO1 10 22 19 17 13 15 15 10 15 12 17 17 11 13 12 22 12 13 9 12 9 19 14 14 20 17 26 11 27 14   A_BO1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   A_AO2 11 23 21 20 14 17 17 11 16 12 19 19 12 15 13 25 13 13 11 13 11 20 16 17 22 18 28 13 29 16   A_BO2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV1 792 1580 1400 1276 988 1185 1206 786 1151 935 1240 1354 831 938 899 1633 941 992 740 832 705 1347 1049 1074 1505 1273 1772 883 1904 1088     BV1 0 0 0 0 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV2 932 1705 1535 1475 1114 1327 1311 900 1248 958 1355 1473 918 1101 1010 1826 1072 1063 904 954 887 1436 1230 1331 1598 1372 1902 1018 2077 1257     BV2 0 0 0 0 0 0 0 0 0 0 0 0 79 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0EDD L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 443300 463282 456094 442469 453019 439444 447505 460971 457535 455554 449125 447105 444106 457742 453643 451741 454990 460525 467173 441723 458933 448133 457226 449402 439806 441188 437942 437184 469094 445128Log_Cost 378599 395672 389508 377869 386876 375239 382115 393642 390774 389033 383525 381849 379274 390911 387448 385768 388585 393312 398994 377239 392001 382741 390494 383808 375580 376758 373982 373321 400685 380121Lugr_icost 105 82 84 100 99 118 108 79 96 94 109 96 95 85 90 100 95 83 87 112 83 96 98 110 108 114 118 104 74 99Luat_icost 133 109 111 126 126 148 141 107 122 122 141 123 120 112 114 129 121 107 110 139 107 121 125 136 137 142 147 133 99 127Sawing_cost 62918 65813 64808 62833 64346 62412 63589 65549 64949 64722 63789 63482 63071 65048 64412 64178 64604 65422 66355 62692 65142 63611 64918 63784 62452 62636 62169 62109 66601 63237   At_cost 1545 1607 1583 1540 1573 1528 1552 1594 1594 1582 1560 1555 1546 1587 1579 1566 1585 1601 1627 1541 1600 1563 1591 1565 1528 1538 1525 1517 1635 1545Log_input 11651 12188 12002 11636 11916 11558 11776 12139 12028 11986 11813 11756 11680 12046 11928 11885 11964 12115 12288 11610 12063 11780 12022 11812 11565 11599 11513 11502 12333 11711GRLumber 7629 7937 7817 7605 7766 7544 7664 7872 7873 7814 7706 7678 7637 7838 7798 7736 7827 7907 8033 7609 7901 7718 7855 7729 7546 7593 7532 7490 8075 7628ATLumber 7621 7929 7809 7597 7758 7536 7656 7864 7865 7806 7698 7670 7629 7830 7790 7728 7819 7899 8025 7601 7893 7710 7847 7721 7538 7585 7524 7483 8067 7620ATDemand 7621 7929 7809 7597 7758 7536 7656 7864 7865 7806 7698 7670 7629 7830 7790 7728 7819 7899 8025 7601 7893 7710 7847 7721 7538 7585 7524 7483 8067 7620   A_AO1 17 12 13 16 15 19 17 12 15 14 17 15 15 13 14 16 15 13 13 18 13 15 15 17 17 18 19 17 11 16   A_BO1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   A_AO2 18 15 15 18 17 21 19 14 16 17 19 17 17 15 15 18 16 14 15 19 14 17 17 19 19 20 21 19 13 18   A_BO2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV1 1270 986 1009 1208 1187 1425 1302 947 1159 1130 1314 1159 1139 1018 1080 1200 1147 998 1047 1354 1005 1158 1186 1320 1305 1377 1427 1253 897 1193     BV1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     AV2 1398 1148 1168 1330 1331 1555 1480 1123 1284 1289 1485 1297 1258 1177 1205 1362 1272 1124 1163 1467 1129 1279 1321 1429 1444 1496 1552 1397 1040 1333     BV2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0134   Table D.6 Pl-S-L with large, small, mixed, high variation, and low variation lumber products demand  LPT B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 449833 449109 449188 449917 449319 449639 449533 447872 447388 449663 448521 448758 447678 449432 446888 449168 450339 449205 450111 448617 449814 451804 448445 449482 447767 450116 447474 447430 450433 447600Log_Cost 384041 383446 383554 384041 383568 383843 383834 382422 381883 383904 382882 383150 382066 383617 381560 383428 384465 383423 384336 382953 384083 385694 382819 383662 382307 384317 382061 382000 384582 382068Lugr_icost 163 149 129 201 178 182 135 131 202 155 179 145 234 207 144 187 167 203 130 190 127 180 182 206 143 144 143 156 147 199Luat_icost 190 177 155 236 210 212 161 154 234 184 208 170 273 243 167 217 197 238 156 224 150 212 212 243 168 171 171 185 173 233Sawing_cost 63875 63775 63790 63875 63802 63839 63840 63608 63515 63856 63695 63734 63550 63803 63464 63776 63945 63781 63924 63691 63892 64147 63674 63810 63594 63919 63544 63534 63965 63544   At_cost 1564 1562 1562 1564 1561 1563 1562 1557 1555 1563 1558 1559 1555 1562 1553 1561 1565 1560 1565 1559 1563 1571 1558 1562 1556 1565 1555 1555 1566 1556Log_input 11829 11810 11813 11829 11815 11822 11822 11779 11762 11825 11795 11803 11768 11815 11753 11810 11842 11811 11838 11795 11832 11879 11791 11817 11777 11837 11767 11766 11845 11767GRLumber 7722 7713 7712 7722 7709 7719 7716 7687 7678 7719 7692 7701 7679 7714 7671 7707 7731 7706 7727 7701 7719 7756 7695 7713 7684 7729 7681 7679 7732 7684ATLumber 7714 7705 7704 7714 7701 7711 7708 7679 7670 7711 7684 7693 7671 7706 7663 7699 7723 7698 7719 7693 7711 7748 7687 7705 7676 7721 7673 7671 7724 7676ATDemand 7714 7705 7704 7714 7701 7711 7708 7679 7670 7711 7684 7693 7671 7706 7663 7699 7723 7698 7719 7693 7711 7748 7687 7705 7676 7721 7673 7671 7724 7676   A_AO1 25 23 20 31 28 28 21 20 31 24 28 23 37 32 23 29 26 32 20 30 20 28 28 32 22 22 22 24 23 31   A_BO1 39 23 9 72 37 52 16 15 76 28 38 25 86 65 40 56 37 65 12 74 17 45 74 68 34 23 18 23 31 79   A_AO2 26 24 21 32 29 29 22 21 32 25 28 23 37 33 23 30 27 32 21 30 20 29 29 33 23 23 23 25 23 32   A_BO2 38 22 8 70 36 50 15 13 74 26 36 23 83 63 37 53 36 63 12 72 16 43 72 66 33 22 17 22 29 77     AV1 1960 1794 1550 2426 2147 2193 1627 1575 2431 1872 2151 1746 2820 2498 1739 2247 2006 2442 1569 2294 1526 2167 2196 2476 1719 1731 1727 1881 1774 2392     BV1 3023 1792 685 5546 2863 4014 1227 1122 5800 2110 2907 1916 6533 5025 3021 4280 2865 5012 956 5642 1306 3453 5662 5190 2631 1778 1390 1759 2367 5997     AV2 1999 1862 1627 2482 2210 2228 1699 1623 2459 1940 2188 1792 2870 2554 1756 2281 2071 2501 1644 2354 1574 2229 2231 2561 1764 1802 1796 1949 1826 2448     BV2 2881 1704 617 5377 2747 3850 1148 1018 5602 2013 2761 1794 6340 4867 2854 4108 2753 4849 883 5473 1198 3328 5463 5055 2493 1688 1307 1669 2246 5821LPTS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 359803 360306 359563 360967 358142 356678 360399 359585 359874 358126 359822 448793 359909 359370 446923 361618 362697 361515 359683 359430 360945 361025 358694 361433 360329 358052 359728 360741 359818 358820Log_Cost 307101 307510 306839 308039 305662 304409 307577 306906 307139 305648 307150 383150 307207 306711 381560 308622 309528 308609 306964 306767 308085 308139 306163 308475 307508 305670 307023 307928 307049 306302Lugr_icost 172 185 206 202 182 181 191 178 186 186 158 163 163 182 160 190 199 152 191 181 167 178 160 183 196 135 178 161 206 148Luat_icost 201 215 239 235 214 211 222 208 217 216 183 188 191 212 186 221 231 178 223 213 195 209 192 215 228 159 208 188 239 176Sawing_cost 51079 51143 51029 51236 50841 50638 51157 51043 51082 50833 51079 63734 51098 51017 63464 51329 51478 51319 51055 51020 51244 51244 50933 51305 51144 50845 51070 51209 51074 50947   At_cost 1250 1252 1250 1254 1244 1239 1252 1250 1251 1244 1251 1559 1250 1248 1553 1257 1261 1257 1250 1249 1254 1255 1246 1256 1252 1244 1249 1255 1250 1247Log_input 9459 9471 9450 9488 9415 9377 9474 9452 9460 9413 9459 11803 9463 9448 11753 9505 9533 9503 9455 9448 9490 9490 9432 9501 9471 9416 9457 9483 9458 9435GRLumber 6175 6184 6171 6192 6144 6118 6183 6171 6176 6145 6179 7701 6175 6164 7671 6206 6225 6208 6172 6169 6194 6198 6151 6202 6183 6144 6170 6195 6172 6159ATLumber 6169 6178 6165 6186 6138 6112 6177 6165 6170 6139 6173 7693 6169 6158 7663 6200 6219 6202 6166 6163 6188 6192 6145 6196 6177 6138 6164 6189 6166 6153ATDemand 6169 6178 6165 6186 6138 6112 6177 6165 6170 6139 6173 7693 6169 6158 7663 6200 6219 6202 6166 6163 6188 6192 6145 6196 6177 6138 6164 6189 6166 6153   A_AO1 33 36 40 39 36 35 37 35 36 36 31 25 32 35 25 37 38 29 37 35 32 35 31 35 38 26 35 31 40 29   A_BO1 12 51 46 46 51 34 46 23 40 45 25 75 12 53 39 23 63 12 31 29 40 12 0 17 53 18 23 6 57 0   A_AO2 34 37 41 40 37 36 38 35 37 37 31 25 33 36 25 37 39 30 38 36 33 36 33 36 39 27 35 32 41 30   A_BO2 11 50 44 44 49 32 44 22 39 44 24 72 11 51 37 22 61 11 29 28 39 11 0 17 51 16 22 6 55 0     AV1 2070 2229 2481 2439 2189 2178 2298 2146 2240 2235 1905 1964 1967 2190 1922 2284 2397 1831 2306 2180 2008 2143 1931 2199 2367 1626 2144 1936 2480 1779     BV1 723 3147 2798 2823 3077 2060 2815 1423 2456 2760 1550 5740 723 3224 2956 1454 3909 756 1886 1766 2483 716 0 1062 3247 1077 1425 386 3470 14     AV2 2111 2268 2515 2475 2250 2219 2336 2186 2282 2273 1929 1977 2012 2230 1962 2322 2432 1877 2344 2241 2052 2204 2023 2260 2402 1669 2185 1984 2514 1855     BV2 665 3041 2699 2721 2991 1973 2713 1352 2368 2660 1456 5509 665 3112 2817 1382 3787 700 1806 1708 2390 680 0 1018 3137 1003 1352 342 3353 0LPT BS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 400518 390985 396557 403402 402000 408412 408983 400763 410347 405856 398729 405258 410483 404228 446929 396711 408695 403923 398072 391692 409048 412396 394106 400617 404037 404948 401591 410735 400039 399724Log_Cost 341897 333770 338491 344330 343165 348551 349096 342120 350246 346493 340352 345917 350429 345077 381560 338645 348859 344758 339796 334301 349121 352088 336360 342038 344923 345655 342814 350542 341507 341227Lugr_icost 160 153 177 191 180 197 209 170 184 160 175 178 162 162 163 169 178 195 176 191 180 149 178 142 167 191 164 206 154 178Luat_icost 190 176 207 224 210 230 243 197 213 185 207 210 188 189 190 198 209 228 207 223 212 176 212 171 194 223 191 241 183 206Sawing_cost 56881 55528 56303 57256 57046 58017 58011 56882 58282 57605 56609 57547 58277 57397 63464 56320 58029 57338 56509 55619 58119 58549 55991 56872 57347 57469 57027 58321 56805 56723   At_cost 1390 1358 1379 1402 1399 1417 1424 1393 1423 1413 1386 1406 1426 1403 1553 1379 1420 1405 1385 1359 1415 1435 1366 1394 1406 1410 1395 1425 1391 1391Log_input 10534 10283 10426 10603 10564 10744 10743 10534 10793 10668 10483 10657 10792 10629 11753 10430 10746 10618 10465 10300 10763 10842 10369 10532 10620 10642 10561 10800 10519 10504GRLumber 6864 6705 6810 6922 6910 6997 7030 6881 7025 6976 6846 6942 7044 6928 7671 6809 7012 6939 6838 6712 6990 7085 6744 6884 6943 6962 6888 7039 6868 6869ATLumber 6857 6698 6803 6915 6903 6990 7023 6874 7018 6969 6839 6935 7037 6921 7663 6802 7005 6932 6831 6705 6983 7078 6737 6877 6936 6955 6881 7032 6861 6862ATDemand 6857 6698 6803 6915 6903 6990 7023 6874 7018 6969 6839 6935 7037 6921 7663 6802 7005 6932 6831 6705 6983 7078 6737 6877 6936 6955 6881 7032 6861 6862   A_AO1 28 27 30 32 31 33 35 29 31 27 30 31 27 27 25 29 30 33 30 34 31 24 31 25 28 33 28 34 26 30   A_BO1 21 41 30 38 26 44 57 35 47 32 58 21 26 63 39 24 31 49 55 28 57 39 57 4 32 39 33 65 35 47   A_AO2 29 27 31 33 32 34 35 29 31 27 31 32 27 27 26 30 31 34 31 35 32 25 33 26 28 33 28 35 27 30   A_BO2 20 38 29 36 25 43 55 34 45 30 57 21 25 60 37 23 30 48 53 27 55 38 56 4 31 38 31 63 34 46     AV1 1922 1839 2133 2306 2164 2371 2520 2047 2212 1924 2112 2142 1951 1952 1959 2036 2148 2350 2126 2297 2173 1789 2142 1716 2015 2300 1978 2481 1854 2142     BV1 1421 2691 2018 2581 1809 3040 3828 2426 3269 2118 3825 1421 1846 4172 2956 1634 2175 3353 3666 1889 3924 2767 3816 292 2147 2690 2222 4337 2330 3144     AV2 2003 1851 2180 2353 2211 2423 2562 2075 2245 1951 2181 2214 1982 1986 1997 2087 2198 2395 2176 2346 2236 1853 2227 1799 2041 2342 2010 2538 1925 2164     BV2 1363 2533 1938 2486 1737 2924 3723 2305 3114 2022 3724 1363 1753 4004 2817 1553 2079 3244 3549 1795 3760 2668 3695 264 2047 2597 2102 4191 2261 3034LPT H 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 478824 430616 433884 442572 462919 451271 456513 465671 456385 463689 429368 456511 462025 438804 459482 444976 471858 460851 467609 436729 463419 423122 457866 463019 434059 446156 408803 475399 424270 460912Log_Cost 408998 367718 370457 377839 395282 385248 390006 397732 389905 396154 366546 389844 394639 374610 392371 379863 402987 393606 399284 372815 395606 361334 390941 395114 370704 381039 348877 405899 362306 393518Lugr_icost 146 117 183 150 214 192 141 120 114 190 171 162 84 188 178 197 99 239 176 164 197 157 120 197 168 155 189 172 93 142Luat_icost 170 136 214 177 248 223 166 141 135 220 197 191 98 221 205 228 123 274 206 194 233 181 146 230 195 180 220 201 114 169Sawing_cost 67836 61152 61523 62879 65562 64043 64610 66061 64640 65501 60974 64723 65596 62268 65127 63147 67014 65118 66314 62041 65776 59978 65073 65882 61484 63231 58112 67475 60287 65490   At_cost 1674 1494 1506 1527 1612 1565 1591 1618 1590 1624 1481 1592 1608 1517 1602 1541 1634 1614 1629 1516 1608 1473 1585 1594 1509 1551 1405 1651 1470 1593Log_input 12562 11324 11393 11644 12141 11860 11965 12234 11970 12130 11291 11986 12147 11531 12060 11694 12410 12059 12280 11489 12181 11107 12051 12200 11386 11709 10761 12495 11164 12128GRLumber 8266 7376 7436 7542 7962 7728 7855 7988 7854 8018 7315 7860 7941 7491 7910 7610 8070 7969 8043 7484 7939 7272 7827 7874 7449 7661 6938 8155 7259 7869ATLumber 8258 7369 7429 7534 7954 7720 7847 7980 7846 8010 7308 7852 7933 7484 7902 7602 8062 7961 8035 7477 7931 7265 7819 7866 7442 7653 6931 8147 7252 7861ATDemand 8258 7369 7429 7534 7954 7720 7847 7980 7846 8010 7308 7852 7933 7484 7902 7602 8062 7961 8035 7477 7931 7265 7819 7866 7442 7653 6931 8147 7252 7861   A_AO1 20 19 29 22 30 28 22 17 17 26 25 23 12 28 25 29 14 35 25 24 28 23 17 29 25 23 29 24 15 20   A_BO1 36 11 55 37 79 63 26 22 17 72 57 43 14 69 45 55 28 82 78 48 72 37 25 75 51 32 45 54 6 40   A_AO2 21 19 29 23 30 28 22 18 17 27 25 24 13 29 25 29 15 35 25 25 29 23 18 30 26 24 30 24 16 21   A_BO2 35 10 54 35 77 60 25 20 17 71 54 42 13 66 44 53 27 81 76 47 69 36 24 71 50 31 42 52 5 38     AV1 1758 1406 2207 1806 2579 2315 1694 1444 1379 2294 2057 1958 1009 2261 2142 2371 1196 2885 2120 1981 2372 1887 1448 2378 2020 1865 2273 2077 1121 1706     BV1 2996 725 3918 2704 5790 4398 1933 1813 1466 5463 3851 3155 1111 4261 3546 3864 1751 6132 5331 3488 5480 2343 1685 5760 3831 2400 2873 4424 351 3120     AV2 1788 1432 2256 1863 2615 2347 1750 1484 1419 2316 2072 2006 1034 2323 2160 2396 1296 2884 2173 2037 2448 1905 1541 2424 2055 1893 2320 2115 1198 1780     BV2 2901 654 3810 2561 5637 4214 1885 1704 1417 5382 3652 3098 983 4099 3459 3707 1657 6058 5189 3364 5274 2264 1580 5451 3723 2342 2675 4258 294 2921LPT L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 443396 463367 456368 442631 453122 439448 447575 461145 457715 455782 449244 447378 444212 457894 453867 451926 455140 460722 467334 441783 459201 459201 457341 449626 439937 441379 438133 437307 469176 445220Log_Cost 378599 395672 389508 377869 386876 375239 382115 393642 390774 389033 383525 381849 379274 390911 387448 385768 388585 393312 398994 377239 392001 392001 390494 383808 375580 376758 373982 373321 400685 380121Lugr_icost 154 125 217 179 151 122 145 163 185 203 169 227 146 162 199 190 167 178 164 144 212 212 155 218 173 207 211 166 116 146Luat_icost 180 151 252 210 177 147 174 195 214 242 200 265 174 187 230 223 199 208 194 168 247 247 183 251 204 240 246 194 139 172Sawing_cost 62918 65813 64808 62833 64346 62412 63589 65549 64949 64722 63789 63482 63071 65048 64412 64178 64604 65422 66355 62692 65142 65142 64918 63784 62452 62636 62169 62109 66601 63237   At_cost 1545 1607 1583 1540 1573 1528 1552 1594 1594 1582 1560 1555 1546 1587 1579 1566 1585 1601 1627 1541 1600 1600 1591 1565 1528 1538 1525 1517 1635 1545Log_input 11651 12188 12002 11636 11916 11558 11776 12139 12028 11986 11813 11756 11680 12046 11928 11885 11964 12115 12288 11610 12063 12063 12022 11812 11565 11599 11513 11502 12333 11711GRLumber 7629 7937 7817 7605 7766 7544 7664 7872 7873 7814 7706 7678 7637 7838 7798 7736 7827 7907 8033 7609 7901 7901 7855 7729 7546 7593 7532 7490 8075 7628ATLumber 7621 7929 7809 7597 7758 7536 7656 7864 7865 7806 7698 7670 7629 7830 7790 7728 7819 7899 8025 7601 7893 7893 7847 7721 7538 7585 7524 7483 8067 7620ATDemand 7621 7929 7809 7597 7758 7536 7656 7864 7865 7806 7698 7670 7629 7830 7790 7728 7819 7899 8025 7601 7893 7893 7847 7721 7538 7585 7524 7483 8067 7620   A_AO1 24 19 32 27 23 19 22 24 27 31 26 35 22 23 30 29 25 26 24 22 31 31 23 32 26 32 33 25 17 22   A_BO1 39 25 73 58 39 12 41 57 52 69 41 86 33 51 68 69 42 49 48 36 89 89 71 72 59 67 63 50 14 26   A_AO2 24 20 33 28 23 20 23 25 28 32 27 35 23 23 30 30 26 26 25 23 32 32 24 33 26 33 33 26 18 23   A_BO2 37 24 70 56 37 11 40 55 50 67 39 84 32 48 66 67 41 47 47 34 87 87 69 70 57 65 61 48 14 24     AV1 1858 1502 2616 2162 1824 1476 1742 1970 2224 2450 2042 2739 1765 1948 2392 2290 2010 2149 1979 1736 2550 2550 1868 2623 2087 2496 2542 2000 1400 1764     BV1 2961 1882 5531 4329 3034 923 3041 4411 4023 5051 3109 6299 2566 3853 5158 5259 3300 3749 3836 2716 6452 6452 5312 5359 4353 4704 4558 3678 1136 1968     AV2 1891 1587 2649 2210 1861 1551 1835 2058 2256 2543 2101 2785 1827 1967 2425 2351 2095 2186 2043 1768 2601 2601 1928 2646 2147 2530 2587 2040 1464 1812     BV2 2840 1786 5323 4201 2897 854 2957 4228 3887 4912 3010 6119 2479 3654 4987 5076 3204 3606 3720 2614 6299 6299 5156 5205 4220 4577 4433 3518 1074 1835135   Table D.7 Pl-S-S with large, small, mixed, high variation, and low variation lumber products demand  SPT B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 449905 449169 449365 449804 449387 449646 449620 448031 447234 449749 448603 448848 447466 449254 446998 449132 450389 449100 450332 448520 450020 451796 448295 449351 447829 450204 447534 447593 450474 447375Log_Cost 384041 383446 383554 384041 383568 383843 383834 382422 381883 383904 382882 383150 382066 383617 381560 383428 384465 383423 384336 382953 384083 385694 382819 383662 382307 384317 382061 382000 384582 382068Lugr_icost 196 179 213 149 210 185 177 204 128 196 216 185 134 126 192 169 191 155 234 144 222 177 111 146 171 185 172 233 167 94Luat_icost 230 208 246 175 246 216 206 239 153 230 253 220 161 146 228 198 222 181 273 173 260 206 133 172 201 218 203 271 195 112Sawing_cost 63875 63775 63790 63875 63802 63839 63840 63608 63515 63856 63695 63734 63550 63803 63464 63776 63945 63781 63924 63691 63892 64147 63674 63810 63594 63919 63544 63534 63965 63544   At_cost 1564 1562 1562 1564 1561 1563 1562 1557 1555 1563 1558 1559 1555 1562 1553 1561 1565 1560 1565 1559 1563 1571 1558 1562 1556 1565 1555 1555 1566 1556Log_input 11829 11810 11813 11829 11815 11822 11822 11779 11762 11825 11795 11803 11768 11815 11753 11810 11842 11811 11838 11795 11832 11879 11791 11817 11777 11837 11767 11766 11845 11767GRLumber 7722 7713 7712 7722 7709 7719 7716 7687 7678 7719 7692 7701 7679 7714 7671 7707 7731 7706 7727 7701 7719 7756 7695 7713 7684 7729 7681 7679 7732 7684ATLumber 7714 7705 7704 7714 7701 7711 7708 7679 7670 7711 7684 7693 7671 7706 7663 7699 7723 7698 7719 7693 7711 7748 7687 7705 7676 7721 7673 7671 7724 7676ATDemand 7714 7705 7704 7714 7701 7711 7708 7679 7670 7711 7684 7693 7671 7706 7663 7699 7723 7698 7719 7693 7711 7748 7687 7705 7676 7721 7673 7671 7724 7676   A_AO1 31 28 33 23 33 29 28 32 20 31 34 29 21 20 30 27 30 24 37 23 35 28 17 23 27 29 27 37 26 15   A_BO1 58 69 86 24 65 47 74 79 15 69 65 68 12 28 53 41 60 32 86 19 79 54 12 28 55 70 72 79 60 7   A_AO2 31 29 34 24 34 30 28 33 21 32 35 30 22 20 31 27 30 25 37 24 36 28 18 23 28 30 28 37 27 15   A_BO2 56 67 83 22 63 45 72 76 14 67 63 66 11 26 52 39 58 30 84 18 77 51 11 27 53 68 69 76 58 6     AV1 2358 2151 2567 1798 2534 2228 2133 2463 1536 2362 2604 2229 1620 1518 2317 2042 2302 1864 2825 1735 2677 2136 1333 1764 2055 2230 2069 2804 2009 1135     BV1 4488 5368 6640 1847 5053 3584 5741 6088 1159 5346 4996 5264 941 2169 4081 3172 4677 2456 6730 1436 6178 4159 945 2202 4273 5424 5517 6082 4666 555     AV2 2417 2186 2593 1842 2590 2270 2170 2520 1613 2421 2659 2318 1695 1539 2404 2080 2334 1906 2873 1816 2732 2171 1401 1807 2121 2290 2133 2853 2047 1179     BV2 4340 5178 6423 1727 4890 3440 5538 5905 1079 5180 4830 5124 867 2019 3968 3023 4497 2323 6535 1367 5989 3993 857 2076 4127 5259 5348 5900 4488 453SPT S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 443383 463498 456252 442556 453206 439594 447588 461160 457672 455711 449221 447210 444311 457934 453807 451894 455115 460696 467296 441906 459028 448253 457330 449516 439832 441332 438071 437331 469310 469310Log_Cost 378599 395672 389508 377869 386876 375239 382115 393642 390774 389033 383525 381849 379274 390911 387448 385768 388585 393312 398994 377239 392001 382741 390494 383808 375580 376758 373982 373321 400685 400685Lugr_icost 148 186 161 143 190 193 154 172 164 171 161 147 195 179 170 174 159 166 147 201 130 156 149 165 124 184 182 176 180 180Luat_icost 172 221 192 171 222 223 178 202 192 202 186 177 225 209 198 207 182 194 173 234 156 182 178 194 148 216 214 208 209 209Sawing_cost 62918 65813 64808 62833 64346 62412 63589 65549 64949 64722 63789 63482 63071 65048 64412 64178 64604 65422 66355 62692 65142 63611 64918 63784 62452 62636 62169 62109 66601 66601   At_cost 1545 1607 1583 1540 1573 1528 1552 1594 1594 1582 1560 1555 1546 1587 1579 1566 1585 1601 1627 1541 1600 1563 1591 1565 1528 1538 1525 1517 1635 1635Log_input 11651 12188 12002 11636 11916 11558 11776 12139 12028 11986 11813 11756 11680 12046 11928 11885 11964 12115 12288 11610 12063 11780 12022 11812 11565 11599 11513 11502 12333 12333GRLumber 7629 7937 7817 7605 7766 7544 7664 7872 7873 7814 7706 7678 7637 7838 7798 7736 7827 7907 8033 7609 7901 7718 7855 7729 7546 7593 7532 7490 8075 8075ATLumber 7621 7929 7809 7597 7758 7536 7656 7864 7865 7806 7698 7670 7629 7830 7790 7728 7819 7899 8025 7601 7893 7710 7847 7721 7538 7585 7524 7483 8067 8067ATDemand 7621 7929 7809 7597 7758 7536 7656 7864 7865 7806 7698 7670 7629 7830 7790 7728 7819 7899 8025 7601 7893 7710 7847 7721 7538 7585 7524 7483 8067 8067   A_AO1 25 30 25 23 31 32 26 27 26 27 26 23 32 29 27 28 26 27 23 32 20 25 23 27 21 30 30 30 28 28   A_BO1 47 72 30 30 58 75 47 42 47 33 53 11 60 47 32 30 54 51 49 56 7 40 21 28 24 31 34 40 83 83   A_AO2 25 31 26 24 31 32 26 28 27 28 26 24 32 30 27 29 26 27 23 33 21 25 24 28 22 31 30 30 29 29   A_BO2 45 70 28 28 55 73 44 39 45 32 50 11 58 44 30 28 52 49 48 54 6 38 20 27 22 30 33 38 81 81     AV1 1781 2245 1942 1725 2290 2319 1854 2076 1978 2061 1942 1769 2343 2160 2050 2100 1910 2001 1775 2418 1572 1874 1801 1982 1489 2215 2192 2118 2169 2169     BV1 3626 6270 2302 2266 4561 5929 3669 3396 3728 2686 4202 913 4697 3710 2461 2365 4313 4050 4017 4274 572 3104 1696 2194 1786 2441 2638 3009 7089 7089     AV2 1816 2322 2019 1798 2333 2349 1871 2126 2017 2127 1956 1862 2371 2204 2089 2182 1919 2046 1825 2466 1640 1917 1872 2046 1556 2278 2248 2187 2204 2204     BV2 3482 6059 2190 2156 4351 5722 3428 3192 3584 2568 3964 867 4527 3516 2339 2243 4138 3872 3861 4152 514 2983 1605 2094 1668 2347 2535 2876 6884 6884SPT BS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 400,617 391,026 396,508 403,380 402,017 408,409 408,933 400,725 410,333 405,920 398,697 405,315 410,554 404,216 447,002 396,750 408,771 403,849 397,933 391,709 408,931 412,429 394,064 400,749 404,094 404,920 401,667 410,705 400,028 399,774Log_Cost 341,897 333,770 338,491 344,330 343,165 348,551 349,096 342,120 350,246 346,493 340,352 345,917 350,429 345,077 381,560 338,645 348,859 344,758 339,796 334,301 349,121 352,088 336,360 342,038 344,923 345,655 342,814 350,542 341,507 341,227Lugr_icost 206 170 155 182 188 195 186 152 174 187 160 205 194 154 196 187 214 160 111 199 126 165 158 206 193 178 198 191 149 199Luat_icost 243 200 180 210 219 228 217 177 208 222 191 240 227 185 230 219 249 188 133 232 149 193 190 239 225 208 233 226 177 235Sawing_cost 56,881 55,528 56,303 57,256 57,046 58,017 58,011 56,882 58,282 57,605 56,609 57,547 58,277 57,397 63,464 56,320 58,029 57,338 56,509 55,619 58,119 58,549 55,991 56,872 57,347 57,469 57,027 58,321 56,805 56,723   At_cost 1,390 1,358 1,379 1,402 1,399 1,417 1,424 1,393 1,423 1,413 1,386 1,406 1,426 1,403 1,553 1,379 1,420 1,405 1,385 1,359 1,415 1,435 1,366 1,394 1,406 1,410 1,395 1,425 1,391 1,391Log_input 10,534 10,283 10,426 10,603 10,564 10,744 10,743 10,534 10,793 10,668 10,483 10,657 10,792 10,629 11,753 10,430 10,746 10,618 10,465 10,300 10,763 10,842 10,369 10,532 10,620 10,642 10,561 10,800 10,519 10,504GRLumber 6,864 6,705 6,810 6,922 6,910 6,997 7,030 6,881 7,025 6,976 6,846 6,942 7,044 6,928 7,671 6,809 7,012 6,939 6,838 6,712 6,990 7,085 6,744 6,884 6,943 6,962 6,888 7,039 6,868 6,869ATLumber 6,857 6,698 6,803 6,915 6,903 6,990 7,023 6,874 7,018 6,969 6,839 6,935 7,037 6,921 7,663 6,802 7,005 6,932 6,831 6,705 6,983 7,078 6,737 6,877 6,936 6,955 6,881 7,032 6,861 6,862ATDemand 6,857 6,698 6,803 6,915 6,903 6,990 7,023 6,874 7,018 6,969 6,839 6,935 7,037 6,921 7,663 6,802 7,005 6,932 6,831 6,705 6,983 7,078 6,737 6,877 6,936 6,955 6,881 7,032 6,861 6,862   A_AO1 36 31 28 33 34 34 32 28 30 33 29 37 34 27 31 34 37 28 20 36 22 29 28 37 35 31 35 33 27 35   A_BO1 59 27 41 45 54 42 29 36 34 46 14 64 56 8 58 53 56 29 9 52 15 36 14 73 48 42 45 22 34 32   A_AO2 37 32 29 34 34 35 33 28 32 34 30 37 35 28 32 35 38 29 21 37 23 30 30 37 35 32 36 34 28 37   A_BO2 57 25 39 43 52 41 28 35 33 45 14 62 54 8 56 51 54 28 8 50 14 35 13 71 46 40 44 21 32 31     AV1 2481 2051 1867 2189 2261 2351 2239 1835 2101 2258 1929 2472 2335 1856 2359 2257 2574 1931 1339 2395 1520 1984 1903 2481 2323 2140 2383 2301 1797 2396     BV1 4098 1774 2811 3199 3776 2991 2144 2510 2354 3289 1006 4615 4041 552 4415 3699 3960 2110 602 3523 1066 2599 942 5213 3430 2947 3196 1603 2347 2190     AV2 2555 2102 1898 2215 2306 2405 2285 1867 2190 2334 2007 2527 2391 1948 2418 2301 2620 1984 1402 2441 1571 2027 2000 2519 2371 2188 2457 2380 1858 2469     BV2 3979 1688 2677 3054 3660 2865 2058 2389 2273 3201 960 4453 3900 514 4268 3570 3822 2019 543 3389 949 2497 892 5067 3312 2847 3097 1540 2242 2114SPT H 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 478,907 430,866 433,808 442,639 462,815 451,230 456,615 465,844 456,563 463,499 429,374 456,488 462,248 438,794 459,562 444,995 472,091 460,695 467,541 436,678 463,249 423,192 458,018 462,937 434,090 446,264 408,744 475,314 424,529 460,975Log_Cost 408,998 367,718 370,457 377,839 395,282 385,248 390,006 397,732 389,905 396,154 366,546 389,844 394,639 374,610 392,371 379,863 402,987 393,606 399,284 372,815 395,606 361,334 390,941 395,114 370,704 381,039 348,877 405,899 362,306 393,518Lugr_icost 184 232 146 180 165 171 190 201 198 102 170 153 187 180 213 203 209 166 145 143 120 186 192 159 181 205 161 133 214 171Luat_icost 215 271 175 214 193 203 219 233 229 118 203 177 218 218 249 240 246 192 170 164 139 221 227 188 213 238 189 155 252 202Sawing_cost 67,836 61,152 61,523 62,879 65,562 64,043 64,610 66,061 64,640 65,501 60,974 64,723 65,596 62,268 65,127 63,147 67,014 65,118 66,314 62,041 65,776 59,978 65,073 65,882 61,484 63,231 58,112 67,475 60,287 65,490   At_cost 1,674 1,494 1,506 1,527 1,612 1,565 1,591 1,618 1,590 1,624 1,481 1,592 1,608 1,517 1,602 1,541 1,634 1,614 1,629 1,516 1,608 1,473 1,585 1,594 1,509 1,551 1,405 1,651 1,470 1,593Log_input 12,562 11,324 11,393 11,644 12,141 11,860 11,965 12,234 11,970 12,130 11,291 11,986 12,147 11,531 12,060 11,694 12,410 12,059 12,280 11,489 12,181 11,107 12,051 12,200 11,386 11,709 10,761 12,495 11,164 12,128GRLumber 8,266 7,376 7,436 7,542 7,962 7,728 7,855 7,988 7,854 8,018 7,315 7,860 7,941 7,491 7,910 7,610 8,070 7,969 8,043 7,484 7,939 7,272 7,827 7,874 7,449 7,661 6,938 8,155 7,259 7,869ATLumber 8,258 7,369 7,429 7,534 7,954 7,720 7,847 7,980 7,846 8,010 7,308 7,852 7,933 7,484 7,902 7,602 8,062 7,961 8,035 7,477 7,931 7,265 7,819 7,866 7,442 7,653 6,931 8,147 7,252 7,861ATDemand 8,258 7,369 7,429 7,534 7,954 7,720 7,847 7,980 7,846 8,010 7,308 7,852 7,933 7,484 7,902 7,602 8,062 7,961 8,035 7,477 7,931 7,265 7,819 7,866 7,442 7,653 6,931 8,147 7,252 7,861   A_AO1 28 40 26 30 27 28 31 32 34 18 31 26 32 31 37 35 34 26 23 27 20 35 32 25 32 35 30 22 39 27   A_BO1 69 81 32 54 23 34 71 79 80 19 27 51 80 25 63 47 74 24 19 38 24 46 71 29 38 66 34 47 78 55   A_AO2 29 41 27 32 27 29 31 33 34 18 32 26 32 33 37 36 34 26 24 28 21 36 33 26 32 35 31 22 40 28   A_BO2 68 78 31 51 22 33 69 76 77 19 25 49 78 23 62 45 71 23 18 35 22 45 68 26 37 63 31 45 76 53     AV1 2222 2796 1763 2169 1988 2063 2287 2417 2384 1230 2050 1845 2249 2175 2568 2450 2517 1995 1746 1718 1446 2238 2313 1911 2180 2469 1943 1606 2582 2063     BV1 6012 6650 2671 4293 1790 2967 6143 6668 7032 1564 1953 4171 7645 2205 5607 3814 6938 1907 1641 2892 1964 3556 6199 2259 2953 5263 2271 3961 6644 4562     AV2 2263 2851 1839 2255 2036 2137 2305 2454 2410 1242 2139 1858 2295 2296 2624 2527 2592 2022 1790 1729 1464 2331 2388 1976 2241 2505 1989 1629 2648 2129     BV2 5897 6422 2551 4072 1726 2837 5984 6433 6831 1510 1802 3989 7428 2111 5451 3632 6699 1818 1532 2676 1829 3457 5964 2046 2837 5039 2074 3746 6410 4374SPT L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Man_cost 443,383 463,498 456,252 442,556 453,206 439,594 447,588 461,160 457,672 455,711 449,221 447,210 444,311 457,934 453,807 451,894 455,115 460,696 467,296 441,906 459,028 448,253 457,330 449,516 439,832 441,332 438,071 437,331 469,310 445,300Log_Cost 378,599 395,672 389,508 377,869 386,876 375,239 382,115 393,642 390,774 389,033 383,525 381,849 379,274 390,911 387,448 385,768 388,585 393,312 398,994 377,239 392,001 382,741 390,494 383,808 375,580 376,758 373,982 373,321 400,685 380,121Lugr_icost 148 186 161 143 190 193 154 172 164 171 161 147 195 179 170 174 159 166 147 201 130 156 149 165 124 184 182 176 180 184Luat_icost 172 221 192 171 222 223 178 202 192 202 186 177 225 209 198 207 182 194 173 234 156 182 178 194 148 216 214 208 209 214Sawing_cost 62,918 65,813 64,808 62,833 64,346 62,412 63,589 65,549 64,949 64,722 63,789 63,482 63,071 65,048 64,412 64,178 64,604 65,422 66,355 62,692 65,142 63,611 64,918 63,784 62,452 62,636 62,169 62,109 66,601 63,237   At_cost 1,545 1,607 1,583 1,540 1,573 1,528 1,552 1,594 1,594 1,582 1,560 1,555 1,546 1,587 1,579 1,566 1,585 1,601 1,627 1,541 1,600 1,563 1,591 1,565 1,528 1,538 1,525 1,517 1,635 1,545Log_input 11,651 12,188 12,002 11,636 11,916 11,558 11,776 12,139 12,028 11,986 11,813 11,756 11,680 12,046 11,928 11,885 11,964 12,115 12,288 11,610 12,063 11,780 12,022 11,812 11,565 11,599 11,513 11,502 12,333 11,711GRLumber 7,629 7,937 7,817 7,605 7,766 7,544 7,664 7,872 7,873 7,814 7,706 7,678 7,637 7,838 7,798 7,736 7,827 7,907 8,033 7,609 7,901 7,718 7,855 7,729 7,546 7,593 7,532 7,490 8,075 7,628ATLumber 7,621 7,929 7,809 7,597 7,758 7,536 7,656 7,864 7,865 7,806 7,698 7,670 7,629 7,830 7,790 7,728 7,819 7,899 8,025 7,601 7,893 7,710 7,847 7,721 7,538 7,585 7,524 7,483 8,067 7,620ATDemand 7,621 7,929 7,809 7,597 7,758 7,536 7,656 7,864 7,865 7,806 7,698 7,670 7,629 7,830 7,790 7,728 7,819 7,899 8,025 7,601 7,893 7,710 7,847 7,721 7,538 7,585 7,524 7,483 8,067 7,620   A_AO1 25 30 25 23 31 32 26 27 26 27 26 23 32 29 27 28 26 27 23 32 20 25 23 27 21 30 30 30 28 30   A_BO1 47 72 30 30 58 75 47 42 47 33 53 11 60 47 32 30 54 51 49 56 7 40 21 28 24 31 34 40 83 67   A_AO2 25 31 26 24 31 32 26 28 27 28 26 24 32 30 27 29 26 27 23 33 21 25 24 28 22 31 30 30 29 31   A_BO2 45 70 28 28 55 73 44 39 45 32 50 11 58 44 30 28 52 49 48 54 6 38 20 27 22 30 33 38 81 64     AV1 1781 2245 1942 1725 2290 2319 1854 2076 1978 2061 1942 1769 2343 2160 2050 2100 1910 2001 1775 2418 1572 1874 1801 1982 1489 2215 2192 2118 2169 2220     BV1 3626 6270 2302 2266 4561 5929 3669 3396 3728 2686 4202 913 4697 3710 2461 2365 4313 4050 4017 4274 572 3104 1696 2194 1786 2441 2638 3009 7089 5272     AV2 1816 2322 2019 1798 2333 2349 1871 2126 2017 2127 1956 1862 2371 2204 2089 2182 1919 2046 1825 2466 1640 1917 1872 2046 1556 2278 2248 2187 2204 2252     BV2 3482 6059 2190 2156 4351 5722 3428 3192 3584 2568 3964 867 4527 3516 2339 2243 4138 3872 3861 4152 514 2983 1605 2094 1668 2347 2535 2876 6884 5062

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0166662/manifest

Comment

Related Items