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A multiple period combined optimization approach to sawmill production planning systems Norton, Scott Erling 1993

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A MULTIPLE PERIOD COMBINED OPTIMIZATION APPROACH TO SAWMILL PRODUCTION PLANNING SYSTEMS by SCOTT ERLING NORTON B.Sc., The University of Puget Sound, 1990  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Forestry)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA July 1993 © Scott Erling Norton, 1993  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of Forestry The University of British Columbia Vancouver, Canada  Date 8 July 1993  DE-6 (2/88)  ABSTRACT  Recent efforts on the development of production planning systems for sawmills have focused on combined optimization type solutions in a steady state market environment. This thesis focuses on developing a multiple period production planning system which responds to expected changes in product value or market demand by changing production policy, sawing policy, or log boom selection. Production periods are tied together by log and lumber inventory. The model was tested using four market scenarios on a large log mill producing export products. In general, it was shown that the model responded to market changes using sawing pattern selection and altered boom distribution and consumption.  ii  TABLE OF CONTENTS  ABSTRACT^  ii  TABLE OF CONTENTS^  iii  LIST OF TABLES^  iv  LIST OF FIGURES^  v  ACKNOWLEDGMENT^ I.  INTRODUCTION^  II.  LITERATURE REVIEW  Integrated Production Planning Models ^ Planning Over Multiple Periods^ III.  IV.  vi 1  3 6  DESCRIPTION OF A MODEL FOR MULTIPLE PERIOD PRODUCTION PLANNING Introduction^ Resource Coordinator^ Objective Function^ Supply Constraints^ Production Constraints^ Marketing Constraints^ Inventory Constraints^ Sawing Model^ Integrating the Models^ Data Acquisition^ Analysis of Iterations^ Conclusion^ Literature Cited^  7 8 11 13 14 14 16 17 17 18 18 22 23  THE EFFECTS OF MARKETS ON MULTIPLE PERIOD PRODUCTION PLANNING Introduction^ Experimental Design^ Case Specific Inputs and Expectations^ Tuning the Model^ Computational Experience^ Results and Analysis^ Conclusion^ Literature Cited^  24 24 26 27 28 28 34 34  V.^CONCLUSION^ BIBLIOGRAPHY  ^  APPENDIX - Solution Reports of Example Cases^  35 38 39 iii  LIST OF TABLES  7  1.  General Mill Information  2.  Solution - Steady State Case  12  3.  Boom Selection - Steady State Case  13  4.  Under Production Example  15  5.  Product Degrade Example  16  6.  Degrade Table  25  7.  Historical Price Fluctuations  26  8.  Solution - Price Increase Case  29  9.  Production and Inventory Policy  29  10.  Solution - Price Decrease Case  30  11.  Solution - Demand Increase Case  30  12.  Production Sales as a Percent of Base Case - Demand Increase Case  31  13.  Boom Selection - Price Decrease Case  32  iv  LIST OF FIGURES 1.  Integrated Model  18  2.  Change in Optimal Value - Steady State Case  19  3.  Relationship Between Production and Shadow Prices  20  4.  Iteration Effects - Inventory Sales  21  5.  Iteration Effects - Production Sales  22  ACKNOWLEDGMENTS  Several people should be acknowledged for their assistance in the preparation of this research. Foremost is the research supervisor, Dr. Thomas C. Maness. His interest and assistance in this project has been invaluable. The author would also like to acknowledge Mr. Jan Aune at MacMillan Bloedel for the funding of this project in cooperation with NSERC and for his readiness to cooperate in any way possible. Finally, Mr. Dallas Foley, our technician, must be thanked for his crack programming assistance.  vi  I.^INTRODUCTION  The environment for forest products manufacturing is changing. New strategies are being developed in the areas of manufacturing process, raw material mix, and marketing. These interrelated areas give the industry greater control over production, but also introduce new variables. The added variables complicate decision making, and require a greater amount of information for analysis. One example of this is in the area of marketing. The forest products industry has seen a significant shift over the past several years from a producer driven to a consumer driven market. Consumers are placing greater emphasis on quality, specialty sizes, and responsive delivery times (Cohen, 1992). The producer needs a way to evaluate the market demand and select the optimal levels of production and raw material required. Production planning systems are tools to be used in decision making. Their purpose is to assist in dealing with the variables and constraints involved in the production process. For the sawmill, an accurate model provides a test bed for recommendations before extensive amounts of time and/or capital are spent. With a system of this type, it is much easier to look at a range of economic scenarios before making a decision. It may also be used to challenge assumptions. To incorporate the time dimension of these problems, a series of single period production planning models may be chained together using inventory as the connection from one period to the next. Because sawmills are exposed to fluctuations in market demand and value, holding inventory of logs and lumber may increase revenues; but costs are incurred as well. These costs include the degrade of inventory and administrative costs such as management and storage. A multiple period production planning system will help determine the impact of these costs on the optimal levels of inventory and the production strategy required to meet the recommendations.  1  There are four major objectives in this study: 1. 2. 3. 4.  Develop a model which can be used for sawmill production planning over multiple periods. Model the impact of lumber inventories on overall profit/loss and production. Develop a method of analysis for determining the optimal log mix over a series of periods. Develop general production strategies for producing in periods of falling or rising prices.  This thesis is written in three main sections. The first focuses on production planning methods developed in the past. The second section describes the formulation and operation of the Multiple Period Planning Model. The final section shows how the model reacts to three different market scenarios. Sections two and three include a list of literature cited. A complete bibliography is located at the end of the thesis.  2  LITERATURE REVIEW  The model presented in this thesis is partially based on the work done by other researchers in the area of production planning systems. This review of their contributions is split into two parts: the integrated production planning model and planning over multiple periods. Integrated Production Planning Models  There have been three different approaches to the integrated production planning model. These models can be described as a combination of linear and dynamic programming techniques. Each of the models is based on the Decomposition Principle (Danzig and Wolf, 1960). The principle basically states that certain types of linear programming problems can be broken into sub-problems. The linear program consists of a coordinating problem which allocates limited resources and a series of subproblems which generate activities for the coordinating problem. The subproblems are optimized with respect to the dual values (or shadow prices) of the resources in the coordinating problem. These shadow prices represent the value of obtaining one more unit of each resource (marginal value). The first model was described by McPhalen (1978) to determine a combination of bucking and sawing policies which maximize the overall value for each class of raw material for a single production facility. Raw material and sawmill resources were allocated by a linear program. The program was constrained by the availability of a limited resource consumed by a cutting strategy. The demand for lumber was modeled as a lower bound on production. This type of market model assumes that an unlimited quantity of any product can be sold without a change in value. The emphasis is placed on producing at least a minimum amount of the product.  3  The most significant development in this model was the use of a subproblem to determine optimal bucking policies. A dynamic algorithm was used to maximize the value recovery of a stem. The product values applied to the algorithm came from the shadow prices of the coordinating linear program. The advantage of using a dynamic subproblem is that it creates only the optimal bucking policy based on current values. An approach based solely on linear programming would require complete enumeration of all the possible bucking policies to achieve a similar result. It may be computationally impossible to calculate all of the possible stem bucking policies. The integrated model also has an advantage over using a dynamic programming (DP) algorithm in isolation. When the DP is a subproblem, the optimal value is based on the marginal values of the products. The marginal value of a product that is under produced will be greater than that of a product that has reached its production quota. As a result, the optimal solution considers market requirements. In contrast, the isolated DP will try to produce the product with the highest sales value; whether or not the product can be sold at the market price. This may result in the over production of high value products. A similar model was developed by Mendoza (1980) to determine the best allocation and bucking of tree length stems to multiple facilities. In this case the product is a mill length log rather than a piece of lumber. The goal of this model was to maximize the value of a stem for the harvester while meeting the demands of the manufacturing facilities for short logs. The model was constrained by the availability of raw material, facility limitations, and market restrictions. In this model the market was constrained from both over production and underproduction. Products were required to meet a given level, but could not exceed an upper limit. This constraint is very restrictive because it does not model the sawmilling  4  environment. Mills must often over or under produce to meet requirements for other products. To maintain feasibility, the production constraints must be very broad. Mendoza used a modified knapsack algorithm to optimize the bucking of logs. The objective of the algorithm was to maximize the sum of the log segment values, minimize waste, and constrain the production of high grade logs from low grade stems. This last constraint is particularly important in a log allocation model because some uses are dependent on the grade of the log. For example, plywood veneer mills require a peeler grade of log for production. Neither McPhalen nor Mendoza used a dynamic algorithm to optimize breakdown methods in the manufacturing facility. This was first done in a model developed by Maness (1991) called the Combined Optimization Model. The model was developed for sawmill production planning. In this model, there are two types of dynamic subproblems: a sawing algorithm and a bucking algorithm. The sawing algorithm is used to determine the sawing pattern which produces lumber with the highest combined value. The product values are the shadow prices of the products in the coordinating problem. The optimal value of each log is then used in the bucking algorithm to determine the optimal bucking for a stem. In this model, the product yield of a given log class is not fixed by empirical study. As with the bucking algorithms included in previous models, the sawing algorithm produces activities for the coordinating LP. The coordinating problem then chooses the combination of activities that result in the optimal solution for the complete model. The optimal set of sawing patterns will change relative to the demands of the market and the availability of raw material. The Combined Optimization Model also handled market constraints in a different manner. Previously, the market for a product was modeled using both upper and lower bounds on production. This model uses four demand levels to represent the market. Each demand level has a separate price. Lumber produced in the first demand group will have a 5  higher value than quantities produced above that demand limit. This method is a piecewise linear representation of the demand and supply model. It does not account for sales that may be lost by missing minimum production requirements. Planning Over Multiple Periods In a study on sawmill shift scheduling, McKillop and Hoyer-Nielsen (1968) developed an LP model to optimize production over several periods. The researchers observed that seasonal market trends could be accommodated by varying production rates or by holding inventory. The goal was to determine what combination of these two variables would maximize profit. The model could be characterized as a series of linear programming models connected together by inventories. Each period was constrained by production capacity in the period, raw material availability, and inventory space for logs and lumber. Any material produced over the market demand was placed in inventory and sold in the next period. The advantage of the multiple period program was the development of market constraints that model production over product sold as inventories. By modeling inventory, the mill could determine optimal production levels over a series of months. This also alleviates the problem of infeasibilities caused by the rigid production requirements used in earlier models. Unlike the models described above, the multiperiod model was strictly a linear program. It does not include the dynamic submodel to select optimal sawing patterns. This approach limits the model's ability to fully represent the production system. Each log class may have many possible sawing patterns, far too many to completely enumerate and list in the linear programming problem. In the McKillop and Hoyer-Nielsen model, short logs were converted to product using a lumber recovery factor. This may be acceptable in a stud mill, but will not work in complex market and product environments.  6  DESCRIPTION OF A MODEL FOR MULTIPERIOD PRODUCTION PLANNING  Introduction Production planning systems were developed to assist decision makers in the process of optimizing production of a mill given the availability of raw material and market demand. Questions in the forest products industry ranged from the optimal bucking and sawing policies for stem classes to determining the optimal allocation of stem segments to multiple facilities. Each of these models makes the assumption that markets and raw material are stable over the planning period. In a model with a planning period under two weeks, this may very well be the case; but it is certainly not true over longer periods. The model described here has been developed to analyze the impact of production planning over multiple periods. The prototype mill is a large-log sawmill which focuses on sawing for grade. For simplicity and clarity, the raw material for the mill is short logs instead of stems. The steps required to include an algorithm for stem bucking are relatively simple. Mill Type: Maximum Hours:  Total Production (MFBM): :  Residue Value (Monne):  Minimum Hours:,  Inventory Cost (S/MFBM):  Sawlines per Hour:,...  Inventory Capacity (MFBM):  Sawing Cost ($/hour): Finishing Cost (S/MFBM): Trim Factor: 6  10^11^12^13  Lumber Lengths  Table 1: General Mill Information In the sections below the model is described both mathematically and by example. The model used in the example is a three period, steady state scenario. In this situation, the inputs used for each period are equal. The production strategy developed corresponds 7  to one where there are no changes in mill capacity, raw material, lumber sizes produced, lumber prices, or market demand. The inputs used to describe the mill are summarized in table 1. Resource Coordinator The production planning system has two components, the resource coordinator which is described below, and a log sawing model. The resource coordinator is a linear programming algorithm designed to choose the optimal level of activities while remaining within resource and market constraints. The formulation is similar to that of Maness and Adams (1991) with modifications to create a multiperiod model. The sawing activity columns are created by a sawing pattern optimization algorithm using shadow prices as the basis for value optimization. The key variables are lumber sold (LumSalesdp), lumber inventoried (LumInvip/), and the log sawing solutions (SawLogd mLc)• The constraints are divided into 4 sections: supply constraints, production constraints, marketing constraints, and inventory constraints. The key constraints are sawing time (equation 6) and raw material availability (equation 2). The model is described as a mathematical formulation below. The equations are described in the section following the formulation. Maximize: {((LumSales did +  EEE d p 1  E InvSales ) * LumPrice  dpi )  — (UnderTarget dpi * UPenalty dpi ) —^{LumInv ipi * InvCost d }}  E {FiberVol FiberPrice } -EE {OverProd *OverPen } -EE {SawTime SawCost _ EE {Finishing „ *FinCost ,„} -EE {BoomUsed *BoomPrice d  d*  P  1  d  m  d  m  d  b  p/  iv  dm *  dm }  d  d  c/5  db  } (1) 8  Subject to: Supply Constraints  E BoomUsed E {LogVol  db  db  (2)  1 3 b^  ,, * BoomUsed ab }^SawLoga.m. = 0 3 (d,L)^(3) mC  Production Constraints  EE {Recovery L C  dmpILC  * SaWLOgdmw —  (Trim. * MillProduction dmpi ) = 0 3 (d,p,l,m) (4)  EE {FiberVol L C  amLC  * SawLo ,,dmLC — FiberProdcution dm = 0 3 (d,m) (5)  E E tsawLo  gdm  SawTime d  MaxTime d  L  ,„ *Hours d„,Lc — SawTime am = 0 3 (d,m)^ (6)  C  „,  3 (d, m)^  (7)  3 (d,m)^  (8)  „,  SawTime dm MinTime  EE {MillProduction  dm  dmpi  } — Finishing  dm  =  0 3 (d,m)^  P 1  (9)  Marketing Constraints  E {MillProduction d  mp i}  LumSales dpi — LumInv fp/ = 0 3 (d,p,1) where d=1,2,...,MaxPeriod-1 t=1^  E {MillProduction  d,„1, /  (10)  } — LumSales dpi = 0 3 (p,1) where d=MaxPeriod^(11)  E {FiberProduction d  m  } — FiberSales  d  =  0 3 (d)  where d=1,2,...,MaxPeriod^(12) LumSales dpi + UnderTarget dpi = Target dpi 3 (d,p,l) where d=1^  (13)  9  E {InvSales d -1  i=i  dipi  } + LumSales dpi UnderTarget dpi = Target dpi 3 (d,p,1) where d=2,...,MaxPeriod-1  (14)  max period-1  E {InvSales  } + LumSales dpi UnderTarget dp, — OverProd = Target dpi  thpi  1=1  3 (4,0 where d=MaxPeriod^(15) Inventory Constraints  EE (Degrade  tpk  1  EEE {LumInv  , * LumInv ipi 1 — InvSales dtqk — LumInv (t+Dqk = 0 3 (d,t,q,k) ^ where t=1,2,...,d-1 (16) ipi  } <= Capacity d 3 (d)  p^1^t  where t=1,2,...,d-1  ^  (17)  Non-negativity on all variables. Subscript definitions: ^ ^ d t^inventory age^L log class period ^ C sawing policy products^m mill k,1  ^  product lengths b boom  Definitions: BoomPricedb  = Cost of boom b in period d ($).  BoomUseddb  = Amount of boom b used in period d (boom units).  Capacityd  = Inventory capacity in period d (MFBM).  Degradevqk FiberPriced  = Percent degrade of product p with length 1 into product q with length k at inventory age t. Value of fiber in period d ($/tonne).  FiberProductiondm  = Fiber produced at mill m in period d (tonnes)  FiberSalesd  = Fiber sold in period d (tonnes)  FiberVoldmLC  = Ratio of fiber recovered from log class L at mill m using sawing policy C in period d (tonnes/m 3 ). The cost of finishing 1 MFBM of product at mill m in period d ($).  FinCostdm  10  Finishingdm HoursdmLC InvCostd InvSalesdq3/ LogVolda  Products to be finished in mill m in period d (MFBM). Time in hours required to execute sawing policy C on log class L at mill m in period d. = Inventory cost in period d ($/MFBM). = Inventory sold at inventory age t of product p with length 1 in period d (MFBM). = Volume of log class L found in boom b in period d (m 3 ).  LumInvip i  Volume of product p with length 1 degraded at inventory age t (MFBM). Value of product p with length 1 sold in period d ($/MFBM).  LumPricedpi LumSalesdpi  = Product p with length 1 sold in period d (MFBM).  MaxPeriod  = Final production period in model.  MaxTimedm  = Maximum operating hours available at mill m in period d.  MillProductiondmp /  = Product p with length 1 produced at mill m in period d (MFBM). • Minimum operating hours available at mill m in period d.  MinTimedm OverPen  Cost of over producing product p with length 1 ($/MFBM).  OverProdp i  Over production of product p with length 1 (MFBM).  Recoverydmp lLC  Ratio of product p with length I recovered from log class L at mill m using sawing policy C in period d (MFBM/m 3 ). Cost of sawing at mill m in period d ($/hour).  SawCostdm  •  SawLogdmif  •  SawTimedm  Volume of log class L cut using sawing policy C at mill m in period d (m3 ). Total sawing hours used at mill m in period d.  Targetdp i  = Product p with length 1 required in period d (MFBM).  Trimm  = Ratio of volume loss at mill m.  UPenaltydp i  Cost of under production of product p with length 1 in period d ($/MFBM). = Product p with length 1 under produced in period d (MFBM).  UnderTargetdp i Objective Function  The objective function is a linear profit maximization equation. Revenue is derived from the sale of lumber and chips. Lumber may be sold either in the period that it is produced (LumSales) or in a future period (InvSales). Chips are always sold in the period  11  produced. In the example scenario, the optimal objective value is $2,880,338. Table 2 shows a summary of the revenue and cost components. Resource costs are divided into several categories: operating cost, raw material cost, and finishing cost. The operating cost is a time related cost based on the number of saw lines required to complete a sawing pattern. Raw material cost is calculated in terms of the price of a log boom. The boom cost incurred in a period is proportional to the amount of the boom used in the period. The finishing cost is a volume based cost charged to all lumber produced in the period that it is produced. A second group of costs, called market costs, is incurred if lumber is under or over produced in any period of the model. All of these costs are volume based. The under production cost may be seen as the expense required to fill an order from another source or as the cost of losing a customer because of unfilled orders. It may be set particularly high if an order must be filled to meet a shipment on a specific date. Unsold lumber placed in inventory is charged with a storage fee which may include administrative costs. The storage fee is also incurred for each period the lumber remains in inventory. A Revenues Production Sales Inventory Sales Chips Costs Raw Material Saw Time Finishing Under Production Inventory Net Revenue  $3,061,219 $0 $216,717  Period B^C $3,053,111 $152,163 $223,315  $3,148,763 $209,743 $225,137  Total  $9,263,093 $361,907 $665,169  ($1,589,077) ($1,494,677) ($1,579,859) ($4,663,612) ($800,000) ($800,000) ($800,000) ($2,400,000) ($182,071) ($61,068) ($60,310) ($60,693) ($53,686) ($47,614) ($160,696) ($59,396) $0 ($1,497) ($1,954) ($3,451) $767,273 $1,017,205 $1,095,860 $2,880,338  Table 2: Solution - Steady State Case An over production cost is incurred only in the final period of the production planning model to represent the cost of carrying inventory into the next planning period.  12  In the example, the over production cost equals the sales price. This is reflected in the inventory cost for period C. Supply Constraints There have been several approaches to the formulation of raw material supply constraints. McPhalen (1978) used a single fixed raw material distribution where the availability of a resource was modeled as a right hand side. In Maness and Adams (1991), raw material input was modeled as a distribution of tree length stems. The LP was permitted to select as much of the stem distribution as required. Period  Boom 1 2 3 4 5 6 7 8 9 10 11 12  A  100.00% 0.00% 0.00% 0.00% 17.91% 0.00% 2.14% 0.00% 65.81% 0.00% 0.00% 0.00%  B^C  0.00% 0.00% 0.00% 0.00% 70.90% 0.00% 48.09% 0.00% 34.19% 0.00% 72.97% 0.00%  0.00% 0.00% 100.00% 100.00% 11.19% 0.00% 49.77% 0.00% 0.00% 0.00% 27.03% 0.00%  Total  100.00% 0.00% 100.00% 48.35% 100.00% 0.00% 100.00% 0.00% 100.00% 0.00% 100.00% 0.00%  Cost  $448,560 $278,530 $274,720 $359,530 $225,370 $394,530 $486,030 $228,760 $448,560 $278,530 $274,720 $359,530  Table 3: Boom Selection - Steady State Case To better model the requirements of a coastal mill, the basic unit for raw material supply used in this model is the boom. The mill may process all or part of a boom. In this model, each boom contains a user defined distribution of short logs classified by top diameter, length, species, source, and grade. The use of a boom may be spread over any period in which it is available (equation 2). The model does not allow for the choice of individual logs from the boom. When a boom is selected for use, all logs from the distribution must be sawn (equation 3). If a fraction of a boom is used in one period, an equal fraction of volume from each representative log class is processed. To model the  13  degrade of booms while in inventory, the boom's volume distribution may be changed for each period. This method of supply formulation yields information about the preferred order for processing booms. Since each boom has a different distribution of logs and log qualities, a particular boom or set of booms may be best for filling current market requirements. For example, Table 3 shows boom selection in the optimal solution. The table may be read as follows: For boom 5, 17.91% of the boom was allocated to period A, 70.90% was allocated to period B, and 11.19% was used in period C. The booms that are chosen are not necessarily the least expensive. Instead, they are booms which maximize. Each period is allocated a different set of booms based on availability, lumber yield, and market requirements. Production Constraints The production constraints model the conversion of raw material into product. Equation 4 models the recovery of a product from a log. The time required to saw a log is modeled in equation 5. Equations 4 and 5 can be interpreted as recovery columns where each column represents the conversion of a particular log class into lumber, by-products, and sawing time. The columns are generated by a sawing subproblem in each iteration of the production planning system. Sawing time is used in equation 6 to constrain the hours of mill operation. The summation of total production (equation 7) is used to determine the finishing costs incurred in the process. Marketing Constraints Several methods have been proposed in the past to model market requirements. McPhalen (1978) used a minimum production constraint; production for each product had to meet a minimum level. Both a minimum and maximum production constraint were used in Mendoza and Bare (1986). Both of these methods place limits on production that do not exist. If minimum production levels for a particular product are not met, the product  14  available can still be sold. It may not be possible to meet minimum product restrictions using the raw material available. Similarly, maximum production levels can be exceeded. Product which cannot be sold at the current price can be inventoried, or it may be sold at a lower price. The production planning system uses targets and costs to regulate production. For each product, a production target is set. If the resource coordinator chooses a solution where the targets are not met for a particular product, costs are incurred. Production under target is modeled in equation 10. The UnderTarget variable is used to pickup the slack between sales and target. A negative cost coefficient in the objective function ensures that under production decreases net revenue. Table 4 shows an example of this method. In the optimal solution, 0.30 MFBM of 16 foot 2x3 are produced in period A. Since that is below the production target of 5.27 MFBM a cost of $89.37 is incurred. While the underproduction cost for one product is small, it accumulates over 201 products in 8 lengths to the total production values shown in the table. Period A Volumes in MFBM^Production^Value^Targets^Penalties  2x3 16' D Clear Total Production  0.30 2836.29  $135 $3,060,950  5.27 4500.87  ($89.37) ($59,408)  Table 4: Under Production Example When lumber is over produced, it is placed in inventory and carried from one period to the next. In the production planning system, any mill production that cannot be sold is placed in inventory (equation 8) and sold in a later period (equation 11). Special cases exist in the first and last periods of the model. In the first period, there is no inventory to sell (equation 10). Starting inventory can be modeled by decreasing the market targets for the first period. In the last period, all production must be sold (equation 9). Unsold ending inventory is combined with unsold production in the OverProd variable (equation 12).  15  Inventory Constraints The resource coordinator tracks inventories by period and by product. This is important in determining the cost of finished goods inventory. As described in the objective function, storage costs are incurred while lumber is in inventory. Equation 13 models a secondary cost, degrade. As lumber sits in the yard, losses in grade and dimension occur due to handling and exposure to the elements. Degrade is characterized by a piecewise linear function. As the inventory age increases (denoted by subscript t), the degrade level increases. These losses are modeled in a series of falldown functions which represent the volume loss to lower grade products. The second part of the equation divides the degraded inventory into product that will be sold to meet production targets in the current period (InvSales) and product that will be held into the next period (LumInv). The amount of lumber that can be inventoried is constrained by equation 14. Volumes in MFBM  Overproduction -Length Loss -Grade Loss Available for sale Sold Remaining Inventory  t= 1  6.35 0.08 0.13 6.14 0.75 5.39  t=2  5.39 0.07 0.11 5.21 5.28 0  Table 5: Product Degrade Example Table 5 shows an example of how this process works for a 8 x 8 13' product produced in period A. At the end of the period in which the product is produced, 6.35 MFBM remain unsold. During the first period in inventory (t = 1), 0.08 MFBM of the product must be trimmed to 8' due to volume loss; and exposure causes a 0.13 MFBM loss to the next grade level. In period B, 6.14 MFBM is available for sale. Since only a portion of the inventory is sold in period B, (0.75 MFBM), the product sits in inventory for a second period (t = 2) and goes through the degrade process again. All inventory is sold in the final period.  16  Sawing Model  The sawing model is a proprietary dynamic sawing algorithm for grade and is based on x-ray scanner data. The inputs to the model are a description of the log and the values, grade rules, and dimensions of valid products. The program processes each log, determines the optimal sawing pattern based on value recovery, calculates sawing time required and the volume of by-products, and outputs the recovery information in the format required for the resource coordinator. Integrating the Models  The method used to integrate the sawing subproblem into the resource coordinator is similar to the one described in Maness and Adams (1991). In general, the objective function of the sawing subproblem is: Maximize  E E E LumSales„ , * gdpg, g  3  period d^  (18)  p^g^1  where ltdpgl is the Lagrangian Multiplier for product p with grade g and length / in period d. In the first iteration of the process, market prices of lumber are used to determine the  value of a particular sawing pattern. The optimal sawing pattern for each log class is transferred into the resource coordinator as a column. In further iterations the Lagrangian Multiplier ( or shadow price ) of the associated constraint (equation 4) in the resource coordinator is used to evaluate each sawing pattern. The multipliers represent the marginal value of the additional production of each product. It is important to note that the shadow prices will be different for each period. This occurs because each period has different raw material mixes, markets, and inventories. As a result, the marginal benefit of producing a particular product is different in each period. To model this, the sawing model is run once in each period for each system iteration. To illustrate this process, figure 1 shows the relationship between the  17  production of a product and the shadow prices of the product at the end of a system iteration. Input Data  ^  Lumber Grades and Sizes Initial Lumber Values Log Descriptions  —110.  Integrated Model^Reports Sawing Pattern Generator A  g*  ro  5" ;1. '1  Raw Material Product Table Mill Constraints Production Targets Product Values Inventory Degrade  Resource Coordinator  Boom Choices Sawing Policies Lumber Sales Production Data Inventory Inventory Sales  Figure 1 At each iteration, a new sawing pattern may be added to the LP matrix for each log class. The resource coordinator determines how much raw material should be allocated to the new patterns for each log class. The iterative process continues until the marginal increase in the objective function is zero. Data Acquisition  It was anticipated early in this project that a model of this type would require a significant amount of data and the ability to easily update the information. An input module based on the familiar interface of the Excel spreadsheet package was developed to expedite data input. The spreadsheet features macros, graphic elements, and buttons to guide the user through the data entry procedure. The data in spreadsheet form was converted to the MPS format using a custom dynamic link library written in C. MPS is a format commonly used for describing mathematical models. Analysis of Iterations  An analysis of the iterations of the model yields interesting information about the optimization process. It also gives insight on how the model works. During the run of the example model, information was gathered on changes in shadow prices, production and inventory levels, sales, and optimal values. A subset of the iterations is analyzed below. 18  The model was run on an MS-DOS based computer equipped with an Intel 486-66 cpu and 32 MB RAM. The XA linear programming package was used to solve the resource coordinator. All program control and data exchange programs were written in C. Using this configuration, the model solved in approximately two weeks at a rate of 3 hours per system iteration. The system described above is considered a minimum configuration for this model.  Change in Optimal Value Steady State Case 3000000 2000000 1000000  c 2 >^0  -a  E -1000000  21^41^61^81^101^121^141^161  o. 0  -2000000 -3000000 -4000000 Iteration Number  Figure 2 The steady state case reached convergence at iteration 163. As shown in figure 2, the optimal value increases significantly in the first five periods; further increases are less dramatic. Changes in the optimal value after iteration 19 are all under $30,000. The number of iterations to convergence is significantly higher than combined optimization models developed in the past. For example, the Maness and Adams model reached convergence at 6 iterations. The cause of the difference between these two models is the difference in the target mill and sensitivity of the sawing program to shadow prices. The sawing model used is infinitely adjustable, so small changes in the shadow prices will cause the calculation of new sawing patterns.  19  ^ ^  The goal of the multiperiod model in the steady state scenario is to maximize net revenue by generating sawing patterns which meet market requirements and by choosing the combination of sawing patterns which produce the required products. In general, this will be done by minimizing finished goods inventory. The price changes driving the generation of sawing patterns are shown in figure 3. For each iteration, the graph shows the shadow price of a product at the beginning of the iteration and the difference between actual and target production of the product at the end of the iteration. At iteration 16, an increase in the shadow price yields an increase in production. The increase in the value of the product makes it more profitable for the model to create patterns with the under produced product. The resource coordinator then chooses these patterns to better meet market requirements. When actual production equals the target (iteration 31), shadow prices begin to decrease.  Relationship Between Production and Shadow Prices 0 -2  700  -4  x-  600  -6  500  2 -8 LL 2 1 - -10^.  400 t co 300  -12 -14 -16  - - - ><- - - over target^—°--- shadow price  U) CO N  200 100  x  i^i -18 ^  E  1111111111111111111 ^0  op 0) 0^N CO d• U) OD N CO 0) 0^N CI szt U) CD ^ N N N N N N N N N N CI Fl Cn Cl C'') CO C)  Figure 3 In a scenario where market demand and prices are constant, the focus of the model is on hitting the market target for each period without under or over producing. This is difficult to accomplish in early iterations because there are few sawing patterns to choose from for each log. As the number of sawing patterns created for a specific market  20  scenario increases, the resource coordinator chooses patterns which better meet targets. Figure 4 shows the effect of improved sawing pattern selection on sales from inventory. Since there is a cost for holding inventory to the next period, inventory sales have a lower value than production sales. Inventory sales decrease as the model improves the planned production process.  Figure 4 The relationship between ending inventory, production sales, and inventory sales can be seen by comparing figure 4 with a graph of production sales (figure 5). As the selection of sawing patterns improves, the resource coordinator is able to sell a greater proportion of production. This also effects the volume of production available for inventory sales in later periods. Since an increasing proportion of production is sold immediately, less is available for sale in later periods. For example, between iteration 14 and 15 there is a decrease in production sales in period B. This results in an increase of the ending inventory of that period. Because of the increase in period C of inventory available, the sale of inventory in period C increases.  21  Iteration Effect- Production Sales 2850 „. 2800 -  •^-  2750 2700 2650 2600 ^Period A  2550 2500 ' 13  ^  18  ^  23  ^  Period B  28  --- ---^Period C  ^  33^38  Iteration  Figure 5 Conclusion  Production planners and decision makers continue to look for tools which will assist in the process of developing production schedules. Models developed in the past focused on developing production strategies for current product prices and market demands, but ignored the future implications of those decisions on inventory and future sales. To respond to this need, a model was formulated which extends production planning in sawmills to multiple periods. With the Multiple Period Production Planning System developed in this paper, decision makers can explore market scenarios which take into account expected trends in product value and demand, and determine the impact of those scenarios on sawmill production and net revenue. A sample run of the model based on a steady state scenario was used to describe the operation of the model and the information generated. The operation and output were consistent with the expectation of constant production levels and minimization of ending inventory. In the next chapter, the Production Planning System will be applied to three different market scenarios to develop a series of production plans for changing markets.  22  This will further test the effectiveness of the system and provide additional information on its capabilities. Literature Cited  Maness, T. C. and D. M. Adams 1991. The combined optimization of log bucking and sawing strategies. Wood and Fiber Science 23(2):296-314. McPhalen, J. C. 1978. A method of evaluating bucking and sawing strategies for sawlogs. M. Sc. Thesis, University of British Columbia, Vancouver, B.C., Canada. Mendoza, G. A. and B. B. Bare 1986. A two-stage decision model for log bucking and allocation. Forest Products Journal 36(10):70-74.  23  IV. THE EFFECTS OF MARKETS ON MULTIPLE PERIOD PRODUCTION PLANNING  Introduction  The focus of the Multiple Period Production Planning System developed in the previous chapter is on maximizing profits by meeting market demands at the highest possible prices. This is done through two sets of key variables, production and inventory. By forecasting the two market components, demand and price, a strategy for each production period can be developed. In this chapter, three different market cases are analyzed using the model. The first and second cases show the changes in production and inventory strategy when lumber prices fluctuate over time. The third case models a change in market demand for a specific product group. These cases are compared to a control which models a situation where market demand and product values remain constant from period to period. Experimental Design  The design parameters for these experimental cases are divided into two areas: general parameters and scenario specific parameters. All inputs are consistent with a modern sawmill located on the coast of British Columbia. The mill modeled in these cases processes full length logs delivered in booms sorted by species and log grade. Each species and grade combination is sawn for a specific market. The processing system includes computer controlled headrig and edger systems. The optimizers at each machine center use data supplied by x-ray scanners to identify defects and obtain maximum value recovery. A market is composed of a list of products to be produced and a production target for each product. The market list chosen for the example cases includes 201 products in 8 product classes and 5 grades. For the purposes of discussion, the product classes will be labeled A through F where A is the smallest thickness and F is the largest. The product  24  mix can be characterized as a selection of large Hemlock products aimed at the Japanese export market. The general mill parameters for each period are shown in the previous chapter. All three periods are 10 days long with a maximum production run of 21.5 hours per day. The minimum production run per day is 16 hours. The periods represent a production run of a specific log class for a specific market. These production runs are normally set one month apart. In these case studies, the underproduction cost is 4% of product value. The inventory cost is set at $7.50 per MFBM to reflect the cost to store the lumber into the next period, one month later. Products which are placed in inventory are often exposed to the elements where degrade may occur both in dimension and product grade. Table 6 shows an example of degrade loss factors for thickness class C. The first line shows a B clear product with a 20 inch width. Two percent of the product in inventory will degrade to C clear by the end of the next production period. The remainder of the line shows length losses. One percent of the 8 foot product in inventory goes to chips. All other lengths are trimmed down one length class. For example, 1.3% of the 13 foot product is trimmed to 12 feet. Product Degrade  Product B CX20 B CX18 B CX16 B CX14 B CX12 B CX10 B CX8 B CX6 B CX5  %Degrade^To  2.00% 2.00% 2.00% 2.00% 2.00% 2.00% 2.00% 2.00% 2.00%  C CX20 C CX18 C CX16 C CX14 C CX12 C CX10 C CX8 C CX6 C CX5  Length Fa//down  8^10^12  1.0% 1.0% 1.0% 1.0% 1.0% 1.0% 1.0% 1.0% 1.0%  1.1% 1.1% 1.1% 1.1% 1.1% 1.1% 1.1% 1.1% 1.1%  1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2% 1.2%  13  14  16  18  20  1.3% 1.3% 1.3% 1.3% 1.3% 1.3% 1.3% 1.3% 1.3%  1.4% 1.4% 1.4% 1.4% 1.4% 1.4% 1.4% 1.4% 1.4%  1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5% 1.5%  1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6% 1.6%  1.7% 1.7% 1.7% 1.7% 1.7% 1.7% 1.7% 1.7% 1.7%  Table 6: Degrade Table Raw material is supplied to the mill in booms sorted by species and log grade. To maintain reasonable run times it is neither possible nor desirable for the sawing algorithm  25  to process each log in each boom. Booms are processed using a sample of 30 log types from 15 top diameter classes and 3 length classes. Diameter classes range from 17 inches to 32 inches. Short logs enter the mill in 13, 16, 18, and 20 foot lengths. Chip production is considered a residual and is not included in the objective function of the proprietary sawing pattern optimization program. The amount of chips produced by a sawing pattern is calculated using the equation below.  (Log Volume (m 3 ) *%Fiber Loss -Lumber Recovery (MFBM / m 3 ) * Log Volume (m 3 )*2.358 (m 3 / MFBM))(1) *Density Hemlock (tonnes / m 3 ) Case Specific Inputs and Expectations The base case is defined and analyzed in the preceding chapter. In general, it represents a situation where all mill and market parameters are constant throughout the three period production planning cycle. The product price inputs and market targets are abstracted from historical mill data. Thickness Class A  B C D,E,F  Price Change (%) Period B^Period C 12.5 7.0 9.7 13.7  7.4 3.3 3.2 2.3  Table 7: Historical Price Fluctuations In case 1, only product prices change between periods. The lumber prices from the previous case are used for period A. Price fluctuations as reported in Madison's Canadian Lumber Reporter between December 18, 1992, and February 26, 1993, are used as indexes for periods B and C. The relative changes in price for each thickness class are shown in Table 7. When prices are increasing, the model would be expected to react by building inventories to take advantage of more favorable selling prices. 26  A second scenario describes a situation where prices are decreasing. The historical market for Hemlock export products has been steadily, but slowly, increasing since the middle of 1990. A slight fall in value in September of 1990 was preceded by steady increases; therefore, it is difficult to place a general decrease in prices in a historical perspective. For the purposes of this experiment, the price changes from table 7 will also be used to index price decreases. In this scenario, the model would be expected to react by decreasing production. There should be no reward for collecting inventories. The third case focuses on changes in market requirements for a specific product group. In this model, market targets are represented as a percentage of total production. Added demand leads to higher production targets for the affected products. As a result, production targets will be higher than production capacity. To illustrate this process, the demand for thickness class C will increase by 3% in period B and by an additional 5% in period C. The model should react to these changes by selecting a different combination of booms or by generating a different set of sawing patterns or both. If the change in demand is large enough, there may also be an increase in inventory to meet future requirements. To offset the increased production of thickness class C, other products will be under produced and under production costs for those products will increase. Tuning the Model Before running the model to convergence, it was necessary to "tune" the inputs used by the model. This step verifies that the inputs are consistent with the mill scenario being modeled. The early runs of the model showed that large quantities of lumber would be produced and inventoried for sale unless the cost of over production was very high in the last period. When the over production costs are low in the final period, the model is able to obtain revenue by selling significant amounts of lumber at very low cost. In reality this lumber would stay in inventory and incur further degrade and inventory costs. To fix  27  this problem, the over production cost was set equal to the sales price. In this way, the building of inventories is discouraged. The final inventories could be included in the next run of the model by decreasing market targets in the next model's first period. Early runs indicated that the grade yields from the sample logs did not match the production output of the target mill. It was not possible to add additional logs to the sample, so the grade sawing function of the sawing algorithm was disabled. To determine product grade, the sawing patterns for each log were applied to a grade distribution table. The grade distributions used in these case studies are based on historical production data. The tuning process permitted the analysis of the sawing algorithm for accuracy. The analysis indicated a significant under production of merch grade products in thickness class B. Further investigation indicated that the cant algorithm was not cutting these products. To remove the effects of this problem on underproduction costs, the merch grade products from thickness class B were removed from the model. Computational Experience As stated in the previous chapter, the base case was run for 162 system iterations. The sawing patterns generated during the base case system iterations were preserved for use as a starting set for each of the other three cases. This has the effect of decreasing run time without altering the validity of the solution. The resource coordinator will continue to select sawing patterns which improve the objective value. To compare the effect of changing markets on inventory and production policy, each of the three cases was run for 14 system iterations. The sawing model was not run in the first system iteration to determine the effect of the starting set of sawing patterns on the objective value. Results and Analysis This section summarizes the results of each of the three example cases. A full report of the solution of each case may be found in the appendix. For comparison, the  28  solution to the base case is found in table 2 in the preceding chapter. The summary is followed by an analysis of the methods used by each case to take advantage of the changes in product price and market demand. Period B^C  A  Revenues Production Sales Inventory Sales Chips Costs Raw Material Saw Time Finishing Under Production Inventory Net Revenue  $2,535,264 $0 $215,116  $3,195,974 $527,953 $222,518  Total  $3,766,336 $767,249 $232,081  $9,497,573 $1,295,201 $669,715  ($1,434,402) ($1,633,929) ($1,651,014) ($4,719,345) ($800,000) ($800,000) ($800,000) ($2,400,000) ($61,294) ($61,753) ($61,581) ($184,627) ($100,856) ($69,554) ($41,231) ($211,641) ($5,217) ($5,629) $0 ($10,845) $348,611 $1,375,581 $2,211,840 $3,936,032  Table 8: Solution - Price Increase Case The price increase case showed a significant increase in the objective value and a clear change in production and inventory policy. A summary of revenues and costs for the final system iteration of the price increase case is shown in table 8. The inventory cost for period C is always zero because there is no carrying cost.  Volumes in MFBM  Total Production Production Sales Inventory Sales New Inventory Ending Inventory  Price Increase A^B  3064.69 2369.15 0.00 695.54 695.54  3087.63 2618.53 413.93 469.10 750.49  Price Decrease A^B  3079.04 3014.15 661.54 64.89 153.32  3066.17 2920.1 0.00 146.07 146.07  3050.12 2895.28 89.42 154.84 211.25  C 3023.21 2974.06 130.75 49.15 129.22  Table 9: Production and Inventory Policy The change in inventory policy matches the expected result. Earlier periods build inventory in order to sell it when prices are higher. In this case, 22.7% of period A production volume goes into inventory compared to 6.6% in the base case. There is also an increase in inventory sales; in period B inventory sales increase by a factor of 5. Similar changes occur in periods B and C. Note that the production plan accepts low revenues  29  and high under production costs in period A in order to take advantage of the higher prices in periods B and C. Table 9 summarizes the inventory and sales policy for this case. Period B^C  A Revenues Production Sales Inventory Sales Chips Costs Raw Material Saw Time Finishing Under Production Inventory Net Revenue  $3,165,708 $0 $220,615  $2,728,179 $81,609 $219,629  $2,693,444 $124,354 $226,718  Total  $8,587,331 $205,963 $666,962  ($1,653,544) ($1,455,676) ($1,573,146) ($4,682,366) ($800,000) ($800,000) ($800,000) ($2,400,000) ($182,790) ($61,002) ($60,464) ($61,323) ($55,249) ($48,246) ($42,464) ($145,959) ($1,584) $0 ($2,680) ($1,095) $662,908 $568,443 $2,046,461 $815,110  Table 10: Solution - Price Decrease Case The solution to the price decrease case is shown in table 10. In general, the production plan is the opposite of the previous case. As shown in table 9, both inventories and inventory sales are lower than the price increase case for all periods. Both under production and inventory costs are lower for all three periods because there is no incentive for collecting inventories. This is also true when comparing this case to the base case. In the base case there is also no incentive for inventories, but the price decrease case adds the additional penalty of lower sales prices for future periods. Period B^C  A Revenues Production Sales Inventory Sales Chips Costs Raw Material Saw Time Finishing Under Production Inventory Net Revenue  $3,082,935 $0 $219,168  $3,090,256 $139,347 $225,242  $3,186,006 $221,904 $229,418  Total  $9,359,196 $361,250 $673,828  ($1,620,395) ($1,500,816) ($1,597,901) ($4,719,111) ($800,000) ($800,000) ($800,000) ($2,400,000) ($61,518) ($60,954) ($61,476) ($183,948) ($58,537) ($53,252) ($158,881) ($47,092) ($1,633) ($2,080) $0 ($3,713) $760,062 $1,037,178 $1,131,381 $2,928,621  Table 11: Solution - Demand Increase Case  30  The scenario focusing on changes in demand requires a detailed look at the production policies for individual lumber thickness classes. Table 11 shows a solution summary that is very similar to the base case. The demand change case has a slightly higher value of production and inventory sales. Table 12, shows how production sales for each product class in case 3 differ from the base case. Thickness classes A, D, and E show very little difference from base case production levels. It is class C which shows the effect of changing demand. The model did not decrease production of other product classes in order to increase the production of thickness class C. A  A B C D E F  0.35% 3.64% 0.51% -0.05% 0.60% 1.89%  B  0.21% 1.45% 3.72% 1.18% 1.26% 0.88%  C  -0.01% 2.38% 9.60% -1.11% 0.46% 0.29%  Total  0.15% 2.16% 4.59% -0.06% 0.74% 1.08%  Table 12: Production Sales as a Percent of Base Case - Demand Increase Case Each of the cases described above reacts to market conditions by setting a production and inventory policy for each period. The model has three ways of altering the production policy: production schedules, raw material consumption and distribution, and sawing pattern creation and selection. Each of these methods will be described in the following sections with examples from the experimental cases. The first method for altering production requires a change in the production schedule. By increasing the time available for production, the mill can either increase the inventory of products to take advantage of future price increases or supply the increase in demand. This method was not used in any of the four cases, the solution recommended the minimum operating schedule, 160 hours for each period. As a result, the total production levels are relatively constant for all cases. The reason for the limit on the production schedule can be seen by looking at the shadow prices on the sawing time constraints. In case 2, the shadow price is $4.11 for period A, $2.58 in period B, and  31  $1.71 in period C. One additional unit of time added to the production schedule incurs additional operating and inventory costs. In this case those costs are higher than the revenue that could be derived. The second method is a change in the consumption and distribution of booms. A particular boom may have a distribution of logs that is better able to meet the market requirements of the case. The cases modeling price fluctuation show an increase in raw material consumption and a change in distribution. The price increase case consumed 452.64 m3 more raw material than the base case. Most of the additional raw material came from boom 12. There were also other changes in the distribution of raw material to production periods. In the base case, boom 5 had been divided between all three periods with the majority going to period B. The same boom is completely consumed in period A in the price increase case. Period  Boom 2 3 4 5 6 7 8 9 10 11 12  A  91.08% 0.00% 0.00% 0.00% 0.00% 0.00% 33.33% 0.00% 57.12% 0.00% 0.00% 0.00%  B^C  8.92% 0.00% 96.10% 0.00% 81.16% 0.00% 49.58% 0.00% 0.00% 0.00% 0.00% 0.00%  0.00% 0.00% 3.90% 0.00% 18.84% 0.00% 17.09% 0.00% 42.89% 0.00% 100.00% 100.00%  Total  100.00% 0.00% 100.00% 0.00% 100.00% 0.00% 100.00% 0.00% 100.00% 0.00% 100.00% 100.00%  Cost  $448,560 $278,530 $274,720 $359,530 $225,370 $394,530 $486,030 $228,760 $448,560 $278,530 $274,720 $359,530  Table 13: Boom Distribution - Price Decrease Case Table 13 shows the boom distribution for the second case. A difference in boom distribution can be seen by comparing the base case to the solution for the price decrease case. In the base case (shown in table 3), 65.81% of boom 9 was consumed in period A and 34.19% was consumed in period B. The price decrease case shifted consumption of boom 9 from period B to period C. This case shows allocation shifts for almost all booms.  32  Unexpectedly, the price decrease case used 154.88 m 3 more raw material than the base case. While it is difficult to determine what caused this increase, it could be theorized that the change in consumption was necessary to take advantage of sawing patterns which reduce inventory. Each of the cases used the third method for changing production policy to develop a production plan, sawing pattern creation and selection. The effect of sawing pattern selection alone can be seen by comparing the results of the first system iteration of the demand increase case with the final iteration of the base case. The solutions of these iterations showed a difference of $39,724.31 in the objective value. Both iterations use the same set of candidate sawing patterns, the same product values for all periods, and the same raw material distributions, so any difference in the objective value is due to a different optimal selection of sawing patterns in the optimal solution. The combined effect of a change in the selection of booms and sawing patterns can be illustrated using the first system iteration of the price increase case. The difference in optimal value between the first case and the base case was $1,055,694. It includes a change in production and inventory policy and a change in market prices for periods B and C. The two components of the difference can be isolated by applying the product values of the price increase case to the production and inventory policy of the base case. The resulting objective value indicates the net revenue obtained if the mill did not respond to the market by changing policies. This analysis indicated that $909,264 of the difference was due to the increase in lumber prices alone. The remainder, $146,430, can be attributed to a change in the production policy. The final solutions of each case used sawing patterns from both the starting set and new patterns created in the system iterations of each case. In period A of the demand increase case, 8% of log volume was cut with new sawing patterns. Patterns from the base case accounted for the remaining 92%. The new sawing patterns account for 7% of log volume in period B and 10% in period C. The price increase case uses a different 33  proportion of sawing patterns. In period A, the logs were processed with the sawing patterns created specifically for this case account for 17% of total log volume. 83% of log volume was processed using the starting set of sawing patterns. Periods B and C show similar relationships. Conclusion The four cases described and analyzed in this chapter indicate that gains in net revenue can be obtained using a multiple period planning system. By considering the implications of changes in the market environment, better sawing decisions for the current period can be made. In general, production and inventory policies were not changed by adding shifts at the sawmill. Instead, the model improved the objective value by selecting a different boom schedule to meet production requirements and a different set of sawing patterns for each case. The analysis of the model shows that the effect of these changes can be significant. Using this analysis, the decision maker will be able to make better production decisions which take into account the entire market environment. Literature Cited Cargo and Export Report. Madison's Canadian Lumber Reporter 42(50). Cargo and Export Report. Madison's Canadian Lumber Reporter 43(4). Cargo and Export Report. Madison's Canadian Lumber Reporter 43(8).  34  V.^CONCLUSION In a time of fluctuating markets and prices, it is necessary for sawmill planners to look farther into the future when developing production plans. By changing production and inventory policies to match market forecasts, it may be possible to realize significant gains in net revenue that would not be attained by maintaining a constant approach. Sawmill production planning model development has focused on single periods where market demands and prices are constant. The most recent example of this is the combined optimization model described by Maness and Adams. Their method uses the decomposition principle to combine a linear program for resource allocation with dynamic bucking and sawing subproblems. To meet the requirements of sawmill decision makers, a new model was formulated for multiple periods. By combining information on mill operation and product mix, with forecasts of raw material availability, expected changes in market demand, and expected price fluctuations; the model is able to determine a production plan over multiple time periods which maximizes net revenue. This model merges the combined optimization approach with the multiple period linear programming method developed by McKillop and Hoyer-Nielsen. The most significant advance in this model is the addition of log and lumber inventories to the production planning decision. The use of inventories models the effect of current production decisions on the ability to meet future market requirements. All lumber produced above the market targets for each period is place into inventory and carried until sold. Inventory costs are calculated on a volume and degrade basis. Information on log inventories may be used to determine optimal boom selection and scheduling over the planning cycle.  35  It was shown through the use of four sample cases that the model responds appropriately to different market environments. The cases were based on a large log sawmill producing lumber for the Japanese export market. A steady state case was used to verify the correct operation of the model. As expected, the solution identified a steady production policy where inventories are minimized and production is sold in the period that it is produced. The same case was used to identify the effects of shadow prices and pattern selection in the system iteration process. A lumber value increase case was put in the historical perspective of price changes in early 1993. To allow the mill to take advantage of expected price increases, the model increased inventories in early periods and deferred product sales to the final period. The use of this production policy resulted in an increase in net revenue of nearly $1 million over the steady state case. The model was also tested in market environments where lumber values fell during the planning cycle and in a situation where product demands changed. In all four cases, the production strategy was altered by changing the selection of sawing patterns and creating new ones. This allowed the mill to alter inventory policy and take advantage of the best market conditions. While this model has successfully fulfilled the objectives of this study, there are some additional steps that must be taken before it becomes a fully useful tool. In a mill which produces over 200 different products in multiple lengths it is very difficult to obtain information on market targets and changes in the market environment for each product. To receive the full benefit of a model of this type, it will be necessary to develop a more rigorous method for estimating lumber prices and market demand. There may also be some rational procedure for aggregating similar products to create a smaller number of product classes. For example, many products are sold in random lengths or widths. Products sold in these bundles could be aggregated at the market constraint level with one 36  market target. Using this method, the sawing model would still solve for each individual product. It is important to avoid over aggregation because it is the market targets which drive product differentiation through shadow prices. Traditionally, lumber sales have been driven by mill production. This model allows the opposite to occur if reliable market information is available. As demonstrated in the example cases, market driven planning can be accomplished by testing a variety of possible market scenarios and selecting an average case, or by analyzing past performance. This type of decision-making tool will allow managers to more accurately prepare for expected market trends by taking a long term view of raw material resources and production levels.  37  BIBLIOGRAPHY  Cargo and Export Report. Madison's Canadian Lumber Reporter 42(50). Cargo and Export Report. Madison's Canadian Lumber Reporter 43(4). Cargo and Export Report. Madison's Canadian Lumber Reporter 43(8). Cohen, D. H. 1992. Adding Value Incrementally: A Strategy to Enhance solid Wood Exports to Japan. Forest products Journal 42(2):40-44. Dantzig, G. B. and P. Wolfe 1960. Decomposition Principle for Linear Programs. Operations Research 8:101-111. Maness, T. C. and D. M. Adams 1991. The Combined Optimization of Log Bucking and Sawing Strategies. Wood and Fiber Science 23(2):296-314. McKillop, W. and S. Hoyer-Nielsen 1968. Planning Sawmill Production and Inventories Using Linear Programming. Forest Products Journal McPhalen, J. C. 1978. A method of Evaluating Bucking and Sawing Strategies for Sawlogs. M. Sc. Thesis, University of British Columbia, Vancouver, B.C., Canada. Mendoza, G. A. 1980. Integrating Stem Conversion and Log Allocation Models for Wood Utilization Planning. Ph. D. Dissertation, University of Washington, Seattle, Washington. Mendoza, G. A. and B. B. Bare 1986. A two-stage decision model for log bucking and allocation. Forest Products Journal 36(10):70-74.  38  APPENDIX  Solution Reports of Example Cases  39  ^  Base Case: Steady State Period A  z„„s„,„,-way.„^RICERUMOMAREZEMMUZIMMOV 958.89 295.76 308.89 725.79 409.64 136.66 2835.62  $590,712.09 $283,812.11 $405,481.57 $932,011.05 $616,744.37 $232,457.54 $3,061,219  1355.86^($3,920.97) 1121.84 ($29,710.68) 338.85^($1,537.22) 816.72^($4,729.68) 584.10 ($10,998.72) 283.50^($8,498.70) 4500.87^($59,396)  - ($1,589,077)  40  Period B  AVIDAMMENEL: 71111112 ..311E.PLOMOUNROINIEUTS -  .--  959.00 423.37 257.24 697.90 384.39 139.26 2861.16  $587,354.29 $406,979.97 $339,780.61 $897,993.10 $585,611.76 $235,391.06 $3,053,111  12.05 11.58 60.58 17.25 29.18 0.42 131.06  $6,863 $9,156 $77,271 $20,362 $37,824 $689 $152,163  1355.86 1121.84 338.85 816.72 584.10 283.50 4500.87  -  ($3,783.38) ($24,466.21) ($1,080.08) ($5,268.92) ($10,731.54) ($8,355.60) ($53,686)  ($53,686)  $3,374,904  ($1,494,677) 160.00  ($5,000)  ($800,000)  3053.40  ($20)  ($61,068)  ^ 68.25263 192.3095  ($7.50)  ( $7 .50 )  ($512) ($1,442) ($2,357,699) $1,017,205  41  Period C  01,11NViiAav, 1:WitiWt1  978.31 483.69 271.39 682.36 400.10 140.56 2956.41  $591,012.92 $462,877.76 $366,156.45 $875,781.04 $615,317.31 $237,617.57 $3,148,763  13.73 17.02 53.77 33.99 58.09 0.00  $41,921 $83,933 $0  176.60  $209,743  1355.86 1121.84 338.85 816.72 584.10 283.50 4500.87  ($3,588.17) ($22,055.38) ($675.52) ($5,295.71) ($7,704.84) ($8,294.48) ($47,614)  $8,254 $14,506  $61,130  ($47,614)  $3,536,029  .:WPOr ($1,579,859) 160.00  ($5,000)  ($800,000)  3015.50  ($20)  ($60,310)  24.66 22.72 55.75 24.23 14.14  EllamandiVETRISERNittnag $128,912^($128,912)  (31 . 31 141.81^  ($2,440,169) $1,095,861  42  ^  Totals 131/0411.4 -‹* •  ts  v my  *^'  2896.20 1202.82 837.52 2106.04 1194.12 416.48 8653.19  ti  $1,769,079.30 $1,153,669.85 $1,111,418.63 $2,705,785.19 $1,817,673.44 $705,466.17 $9,263,093  4067.59 3365.52 1016.56 2450.16 1752.29 850.49 13502.61  ($11,292.52) ($76,232.26) ($3,292.83) ($15,294.31) ($29,435.10) ($25,148.78) ($160,696)  25.78^$15,117 28.59^$23,662 114.35^$138,401 51.24^$62,282 87.27^$121,757 0.42^$689 307.66^$361,907  ($160,696)  $10,129,472  . ($4,663,612) 480.00  ($2,400,000)  9103.55  ($182,071)  460.11 141.81  ($3,451) $0 ($7,249,134) $2,880,338  t t^1'00^t.^e-t tt t^t^r'^-t  24.66 22.72 55.75 24.23 14.14 0.31 141.81  43  Case 1: Increasing Lumber Prices Period A  $575,778.22 $79,743.36 $392,380.13 $846,862.13 $469,985.15 $170,514.54 $2,535,264  1355.86 1121.84 338.85 816.72 584.10 283.50 4500.87  ($5,776.68) ($47,575.18) ($2,582.89) ($10,154.20) ($21,027.99) ($13,739.47) ($100,856)  ($100,856)  n  $2,649,523  ($1,434,402) ($800,000) ($61,294)  ($5,217) ($2,300,912) $348,611  44  Period B ........................ ...... ..  903.39 $635,699.92 266.33 $275,726.50 249.82 $363,160.14 689.85 $1,007,110.72 372.40 $649,486.67 136.74 $264,789.89 2618.53 53,195,974 53.47 131.98 71.22 45.10 95.82 16.34 413.93  1355.86 1121.84 338.85 816.72 584.10 283.50 4500.87  ($5,532.81) ($35,828.38) ($1,266.64) ($6,263.77) ($10,070.29) ($10,591.81) ($69,554)  $29,626 $142,482 $98,810 $62,239 $163,600 $31,198 $527,953  4450.36 ($69,554)  160.00  ($5,000)  ($800,000)  3087.63  ($20)  ($61,753)  ($7.50) ($7.50)  ($2,110) ($3,518)  $3,876,891  126.28 341.00 86.72 90.24 89.96 16.29 281.3931 469.0988  ($2,501,310)  Mg%  $1,375,581  45  ^  Period C  zzlIgiAzzrtAlItyezzaws0;  era  -  968.16 $703,084.22 524.01 $582,197.57 270.38 $412,474.40 678.91 $1,019,927.25 428.38 $763,227.07 144.29 $285,425.05 3014.15  $3,766,336  1355.86 1121.84 338.85 816.72 584.10 283.50  ($4,145.24) ($14,616.71) ($758.61) ($4,757.39) ($6,846.24) ($10,106.36)  4500.87  ($41,231)  ^112.48^$48,172 322.96^$367,376 58.31^$75,371 76.67^$107,313 75.12^$137,818 16.00^$31,198 ^661.54^$767,249  23.76 30.72 47.99 24.41 26.03 0.40 153.32 ^$165,129^($165,129) s  46  * •  esiar i • :::".mumnr7.7.3aeso..44.....7.....I....§..."„1:ttatzta-41:::::11:19::::::::.::as ,,  -.  2776.20 881.78 818.07 2021.34 1123.24 381.21 8001.83  $1,914,562.36 $937,667.43 $1,168,014.66 $2,873,900.10 $1,882,698.89 $720,729.47 $9,497,573  165.94 454.94 129.53 121.77 170.94 32.35 1075.47  $77,798 $509,858 $174,181 $169,551 $301,418 $62,396 $1,295,201  13394.42  $669,715 $11,462,490  ,,  4067.59 3365.52 1016.56 2450.16 1752.29 850.49 13502.61  ($15,454.73) ($98,020.26) ($4,608.14) ($21,175.36) ($37,944.53) ($34,437.64) ($211,641)  ($211,641)  $11,250,849 •"'":?.K:;;;;;;.;;;;:.  ($4,719,345) 480.00  ($2,400,000)  9231.36  ($184,627)  1446.03 153.32  ($10,845) $0 ($7,314,817) $3,936,032  23.76 30.72 47.99 24.41 26.03 0.40 153.32  47  ^  Case 2: Decreasing Lumber Prices Period A n  ••  ^959.15 $591,036.82^1355.86^($3,907.68) 342.22 $330,399.20^1121.84 ($27,869.16) 312.77 $410,505.82^338.85^($1,338.72) 726.32 $930,850.83^816.72^($4,775.57) 432.82 $654,430.60^584.10^($9,495.30) 146.83 $248,484.51^283.50^($7,862.80) 2920.10 $3,165,708^4500.87^($55,249)  48  Period B ,...aaae •  967.98 430.32 290.96 669.65 398.39 137.99 2895.28  $516,686.62 $386,531.75 $347,267.53 $751,920.42 $523,593.63 $202,179.12 $2,728,179  9.87 10.54 27.52 23.48 17.89 0.12 89.42  $5,030 $6,822 $30,285 $20,804 $18,533 $134 $81,609  1355.86 1121.84 338.85 816.72 584.10 283.50 4500.87  ($3,250.13) ($22,423.31) ($936.66) ($5,347.27) ($9,033.01) ($7,255.81) ($48,246)  4392.58 ($48,246) ,ti,:••••••••••••••  iLlabant:Naili:::0 ( $1,455,676 )  160.00  ($800,000)  3050.12  ($61,002)  56.4135 154.8358  $2,981,170  ($7.50) ($7.50)  ($423) ($1,161) ($2,318,262) $662,908  49  Period C  „ „IIEBEL„„.„..aeaaKzea„—„Lhattniti: . 72REMINIMEK 983.81 502.79 284.79 645.58 423.20 133.89 2974.06  $479,912.89 $429,574.82 $333,235.95 $708,796.12 $550,456.12 $191,468.21 $2,693,444  13.30 16.41 37.17 31.36 32.12 0.39  $6,595 $13,495  130.75  1355.86 1121.84 338.85 816.72 584.10 283.50 4500.87  ($2,800.15) ($19,188.13) ($739.29) ($5,783.22) ($6,619.13) ($7,333.76) ($42,464)  $36,519 $31,684  $35,507 $556 $124,354  ($42,464)  Papr:.*:Fs::"" tt  -44k.ZZ.ZZZ&;:k.kaggsWM-Z.4402...‘iatt.C.:IZZZ::  $3,002,053  „ AMMV:RUITIO.W.;  ^($1,573,146)  20.11 26.60 54.67 21.72 5.90 °. 24 129.22  $99,317^($99,317)  50  ^  REF  SAS  2910.93 1275.33 888.52 2041.55 1254.40 418.71 8789.44  $1,587,636.32 $1,146,505.77 $1,091,009.30 $2,391,567.37 $1,728,480.35 $642,131.84 $8,587,331  4067.59 3365.52 1016.56 2450.16 1752.29 850.49 13502.61  ($9,957.96) ($69,480.59) ($3,014.67) ($15,906.07) ($25,147.44) ($22,452.37) ($145,959)  ^23.17^$11,624 26.95^$20,317 64.69^$66,804 54.85^$52,488 50.00^$54,039 0.51^$691 220.18^$205,963  ($145,959)  ($4,682,366) 480.00  $9,314,297 ,,,,,,,,,,,,  ($2,400,000)  9139.49  ($182,790)  357.32 129.22  ($2,680) $0 ($7,267,836) $2,046,461  20.11 26.60 54.67 21.72 5.90 0.24 129.22  51  Case 3: Product Demand Change  Period A  AtiVIgr'llattiONEKISMISrr %P.;\^Wzki  962.28 $592,141.07 306.53 $294,356.24 310.48 $407,668.09 725.42 $931,012.44 412.11 $620,920.92 139.24 $236,835.83 2856.07 $3,082,935  1355.86 1121.84 338.85 816.72 584.10 283.50 4500.87  ..... ...  ($3,865.10) ($29,294.72) ($1,450.83) ($4,769.14) (S10,832.05) ($8,324.83) ($58,537)  $0  $0 $0 $0  $0  ($58,537) Mi•Mr`r•r..•`," MMM.^ •  sm.  $3,243,566  (621,1144.114.11EPalarkg  ($1,620,395) \\*.:11 :06ktrtv‘k.***kstss'in.:' s :\\ %1^.‘  160.00  ($5,000)  ($800,000)  3073.80  ($20)  ($61,476)  217.7286  ($7.50)  ($1,633) ($2,483,504) $760,062  52  Period B SS,  I  ' ' ::::::::::::::::::::::::: ktaZAAZA*Zakt.kka  •  : ::::::: ::::::::::::::::::::::::::::::::::  961.04 429.49 266.80 706.13 389.24 140.49 2893.20  $586,569.74 $412,141.13 $351,046.04 $907,696.50 $595,287.15 $237,515.08 $3,090,256  13.71 11.25 61.04 11.19 25.56 0.06 122.81  $8,564 $9,163 $79,137 $12,813 $29,592 $78 $139,347  1355.86 1121.84 349.02 816.72 584.10 283.50 4511.04  ($3,748.20) ($24,261.24) ($1,088.42) ($5,184.06) ($10,674.37) ($8,295.46) ($53,252)  ($53,252)  $3,401,593  makmaw\rWWW:MMAWPRM.WERWM:% %.aaamma: :gamksv..fti. ...„...,...:=4;:o§...4w.......k4m&a:.;s..;.:;,.„,;,::as. ..  .  ,  ,  ($1,500,816)  53  :AMBEIL  "  978.24 $589,749.76 495.22 $472,485.25 297.44 $398,104.11 674.81 $869,299.55 401.93 $618,055.11 140.97 $238,312.34 2988.60 $3,186,006 17.96 12.41 53.03 41.79 61.72 0.00  186.92  1355.86 1121.84 366.46 816.72 584.10 283.50 4528.48  ($3,554.31) ($21,846.57) ($792.08) ($5,273.38) ($7,359.37) ($8,266.79) ($47,092)  $10,383 $10,236 $62,447  $48,968 $89,866 $3  $221,904  ($47,092)  $3,590,235  ($1,597,901)  26.62 24.09 58.68 23.39 15.83  54  ^  Totals  OK. * EMBEEKrIEMENLJ .:INIERNOWASISSIMMESEa...,.: ' --  2901.56 $1,768,460.56 1231.24 $1,178,982.63 874.73 $1,156,818.23 2106.36 $2,708,008.49 1203.28 $1,834,263.18 420.70^$712,663.25 8737.87^$9,359,196 31.67  0.06  $18,947 $19,399 $141,584 $61,781 $119,458 $81  309.73  $361,250  23.66 114.07 52.98  87.29  13476.72  4067.59 3365.52 1054.34 2450.16 1752.29 850.49 13540.39  ($11,167.61) ($75,402.52) ($3,331.33) ($15,226.57) ($28,865.80) ($24,887.08) ($158,881)  $673,828  $10,394,275  ($158,881) •^-^  $10,235,394 ........  ($4,719,111) 480.00  ($2,400,000)  9197.41  ($183,948)  495.12  ($3,713)  149.04  $0 ($7,306,773) $2,928,621  t • •^-'t^•^• - t^' t'  26.62 24.09 58.68 23.39 15.83 0.43  149.04  55  


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