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A MULTIPLE PERIOD COMBINED OPTIMIZATION APPROACHTO 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 conformingto 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 ColumbiaVancouver, 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^ vi
I. INTRODUCTION^ 1
II. LITERATURE REVIEWIntegrated Production Planning Models^ 3Planning Over Multiple Periods^ 6
III. DESCRIPTION OF A MODEL FOR MULTIPLE PERIOD PRODUCTION PLANNINGIntroduction^ 7Resource Coordinator^ 8Objective Function^ 11Supply Constraints^ 13Production Constraints^ 14Marketing Constraints^ 14Inventory Constraints^ 16Sawing Model^ 17Integrating the Models^ 17Data Acquisition^ 18Analysis of Iterations^ 18Conclusion^ 22Literature Cited^ 23
IV. THE EFFECTS OF MARKETS ON MULTIPLE PERIOD PRODUCTION PLANNINGIntroduction^ 24Experimental Design^ 24Case Specific Inputs and Expectations^ 26Tuning the Model^ 27Computational Experience^ 28Results and Analysis^ 28Conclusion^ 34Literature Cited^ 34
V.^CONCLUSION^ 35
BIBLIOGRAPHY^ 38
APPENDIX - Solution Reports of Example Cases^ 39
iii
LIST OF TABLES
1. General Mill Information 7
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. Develop a model which can be used for sawmill production planning overmultiple periods.
2. Model the impact of lumber inventories on overall profit/loss andproduction.
3. Develop a method of analysis for determining the optimal log mix over aseries of periods.
4. Develop general production strategies for producing in periods of falling orrising 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
Mill Type:
Maximum Hours: :
Minimum Hours:,
Sawlines per Hour:,...
Sawing Cost ($/hour):
Finishing Cost (S/MFBM):
Trim Factor:
6
Total Production (MFBM):
Residue Value (Monne):
Inventory Cost (S/MFBM):
Inventory Capacity (MFBM):
10^11^12^13
Lumber Lengths
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.
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 (SawLogdmLc)• 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:EEE {((LumSales did + E InvSales ) * LumPrice dpi )d p 1
— (UnderTarget dpi * UPenalty dpi ) —^{LumInv ipi * InvCost d }}
E {FiberVol d * FiberPriced }
-EE {OverProd p/ *OverPen iv }P 1
-EE {SawTime dm * SawCost dm }d m
_ EE {Finishing d„ *FinCost d,„}d m
-EE {BoomUsed c/5 *BoomPrice db }d b
(1)
8
Subject to:Supply Constraints
E BoomUsed db 1 3 b^ (2)
E {LogVol db,, * BoomUsed ab }^SawLoga.m. = 0 3 (d,L)^(3)m C
Production Constraints
EE {Recovery dmpILC * SaWLOgdmw — (Trim. * MillProduction dmpi ) = 0L C
3 (d,p,l,m) (4)
EE {FiberVol amLC * SawLo ,,dmLC — FiberProdcution dm = 0L C
3 (d,m) (5)
E E tsawLogdm,„ *Hoursd„,Lc — SawTime am = 0 3 (d,m)^ (6)L C
SawTime d„, MaxTime d„, 3 (d, m)^ (7)
SawTime dm MinTime dm 3 (d,m)^ (8)
EE {MillProduction dmpi } — Finishing dm = 0 3 (d,m)^ (9)P 1
Marketing Constraints
E {MillProduction dmpi} LumSales dpi — LumInv fp/ = 0 3 (d,p,1)
where d=1,2,...,MaxPeriod-1t=1^ (10)
E {MillProduction d,„1,/ } — LumSales dpi = 0 3 (p,1)
where d=MaxPeriod^(11)
E {FiberProduction dm } — 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
d -1E {InvSales dipi } + LumSales dpi UnderTarget dpi = Targetdpi 3 (d,p,1)i=i
where d=2,...,MaxPeriod-1 (14)
max period-1
E {InvSales thpi } + LumSales dpi UnderTarget dp, — OverProd = Target dpi1=1
3 (4,0where d=MaxPeriod^(15)
Inventory Constraints
EE (Degrade tpk, * LumInv ipi 1 — InvSales dtqk — LumInv (t+Dqk = 0 3 (d,t,q,k)1
where t=1,2,...,d-1^
(16)
EEE {LumInv ipi } <= Capacity d 3 (d)p^1^t
where t=1,2,...,d-1^
(17)
Non-negativity on all variables.
Subscript definitions:
d^period^
t^inventory age^L log class
products^m mill^
C sawing policy
k,1^product lengths b boom
= Cost of boom b in period d ($).
= Amount of boom b used in period d (boom units).
= Inventory capacity in period d (MFBM).
= Percent degrade of product p with length 1 into product q withlength k at inventory age t.Value of fiber in period d ($/tonne).
= Fiber produced at mill m in period d (tonnes)
= Fiber sold in period d (tonnes)
= Ratio of fiber recovered from log class L at mill m usingsawing policy C in period d (tonnes/m 3 ).The cost of finishing 1 MFBM of product at mill m in period d($).
Definitions:
BoomPricedb
BoomUseddb
Capacityd
Degradevqk
FiberPriced
FiberProductiondm
FiberSalesd
FiberVoldmLC
FinCostdm
10
Finishingdm
HoursdmLC
InvCostd
InvSalesdq3/
LogVolda
LumInvipi
LumPricedpi
LumSalesdpiMaxPeriod
MaxTimedm
MillProductiondmp/
MinTimedm
OverPen
OverProdpi
RecoverydmplLC
SawCostdm
SawLogdmif
SawTimedm
Targetdpi
Trimm
UPenaltydpi
UnderTargetdpi
Products to be finished in mill m in period d (MFBM).
Time in hours required to execute sawing policy C on log classL at mill m in period d.
= Inventory cost in period d ($/MFBM).
= Inventory sold at inventory age t of product p with length 1 inperiod d (MFBM).
= Volume of log class L found in boom b in period d (m3 ).Volume of product p with length 1 degraded at inventory age t(MFBM).Value of product p with length 1 sold in period d ($/MFBM).
= Product p with length 1 sold in period d (MFBM).= Final production period in model.
= Maximum operating hours available at mill m in period d.
= Product p with length 1 produced at mill m in period d(MFBM).
• Minimum operating hours available at mill m in period d.
Cost of over producing product p with length 1 ($/MFBM).
Over production of product p with length 1 (MFBM).
Ratio of product p with length I recovered from log class L atmill m using sawing policy C in period d (MFBM/m3 ).
• Cost of sawing at mill m in period d ($/hour).• 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.
= Product p with length 1 required in period d (MFBM).
= Ratio of volume loss at mill m.
Cost of under production of product p with length 1 in periodd ($/MFBM).
= Product p with length 1 under produced in period d (MFBM).
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.
APeriod
B^C Total
RevenuesProduction Sales $3,061,219 $3,053,111 $3,148,763 $9,263,093Inventory Sales $0 $152,163 $209,743 $361,907Chips $216,717 $223,315 $225,137 $665,169
CostsRaw Material ($1,589,077) ($1,494,677) ($1,579,859) ($4,663,612)Saw Time ($800,000) ($800,000) ($800,000) ($2,400,000)Finishing ($60,693) ($61,068) ($60,310) ($182,071)Under Production ($59,396) ($53,686) ($47,614) ($160,696)Inventory ($1,497) ($1,954) $0 ($3,451)
Net Revenue $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.
Boom A
Period
B^C Total Cost
1 100.00% 0.00% 0.00% 100.00% $448,5602 0.00% 0.00% 0.00% 0.00% $278,5303 0.00% 0.00% 100.00% 100.00% $274,7204 0.00% 0.00% 100.00% 48.35% $359,5305 17.91% 70.90% 11.19% 100.00% $225,3706 0.00% 0.00% 0.00% 0.00% $394,5307 2.14% 48.09% 49.77% 100.00% $486,0308 0.00% 0.00% 0.00% 0.00% $228,7609 65.81% 34.19% 0.00% 100.00% $448,56010 0.00% 0.00% 0.00% 0.00% $278,53011 0.00% 72.97% 27.03% 100.00% $274,72012 0.00% 0.00% 0.00% 0.00% $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 AVolumes in MFBM^Production^Value^Targets^Penalties
2x3 16' D Clear 0.30 $135 5.27 ($89.37)Total Production 2836.29 $3,060,950 4500.87 ($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 t= 1 t=2
Overproduction 6.35 5.39-Length Loss 0.08 0.07-Grade Loss 0.13 0.11Available for sale 6.14 5.21Sold 0.75 5.28Remaining Inventory 5.39 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„g, * gdpg, 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
Input Data^Integrated Model^Reports
Lumber Grades and SizesInitial Lumber ValuesLog Descriptions —110.
SawingPatternGenerator
g*ro
A
5"
'1
;1.
Raw MaterialProduct TableMill ConstraintsProduction TargetsProduct ValuesInventory Degrade
Boom ChoicesSawing PoliciesLumber SalesProduction DataInventoryInventory Sales
ResourceCoordinator
production of a product and the shadow prices of the product at the end of a system
iteration.
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
Change in Optimal Value
Steady State Case
21^41^61^81^101^121^141^161
Iteration Number
3000000
2000000
1000000c2>^0-aE -1000000o.0
-2000000
-3000000
-4000000
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.
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
x-
-6
2 -8LL
^
12- -10^.
-12
-14
-16x
^
-18 i^iU) CO N
- - - ><- - - over target^—°--- shadow price
1111111111111111111^0op 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)
0
-2
-4
700
600
500
400 tco
300 E
200
100
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
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 -
2550 - ^Period A Period B ---^Period C---
2500 '13^
18^
23^
28^
33^38Iteration
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 andsawing 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 andallocation. 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.
ProductProduct Degrade
%Degrade^ToLength Fa//down
8^10^12 13 14 16 18 20B CX20 2.00% C CX20 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.6% 1.7%B CX18 2.00% C CX18 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.6% 1.7%B CX16 2.00% C CX16 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.6% 1.7%B CX14 2.00% C CX14 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.6% 1.7%B CX12 2.00% C CX12 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.6% 1.7%B CX10 2.00% C CX10 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.6% 1.7%B CX8 2.00% C CX8 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.6% 1.7%B CX6 2.00% C CX6 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.6% 1.7%B CX5 2.00% C CX5 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.6% 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.
ThicknessClass
Price Change (%)Period B^Period C
A 12.5 7.4B 7.0 3.3C 9.7 3.2D,E,F 13.7 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.
APeriod
B^C TotalRevenues
Production Sales $2,535,264 $3,195,974 $3,766,336 $9,497,573Inventory Sales $0 $527,953 $767,249 $1,295,201Chips $215,116 $222,518 $232,081 $669,715
CostsRaw Material ($1,434,402) ($1,633,929) ($1,651,014) ($4,719,345)Saw Time ($800,000) ($800,000) ($800,000) ($2,400,000)Finishing ($61,294) ($61,753) ($61,581) ($184,627)Under Production ($100,856) ($69,554) ($41,231) ($211,641)Inventory ($5,217) ($5,629) $0 ($10,845)
Net Revenue $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
Price IncreaseA^B
Price DecreaseA^B C
Total Production 3064.69 3087.63 3079.04 3066.17 3050.12 3023.21Production Sales 2369.15 2618.53 3014.15 2920.1 2895.28 2974.06Inventory Sales 0.00 413.93 661.54 0.00 89.42 130.75New Inventory 695.54 469.10 64.89 146.07 154.84 49.15Ending Inventory 695.54 750.49 153.32 146.07 211.25 129.22
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.
APeriod
B^C Total
RevenuesProduction Sales $3,165,708 $2,728,179 $2,693,444 $8,587,331Inventory Sales $0 $81,609 $124,354 $205,963Chips $220,615 $219,629 $226,718 $666,962
CostsRaw Material ($1,653,544) ($1,455,676) ($1,573,146) ($4,682,366)Saw Time ($800,000) ($800,000) ($800,000) ($2,400,000)Finishing ($61,323) ($61,002) ($60,464) ($182,790)Under Production ($55,249) ($48,246) ($42,464) ($145,959)Inventory ($1,095) ($1,584) $0 ($2,680)
Net Revenue $815,110 $662,908 $568,443 $2,046,461
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.
APeriod
B^C Total
RevenuesProduction Sales $3,082,935 $3,090,256 $3,186,006 $9,359,196Inventory Sales $0 $139,347 $221,904 $361,250Chips $219,168 $225,242 $229,418 $673,828
CostsRaw Material ($1,620,395) ($1,500,816) ($1,597,901) ($4,719,111)Saw Time ($800,000) ($800,000) ($800,000) ($2,400,000)Finishing ($61,476) ($61,518) ($60,954) ($183,948)Under Production ($58,537) ($53,252) ($47,092) ($158,881)Inventory ($1,633) ($2,080) $0 ($3,713)
Net Revenue $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 B C Total
A 0.35% 0.21% -0.01% 0.15%B 3.64% 1.45% 2.38% 2.16%C 0.51% 3.72% 9.60% 4.59%D -0.05% 1.18% -1.11% -0.06%E 0.60% 1.26% 0.46% 0.74%F 1.89% 0.88% 0.29% 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.
Boom A
Period
B^C Total Cost
91.08% 8.92% 0.00% 100.00% $448,5602 0.00% 0.00% 0.00% 0.00% $278,5303 0.00% 96.10% 3.90% 100.00% $274,7204 0.00% 0.00% 0.00% 0.00% $359,5305 0.00% 81.16% 18.84% 100.00% $225,3706 0.00% 0.00% 0.00% 0.00% $394,5307 33.33% 49.58% 17.09% 100.00% $486,0308 0.00% 0.00% 0.00% 0.00% $228,7609 57.12% 0.00% 42.89% 100.00% $448,56010 0.00% 0.00% 0.00% 0.00% $278,53011 0.00% 0.00% 100.00% 100.00% $274,72012 0.00% 0.00% 100.00% 100.00% $359,530Table 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 CitedCargo 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 WoodExports 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 andSawing Strategies. Wood and Fiber Science 23(2):296-314.
McKillop, W. and S. Hoyer-Nielsen 1968. Planning Sawmill Production and InventoriesUsing Linear Programming. Forest Products Journal
McPhalen, J. C. 1978. A method of Evaluating Bucking and Sawing Strategies forSawlogs. M. Sc. Thesis, University of British Columbia, Vancouver, B.C., Canada.
Mendoza, G. A. 1980. Integrating Stem Conversion and Log Allocation Models forWood 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 andallocation. Forest Products Journal 36(10):70-74.
38
APPENDIX
Solution Reports of Example Cases
39
Base Case: Steady StatePeriod A
z„„s„,„,-way.„^RICERUMOMAREZEMMUZIMMOV
958.89295.76308.89725.79409.64136.66
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
^ 68.25263192.3095
($53,686)
($1,494,677)
$3,374,904
($800,000)
($61,068)3053.40
($5,000)
($20)
160.00
($512)($1,442)
($7.50)( $7 .50 )
AVIDAMMENEL:- 71111112.--..311E.PLOMOUNROINIEUTS-959.00423.37257.24697.90384.39139.26
2861.16
12.0511.5860.5817.2529.180.42
131.06
$587,354.29$406,979.97$339,780.61$897,993.10$585,611.76$235,391.06
$3,053,111
$6,863$9,156
$77,271 $20,362$37,824
$689$152,163
1355.86 ($3,783.38)1121.84 ($24,466.21)338.85 ($1,080.08)816.72 ($5,268.92)584.10 ($10,731.54)283.50 ($8,355.60)
4500.87 ($53,686)
($2,357,699)
$1,017,205
Period B
41
($47,614) $3,536,029
($5,000)
($20)
160.00
3015.50
($800,000)
($60,310)
Period C
01,11NViiAav, 1:WitiWt-
1355.86 ($3,588.17)1121.84 ($22,055.38)338.85 ($675.52)816.72 ($5,295.71)584.10 ($7,704.84)283.50 ($8,294.48)
4500.87 ($47,614)
1 978.31483.69271.39682.36400.10140.56
2956.41
13.7317.0253.7733.9958.090.00
176.60
$8,254$14,506$61,130$41,921$83,933
$0$209,743
.:WPOr
($1,579,859)
24.6622.7255.7524.2314.14
(31 . 31 EllamandiVETRISERNittnag141.81^ $128,912^($128,912)
($2,440,169)
$1,095,861
$591,012.92$462,877.76$366,156.45$875,781.04$615,317.31$237,617.57
$3,148,763
42
Totals
480.00
9103.55
460.11141.81
($3,451)$0
v my *^'
2896.20 $1,769,079.30 4067.591202.82 $1,153,669.85 3365.52837.52 $1,111,418.63 1016.56
2106.04 $2,705,785.19 2450.161194.12 $1,817,673.44 1752.29416.48 $705,466.17 850.49
8653.19 $9,263,093 13502.61
131/0411.4-‹* • ts
($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)
ti
$10,129,472
.
($4,663,612)
($7,249,134)
$2,880,338
t t^1'00^t.^e-t tt t^t^r'^-t
24.6622.7255.7524.2314.140.31
141.81
($2,400,000)
($182,071)
43
Case 1: Increasing Lumber PricesPeriod A
($100,856) $2,649,523
$575,778.22 1355.86$79,743.36 1121.84
$392,380.13 338.85$846,862.13 816.72$469,985.15 584.10$170,514.54 283.50
$2,535,264 4500.87
($5,776.68)($47,575.18)
($2,582.89)($10,154.20)($21,027.99)($13,739.47)
($100,856)
n($1,434,402)
($800,000)
($61,294)
($5,217)
($2,300,912)
$348,611
44
...... ..........................
160.00
3087.63
126.28 341.00
86.72 90.24 89.96 16.29
281.3931469.0988
($5,000) ($800,000)
($20) ($61,753)
($7.50)($7.50)
($2,110)($3,518)
903.39 $635,699.92 1355.86266.33 $275,726.50 1121.84249.82 $363,160.14 338.85689.85 $1,007,110.72 816.72372.40 $649,486.67 584.10136.74 $264,789.89 283.50
2618.53 53,195,974 4500.87
($5,532.81)($35,828.38)
($1,266.64)($6,263.77)
($10,070.29)($10,591.81)
($69,554)
53.47131.9871.2245.1095.8216.34
413.93
4450.36
$29,626$142,482
$98,810$62,239
$163,600$31,198
$527,953
($69,554) $3,876,891
($2,501,310)
$1,375,581Mg%
Period B
45
-zzlIgiAzzrtAlItyezzaws0; era
968.16524.01270.38678.91428.38144.29
3014.15
$703,084.22$582,197.57$412,474.40
$1,019,927.25$763,227.07$285,425.05$3,766,336
1355.861121.84338.85816.72584.10283.50
4500.87
($4,145.24)($14,616.71)
($758.61)($4,757.39)($6,846.24)
($10,106.36)($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 s^$165,129^($165,129)
Period C
46
* • esiar,i, • -.:::".mumnr7.7.3aeso..44.....7.....I....§...",„1,:ttatzta-41:::::11:19::::::::.::as2776.20
881.78818.07
2021.341123.24381.21
8001.83
165.94454.94129.53121.77170.9432.35
1075.47
13394.42
$1,914,562.36 4067.59 ($15,454.73)$937,667.43 3365.52 ($98,020.26)
$1,168,014.66 1016.56 ($4,608.14)$2,873,900.10 2450.16 ($21,175.36)$1,882,698.89 1752.29 ($37,944.53)
$720,729.47 850.49 ($34,437.64)$9,497,573 13502.61 ($211,641)
$77,798$509,858$174,181$169,551$301,418
$62,396$1,295,201
$669,715$11,462,490 ($211,641) $11,250,849
•"'":?.K:;;;;;;.;;;;:.
($4,719,345)
($2,400,000)
($184,627)
($10,845)$0
480.00
9231.36
1446.03153.32
($7,314,817)
$3,936,032
23.7630.7247.9924.41 26.03
0.40153.32
47
• •
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)
Case 2: Decreasing Lumber PricesPeriod A
48
160.00 ($800,000)
3050.12 ($61,002)
56.4135154.8358 ($7.50)
($7.50) ($423)($1,161)
Period B
,...aaae •
967.98430.32290.96669.65398.39137.99
2895.28
9.8710.5427.5223.4817.890.12
89.42
4392.58
$516,686.62$386,531.75$347,267.53$751,920.42$523,593.63$202,179.12$2,728,179
$5,030$6,822
$30,285$20,804$18,533
$134$81,609
1355.861121.84338.85816.72584.10283.50
4500.87
($3,250.13)($22,423.31)
($936.66)($5,347.27)($9,033.01)($7,255.81)
($48,246)
($48,246) $2,981,170
iLlabant:Naili:::0( $1,455,676 )
,ti,:••••••••••••••
($2,318,262)
$662,908
49
($42,464) $3,002,053
20.11 26.60 54.67 21.72
5.90 °.24
129.22 $99,317^($99,317)
Period C
„ „IIEBEL„„.„..aeaaKzea„—„Lhattniti: . 72REMINIMEK
983.81 $479,912.89 1355.86502.79 $429,574.82 1121.84284.79 $333,235.95 338.85645.58 $708,796.12 816.72423.20 $550,456.12 584.10133.89 $191,468.21 283.50
2974.06 $2,693,444 4500.87
($2,800.15)($19,188.13)
($739.29)($5,783.22)($6,619.13)($7,333.76)
($42,464)
13.3016.41 37.1731.36 32.12
0.39130.75
$6,595$13,495$36,519$31,684$35,507
$556$124,354
-44k.ZZ.ZZZ&;:k.kaggsWM-Z.4402...‘iatt.C.:IZZZ::tt
^($1,573,146)
Papr:.*:Fs::"" „ AMMV:RUITIO.W.;
50
480.00
9139.49
357.32129.22
($2,400,000)
($182,790)
($2,680)$0
REFSAS
2910.93 $1,587,636.32 4067.591275.33 $1,146,505.77 3365.52888.52 $1,091,009.30 1016.56
2041.55 $2,391,567.37 2450.161254.40 $1,728,480.35 1752.29418.71 $642,131.84 850.49
8789.44 $8,587,331 13502.61
^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
($9,957.96)($69,480.59)
($3,014.67)($15,906.07)($25,147.44)($22,452.37)
($145,959)
($145,959) $9,314,297
,,,,,,,,,,,,($4,682,366)
($7,267,836)
$2,046,461
20.11 26.6054.67 21.72
5.900.24
129.22
51
Period ACase 3: Product Demand Change
AtiVIgr'llattiONEKISMISrr ...
.....
%P.;\^Wzki
962.28 $592,141.07306.53 $294,356.24310.48 $407,668.09725.42 $931,012.44412.11 $620,920.92139.24 $236,835.83
2856.07 $3,082,935
1355.86 ($3,865.10)1121.84 ($29,294.72)338.85 ($1,450.83)816.72 ($4,769.14)584.10 (S10,832.05)283.50 ($8,324.83)
4500.87 ($58,537)
$0
$0$0$0
$0
($58,537) $3,243,566
•Mi•Mr`r•r..•`," MMM.^ (621,1144.114.11EPalarkgsm.
($1,620,395)
\\*.:11:06ktrtv-‘k.*** skstss'in.:':\\
%1^.‘ 3073.80
160.00
217.7286
($5,000) ($800,000)
($20) ($61,476)
($7.50) ($1,633)
($2,483,504)
$760,062
52
Period B
($53,252) $3,401,593
SS,
•
961.04 $586,569.74 1355.86429.49 $412,141.13 1121.84266.80 $351,046.04 349.02706.13 $907,696.50 816.72389.24 $595,287.15 584.10140.49 $237,515.08 283.50
2893.20 $3,090,256 4511.04
I
($3,748.20)($24,261.24)
($1,088.42)($5,184.06)
($10,674.37)($8,295.46)
($53,252)
13.71 11.2561.0411.1925.56
0.06122.81
$8,564$9,163
$79,137$12,813$29,592
$78$139,347
makmaw\rWWW:MMAWPRM.WERWM:%.. %..aaamma:,:gamksv..fti.,...„...,...:=4;:o§...4w.......k4m&a:.;s..;.:;,.„,;,::as.
($1,500,816)
'''' :::::::::::::::::::::::::ktaZAAZA*Zakt.kka
::::::::::::::::::::::::::::::::::::::::::
53
:AMBEIL978.24495.22297.44674.81401.93140.97
2988.60
$589,749.76$472,485.25$398,104.11$869,299.55$618,055.11$238,312.34
$3,186,006
"
1355.86 ($3,554.31)1121.84 ($21,846.57)366.46 ($792.08)816.72 ($5,273.38)584.10 ($7,359.37)283.50 ($8,266.79)
4528.48 ($47,092)
17.9612.41 53.0341.7961.720.00
186.92
$10,383$10,236$62,447$48,968$89,866
$3$221,904
($47,092) $3,590,235
($1,597,901)
15.83
54
26.6224.0958.6823.39
Totals
480.00
9197.41
495.12149.04
OK. * EMBEEKrIEMENLJ--.:INIERNOWASISSIMMESEa...,.: '2901.56 $1,768,460.561231.24 $1,178,982.63874.73 $1,156,818.23
2106.36 $2,708,008.491203.28 $1,834,263.18
^
420.70^$712,663.25
^
8737.87^$9,359,196
4067.59 ($11,167.61)3365.52 ($75,402.52)1054.34 ($3,331.33)2450.16 ($15,226.57)1752.29 ($28,865.80)850.49 ($24,887.08)
13540.39 ($158,881)
31.6723.66
114.0752.9887.29
0.06309.73
13476.72
$18,947$19,399
$141,584$61,781
$119,458 $81
$361,250
$673,828$10,394,275 ($158,881) $10,235,394
•^-^ ........
($4,719,111)
($7,306,773)
$2,928,621t • •^-'t^•^• - t^' t'
26.6224.0958.6823.39 15.83 0.43
149.04
($2,400,000)
($183,948)
($3,713)$0
55
