Wood-D Industries has just signed contracts for the clear felling of three large forest tracts of second-growth radiata pines. The harvested trees will supply the firm's sawmill and chipboard plant. Some of the logging output is also available for export.
All trees harvested are cut on location into sections 20 feet long, referred to as first cuts, second cuts, etc. On the basis of a detailed survey of each forest, the firm's chief forester estimated the average composition of each forest's total output as shown below:
Log Cuts Forest A Forest B Forest C --------------------------------------------------------- First and Second 42% 46% 45% Third and Fourth 40% 41% 45% Fifth and over 18% 13% 10% ---------------------------------------------------------The average daily output is 128 HC for forest A, 192 HC for forest B, and 200 HC for forest C (1 HC = 100 cubic feet = 2.832 cubic meter).
The log cuts are sorted onto logging trucks for transportation to either the sawmill or the chipboard plant, the two facilities being at different locations. Transportation costs per HC are given in the following table:
Transportation Sawmill Chipboard ------------------------------------------- Forest A $4 $7 Forest B $3 $5 Forest C $6 $3 -------------------------------------------Handling costs per HC is given in the table below:
Handling Sawmill Chipboard ------------------------------------------- First and Second $2.5 $1.2 Third and Fourth $3.5 $1.5 Fifth and over $5.0 $2.0 -------------------------------------------At the sawmill, logs are sawn into three grades of finished products: clear grade, dressing grade, and construction grade. A substantial fraction of the incoming volume of wood ends up as scrap and sawdust. The table below shows the average log conversion factors at the sawmill, as well as the average processing rates. Excluding breakdowns, the productive capacity at the mill averages 360 minutes per day.
Sawn Timber Scraps Sawdust Processing
Log Cuts MBF/HC HC/HC HC/HC Time per HC
------------------------------------------------------------------------
First and Second 0.72 0.26 0.14 1.8
Third and Fourth 0.66 0.30 0.15 2.6
Fifth and over 0.54 0.39 0.16 3.9
------------------------------------------------------------------------
From sample logs processed at the sawmill, average yields for each grade of
sawn timber were determined. They are summarized in the following table.
Clear Dressing Construction
---------------------------------------------------------
Log Cuts/Forest A B C A B C A B C
-------------------------------------------------------------------------------
First and Second 35% 28% 20% 48% 42% 30% 17% 30% 50%
Third and Fourth 10% 3% 25% 18% 9% 30% 72% 88% 45%
Fifth and over 0 0 5% 5% 0 30% 95% 100% 65%
-------------------------------------------------------------------------------
The ex-mill wholesale price per MBF is $150 for clear grades, $110 for dressing
grades and $80 for construction grades. Scraps at the sawmill are transferred
by truck to the chipboard plant for chipping. The transportation cost is $4
per HC. Sawdust is used as fuel in the mill's drying kiln and saves $12 in
other fuel costs per HC.
At the chipboard plants, logs and scraps are chipped. The chips are then mixed with additive glues, filled into 4' by 8' forms and then compressed into boards of various thicknesses. The whole process is highly automated. Each HC of wood yields 0.76M3/4 of chipboard (1M3/4 = 1000 square feet of 3.4 thickness = 1.77 cubic meters). The plant can produce upto 112M3/4 of chipboard per day. Chipboard prices ex-factory are $105 per M3/4.
In light of predicted demand and desired stock levels, certain minimum daily output rates of finished product are set up by Wood-D management for given planning period. These are 31 MBF of clear grades, 35 MBF of dressing grade, 48 MBF of construction grade, and 96 M3/4 of chipboard. Export prices valid during the same planning period are $95 per HC for first and second cuts, and $88 per HC for third and fourth cuts. Fifth or higher cuts are not exported.
The problem is to determine the optimal daily operating policy during the planning period in question.
Submitted by Prof. Ariela Sofer, Operations Research Dept., George Mason University Based on a problem from the book "Introduction to Operations Research Techniques" by Hans G. Daellenbach, John A. George, and Donald C. McNickle, 1983