Understanding the relationship of lumber yield and cutting bill requirements: a statistical approach
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Secondary hardwood products manufacturers have been placing heavy emphasis on lumber yield improvements in recent years. More attention has been on lumber grade and cutting technology rather than cutting bill design. However, understanding the underlying physical phenomena of cutting bill requirements and yield is essential to improve lumber yield in rough mills. This understanding could also be helpful in constructing a novel lumber yield estimation model.
The purpose of this study was to advance the understanding of the phenomena relating cutting bill requirements and yield. The scientific knowledge gained was used to describe and quantify the effect of part length, width, and quantity on yield. Based on this knowledge, a statistics based approach to the lumber yield estimation problem was undertaken. Rip-first rough mill simulation techniques and statistical methods were used to attain the study's goals.
To facilitate the statistical analysis of the relationship of cutting bill requirements and lumber yield, a theoretical concept, called cutting bill part groups, was developed. Part groups are a standardized way to describe cutting bill requirements. All parts required by a cutting bill are clustered within 20 individual groups according to their size. Each group's midpoint is the representative part size for all parts falling within an individual group. These groups are made such that the error from clustering is minimized. This concept allowed a decrease in the number of possible factors to account for in the analysis of the cutting bill requirements - lumber yield relationship. Validation of the concept revealed that the average error due to clustering parts is 1.82 percent absolute yield.
An orthogonal, 220-11 fractional factorial design of resolution V was then used to determine the contribution of different part sizes to lumber yield. All 20 part sizes and 113 of a total of 190 unique secondary interactions were found to be significant (a = 0.05) in explaining the variability in yield observed. Parameter estimates of the part sizes and the secondary interactions were then used to specify the average yield contribution of each variable. Parts with size 17.50 inches in length and 2.50 inches in width were found to contribute the most to higher yield. The positive effect on yield due to parts smaller than 17.50 by 2.50 inches is less pronounced because their quantity is relatively small in an average cutting bill. Parts with size 72.50 by 4.25 inches, on the other hand, had the most negative influence on high yield. However, as further analysis showed, not only the individual parts required by a cutting bill, but also their interaction determines yield. By adding a sufficiently large number of smaller parts to a cutting bill that requires large parts to be cut, high levels of yield can be achieved.
A novel yield estimation model using linear least squares techniques was derived based on the data from the fractional factorial design. This model estimates expected yield based on part quantities required by a standardized cutting bill. The final model contained all 20 part groups and their 190 unique secondary interactions. The adjusted R2 for this model was found to be 0.94. The model estimated 450 of the 512 standardized cutting bills used for its derivation to within one percent absolute yield. Standardized cutting bills, whose yield level differs by more than two percent can thus be classified correctly in 88 percent of the cases. Standardized cutting bills whose part quantities were tested beyond the established framework, i.e. the settings used for the data derivation, were estimated with an average error of 2.19 percent absolute yield. Despite the error observed, the model ranked the cutting bills as to their yield level quite accurately. However, cutting bills from actual rough mill operations, which were well beyond the framework of the model, were found to have an average estimation error of 7.62 percent. Nonetheless, the model classified four out of five cutting bills correctly as to their ranking of the yield level achieved. The least squares estimation model thus is a helpful tool in ranking cutting bills for their expected yield level. Overall, the model performs well for standardized cutting bills, but more work is needed to make the model generally applicable for cutting bills whose requirements are beyond the framework established in this study.
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