Linear programming and optimizing lumber quality composition in secondary hardwood Tdimension mills

Linear programming has been widely applied in the secondary hardwood dimension manufacturing industries to solve the least-cost lumber grade mix problem, which refers to the search for minimal raw material costs. Most existing models are based on the assumption of a simple linear relationship between lumber grade mix and yield. However, this crucial assumption has never been verified. In this study, the results from a fivefactor mixture design statistically proved that none of the cutting bills tested has a simple linear relationship between yield and different lumber grade mixes. It was observed that cutting bill characteristics and lumber quality affect the relationship between yield and lumber grades. Cutting bills that require wider and\or longer parts tend to behave non-linearly. In addition, the more dissimilar lumber grade qualities that are processed, the more likely is the occurrence of a non-linear response. The inability to predict the relationship between yield and lumber grades, coupled with the high percentage of nonsimple linear relationships observed in this study, brings into question the validity of the linearity assumption applied in previous linear programming models. Further efforts are needed to construct a new least-cost lumber grade mix model that does not rely on the assumption of a simple linear relationship between lumber grade mix and yield.