Second-Order Polynomial Model to Solve the Least-Cost Lumber Grade Mix Problem

Abstract

Material costs when cutting solid wood parts from hardwood lumber for secondary wood products manufacturing account for 20 to 50 percent of final product cost. These costs can be minimized by proper selection of the lumber quality used. The lumber quality selection problem is referred to as the least-cost lumber grade mix problem in the industry. The objective of this study was to create a least-cost optimization model using a design that incorporates a statistical approach to address shortcomings of existing models using linear optimization methods. The results of this study showed that optimal solutions tend to use as much low-quality lumber as possible to minimize costs. Comparison ofresults from this new least-cost grade mix model with other existing least-cost lumber grade mix models has shown that the new model results in lower-cost solutions. In rough mills of the secondary wood products industry, secondary hardwood manufacturers cut hardwood boards into parts of specified-sizes, qualities, and quantities according to customer orders, called cutting bills (Buehlmann et al. 1999). This is an economically important process since raw material costs contribute up to 70 percent of total product cost (Carino and Foronda 1990, Wengert and Lamb 1994, Mitchell et al. 2005). Cutting lumber into smaller components is a typical cutting stock problem (Dyckhoff 1990). The problem exists in many industries, such as in the paper, glass, or car manufacturing industries. In the 1960s, Gilmore and Gomory (1961, 1963, 1965, 1966) proposed a series of solutions to the cutting stock problem using linear and dynamic programming. Since then, numerous additional research efforts have been made to address the problem in different numbers of dimensions, such as in one, two, three,