Cutting to order in the rough mill: A sampling approach

A cutting order is a list of dimension parts along with demanded quantities. The cutting-order problem is to minimize the total cost of filling a cutting order from a given lumber supply. Similar cutting-order problems arise in many industrial situations outside of forest products. This paper adapts an earlier, linear programming approach that was developed for uniform, defect-free stock materials. The adaptation presented here allows the method to handle nonuniform stock material (e.g., lumber) that contains defects that are not known in advance of cutting. The main differences are the use of a random sample to construct the linear program and the use of prices rather than cutting patterns to specify a solution. The primary result of this research is that the expected cost of filling an order under the proposed method is approximately equal to the minimum possible expected cost for sufficiently large order and sample sizes. A secondary result is a lower bound on the minimum possible expected cost. Computer simulations suggest that the proposed method is capable of attaining nearly minimal expected costs in moderately large orders.