A Probabilistic Approach to Cost Optimal Timber Grading

Authors

Jochen Köhler and Michael H. Faber

Abstract

Timber is by nature a very inhomogeneous building material. On a large scale the material properties are a product of e.g. the specific wood species and the geographical location where the wood has been grown. The material properties of timber may be ensured to fulfill given requirements only by quality control procedures – hereafter referred to as grading. For mechanical grading various schemes have been developed using different principles, however, the basic idea behind them all is that the relevant material properties such as, e.g. the bending strength, are assessed indirectly by means of other indicative properties (indicators) observed during a grading procedure such as e.g. the density or the modulus of elasticity, see also Madsen (1992) and Walker et al. (1993). The allocation into different grades is takes basis in acceptance criteria, which formulated in terms of the indicators. An acceptance criterion can be implemented by means of the settings of a grading machine, whereby a given grade is allocated to timber for which the indicators have values belonging to defined intervals. The timber graded in this way has to match given requirements implicitely defining the indicator value intervals. As an example, the European standard EN 338 defines grades in terms of the lower 5% fractile value of the graded timber bending strength, the mean value of the bending modulus of elasticity and the mean value of the density. These requirements can be implemented by adjusting the settings of the applied grading machine. The acceptance criterion, i.e. the grading machine settings, leading to normative grades are normally obtained and calibrated by performing many test of the graded timber.

Due to the special way timber material properties are ensured by means of grading in the production line, special considerations must be made when modeling their probabilistic characteristics. Previous work on this subject is reported in e.g. Glos (1981) and Rouger (1996). In Pöhlmann and Rackwitz (1981) a bi-variate Normal distribution model is suggested in order to describe the probabilistic characteristics of the graded timber. In Faber et al. (2003) a Bayesian approach is proposed allowing for a generalization of the bi-variate Normal distribution model such that the prior probability distribution function of the un-graded timber material properties may be chosen freely in accordance with statistical evidence.

The selection of a grading procedure, i.e. the type of grading machine and the acceptance criteria, could be made based on cost benefit considerations. Different procedures have different costs and different efficiency characteristics. In the present paper it is demonstrated how an optimal (in terms of monetary benefit) set of timber grades can be identified. Therefore, the approach suggested by Faber et al. (2003) is utilized to identify timber grades and quantify the probabilistic characteristics of their relevant material properties. This requires that the probabilistic characteristics of the relevant material properties of the ungraded timber are known together with their correlation to the indicator. An optimization problem can be defined for identifying the grading procedure, leading to the optimal set of timber grades.

Published in/by

Proceedings of CIB-W18, Paper No. 36-5-2, Colorado, USA, August 11-14, 2003.