A branch-and-price approach for harvest scheduling subject to maximum area restrictions

Recently, research on exact methods has been undertaken to solve forest management problems subject to constraints on the maximum clearcut area by using the area restriction model approach. Three main basic integer programming models for these problems have been discussed in the literature: the so-called cluster, path and bucket formulations. Solving these models via branch-and-bound, where all variables and constraints are used a priori, is adequately suited for real problems of a small to medium size, but is not appropriate for larger problems. In this paper, we describe a branch-and-price approach for the cluster model, and we show that this formulation dominates the bucket model, by completing the results of the dominance relationships between the bounds of the three models. Branch-and-price was tested on real and hypothetical forests ranging from 45 to 2945 stands and temporal horizons ranging from three to twelve periods were employed. Results show that the solutions obtained by the proposed approach stood within 1% of the optimal solution and were achieved in a short computation time. It was found that branch-and-bound was unable to produce solutions for most forests from 850 stands with either eleven or an average number of stands per clearcut greater or equal than eight.