Optimal Forest Harvesting with Ordered Site Access

Abstract

A long‐standing problem in forestry management is the optimal harvesting of a growing population of trees to maximize the resulting discounted aggregate net revenue. For an ongoing forest, the trees are harvested and replanted repeatedly; for a once‐and‐for‐all forest, there is no replanting after a single harvest. In this paper, we outline a new formulation for the optimal‐harvest problem which avoids difficulties associated with functional‐differential equations or partial differential equations of state in the relevant optimal‐control problem encountered in recent studies of the ongoing‐forest problem. Our new formulation is based on the observation that tree logging is necessarily ordered by practical and/or regulatory considerations (e.g., it is illegal to cut the younger trees first in some jurisdictions); random access to tree sites does not occur in practice. The new formulation is described here for the simpler problem of a once‐and‐for‐all forest. New results for nonuniform initial age distributions and variable unit harvest costs for this simpler problem are reported herein; results for an ongoing forest will be reported in [10]. The new model is also of interest from a control‐theoretic viewpoint, as it exhibits the unique feature of having time as a state variable, in contrast to its usual role as an independent variable in conventional control problems.