Use of change-constrained programming to account for stochastic variation in the A-matrix or large-scale linear programs: A forestry application

Linear programming (LP) is widely used to select the manner in which forest lands are managed. Because of the nature of forestry, this application has several unique characteristics. For example, the models consider many different management actions that take place over many years, thus resulting in very large LP formulations with diverse data. In addition, almost none of the data are known with certainty. The most pervasive occurrence of stochastic information is in the production coefficients, which indicate the uncertain response of the managed forest ecosystem to various management options. A ‘chance-constrained’ approach to handling this uncertainty would often be appropriate in forestry applications-managers and decision makers would like to specify a probability with which uncertain constraints are met. Unfortunately, chance-constrained procedures for A-matrix uncertainty produce nonlinear programming problems, which cannot currently be solved for large-scale forestry applications. This paper utilizes a Monte Carlo simulation approach (a linear program is repeatedly solved with randomly perturbed A-matrix coefficients) to describe the distribution of total output when the individual production coefficients are random. An iterative procedure for ‘chance-constraining’ feasibility is developed and demonstrated with this sort of random A-matrix. An iterative approach is required because the mean and variance of total output are unknown functions of the random A-matrix coefficients and the level of output required. This approach may have applications in other fields as well.