Harvest Scheduling Subject to Maximum Area Restrictions: Exploring Exact Approaches

Abstract

We consider a spatial problem arising in forest harvesting. For regulatory reasons, blocks harvested should not exceed a certain total area, typically 49 hectares. Traditionally, this problem, called the adjacency problem, has been approached by forming a priori blocks from basic cells of 5 to 25 hectares and solving the resulting mixed-integer program. Superior solutions can be obtained by including the construction of blocks in the decision process. The resulting problem is far more complex combinatorially. We present an exact algorithmic approach that has yielded good results in computational tests. This solution approach is based on determining a strong formulation of the linear programming problem through a clique representation of a projected problem.

Optimal Forest Stand Aggregation and Harvest Scheduling Using Compactly Formulated Integer Programming

“…Constraints include land accounting, harvest flow, and spatial restrictions to avoid harvesting two adjacent stands during the same period. Unlike the methods introduced by McDill et al (2002), Murray and Weintraub (2002), Goycoolea et al (2005), and Rebain and McDill (2003), we build on the Model I formulation and concept of adjacency. Our approach consists of four steps: 1) create candidate "hyper units"-predefined before solution search starts-for forest stand aggregation based on each forest stand (distinguish this as "base unit" hereafter) and define the decision variable matrix considering the base and hyper units, 2) develop adjacency constraints for the set of base and hyper units, 3) develop the extended land accounting constraints by introducing an overlapping relationship among the base and hyper units, and 4) formulate a forest stand aggregation harvest scheduling problem to allow for multiple harvests.…”

“…Note that overlapping is not considered as adjacency unlike Goycoolea et al (2005). An element of A u-u is defined by:…”

“…Murray (1999) referred this problem as an area restriction model (ARM) as opposed to conventional adjacency problems. Several studies have addressed ARM-type problems with an exact method, including McDill et al (2002), Murray and Weintraub (2002), Rebain and McDill (2003), Goycoolea et al (2005). These efforts, however, have limited ability to address harvest scheduling with multiple harvests over the planning horizon.…”

“…The concept of Model I (Johnson and Scheurman 1977) is applied to deal with multiple harvests over time in the integer programming formulation as well as the concept of the general management unit approach (McDill et al, 2002;Goycoolea et al, 2005) to enumerate feasible aggregated units (called "hyper units" hereafter) for forest stand aggregation. The matrix notation is used as the general expression of compactness to define decision variables in Model I as well as adjacency among forest units and "hyper units".…”

Optimization Model of Selective Cutting for Timber Harvest Planning Based on a Simulated Annealing Approach

“…However, because maintaining other forest values has become increasingly important, various IP-and MIPbased models incorporating spatial constraints have been reported in the literature (Goycoolea and Murray, 2005;Vielma et al, 2007;Constantino et al, 2008;Ohman and Wikstrom, 2008). Nevertheless, the performance of these solution techniques is restricted by the complexity of the problem.…”