Harvest scheduling with area-based adjacency constraints

Adjacency constraints in harvest scheduling models prevent the harvest of adjacent management units within a given time period. Two mixed integer linear programming harvest scheduling formulations are presented that include adjacency constraints, yet allow the simultaneous harvest of groups of contiguous management units whose combined areas are less than some predefined limit. These models are termed Area Restriction Models, or ARMs, following Murray. The first approach, the Path Algorithm, generates a set of constraints that prevent concurrent harvesting of groups of contiguous stands only when the combined area of a group exceeds the harvest area restriction. The second approach defines the set of Generalized Management Units that consist of groups of contiguous management units whose combined areas do not exceed the maximum harvest area limit. This formulation of the model can recognize direct cost savings – such as sale administration costs or harvest costs – or higher stumpage prices that may be realized by jointly managing stands. Example problems are formulated and solved using both ARM approaches and compared with models that restrict concurrent harvests on all adjacent units, regardless of area (termed Unit Restriction Models, or URMs, again following Murray). The ARM formulations usually result in larger models and take longer to solve, but allow for higher objective function values than otherwise similar URM formulations. While the proposed ARM approaches should be applicable to more general problems, the examples are constructed so that the largest number of contiguous stands that can be harvested jointly is three. Strategies for reducing the size of the ARM formulations are discussed and tested.