Spatially constrained harvest scheduling for multiple harvests by exact formulation with common matrix algebra

Exact formulations currently developed for spatially constrained harvest scheduling problems mostly consider only a single harvest over time for individual forest units. We propose a new method for formulating the scheduling problem of allowing multiple harvests over time by using common matrix algebra. We combine the concept of Model I formulation, which defines treatments to overcome issues of multiple harvests, with that of adjacency constraints for treatments. Conflicting harvests over space and time are resolved by introducing two kinds of adjacency matrices. One is an ordinary spatial adjacency matrix for the forest unit location, and the other is a newly introduced activity adjacency matrix to identify concurrent harvesting activities in a set of possible treatments for one forest unit. The Kronecker product of these two adjacency matrices is used to generate the entire adjacency constraint for treatments among all forest units to avoid adjacent harvests. The advantage of our approach is that it relies on the concept of the Model I formulation to satisfy spatial restrictions and identify decision variables for treatments of all forest units systematically using common matrix algebra, so that conversion and extension of existing non-spatial forest planning models (e.g., FORPLAN) to consider multiple harvests and green-up constraints can easily be achieved in a spatially explicit manner.