# Spatially explicit forest harvest scheduling with difference equations

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## Abstract

Spatially explicit harvest scheduling models optimize the layout of harvest treatments to best meet management objectives such as revenue maximization subject to a variety of economic and environmental constraints. A few exceptions aside, the mixed-integer programming core of every exact model in the literature requires one decision variable for every applicable prescription for a management unit. The only alternative to this “brute-force” method has been a network approach that tracks the management pathways of each unit over time via four sets of binary variables. Named after their linear programming-based aspatial predecessors, Models I and II, along with Model III, which has no spatial implementation, each of these models rely on static volume and revenue coefficients that must be calculated pre-optimization. We propose a fundamentally different approach that defines stand volumes and revenues as variables and uses difference equations and Boolean algebra to transition forest units from one planning period to the next. We show via three sets of computational experiments that the new model is a computationally promising alternative to Models I and II.

## Keywords

Forestry Harvest scheduling Integer programming Spatial optimization## Notes

### Acknowledgements

This work was funded by the University of Washington’s Precision Forestry Cooperative and the USDA National Institute of Food and Agriculture (NIFA) under grant number WNZ-1398. The New Zealand data were kindly made available by Geoff Thorp of the Lake Taupo Forest Trust, and Chas Hutton, John Hura and Colin Lawrence of the New Zealand Forest Managers Ltd. We also thank Dr. Pete Bettinger of the University of Georgia’s Warnell School of Forest Resources for providing us with the spatial and the growth and yield data for the Loblolly pine experiment. Lastly, thanks to Drs Bruce Bare, University of Washington, Seattle and Thomas Lynch, Oklahoma State University for their valuable pre-submission reviews.

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