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One aim of spatially explicit timber supply modeling is to devise a harvest schedule that maximizes the volume of timber cut by "optimizing" a harvest flow profile subject to certain modeling choices and management constraints (e.g., environmental limitations, economic thresholds, adjacency restrictions, transportation access, and harvest shortfalls). The schedule often extends for 200 years or more, specifying annual harvests in spatially explicit form for a fixed geographic area (e.g., a large forested region of British Columbia).
An example of a harvest flow profile is given by the piecewise-linear function in Figure 1. This function depends on the parameters t1, t2, and v2; v1 is fixed at the outset, and so the harvest flow starts at a given level, remains at that level for some time, and then gradually decreases to a constant sustainable level.
Figure 1. Example of a harvest flow profile for timber supply modeling.
The management constraints, being spatial in nature, are cumbersome to express algebraically and/or are highly nonlinear. We thus choose to incorporate these constraints into a spatial simulation model, and handle them during simulation by testing whether the harvesting of a block (a basic spatial harvesting unit) violates a constraint, or by removing blocks from consideration during certain periods because they have recently been cut or are adjacent to blocks that have been cut.
For optimizing harvest flow, we use the results of these simulations to determine a maximal harvest flow profile that is consistent with the management constraints embodied in the simulation model. However, maximization of the harvest flow profile alone generally yields solutions containing shortfalls. An example of the results of a simulation for which shortfalls occur for some time periods is shown in Figure 2.
Figure 2. Results of a simulation using a target harvest flow profile (solid line). The small dots indicate those years when the actual volume harvested Hi reached or exceeded the target harvest volume Ti and the boxes indicate years when shortfalls occurred.
For planning reasons, harvest flow profiles which result in few (or small) shortfalls are preferable to those which may give a somewhat greater overall harvested volume but which result in many (or large) shortfalls for some time periods. This practical consideration dictates that a more sophisticated objective function than just the harvest flow profile is required. One example would be an objective function that takes into consideration the target volumes, the actual volumes harvested, and which penalizes any shortfalls.
One objective function to be maximized which expresses this desired outcome is the following:
where:
x is the vector of parameters (e.g., t1, t2, and v2),
Ti is the target harvest volume for the ith time period,
Hi is the actual volume harvested for the ith time period,
P is a nonnegative penalty factor for shortfalls,
and the notation (�)+ denotes that (a)+ = a if a > 0, and 0 otherwise.
An example of this objective function, generated for a two-parameter model, where t1 and v1 are fixed and t2 and v2 are varied, is shown in Figure 3.
Figure 3. Example of the objective function f(x) computed by running simulations for a range of values for the parameters t2 and v2.
The function f(x) is smoothly-varying for the leftmost region of the parameter space, but becomes erratic as the parameter values increase, owing to the occurrence of shortfalls. This erratic behaviour is a consequence of the fact that closely related parameter values can produce significantly different harvest flow volumes.
Our approach to solving this problem uses the Nelder-Mead simplex optimization algorithm with a starting point set in the shortfall-free region of the objective function. This approach deals effectively with the erratic nature of the volume functions and provides an effective way of optimizing the harvest flow while restricting the level of shortfalls.
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Optimization in Timber Supply Modeling
