Spatially explicit forest harvest scheduling with

Authors: Rachel St John Sándor F Tóth

Publish Date: 2013/01/19

Volume: 232, Issue: 1, Pages: 235-257

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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 mixedinteger 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 “bruteforce” 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 programmingbased 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 preoptimization 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 IIThis 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 WNZ1398 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 presubmission reviewsWe present Model I the first benchmark model differently from existing literature in that we use a prescriptionbased rather than harvest timingbased formulation The decision variables denote the choice whether a sequence of actions eg harvests should be applied to a management unit or not Traditionally integer versions of Model I have been presented with variables that represented “cut or not cut” decisions for each unit The more general prescriptionbased formulation was needed in our experiments because both Models II and IV allow any number of harvests to be applied to a given stand over a particular planning horizon Among other things one consequence of this generalization of Model I is that the prescription variables need to be mapped to harvest timingbased cluster variables for the Cluster Packing formulation to work see Constraints A8–A9 Finally our presentation of Model II is also new in that here an ARMbased extension is used Snyder and ReVelle’s 1996 1997 Model II construct was URMbasedTo define the integer version of Johnson and Scheurman’s 1977 Model I we let S denote the set of management units t=12…T the time periods in the planning horizon and k the minimum rotation age We note that based on the assumption that harvests can occur only in the midpoints of the planning periods the number of times a unit can potentially be harvested over the planning horizon is T/k Model I requires the definition of the set of all possible prescriptions that can be assigned to a management unit P=0…010…0… where every prescription p∈P is a vector of length T The elements of vector p represent the binary decisions of whether the management unit should be harvested in a particular planning period or not The first element of the vector corresponds to period 1 the second to period 2 and so on A value of one indicates that a harvest is to occur in the corresponding period whereas zero indicates that no harvest should occur Let 0–1 variable x sp represent the decision whether unit s should follow prescription p If it should x sp =1 0 otherwise Decision variables are created only for those prescriptions that would not lead to premature harvests In other words all prescription variables with first harvests that occur before the unit reaches its minimum rotation age are excluded from the model during preprocessingTo give a formal definition of Model II we let variable sets b st ℓ st ∈01 ∀st denote the decision whether unit s should be cut in period t the first time and whether it should be cut in period t the last time respectively If unit s is to be cut the first time in period t then b st =1 0 otherwise If unit s is to be cut the last time in period t then ℓ st =1 0 otherwise Further we let variable set g st′t ∈01 ∀st denote the decision whether unit s should be harvested in period t after it had last been cut in period t′ If it is then g st′t =1 0 otherwise Lastly we let n s ∈01 ∀st represent the “donothing” scenario If unit s is not to be cut during the planning horizon then n s =1 0 otherwise Strictly speaking variable set n s ∈01 is not needed for Model II except for the convenience of keeping track of uncut forest tractsWe let ω a denote the unit area volume of a management unit at age a in periods available from the original data and we let omega a denote the volume at age a coming from the logistic curve 1 For simplicity we assume that the growth of each management unit is the same volumes ω a accounting variables omega a and the five parameters that are optimized for best fit have no s sub or superscripts This of course does not mean that Function 1 cannot be fitted for each management unit individuallyConstraints B2 and B3 calculate the absolute deviation δ a between omega a and ω a with the help of objective function B1 which maximizes the sum of the deviations Constraint B4 sets the value of the fitted omega 1 to be equal to ϕ min Constraints B5 and B6 work together to determine if the fitted volume at a particular age is above or below the inflection point β If the volume is strictly above the inflection point then y a =1 Indicator y a is zero otherwise Note that the goal program will not allow the inflection point to be equal to the volume in any particular period unless using the difference equation for the taper B9–B10 vs the exponential segment of the curve B7–B8 does not make any difference in the objective function value Recall that in the goal program both the inflection point and the fitted volumes are variables Constraints B5 through B10 establish the relationship between the five function parameters ϕ min ϕ max γ exp γ taper and β that are to be optimized and the fitted volumes omega a forall a in A Along with inequalities B5–B6 Constraints B7–B10 capture Function 1 Lastly Constraints B11–B13 define the domains of the variablesWhile GP B1–B13 is a quadratically constrained nonconvex1 and nonsmooth2 optimization problem it is trivial to solve since it has only five decision variables ϕ min ϕ max γ exp γ taper and β and only A accounting variables for the fitted volumes omega a the deviations δ a and the indicators y a each Moreover there is only one pair of goal constraints per data point B2–B3 and only a few extra constraints to account for the relationship between the volumes in consecutive age classes B4–B10 GP B1–B13 can be converted to a smooth optimization problem by optimizing for each binary combination of y a ’s and selecting the combination that leads to the smallest total deviation In this study we used MS Excel Solver to fit Function 1In this appendix we discuss how Model IV can incorporate intermediate treatment decisions that are hard to capture in Models I and II parsimoniously Should a management unit be thinned in a particular planning period with a given intensity or not Should fertilization be applied to increase site productivity or should the trees be pruned Thinning reduces the volume of the unit at the time of treatment and puts it on a steeper growth trajectory for merchantable timber Thinings are applied in an attempt to maximize revenues or to improve forest health or both In addition to stimulating growth fertilization can also increase site productivity and pruning can change the quality of timber productsThe potential timings of treatments introduce additional prescriptions that are applicable to a management unit leading to an explosion of 0–1 variables in Model I While the explosion is less dramatic in Model II it is still an exponential function of the number of periods where intermediate treatments can occur This is because additional sets of “first” “last” and “intermediate harvest” variables are needed to link harvest decisions to intermediate treatments in prior or subsequent periods If in addition to the timing the intensity of the treatments is also variable then the combinatorial explosion is even more significantThe number of decision variables per unit that need to be introduced in Model IV to account for thinning decisions with set intensities is equal to the number of planning periods that are eligible for thinning The number of extra constraints Constraints C5–C8 is four times the number of eligible periods If the range of volumes that correspond to planning periods eligible for thinning falls exclusively below or above the inflection point then either C5–C6 or C7–C8 may be dropped Lastly alternative thinning intensities as well as fertilization and pruning decisions may be modeled the same way as thinning decisions For fertilization and pruning coefficient α will be one since no merchantable volume is removed from the unit The posttreatment growth rates in volume for fertilization or revenues for pruning will drive the transition of the units from one period to the next in accordance with Constraints C5–C8

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