Promoting Large, Compact Mature Forest Patches in Harvest Scheduling Models

Abstract

Spatially explicit harvest scheduling models that can promote the development of dynamic mature forest patches have been proposed in the past. This paper introduces a formulation that extends these models by allowing the total perimeter of the patches to be constrained or minimized. Test run results suggest that the proposed model can produce solutions with fewer, larger, and more compact patches. In addition, patches are more likely to be temporally connected with this formulation. Methods for identifying the tradeoffs between the net present value of the forest and the size and perimeter of the evolving patches are demonstrated for a hypothetical forest.

Appendix

1.1 The Full Model (TOTALMIN)

$$ \min {\sum\limits_{t \in T} {\mu _{t} } } $$

(A1)

subject to:

$$ X_{{m0}} + {\sum\limits_{t = h_{m} }^T {X_{{mt}} } } \leqslant 1\quad \quad {\text{for}}\;m \in M $$

(A2)

$$ {\sum\limits_{m \in M_{{ht}} } {v_{{mt}} \cdot A_{m} \cdot X_{{mt}} } } - H_{t} = 0 {\quad\text{for}}\;t = 1,\,2, \ldots ,\,T $$

(A3)

$$ b_{{lt}} H_{t} - H_{{t + 1}} \leqslant 0\quad \quad {\text{for}}\;t = 1,2, \ldots ,\!T - 1 $$

(A4)

$$ \, - b_{{ht}} H_{t} + H_{{t + 1}} \leqslant 0\quad \quad {\text{for}}\;t = 1,2, \ldots ,\!T - 1 $$

(A5)

$$ {\sum\limits_{m \in M_{p} }\! {X_{{mt}} \leqslant \!{\left| {M_{p} } \right|}\! -\! 1} }{\quad\text{for}}\,\,{\text{all}}\;p \in P\;{\text{and}}\;t = k_{p} , \ldots ,\!T $$

(A6)

$$ {\sum\limits_{j \in J_{{mt}} } {X_{{mj}} - O_{{mt}} \ge 0} } {\,\,\text{for}}\;m \in M,\;{\text{and}}\;{\text{all}}\;t\;{\text{such}}\;{\text{that}}\;J_{{mt}} \ne \emptyset <$> <$> \kern216 pt {\text{(A7)}} $$

(A7)

$$ {\sum\limits_{j \in J_{{mt}} } {X_{{mj}} \!-\! {\left| {J_{{mt}} } \right|}O_{{mt}} \!\leqslant\! 0} } \,{\text{for}}\;m \!\in\! M,\;{\text{and}}\;{\text{all}}\;t\;{\text{such}}\;{\text{that}}\;J_{{mt}} \!\ne\! \emptyset <$> <$> \kern216 pt {\text{(A8)}} $$

(A8)

$$ {\sum\limits_{m \in MC_{c} }\!\!\! {O_{{mt}} } } -\! {\left| {MC_{c} } \right|}B_{{ct}} \ge\! 0{\;\text{for}}\;c \!\in\! C,\;{\text{and}}\;t \!\!=\!\! 1,2, \ldots ,\!T\kern-2pt$$

(A9)

$$ {\sum\limits_{m \in MC_{c} }\!\!\!{O_{{mt}} } } - B_{{ct}} \leqslant {\left| {MC_{c} } \right|} - 1\,{\text{for}}\;c \in C,\;{\text{and}}\;t \!=\! 1,2, \ldots ,T <$> <$> \kern211pt {\text{(A10)}} $$

(A10)

$$ {\sum\limits_{c \in C_{m} }\! {B_{{ct}} \!-\! {\text{BO}}_{{mt}} \ge 0} }{\;\;\text{for}}\;m \in M,\;{\text{and}}\;t \!=\! 1,2, \ldots ,\!T $$

(A11)

$${\sum\limits_{c \in C_{m} } {B_{{ct}} - {\left| {C_{m} } \right|}{\text{BO}}_{{mt}} \leqslant 0} }\,\,{\text{for}}\;m \in M,\;{\text{and}}\;t = 1,\,2, \ldots ,\,T$$

(A12)

$$ {\sum\limits_{m \in M} {A_{m} {\text{BO}}_{{mt}} } } \ge \lambda \quad \quad {\text{for}}\;t = 1,2, \ldots ,\!T $$

(A13)

$${\sum\limits_{m \in M} {P_{m} {\text{BO}}_{{mt}} } } - 2{\sum\limits_{pq \in N} {{\text{CB}}_{{pq}} \Omega ^{t}_{{pq}} } } = \mu _{t} \quad \quad {\text{for}}\;t = 1,\,2, \ldots ,\,T$$

(A14)

$$ {\text{BO}}_{{pt}} + {\text{BO}}_{{qt}} - 2\Omega ^{t}_{{pq}} \geqslant 0\quad \quad {\text{for}}\;t = 1,\,2, \ldots ,\,T,\;pq \in N $$

(A15)

$$ {\text{BO}}_{{pt}} + {\text{BO}}_{{qt}} - \Omega ^{t}_{{pq}} \leqslant 1\quad \quad {\text{for}}\;t = 1,\,2, \ldots ,\,T,\;pq \in N $$

(A16)

$$\begin{array}{*{20}l} {{{\sum\limits_{m \in M} {A_{m} } }\left[ {{\left( {{\text{Age}}^{T}_{{0t}} - \overline{{{\text{Age}}}} ^{T} } \right)}X_{{0t}} } \right.} \hfill} \\ {{\left. { + {\sum\limits_{t = h_{m} }^T {{\left( {{\text{Age}}^{T}_{{mt}} - \overline{{{\text{Age}}}} ^{T} } \right)}X_{{mt}} } }} \right] \geqslant 0} \hfill} \\ \end{array} $$

(A17)

$$X_{{mt}} \in {\left\{ {0,1} \right\}}\quad \quad {\text{for}}\;m \in M,\;{\text{and}}\;t = 0,\;h_{m} ,\;h_{m} + 1, \ldots ,\,T$$

(A18)

$$ B_{{ct}} \in {\left\{ {0,1} \right\}}\quad \quad {\text{for}}\;c \in C,\;t = 1,2, \ldots ,\!T $$

(A19)

$$ O_{{mt}} ,{\text{ BO}}_{{mt}} \!\in\! {\left\{ {0,1} \right\}}{\;\;\text{for}}\;m \in M,\;{\text{and}}\;t = 0,1, \ldots ,\!T $$

(A20)

$$ \Omega _{{pq}} \in {\left\{ {0,1} \right\}}\quad \quad {\text{for}}\;pq \in N $$

(A21)

where the variables are defined as follows:

X _mt :

A binary decision variable whose value is 1 if management unit m is to be harvested in period t for t = h _m , h _m+1,...,T. In other words, X _mt represents a harvesting prescription for management unit m. When t = 0, the value of the binary variable is 1 if management unit m is not harvested at all during the planning horizon (i.e., X _m0 represents the “do-nothing” alternative for management unit m). Note: in constraint sets (A7) and (A8), index j is used, in addition to t, to denote harvest periods. The new identifier is needed in these constraints because t is already used to define the period for which O _mt applies;

λ :

The minimum area of mature forest habitat patches over all periods;

μ _t :

The total perimeter of mature forest habitat patches in period t;

H _t :

A continuous variable indicating the total volume of sawtimber in m³ harvested in period t;

O _mt :

A binary variable whose value may equal 1 if management unit m meets the minimum age requirement for mature patches in period t; that is, the management unit is old enough to be part of a mature patch;

B _ct :

A binary variable whose value is 1 if all of the stands in cluster c meet the minimum age requirement for mature patches in period t; that is, the cluster is part of a mature patch;

BO _mt :

A binary variable whose value is 1 if management unit m is part of a cluster that meets the minimum age requirement for large mature patches; that is, the management unit is part of a patch that is big enough and old enough to constitute a large mature patch;

$\Omega ^{t}_{{pq}}$ :

A binary variable whose value is 1 if adjacent management units p and q are both part of a cluster that meets the minimum age requirement for large mature patches in period t;

and the parameters are as follows:

M :

The set of management units in the forest;

A _m :

The area of management unit m in hectares;

T :

The number of periods in the planning horizon;

h _m :

The first period in which management unit m is old enough to be harvested;

M _ht :

The set of management units that are old enough to be harvested in period t;

v _mt :

The volume of sawtimber in m³/ha harvested from management unit m if it is harvested in period t;

b _lt :

A lower bound on decreases in the harvest level between periods t and t + 1 (where, for example, b _lt  = 1 requires nondeclining harvest; b _lt  = 0.9 would allow a decrease of up to 10%);

b _ht :

An upper bound on increases in the harvest level between periods t and t + 1 (where, for example, b _ht  = 1 allows no increase in the harvest level; b _ht  = 1.1 would allow an increase of up to 10%);

M _p :

The set of management units in path p;

P :

The set of all paths, or groups of contiguous management units, whose combined area is just above the maximum harvest opening size (the term “path,” as used in this paper, is defined in the following discussion);

k _p :

The first period in which the youngest management unit in path p is old enough to be harvested;

J _mt :

The set of all prescriptions under which management unit m meets the minimum age requirement for mature patches in period t;

MC _c :

The set of management units that compose cluster c;

C :

The set of all clusters or groups of contiguous management units whose combined area is just above the minimum large mature patch size (the term “cluster,” as used in this paper, is defined in the following discussion);

C _m :

The set of all clusters that contain management unit m;

P _m :

The perimeter of management unit m in meters;

N :

The number of pairs of management units in the forest that are adjacent;

CB _pq :

The length of the common boundary between the two adjacent stands p and q in meters;

${\text{Age}}^{T}_{{mt}}$ :

The age of management unit m at the end of the planning horizon if it is harvested in period t; and

$\overline{{{\text{Age}}}} ^{T}$ :

The target average age of the forest at the end of the planning horizon.

Equation (A1) minimizes the sum of the perimeters that border all of the large mature forest patches that evolve over the entire planning horizon. Forest planning models generally consider management actions and the consequent state of the forest over a finite time period known as the planning horizon. The planning horizon is then subdivided into discrete planning periods, and it is assumed that all of the activities that occur within a given planning period happen at one point, typically the midpoint, of the period. In the example problems discussed in the paper, the planning horizon is 60 years, with three 20-year planning periods.

Constraint set (A2) consists of logical constraints that allow only one prescription to be assigned to a management unit, including a do-nothing prescription. Harvest variables (X _mt ) are only created for periods where the stand is old enough to be harvested. Constraint set (A3) consists of harvest accounting constraints that assign the harvest volume for each period to the harvest variables (H _t ). Constraint sets (A4) and (A5) are flow constraints that restrict the amount by which the harvest level is allowed to change between periods. In the example problems in this paper, harvests were allowed to increase by up to 15% from one period to the next or to decrease by up to 3%.

Constraint set (A6) consists of adjacency constraints generated with the path algorithm [13]. They limit the maximum size of a harvest opening, a restriction often imposed for legal or policy reasons, by prohibiting the concurrent harvest of any contiguous set of management units whose combined area just exceeds the maximum harvest opening size. The exclusion period imposed by these constraints equals one planning period, but the constraints can be modified easily to impose longer exclusion periods in integer multiples of the planning period. A “path” is defined for the purposes of the algorithm as a group of contiguous management units whose combined area just exceeds the maximum harvest opening size. These paths are enumerated with a recursive algorithm described in [13]. A constraint is written for each path and period in which all of the management units in the path are old enough to be harvested. (In the initial periods of the planning horizon, some of the management units in a path may not be mature enough to be harvested.) The constraints prevent the concurrent harvest of all of the management units in that path since this would violate the maximum harvest opening size.

Constraint sets (A7)–(A13) are the mature patch size constraints. Constraint sets (A7) and (A8) determine whether or not management units meet the minimum age requirement for mature patches. These constraints sum over all of the prescription variables (X _mj ) for a management unit under which the unit would meet the age requirement for mature patches in a given period. O _mt equals 1 if and only if one of these prescriptions has a value of 1, indicating that the management unit will be old enough in that period to be part of a large mature patch. As an example, if the initial age class of management unit m is 3 (41–60 years) and the minimum age requirement for mature patches is age class 4 (61–80 years), then only under prescriptions X _m0 = 1 or X _m3 = 1 (J _m2 = {0, 3}) can this unit become old enough by period 2 to be part of a mature patch. One pair of these constraints is written for each management unit in each period in which it is possible for the management unit to meet the age requirement for a mature patch (i.e., when $J_{{mt}} \ne \emptyset$). For example, if the initial age class of management unit m is 1 or 2 (0–40 years) in the above example, then under no prescription will this unit become old enough by period 2. O _m2 will never turn on in this case.

Constraint sets (A9) and (A10) determine whether or not a cluster of management units meets the minimum age requirement for mature patches. All possible clusters are enumerated using a recursive algorithm described in [23]. A cluster meets the age requirement for mature patches in period t if all of the management units that compose that cluster meet the age requirement, as indicated by the set of O _mt variables for the management units in that cluster. B _ct takes a value of 1 if and only if cluster c meets the age requirement in period t. These pairs of constraints are written for each cluster in each period including those in which for one or more stands in the cluster, the set J _mt is empty. It is clear, however, that in these periods, the cluster in question cannot meet the age requirement under any harvest scheduling scenario. We relied on the IP solver’s preprocessor to identify and eliminate these constraints.

Constraint sets (A11) and (A12) determine whether or not individual management units are part of a cluster that meets the minimum age requirement for mature patches, i.e., whether a management unit is part of patch that is big enough and old enough. Since the clusters overlap, this constraint set is necessary to properly account for the total area of large mature patch habitat. These constraints set the values of the variables that indicate whether a management unit is part of a patch that meets the minimum age and size requirement for large mature patches in period t. BO _mt  = 1 if and only if at least one of the clusters it belongs to meets the age requirement in that period. Constraint set (A13) specifies that the total mature patch area for each period must be larger than λ in all periods.

Constraint sets (A14)–(A16) work together. Constraints in (A14) calculate the total perimeter of all groups of stands that fulfill the minimum age and area requirements for large mature forest patches in period t, and assign this value to the accounting variables μ _t . The total perimeter ($\,{\sum\limits_{t \in T} {\mu _{t} } }$) is minimized by objective function (1). Constraints (A15) and (A16) control the values of the binary variables $\Omega ^{t}_{{pq}}$, which replace the otherwise nonlinear cross-product terms ${\left( {\Omega ^{t}_{{pq}} = {\text{BO}}_{{pt}} {\text{BO}}_{{qt}} } \right)}$ in (A14). Notice that constraint set (A16) is not necessary if objective function (A1) is minimization. On the other hand, if maximizing the edge habitat is the objective, then constraint set (A16) would be necessary and (A15) could be dropped.

Constraint (A17) is an ending age constraint. It requires the average age of the forest at the end of the planning horizon to be at least $\overline{{{\text{Age}}}} ^{T}$ years, preventing the model from overharvesting the forest. In the example problems in this paper, the minimum average ending age was set at 40 years, or one half the optimal economic rotation.

Constraint sets (A18)–(A21) identify the stand prescription, mature patch size, and the cross-product linearization ${\left( {\Omega ^{t}_{{PQ}} } \right)}$ variables as binary.