Forest harvest scheduling with clearcut and core area constraints

Abstract

Many studies regarding environmental concerns in forest harvest scheduling problems deal with constraints on the maximum clearcut size. However, these constraints tend to disperse harvests across the forest and thus to generate a more fragmented landscape. When a forest is fragmented, the amount of edge increases at the expense of the core area. Highly fragmented forests can neither provide the food, cover, nor the reproduction needs of core-dependent species. This study presents a branch-and-bound procedure designed to find good feasible solutions, in a reasonable time, for forest harvest scheduling problems with constraints on maximum clearcut size and minimum core habitat area. The core area is measured by applying the concept of subregions. In each branch of the branch-and-bound tree, a partial solution leads to two children nodes, corresponding to the cases of harvesting or not a given stand in a given period. Pruning is based on constraint violations or unreachable objective values. The approach was tested with forests ranging from some dozens to more than a thousand stands. In general, branch-and-bound was able to quickly find optimal or good solutions, even for medium/large instances.

Apendix: Determining core area

To determine core area, the forest is classified into subregions with the geographic information system ArcGis 9.2. A surrounding impact zone of a given width for each stand is created, using the tool Buffer, available in ArcToolbox $\setminus$ Analysis Tools $\setminus$ Proximity. Then, subregions are provided by ArcToolbox $\setminus$ Analysis  Tools $\setminus$ Overlay. For each subregion, the defining set and the area are displayed by default. Centroids, useful to distinguish subregions with the same defining set and the same area (see subregions $C_4$ and $C_5$ in Fig. 8 and Example 2), are also computed, using the tools Feature to Point and Add XY Coordinates from ArcToolbox $\setminus$ Data  Management  Tools $\setminus$ Features.

Example 2

Figure 8 provides another example of a mature forest with three stands, A, B and C, before and after intersecting the stands and impact zones. The sets of stands determining if subregions $C_4, C_5$ are core area are: $\mathcal {I}_{C_4} = \mathcal {I}_{C_5}=\{C,A,B\}$.

Fig. 8

a Three-stand mature forest. b Subregions

The choice of harvesting a stand (a left branch in the branch-and-bound tree) requires updating the total area and the total core area of habitats. For node k, let $H^k_t$ be a list that contains the existing habitats in period t. Let $\tau _{h}$ be the total area and $\gamma _{h}$ be the total core area of habitats $h \in H^k_t$. At node $k+1$, where stand $i_k$ is selected to be harvested in period $t_k$, updates to $H^k_{t_k}$, $\tau _{t_k}^k$ and $\gamma _{t_k}^k$ are done according to the following three possibilities:

• (a) Stand $i_k$ belongs to a habitat $h \in H^k_{t_k}$ and harvesting $i_k$ leads to one smaller patch $h'$ (Fig. 9a). Let $\textsf {s} _{i_k}$ be the area of $i_k$ and R be the amount of core area removed, i.e., the core area of h that was inside $i_k$ plus the new edge in $h'$ caused by harvesting $i_k$; then:

  • If $h'$ meets the minimum core area $\textsf {C} ^ \textsf {min}$ requirement for a habitat, then:

    $$\begin{aligned}&H^{k+1}_{t_k} = H^{k}_{t_k} \setminus \{h\} \cup \{h'\} \\&\gamma _{h'} = \gamma _{h} - R \\&\tau _{h'} = \tau _{h} - \textsf {s} _{i_k} \end{aligned}$$

  • Otherwise, h is removed: $H^{k+1}_{t_k} = H^{k}_{t_k} \setminus \{h\}$.

• (b) Stand $i_k$ belongs to habitat $h \in H^{k}_{t_k}$ and harvesting $i_k$ splits up h into a set of new patches $N = \{h_1, \ldots , h_m\}$ (Fig. 9 b). In this case, it is necessary to calculate the area and the core area of each new patch $h' \in N$, as in (a).

  • New habitats are the newly formed patches that meet the requirement for a habitat:

    $$\begin{aligned} H^{k+1}_{t_k} = H^{k}_{t_k} \setminus \{h\} \cup \{h' \in N : \gamma _{h'} \ge \textsf {C} ^ \textsf {min} \} \end{aligned}$$

• (c) Stand $i_k$ (belonging or not to a habitat $h \in H^{k}_{t_k}$) causes edge effects on other habitats $h_1,\ldots ,h_m$ (Fig. 9c). In this case, the core area of each affected habitat $h'$ is updated by subtracting the new edge caused by harvesting $i_k$.

  • Each $h'$ that becomes non-habitat is removed from $H^{k+1}_{t_k}$.

Fig. 9

Mature forest with four stands, where patch $\{A, B, C\}$ is habitat. After harvesting stands A (left), B (center) or D (right), the dotted and dark shaded areas are subtracted from the habitat core area, and the dotted area is subtracted from the habitat area