Exploring the tradeoffs among forest planning, roads and wildlife corridors: a new approach

Abstract

Protecting wildlife corridors is a common management problem in regions of industrial forestry. In boreal Canada, human disturbances have negatively affected woodland caribou populations (Rangifer tarandus caribou), which prefer to function in large undisturbed areas. We present a linear programming model that allocates a fixed-width corridor between isolated caribou ranges and estimates its impact on harvest activities. Our corridor placement problem minimizes total resistance for caribou passing through the corridor, which is protected by a prohibition on all economic activities. We link this corridor placement problem with a harvest planning problem that maximizes the net revenues from harvest minus the cost of building and maintaining forest access roads. We depict gradual expansion of the forest road network over time as a multi-temporal network flow problem. We applied our approach to explore corridor options for connecting caribou populations in the Lake Superior Coast Range, with the Nipigon and Pagwachuan Ranges in the Kenogami-Pic Forest, in northern Ontario, Canada. Our results revealed two locations where corridor placement is cost-effective. Optimal corridor placement depends on the perception of the severity of the impact of roads on caribou populations and decision-making objectives. When the negative impact of roads is perceived to be high and/or maximizing harvest revenues is important, the optimal corridor location is in the eastern part of the study area. However, it is optimal to place the corridor in the western part of the area when the negative impact of roads is perceived to be small or the shortest corridor is desired.

Appendices

Appendix 1

Model initialization

Hard combinatorial complexity makes it difficult to find feasible solutions for the full problem with large datasets. We used the following initialization procedure to warm start the full problem in Eq. (30) in the main text. First, we solved the corridor placement problem without harvest planning, i.e.:

$$ \min \sum\limits_{n = 2}^{N - 1} {\left( {w_{n} \sum\limits_{t = 1}^{T} {b_{n1t} } } \right)} $$

(34)

subject to constraints (2–7) in the main text (see symbol definitions in Table 1 in the main text).

We then used the w_n values (which define optimal corridor placement in problem (34)) as a fixed parameter, w’_n, to solve the harvest planning problem with a fixed corridor.

To initialize the road construction model, we modified the problem formulation by introducing separate time sets for the harvest planning problem and the road construction sub-problem, t ∈ T and t’ ∈ T’, respectively. Harvest is always allocated over the full planning horizon T, but road construction may be planned over a shorter time span T’, T’ ≤ T. All equations in the road construction sub-problem use the time set T’ (a subset of the time set T), i.e.:

$$ \max \sum\limits_{n = 2}^{N - 1} {\left[ {\sum\limits_{i = 1}^{I} {(R_{ni} x_{ni} )} - \sum\limits_{t^{\prime} = 1}^{T^{\prime}} {\sum\limits_{m = 2}^{N - 1} {(v_{mnt^{\prime}} [c_{mn} + j_{mn} (T - t)])} } - f_{1} \sum\limits_{t^{\prime} = 1}^{T^{\prime}} {P_{{1_{nt} ^{\prime}}} } } \right]} - f_{2} \sum\limits_{t^{\prime} = 1}^{T^{\prime}} {P_{{2_{t} ^{\prime}}} } $$

(35)

s.t.: harvest planning constraints (11–13),(19) in the main text and:

$$ y_{1nt^{\prime}} \, = \, 0 \, \forall \, t^{\prime} \in T^{\prime}, \, n = 0,2,...,N - 1 $$

(36)

$$ y_{nNt^{\prime}} \, = \, 0 \, \forall \, t^{\prime} \in T^{\prime},n = 0,2,...,N - 1 $$

(37)

$$ \sum\limits_{{m = 2,\{ 0\} }}^{{(N - 1)_{n}^{ - } }} {y_{mnt^{\prime}} } - \sum\limits_{m = 2}^{{(N - 1)_{n}^{ + } }} {y_{nmt^{\prime}} = \sum\limits_{{m = 2,\{ 0\} }}^{{(N - 1)_{n}^{ - } }} {v_{mnt^{\prime}} } } \, \forall \, t^{\prime} \in T^{\prime},n,m = \{ 0\} ,2,...,N - 1 $$

(38)

$$ y_{nmt^{\prime}} \le Uv_{nmt^{\prime}} \, \forall \, t^{\prime} \in T^{\prime}, \, n,m = \{ 0\} ,2,...,N - 1 \, $$

(39)

$$ v_{nmt^{\prime}} \le y_{nmt^{\prime}} \, \forall \, t^{\prime} \in T^{\prime}, \, n,m = \{ 0\} ,2,...,N - 1 \, $$

(40)

$$ v_{0nt^{\prime}} \le \Gamma_{n} + \sum\limits_{u^{\prime} = 1}^{t^{\prime} - 1} {\sum\limits_{{m = 2,\{ 0\} }}^{N - 1} {v_{nmu} } } \, \forall \, t^{\prime} = 2,...,T^{\prime},u^{\prime} \in T^{\prime},n = \{ 0\} ,2,...,N - 1, \, \Gamma_{n} \in \{ 0,1\} $$

(41)

$$ \sum\limits_{u^{\prime} = 1}^{t^{\prime}} {\left[ {\sum\limits_{\substack{ m = 2,\{ 0\} \\ } }^{N - 1} {v_{mnu^{\prime}} } } \right]} \, \ge \sum\limits_{i = 1}^{I} {(x_{ni} W_{nit^{\prime}} )} \, \forall \, u^{\prime} \in T^{\prime},t^{\prime} \in T^{\prime},n = 2,...,N - 1, \, W_{nit^{\prime}} \in \{ 0,1\} |\Gamma_{n} = 0, $$

(42)

$$ \sum\limits_{t^{\prime} = 1}^{T^{\prime}} {v_{nmt^{\prime}} \le 1} \, \forall \, n,m \in 2,...,N - 1 $$

(43)

$$ \sum\limits_{{m = 2,\{ 0\} }}^{{(N - 1)_{n}^{ - } }} {v_{mnt^{\prime}} } \le 1 \, \forall \, t^{\prime} \in T^{\prime}, \, n = \{ 0\} ,2,...,N - 1 $$

(44)

$$ v_{nmt^{\prime}} + v_{mnt^{\prime}} \le 1 \, \forall \, n,m \in N $$

(45)

$$ v_{{_{nmt^{\prime}} }} \le 2 - \Gamma_{n} - \Gamma_{m} \, \forall \, n,m = 2,...,N - 1,t^{\prime} \in T^{\prime}|\Gamma_{n} = 1 \wedge \Gamma_{m} = 1 $$

(46)

$$ \sum\limits_{i = 1}^{I} {\left( {x_{ni} \sum\limits_{t^{\prime} = 1}^{T^{\prime}} {V_{nit^{\prime}} } } \right)} = 0 \, \forall \, n = 2,...,N - 1|\xi_{n} = 0 $$

(47)

$$ P_{1nt^{\prime}} \ge \sum\limits_{{m = 2,\{ 0\} }}^{N - 1} {v_{mnt^{\prime}} } - \sum\limits_{k = 2}^{N - 1} {v_{nkt^{\prime}} - \sum\limits_{i = 1}^{I} {(x_{ni} W_{nit^{\prime}} )} } \, \forall \, t^{\prime} \in T^{\prime},n = 2,...,N - 1 $$

(48)

$$ P_{2t^{\prime}} \ge \sum\limits_{{m = 2,\{ 0\} }}^{N - 1} {\sum\limits_{{n = 2,\{ 0\} }}^{N - 1} {(v_{mnt^{\prime}} )} } - B_{t^{\prime}} \, \forall \, t^{\prime} \in T^{\prime} $$

(49)

$$ \sum\limits_{i = 1}^{I} {\left( {x_{ni} \sum\limits_{t^{\prime} = 1}^{T^{\prime}} {W_{nit^{\prime}} } } \right) \le \lambda (1 - w^{\prime}_{n} )} \, \forall \, n = 2,...,N - 1 $$

(50)

$$ \sum\limits_{t^{\prime} = 1}^{T^{\prime}} {\left( {\sum\limits_{{n = 2,\{ 0\} }}^{N - 1} {v_{mnt^{\prime}} } } \right) \le 1 - w^{\prime}_{n} } \, \forall \, n = 2,...,N - 1 $$

(51)

where w’_n is a binary parameter equal to the optimal w_n values from the corridor placement solution of the problem (34). The formulation in equations (34–53) above is similar to the harvest planning with road construction formulation in Eqs. (10–28) in the main text, except that a separate time domain T’ is used in equations defining the road construction sub-problem. Using a shorter time domain for the road construction sub-problem, while solving the harvest allocation for the full planning horizon T with an account for road building cost over a time span T’), reduces the numeric complexity of the problem and makes it possible to find feasible solutions via a set of T consecutive optimizations with a stepwise increase of the time horizon T’ from 1 to T, as described below.

The initialization was started by setting the road construction horizon T’ to one period (i.e., the first period, T’ = 1), but solved the harvest allocation problem for the full horizon T. After finding the harvest solution for T periods and optimal road construction pattern for period 1, we set the time domain T’ to two periods and solved the problem again using the decision variables x_ni, v_nmt’ and y_nmt’ from the solution with T’ = 1 as a warm start. After saving the optimal solution with T’ = 2, we increased the T’ value to three periods and solved the model again using x_ni, v_nmt’ and y_nmt’ from the solution with T’ = 2 as a warm start, and so on until we solved the model for T’ = T periods. To speed up the solution we included, at each solution step starting from T’ = 2, two more constraints which fixed the road construction decision variables v_nmt’ and y_nmt’ to their initialized values for the planning periods 1,..,T’-1 so that at each initialization step the model only needed to find the optimal road construction network for one period t’ = T’, i.e.:

$$ y_{nmt^{\prime}} = y^{\prime}_{nmt^{\prime}} \, \forall \, t^{\prime} = 1,...,T^{\prime} - 1, \, n,m = \{ 0\} ,2,...,N - 1 \, $$

(52)

$$ v_{nmt^{\prime}} = v^{\prime}_{nmt^{\prime}} \, \forall \, t^{\prime} = 1,...,T^{\prime} - 1, \, n,m = \{ 0\} ,2,...,N - 1 \, $$

(53)

where y’_nmt’ and v’_nmt’ are fixed parameters equal to the optimal values of decision variables y_nmt’ and v_nmt’ in the solution in the previous step.

After solving the problem repeatedly for a sequence of 1 to T’ horizons, the optimal solution depicted a short-sighted road planning policy where harvesting was optimized over the entire horizon T but the road construction network was optimized only within a single planning period t. We then used the set of decision variables from the last solution for T’ = T to warm start the full problem (30) in the main text. A similar procedure but without the corridor placement sub-problem (34) was used to solve the harvest-only problem with optimal road construction, In the harvest-only problem, the w_’n values were fixed to zeroes. We composed the model in the General Algebraic Modeling System (GAMS) [93] and solved it with the GUROBI linear programming solver [94].