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Optimal selection of forest patches using integer and fractional programming

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Abstract

Forest fragmentation occurs when large and continuous forests are divided into smaller patches. This fragmentation may result from natural processes, urban development, agricultural use, and timber harvesting. Many studies have shown that forest fragmentation have led to global biodiversity loss, hence forestry management needs to explicitly incorporate spatial ecology objectives. Given a set of forest patches distributed on a landscape, the fragmentation can be measured by many indicators. In this paper, we consider the three usual following indicators: the mean proximity index, the mean nearest neighbour distance, and the mean shape index. In a fragmented forest landscape, a natural objective is to select a subset of patches satisfying some constraints such as area constraint, and optimal regarding these indicators with the aim of protecting biodiversity. These optimisation problems have been already studied in the literature by heuristic methods. However, these algorithms which generally are fast and provide good solutions, have significant drawbacks. In this paper, we propose an original 0–1 linear programming formulation of the search for a subset of patches minimising the sum of the distances between every patch and its closest neighbour. Using this formulation, we show that it is possible to efficiently optimise forest patch selection in a landscape with regards to the previous metrics. The optimising procedure is based on integer fractional programming and integer linear programming. The mathematical programming models are simple. The implementations are immediate by using a mathematical programming language and integer linear programming software. And the computational experiments, carried out on simulated landscapes comprising up to 200 patches, show that the performance of this approach is excellent: A few seconds of computation are sufficient to find an optimal solution to each patch selection problem.

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Correspondence to Alain Billionnet.

Appendix: Fractional programming

Appendix: Fractional programming

In this appendix, we briefly recall basic definitions in fractional programming and the Dinkelbach algorithm, a classical and efficient algorithm for solving fractional programs. Let f(x) and g(x) denote real-valued functions which are defined on R n, the set of vectors with n components in R. The program FP: \( \max \left\{ {{\frac{f(x)}{g(x)}}:x \in S \subset R^{n} } \right\} \)is called a fractional program. We assume that f(x) > 0 for at least one feasible solution of FP, and g(x) > 0 for all feasible solutions of FP. If in FP, S is a subset of {0,1}n, then FP is called a fractional combinatorial program. An important particular case of fractional (combinatorial) optimisation arises when functions f(x) and g(x) are linear, and S is defined by linear constraints. In this case, we get a linear fractional (combinatorial) program. Let us associate to FP the auxiliary program FPλ where λ is a real parameter: max{f(x) − λg(x):x ∈ S}. Let r(λ) be the optimal value of FPλ, and x λ be an optimal solution of this program. One can prove that r(λ) = 0 if and only if (λ, x λ) is an optimal solution of FP, i.e. if and only if λ is the optimal value of FP and x λ is an optimal solution of this program. We can then formulate FP as follows: find λ ∈ R such that r(λ) = 0. From this formulation, we can solve program FP by applying classical methods for finding a root of a function. Some properties of the function r(λ) make that it is easy to find a root of this function. The Dinkelbach method for FP (Dinkelbach 1967) is an application of the classical Newton method to the problem of finding a root of r(λ). We recall below the algorithm.

1.1 Dinkelbach algorithm

  1. 1.

    λ ← λ0

  2. 2.

    compute x λ, a solution of FPλ

  3. 3.

    if f(x λ) − λg(x λ) ≠ 0 then λ ← f(x λ)/g(x λ); go to 2

  4. 4.

    else x λ is an optimal solution of FP

  5. 5.

    endif

If FP is a linear fractional (combinatorial) program, then the optimisation problem of step 2 consists in solving a (0–1) linear program. The reader can consult Schaible (1995) for further results in fractional programming and Radzik (1998) for further results in combinatorial fractional programming.

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Billionnet, A. Optimal selection of forest patches using integer and fractional programming. Oper Res Int J 10, 1–26 (2010). https://doi.org/10.1007/s12351-009-0062-6

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