# 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.

### Keywords

Landscape ecology Forest fragmentation Fractional programming Integer programming Simulation experiments### References

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