# 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

- Beale EML (1988) Introduction to optimization. Wiley, New-YorkGoogle Scholar
- Bell S, Apostol D (2008) Designing sustainable forest landscapes. Taylor & Francis, LondonGoogle Scholar
- Burjorjee K (2007) A fast simple genetic algorithm. Available at http://www.mathworks.com/matlabcentral/fileexchange/15164
- CPLEX (2007) ILOG CPLEX 10.2.0 Reference Manual. ILOG CPLEX Division, Gentilly, FranceGoogle Scholar
- Dinkelbach W (1967) On nonlinear fractional programming. Manage Sci 13:492–498CrossRefGoogle Scholar
- Fourer R, Gay DM, Kernighan BW (1993) AMPL, a modeling language for mathematical programming. Boyd & Fraser Publishing Company, DanversGoogle Scholar
- Hargis CD, Bissonette JA, David JL (1998) The behaviour of landscape metrics commonly used in the study of habitat fragmentation. Landsc Ecol 13:167–186CrossRefGoogle Scholar
- Hargis CD, Bissonette JA, Turner DL (1999) The influence of forests fragmentation and landscape pattern on American martens. J Appl Ecol 36:167–186CrossRefGoogle Scholar
- Hof J, Bevers M (1998) Spatial optimization for managed ecosystems. Columbia University Press, New YorkGoogle Scholar
- Hof J, Bevers M (2002) Spatial optimization in ecological applications. Columbia University Press, New YorkGoogle Scholar
- Hof J, Joyce LA (1993) A mixed-integer linear programming approach for spatially optimizing wildlife and timber in managed forest ecosystems. For Sci 39:816–834Google Scholar
- Lindenmayer DB, Cunningham RB, Pope ML, Donnelly CF (1999) A large-scale “experiment” to examine the effects of landscape context and habitat fragmentation on mammals. Biol Conserv 88:387–403CrossRefGoogle Scholar
- Marks BJ, McGarigal K (1994) Fragstats: Spatial pattern analysis program for quantifying landscape structure. Technical report, Forest Science Department, Oregon State UniversityGoogle Scholar
- Matlab (2007) Matlab 7.0.4. The MathWorks, NatickGoogle Scholar
- Mitchell M (1998) An introduction to genetic algorithms. MIT Press, CambridgeGoogle Scholar
- Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley, New YorkGoogle Scholar
- Radzik T (1998) Fractional combinatorial optimization. In: Du D-Z, Pardalos PM (eds) Handbook of combinatorial optimization, vol 1. Springer, Heidelberg, pp 429–478Google Scholar
- Salkin HM, Mathur K (1989) Foundations of integer programming. North-Holland, AmsterdamGoogle Scholar
- Schaible S (1995) Fractional programming. In: Horst R, Pardalos P (eds) Handbook of global optimization. Kluwer, Dodrecht, pp 495–608Google Scholar
- Sisk TD, Haddad NM (2002) Incorporating the effect of habitat edges into landscape models: effective area models for cross-boundary management. In: Liu J, Taylor WW (eds) Integrating landscape ecology into natural resource management. Cambridge University Press, Cambridge, pp 208–240CrossRefGoogle Scholar
- Venema HD (2005) Forest structure optimization using evolutionary programming and landscape ecology metrics. Eur J Oper Res 164:423–439CrossRefGoogle Scholar
- Walters JR (1998) The ecological basis of avian sensitivity to habitat fragmentation. In: Marzluff J, Sallabanks R (eds) Avian conservation: research and management. Island Press, WashingtonGoogle Scholar
- Weintraub A, Murray AT (2006) Review of combinatorial problems induced by spatial forest harvesting planning. Discrete Appl Math 154(5):867–879CrossRefGoogle Scholar
- Weintraub A, Romero C, Bjorndal T, Epstein R (eds) (2007) Handbook of operations research in natural resources. Springer, HeidelbergGoogle Scholar
- Wolsey LA (1998) Integer programming. Wiley-Interscience, New YorkGoogle Scholar