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.
Similar content being viewed by others
References
Beale EML (1988) Introduction to optimization. Wiley, New-York
Bell S, Apostol D (2008) Designing sustainable forest landscapes. Taylor & Francis, London
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, France
Dinkelbach W (1967) On nonlinear fractional programming. Manage Sci 13:492–498
Fourer R, Gay DM, Kernighan BW (1993) AMPL, a modeling language for mathematical programming. Boyd & Fraser Publishing Company, Danvers
Hargis CD, Bissonette JA, David JL (1998) The behaviour of landscape metrics commonly used in the study of habitat fragmentation. Landsc Ecol 13:167–186
Hargis CD, Bissonette JA, Turner DL (1999) The influence of forests fragmentation and landscape pattern on American martens. J Appl Ecol 36:167–186
Hof J, Bevers M (1998) Spatial optimization for managed ecosystems. Columbia University Press, New York
Hof J, Bevers M (2002) Spatial optimization in ecological applications. Columbia University Press, New York
Hof J, Joyce LA (1993) A mixed-integer linear programming approach for spatially optimizing wildlife and timber in managed forest ecosystems. For Sci 39:816–834
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 88:387–403
Marks BJ, McGarigal K (1994) Fragstats: Spatial pattern analysis program for quantifying landscape structure. Technical report, Forest Science Department, Oregon State University
Matlab (2007) Matlab 7.0.4. The MathWorks, Natick
Mitchell M (1998) An introduction to genetic algorithms. MIT Press, Cambridge
Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley, New York
Radzik T (1998) Fractional combinatorial optimization. In: Du D-Z, Pardalos PM (eds) Handbook of combinatorial optimization, vol 1. Springer, Heidelberg, pp 429–478
Salkin HM, Mathur K (1989) Foundations of integer programming. North-Holland, Amsterdam
Schaible S (1995) Fractional programming. In: Horst R, Pardalos P (eds) Handbook of global optimization. Kluwer, Dodrecht, pp 495–608
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 208–240
Venema HD (2005) Forest structure optimization using evolutionary programming and landscape ecology metrics. Eur J Oper Res 164:423–439
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, Washington
Weintraub A, Murray AT (2006) Review of combinatorial problems induced by spatial forest harvesting planning. Discrete Appl Math 154(5):867–879
Weintraub A, Romero C, Bjorndal T, Epstein R (eds) (2007) Handbook of operations research in natural resources. Springer, Heidelberg
Wolsey LA (1998) Integer programming. Wiley-Interscience, New York
Author information
Authors and Affiliations
Corresponding author
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.
λ ← λ0
-
2.
compute x λ, a solution of FPλ
-
3.
if f(x λ) − λg(x λ) ≠ 0 then λ ← f(x λ)/g(x λ); go to 2
-
4.
else x λ is an optimal solution of FP
-
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12351-009-0062-6