Other Spatial Aspects
5.5 Optimal Reserve in the Case of Non-Disjoint Candidate Zones
5.5.5 A Second Mathematical Programming Formulation
As before, the candidate zones for protection are considered to belong to a region represented by a grid of nr rows and nc columns. We put M = {1,…, nr} and N= {1,…, nc}. All the cells in this grid are identical squares whose side length is equal to one unit. Each candidate zone is made up of a set of cells in the grid, all in one piece. The number of cells in zoneziis denoted byni. Each cell is described by a pair composed of its row index and column index. In the example in figure5.13there are 17 rows and 24 columns and zonez7contains the 12 cells (10, 3), (10, 4), (10, 5), (10, 6), (11, 3), (11, 4), (11, 5), (11, 6), (12, 3), (12, 4), (12, 5), and (12, 6). In this new formulation of the problem, we use, as in the previous formulation, the Boolean
z2 z5
z1 z4
z3
z6 z8 z10
z9 z7
z12
z14 z15
z13
z11
FIG. 5.14–Each of the zonesz1,z5,z7,z10,z12, andz15can participate in the protection of the 4 species s1, s2, s3, and s4, each of the zones z2,z4,z8,z13, andz14 can participate in the protection of the 3 speciess5,s6, ands7, and each of the zonesz3,z6,z9, andz11can participate in the protection of the 3 speciess8,s9, ands10. The optimal solution is to decide to select the grey zonesz1,z5,z7,z8,z10,z11,z12,z13, andz15, and, therefore, not to select zonesz2,z3,z4,z6, z9, andz14. However, the choice of the zones selected for protection implies that a fraction of zones z2, z3, z6, z9, and z14 is also protected. For this solution, the weighted number of protected species is 37.55 and the total area of protected zones is 94.
variablexiwhich is equal to 1 if and only if we decide to select zoneziand we also use the Boolean variable trc which is equal to 1 if and only if the cell (r,c) is selected (taking into account the decisions made regarding the zones to be selected). Note that it is not necessary to define variables trcon all the grid cells representing the region in question; it is sufficient to define them on all the cells belonging to at least one candidate zone. We denote by RC all the pairs ðr;cÞ 2M N such that the cell (r,c) belongs to at least one zone. So, RC¼ fðr;cÞ 2MN : 9zi 2Z such thatðr;cÞ 2zig. The notation“ðr;cÞ 2zi”means that the cellðr;cÞ– ris the row index andcis the column index of this cell– is included in the zonezi. Note also that in this formulation, variableαiused in the previous formulation is no longer necessary. The linear program in Boolean variables P5.11solves the problem.
P5:11:
In the expression of the economic function of P5.11, the quantityP
ðr;cÞ2ziðtrc=aiÞ represents the proportion of the area of zone zi that is protected. The economic function–to be maximized–therefore represents the weighted number of protected species. Constraint 5.11.1 expresses the area constraint since the total protected area is equal toP
ðr;cÞ2RC trc. Constraints 5.11.2 express the fact that if it is decided to select zonezi–xi = 1–then all the cells in this zone are protected. In other words, if xi = 1, thentrc = 1 for all the cells (r,c) ofzi. According to constraints 5.11.3, a cell is selected if at least one of the zones containing it is selected. Constraints 5.11.4 and 5.11.5 specify the Boolean nature of variables xiandtrc.
5.5.6 Computational Experiments
The formulation of the problem by program P5.11is much easier than by program P5.10 – and its linearization. Indeed, the formulation P5.10 requires the list of the zones, the area of each zone, but also the list of all the intersections of zones, 2 to 2, 3 to 3, etc. Formulation P5.11only requires the list of zones and, for each zone, the list of the cells that compose it. In both formulations, it is also necessary to know, of course, the list of the species living in each zone. Table5.2gives some computational results with the formulation P5.11. The zones are rectangles distributed in a grid, as in figure5.13. The coordinates, in the grid, of the cell located at the top left of each rectangle are drawn at random. The lengths of each side of the rectangles are random integers drawn uniformly between 1 and 50. The number of species is set at 200 and the presence of a given species in a given zone is also randomly selected with a certain probability.
Other Spatial Aspects 103
TAB. 5.2– Resolution of P5.11: Some computational results on large-sized instances.
Dimension of the grid (nr×nc)
Total number of
candidate zones (n)
Total area of candidate
zones
Probability of occurrence of speciesskin the
zonezi
Maximal area of protection
(Amax)
Solution value
Number of zones selected
Total area
CPU time (s)
Number of nodes in the search
tree
500×500 1,000 212,909 0.01 4,000 304.4 88 3,999 228 0
500×500 1,000 212,909 0.1 4,000 2,585.4 104 4,000 236 222
1,000×1,000 1,000 479,858 0.01 4,000 275.4 89 3,998 253 143
1,000×1,000 1,000 479,858 0.1 4,000 2,283.9 105 3,998 236 0
1,000×1,000 2,000 712,883 0.1 4,000 3,234.4 138 4,000 973 0
1,000×1,000 2,000 712,883 0.1 8,000 4,723.1 200 8,000 1,147 0
1,000×1,000 2,000 712,883 0.01 8,000 560.3 170 7,999 862 0
DesigningProtectedAreaNetworks
The results presented in table5.2show that large-sized instances of the problems can be solved relatively quickly. The longest instance to resolve requires about 19 min of CPU time. It can also be seen that the number of nodes developed in the search tree by the solver is low and often even zero. This is partly due to the fact that the value of the optimal solution of the continuous relaxation of program P5.11, which is obtained by replacing xi2 f0;1g and trc2 f0;1g by 0xi1 and 0trc1, respectively, is not far from the value of the optimal solution of P5.11.
References and Further Reading
Batisse M. (1982) The biosphere reserve: a tool for environmental conservation and management, Environ. Conserv.9, 101.
Batisse M. (1990) Development and implementation of the biosphere reserve concept and its applicability to coastal regions,Environ. Conserv.17, 111.
Billionnet A. (2010) Solving a cut problem in bipartite graphs by linear programming: application to a forest management problem,Appl. Math. Model.34, 1042.
Billionnet A. (2011) Spatial optimization of wildlife populations with probabilistic habitat connections,For. Sci.57, 336.
Clemens M.A., ReVelle C.S., Williams J.C. (1999) Reserve design for species preservation,Euro.
J. Oper. Res.112, 273.
Ebregt A., De Greve P. (2000)Buffer zones and their management - policy and best practices for terrestrial ecosystems in developing countries. National Reference Centre for Nature Manage-ment, International Agricultural Centre, Wageningen, the Netherlands, p. 64.
Fletcher R.J. Jr. (2005) Multiple edge effects and their implications in fragmented landscapes,J.
Anim. Ecol.74, 342.
Hamaide B., Williams J.C., ReVelle C.S. (2009) Cost-efficient reserve site selection favoring persistence of threatened and endangered species,Geogr. Analysis41, 66.
Hof J., Bevers M. (1998)Spatial optimization for managed ecosystems. Columbia University Press.
Holzkämper A., Lausch A., Seppelt R. (2006) Optimizing landscape configuration to enhance habitat suitability for species with contrasting habitat requirements,Ecol. Model.198, 277.
Laurance W.F., Yensen E. (1991) Predicting the impacts of edge effects in fragmented habitats, Biol. Conserv.55, 77.
Lindenmayer D., Franklin J., Eds (2002) Conserving forest biodiversity - a comprehensive multiscaled approach. Island Press.
Martino D. (2001) Buffer zones around protected areas: a brief literature review,UCLA, Electro.
Green J.1, 20.
Millspaugh J.J., Thompson F.R., Eds. (2008)Models for planning wildlife conservation in large landscape. Elsevier.
Nemhauser G., Wolsey L.A. (1988)Combinatorial optimization. Wiley-Interscience.
Schonewald-Cox C., Bayless J. (1986) The boundary model: a geographical analysis of design and conservation of nature reserves,Biol. Conserv.38, 305.
Williams J.C., ReVelle C.S. (1996) A 0-1 programming approach to delineating protected reserves, Environ. Plan. B: Plan. Des.23, 607.
Williams J.C., ReVelle C.S. (1998) Reserve assemblage of critical areas: a zero-one programming approach,Euro. J. Oper. Res.104, 497.
Woodroffe R., Ginsberg J.R. (1998) Edge effects and the extinction of populations inside protected areas,Science280, 2126.
Other Spatial Aspects 105