Connectivity of Zones Outside the Reserve

To determine an optimal reserve that takes this constraint into account, we can proceed as follows: (1) solve P_4.6without taking it into account, (2) if the constraint is satisfied by the solution obtained, then this solution is the optimal solution to the problem, (3) if not, solve P_4.6 again, but with an additional constraint to prohibit the obtained configuration. The process is repeated until an admissible reserve is obtained. Computational experiments have shown that, at least under our experi-mental conditions, an admissible reserve is obtained directly– i.e., by solving P_4.6

Compactness 75

once – in more than one case out of three and that if this is not the case, a few iterations are sufficient to obtain an admissible solution. Let us look more precisely at a way of implementing point (3). Given a reserve, a set of “isolated” zones is defined as a set of candidate zones, not belonging to the reserve, in one piece – connected – that cannot be reached from outside the grid without crossing the reserve and that is maximal in the inclusion sense. A set of “isolating” zones is associated with any set of isolated zones as follows: any reserve containing all the zones of the set of isolating zones is admissible only if it contains all the zones of the set of isolated zones. In fact, we associate to any subset of isolated zones a set of isolating zones, minimal in the inclusion sense. IfZ^a denotes a set of isolated zones and Z^b, the set of isolating zones associated with Z^a, then any admissible reserve must satisfy constraint C_4.11. It should be noted that the reserve that had been obtained is no longer admissible.

C4:11: X

i:z_i2Z^a

xi j jZ^a 1 X

i:z_i2Z^b

ð1xiÞ

! :

Indeed, if all the zones of the set of isolating zones are selected, constraint C_4.11 becomes P

i:z_i2Z^axi jZ^aj and imposes that all the zones of the set of isolated zones are selected in the reserve. On the contrary, if qð 1Þ zones of the set of isolating zones are not selected, this constraint becomes P

i:z_i2Z^a xi jZ^aj ð1qÞ and is then inactive– always satisfied.

In summary, if the solution obtained by P_4.6 corresponds to a reserve with at least one set of isolated zones, it is necessary to add constraint C_4.11to the program and solve it again. The process must then be iterated – keeping the constraints already added – until a reserve is obtained without a set of isolated zones. Let us again take the example of figure 4.9and assume that the resolution of P_4.6results in the reserve of this figure. It is then necessary to add to P_4.6 the constraint x45þx462ð1þx35þx36þx44þx47þx55þx566Þ.

4.5.3 Computational Experiments

In order to test the effectiveness of the approach, we considered different instances of the problem and solved them with the mathematical program P_4.6. We considered hypothetical instances constructed from a set of zones forming a grid of 2020 identical square zones whose length of sides is equal to one unit, and 100 hypo-thetical species, which are divided into 3 groups whose weight in the economic function is equal to 10, 5, and 1, respectively.

Group I (species numbered 1 to 20): These species are rare; they are present in only 10% of the candidate zones and their presence is randomly selected.

Group II (species numbered 21 to 50): These species are relatively rare; they are present in only 20% of the candidate zones and their presence is randomly selected.

Group III (species numbered 51 to 100): These species are relatively common;

they are present in 30% of the candidate zones and their presence is randomly selected.

For each species present in a zone, its population size in that zone is chosen at random according to the uniform law, between 5 and 10 units. In order to simplify the presentation, the minimal size of the total population necessary for the survival of each of the 100 species considered is set to 25. The distance between two adjacent zones is equal to the distance between their centres, i.e., one unit. The costs asso-ciated with each zone are randomly selected, according to the uniform law, between 5 and 20 units. With regard to compactness, 5 values ofρare considered, 4, 5, 6, 7, and 8, and for each of these values, 3 values of the available budget, B, are con-sidered, 150, 300, and 450. We also randomly select 3 zones that must necessarily belong to the reserve and, on the contrary, 20 zones that cannot be included in it.

With these values, the reserves obtained allow for the protection of 0–100 species and the value of the economic function–the weighted number of protected species– varies from 0 to 400. The computation results are presented in table4.6. All the instances considered could be solved. Whenρ = 3, there are no admissible reserves, regardless of the value ofBconsidered, and the resolution of P_4.6for these 3 values of Brequires less than one second of computation. As expected, the resolution of P_4.6is very fast for small radius values,ρ, and slower for large values. Indeed, the number of Boolean variables t_ijh –associated with zonez_ijand levelh–increases rapidly with the value ofρ; in our experiments, this number is equal to 400(ρ+ 1). Several hours of computation are required to solve the problem when ρ= 8 andB = 300. It can also be seen that, for any fixed value of the radius, the CPU time increases with the budget up to a certain value and then decreases. The resolution of program P_4.6 provides, for 9 instances out of 15, a reserve with at least one enclave–zones outside the reserve and isolated. The results presented in table 4.6 show that, for these 9 instances, only a few iterations are sufficient to determine a reserve without an enclave. They also show that taking into account the connectivity constraint for the zones outside the reserve deteriorates only slightly the value of the economic func-tion. Figure4.12a shows the reserve obtained by solving P_4.6 with ρ = 6 and B = 300 without taking into account the connectivity constraint for the zones outside the reserve. We see that zone z₆₆is isolated. Figure 4.12b shows the optimal reserve without an enclave. Respecting the “no enclave” constraint does not sig-nificantly penalize the value of the solution: it decreases by only 5 units out of 310– less than 2%. On the other hand, the structure of the reserve has been profoundly modified. A table such as table 4.6can help a decision-maker to choose the level of compactness of the envisaged reserve. In this example, if he/she can only use a budget of 150 units, he/she can afford to look for a very compact reserve. Indeed, in this case, the compactness constraint does not influence the optimal value of the weighted number of protected species. In contrast, if he/she has a larger budget, such as 300 units, he/she must deal with a compromise between the compactness and the weighted number of protected species.

Compactness 77

TAB. 4.6–Results obtained by solving program P4.6for hypothetical instances, constructed from a grid 2020 with 100 hypothetical species, whenhk ¼25 for allk2S.

ρ B Number of protected species in each group

(I, II, III)

Economic function

value

CPU time (s)

Presence of enclaves in the obtained reserve

Number of iterations to obtain a reserve without enclaves

Final value of the economic

function

Additional CPU time

(s)

4 150 1, 11, 37 102 1 No – – –

300 7, 25, 50 245 1 Yes 4 235 3

450 8, 30, 50 280 1 Yes 2 270 1

5 150 1, 11, 37 102 3 No – – –

300 10, 28, 50 290 11 Yes 1 290 4

450 16, 30, 50 360 1 Yes 3 350 3

6 150 1, 11, 37 102 8 No – – –

300 11, 30, 50 310 174 Yes 2 305 292

450 20, 30, 50 400 3 Yes 5 390 28

7 150 1, 11, 37 102 24 No – – –

300 13, 28, 50 320 1,046 Yes 1 320 881

450 20, 30, 50 400 3 Yes 2 400 7

8 150 1, 11, 37 102 207 No – – –

300 13, 30, 50 330 10,267 Yes 3 325 27,292

450 20, 30, 50 400 5 No – – –

DesigningProtectedAreaNetworks

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(a) (b)

1 2 3 4 5 6 7 8 9 10 11 20 1 2 3 4 5 6 7 8 9 10 11 20 1 ¹ 2 2 3 3 4 4 5 ⁵ 6 6 7 7 8 ⁸ 9 9 10 ¹⁰ 11 11 12 ¹² 13 13 14 ¹⁴ 15 15 20 20 FIG. 4.12–Two optimal connected reserves, with a radius of 6 or less and a cost of 300 or less.

(a) The centre of the reserve is zonez96. 11 species in Group I and all the species in Groups II and III are protected. The value of the solution is 310 but zone z66, which does not belong to the reserve, is isolated by zonesz56,z67,z76, andz65. (b) The centre of the reserve is zone z95. 11 species in Group I and all the species in Groups II and III, except one species in Group II, are protected. The value of the solution is 305 and there is no enclave in this reserve.

Compactness 79

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Chapter 5