Connectivity Properties in Forest Exploitation

5.4.1 Optimal Protection of Two Species

The problem studied in this section is similar to the one studied in section 5.3, in that it consists in defining the exploitation of a forest region in order to protect certain species present in that region. We consider a set, Z, of square and identical forest zones represented by a grid of nr rows and nc columns and two species,s₁and s₂, living in these zones. Denote by z_ij the zone at the intersection of row i and column j, and Z the set of index pairs associated with the zones, i.e., Z¼

f1;:::;nrg f1;:::;ncg. The problem is to determine the zones to be cut and the

zones to be left as they are in order to maximize the weighted sum of the population sizes of speciess₁ands₂. The weightw₁is assigned to the population size of species s₁and weightw₂to the population size of speciess₂. The total expected population size of species s₁ in each cut (resp. uncut) zone z_ij is equal to n_ij (resp. 0). The calculation of the population size of speciess₂differs from that of section5.3. The habitat of this species is composed of uncut zones but in each zone, its population size depends on the connection of this zone with the other uncut zones, more pre-cisely on the probability that this zone is connected to at least one other uncut zone.

Several studies aiming to optimize landscape configuration take into account this type of dependence between zones. Hof and Bevers (1998) propose a simple linear approximation of the population size of species s₂for a particular case of the con-nection probabilities. We propose here a general method, which can be used with any set of connection probabilities, to estimate with great accuracy the population size of this species. As in section 5.3, with each zone z_ij is associated a Boolean variable,xij, which is equal to 1 if and only if the zone is uncut. The population size of species s₁in the set of considered zones is then equal toP

ði;jÞ2Z n_ijð1x_ijÞ. The population size of species s₂is more difficult to estimate. As mentioned above, its habitat consists of uncut zones. The population size of this species is equal to P

ði;jÞ2Z pijPRijxij where PRij ð0PRij1Þ refers to the connectivity of zone z_ij with other uncut zones, andπijis the population size of speciess₂in zonez_ijwhen PRij ¼1 andxij ¼1. Two zones,z_ijandz_kl, are considered to be connected with a certain probability, denoted by pr_ijkl ð0pr_ijkl\1Þ. It is further assumed that all these probabilities are independent. The connectivity of zonez_ijis measured by the probability that this zone is connected to the other uncut zones and we assume that this probability is equal to the probability that this zone is connected to at least one other uncut zone. The connectivity of zone z_ij, PR_ij, is therefore equal to 1 Q

ðk;lÞ 2Z1pr_ijklx_kl

with pr_ijij¼0. It is further assumed that speciess₂can only survive in the set of considered zones if at least TH of these zones are not cut.

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5.4.2 Illustration of the Problem

Let us consider a forest region represented by a grid of 55 square and identical zones (figure 5.10). In each zone, the values of πij and n_ij are specified, πij being placed above n_ij. A feasible solution is shown in this figure in which the grey zones are the uncut zones (x_ij= 1).

Given zonez_ij, suppose that pr_ijkl¼0:5 if zonez_klbelongs to the set of zones that immediately surround zonez_ij, that pr_ijkl¼0:15 if zonez_klbelongs to the set of zones that surround the previous set of zones, and that pr_ijkl¼0 for the other zones z_kl. Figure 5.11 shows the values of pr_44kl. The values of PRij for the uncut zones in figure5.10are presented in figure 5.12.

For the solution presented in figure 5.10, the population size of species s₁, P

ði;jÞ2Z n_ijð1x_ijÞ, is equal to 815 and that of speciess₂,P

ði;jÞ2ZpijPR_ijx_ij, is equal to 66.66.

FIG. 5.10–A hypothetical forest region consisting of 25 zones where the values ofπijandnij, πijabovenij, are specified. A solution with 9 cut zones and 16 uncut zones.

FIG. 5.11– Values of pr_44klfor allðk;lÞ 2 f1;:::;5g².

5.4.3 Mathematical Programming Formulation

Using variable σ1(resp.σ2) to represent the population size of speciess₁(resp. s₂) and the Boolean variable bto express the constraint on the number of zones that must be uncut so that species s₂ can survive, Hof and Bevers (1998) propose to formulate the problem as the mixed-integer non-linear program P_5.6.

P5:6:

γ is a constant that must be greater than or equal to the maximal value that variable σ2 can take. We can set, for example, c¼P

ði;jÞ2Zp^ij. Thus, if b= 0, constraint 5.6.4 forces variableσ2to take the value 0 and ifb= 1 this constraint is inactive. Because of constraint 5.6.5, the Boolean variable b takes the value 0 if P

ði;jÞ2Zx_ij\TH and the value 1 – at the optimum – in the opposite case. The economic function and all the constraints are linear, except constraints 5.6.2 and 5.6.3. Hof and Bevers (1998) solve P_5.6in an approximate way using a simple linear approximation of the population size of species s₂for a particular case of the con-nection probabilities. We propose below a general method for solving P_5.6, also in an approximate way, but valid whatever the definition of the connection probabilities.

In addition, the population size of speciess₂is evaluated with a great accuracy.

FIG. 5.12– Values of PR_ijassociated with the solution of figure5.10.

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Using the same technique as in section7.5of chapter7, an approximate solution of program P_5.6and an upper bound of its optimal value can be obtained by solving a mixed-integer linear program. To do this, first rewrite P_5.6as P_5.7by replacing, in P_5.6, the product of variables PR_ijx_ij with variable e_ij. Because of the objective function to be maximized, constraints 5.7.3 and 5.7.4 imply eij ¼PRijxij at the optimum.

Using the properties of the logarithmic function and taking into account that variables x_kl,ðk;lÞ 2Z, are Boolean, log Q

P_5.8 is not yet a linear program – in mixed-integer variables – because of the expression logð1eijÞthat appears in constraints 5.8.3. Using the same technique as in section 7.5 of chapter 7, a relaxation of program P_5.8 is obtained by replacing

constraints 5.8.3 by the constraints _u¹ the optimal value of P_5.6. To obtain a good approximate solution of P_5.6,qmust be large enough but, the largerqis, the greater the number of constraints 5.9.3 is.

5.4.4 Example

provide an – approximate – solution to the problem, with a value of 2,237.8.

The relative error made in retaining this feasible solution rather than an optimal solution is, therefore, at most equal to 1:7 10⁴. Note that, in this approximate solution, the true value of the population size of speciess₂is 43.3401 while the value of variableσ2is 43.3587.

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5.5 Optimal Reserve in the Case of Non-Disjoint