4.3 Some Problems of Selecting Compact Reserves and their Mathematical Programming Formulation
4.3.1 Problem I: Protection, at the Lowest Cost, of at Least Ns Species of a Given Set, with a Compactness
Constraint
This problem can be formulated as the mathematical program P4.1 in which ρ designates the value that the compactness indicator of the selected reserve must not exceed. As in all the programs we have studied, the Boolean variablexiis equal to 1 if and only if zone ziis selected to form the reserve.
P4:1 :
TAB. 4.2– Three reserve selection problems with a compactness objective.
Problem no.
Problem statement Formulation
I Selection of a reserve,R, of minimal cost, allowing to protect, at a minimum, a given number of species, Ns, and such that the compactness indicator, Comp(R),
is lower than or equal to a given value,ρ.
min CðRÞ
II Selection of a reserve,R, of cost less than or equal to a given value,B, allowing the greatest possible number of species to be protected, and whose value of the compactness indicator, Comp(R), is less than or
equal to a given value,ρ.
max Nb1ðRÞ
III Selection of a reserve,R, of cost less than or equal to a given valueB, allowing to protect, at a minimum, a given number of species, Ns, and minimizing the
value of the compactness indicator, Comp(R).
min CompðRÞ
The economic function to be minimized represents the cost of the reserve.
According to constraints 4.1.1, variableykcan take the value 1 if and only if at least one zone inZkis protected. Constraint 4.1.2 expresses that the number of protected species must be greater than or equal to Ns. Constraint 4.1.3 imposes a compactness index less than or equal to ρ. Note that if we seek to protect all the species – Ns =m – we can replace constraints 4.1.1 and 4.1.2 by the single family of constraintsP
i2Zkxi1; k2S. If we want to obtain, among the optimal solutions of P4.1, a solution that maximizes the number of protected species, we only need to subtract from the economic function to be minimized the quantityeP
k2Sykwhereε is a sufficiently small constant. Similarly, if one wants to obtain, among the optimal solutions of P4.1, a solution that minimizes the value of the compactness criterion, it is sufficient to add to the economic function to be minimized the quantityeCompðRÞ whereεis a sufficiently small constant. Let us now study constraint 4.1.3 according to the criterion retained to measure compactness. Recall thatR¼ fzi :i¼1;:::;n; xi ¼1g.
Criterion No. 1. The compactness of a reserve is measured by the diameter of the reserve,i.e., by the maximal distance between two zones of the reserve (see appendix at the end of the book). In this case, P4.1 solves the problem by replacing the – generic– constraint 4.1.3 with one of the specific constraint sets C4.1or C4.2:
C4:1:xiþxj1 ði;jÞ 2Z2; i\j; dij[q;
C4:2:xiþ X
j2Z;j[i;dij[q
xj1þMð1xiÞ i2Z:
Constraints C4.1express that if the distance between any two zones,ziandzj, is greater than ρ then these two zones cannot both be part of the reserve. In other words, in this case, variables xi and xj cannot simultaneously take the value 1.
According to constraints C4.2, if zoneziis selected–xi= 1–then none of the zones located at a distance greater thanρfromzican belong to the reserve. In case zonezi is not retained–xi = 0–the corresponding constraint is inactive provided that the constantM is chosen large enough.
Criterion No. 2. Let us now consider the case where the compactness of a reserve,R, is measured by the total perimeter of the reserve divided by its total area:
CompðRÞ ¼ P
i2Rli2P
ði;jÞ2R2;i\j lij
.P
i2R ai. In this case, the problem con-sidered can be solved by program P4.1by replacing the–generic–constraint 4.1.3 by the specific constraint C4.3:
C4:3: P
i2Zlixi2P
ði;jÞ2Z2;i\j lijxixj P
i2Z aixi
q:
Compactness 61
The perimeter of the reserve is calculated by summing the perimeters of all the zones that constitute the reserve,P
i2Zlixi, and by subtracting twice the sum of the lengths of the borders common to each pair of zones of the reserve, 2P
ði;jÞ2Z2;i\jlijxixj. Note that, in the latter expression, many termslijare equal to 0.
The quantityP
i2Zaixirepresents the sum of the areas of each zone constituting the reserve,i.e., the total area of the reserve. Constraint C4.3is equivalent to constraint C4.4in which the first member is quadratic and the second member is linear:
C4:4:X
It is possible to replace C4.4with equivalent linear constraints (see appendix at the end of the book). To do this, each productxixj is replaced in C4.4by variableuij and 2 families of linearization constraints are added to force variablesuijto be equal to the productsxixj, at the optimum of the obtained program. Finally, the problem can be solved by program P4.1by replacing the–generic–constraint 4.1.3 by the set of specific constraints C4.5:
C4:5 :
Note that if the compactness criterion is the perimeter of the reserve and not the perimeter-to-area ratio, it is sufficient to replace the first constraint of C4.5by the constraint P
i2Zlixi2P
ði;jÞ2Z2;i\jlijuijq where ρ now refers to the maximal allowed perimeter.
Criterion No. 3. Let us now consider the case where the compactness of a reserve is measured by the sum of the distances between all the pairs of zones of the reserve:
CompðRÞ ¼P
ði;jÞ2R2;i\jdij. In this case, the problem considered can be solved by program P4.1by replacing the–generic–constraint 4.1.3 by the specific constraint C4.6:
C4:6: X
ði;jÞ2Z2;i\j
dijxixjq:
Constraint C4.6is quadratic since it involves the productsxixj. As in the case of C4.3, it is possible to replace this constraint by a set of linear constraints (see appendix at the end of the book). To do this, each productxixjis replaced in C4.6by variableuijand 2 sets of linearization constraints are added to force variablesuijto be equal to products xixj – at the optimum of the obtained program. Finally, the problem can be solved by program P4.1by replacing constraint 4.1.3 by the set of constraints C4.7:
C4:7 :