Optimal Protection of Two Species

We consider a set, Z, of forest zones – or parcels – that are square and identical, represented by a grid of nr rows and nc columns and two species,s₁ands₂. Denote byz_ijthe zone at the intersection of rowiand columnj,lthe side length of the zones and Z, the set of index pairs associated with the zones, i.e., Z ¼ f1;:::;nrg f1;:::;ncg. The habitat of speciess₁is mainly in cut zones and the habitat of species s₂is mainly in the edges between cut and uncut zones. For example, the goshawk population likes this edge habitat, in the vicinity of which there are open zones where it can hunt small mammals living in the same habitat. To simplify the pre-sentation, it is considered that all the zones represented by the grid are initially uncut and that the zone outside the grid is a cut zone. The total expected popu-lation size of speciess₁in each cut (resp. uncut) zonez_ijis equal ton_ij(resp. 0). The total expected population size of species s₂ is equal to gL where g refers to the expected population size of speciess₂for each kilometre of edge andLto the total edge length taking into account the cuts made. 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 total population sizes of species s₁ and s₂. The weighting reflects the different importance given to the two species. The weight w₁ is assigned to the population of species s₁and weight w₂to the population of speciess₂.

5.3.2 Mathematical Programming Formulation

First, let us give the formulation proposed by Hof and Bevers (1998). These authors associate to each zonez_ijthe Boolean variablex_ijwhich is equal to 1 if and only if the zone is not cut. They also associate to each zonez_ijthe additional positive or zero variabled_ij, which represents the number of sides of this zone–from 0 to 4–that do not form part of the edge when this zone is uncut; in the case where zonez_ijis cut, variable d_ij is equal to 0. Finally, these authors formulate the problem as the mixed-integer linear program P_5.2.

P5:2: adjacent to zonez_ij. Remember thatw₁andw₂are the weighting coefficients andlis the side length of each parcel. The first part of the economic function expresses the total weighted population size of speciess₁. Indeed, the total population size of this species in zonez_ijis equal ton_ijif the parcelz_ijis cut–x_ij= 0–and to zero if the parcel is not cut– x_ij= 1. The second part of the economic function expresses the weighted total population size of speciess2since the total length of the edge can be

Other Spatial Aspects 87

calculated by summing, on all uncut zones, the zone’s contribution to this edge.

We can verify that with the definition of variabled_ij, the contribution of the uncut zonez_ijto the length of the edge is equal to 4x_ij−d_ij. The total length of the edge is therefore equal tolP

ði;jÞ 2Zð4x_ijd_ijÞand the total population size of speciess₂is therefore equal to this last quantity multiplied by g. Let us now examine the behaviour of the positive or zero variabled_ijin relation to the Boolean variablex_ij. Because of the economic function to be maximized, variabled_ijtakes, at the opti-mum of P_5.2, the smallest possible value,i.e., because of constraints 5.2.1 and 5.2.2, the value max P

ðk;lÞ2Adjijx_klAdj_ijð1x_ijÞ; 0

n o

. If zonez_ijis not cut–x_ij= 1– variabled_ijis equal toP

ðk;lÞ2Adjijxkl,i.e., the number of uncut zones adjacent toz_ij. If zone z_ij is cut – x_ij= 0 – d_ij ¼max P

ðk;lÞ2Adjijx_klAdj_ij;0

n o

, which implies d_ij= 0. This means that, if zonez_ijis cut, its contribution to the edge length is equal to 0. Finally, the quantity 4x_ij−d_ijis well equal to the number of sides of zonez_ijthat are part of the edge when this zone is uncut and to 0, when this zone is cut.

We propose below an alternative formulation of the problem, based on the fol-lowing observation: an edge separating two zones, z_ij and z_kl, is to be taken into account in the calculation of the edge length if and only if zonez_ijis cut while zonez_kl is not or if it is the opposite. In order not to count the same edge several times, only the following two adjacent zones are considered for any zonez_ij: the one located“to the right”ofz_ijand the one located“under”z_ij. As in the previous formulation, with each zonezijis associated the Boolean variablexijwhich is equal to 1 if and only if the zone is uncut. The problem can then be formulated as the non-linear program in Boolean variables P_5.3.

The first part of the economic function is identical to that of P_5.2. Let us look at the second part. Consider two zones,z_ijandz_kl,z_klbeing adjacent toz_ijand located to the right or below it. Let us check that the quantity is indeed equal to the number of sides to be taken into account– 0 or 1 – in the edge possibly generated by the adjacency of zonesz_ijand z_kl. This is indeed the case since if these 2 zones are not cut, this quantity is equal to 1 + 1−2 = 0, if these two zones are cut, it is equal to 0 + 0−0 = 0 and, finally, if one of the zones is cut and the other not, it is equal to 1 + 0−0 = 1 or 0 + 1−0 = 1. The quantityP

ði;j;k;lÞ2Pðxijþxkl2xijxklÞis therefore well equal to the number of sides belonging to the edge and coming from the adjacency of all the pairs of zones. To count the total number of sides belonging to the edge, it is still necessary to take into account the uncut zones that are adjacent to the outside of the grid. Since the zone outside the grid is considered a cut zone, it

is easy to verify that the number of sides belonging to these zones and forming part of the edge is equal toP

ði;jÞ2Qx_ijþP

ði;jÞ2Ux_ij. It must be taken into account that if the zones z_ij,ði;jÞ 2U, are not cut, two of their sides belong to the edge between these zones and the outside of the grid.

The advantage of this formulation is that the matrix of constraints associated with its classic linearization is totally unimodular (TU), which is not the case with the matrix of constraints associated with program P_5.2(see appendix at the end of the book). The classic linearization of P_5.3consists in replacing the productsx_ijx_klby variablesy_ijkland adding the linear constraints 1x_ijx_klþy_ijkl0 andy_ijkl0 to force the equality y_ijkl=x_ijx_klat the optimum (see appendix at the end of the book). We show that the constraint matrix of this linearization is TU, based on the fact that the vertex-edge incidence matrix of a bipartite graph is TU (see, for example, Nemhauser & Wolsey, 1988). This formulation, therefore, allows large-sized instances of the problem to be solved without difficulty. Let us examine this second formulation of the problem; it is written as the mixed-integer linear program P_5.4.

P5:4:

Let us study the matrixAassociated with the set of constraints C_5.1below and derived from constraints 5.4.1 and 5.4.3.

C5:1: x_ijþx_kl y_ijkl1 ði;j;k;lÞ 2P x_ij1 ði;jÞ 2 Z

The matrix Ais composed of the three sub-matrices, B,C, and D: A¼^{B C}_D^j (figure5.5). LetG¼ ðZ;EÞbe the graph defined as follows: with each zonez_ijofZis associated a vertex, and two vertices, (i,j) and (k,l), are connected by an edge if and only if the two zones z_ij and z_kl have a common side. This graph is a grid and, therefore, a bipartite graph (figure 5.6). Matrix B is the transposed matrix of the vertex-edge incidence matrix of the graph; it is therefore TU. Each column in matrix C has a single non-zero element that is equal to −1. Using calculation of matrix determinants (expansion by cofactors), it can be shown that the determinant of any square sub-matrix of (B, C) belongs to {−1, 0, 1}. (B,C) is therefore TU. Similarly, each row ofDhas a single non-zero element that is equal to 1; the matrix ^{B C}_D^j

is therefore TU. Recall that the minor,M_ij, of a square matrixMis the determinant of the matrix obtained by eliminating the ith row and the jth column of M. The cofactor,C_ij, of the matrixMis defined byC_ij= (−1)^i+jM_ij.

Other Spatial Aspects 89

In conclusion, since (1) matrix Aassociated with the constraints of P_5.4 is TU and (2) the vector of the second members of the constraint set C_5.1 is an integer vector, the problem considered can be formulated as the linear program in real variables P_5.5which corresponds to program P_5.4in which the integrality constraint has been relaxed.

x11 x_ij

x

_kl

x

_mn y1112 y_ijkl y_{m,n 1,m,n}

-1

1 1 -1

B C

_-1

1

1

1

D

1

FIG. 5.5–The non-zero terms of the matrix,A,i.e., the matrix corresponding to the set of inequalities C5.1:xijþxklyijkl1,ði;j;k;lÞ 2P, andxij1,ði;jÞ 2Z.

1 2 3 4 5 6 7

1 2 3 4 5

FIG. 5.6–Graph associated with a grid with 5 rows and 7 columns. The edges are represented by bold lines and the vertices by black circles.

P5:5:

maxw1

P

ði;jÞ 2Z

nijð1xijÞ

þw2gl P

ði;j;k;lÞ 2P

ðxijþxkl2yijklÞ þ P

ði;jÞ 2Q

xijþ P

ði;jÞ 2U

xij

!

s.t:

1xijxklþyijkl0 ði;j;k;lÞ 2P ð5:5:1Þ yijkl0 ði;j;k;lÞ 2P ð5:5:2Þ 0x_ij1 ði;jÞ 2 Z ð5:5:3Þ

8>

>>

>>

>>

>>

><

>>

>>

>>

>>

>>

:

5.3.3 Examples

Example A. Consider the instance represented by a grid of 55 square and iden-tical zones (figure 5.7a). The valuesn_ijare indicated in each zone of the grid. The side length of each zone is equal to 3 units, the weights associated with speciess₁and s₂are equal to 2 and 1, respectively, and the coefficientgis equal to 1.26157. The solution is given in figure 5.7b in which the uncut zones are shown in grey. In this solution, 5 zones are uncut, the number of speciess₁is equal to 191, the number of speciess₂is equal to 60.56, the number of sides belonging to the edge is equal to 16 and the value of the economic function is equal to 442.56.

Example B. Consider a second instance represented by a grid of 1010 identical square zones and presented in figure5.8. The valuesn_ijare indicated in each zone of the grid. The side length of each zone is equal to 3 units, the weights associated with speciess₁ands₂are respectively equal to 1 and 5, and the coefficientgis equal to 1.26157.

The optimal solution for this instance is given in figure5.9a in which the uncut zones are shown in grey. In this solution, 21 zones are uncut, the number of speciess1is

(a)

1 2 3 4 5

(b)

1 2 3 4 5

1 10 10 10 1 10 ¹

2 10 10 1 1 10 ²

3 10 10 1 10 10 ³

4 1 10 10 10 10 ⁴

5 1 10 10 10 10 ⁵

FIG. 5.7 – A hypothetical forest massif represented by a grid of 55 square and identical zones. (a) The valuesnijare given in each zone. (b) Cut–white–and uncut–grey–zones in an optimal solution.

Other Spatial Aspects 91

6,630, the number of speciess₂is 317.92, the number of sides belonging to the edge is 84 and the value of the economic function is 8,219.58.

Example C. Now consider the same instance as in Example B above but with the following additional constraint: the number of uncut zones must be greater than or equal to 60. Adding this constraint causes the loss of the TU property of the con-straint matrix associated with this variant of the initial problem. To solve it, it is therefore necessary to solve the mathematical program P_5.4 to which is added the constraint P

ði;jÞ 2Zx_ij60: The optimal solution of this instance is given by figure5.9b in which the uncut zones are represented in grey. In this solution, 60 zones

1 2 3 4 5 6 7 8 9 10 1 84 68 97 98 64 89 82 71 74 76 2 87 83 98 75 60 90 78 67 92 94 3 84 68 70 81 67 61 73 92 86 90 4 79 62 86 79 73 84 76 98 84 90 5 62 72 66 72 92 80 71 91 87 70 6 85 77 63 93 90 94 76 81 99 98 7 76 63 66 84 94 93 72 92 79 65 8 76 63 92 69 60 88 79 93 66 73 9 92 82 77 72 77 81 89 95 80 80 10 88 89 83 86 69 78 91 64 94 92

FIG. 5.8–A hypothetical forest massif represented by a grid of 1010 square and identical zones. The valuesnijare given in each zone.

FIG. 5.9–(a) Cut zones–in white–and uncut zones–in grey–in an optimal solution of the instance described in figure5.8; (b) Cut zones–in white–and uncut zones–in grey–in an optimal solution of the instance described in figure5.8in the case where the number of uncut zones must be greater than or equal to 60.

are uncut, the number of species s₁is 3,272, the number of speciess₂is 696.39, the number of sides belonging to the edge is 184 and the value of the economic function is 6,753.93.

Dans le document EDP Open (Page 119-125)