Problem Relaxation and Determination of an Approximate Solution

The approximate resolution of P_7.11 that we propose can be interpreted as an approximation of the logarithmic function, which is a concave function, by a piecewise linear function greater than or equal to the logarithmic function at all points (see appendix at the end of this book and figure7.5). The advantage of this approach is that it provides not only an approximate solution to the problem but also a relaxation of the initial problem and thus an upper bound of the true value of the optimal solution. In other words, the method provides–using an integer linear programming solver – an approximate solution as well as some guarantee on the value of this solution.

Relaxation of P_7.11: To build a relaxation of P_7.11, the idea is to replace, within the constraints 7.11.1 and for all k2S, the logarithmic function, logak, by a piecewise linear function,fða^kÞ, greater than or equal to loga^k for alla^k such that 0\ak1 (figure7.5).

This substitution leads to program P_7.13. This is a relaxation of P_7.11 that includes a piecewise linear function. Indeed, any feasible solution of P_7.11is also a feasible solution of P_7.13since, if the constraint, loga^k ¼P

i2I_k^\1logð1qikÞxi,k2S, is satisfied, then the constraintfðakÞ P

i2I_k^\1logð1q_ikÞx_i,k2S, is also satisfied.

In addition, the economic functions of P_7.11and P_7.13are identical. Note that P_7.13 can easily be converted into a linear program–in Boolean variables–since function fis concave and appears in the left member of a“greater than or equal”constraint (see appendix at the end of the book).

P7:13:

Let ðx;a;bÞ be an optimal solution of P_7.13. An approximate solution to the initial problem – a reserve – is given by x; its value, i.e., the expected weighted number of species protected by this reserve, is equal to P

k2Swk

1Q

i2Zð1qikxiÞ

h i

. An upper bound of the true optimal value of the problem considered is given by the optimal value of P_7.13,i.e.,P

k2Sw_kð1bkÞ. The relative gap between the value of the optimal solution and the value of the approximate solution is, therefore, less than or equal to P

k2Swkð1bkÞ

Let us now see how to construct the piecewise linear functionfsuch thatfðakÞis greater than or equal to loga^k for all a^k such as 0\ak1.

Lemma 7.1. For all a[0 andb[0, logaþ¹_aðbaÞ logb.

The proof is immediate since 1=a is the value of the derivative of the function logy at pointa, and the function logyis concave.

Lemma 7.2. Letube a vector ofR^V such that 0\u₁\u₂\ \u_V ¼1, and letfbe the piecewise linear function composed of theVsegments tangent to the curve logak

at theVpoints of abscissau₁;u₂;:::;u_V. The following 5 properties are satisfied:

(i) The abscissa of the breaking points offarebpv¼^log₁₌^u_u^vþ1loguv

v1=u_vþ1,v¼1;:::;V 1.

(ii) Fora^k such that 0\akbp1,fða^kÞ ¼_u¹

1a^kþlogu11.

(iii) For a^k such that bpva^kbpvþ1, fða^kÞ ¼_u¹

vþ1a^kþloguvþ11,

v¼1;:::;V 1.

(iv) The functionfðakÞis concave.

(v) fðakÞ logðakÞfor allak20;1.

Proof. The proof of (i) is immediate; it results from elementary calculations.

The proofs of (ii) and (iii) are direct consequences of lemma7.1.

The successive slopes of the function f are 1=u₁[1=u₂[ [1=u_V. The functionfis therefore concave.

The proof of (v) is a direct consequence of lemma7.1.

In summary, the expression logak, which appears in P_7.11, is upper approximated in P_7.13byfðakÞwherefis the piecewise linear function defined in lemma7.2and illustrated in figure 7.5. Program P_7.11 can, therefore, be reformulated – in an approximate way – by a mixed-integer linear program by adding a single set of constraints. This gives program P_7.14.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-7 -6 -5 -4 -3 -2 -1 0

k

log k

f( k)

FIG. 7.5–An upper approximation of logakfor 0\ak1 (dotted curve) by a piecewise linear function,fðakÞ(solid line curve).

Species Survival Probabilities 145

P7:14 :

max P

k2S

w_kð1bkÞ

s.t:

ð7:11:2Þ;ð7:11:3Þ;ð7:11:4Þ;ð7:11:5Þ

ak

uv þlogu_v1 P

i2I_k^\1

x_ilogð1q_ikÞ k2S;v¼1;:::;V ð7:14:1Þ

8>

>>

>>

<

>>

>>

>:

Another way to proceed is to use a modelling language that automatically performs this conversion. This is the case, for example, with the AMPL language (Foureret al., 1993). Thus, using the syntax of this language, the function fðakÞis defined as follows:

< < {v in 1..V-1} bp[v]; {v in 1..V} 1/u[v] > > (fα[k], 1).

The expression between < < and > > describes the piecewise linear function, and is followed by the name of the variable concerned. Here, this variable, which representsfðakÞ, is denoted by fα[k]. There are two parts in this expression, the list of the abscissa of the breaking points, bpv, v¼1;. . .;V1, corresponding to the changes in the slope of the function, and the list of the slopes, 1=uv,v¼1;. . .;V. The two lists are separated by a semicolon. The first slope is the slope before the first breaking point and the last slope is the slope after the last breaking point. For the function to be perfectly defined, it is also necessary to specify at what point it takes the value 0. Herefð1Þ ¼0.

An optimal solution of P_7.14provides a feasible solution to the problem,i.e., a set of zones to be protected to form the reserve. By definition, the value of this approximate solution is equal toP

k2Swk½1Q

i2Zð1qikxiÞwherexi is the value of variablex_i in an optimal solution of P_7.14. Since P_7.14is a relaxation of P_7.11, the optimal value of P_7.14gives an upper bound of the optimal value of P_7.11and, therefore, an upper bound of the value of the optimal solution of the reserve selection problem consid-ered. Thus, the formulation P_7.14allows us to obtain, with the help of integer linear programming, an approximate solution to our problem but also an upper bound on the gap between the value of this solution and the value of an optimal solution. To obtain a good approximation of logak by the piecewise linear function defined in lemma7.2,Vmust be large enough. However, the largerVis, the larger the size of program P_7.14is. The results of the experiments presented in section7.5.3show that by choosing the vectorucarefully–see the example in section7.5.3–we obtain an approximate solution whose value is very close to the value of the optimal solution.

Thus, program P_7.14provides a solution to the problem, regardless of the probability values, and provides a guarantee on the quality of the solution obtained.

Remark. Again, the problem considered in this section7.5can be interpreted in a slightly different way–as some authors have done–assuming that the presence of species s_k in zone z_i is defined by a probability denoted by q_ik. The problem sidered above then becomes: determine a reserve that respects a budgetary con-straint and maximizes the expected number of species present in this reserve. Thus, in the first interpretation we are interested in the mathematical expectation of the weighted number of species that will survive in the reserve and in the second, in the

mathematicalexpectationoftheweightednumber ofspecies thatareprotected,at leastinsomeway,i.e., presentinthereserve.

7.5.3 Example

Weillustratethesolutiontotheproblemstudiedinsection7.5,whichwerecallhere:

determine a reserve of cost less than or equal to a certain value, B, and which maximizes theexpected number of species that willsurvive in thisreserve. A hy-potheticalsetofcandidatezones,representedbyagridof8 8squareandidentical zonesand describedin section7.4.2, isconsidered. Inthisexample,10 species, s₁, s₂,…,s₁₀, are concerned. The description of this example concerns the list of the candidatezones,z_ij,i= 1,…, 8,j= 1,…,8,thesurvivalprobabilitiesofthespecies inthedifferentzonesiftheyareprotected,qijkforthesurvivalofspeciess_k inzonez_ij, andfinallythecostofprotectingzones,c_ij forzonez_ij.Notethat,inthisexample,all thesurvivalprobabilitiesarestrictlylessthan1.Theweightofspeciess_k isdenoted byw_k and,inthisexample,w¼ ð1; 1; 4; 2; 1; 2; 4; 2; 1; 2Þ.Allsurvivalprobabilitiesin unprotectedzonesarezero.RememberthatZ_k referstothesetofcandidatezonesin whichthesurvivalprobabilityofspeciess_k isstrictlypositiveincaseofprotection – Zk ¼ fzij 2Z : qijk[ 0g – andwedenotebyZ_k thesetofindexpairsassociatedwith the zones of Z_k. Different values of the available budget, B, are considered. The valuesobtainedfortheexpectedweightednumberofspeciesrangefrom37to98%of thelargestpossiblevalueoftheexpectedweightednumberofspecieswhichisequal to P10

k¼1wk ¼20. The vector u (see section 7.5.2) is chosen as follows: uv¼ u₁^{ðVvÞ=ðV1Þ}forv¼1;:::;V withu1¼0:01 andV ¼20. Choosinguv ¼u₁^{ðVvÞ=ðV1Þ},

v¼1;:::;V, produces a regular decrease in the slope of the piecewise linear

function f defined in lemma 7.2. since ¹⁼₁^u₌^v_u^þ1

v ¼_u^uv

vþ1¼ ^{u1ðVvÞ=ðV1Þ}

u₁^{ðVv1Þ=ðV1Þ}= u₁^{½ðVvÞ=ðV1ÞðVv1Þ=ðV1Þ}¼u₁^1=ðV1Þ. The experimental results are presented in table7.6. For example, letðx;a;bÞbe an optimal solution of P_7.14whenB¼50. In this case, the expected weighted number of species,P

k2Sw_k 1Q

i2Zð1q_ikx_iÞ

h i

, is equal to 18.59, the value of the optimal solution of P7.14,P

k2Swkð1bkÞ, which is an upper bound of the optimal value of the problem considered, is also equal to 18.59, and the associated relative gap is equal to 0.02% – expressing the different results with a precision of two decimal places. Note that the expected weighted number of species is 19.69 if all the zones are selected. The solution corresponding to B = 50 is represented by figure7.6a: 14 zones are selected at a cost equal to the available budget, i.e., 50 units.

The results in table7.6show that, in this instance, the approach is very effective in solving the problem under consideration. Indeed, the relative gap is always less than 0.1%. It should also be noted that all the solutions are obtained almost instantaneously. However, the reserves obtained can be very fragmented when there is no compactness constraint. If we introduce a compactness constraint consisting in prohibiting the distance between two zones of the reserve from being more than

Species Survival Probabilities 147

3 units, we obtain, when the budget is equal to 50, the solution described in the 4th row of table 7.6(50*) and by figure7.6b. In this case, the compactness constraint decreases the value of the solution by about 30%.

7.5.4 Computational Experiments on Large-Sized Instances

In order to test the effectiveness of the approach, various large-sized artificial instances were tested. In these instances, 300 species are concerned and the set of candidate zones is represented by a grid of 2020 square and identical zones (figure 7.7). The zones are designated byzij whereirepresents the row index of the zone and jits column index. The 300 species considered are divided into 4 groups and a weight is assigned to each species. All species in the same group have the same weight.

(a) B 50, no compactness

constraints. (b) B 50, with a compactness constraint.

FIG. 7.6–An optimal reserve with or without a compactness constraint when the available budget,B, is equal to 50 (see rows“50”and“50*”of table7.6).

TAB. 7.6–Results concerning the resolution by P7.14of the problem studied in this example.

The instance under consideration is described in section7.4.2 (88 zones and 10 species), w¼ ð1;1;4;2;1;2;4;2;1;2Þ,u₁= 0.01, andV= 20.

B Number of

selected zones

Budget used

Expected weighted number

of species

Upper bound

Relative gap (%)

Associated figure

10 4 10 11.10 11.11 0.08 –

30 11 30 17.04 17.05 0.05 –

50 14 50 18.59 18.59 0.02 7.6a

50* 8 43 13.04 13.05 0.08 7.6b

70 17 70 19.21 19.23 0.05 –

150 29 150 19.67 19.69 0.10 –

200 35 180 19.69 19.70 0.07 –

– Group I (species numbered from 1 to 50): This group includes species with a critical extinction risk. The weight of the species in this group is set at 8.

– Group II (species numbered from 51 to 100): This group includes species with a certain extinction risk. The weight of the species in this group is set at 4.

– Group III (species numbered from 101 to 150): This group includes species that are relatively rare but do not currently present an extinction risk. The weight of the species in this group is set at 2.

– Group IV (species numbered from 151 to 300): This group includes relatively common species that do not currently present an extinction risk. The weight of the species in this group is set at 1.

In these experiments, the cost of protecting a zone is generated randomly, in a uniform way, within the set of values {1, 2,…, 10}. Six values of the available budget,B, are considered: 20, 40, 60, 80, 100, and 120. These values ofBwere chosen in order to obtain an expected weighted number of species ranging from 60 to 100%

of the largest possible value of the expected weighted number of species, P

k¼1;300wk ¼850. The probabilities qijk – the survival probability of species s_k in zonez_ijif it is protected–are drawn at random as follows: for each tripletði;j;kÞ, a number is generated at random in a uniform way from the setf1; 2;:::; 20g. If this number is less than or equal to 18, then qijk¼0 otherwiseqijk is randomly drawn uniformly from the set of values f0:1; 0:2;:::; 0:9g. As in the example in section 7.5.3, the vectoruis chosen as follows:uv ¼u₁^{ðVvÞ=ðV1Þ}forv¼1;:::;V with u₁¼0:01 andV ¼20. Note that to obtain even more accurate approximations, the values ofu₁ can be chosen according to the value ofB. Indeed, when the available budget is large, many survival probabilities of the species in the reserve are close to 1 in an optimal solution, which implies that many extinction probabilities (a^k, see section 7.5.1) are close to 0. To obtain a good approximation of the logarithmic function by the piecewise linear function defined in lemma7.2, it may therefore be interesting to decrease the value of u₁when the value ofB increases.

The experimental results are presented in table 7.7 for different values of the available budget, B.

For example, let ðx;a;bÞ be an optimal solution of P_7.14 whenB = 60. In this case, the expected weighted number of protected species, P

k2Swk 1Q

i2Z

h

1 2 3 20 1

2 3

20

FIG. 7.7–A set of 400 candidate zones represented by a grid of 2020 square and identical cells.

Species Survival Probabilities 149

TAB. 7.7–Results regarding the resolution of the problem by P7.14for the example described in this section7.5.4(2020 zones, 300 species, u₁= 0.01, andV= 20).

B Number of

selected zones

Budget used

Expected weighted number of species

Upper bound

Relative gap (%)

CPU time (s)

Number of nodes in the search tree

Associated figure

20 19 20 598.59 599.14 0.09 6 641 –

40 31 40 745.54 745.98 0.06 24 4,903 –

60 42 60 802.16 802.62 0.06 59 15,060 7.7a

60* 26 60 686.56 687.08 0.08 45 4,152 7.7b

80 52 80 825.47 826.04 0.07 217 92,935 –

100 59 100 836.41 837.20 0.09 549 407,188 –

120 67 120 842.33 843.10 0.09 538 416,902 –

DesigningProtectedAreaNetworks

ð1qikxiÞ, is equal to 802.16, the value of the upper bound,i.e., the value of the optimal solution of P_7.14, P

k2Swkð1bkÞ, is equal to 802.62, and the associated relative gap, 100 P

k2Sw_kð1bkÞ P

k2Sw_k 1Q

i2Zð1q_ikx_iÞ

h i

divided by P

k2Swk 1Q

i2Zð1qikxiÞ

h i

, is equal to 0.06%. Also in the case whereB= 60, the CPU time required to solve the problem by P_7.14is equal to 59 s, and 15,060 nodes are developed in the search tree. The solution – without any compactness con-straints – is presented in figure7.8a: 42 zones are selected and the corresponding cost is equal to the value of the budget. Figure 7.8b represents the solution obtained by adding a compactness constraint that imposes a maximal distance of 9 units between two zones.

The results in table7.7show that in this example–300 species and 400 zones– the approach is effective in solving the problem under consideration. Indeed, the average computation time is about 205 s and the relative gap is always less than 0.1%. The larger B is, the longer the computation time required is. It can also be noted that the value of B clearly influences the value of the optimal solution:

increasing Bfrom 40 to 100 increases the expected weighted number of species by about 12%. It should also be noted that the solution presented in figure 7.8a is very fragmented since some zones are very far from each other. The diameter of this reserve is equal to approximately 23 units,i.e., the distance between zonesz_3,20and z_16,1, if the length of the side of each zone on the grid is equal to one unit and if the distance between two zones is measured by the distance between the centres of these two zones. If we impose the compactness constraint consisting in prohibiting the distance between two zones of the reserve from being greater than 9 units, we obtain, for B= 60, the solution described in the 4th row of table 7.7 (60*) and by figure7.8b. We can see that this compactness constraint decreases the value of the solution by about 15%. For very large problems, it would probably be difficult to obtain the optimal solution within a reasonable computation time. A heuristic approach should then be used. A simple way to do this would be to use a truncated branch and bound procedure, a procedure that is easy to implement by limiting the

(a) Expected weighted number of species:

802.16 (42 zones) (b) Expected weighted number of species:

686.56 (26 zones)

FIG. 7.8–Optimal reserves for a budget of 60 units. (a) Without a compactness constraint.

(b) With a compactness constraint imposing a maximal distance of 9 units between two zones.

Species Survival Probabilities 151

computation time allocated to the solver to address the problem, as we did in chapter 3, section 3.8, to determine“good”connected reserves.

7.6 Consideration of Uncertainties Affecting Species

Dans le document EDP Open (Page 176-184)