Set that will Survive there
7.6 Consideration of Uncertainties Affecting Species Survival Probabilities
7.6.4 A Variant of the Previous Problem: A Robust, Least-Cost Reserve that Ensures a Given Survival
Probability for All Species Considered
A variant of the problem studied in the previous section is to determine a least-cost reserve ensuring, for each species ofS, a survival probability greater than or equal to a certain threshold value, regardless of the values taken by the survival probabilities in each zone, in the set of possible values. This reserve can be qualified as an optimal robust reserve in the sense that there are no other reserves, of lower cost, ensuring that all species have a survival probability greater than or equal to the threshold value, regardless of the values taken by the survival probabilities in each zone, in the set of possible values. This reserve can be determined by the mixed-integer linear program P7.17. optimal reserve includes zoneszisuch thatxi¼1, its cost is equal toP
i2Zcixi and
Species Survival Probabilities 157
the worst case is defined by the worst-case survival probabilities in themk zones of the reserve corresponding to themkhighest values of the expressiondik=ð1qikÞwith mk¼minfCk;jZk\fi:xi ¼1gjg.
In the particular case whereCk¼bgkj j=Zk 100c, for allk, anddik ¼rkqik=100 for alliand for all k, program P7.17becomes program P7.18.
P7:18:
7.6.5.1 Landscape Represented by a 8× 8 Grid
In this section, we illustrate the results of the previous sections on a set of hypo-thetical zones represented by a grid of 88 square and identical zones. In this example, 10 species are concerned. The data are presented in figure7.9. The zones are designated byzij whereirepresents the row index andjrepresents the column index of these zones. On each zone is indicated (1) the list of species whose nominal survival probability is positive if the zone is protected and (2) the corresponding nominal survival probability, denoted by qijk for species sk in zone zij. The cost associated with protecting each zone is indicated in the lower right corner of the corresponding zone. All the survival probabilities in unprotected zones are assumed to be zero. To facilitate the analysis of this example, we give in table 7.8the com-position of the sets Zk, for all k2S. We consider the problem discussed in section 7.6.4and recalled below.
Problem. Determine a least-cost reserve that guarantees a survival probability greater than or equal to 0.8 and then 0.9 for all species considered, which corre-sponds to qk ¼0:8 then qk¼0:9, for all k, regardless of the values taken by the survival probabilities in the set of possible values.
Definition of the uncertainty affecting the species survival probabilities in each zone. The level of uncertainty,Ck, is defined as a percentage, gk, of the number of zones of Zk. Remember that Zk refers to the subset of zone ofZ whose protection provides a non-zero nominal survival probability – in the corresponding zone – of speciessk. We therefore haveCk ¼bgkj j=Zk 100cwhereb ca refers to the integer part ofa. For example, figure7.9shows non-zero nominal survival probabilities of species s3 in zones z12, z26, z34, z55, z65, z75, z83, and z85. So we have Z3¼ fz12;z26;z34;z55;z65;z75;z83;z85g andj j ¼Z3 8. Ifg3 ¼30, then the maximal number
of zones where the survival probability of this species may differ from its nominal value is equal toC3 ¼b308=100c ¼2. In other words, the survival probabilities of s3may differ from their nominal value in up to 2 of the 8 zones ofZ3. We therefore admit that, among these 8 probabilities, 2 (at most) may be erroneous but we do not know which ones. Note that since we have to consider the “worst-case”, these two probabilities will be equal toqikdikinstead ofqik. In these experiments we consider that gk¼g, for all k, and we consider 4 different values of g: 0, 20, 30, and 100.
FIG. 7.9–A set of 64 candidate zones for protection represented by a grid of 88 square and identical zones. 10 species s1,s2,…,s10 are concerned. The corresponding nominal survival probabilities,qik, and the protection costs are indicated in each zone. Consider, for example, zonez56. Speciess7ands8are concerned. The nominal survival probabilities of these 2 species in this zone, if protected, are 0.5 and 0.8, respectively. The cost of protecting this zone is equal to 6.
Species Survival Probabilities 159
We also assume that dijk ¼rkqijk=100 with rk ¼ r, for all k, and we consider two values ofr: 10 and 20.
The results obtained are presented in table 7.9. Some optimal robust reserves, corresponding to the instances of this table, are presented in figure7.10.
TAB. 7.8– List of zones whose protection provides speciessk, k= 1,…,10, with a positive nominal survival probability.
sk Zk
s1 z11z16z25z83
s2 z17z18z43z61z71z84z86 s3 z12z26z34z55z65z75z83z85 s4 z17z44z54z62z67z75z84
s5 z14z24z57z68z87
s6 z15z78z84z88
s7 z18z26z37z55z56
s8 z11z14z56z73z76
s9 z22z23z36z42z46z62
s10 z11z15z18z34
TAB. 7.9 – Results for the example in figure7.9when the threshold of survival probability required for each species,ρ, is equal to 0.8 or 0.9, and for different values ofgandr.
ρ r g Cost of the optimal
robust reserve
Number of zones in the reserve
Associated figure
0.8 10 0 40 11 Figure7.10a
20 47 13 Figure7.10b
30 51 15
100 53 14
20 0 40 11
20 50 14 Figure7.10c
30 53 14
100 70 16
0.9 10 0 69 15 Figure7.10d
20 77 16 Figure7.10e
30 80 17
100 – –
20 0 69 15
20 84 20 Figure7.10f
30 – –
100 – –
–: No feasible solution.
The detailed results corresponding to figure 7.10b are presented in table 7.10.
Given the value ofg, there can be no errors for speciess1,s6, ands10. For the other species, errors may only concern one zone.R* refers to the set of zones selected to form the optimal robust reserve,PkðRÞ, the survival probability of speciesskin that reserve in the worst case and P~kðRÞ, the survival probability of species skin that reserve, calculated with the nominal survival probabilities in each zone.
(a) 0.8, 0 (b) 0.8,r 10, 20 (c) 0.8,r 20, 20
(d) 0.9, 0 (e) 0.9,r 10, 20 (f) 0.9,r 20, 20
FIG. 7.10–Optimal robust reserves for some instances of table7.9.
TAB. 7.10– Details of the results corresponding to figure7.10b.
Species (sk)
Zk
j j gj jZk
100
j k
R\Zk Zone where the survival probability of speciessk
takes its worst value,i.e., qijkr qijk=100
PkðRÞ P~kðRÞ
s1 4 0 z16z83 – 0.8 0.8
s2 7 1 z18 z18: 0.9!0.81 0.81 0.9
s3 8 1 z55z83 z55: 0.8!0.72 0.916 0.94
s4 7 1 z62z67 z62: 0.8!0.72 0.916 0.94
s5 5 1 z14z57z87 z87: 0.7!0.63 0.8668 0.892
s6 4 0 z15z78 – 0.8 0.8
s7 5 1 z18z55z56 z18: 0.6!0.54 0.885 0.9
s8 5 1 z14z56 z56: 0.8!0.72 0.888 0.92
s9 6 1 z22z62 z22: 0.8!0.72 0.916 0.94
s10 4 0 z15z18 – 0.94 0.94
Species Survival Probabilities 161
Now let us consider the instance where the threshold value chosen for each speciesskis always defined byq¼0:8 but without taking into account the uncer-tainty about the species survival probabilities in each zone. The optimal reserve obtained costs 40 units and is shown in figure 7.10a. The survival probabilities of each species in this reserve, calculated from the nominal values of the survival probabilities in each zone, are given in table 7.11a.
Let us always consider the reserve in figure 7.10a but now take into account the uncertainty about the survival probabilities in each zone when g¼20 and r¼20.
This means that in at most b20j jZk =100c zones, the true values of the survival probabilities of speciesskare only equal to 80% of their nominal value,qijk. In this context, the worst case corresponds to the species survival probabilities in the reserve given in table7.11b. As expected, all these probabilities are less than or equal to those in table7.11a, but the survival probabilities of speciess2,s4,s5,s7,s8, ands9 fall below the fixed threshold value, 0.8. The reserve under consideration is therefore not at all robust. Remember that in this context of uncertainty, the optimal robust solution is given by the reserve in figure7.10c, the cost of which is equal to 50. Thus, we can say that, for this hypothetical example, protection against uncertainty defined byg¼20 andr = 20, increases the cost of the optimal reserve by 25%.
The problem discussed in this section 7.6.5.1 can be addressed in a slightly different way. We can look, for example, at the maximal error that can be made in estimating the probabilityqijkin such a way that there is a reserve that satisfies the required performance–survival probabilities of each species greater than or equal to a certain threshold value. We will see that the cost of the optimal robust reserve associated with this problem increases with uncertainty on qijk until there is no longer a reserve that meets the required performance.
TAB. 7.11 – Survival probabilities of the species in the reserve of figure7.10a. (a) No uncertainty about the survival probabilities,qijk. (b) Results in the worst case and when the uncertainty is defined byg¼20 andr¼20; probabilities that differ from their nominal values are in bold.
(a) (b)
sk Species survival probability in the reserve
sk gj jZk
100
j k
Species survival probability in the reserve, in the worst case and wheng¼20 and
r¼20 s1 1−(1–0.6)(1–0.5) = 0.8 s1 0 1−(1–0.6)(1–0.5) = 0.8
s2 1−(1–0.9) = 0.9 s2 1 1−(1–0.72) = 0.72
s3 1−(1–0.8)(1–0.7) = 0.94 s3 1 1−(1–0.64)(1–0.7) = 0.892 s4 1−(1–0.7)(1–0.4) = 0.82 s4 1 1−(1–0.56)(1–0.4) = 0.736 s5 1−(1–0.4)(1–0.7) = 0.82 s5 1 1−(1–0.4)(1–0.56) = 0.736
s6 1−(1–0.6)(1–0.5) = 0.8 s6 0 1−(1–0.6)(1–0.5) = 0.8
s7 1−(1–0.6)(1–0.5) = 0.8 s7 1 1−(1–0.48)(1–0.5) = 0.74
s8 1−(1–0.8) = 0.8 s8 1 1−(1–0.64) = 0.64
s9 1−(1–0.8) = 0.8 s9 1 1−(1–0.64) = 0.64
s10 1−(1–0.7)(1–0.8) = 0.94 s10 0 1−(1–0.7)(1–0.8) = 0.94
Let us look again at the landscape described in figure7.9, set ρto 0.9,g to 30, and look for the highest value ofrfor which there is a robust reserve,i.e., a reserve that ensures that all the species have a survival probability in the reserve that is greater than or equal toρin the worst case. This problem can be directly formulated by replacing in program P7.18 the economic function with the expression r to be maximized– in this case,rbecomes a variable. However, the optimization problem obtained is difficult because the constraints 7.18.2 become non-linear: likþ kk log 1 qik1100r
logð1qikÞ
xi. Note that in this case rk does not depend onkand is therefore denotedr. Another way to determine this limit value of ris to solve P7.18, iteratively, by gradually increasing the value ofruntil there is no longer any feasible reserve. In this case, the survival probability, in any reserve, of at least one species falls below the desired threshold value, in the worst case. Table7.12 presents the results obtained by this approach: there are robust reserves as long as ris less than or equal to 12 and there are no more such reserves ifris greater than or equal to 13.
Similarly, it may be interesting whenρandrare fixed to determine the largest value ofgfor which a robust reserve exists. Here again, the optimization problem is difficult since it consists in solving P7.18after having replaced the economic function by variablegto be maximized. Whenrandρare fixed, all the constraints of P7.18are linear except constraints 7.18.1 which contain the termbgj j=Zk 100ckk. Heregk does not depend onkand is therefore denotedg. As before, one way to determine this limit value is to solve P7.18iteratively by gradually increasing the value ofg until there is no longer any feasible robust reserve. In this case, the survival probability of at least one species falls below the required threshold, in the worst case, for any reserve. Table7.13presents the results obtained by this approach whenq¼0:9 and r ¼20. It can be seen that, under these conditions, there are robust reserves as long asg24 and that there are no more such reserves wheng25.
7.6.5.2 Large-Sized Instances
In this section we present the results obtained regarding the resolution of P7.18on large-sized instances. The landscapes studied are characterized by the following 4 parameters: the number of zones, the cost of protection of each zone, the number of TAB. 7.12 –Cost of an optimal robust reserve for the landscape in figure7.9whenρ= 0.9, g¼30, and for different values ofr. For example, ifris between 1 and 6, the optimal robust reserve costs 75 units and has 16 zones whenr2 f1;4;5;6gand 17 zones whenr2 f2;3g.
r Cost of an optimal
robust reserve
Number of zones in the reserve for the different values ofr
r¼0 69 15
r2f1;2;3;4;5;6g 75 16, 17, 17, 16, 16, 16
r2f7;8;9g 77 16, 16, 16
r2f10;11;12g 80 17, 17, 17
r13 No feasible robust
reserve
–
Species Survival Probabilities 163
species concerned and the nominal survival probability of these species in each zone.
In these experiments, 50 species are present in the studied landscape and we con-sidered two cases for each of the other 3 parameters, thus obtaining 8 types of landscapes. The landscape is represented by a grid of 1515 identical square zones or a grid of 2020 identical square zones; the costs of protecting the zones are drawn at random, uniformly, between 1 and 10 or between 1 and 20; finally, in a first case, each species appears randomly in 5% of the zones and the nominal survival probabilities in the zones are drawn at random, uniformly, from the set f0:5;0:6;0:7;0:8g; in a second case, each species appears randomly in 4% of the zones and the nominal survival probabilities in the zones are drawn at random, uniformly, from the setf0:7;0:8g. The characteristics of these 8 landscape types are summarized in table 7.14. When a species does not appear in a zone, its survival probability in that zone is zero.
For each of the 8 types of landscape we considered 5 different landscapes by modifying the germ of the random number generator thus obtaining 40 different landscapes.
Now let us see how the values of Ck;qk, and dijk are defined. As in the experi-ments in section7.6.5.1, Ck¼bgj j=Zk 100c, qk ¼q, k2S, and dijk ¼r qijk=100, ði;jÞ 2Z;k2S. For each of the 40 landscapes, we studied solutions for the following values of the different parameters: q2f0:85;0:90;0:95g, g2 f10;20g, and r 2f20;30;100g. For each of the 40 landscapes we also studied the case where there is no uncertainty by solving P7.18 with q2f0:85;090;0:95g and g¼0. We thus resolved a total of 840 instances of P7.18.
All the instances were resolved quickly except some instances of size 20 20 when the threshold value is equal to 0.95. In this case, the resolution of several instances requires a few hundred seconds, the most difficult instances corresponding to g¼20 and r¼20 or 30. All the instances corresponding to landscapes of size 20 20 have a feasible solution. For the instances corresponding to landscapes of size 15 15, there is no feasible solution for one of the 5 germs of the random number generator whenq¼0:95 and when each species appears randomly in 4% of the zones with a nominal survival probability drawn at random from the set f0:7;0:8g – landscapes of types 2 and 4. The other instances for which there is no feasible solution are presented in table7.15.
TAB. 7.13– Cost of an optimal robust reserve for the landscape in figure7.9whenq¼0:9 andr¼20, and for different values ofg.
g Cost of an optimal robust
reserve
Number of zones in the reserve
g2f0;:::;12g 69 15,…,15
g2f13;14g 70 17, 17
g2f15;16g 75 18, 18
g2f17;18;19g 76 19, 19, 19
g2f20;21;22;23;24g 84 20, 20, 20, 20, 20
g25 No feasible robust reserve
Experiments have shown that the cost of protection against uncertainty is often high. Table 7.17 summarizes the average percentage increase in the cost of an optimal robust reserve–taking into account uncertainty hypotheses–compared to the cost of an optimal reserve when there is no uncertainty, for all the instances of size 20×20–landscapes of types 5, 6, 7, and 8. The average increase is calculated on the 5 instances associated with the 5 values of the generator germ. Consider, for example, the landscapes of type 8 when the threshold value of the survival proba-bility, ρ, is equal to 0.90. The results obtained forg¼30 andr ¼10 are given in table 7.16. In this case, the average increase in the cost due to protection against uncertainty is about 55%.
Let us look at table7.17when the threshold value of the survival probability is equal to 0.85. We see on this table that, whenr ¼10 andg = 30 or 100, there is no TAB. 7.14–Description of the 8 types of landscape considered in the experiments. In all the cases, 50 species are concerned.
Type Dimension of the grid (nn)
Set of values in which the costs,
cij, are generated
Probability of presence of each
species in each zone
Set of values in which the nominal survival probabilities,qijk, are
generated
1 1515 f1;2;:::;10g 5ðnnÞ=100 f0:5;0:6;0:7;0:8g
2 1515 f1;2;:::;10g 4ðnnÞ=100 f0:7;0:8g
3 1515 f1;2;:::;20g 5ðnnÞ=100 f0:5;0:6;0:7;0:8g
4 1515 f1;2;:::;20g 4ðnnÞ=100 f0:7;0:8g
5 2020 f1;2;:::;10g 5ðnnÞ=100 f0:5;0:6;0:7;0:8g
6 2020 f1;2;:::;10g 4ðnnÞ=100 f0:7;0:8g
7 2020 f1;2;:::;20g 5ðnnÞ=100 f0:5;0:6;0:7;0:8g
8 2020 f1;2;:::;20g 4ðnnÞ=100 f0:7;0:8g
TAB. 7.15– Instances without feasible solutions.
Landscape type
q r g Number of
instances without feasible solutions
(out of the 5 considered)
2 0.85 20 100 1
4 0.85 20 100 1
2 0.90 10 100 1
2 0.90 20 100 1
4 0.90 10 100 1
4 0.90 20 100 1
1 0.95 20 100 1
2 0.95 20 100 1
4 0.95 20 100 2
Species Survival Probabilities 165
increase in cost for the landscapes of types 6 and 8. However, for the landscapes of types 5 and 7 and for the same values ofgandr, the average cost increases by about 20%. The largest average increase, again for this same threshold value of survival probability, occurs for the landscapes of type 8 whenr= 20 andη=100 and is about 57%. Regardless of the landscape considered, the largest average increase in the TAB. 7.16–Increase in the costs of an optimal reserve, due to protection against uncertainty, for 5 landscapes of type 8, wheng¼30,r¼10, andq¼0:90.
No. of the instance
Minimal cost when there is no uncertainty
Minimal cost when uncertainty is defined byg¼30 andr¼10
Cost increase
(%)
1 119 188 57.98
2 90 147 63.33
3 114 180 57.89
4 124 183 47.58
5 99 148 49.49
Av 55.26
TAB. 7.17–Cost increases due to uncertainty for the 4 types of landscape of size 2020 and for 3 values of the parameterρ: 0.85, 0.90, and 0.95.
Percentage increase in cost due to uncertainty Survival
probability to be ensured for each species (ρ)
Definition of the considered
uncertainty
Landscape of type 5
Landscape of type 6
Landscape of type 7
Landscape of type 8
0.85 r¼10;g¼30 + 21.3 + 0 + 20.1 + 0
r¼10;g¼100 + 21.3 + 0 + 20.1 + 0
r¼20;g¼30 + 51.6 + 46.5 + 50.6 + 55.3
r¼20;g¼100 + 51.6 + 49.3 + 50.6 + 56.5
0.90 r¼10;g¼30 + 32.8 + 46.8 + 32.1 + 55.3
r¼10;g¼100 + 32.8 + 49.3 + 32.1 + 56.5
r¼20;g¼30 + 61.1 + 72.5 + 63.4 + 76.4
r¼20;g¼100 + 61.1 + 72.5 + 64.5 + 77.0
0.95 r¼10;g¼30 + 28.1 + 18.2 + 31.1 + 17.2
r¼10;g¼100 + 28.1 + 19.2 + 31.2 + 17.4
r¼20;g¼30 + 60.8 + 57.5 + 63.0 + 58.8
r¼20;g¼100 + 62.1 + 63.4 + 67.0 + 66.5
table occurs for the instances defined byρ= 0.90,r¼20, andη=100. Indeed, in this case, this average increase in cost for the 4 types of landscape is approximately between 61 and 77%.
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