Designing Nature Reserves With Connectivity and Buffer...

Designing Nature Reserves With Connectivity andBuffer Requirements

E. Alvarez-Miranda1 M. Goycoolea2 I. Ljubic3 M. Sinnl4

1 Universidad de Talca, Curico, Chilee2 Universidad Adolfo Ibanez, Santiago, Chile, Chile

3 ESSEC Business School of Paris, France4 University of Vienna, Austria

OR 2015, September 1-4, Vienna, Austria

Motivation

dramatic loss of biodiversity inthe last decades

over 20,000 species arethreatened with extinction(IUCN red list)

immense efforts by internationalorganizations, governments, . . .

establish protected areas forwildlife.

Figure : (IUCN red list)Markus Sinnl (University of Vienna) Designing Nature Reserves OR 2015, September 1-4, Vienna 2

Modelling

development of mathematical models

widely explored research area [5, 7, 10, 11, 15, 16, 17, 18]

reserves that respect ecological, economical, . . . , requirements [4]

most basic problem: Reserve Set Covering Problem (RSC) [6, 14]

set V of land sites (land units, parcels)

a set of species S

sets of land sites Vs ⊂ V (one for each specie s ∈ S)

xi = 1 iff site i ∈ V is selected

(RSC) min z =∑i∈V

xi (RSC.1)

s.t.∑i∈Vs

xi ≥ 1, ∀s ∈ S (RSC.2)

x ∈ {0, 1}|V | (RSC.3)

Markus Sinnl (University of Vienna) Designing Nature Reserves OR 2015, September 1-4, Vienna 3

Modelling

RSC: no spatial considerations

spatial requirements [4, 19]I reserve size or compactnessI number of reservesI connectivity

avoids habitat fragmentation → improves the conditions for sustainableecosystems [3]

I presence of core and buffer areasallows the development of so-called biosphere reserves [1, 2] →promoting the long-term viability of critical species

I proximityI shape

our contribution: for the first time, combine connectivityrequirements and core/buffer zones


Different Reserve Design Problems

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empty

core

(a) RSCSolution

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empty

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(b) ConnectedRSC Solution

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empty

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core

(c) RSCSolution withbuffer zones

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(d) ConnectedRSC Solutionwith bufferzones

Figure : Representation of solutions of different reserve design problems (denotes core land sites, denotes buffer land sites)


Outline

1 Introduction

2 ILP FormulationsMinimum Cost Connected Reserve with Buffer RequirementsProblem (MCCRB)Maximal Suitability Subject to a Budget Constraint (MSBC)

3 Computational Results

4 Real-Life Case-Study

5 Conclusion


Notation

graph G = (V ,E )

set of species S = S1 ∪ S2

S1 are endangered species (must be in the core)

hs ≥ 1: number of land sites we need to cover for s ∈ S

cost function c : V → R≥0

xi = 1, i ∈ V , iff i part of the reserve

zi = 1, i ∈ V , iff i part of the core

yi = 1, i ∈ V , iff i is root

root: used to model connectivity

Q: buffer thickness


Connectivity

connectivity by node separators

i j

CjCi

N

Figure : N: Node separator between i and j [9]

N (k , `): family of all (k , `) separators.

N` = ∪k 6=`N (k, `)

for N ∈ N`: WN,` = {i ∈ V \N | ∃(i , `) path P in G −N} ∪ {`}, i.e.,all nodes reachable by ` after removal of N


Buffer

Definition (Neighborhood)

For a given integer q ≥ 0 and a given land site i ∈ V , the q-neighborhoodset of i , δq(i), is defined as

δq(i) = {j ∈ V | the min number of hops between i and j is at most q} .

used to model buffer requirements

if i in core, then all j ∈ δq(i) mus be taken

we focus on buffer size one in the following

define δ(i) = δ1(i)

δ(i) are adjacent nodes and i itself

further work: general precedence set instead of neighborhood

many results (should be) generalizable


Minimum Cost Connected Reserve with BufferRequirements Problem (MCCRB)

min∑i∈V

cixi (MCB.1)

s.t.∑i∈Vs

zi ≥ hs , ∀s ∈ S1 (MCB.2)

∑i∈Vs

xi ≥ hs , ∀s ∈ S2 (MCB.3)

zi ≤ xj , ∀j ∈ δQ(i), ∀i ∈ V (MCB.4)∑i∈N

zi +∑

j∈WN,`

yj ≥ z`, ∀N ∈ N`, ∀` ∈ V (MCB.5)

∑i∈N

xi +∑

j∈WN,`

yj ≥ x`, ∀N ∈ N`, ∀` ∈ V (MCB.6)

∑j∈V

yj = 1 (MCB.7)

yi ≤ zi , ∀i ∈ V (MCB.8)

(x, z, y) ∈ {0, 1}3×|V | (MCB.9)


Valid Inequalities

Species Cuts

Lifting of CutsI also of (MCB.2) and (MCB.3)

Root-Asymmetry based

Flow-Balance based

derived with Q = 1 and hs = 1 in mind

should be strengthenable for different Q, hs


Species Cuts

Proposition

(Core-Species Cut) For a given specie s ∈ S, let Gs = (V ′,Es) be anauxiliary graph created with V ′ = V ∪ {ρ} and Es = E ∪ {{i , ρ} | i ∈ Vs}.Given this graph, the following constraints must hold∑

i∈Nzi +

∑j∈WN,ρ

yj ≥ 1, ∀N ∈ Nρ, (C-SC)

for every specie s ∈ S.

similar cut (B − SC ) in x-variables also for Buffer species

alternative way to establish connectivity (instead of(MCB.5),(MCB.6))


Lifting of Core-Connectivity Cuts

idea: exploit zi ≤ xj

Proposition

For a given i ∈ V , let N ∈ Ni be an i-separator. Let N ′ ⊂ N be a subsetof nodes such that N ′ ⊆ δ(j ′) for some j ′ ∈ N. The following inequalities

xj′ +∑

j∈N\N′

zj +∑

l∈WN,i

yl ≥ zi

are lifted version of (MCB.5) and hold ∀N ∈ Ni and ∀i ∈ V

useful for |δ(j)| ≥ 2

more involved version exists/used (based on multiple j ′)

similar lifting also for (C − SC )


Lifting of Buffer-Connectivity Cuts

idea: exploit zi ≤ xj again

Proposition

For a given i ∈ V , let N ∈ Ni be an i-separator. Let j ′ ∈ V , be a nodewith |δ(j ′) ∩ N| = |δ(j ′)| . The following inequalities∑

j∈N

xj +∑

l∈WN,i

yl ≥ xi + (|δ(j ′) ∩ N| − 1) zj′

are lifted version of (MCB.6) and hold ∀N ∈ Ni and ∀i ∈ V

more involved version exists/used (based on multiple j ′)

similar lifting also for (B − SC )


Maximal Suitability Subject to a Budget Constraint(MSBC)

habitat suitability function w : V → RB: given budget

max z ′ =∑i∈V

wizi (MSBB.1)

s.t.∑i∈Vs

cixi ≤ B (MSBB.2)

(MCB.2)-(MCB.8) must be satisfied (MSBB.3)

(x, z, y) ∈ {0, 1}3×|V | (MSBB.4)

Species Cuts, Lifting, Root-Asymmetry based ineq. still valid

Flow-Balance based ineq. not valid


Outline

1 Introduction




5 Conclusion


Implementation Details and Instances

CPLEX 12.6.1, default settings

branch-and-cut: separation of connectivity/species cuts

primal heuristic

Inte Xeon 2.3 GHz, 3 GB RAM given, time limit of 1800 seconds

instances generated following [8] and [15]I L× L grid graph G = (V ,E )I integer random node costs c (uniformly from [1, 100])I integer random node suitabilities w

2 variants: uniformly from [1, 100]; correlated, i.e, from c + [−50, 50]I core species: |S1| sets of |Vs | randomly taken nodesI buffer species: |S2| sets of |Vs | randomly taken nodesI parameters used:

L = {10, 20, 30}, |S1| = {2, 3, 5}, |S2| = 2|S1|, |Vs | = {3, 6, 9}I 10 instances are created for any parameter combination and

uniform/correlatedI B for MSBC: best MCCRB solution times multiplicator


LP-gaps

Figure : Boxplot of LP-gaps of root node for different settings.

N F1 F2 FS RV SC CL

010

20

30

40

setting

gap


MCCRB

Table : L: 10, 20, 30 ; t[s]: runtime; bbnodes: branch-and-bound nodes; gap:LP-gap at the timelimit.

basic LP fullL S1 |V1| t[s] bbnodes gap t[s] bbnodes gap t[s] bbnodes gap

10 2 3 0 2.35 0.00 0 0.50 0.00 0 0.15 0.0010 2 6 0 6.60 0.00 0 1.30 0.00 0 0.55 0.0010 2 9 0 17.85 0.00 0 1.25 0.00 0 1.15 0.0010 4 3 1 37.50 0.00 1 8.05 0.00 1 6.10 0.0010 4 6 1 30.70 0.00 0 2.20 0.00 1 3.50 0.0010 4 9 1 21.05 0.00 0 3.00 0.00 1 2.20 0.0010 5 3 1 20.75 0.00 0 4.80 0.00 1 3.10 0.0010 5 6 91 77.05 1.47 1 13.55 0.00 1 18.05 0.0010 5 9 2 103.70 0.00 1 14.95 0.00 1 12.65 0.0020 2 3 18 60.80 0.00 5 10.50 0.00 11 5.90 0.0020 2 6 5 17.10 0.00 1 2.85 0.00 3 0.70 0.0020 2 9 99 36.75 1.14 1 0.85 0.00 1 0.25 0.0020 4 3 565 641.95 1.47 113 118.30 0.00 103 75.50 0.0020 4 6 455 389.20 0.76 38 21.80 0.00 53 21.05 0.0020 4 9 363 302.00 2.12 42 34.25 0.00 50 22.55 0.0020 5 3 890 946.65 4.01 88 79.10 0.00 110 73.80 0.0020 5 6 646 433.85 3.22 43 41.00 0.00 74 37.25 0.0020 5 9 874 387.20 6.92 80 42.25 0.00 90 41.35 0.0030 2 3 789 190.05 6.00 188 44.20 0.00 266 20.40 0.0030 2 6 565 72.40 3.89 89 14.45 0.00 187 14.15 0.0030 2 9 330 56.65 0.68 12 7.75 0.00 17 2.50 0.0030 4 3 1699 204.55 24.10 1250 153.45 4.98 1304 81.60 7.0030 4 6 1546 84.00 18.83 631 59.60 1.07 699 31.90 1.2330 4 9 1655 95.55 26.71 858 58.30 1.85 876 34.30 5.0830 5 3 1800 167.40 25.78 1177 182.80 5.46 1347 110.05 5.8830 5 6 1800 95.20 28.93 1001 77.35 4.85 1220 48.75 4.9730 5 9 1800 77.35 29.73 955 54.75 3.71 1163 36.70 5.84


MSBC: Uniform Instances

Table : L: 10, 20, 30; t[s]: runtime; bbnodes: branch-and-bound nodes; gap:LP-gap at the timelimit.

multi. of B → 1.1 1.2 1.3L S1 |V1| t[s] bbnodes gap t[s] bbnodes gap t[s] bbnodes gap

10 2 3 1 10.70 0.00 1 35.70 0.00 1 51.60 0.0010 4 3 1 53.90 0.00 1 59.20 0.00 1 55.40 0.0010 5 3 1 40.20 0.00 1 48.80 0.00 1 51.00 0.0010 2 6 0 8.40 0.00 1 15.40 0.00 1 24.10 0.0010 4 6 1 67.70 0.00 1 66.30 0.00 2 116.60 0.0010 5 6 1 40.50 0.00 1 69.80 0.00 1 63.90 0.0010 2 9 1 18.40 0.00 1 52.40 0.00 1 51.30 0.0010 4 9 1 29.90 0.00 1 35.10 0.00 1 47.70 0.0010 5 9 2 56.40 0.00 2 70.00 0.00 2 89.70 0.0020 2 3 112 289.20 0.00 296 502.60 0.18 299 620.30 0.0020 4 3 750 863.10 8.15 671 1114.20 3.14 570 1190.70 3.2420 5 3 971 1116.40 5.77 731 1221.80 6.18 648 1510.70 2.9620 2 6 31 66.10 0.00 74 160.50 0.00 76 273.60 0.0020 4 6 143 530.30 0.00 189 575.40 0.00 188 758.10 0.0020 5 6 895 993.10 13.95 1097 1419.00 8.96 919 1422.40 5.5920 2 9 37 82.60 0.00 48 141.70 0.00 95 263.10 0.0020 4 9 341 615.10 1.04 607 842.10 2.89 321 577.80 1.8420 5 9 404 743.30 0.00 630 837.30 3.87 766 1335.00 4.25


MSBC: Correlated Instances

Table : L: 10, 20; t[s]: runtime; bbnodes: branch-and-bound nodes; gap: LP-gapat the timelimit.

multi. of B → 1.1 1.2 1.3L S1 |V1| t[s] bbnodes gap t[s] bbnodes gap t[s] bbnodes gap

10 2 3 1 22.80 0.00 1 25.40 0.00 1 36.40 0.0010 4 3 1 59.10 0.00 1 44.50 0.00 1 74.60 0.0010 5 3 181 48.70 2.48 2 95.00 0.00 1 86.50 0.0010 2 6 1 10.50 0.00 1 17.50 0.00 1 40.20 0.0010 4 6 1 35.40 0.00 14 45.90 0.00 1 50.00 0.0010 5 6 2 86.70 0.00 2 99.50 0.00 2 122.20 0.0010 2 9 1 30.40 0.00 1 39.60 0.00 182 73.00 7.2910 4 9 1 45.20 0.00 1 46.50 0.00 1 36.30 0.0010 5 9 2 2 81.40 0.00 2 85.80 0.00 2 111.30 0.0020 2 3 183 414.10 0.00 248 626.50 0.00 292 663.20 0.0020 4 3 826 854.80 9.09 459 681.70 1.39 471 927.40 1.1120 5 3 616 979.60 5.43 725 1080.10 5.44 713 1832.30 0.4220 2 6 81 275.40 0.00 83 295.10 0.00 98 350.90 0.0020 4 6 185 404.60 0.00 555 748.80 0.61 489 827.00 0.0020 5 6 284 330.60 7.37 237 418.10 0.00 352 611.40 0.0020 2 9 22 54.70 0.00 49 140.30 0.00 91 213.10 0.0020 4 9 171 359.20 0.00 147 365.00 0.00 232 447.00 0.0020 5 9 323 605.60 0.00 481 855.60 0.00 541 765.90 1.42


Outline

1 Introduction




5 Conclusion


Oregon Data Setvariants of data set widely used in literaturebased on US-GAP data from [13]we concentrated on forest part in western Oregon129 mammals, 5 of it core (endangered/vulnerable status [12])1245 land parcelscost function: area


Results

(a) no buf., no con. (b) buf., no con. (c) no buf., con.(35% gap)

(d) buf., con.(20% gap)


Not Considering Buffer Species Constraints

shape/compactness of the reserves not very pleasing

many buffer species in many land parcels

→ throw out the buffer species constraints

(a) no buf., no con.80 species in core0 species in buffer80 total

(b) buf., no con.80 species in core80 species in buffer80 total

(c) no buf., con.88 species in core0 species in buffer88 total

(d) buf., con.86 species in core91 species in buffer91 total


MSBC (Not Considering Buffer Species Constraints)

suitability of i : number of species occurring there

(a) B=1.594 species in core95 species in buffer95 total

(b) B=295 species in core99 species in buffer99 total


Conclusion

What we did . . .I . . . introduced Natural Reserve Design Problem considering both

connectivity and buffer requirementsI presented valid inequalitiesI computational studyI case study on real-life instance.

Directions for future work . . .I . . . iterative refinement procedureI . . . Benders decomposition, extended formulation, . . .I . . . study the general precedence versionI . . . look at other variantsI . . . what problem/constraints really capture the real-life issues?


Thank You for Your Attention!

Questions?

Designing Nature Reserves With Connectivity and BufferRequirements

E. Alvarez-Miranda1 M. Goycoolea2 I. Ljubic3 M. Sinnl4

http://homepage.univie.ac.at/markus.sinnl/wp-content/uploads/

2015/09/or2015.pdf

OR 2015, September 1-4, Vienna, Austria


http://homepage.univie.ac.at/markus.sinnl/wp-content/uploads/2015/09/or2015.pdf

http://homepage.univie.ac.at/markus.sinnl/wp-content/uploads/2015/09/or2015.pdf

Literature I

[1] M. Batisse. The biosphere reserve: A tool for environmentalconservation and management. Environmental Conservation, 9:101–111, 6 1982.

[2] M. Batisse. Development and implementation of the biospherereserve concept and its applicability to coastal regions. EnvironmentalConservation, 17:111–116, 1990.

[3] P. Beier and R. Noss. Do habitat corridors provide connectivity?Conservation Biology, 12(6):1241–1252, 1998.

[4] A. Billionnet. Mathematical optimization ideas for biodiversityconservation. European Journal of Operational Research, 231(3):514–534, 2013.

[5] H. Cayton, N. Haddad, N. McCoy, et al. Conservation Corridor:Technical Papers and Methods, 2015. URLhttp://conservationcorridor.org/corridor-toolbox/

technical-papers-and-methods/.


http://conservationcorridor.org/corridor-toolbox/technical-papers-and-methods/

http://conservationcorridor.org/corridor-toolbox/technical-papers-and-methods/

Literature II

[6] R. Church, D. Stoms, and F. Davis. Reserve selection as a maximalcovering location problem. Biological Conservation, 76(2):105–112,1996.

[7] M. Clemens, C. ReVelle, and J. Williams. Reserve design for speciespreservation. European Journal of Operational Research, 112(2):273–283, 1999.

[8] B. Dilkina and C. Gomes. Synthetic corridor problem generator, 2012.URLhttp://www.cs.cornell.edu/~bistra/connectedsubgraph.htm.

[9] M. Fischetti, M. Leitner, I. Ljubic, M. Luipersbeck, M. Monaci,M. Resch, D. Salvagnin, and M. Sinnl. Thinning out Steiner trees: anode-based model for uniform edge costs, 2014. URLhttp://dimacs11.cs.princeton.edu/workshop/

FischettiLeitnerLjubicLuipersbeck.pdf.


http://www.cs.cornell.edu/~bistra/connectedsubgraph.htm

http://dimacs11.cs.princeton.edu/workshop/FischettiLeitnerLjubicLuipersbeck.pdf

http://dimacs11.cs.princeton.edu/workshop/FischettiLeitnerLjubicLuipersbeck.pdf

Literature III

[10] K. Ohman and T. Lamas. Reducing forest fragmentation in long-termforest planning by using the shape index. Forest Ecology andManagement, 212(1–3):346–357, 2005.

[11] H. Onal and R. Briers. Selection of a minimum boundary reservenetwork using integer programming. Proceedings of the Royal Societyof Londn B: Biological Sciences, 270(1523):1487–1491, 2003.

[12] Oregon Department of Fish and Wildlife. Threatened, Endangered,and Candidate Fish and Wildlife Species.http://www.dfw.state.or.us/wildlife/diversity/species/

threatened_endangered_candidate_list.asp, 2015. [Online;accessed 19-July-2015].

[13] Pacific Northwest Landscape Assessment & Mapping Program.Wildlife Models, 2015. URLhttp://www.pdx.edu/pnwlamp/wildlife-models.


http://www.dfw.state.or.us/wildlife/diversity/species/threatened_endangered_candidate_list.asp

http://www.dfw.state.or.us/wildlife/diversity/species/threatened_endangered_candidate_list.asp

http://www.pdx.edu/pnwlamp/wildlife-models

Literature IV

[14] R. Pressey, H. Possingham, and J. Day. Effectiveness of alternativeheuristic algorithms for identifying indicative minimum requirementsfor conservation reserves. Biological Conservation, 80(2):207–219,1997.

[15] J. Williams. Optimal reserve site selection with distance requirements.Computers & Operations Research, 35(2):488–498, 2008.

[16] J. Williams and C. ReVelle. A 0-1 programming approach todelineating protected reserves. Environment and Planning B:Planning and Design, 23(5):607–624, 1996.

[17] J. Williams and C. ReVelle. Reserve assemblage of critical areas: Azero-one programming approach. European Journal of OperationalResearch, 104(3):497–509, 1998.

[18] J. Williams, C. ReVelle, and S. Levin. Using mathematicaloptimization models to design nature reserves. Frontiers in Ecologyand the Environment, 2:98–105, 2004.


Literature V

[19] J. Williams, C. ReVelle, and A. Levin. Spatial attributes and reservedesign models: A review. Environmental Modeling & Assessment, 10(3):163–181, 2005.


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