Haluk
Eli
¸s
a ,∗,
Barbaros
Tansel
a ,1,
Osman
O
˘guz
a,
Mesut
Güney
c,
Ramez
Kian
b a Department of Industrial Engineering, Bilkent University, Ankara, 06800, Turkeyb Nottingham Business School, Nottingham Trent University, Nottingham NG1 4FQ, UK c Independent
a
r
t
i
c
l
e
i
n
f
o
Article history: Received 21 February 2019 Accepted 1 July 2020 Keywords:Terrain guarding problem Blocking path problem Border security
Network interdiction problem K-barrier coverage problem Finite dominating sets
a
b
s
t
r
a
c
t
Locatingaminimumnumberofguardsonaterrainsuchthateverypointontheterrainisguardedbyat leastoneoftheguardsisknownastheTerrainGuardingProblem(TGP).Inthispaper,arealisticexample oftheterrainguardingproblemisstudied,involvingthesurveillanceofaruggedgeographicalterrainby meansofthermalcameras.AnumberofissuesrelatedtoTGPareaddressedwithinteger-programming modelsproposedtosolvetheproblem.Also,asensitivityanalysisiscarriedoutinwhichfivefictitious terrainsarecreatedtoseethe effectofthe resolutionoftheterrain,and ofterraincharacteristics,on coverageoptimizationand therequirednumberofguards.Finally,anew problem,whichiscalledthe BlockingPathProblem(BPP),isintroduced.BPPisaboutguardingapathontheterrainwithaminimum numberofguardssuchthatthepathblocksallpossibleinfiltrationroutes.Adiscussionisprovidedabout therelationofBPPtotheNetworkInterdictionProblem(NIP),whichhasbeenstudiedextensivelybythe operationsresearchcommunity,andtothek-BarrierCoverageProblem,whichhasbeenstudiedunderthe SensorDeploymentProblem.BPPissolvedviaaninteger-programmingformulationbasedonanetwork paradigm.
© 2020ElsevierLtd.Allrightsreserved.
1. Introduction
Issuesrelatedtohomelandsecurityintoday’sworldhavebeen receiving increasing attention from governments and military. A particularissueofinterestinthiscontextistheproblemof mon-itoringacriticalgeographicalregiontoachieveasatisfactorylevel of surveillance,without havingto deployan excessive amountof resources tocarryout thewatch.Thiscanbeachievedby mathe-maticalprogrammingandterrainmodelingtechniquesdiscussedin thispaper.Watchtowersarelocatedonterrainstodetectfires[23] , militaryunitsaredeployedtowatchtheterraintoprevent infiltra-tion,andrelaystationsareplacedontheterrainsuchthatnodead zoneispresenttomaintainuninterruptedcommunication[14] .In this paper, any entity that is capable of observing or sensing a piece of land oran object on the land isreferred toas a guard. Thus,watchtowers,militaryunitsandrelaystationsareguardsand so are sensors, observers (human beings), cameras and the like.
R This manuscript was processed by Associate Editor Wilco van den Heuvel. ∗ Corresponding author.
E-mail addresses: helis@bilkent.edu.tr (H. Eli ¸s ), ooguz@bilkent.edu.tr (O. O ˘guz), mstdzgny@gmail.com (M. Güney), ramez.kian@ntu.ac.uk (R. Kian).
1 Posthumously
Observing,seeing, sensing, covering and guarding will mean the same. Locating a minimum number of guards on a terrain such that every pointon the terrain isguarded by atleast oneof the guardsisknownastheTerrainGuardingProblem(TGP).Our inter-estinthispaperistheTGPonthree-dimensionalterrains.Tosolve TGPforarealthree-dimensionalterrainofinterest,theterrain sur-face has to be approximated, or represented, as a mathematical objectso thatcertain analyses can be carriedout. Thedigital (or mathematical)representationofarealterrainsurfaceisknownas aDigitalElevationModel(DEM)[13,32] .LetS∈R3 bethesurface ofarealterrainofinterest,forwhichaDEMistobeconstructed. Let P=
{
p1,...,pn}
be the set of n points sampled fromS with knownx,y andz (elevation value)coordinates ina fixed coordi-nate system. We assume that the pointsin P are sampled such thatthey representSsufficiently.Let p∗i∈R2 betheprojection ofpi ontotheEuclideanx− yplane,andP∗=
{
p∗1,...,p∗n}
betheset ofsuchpoints.ThepointsinPandP∗arereferredtoasvertices(or gridpoints).ConsidertheconvexhullDofverticesinP∗.Fora∈R3 letxc(a),yc(a)andzc(a)denotethex,yandzcoordinatesofa, re-spectively. The DEM T˜ that approximates S is characterized by a functiondefinedoverD, f:D−→R+,suchthat f(
p∗i)
=zc(
pi)
,i=1,...,n. Then, T˜=
{
(
x,y,z)
:z= f(
x,y)
,(
x,y)
∈D}
. Note that our definitionallowspointsinP∗tobeconnectedbyedges(orcurves)https://doi.org/10.1016/j.omega.2020.102303 0305-0483/© 2020 Elsevier Ltd. All rights reserved.
Fig. 1. The grid cells on the plane.
suchthat the lines donot intersect exceptatvertices. When the verticesareconnectedbyedges asdescribed,Disthensaidtobe partitionedintoregions.
Twowidely used approximationsforareal terrainare regular squaregrid(RSG)andtriangulatedirregularnetwork(TIN)[13] .To obtainanRSG,pointsaresampledfromSatregularintervalsand theirprojectiononthex− yplanecompriseswhatiscalledaDEM mesh,i.e., grids ofsquare shapesformed among the four neigh-boringpoints. EachpointinP∗ isassumedto beatthe centerof acell(orpixel)(Fig. 1 ).Eachcellisthenelevatedtotheheightof thepointatthecenter.Theterrainconstructedassuchiscalleda steppedmodelofRSG[10,13] (Fig. 2 (a)). TINisobtainedsimilarly, butasopposedto RSG,certain pointssuch aspeaks andpits are includedinthevertexsetduetotheir importanceandtherestof thepointsaresampledirregularlyfromtherealterrain.Next,the projectionsofthesepointsaretriangulatedon theplane [5] ,and thetriangulationiselevatedtotheheightofeachvertextoobtain aTIN(Fig. 2 (b)).
There are differentviews as to whetherRSG or TIN is a bet-ter approximation of the terrain. [23] consider TIN to be a bet-ter representationsince construction can be done atan irregular sampleofpoints, whichallows criticalpointsontheterrain such aspits, peaksandpointsonridgestobe selected for approxima-tion,in contrast to the regular samplingdone in RSG. [17] state thatneither oftherepresentations isbetter thanthe other while [33] arguethatRSGisconceptuallybetter,morecompactand eas-iertoimplement thanTIN. The surfaceina grid maybe created byinterpolating theelevationsof thefourpointsthat are on the cornersofthegridsquareaspointedoutby [20] ,orasdiscussed by[32] ,byusingahigherorderpolynomial.Intheapplication ex-amplegiveninSection 4 ,P∗ iscomposedofgridsbutthesurface isobtainedbybilinearinterpolationusingMATLABsoftware. Bilin-earinterpolationseemsbetterthan bothRSGandTIN because,to approximatetheheight ofacertain pointon therealterrain, the realelevations of four pointson the corners of the grid cell are usedinstead ofthree in TINsandone in RSG. Inthe application study,ourapproachaimsatguardingthesetofinterpolation ver-tices(gridpoints)bytheinterpolationverticesandthisresultsin verygoodcoverage.RSGisalsoreferredtoasgridsforsimplicity.
For most real world studies involving a geographical region, which may be a terrain or a network of cities, the data about the spatial and the nonspatial features of the region of interest are likelyto be needed. The requireddatum could be the height of a point or the population of a city. A Geographical Informa-tion System(GIS) is a decisionsupport system whichstores, ed-itsanddisplaystherawdataonageographicalregion.Itcan per-formcertain analyses andvisualize the resultsof those analyses, [7,10,34] . ArcGIS, MATLAB, Google Earth, andMapInfo are exam-ples of softwarepackages equippedwith GIS. Locationproblems, ingeneral terms,involvelocating facilities (firestations,hospitals, etc.)toserve customersina location spacesuchasa networkof cities.Somelocationsmaynotbedesirable,forinstance,iftheyare nearriverbedswherethereisafloodrisk.Thisandsimilar infor-mation canbeobtainedfromaGIS andvisualizedfora decision-maker. Ina realworld study,a GIS providesthe dataon the ter-rain,performs viewshed analyses anddisplaysthoseregions visi-bletotheobserver.AdetailedoverviewofGISfunctionsandofthe linksbetweenGISandlocationproblemsispresentedin[7,10,34] . The major contributions of our paper can be summarized as follows;
(i) Thepaperserves asa completeguidetoguarda(real) ter-rainforbothscientistsandpractitioners,whichtheexisting litera-turedoesnotprovide.Wecarefullylaydowntheassumptionsand thetheory underlyingtheproblem, anddiscussallaspects ofthe problemstartingfromarigorousdefinitionoftheterrainitself,as discussed above, and a rigorous definition of the Terrain Guard-ingProblem,topointingoutall thefactorsinvolvedinguardinga realterrain,toacarefulinterpretationanddiscussionofthe mod-elsthatareusedtosolvetheproblem.
(ii)Ourpapershowsthat thereisa directconnectionbetween thelocationscienceandtheTerrainGuardingProblem,whichhas notbeenestablishedbeforeintheliterature.Thisconnectionisdue to theconcept of finitedominatingsets (FDS).An FDS isa finite setoffeasiblepointsthatcontainsanoptimalsolutioninan opti-mizationproblemwherethefeasiblesetiscomposedofinfinitely (possiblyuncountably)manypoints.AlthoughFDS’shavefirstbeen shown and discussed for location problems, the concept applies tomanyoptimizationproblemsincludingthelinearprogramming problem, where the setof extremepoints isan FDS.In the TGP, theguardlocationsaretobechosenfromtheterrain,whichisan uncountablepoint set, andno FDShas been found forthe prob-lemyet.Thus,weemphasizethat thereisnoresearch,appliedor theoretical, includingours,that solves the three-dimensional Ter-rainGuardingProblemtooptimalityoneithertriangulated irregu-larnetworks(TIN)orthegridDEMs.
(iii) Inthispaper,we studyTGP ongrids. TGPongrids is de-finedtobelocatingminimumnumberofguardsonthegridpoints
the algorithms used by the above mentioned authors andothers that use similarapproaches are heuristicsanddonot necessarily yieldoptimalsolutions.Also,thesestudiesdonotdiscussthe per-formance oftheir algorithm withrespect tothe optimalvalue to assesstheworthoftheirsolutions.Incontrast,weconsiderallgrid pointsasguardlocationsanduseanexactalgorithm providedby theoptimizationpackagetoobtainoptimalsolutions.
(iv)Thispaperisthefirsttopoint outthatthereisaclose re-lation betweenTGP andthe Sensor Deployment Problem,which, webelieve,willstimulatefurtherresearchonbothproblems. Sen-sor DeploymentProblem isaboutlocating sensors witha certain range ona sensorfield, whichis likelyto be aterrain, such that the sensors sense (guard)thefield andthe numberof sensorsis minimized.
(v)The applicationstudyinSection 4 introduces thereaderto thekindofsoftwareinvolvedinguardingarealterrain.Thevisual presentationsshowthelocationsofgridpointsontheterrainand thishelpsthereadertobetterunderstandwhatitmeanstocover alandbygridpoints.Itisapparentfromourfiguresthatbetween grid pointstheterrain elevationsdonot presentsuddenchanges, which,apart fromthe research resultsthatsupport ourchoiceof resolution,indicatesthattheresolutionusedinthepaperisa sat-isfactory one.MoststudiesonTGPdonotprovideenough visuals forthereadertobetterassessthebenefitoftheseapplication stud-ies.
(vi) It isgenerally assumedthat more rugged terrains require moreguards.ThesensitivityanalysiswepresentinSection 5 tests thisassumptionandshowsthattwoterrainsmayrequirethesame numberofguardsirrespectiveofterraincharacteristics(howsteep or smooth they are). The results also illustrate the trade-offs in-volved inchoosinga resolutioninguardinga realterrain.Results oftheexperimentindicatethata75percentdecreaseinthe reso-lutionoftheterrainresultsina20percentlossincoverage.
(vii) We also introduce a novel problem, called the Blocking Path Problem (BPP), and discussits relation to well-known Net-workInterdictionandk-BarrierCoverageproblems.Theproblemis solved tooptimality usinganetwork model.The blockingpathis a path such that every point on the path isguarded by a guard thatislocatedsomewhereontheterrainandthepathblocksany routethatintrudersmayusewhenthey infiltratethroughthe ter-rain.TheBlockingPathProblem isaboutlocatingminimum num-ber of guards on the land such that a blocking path is created. The BlockingPath Problemprovides adifferentsolutionapproach to the problemof disablingthe routesused by smugglers inthe NetworkInterdictionProblem.
The rest of this paperis organized as follows.Section 2 pro-videsaformaldescriptionoftheTerrainGuardingProblem,which isfollowedbytheliteraturereviewinSection 3 .InSection 4 ,areal worldapplicationoftheTGPispresentedwhileasensitivity anal-ysisiscarriedoutinSection 5 toidentifytheeffectofterrain reso-lutionandterraincharacteristicsontheoptimalnumberofguards andonthecoverageofterrains.InSection 6 ,weintroduceand dis-cusstheBlockingPathProblem,thenwecompareitwiththe
Net-Theline-of-sight (LOS) originatingfroma point ina given di-rectionisthesetofpointsoftheformx+
λ
d,λ
≥ 0,wherexisa pointinR3,disanonzerodirectioninR3 andλ
isanon-negative realnumber.GiventhevisibleregionV,regionF,andthesurfaceTthatformstheborderbetweenVandF,letx1andx2inR3betwo pointssuch thattheir projection on thex− yplane is inD. Con-sider the line segment LS
(
x1,x2)
:=x1+
λ
(
x2− x1)
:λ
∈[0,1] connecting the pointsx1 andx2. We say x2 is visible fromx1 ifLS(x1,x2) is asubset of V, andx2 is not visiblefromx1 if LS(x1, x2)∩F = ∅.As it is commonly assumedin the literature, visibil-ityisasymmetricconcept,i.e., ifx2 isvisiblefromx1 thenx1 is visiblefromx2.Wealsosaythatx1 sees/guards/coversx2 ifx1 is visiblefromx2(Fig. 3 ).Avisibilityfunctionisdefinedasfollows,
v
(
x1, x2)
=v
(
x2, x1)
=1, if LS
(
x1, x2)
⊆ V0, otherwise. (1)
LetxbeapointonTandVS(x)denotethe“viewshed” ofx,i.e.,
VS
(
x)
={
y∈T:v
(
x,y)
=1}
.LetX={
x1,...,xk}
beasetofpoints onT.XguardsorcoversTifeverypointonTisguardedbyatleast oneoftheguardslocatedatpointsinX.Weexpressguardingofa pointybyasetXbythefunction,V IS
(
y, X)
=1, if
∃
xj∈X, s.t.v
(
y, xj)
=10, otherwise. (2)
Inthe TerrainGuardingProblem (TGP), thegoal istofind the minimumcardinalitysetXwhoseelementsbelong toTsuchthat
XguardsT.TGPisformallydefinedasfollows,
(TGP)min
|
X|
(3)s.t.
V IS
(
y, X)
=1,∀
y∈T (4)X⊆ T. (5)
Notethat,bydefinitionoftheproblem,theguardshaveinfinite rangeandthereisonlyone typeofguard.However,TGPcan eas-ilybe extended to includeseveral typesof guards withdifferent rangesandcosts, andtoincludecriticalpointsthat requiretobe guardedbymorethanoneguard.
3. Literaturereview
De Floriani et al. [12] are the first to investigate the terrain guarding problem. They show that the terrain guarding problem canbesolvedby aset-coveringformulation.Althoughthey posed theproblemandproposed asolutionforTGP onTINstheir solu-tionapproachisalsoapplicable toTGPongrids. Theychoose the verticesofthetrianglesaspotentialguardlocations.However,the choiceofthesetofverticesaspotential guardlocationsdoesnot guaranteethatthesolutionobtainedisoptimalsincethesetof ver-ticeswasnotprovedtobea‘finitedominatingset’(FDS).
Asdiscussedintheintroduction,anFDSisafinitesetofpoints thatcontainsanoptimalsolutiontoanoptimizationproblemwith (possibly)an uncountablefeasibleset.Itisimportanttonotethat evenif an exact algorithm isused to solve TGPthe solution ob-tainedisnot optimalunlessthesetofpointsconsideredasguard locationsisshowntobeanFDS.Theconceptwasusedby[25] and othersthereafterinlocation problems,butthe term‘finite domi-natingset’wasfirstintroducedby[27] fornetworklocation prob-lems and later has been extensively used in location science. In locationproblems,ingeneralterms,thereisanumberoffacilities (serviceproviders)tobe locatedonasurfacesuch thatthe facili-tiesmeetthedemandofthecustomers(servicereceivers)andthe numberoffacilitiesisminimized[15 ,19 ,29 ,37] .Inthissense, TGP isalocationproblemwhereguards,insteadoffacilities,arelocated on a terrain to cover the terrain, where each point on the ter-rainmaybeconsideredasacustomerwhosedemandcorresponds tobeingcoveredby aguard.[11] show thattheTerrain Guarding ProblemisNP-hard.
Goodchild andLee[23] discussthelocation setcovering prob-lem (LSCP) and the maximal covering location problem (MCLP) models to placewatchtowers on a terrain for detectionof forest fires.LSCP, firstintroducedby [40] ,isinfact aset-covering prob-lemwithin alocationcontext.Theirstudyinvolveslocating emer-gencyservicefacilities(firestations,ambulances,hospitalsetc.)to servecustomerswithin amaximumresponsetime s.Each poten-tialsitecancovercertaincustomers,whichisdeterminedthrough a preprocessingof the data.The goal is to provide serviceto all customerswitha minimumnumberof facilities.We useLSCPto minimizethenumberofguards tobe locatedonthe terrain,and givethedetailsofthemodelinSection 4 .MCLPwasintroducedby [9] .It issimilar tothelocation setcovering problem, butinstead of minimizing the number of facilities required to serve all, the numberofcustomersprovidedwithservicearemaximizedwitha givennumberoffacilities.Thenumberoffacilities isfixeddueto budgetaryconstraints.InSection 4 ,weusethismodeltomaximize theportionsoftheterraincoveredbyagivennumberofguards.
Eidenbenz [16] utilizes algorithms developed for set covering problem to devise approximation algorithms for TGP on TINs. [6] use resultsfrom covering planar graphs to obtain vertexand edge guards to cover the terrain. [23] and [31] present heuris-ticsfortheproblem.FortheTGPongrids,[21,33] ,and[39] apply heuristicmethodstosolvetheproblem.Baoetal.[4] locate watch-towerstodetectforestfiresandtheyuselocationsetcoveringand maximalcoveringlocation modelstoobtainexactsolutions. How-ever,onlyasubset ofavailable points(gridpointsinthemesh) is chosenaspotentialguardlocationsin[4,21,33] ,whichimpliesthat theirsolutionisnotoptimal.Also,thesestudiesdonotdiscussthe performance oftheir heuristicswithrespect to theoptimal solu-tion.Inourapplication,allpointsareconsideredaspotentialguard locationsandwelettheoptimizationmodel,whichusesanexact algorithm,choosethebestsitestodeploytheguards.
Sensor deployment isabout placing sensors on a sensor field (ontheterrain),whichiscomposedofequallyspacedgridpoints, suchthat the gridpointson thefield are coveredby thesensors (Fig. 4 ). A sensor is said to cover a given grid point if the grid
Fig. 4. A sensor field and the area covered by a sensor placed at a grid point.
point iswithinthe rangeofthesensor. Then,theSensor Deploy-mentProblemistolocatethesensors,whichhavedifferentranges andcosts,onthesensorfieldsuchthatthefieldiscoveredbythe sensorsandthetotalcostofthesensorsisminimized[8,38] .Note that,byourdefinition,theSensorDeploymentProblemisaversion oftheTerrainGuardingProblemwheretheguardsaresensorsand haveacertain range.ThemaindifferencebetweentheSensor De-ploymentProblemandtheTGPaswedefined,isthatintheSensor DeploymentProblemtheshapeoftheterrain,i.e.,theline-of-sight betweenanytwopoints,isnottakenintoaccount.Inguardingreal terrains, however,ashasbeen done inour applicationstudy,the line-of-sight betweentwo pointsneeds tobe checkedagainst the terrain elevationtodecide ifa guardcovers apoint even though thepointmaywellbewithintherangeoftheguard.
4. Arealterrainguardingcasestudy
Inthissection, we aimto solvethe TerrainGuarding Problem ona realterrain. Welocateguardsongrid pointsandthegoalis to coverall grid pointson the terrain, whosesurface is approxi-matedbybilinearinterpolation.Therealisticexampleofthe prob-lem,whichinvolvesthesurveillanceofa ruggedgeographical ter-rainbymeansofthermalcameras,i.e.,guards,isstudiedby apply-ing locationmodels.The terrainstudied is50kmlongby 1.6km wide(Fig. 5 ).Theregionusedislongandthinsinceitrepresentsa sectionofabordernexttotheborderlinebetweentwo neighbor-ing countries, whereillegal intruders canbe apprehended. While the border andthe two neighboring countries are fictitious, the terrainisreal.
Smuggling, illegal immigrants and terrorism are among the biggest threats to national security, and these activities occur mostlyacrossbordersatnight,whenvisibilityisdiminished. Bor-der security has gained more importance since 9/11 [36] . Bor-dersecuritycanbemaintainedusingseveralresources:unmanned aerialvehicles(UAVs)[44] ,balloons,watchtowers,patrollingunits, thermal orother types ofcameras/sensors etc. Watchtowersand balloonsareeasyforanintrudertospotand,therefore,are them-selvestargetsforasimple(rocketorsimilar)attack. Theeffectof inclementweatherconditionsonUAVs’surveillancecapabilityand high accident rates are major drawbacks in implementing UAVs forbordersurveillance[24] .Thesefactorsmakethermalcamerasa betteroptionforbordersecurity. Thermalcamerassense theheat emitted byany agenton theland, i.e., human,animalorvehicle, andarenotaffectedby darkness.Thesedevicescaneitherbe op-eratedby trainedpersonnelormadetooperateautonomously.In this study, cameras are operated by security personnel who can hide fromintruders insidesuitablydugtrenches. Athermal cam-era atsiteican detectan eventatsitej ifthereis adirect line-of-sight betweenpointsi and j, andj is within the range ofthe camera placedati.Personnelusing camerasobserve aregion
as-Fig. 5. Views of the terrain from different angles. Pictures were obtained from Google Earth.
Fig. 6. The reverse triangular shaped field shows the area assigned to a camera for surveillance. The surveillance pattern is from far to near and changes from left to right and right to left as the guard looks closer.
Fig. 7. The image seen by a thermal camera. Picture is taken from the Internet.
signedtothemaccordingtoapre-determinedsurveillancepattern. OnesuchpatternisillustratedinFig. 6 .
We plantolocate thermalcamerasonarugged borderregion thatisboundedwithinarectangularregionwhenprojectedonthe x-y plane. The region is covered under night conditions (Fig. 7 ).
Thepicturesoftheterrainusedinthisstudyweretakenfromthe GoogleEarthsoftware.Thedigitalelevationdataoftheregionwas converted usingNetCAD, a GIS software,into a data format that GoogleEarthcanread.
Arealworldguardingproblemmustnecessarilyinvolvetheuse ofaGIS.[35] considerplacingcamerasonacampusareathat dis-cretizethecontinuous3-Dspacewithregulargridsquaresofsize 3ftby3ftusingArcGIStomodeltheproblemasinteger program-ming.[42] modelthesecurityofaportandsolvetheproblemby abranchandcutalgorithm,usingArcGISforvisibilityanalyses.
Theintervalbetweenthegridpointsisgenerallyfixedandthe squareformedbyfourneighboringpointsisalsoknownasapixel [26] . An important issue in guarding a real terrain is the pixel size,i.e.,thedistancebetweengridpoints,asthedistancebetween grid points increases DEM is less representative of the real ter-rain. In Section 5 , we explore the effect of pixel size on terrain guarding withan experiment. As [26] states,“although no abso-luteidealpixel sizeexists forterrain analysis, andtherightpixel sizeisrelativetotheapplicationtypeandprojectobjectives,20to 200marestandardgridresolutionsinmostcases.” Earlierstudies ([22,33,41] ) usea resolution of10m.,which leadstoa better ap-proximationoftheterrain. Inourapplication, pointsare sampled atevery 200 mstartingfromthe corners ofthe region(i.e., grid resolution is200 m) and2000 grid pointsare obtained for cov-erageanalysis (Fig. 8 ). In determining ifany two grid pointssee each other,line-of-sight algorithms(Xdraw, R3,R2 etc.)generally choosesomepointsalong theline-of-sight,estimate theheightof thesepointsanddecidethattwopointsseeeachotherifthe line-of-sightgoesabovetheheightoftheselectedpointsorotherwise decidethattheyarenotintervisible[18 ,28 ,33] .Thus,such line-of-sightalgorithmsincludingtheoneusedinourstudyare approxi-mations.WehaveusedMATLAB astheGIS tool.Itusesabilinear interpolationto approximatethesurfacewithineach grid cellfor
Fig. 9. The sectors around a potential guard location.
line-of-sight (LOS) calculations. Visibility analyses are performed usingMATLAB’s viewshed tool function on the 2000grid points. Theline-of-sightisrestrictedby7500m,therangeofthethermal camera.
Another factor affecting coverage is the relationship between thesensor’scoverage radiusandgridspacingsince agridspacing greaterthan thesensor’s coverage radiuswouldobviously be ab-surd.[2] exploretherelationshipbetweenthesensor’scoverage ra-diusandgridspacingusingtwo three-dimensionalroomsintheir experiments,andshowthatastheratioofgridspacingtothe sen-sor’scoverage radiusdecreases theportionofthetargetedregion coveredbythesensorsincreases.Specifically,whengridspacingis one-fifthofthesensor’scoverage radius,thecoverage obtainedis 99%.Inotherwords,when allgridpointsarecovered,99% ofthe targetedrealspaceiscovered.Theyrecommendthatthegrid spac-ingbemuchsmallerthanthesensor’scoverageradius.Inourcase, theratioofgridspacingtothesensor’scoverageradiusis approx-imately1/37,whichisverysmallincomparison.Thus,considering thesmall ratioin our realistic case, a very good coverage of the realterrainisexpected.
A‘visibilitymatrix’A=
ai jN×N iscreatedasfollows.Ifthekth gridpointisvisiblefrommthgridpoint,andthedistancebetween themislessthanorequaltotherangeofthethermalcamera,then theak,mentryofthematrixequals1,andequals0,otherwise.This procedureisrepeatedforallpairsofgridpointstoobtainA.Fora terrainwithngridpointsthetimecomplexitytocreateAisO(n2). Letusplacethex− yaxisonpointpsuchthattheyaxisbears north(neighboring country) andp is atthe origin(Fig. 9 ). Axes dividethe region 360◦ around p into fourequal sectors. Assume that a singleguard is placed at p and the guard is assigned all sectorsforsurveillance.Then,whiletheguardwatches,say,sector
I,theother sectors, especially sector IIIinthe opposite direction, will remain unsurveilledfor a considerableamount of time (this timecanbeatleast10minutes,whichisnotdesirable).Therefore, apolicyisusedsuchthatthenumberofcamerastobeplacedatp
istwoifaguardatpseesatleast30locations(basedonthefield study) in atleast two of the foursectors, otherwise one camera isused.Whentwocamerasareneeded, theyareplacedinsucha waythattheyshareallvisiblepartsequally(Fig. 10 ).
Notethat,insteadofdictatingthenumberofcameraswecould letthe optimizationmodeldecidehowmanycamerastoplaceat acertain spot.However, iftherewere morethantwocamerason a spot,then asingle(rocket)attack on thespotcoulddestroy all ofthe sensors,leavingmany partsofthe regionunguarded fora considerableamountoftime,quite apartfromthecostofhuman livesandthesensors.Toovercomethisproblem, themodelcould againbeallowed tochoosethenumberofsensorsatapoint sim-plybydefiningadecisionvariableandrestrictingittobelessthan orequalto2.However,thismightresultinplacingonlyone cam-eraatacertainlocationwhichhastoviewmanylocationsin dif-ferent directions (for example 100 points, 50 in sector I and 50 insector III), andthiswouldmakethe surveillancelesseffective. Duringthepre-processingphase,wethereforeselectcertain loca-tionstobetwo-sensornodesorlocationsduetotheirlargevisible regions. We considerthe cost (weight)vector ascT=
(
c1,...,cN
)
where ci=1 or2,∀
i, andN isthe number ofgrid points. Then, guardingtheregionwithminimumnumberofcamerasunderthe statedsurveillancepolicycanbemodeledbythe‘locationset cov-ering’formulationmentionedinSection 3 ,whoseinteger program-mingformulationisgivenbelow,Model(I): min
N j=1 c jy j (6) s.t. N j=1 a i jy j≥ 1,
∀
i =1, . . . , N (7) y j∈{
0, 1}
,∀
j =1, . . . , N (8) whereyj equals 1ifasensoris placedatsitej,and0otherwise. Thetermaij istheentryforthevisibilitymatrixAwithrowi cor-respondingtoa pointtobe coveredandcolumnj toapoint that isacandidateforsensorlocation.Thefirstconstraintensuresthat allgridpointsarecovered.Another problem of interest is maximizing the area covered withinanavailablebudget,i.e.,withafixednumberofguards. Re-gardingbordersecurity,directorsmaywishtoestablishtheir com-mandpostsatsiteswherethearea seenismaximized.The prob-lemof maximizing thearea covered bypguards isthe ‘maximal
Fig. 10. (a) Black regions are dead-zones that cannot be seen by the camera (b) Two cameras share the visible region equally. One camera is assigned the northwestern part and the other is assigned the southeastern part.
Fig. 11. There are two cameras on red points and one camera on green points in (a), (b) and (c). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
coverage location problem’, whose integer-programming formula-tionisgiveninthefollowing,
Model(II): max
N i=1 k i (9) s.t. k i≤ 1,
∀
i =1, . . . , N (10) k i≤ N j=1 a i jy j,∀
i =1, . . . , N (11) N i=1 c jy j= p (12) k i≥ 0,∀
i =1, . . . , N (13) y j∈{
0, 1}
,∀
j =1, . . . , N. (14)Thevariablekiequals 1iflocation iiscovered andis0 other-wise.Thevariableyjisasdefinedinmodel(I).Notethat kiisnot restrictedtobeabinaryvariable(0≤ ki≤ 1)since,atanoptimal solution,kimustbeeither0or1.
Thesetwoproblemswere solvedby CPLEXsolverusingGAMS softwarefortherealterrainincludingN=2000gridpoints.Model (I),thelocationsetcovering,prescribesatotalof83cameras. The problemissolvedin3.8seconds.Thelocationsofthecamerasare shown in Fig. 11 (a), (b) and (c). The red points in Fig. 11 have two camerasand thegreen ones haveonly one camera. For p=
1,...,83 the maximal coverage problemis solved for model(II); theresultsare presentedin Table 1 .The graphicalrepresentation of coverage versus the number of cameras (Fig. 12 ) shows that thecoverage ofthe regionincreaseswitheach additionalcamera butata decreasingrate,inlinewiththeprinciple ofdiminishing marginalreturnsfromeconomic theory.Theproblemissolved in
Table 1
Maximum number of points that can be covered with p cameras.
p points covered p points covered p points covered p points covered p points covered
1 49 11 1311 21 1643 31 1800 42 1894 2 347 12 1367 22 1663 32 1811 45 1912 3 394 13 1400 23 1684 33 1820 48 1928 4 692 14 1455 24 1703 34 1831 55 1959 5 729 15 1488 25 1724 35 1840 60 1975 6 943 16 1532 26 1740 36 1849 65 1986 7 980 17 1556 27 1753 37 1858 70 1992 8 1134 18 1580 28 1764 38 1865 75 1996 9 1170 19 1601 29 1777 39 1873 80 1998 10 1278 20 1622 30 1788 40 1880 83 2000
Fig. 12. Diminishing marginal returns obtained by using additional cameras.
lessthan4minutesforpsmallerthan 80.But,forp=80ittakes about150minutesandforp=83ittakes534minutestosolvethe problem.
Sixty camerascover1975 points, yettwenty-three more cam-eras are needed to cover the remaining 25 points. A decision-makermaynotwantto incurtheadditionalcost oftwenty-three cameras,whichcanbeashighas$300,000,tocoveronly25more points.Thelocationsofcamerasontheterrainfor p=1,4and16 areillustrated in Figs. 13 ,14 ,and 15 ,respectively. There are sec-tions of the terrain still unguarded as illustrated in Fig. 15 . We notethat the unguardedlocationsacrossthe widthoftheregion inFig. 15 createapathwhichillegalintruderscanusetocrossthe borderwithoutdetection.InSection 6 ,weproposeasolutionthat allowstoblockanysuchpathwithoutrequiringtocoverthewhole terrain.
Toinvestigatehowourapproachextendstolargerdatasetswe haveobtainedseveraladditionalterrain datasets fromtheUnited StatesGeologicalSurveywebsite.Wehavesolved TGPonfivereal terrains,whichhave4900,10,000,14,400,20,164,and30,276grid points(Fig. 16 ). An Intel (R) Core(TM) i5-7200 quad core CPU@ 2.50GHzand2.71 GHz PCwith 8.00GBRAM and64bit operat-ing system is used for creating the visibility matrix and solving location set covering problems. In Table 2 we present the solu-tiontimesobtainedbyILOGCPLEX12.8implementedinC++using CPLEXConcertTechnology,the time tocreatethe LOSmatrix for eachterrainobtainedbyMATLABR2018b,andthememorysizeof thecreatedLOSmatrixforeach dataset,separately.Wenote that whenthesizeofthedatasetincreasesby6.2times,from4900to 30,276, thesolution time for the optimizerto solve the problem increasesby30times,thetimetocreatetheLOSmatrixincreases by21 times,and thememory usedto thecreate theLOS matrix
increases by 38times (see Fig. 17 ). When we haveattempted to solve the problem on a dataset with40,000 pointsthe problem could not be solved due to the memory problems (the memory sizeofthecreatedvisibilitymatrixisover3GB).
Aswehaveobservedhere,thechallengesforlargerdatasetsare threefold: (i) the built-in limitations of the softwarepackages in handling large arraysof dataregardless of the available memory oncomputer;(ii)slowoptimizationduetothecomplexitytimeof integerprogrammingingeneral;and(iii)hardwarebarriers; even-thoughitisbecominglesssignificantinrecentyearswiththe de-velopedtechnologiesandtheirongoingfastgrowth,itisalways a practicalissuedependingontheusersandtheirequippedfacilities. We note that earlier studies such as [22,33,41] , have solved theTerrainGuardingProblem onterrainswithtensofmillionsof pointsinview ofthefact thatthesestudies consideronlya sub-setofavailablepointsaspotentialguardlocationsanduse heuris-ticalgorithms,whichrun inpolynomialtime, tolocateguardson theterrain.Althoughthesestudieshandlelargerdatasets,itisnot knownhowgoodtheirsolutioniswithrespecttotheoptimal so-lution.
Onthe other hand,thelargestterrain forwhichwe can solve theproblemhas30,276points,whichisrelativelysmallcompared to those studies mentioned above. However, the obtained solu-tion is guaranteed to havethe minimumcost asthe problem is solved to optimality. Considering that the cost of a singleguard can be as high asseveral thousand dollars, our studyillustrates tothedecision-makertheinherenttrade-off intheTerrain Guard-ingProblem:onecanuseheuristicstosolvemuchlargerproblems withoutanyinformationastohowcostly theirsolutionis or im-plement,aswe did,an exact algorithmto obtaina solutionwith an exact knowledge of the minimum cost but settle for smaller terrains.
Inaddition tothe othercontributions ofourstudystated ear-lier, to thebest ofour knowledge thispaperis thefirst to show the largest terrain size for which the problem can be solved to optimalitywithconventionalhardwareandoptimizationpackages, whichhelpsthereadertoassesswhenandifheuristicsisuseful. Ourresults alsodocument the relationshipbetween theproblem sizeandthecomputationtimeofthevisibilitymatrixandbetween the problem size and the memory required to store the visibil-itymatrix.Ourapproachalso offersalternative waystosolve the problem;itcanbeutilizedasaheuristicinwhichtheterrainof in-terestcanbedividedintosmallerregionswithupto30,000points, forwhich theTGP issolved to optimality, andthen combinethe solutions obtainedassuch toobtain asolution forthelarger ter-rainofinterest.
Fig. 13. The locations of the camera when p = 1 . Red points represent camera locations and green points represent points covered by the camera (a) top view (b) angled view. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 14. The locations of cameras when p = 4 . Red points represent camera locations and green points represent points covered by cameras (a) top view (b) angled view. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 15. The locations of cameras when p = 16 . Red points represent camera locations and green points represent points covered by cameras (a) top view (b) angled view. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
5. Sensitivityanalysis
As pixel size decreases, i.e., as the resolution increases, the numberofgrid pointsthatrepresentthe terrainincreases,which leadsto a better representationof theterrain, andthe resultsof the coveragemodel becomemorereliable. However,asthe num-berofgridpointsincreasesthecomputationtimeforcreatingthe visibility matrixincreases andcomputer memoryis not likely to suffice.Ontheotherhand,representingaterrainwithfewerpoints willresultinlessreliablesolutionsinthatmoreoftherealterrain islikelytobeunguarded.Inthissection,therearetwogoals:to in-vestigatethe effectofterraincharacteristicsandoftheresolution oftheterrainoncoverage optimization.Tothisend, fivefictitious square terrainsarecreatedwithequalnumbersofgridpointsbut thesteepness(standarddeviationsoftheheightsofthepoints)in theterrainsaredifferent(Figs. 18 and19 ).
Allterrainsarerepresentedby33× 33grids.Theheightofeach grid pointineach terrainwasgeneratedusinganormal distribu-tion witha meanof1000 musing MATLAB.The standard devia-tionsofheightsinterrain1throughterrain5are200m.,150m., 100m.,50m.,and10m.,respectively.Terrain1isthemostrugged terrain whileterrain 5 is the smoothest. The highest andlowest points interrain 1 have heightsof 1633.25 m and251.56 m, re-spectively.Interrain5,themaximumheightis1033.25mandthe minimumheightis962.77m.
Table 2
Computational time for obtaining visibility matrices ( A ) and solving Model (I) with the corresponding matrices, and their sizes on memory.
Size of the dataset Optimization Computation of A Memory size of (in minutes) (in minutes) A (KB)
4900 1.502 3.483 46,900 10,000 4.808 10.967 195,323 14,400 10.253 20.000 405,015 20,164 19.167 36.917 794,135 30,276 45.893 74.300 1,790,135 40,000 - 112.3 3,125,040
Letusassume thattheregionrepresentedwith33 × 33grid pointsisthebestrepresentation(‘true’representation)possiblein termsofa reasonablesolution timeandthe elevationdata avail-ability.Also,5 × 5,9 × 9,and17 × 17gridsareusedascoarse representations.The17 × 17gridrepresentationisobtainedfrom the33 × 33gridby deletingeven-numberedrowsandcolumns (Fig. 20 ).The9 × 9and5 × 5gridsareobtainedsimilarlyfrom 17 × 17and9 × 9,respectively.Withthisconstruction,apoint inacoarser representationwillalsobe apoint infiner represen-tations.Forexample,apointin5 × 5gridwillalsoexistinthe otherthreefinerrepresentations.Thus,wewillbeabletodiscover whatportionofthefinestrepresentation(33 × 33)iscoveredby aguardlocatedin,forexample,5 × 5representation.
Fig. 16. Real terrains chosen for computational test of Model (I).
Fig. 17. (a) Computational times for obtaining visibility matrices and optimizing the derived instances; (b) Memory sizes of the visibility matrices on computer.
Becauseweareinterestedonlyininvestigatingtheeffectofthe terraincharacteristicsandoftheresolutionoftheterrain on cov-erage,itisassumedthatallgridpointsarewithintherangeofthe guards,thereby eliminatingtheeffect,ifany,oftheguard’srange oncoverageresults.Thevisibilitymatrixwascalculatedandthe lo-cationsetcoveringmodelsweresolvedforeachrepresentationand foralltypes ofterrain.The minimumnumberof guardsforeach representationandthefivetypesofterrainispresentedinTable 3 . Theresultsindicate thatwhen one trades‘correct’ representation
forlesscomputational burden (33 × 33 vs. 5 × 5), one uses threeguardswhileinfact,sixty-fivemustbeusedtoguardTerrain 1,andone usestwoguardswhileinfact,sixty-twomustbeused toguardTerrain5.
Table 4 givesthepercentageofthe1089gridpoints(33 × 33) that are covered ifthe guardlocationsobtained forthe5 × 5, 9 × 9, and17 × 17representations –forall typesof terrain– are usedto coverthe 33 × 33 representation.Forexample,for Terrain1,ifthe threeguardlocations foundforthe 5 × 5
Grid representation Terrain 1 Terrain 2 Terrain 3 Terrain 4 Terrain 5
5 × 5 36.1 30.1 27.3 49.4 54
9 × 9 60.9 48.6 60 62.2 65.5
17 × 17 79.5 80.8 78 82.3 82.3
Fig. 20. The red points are selected from the 33 × 33 grid to obtain a 17 × 17 grid. A 9 × 9 and 5 × 5 grid is obtained similarly from 17 × 17 and 9 × 9 grids, respectively. (For interpretation of the references to colour in this figure leg- end, the reader is referred to the web version of this article.)
resentationareusedto coverthe33 × 33grid,only36%ofthe 33 × 33 representation is covered (see Table 4 ), which would leave 64%oftheterrain unguarded. The guardlocationsfoundin 9 × 9 and 17 × 17 cover 61% and 80% of the 33 × 33 grid,respectivelyforTerrain1.Theseresultssuggestthatwhenthe number of grid points is decreased by about 75% (17 × 17 vs 33 × 33), on average 20% of the coverage are lost for all ter-rains.Acleartrade-off is observedherebetweenthe’correct’ rep-resentation(accordingly thecomputational resources requiredfor thesolutioninlargerdatasets)andthe‘correct’numberofguards requiredtocovertheterrain.
The results in Tables 3 and 4 indicate that, considering the shapes ofall terrainsand approximate representations used,
ter-rains do not show much difference in terms of the number of guards neededor in terms of the percentageof the terrain cov-ered by the guards obtained from coarser resolutions. Although Terrain 5 is almost flat, it requires almost the same number of guardsasTerrain1forall representations(seeTable 3 ).These re-sultsmight seem counter intuitive atfirst asone expectsto use moreguardsformorerugged terrains.However,astheterrainsin Fig. 21 illustrate,onlytwoguardsare neededtocovertherugged terrain,whereas fourguards arerequired tocover theflatter ter-rain.Thekeyfactorindeterminingthenumberofguardstocover aterrain is not theshape ofthe terrain butthe viewshedof the pointsof the terrain. Obviously,building a relatively short tower onthesmoothterrain candecreasetheoptimalvalueto 1butas discussedinSection 4 ,itisnotdesirabletoinstalltowers.
To summarize; the number of guards needed to completely covera terrain, and alsothe coverage percentage ofa terrain by theguardsthat arefoundforterrainswithlowerresolutionsboth depend very muchon the topology ofthe terrain which is char-acterizedby the visibility matrix which,surprisingly, is indepen-dentofthesteepness(smooth/rugged) oftheterrain accordingto oursensitivityanalysiswithsimulateddata.So,the visibility ma-trixplaysthekeyroleinprescribingthepropersolution.
6. Blockingpathproblem
Aninfiltrationrouteistheonethatintrudersusetosneakinto thehomeland.Ablockingpathis,inasense,aline-of-defensethat blocksaninfiltrationroutewhichintrudersarelikelytouseto en-terthehomeland.Whenintruderspassthroughthepath,aguard
Fig. 22. White points are not guarded while green points are. Intruders may use unguarded locations to infiltrate into the homeland. (For interpretation of the ref- erences to colour in this figure legend, the reader is referred to the web version of this article.)
detectsthemandpreventivemeasuresaretaken. [3] findthebest infiltration route for an intruder. A possible infiltration route in thecontext of this studyconsists ofa connected path of uncov-eredpointsfrom one side of theregion to the other side across thewidthofthe region,inotherwords,fromtheneighbor coun-trytothehomeland(Fig. 22 ).TheBlockingPath Problem(BPP)is tofindtheminimumnumberofguardslocatedontheterrainsuch thatthereisablockingpath,onwhichall pointsonthepathare coveredbytheguards,thatextendsacrossthelengthoftheregion, andthus,anyinfiltrationrouteisblocked.Asimilarproblemisthe
k-Barrier CoverageProblem in which an infiltration route is cov-eredbyatleastksensors[30] .Butinthisproblem,itisassumed thatsensor(guard)locationsareinthebarrier(path)andthe sen-sorscancoveranypointwithintheirrange.Inourapproach, sen-sorscan be locatedanywhere on theterrain. Further,we assume asensorcancoverapointifboththepointiswithin thesensor’s rangeand the line-of-sight betweenthe sensor and the point is notblockedbytheterrain.Therefore,theproblemgivenin[30] is arestrictedversionofourproblem.
The Network Interdiction Problem (NIP) is another problem closelyrelatedtoBPP.InNIP,thereisanetworkwithasourcenode
s∗andsinknodet∗.Theenemy(follower)wishestomoveasmuch ofacommodityaspossiblefroms∗tot∗andisconstrainedbyarc capacities. As discussed in [43] the interdictor (the leader) tries todestroy arcsto prevent the flow ofcommodity butdestroying anarchasaknowncostandthereisabudget sothatdestroying all thearcs is not possible.[43] proposesa solution to this prob-lembysendingamaximumflowfroms∗tot∗andthendestroying thearcsthatcorrespondtotheminimumcapacitys∗− t∗cut. Con-siderarectangularregionofgridpointssimilarinshapetothegrid representationoftheborderregionweusedforreal-worldterrain guardingexample.TotransformourproblemintoNIP,wecan de-finea network such as theone in Fig. 23 . We assume that each gridpoint isanode inthedirectednetworkG=
(
N,A)
,withthe nodesetN andthearcsetA.Theregionconsistsofurowsandvcolumnssuchthatvisstrictlygreaterthanu.Letp(i,j)denotethe positionofthenode intheithrowand jthcolumnofthe region. Then, for each node in the network an arc can be created such that there is a directed arc from p(i, j) to p
(
i,j− 1)
, p(
i,j+1)
, p(
i− 1,j)
, p(
i− 1,j− 1)
and p(
i− 1,j+1)
, where they are de-fined.Theinfiltrationroute startsats∗andendsint∗.InNIP,the interdictortries to destroy some of these arcs such that the re-mainingarcshavetheminimumtotalcapacity.The main difference betweenNIP and BPP isthat the cost of destroyinganarcisknowninNIPasaprobleminputwhereasthe cost of creating a blocking arc is not known in BPP in advance. NotethatinBPP,insteadofdestroyingthearcsweblockallroutes froms∗tot∗.
Fig. 23. The network used to illustrate NIP.
Fig. 24. The network used to model a blocking path on the terrain.
Letususethesamenetwork butwitharcs definedsomewhat differently to be able to createa blockingpath (Fig. 24 ). Fori= 1,. . .,uandj=2,...,
v
,directedarcsarecreatedfromthenodeinp(i, j) to nodesin p
(
i,j+1)
, p(
i+1,j)
, p(
i+1,j+1)
, p(
i− 1,j)
andp
(
i− 1,j+1)
,incasethesenodesexist.Ifp(i,j)istheposition ofanode inthefirstcolumn, arcsaredirected onlytothenodes incolumn2,i.e.,to p(
i− 1,2)
, p(
i+1,2)
andp(i,2),wherethey aredefined.Eachnodeinpositionp(i,j)isindexedbytheformula((
j− 1)
× 4)
+ i.LetQandQ¯ betwo setsthat consist ofthegridpointsinthe firstcolumnandthelastcolumn,respectively.Apathfromanode inQtoanodeinQ¯,whichiscalledthe‘blockingpath’,issuchthat eachnodeonthepathiscoveredbyaguard(locatednot necessar-ilyonthepath).TheblockingpathinFig. 25 blocksanyinfiltration routethatstartsfromanodeinrowuandendsinanodeinrow
Fig. 25. A blocking path (painted red) blocks any infiltration route (two possible routes are painted green). (For interpretation of the references to colour in this fig- ure legend, the reader is referred to the web version of this article.)
Fig. 27. The blocking path is shown for a part of the region.
1.The BlockingPath Problemisto findablockingpathsuch that thepathextendsfromthefirstcolumnofthegridandendsinthe lastcolumn,andeachgridpointonthepathiscoveredbyaguard suchthatthetotalnumberofguards(cameras)isminimized.Ifan arc(i,j)isincludedinthepaththennodesiandjmustbecovered byaguardlocatedinanyoneofthegridpoints.Thus,thecostof coveringnodesiandj(i.e.,creatingtheblockingarc(i,j))depends onthe locationofguards thatcovereach ofiandj.Asdiscussed earlier, thisis in contrastto NIP, where the cost associated with eacharcisaprobleminput.
TosolveBPP, twoartificialnodesdenotedbysandtareadded to the network such that there is a directed arc from s to each nodeinQandadirectedarcfromeachnodeinQ¯tot(Fig. 26 );the previousnumberingofnodesarekeptafteraddingsandt.Withs
andtincluded,thetotalnumberofnodesandarcsbecome
|
N|
+2 and|
A|
+2×|
Q|
,respectively.Thebinaryvariablexij corresponds totheflowoneacharcasfollows,x i j=
1, ifarc
(
i, j)
is intheblockingpath 0, otherwise.Aspresentedinmodel(I)inSection 4 ,aijequals 1ifthenode (gridpoint)iiscoveredby aguardatnodej andis0,otherwise. The termcj,asbefore,denotes thenumberofguards(thecost of node j)that areneededtobeplaced atj,whichisdeterminedby the preprocessing described in Section 4 andtakes on values of either1or2.Thebinaryvariableyj is1ifaguardislocatedatj and0,otherwise.
TheformulationforBPPisgivenasfollows,
(BPP) min n k=1,k=s,t c ky k (15) s.t. j:(i, j)∈A x i j− j:(j,i)∈A x ji=
⎧
⎪
⎨
⎪
⎩
0, for i = s, t 1, for i = s −1, for i =t (16) x i j≤ k=s,t a iky k,∀
i, s.t. i, j = s, t (17) y j, x i j∈{
0, 1}
. (18)A unit flow is sent from s to t through the network. Eq. (16) givesthebalance-of-flowconstraintsforeachnode,which areverysimilartotheconstraintsintheshortestpathformulation givenin [1] . Note that a feasible flow satisfyingEq. (16) gives a pathfromstot,andtherefore,apathfromanodeinQtoanode inQ¯,asdesired.Eachnodeinthispathiscoveredbyaguardsince Eq. (17) ensures that if (i, j) is in the blocking path, i.e., xi j=1, then i is covered by a guard located at a node in the network. Aconstraint like Eq. (17) is not added forcovering node j when
xi j=1,i.e.,aconstraintoftype x i j≤
k=s,t
a jky k
∀
j s.t. i, j =s, t. (19) The reasonforthisis asfollows; since Eq. (16) requires that the incomingunitflowmustexitj,itmustholdthatxjq=1forsome node q in the network. Eq. (17) also applies to xjq and ensures thatj iscovered,whichshowsthat Eq. (19) isredundant. Remov-ing Eq. (19) fromthe modeleliminatesalmost 11,000 constraints in our real-world problem and helpssolver to obtain a solution ina reasonableamountof time (almost20hrs with 16threads). Wenotethat asolvetime ofthislength isquitereasonablesince theproblemisstaticinnatureandthereforedoesnotneedtobe solvedonafrequentbasis.BPPformulationis appliedtoourrealterrain representingthe borderregionbetweentwocountries.Theproblemissolvedusing CPLEX12.8anditis revealedthat seventeencamerasareneeded tocreatetheblockingpathintheregion(Fig. 27 ).
7. Directionsforfutureresearch
IntheTerrainGuardingProbleminvestigatedinthispaper,the goalwastodetectanyincomingmovement.Tothatend, oncewe
solved theLSCPor BPP, theguard locationswere set asfixed. In asense,ourproblemisstatic. Now,supposeinadditionto detec-tion, we also want to trackthe intruders within the region and apprehend them before they leave the border region. This prob-lem is rather dynamic. In this new version of the problem, the guardsmust be directed to a location where intrudersare likely to go through such that the guards intercept the intruders. The determination of the locations of security units that will catch the intruders, or a certain percent of them, and the determina-tion of the optimal routes of these units are topics for further research.
Besides, in most real cases the intruders are likely to prefer some routes, such as river beds, over others for infiltration due tosupposedlylowerprobabilitiesofbeingdetectedinsuchroutes. Whenthisisthecase,insteadofcreatingablockingpath extend-ingalong the region,only those pointsoncritical routesmaybe covered.Further,insuchrouteswhereintrudersaremostlikelyto use,the resolutionof the terrain maybe increasedto better ap-proximatetheterrain,andthus,toobtainsolutionsthatare more realisticandthusreliable.
An interesting inquiry,extendingthe briefdiscussionprovided inthispaper,iswhetherRSGorTINisabetterapproximationofa realterrain.Anexperimentmaybedesignedthatinvolvesreal ter-rainstoinvestigatethisresearchquestion,inwhichtheelevations ofsufficientlymanypointschosenrandomly fromtherealterrain may be compared to the corresponding elevations estimated by eachDEM to give an indicationofwhich DEM providesthe least error.Comparingthe curvatureoftherealsurfaceandthe curva-tureoftheestimatedsurfacesonselectedregions mightalso pro-ducemeaningfulresults.
Finally,asdiscussedearlier,noFDShasbeenfoundforTGP ei-theron TINsoronRSGs.Withthelackofan FDS,an optimal so-lutionobtainedfrom a setof specific locations(either the set of verticesand/or edges onTIN ortheset ofgridpointson RSG) is onlyanapproximation totheproblem. Thus,futureresearch may focusonfindingFDS’sforbothterraintypes.
CRediTauthorshipcontributionstatement
Haluk Eli¸s: Conceptualization, Methodology, Formal analysis, Investigation,Software,Resources,Writing-originaldraft,Writing - review & editing.Barbaros Tansel: Conceptualization, Method-ology, Supervision, Writing - review & editing. Osman O˘guz:
Methodology, Supervision, Writing - review & editing. Mesut Güney:Software,Validation, Visualization.RamezKian: Software, Validation,Visualization.
Acknowledgements
Wewouldliketothanktheeditorsandtheanonymousreferees fortheir insightfulcomments, which helped improvegreatly the qualityofthispaper.WewouldalsoliketothankMr.EkremUçar, Mr.MustafaErdo˘gan,Mr.BahadrAktu˘g,Mr.OkanOnarım,andMr. Mehmet Simav for providing the terrain data and fortheir help withtheuseofGIS.
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