ABSTRACT B Optimal Unmanned Vehicle Allocation for ...faculty.nps.edu/emcrapar/docs/Volkan_19.pdfVehicle Allocation for Border Surveillance Volkan S¨ozen Turkish Army [email protected]

ABSTRACT

Border surveillance is an importantconcern for nations wishing to detectand intercept intruders. Unmanned

aerial vehicles (UAVs) allow for moderniza-tion of border surveillance efforts, improv-ing performance while reducing cost. WhileUAVs carry a number of advantages overtraditional means of border surveillance,they also present new operational chal-lenges. This paper formulates mathematicalmodels designed to find the best way toutilize a given fleet of UAVs by decidingtheir routes, altitudes, and speeds in order tomaximize the probability of detecting in-truders. These models can aid a decisionmaker in effectively acquiring and employ-ing a UAV fleet for border surveillance.

INTRODUCTIONThe problem of defending a territory

against intruders has existed for millennia,and modern nations face no shortage of po-tential intruders. These intruders caninclude terrorists, drug traffickers, smug-glers, illegal immigrants, and others whorepresent a threat to national interests. Afirst step toward preventing intrusion isdetecting potential intruders. While theproblem of detecting intruders is not new,the ecosystem of threats and countermea-sures continues to expand and evolve. Un-manned aerial vehicles (UAVs) representone emerging tool in this domain. Avarietyof advantages motivate the use of UAVs forborder surveillance: many UAVs can becontrolled by a single operator, reducingpersonnel costs; they are very fast and canpatrol large regions; and they have widerregions of view than conventional surveil-lance methods, thus potentially increasingthe probability of detecting intruders.UAVs’ mobile nature means that they canbe deployedmore flexibly and reconfiguredmore easily that many other forms of bordersurveillance, and it can also give thema stealth advantage. These advantages andothers have made UAVs an important com-plement to human expertise in the bordercontrol domain (Preston, 2014). As notedby Haddah and Gertler (2010), there aretwo types of UAVs: autonomous dronesand remotely piloted vehicles (RPVs). Bothtypes are unmanned, but drones are

preprogrammed for their flight andmission,whereas RPVs are actively controlled by anoperator at the ground station. Both aregood candidates for border surveillance,but both need good planning in order torealize their full potential.

This paper gives guidance to UAVoper-ators and mission planners on how to opti-mally employ their UAVs while conductingsurveillance on a border in order to maxi-mize the probability of detecting intruders.Throughout the paper, we refer to UAVsas searchers and intruders as targets.

This paper is organized as follows. Wefirst review a selection of the small but grow-ing literature on the use of UAVs for bordersurveillance. We then consider a fundamen-tal border patrol problem referred to as the‘‘barrier patrol problem’’ by Wagner et al.(1999) and ‘‘patrollinga channel’’ byWashburn(2002). We derive a more accurate detectionprobability formula than currently exists inthe literature and validate our formula usingMonte Carlo simulation. Building upon thisresult, we expand our analysis to includemultiple searchers. We determine the bestway to allocate a border amongmultiple het-erogeneous searchers, and we study thechange in optimal allocation depending onthe differences in the searchers’ characteris-tics. Finally, we demonstrate a methodologyfor determining the optimal way to operatemultiple searchers when the performance ofthe searchers’ sensors degrades with increas-ing searcher speed. This methodology calcu-lates both the allocation and speeds requiredto achieve the maximum detection probability.

BRIEF LITERATURE REVIEWAlthoughUAVs are relative newcomers

to the border surveillance domain, efforts toanalyze and optimize their operations pre-date their actual deployment. For example,Girard et al. (2004) propose a hierarchicalcontrol architecture for UAV teams, whileMatveev et al. (2011) develop a guidance-control law to allowUAVwith afixed camerato follow and monitor a border in the pres-ence of obstacles. Szechtman et al. (2007)compare the performance of a UAV to a sta-tionary sweeping camera against intruderswho appear according to a Poisson processin locations determined by a specified distri-bution.Ozcan (2013) takes a simulation-basedapproach to examine the effectiveness of

OptimalUnmannedAerialVehicleAllocationfor BorderSurveillance

Volkan Sozen

Turkish [email protected]

E. M. Craparo

Naval Postgraduate [email protected]

APPLICATION AREAS:Unmanned systems,Homeland defense andcivil support

OR METHODOLOGIES:Optimization,Simulation

Military Operations Research, V21 N3 2016, doi 10.5711/1082598321319 Page 19Military Operations Research, V21 N3 2016, doi 10.5711/1082598321319 Page 19

UAVs in a particular scenario. After performinga sensitivity analysis on various parameters, sheconcludes that the UAV’s detection and classifi-cation performance as well as the target’scounter-detection capabilities are the most im-portant factors impacting detection probability.

A body of work also examines various sensormodels, which are not the focus of this paper butare an important component of the search and de-tection problem. For example, Soza & Company(1996), Wagner et al. (1999), Washburn (2002),Sozen (2014), and Haddah and Gertler (2010)study lateral range curves and possible approxi-mations of them. Additionally, Wagner et al.(1999) explains how lateral range curves aredetermined for a particular sensor. They per-form preliminary analyses on the approxima-tions, mentioning the differences between theactual sensor model and its approximations.

THE BORDER PATROL PROBLEMWe consider the basic problem of monitoring

a straight-line border over flat terrain without anyline-of-sight issues. Our hypothetical border isbounded by two barriers that are perpendicularto the border. These barriers may represent actual(physical) barriers or imaginary barriers denotinga region of interest. We have one or more searchersavailable and would like to employ them in sucha way as to maximize our likelihood of detectingan entity attempting to cross the border; this entityis henceforth referred to as ‘‘the target.’’ We as-sume that the target moves with constant speedperpendicular to and toward the border, whilethe searchers patrol the border by moving backand forth at constant speed, reversing direction in-stantaneously at the endpoints of their patrol re-gions. For simplicity, we assume that the targetcannot see the searchers and takes no action toevade them. Future work may consider an intelli-gent target that candetect and evade the searchers.

Border Patrol with a SingleSearcher

As a first step toward optimally deployingone or more searchers, we first determine theprobability of detecting the target with a singledeployed searcher. We assume that the searcher

is equipped with a sensor capable of detectingthe target within a finite radius. For simplicity,we consider a definite range or ‘‘cookie-cutter’’sensor, meaning that if the distance betweenthe target and the searcher is less than the detec-tion radius, detection occurs with probability 1;otherwise, detection does not occur. Such a sen-sor represents an approximation of, for instance,a vision-based sensor operating in the absence ofobstacles. Other sensor models are possible, anda similar analysis could be performed for sensorsthat are not oriented in a downward-facing pos-ture, or sensors for which a definite range ap-proximation is not appropriate.

Figure 1 illustrates our problem setup. Thelength of the border is L units, and the searchermoves back and forth along the border with con-stant speed v. The searcher’s detection radius isR, and we assume that this radius dictates thesearcher’s trajectory along the border: when thesearcher is at a distance of R units away from ei-ther edge of the border, it turns andmoves in theopposite direction. A target attempts to passthrough the border at constant speed u, and ifthe target’s position is within distance R of thesearcher’s position at any point in time, the targetis detected. Otherwise, it is not detected.

Analytical ModelIn order to calculate the probability that the

searcher will detect the target, we utilize target-stationary geometry. Target stationary geometry

Figure 1. Simple border patrol problem setup.

OPTIMAL UNMANNED AERIAL VEHICLE ALLOCATION FOR BORDER SURVEILLANCE

Page 20 Military Operations Research, V21 N3 2016

uses a coordinate system that moves with thetarget rather than a stationary coordinate sys-tem (Eagle, 2013). To use target-stationary ge-ometry, we simply add a vector –u to everyspeed vector in our problem. That is, we adda vector with the same magnitude as the tar-get’s speed vector, but in the opposite direc-tion. After performing the reference geometrytransformation, the transformed speed of thetarget (ue ) is 0, and the transformed speed ofthe searcher (ve) changes depending on the di-rection of the searcher’s movement. In termsof our problem setup shown in Figure 1, if thesearcher is moving to the right, its correspond-ing speed vector will be the vector shown inFigure 2(a). Likewise, if the searcher is movingto the left, its speed vector will change asshown in Figure 2(b).

Following this coordinate transformation,the border patrol problem shown in Figure 1can be visualized as shown in Figure 3(a). Inthis figure, the searcher follows the dashed-dotted lines according to the speed vectorsshown in Figure 2, and its detection radius is in-dicated by solid lines. Two targets, depicted assolid dots, are stationary. The searcher moves inthe infinitely long region bounded by the twobarriers. Thus, when using target-stationary ge-ometry, the border patrol problem is transformedinto a channel search problem in which thesearcher looks for stationary targets in the infi-nitely long channel bounded by the barriers.

Figure 3(b) shows the detection region inthe target-stationary case. If a target is in theshaded region, it is detected. Likewise, if it isnot in the shaded region, it is not detected. Inthis example, target 2 from Figure 3(a) is de-tected, and target 1 is not.

Assuming a uniform target density, theprobability of detection can be calculated as

the ratio of the shaded area in the infinitely longchannel to the area of the channel itself. As Fig-ure 3(b) indicates, the shaded area follows a con-sistent pattern. In particular, between each ofthe searcher’s turning points, the areas of theshaded regions are equal to each other. More-over, the vertical distance between any two con-secutive turning points is the same. Thus, wecan calculate the detection probability by con-sidering a region like that shown in Figure 4(a), which is simply the region between two con-secutive turning points. We can find the proba-bility of detection by computing the ratio ofthe shaded area to the area of the outlined rect-angle in Figure 4(a).

We calculate the area of the shaded regionin Figure 4(a) by dividing it into separate re-gions as shown in Figure 4(b). The area ofthe shaded region inside the rectangle in Fig-ure 4(b) can be found by calculating the areasof the two wedge-shaped regions on theends of the region and the inner rectangle(shaded areas). By adding these two areas and

Figure 2. Transformed speed of the searcher in target-stationary geometry.

Figure 3. The border patrol problem in target-stationary geometry.

Figure 4. Area of coverage between two searcherturning points in target-stationary geometry.


Military Operations Research, V21 N3 2016 Page 21Military Operations Research, V21 N3 2016 Page 21

subtracting the shaded regions outside of therectangle, we obtain the area of the originalshaded region.

Washburn (2002) determined an upperbound on this area by calculating the areashown in Figure 5(a). Wagner et al. (1999) ar-rived at a different approximation of this areaby removing the regions lying outside the rect-angle, as shown in Figure 5(b).

Using his approximation, Washburn (2002)obtains the following upper bound on the detec-tion probability Pd:

Pd#min 1;2R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv21 u

2pLu

( ): (1)

Likewise, Wagner et al. (1999) approximate thedetection probability as:

Pd 512

L

R2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv

u

� �21 1

r2 1

!2R2

LðL2 2RÞ ifRv# uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL L2 2Rð Þp

1 otherwise:

8><>: (2)

By calculating the area of the shaded regionexactly, we calculate the detection probabilitywithout any approximations. The width w ofthe rectangles in Figure 4 is the vertical distancethe searcher travels between turns in target-stationary geometry, or, equivalently, the dis-tance the target travels between the searcher’sturns in the original coordinate system. Hence,

w5 uL2 2R

v5 ðL2 2RÞu

v: (3)

Depending on the width w, we may havedifferent shapes of the areas to be calculated.Figure 6(a) shows the geometry for large w,i.e., w . Rcos(u), where u is the angle whosetangent is the ratio of the target’s speed to thesearcher’s speed (see Figure 2). Figure 6(b)shows the geometry for small w, i.e., w #Rcos(u). To aid our calculations in this case,we introduce angle a, where a ¼ arcsin(w/R).

For largew, we can find the area of rectangleEFGH in Figure 6(a) by multiplying its width(EH) by its length (EF, which equals IK). Itswidth is 2R, and its length isffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

w2 1 ðL2 2RÞ2

q;

thus, its area is

2R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw

2 1 ðL2 2RÞ2q

:

The area of the two wedge-shaped regions is

2 pR2

p22 u

2p

0@ 1A5R2 p

22 u

� �:

The area of the triangular areas (e.g., triangle HIJin Figure 6(a)) is 2(xR/2) ¼ xR ¼ Rcot(u)R ¼R2 cot u.

By combining these areas, we compute theprobability of detection as

2Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiw

2 1 ðL2 2RÞ2q

2R2cotu1R

2 p22 u

� �wL

:

(4)

For the case when w # Rcos(u), it is sim-plest to compute the areas ADE and BCG(which are equal by symmetry) and subtractthese areas from the area of the rectangleABCD. The area ADE can be computed bycalculating the area of the trapezoid AEFDand subtracting the wedge-shaped area DEFfrom it.

The area of trapezoid AEFD is (2R – Rcos a)w/2. The area of semicircle DEF ispR2(a/2p) ¼R2(a/2). By using these, we cal-culate the area of region ADE as (2R – R cosa)w/2 – R2(a/2).

Figure 5. Approximations of the area of coveragefrom (a) Washburn (2002) and (b) Wagner et al. (1999).



Combining these areas, we compute theprobability of detection as

wL2 22R2Rcosa

2 w2R

2a

2

� �wL

5 12ð2R2RcosaÞ w2R

2a

wL: (5)

Finally, in order to obtain the probability ofdetection for the simple border patrol problem,we combine Equations (4) and (5). Furthermore,we express each of the terms w, a, and u as ex-pressions of the original parameters of the prob-lem setup: the border length L, detection radiusR, target speed u, and searcher speed v. The re-sult appears in Equation (6).

Pd 5

2R

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv

u

� �21 1

r1R2v

p22 arctan

u

v

� �2

v

u

� �ðL2 2RÞuL if

Rv2

ðL2 2RÞuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 1 v

2p , 1

11R2v arcsin

ðL2 2RÞuRv

� �ðL2 2RÞuL 2

2R

L1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2v2 2 ðL2 2RÞ2u2

qLv

otherwise:

8>>>>>><>>>>>>:(6)

Figure 7 compares the detection probabili-ties given by Equations (1), (2), and (6) for vari-ous searcher speeds. In this figure, the border

length L is 50 distance units, the detection ra-dius R is 6 distance units, and the speed of thetarget u is 5 speed units (distance units/time

Figure 6. Qualitatively different geometries result from (a) large w and (b) small w. We handle each case separately.



unit). We observe that for these parametervalues, the three formulas give nearly the sameresult for low searcher speeds. The upper boundWashburn (2002) obtained starts to differslightly when the searcher speed is approxi-mately twice the speed of the target. The for-mula of Wagner et al. (1999) starts to differwhen the searcher’s speed exceeds four timesthe target’s speed. We notice from Equations(1), (2), and (6) that increasing the accuracy ofthe formulations increases their complexity.For simplicity, one can choose the most appro-priate formula to use depending on the charac-teristics of a particular problem. However, it isnoteworthy that both of the approximationsprovide optimistic estimates of the detectionprobability for these particular parametervalues; this may be undesirable in practice.

Monte Carlo SimulationWe now examine the border patrol problem

by means of a Monte Carlo simulation model.Our purpose here is twofold: first, we wish tovalidate our analytical formula. Second, we es-tablish a baseline simulation model that we willaugment in later sections. We use MATLAB(2012b) for all simulations. Our initial simula-tion setup is as follows:

1. The searcher’s initial position is at a dis-tance of R units from the left edge of the

border, and its initial direction of move-ment is to the right.

2. We use a time-step model for our simulation.In the time-step model, we calculate the posi-tions of the searcher and the target and makenecessary computations to see if the targetis detected after each time step increment.This temporal discretization introduces asmall error resulting from the fact that detec-tion may not occur at any time step, but oc-curs between the time steps. We choose the

time step as �t5R

25ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 1 v2

p ; this time step

reduces the error below 6.7 3 1023 percent.Sozen (2014) explains how the time-step ischosen and the calculation for the error. Weuse the same time step throughout the time-step simulations in this paper.

3. The simulation is run until the searchermakesa full cycle, i.e., comes back to its initial posi-tion and initial direction of travel. This simula-tion end time is referred to as tmax.

4. Targets are generated randomly using a two-dimensional uniform distribution over a rect-angle between the two barriers. Each target’sinitial vertical position is randomized so as toensure that it passes through the horizontalaxis of the searcher’s movement before theend of the simulation. The targets’ initial hor-izontal position is between 0 and L.

5. We simulate the movement of n targets andrecord the number of targets detected as k.

Figure 7. Comparison of detection probability formulas of Washburn, Wagner et al., and this paper.



This approach generates the same results asgenerating one target and running n differentsimulations, however, the simulation run-time improves considerably. We then useMATLAB’s ‘‘binofit’’ function to fit theMonte Carlo simulation’s results to a bino-mial distribution, and we compute the esti-mated probability of detection along withits 95 percent confidence interval.

Figures 8 and 9 show the probability of de-tection versus the speed of the searcher whenthe border length L is 200 distance units, the de-tection radius is 6 distance units, and the speedof the target u is 5 speed units. Figure 8 showsthe results of the Monte Carlo simulation with1,000 replications, and Figure 9 shows the re-sults of the Monte Carlo simulation with 1 mil-lion replications. In both figures, the solid lineshows the probability of detection obtainedfrom Equation (6). The dashed line shows theestimated probability of detection from theMonte Carlo simulation, and the dotted linesshow the upper and lower 95 percent confi-dence interval of the estimated probability ofdetection.

From Figure 9 we can see that the estimatedprobability of detection obtained from theMonte Carlo matches the probability of detec-tion obtained from Equation (6) very closely.

Recall that we assumewhen the searcher isR distance units away from any edge of theborder, it turns in the opposite direction. Al-though all of the other attributes of the prob-lem setup may be considered as fixed inputs,

the turning distance is clearly within the con-trol of the operator. Sozen (2014) studies theturning distance for 10 scenarios. In all 10 sce-narios, the estimated detection probability bPd

is either insensitive to the turning distance,or it is maximized or nearly maximized at thedetection radius R. Based on these results, wefix the turning distance to R for the remainderof this paper.

BORDER PATROL WITH MULTIPLESEARCHERS

We now build upon the results of the previ-ous section to study the problem of optimallyemploying multiple heterogeneous searchers.Our goal is to provide fundamental insights tooperators of border patrol UAVs, as well as tooutline a basic procedure by which an operatorcould perform an analysis that is suitable to hisor her needs.

First, we consider the problem of optimallyallocating a border among multiple searchers. Iftwo searchers are available, the operatormay ei-ther employ both searchers to patrol the entireborder, or the operator may divide the borderinto two disjoint segments and assign eachsearcher to patrol one segment. We refer to thefirst option as the ‘‘common path’’ allocationand the second option as the ‘‘disjoint path’’allocation.

In the following subsections we examinethe probability of detection when using the dis-joint and common path allocations. For simplic-ity, we consider only two searchers in bothcases, but a similar methodology can beemployed when more than two searchers areavailable.

Disjoint PathFigure 10 shows the border patrol problem

with two searchers and one target. Thesearchers allocate the border into two disjointregions with lengths L1 and L2 ¼ L – L1.

Analytical Solution. The disjoint path problem inFigure 10 decomposes into two separate borderpatrol problemswith single searchers as studiedin the previous section. Based on this insight, we

Figure 8. Comparison of the detection probabilityestimated from a Monte Carlo simulation with 1,000replications and from Equation (6).



compute the probability of detection using thelaw of total probability:

Pd 5Pds1Ptr1 1Pds2Ptr2 (7)

where Pdsi is the conditional probability thatsearcher i detects the target, given the target isin its region, and Ptri is the probability that thetarget is in the region of searcher i.

We can compute Pdsi by substituting the ap-propriate values for searcher i into Equation (6):

Pdsi 5PdðLi;Ri; u; viÞ: (8)

Since we assume that the horizontal po-sition of the target is uniformly distributed

along the border, we have Ptri ¼ Li/L. Thus,we have

Pd 5Pds1L1L1Pds2

L2L

5PdðL1;R1; u; v1ÞL1L1PdðL2;R2; u; v2ÞL2

L:

(9)

In general, with n searchers and one target,we have

Pd 5Xni5 1

PdsiLiL5

1

L

Xni5 1

PdðLi;Ri;u; viÞLi: (10)

Monte Carlo Simulation. We also evaluate the dis-joint path problem by means of a Monte Carlosimulation. The following setup applies to oursimulation:

1. The searchers’ initial positions depend ontheir detection radii and their allocated re-gions. The initial position of the ith searcheris distance Ri away from the edge of its allo-cated region. Its direction of movement is to-ward the opposite edge, and its turningdistance is Ri distance units from the end ofits allocated region. If Ri $ Li/2 for anysearcher, the searcher remains stationary at

Figure 9. Comparison of the detection probability estimated from aMonte Carlo simulation with 1 million rep-lications and from Equation (6).

Figure 10. Problem setup for the disjoint pathallocation.



the midpoint of its allocated region. (In prac-tice, a fixed-wing UAV could not remaintruly stationary and would need to executea loiter pattern. It may be possible for a rotary-wing aircraft or lighter-than-air UAV to remainstationary.)

2. Recall that in the single searcher case, we runour simulation until the searcher arrives at itsinitial starting position. Depending on the al-located regions and characteristics of thesearchers, such a policy may result in a verylong runtime in the disjoint path case. Thus,we limit our simulation end time to 25 timesthe maximum time required for any searcherto make a full cycle in its allocated region.Sozen (2014) explains the rationale for thischoice of multiplier.

3. We generate targets and calculate detectionprobabilities as in the previous section.

Figure 11 shows the probability of detectionPd as calculated analytically using Equation (9),as well as the estimated valuebPd as determinedvia Monte Carlo simulation with one millionreplications. Detection probability is expressedas a function of the ratio of the border length al-located to the first searcher, L1, to the total bor-der length L. This ratio is varied from 0 to 1with 0.01 increments. In this specific case L is

200 distance units, detection radii R1 and R2

are 12 and 6 distance units, respectively; the tar-get speed u is 5 speed units, and the speeds ofthe searchers, v1 and v2, are each 20 speed units.

Comparing the analytical solution to theMonte Carlo simulation, we see a discrepancyat very high or low values of L1/L. When L1/Lis very large or very small, this means that theregion allocated to one of the searchers is verysmall. Figure 12 shows the case when the regionallocated to the first searcher is very small. Inthis case we have R1 . L1/2, so the firstsearcher remains in the middle of its allocatedregion and covers some portion of the area allo-cated to the second searcher. This extra region isshaded with horizontal stripes in Figure 12 andis not accounted for in the analytical solution.Because of this extra region, the Monte Carlosimulation produces a higher detection proba-bility value than the analytical solution whenthe allocation to any searcher is less than twotimes the detection radius of that searcher.

Common PathFigure 13 shows the border patrol problem

with two searchers sharing the same path andattempting to detect a single target. Although

Figure 11. Detection probability utilizing a disjoint path allocation.



the searchers share the same path, their turningpoints may differ due to their differing detec-tion radii. For clarity, we introduced a slight ver-tical displacement in the paths that thesearchers follow, although in reality we assumethat there is no vertical displacement.

The common path allocation is difficult toevaluate analytically; thus, we utilize MonteCarlo simulation to analyze it. Our simulationin this section is based on the Monte Carlo sim-ulation for the disjoint path case, with modifica-tions to the searchers’ initial positions and theirturning points.

The initial starting point of the searcherscould be chosen to be the same point, i.e., theycould initially start their movement from ex-actly the same location. However, if their detec-tion radii and speeds were identical, then theywould move together and would perform nobetter than a single searcher. For this reason,

we choose to initialize searcher i’s location asL(i – 1)/n 1 Ri, where n is the number ofsearchers. The searchers’ initial direction ofmovement is toward the right. As before, eachsearcher’s turning distance is simply its detec-tion radius. Figure 14 shows our initial setupfor the common path simulations; dashed linesshow the turning points of the searchers.

The simulation is run until all searcherscome back to their initial starting point. In orderto reduce the runtime, the simulation end timeis limited to 100 times the maximum time ittakes for any searcher to make a full cycle.

We perform one million Monte Carlo simu-lations for the common path case with the samesearcher and target variable settings we consid-ered in the disjoint path problem (Figure 11).We obtain an estimate and 95 percent confidenceinterval for the probability of detection. This esti-mate is shown in Figure 15, alongwith the resultspreviously obtained for the disjoint path case.(Due to the large number of replications, the con-fidence interval is difficult to detect in the figure.)

Since there is no allocation of borders in thecommon path problem, the estimated probabil-ity of detection is constant. It is plotted over theresults from Figure 11 in order to provide a com-parison between the disjoint and common pathallocations.

Figure 15 shows that the common path de-tection probability is higher than the disjointpath probability for nearly half of the L1/Lvalues. This means that if we do not allocatethe border to the two searchers properly forthe disjoint path case, we may end up with

Figure 12. Disjoint path extreme case.

Figure 13. Problem setup for the common pathallocation.

Figure 14. Searchers’ initial positions and move-ment vectors in the common path simulations.



a worse probability of detection than we wouldobtain without allocating the region at all. How-ever, by intelligently allocating the border, weare able to obtain a higher detection probabilitythan is possible with the common path alloca-tion. We now consider the problem of optimallyallocating the border so as to maximize detec-tion probability.

OPTIMAL ALLOCATION OF REGIONSIn this section we wish to determine an op-

timal allocation of a border between twosearchers. Although we consider only twosearchers, our general approach can be ex-tended to accommodate n . 2 searchers.

In the optimal allocation problem, our goalis to determine L1 and L2 such that the overallprobability of detection is maximized. That is,given values for L, R1, R2, u, v1, and v2, we wishto determine L1 and L2 such that L2 ¼ L – L1 andthe probability of detection is maximized:

max PdðL1;R1; u; v1ÞL1L1PdðL2 L1;R2; u; v2ÞL2 L1

LL1 st 0# L1 # L

(11)

This is a constrained optimization problemwithone decision variable and a convex feasible re-gion. The concavity of the objective function,

given by Equation (9), is difficult to evaluate an-alytically. Sozen (2014) performed an extensivecomputational evaluation andwas unable to finda counterexample showing that the objective func-tion is nonconcave, thus raising the possibilityof finding a globally optimal solution by locat-ing a stationary point of the objective function.However, other good methods for solving theabove optimization problem include conduct-ing a line search on L1 and evaluating potentialallocations by means of a simulation model.

Monte Carlo SimulationWe now perform a Monte Carlo simulation

study to compare various border allocations toa common path approach for a variety of scenar-ios. Specifically, we vary the detection radii andspeeds of the searcherswhile fixing the length ofthe border to 200 distance units and the speed ofthe target to 5 speed units. In each scenario, wevary the allocation of border to the first searcherin 1 percent increments, each with one millionreplications, in order to observe the change inthe probability of detection. We then comparethis detection probability to the detection prob-ability obtained by a common path solution.

Figure 16 illustrates the outcome of six rep-resentative setups. Each subfigure in Figure 16is generated with the same logic as Figure 15,

Figure 15. Common path Monte Carlo simulation results.



Figure 16. Monte Carlo simulation results for various searcher characteristics.



where the subfigures show the probability of de-tection versus the fraction of the border alloca-tion to the first searcher. The dashed line showsthe analytical disjoint path detection probability(Equation (9)), the solid line shows the estimatedprobability of detection for the disjoint path caseobtained from the Monte Carlo simulation, andthe dashed-dotted line shows the estimatedprobability of detection for the common pathcase obtained from the Monte Carlo simulation.We also plotted 95 percent confidence intervalsfor the estimated probabilities of the commonpath and disjoint path cases, but due to the largenumber of replications, they are difficult to detectin the figures.

In Figure 16(a) we see that the optimal allo-cation to the first searcher is very low since itsdetection radius is inferior to that of the secondsearcher, while their speeds are equal to eachother. It is also interesting to note that the opti-mal detection probability is very close to thecommon path probability of detection. This oc-curs because the optimal allocation to the sec-ond searcher is nearly the entire border.

In Figure 16(b), the first searcher’s capabil-ities are still inferior to those of the secondsearcher, but not as much as in Figure 16(a).Thus, more of the border is allocated to the firstsearcher. The difference between the commonpath probability and optimal allocation proba-bility has also increased. Further increasingthe speed of the first searcher (Figure 16(c)), re-sults in a further increase in its allocation, andthe gap between the common path and optimalallocation probability widens.

When we increase the speed of the firstsearcher such that the properties of both searchersare equal, we expect to have the same border allo-cation to both searchers. We might also expect tosee a larger gap between the common path andoptimal allocation probabilities compared to theprevious cases. As Figure 16(d) indicates, our firstexpectation is correct, but our second is not. Thisoccurs mainly because of improved performancein the common path case. In particular, when thetwo searchers have the same characteristics, theycan operate in a cyclical fashion. Specifically, thetwo searchers cross each other at the same placeevery time they transit the border. These crossingsoccur when they are moving in opposite direc-tions, causing them to have relatively brief

overlap periods. Shorter crossing times meansless coincident area covered by the searchers,resulting in a smaller gap between the commonpath and optimal allocation probabilities.

In Figure 16(e) and Figure 16(f) one searcheris inferior, but by a smaller margin than in Fig-ure 16(a) though Figure 16(c). The differencesbetween the common path and optimal alloca-tion probabilities are larger here than the otherfour scenarios.

In general, we observe that if one of thesearchers is superior to the other, the allocationof the border to the inferior searcher is low, andthe difference between the common path andoptimal allocation probabilities is lower thanone might expect. Decreasing the difference inthe capabilities of the searchers results ina higher allocation to the inferior searcher andcan widen the gap between the common pathand optimal allocation probabilities. However,due to the perfect coordination between thesearchers when the searchers have the samecharacteristics, the gap between the commonpath and optimal allocation probabilities canbe low in some cases.

Based on our results, we conclude that thebest possible detection probability is obtainedby an optimal implementation of the disjointpath approach. However, if a user cannot deter-mine the optimal allocation, or if the character-istics of the searchers are unknown or subjectto change, the user may choose the commonpath approach in order to avoid the risk ofchoosing a poor allocation. If the common pathis utilized, the UAVs should be employed insuch away as to reduce the timewhen they passor cross each other to further increase the detec-tion probability.

Varying Detection Radius asa Function of Speed

In practice, it is likely that the detection ca-pability of a searcher depends on its speed. Forexample, with a vision-based sensor, the qual-ity of the video would be reduced if the UAVwere to travel at a very fast speed. In this sub-section, we model this degradation in sensorquality as a reduction in detection radius. Wenow consider the problem of selecting optimalspeeds and allocations of our searchers when



their detection radii vary as a function ofspeed.

Single Searcher. As before, we first consider a sin-gle searcher. In order to make a comparison withthe constant detection radius case (Figure 9), weutilize the same parameters as before, with theexception of the detection radius. We model thedetection radius as a monotonically decreasingfunction of the searcher speed; in particular, wechoose R5 6e2v=60. The same approach couldbe employed for a different sensor model.

Figure 17 shows the detection radius asa function of speed (left) and the correspondingprobability of detection (right). As the figure in-dicates, the probability of detection first in-creases with increasing speed; then it starts todecrease after reaching a maximum.

In general, the probability of detection in-creases with both increasing speed and increas-ing detection radius. However, when detectionradius decreases with increasing speed, the per-formance benefit obtained from a higher speedmust be weighed against the impairmentcaused by a smaller detection radius. For theparticular example in Figure 17, at lowerspeeds, the probability of detection increaseswith increasing searcher speed. After the maxi-mum probability of detection point, effect of de-creasing detection radius dominates the effect ofincreasing speed.

Multiple Searchers.We now consider joint selectionof speed and border allocation in the two-searcher problem. For our computational exper-iments we consider the same border length andtarget speed as in the previous case; in particular,we have L ¼ 200 and u ¼ 5 units. For simplicity,we hold one searcher’s detection radius constantat 6 units while allowing the other searcher’s de-tection radius to vary as a function of its speed. Inparticular, we let R1 5 6e2v1=60 and set the speedof the second searcher to 100 speed units. Then,we vary the speed of the first searcher and ob-serve the optimal allocation and the probabilityof detection at the optimal allocation.

We can see in Figure 18(a) that the allocationto the first searcher starts with 0.06 when itsspeed is 0; in other words, when it is stationary.An allocation of 0.06 means that 0.063 200¼ 12units are allocated to the first searcher, which isdouble its detection radius when it is stationary.In this case, the probability of detection of thefirst searcher in its region is 1.

Although increasing v1 decreasesR1, the op-timal allocation to the first searcher neverthelessincreases as v1 increases from zero. After somepoint (v1¼ 57.2 in Figure 18(a)) the optimal allo-cation to the first searcher starts to decrease.

We have a similar relationship for the prob-ability of detection as a function of the speed ofthe first searcher. When we start with thefirst searcher being stationary and increase

Figure 17. Impact of varying detection radius as a function of speed.



v1 infinitesimally, we notice a reduction in theprobability of detection. Further increasing v1results in an increase. This behavior shows thatthe probability of detection for this problem isnon-concave. Further increasing v1 causes a re-duction in probability of detection after v1 ¼57.8. Interestingly, the peaks of two curves inFigure 18(a) and Figure 18(b) do not occur atthe same v1. That is, the speed that resultsin the largest allocation to the first searcher isnot the speed that results in the highest aggre-gate detection probability. This phenomenonpersists across varying problem parameters.

Multiple Searchers with Varying Detection Radii Depend-ing on Speed. We now consider the case in whichthe detection radii of both searchers are func-tions of their corresponding speeds. We usethe same function for the detection radius ofthe first searcher that we used previously,R1 5 6e2v1=60, and we use a slightly differentfunction for the second searcher in order to ob-serve the difference in the results. Specifically,we use R2 5 6e2v2=90. These functions appear inFigure 19.

Figure 20(a) shows the optimal allocation tothe first searcher with searcher speeds varyingfrom 0 to 100 speed units in increments of 0.1units. For each searcher speed setting, the detec-tion probability resulting from an optimal allo-cation appears in Figure 20(b).

As Figure 17 indicates, when one searcheris stationary its allocation is equal to twice itsdetection radius. In the region where bothsearchers have moderate speeds, increasing v1up to around 60 speed units while keeping v2constant results in an increase in the optimal al-location to the first searcher and in the proba-bility of detection. After that value, both theallocation to the first searcher and the detec-tion probability start to decrease. This is notthe case for v2. When v1 is kept constant andv2 is increased up to around 70 speed units,the allocation to the first searcher decreases,which means that the allocation to the sec-ond searcher increases. After that value, the

Figure 18. Optimal border allocation and detection problemwhen the first searcher’s detection radius varies asa function of speed.

Figure 19. Detection radii of the two searchers.



allocation stays nearly constant. However, thedetection probability continues to increase.This can be understood by noting that the de-tection radius of the second searcher does notdecrease as much as that of the first searcher.The positive effect of increasing the secondsearcher’s speed tends to dominate the nega-tive effect of the reduction in the detection ra-dius for speeds up to 100 speed units for thesecond searcher. For this particular case, themaximumprobability of detection is 56 percentwhich is achieved when v1 ¼ 58.3 and v2 ¼ 88.4speed units. At these speed values, the optimalallocation to the first searcher is about 40 per-cent of the border length.

SUMMARYThis paper develops both analytical and

Monte Carlo simulation models for the simpleborder patrol problem in which a singlesearcher equipped with a cookie cutter sensorpatrols a straight border to detect intruders.Our analytical formula improves the detectionprobabilities computed in the literature by

using highly accurate geometrically-basedcalculations.

After computationally verifying ourmodelsfor the simple border patrol problem, we addcomplexity to the problem by introducing an-other searcher. We propose two different strate-gies for the multiple-searcher problem: thedisjoint path and the common path allocation.We develop both analytical and Monte Carlosimulation models for the disjoint path problemby building upon the single searcher case. Wealso develop a Monte Carlo simulation modelfor the common path problem.

When we compare the results of the twomultiple-searcher cases, we see the importanceof optimally allocating the border to the twosearchers. Although the disjoint path deploy-ment can provide a better detection probabilitythan the common path, it is important to allo-cate the border intelligently between the twosearchers. Otherwise, the detection probabilitymay be worse than that resulting from a com-mon path deployment.

We also analyze the effect of degrading de-tection performance with increasing searcherspeed. We perform an analysis for a single

Figure 20. Impact of varying both searchers’ detection radii.



searcher, then extend this analysis to multiple-searcher problems. Besides analyzing the speedsthat result in maximum detection probability,we determine the optimal allocation to maxi-mize the detection probability.

REFERENCESEagle, J. 2013. Search and Detection Theory.

Lecture notes, course OA3602, NavalPostgraduate School, Monterey, CA(September 27).

Girard, A. R., Howell, A. S., and Hedrick, J. K.2004. Border Patrol and SurveillanceMissionsUsing Multiple Unmanned Air Vehicles, Pro-ceedings 43rd IEEE Conference on Decision andControl, Atlantis, Paradise Island, Bahamas.

Haddah, C. C., and Gertler, J. 2010. HomelandSecurity: Unmanned Aerial Vehicles and BorderSurveillance. CRS Report No. RS21698, Con-gressional Research Service; http://www.fas.org/sgp/crs/homesec/RS21698.pdf(accessed August 20, 2016).

Matveev, A. S., Teimoorib, H., and Savkinb, A. V.2011. A Method for Guidance and Controlof an Autonomous Vehicle in Problems ofBorder Patrolling and Obstacle Avoidance,Automatica, Vol 47, No 3, 515–524.

Ozcan, B. Y. 2013. Effectiveness of Unmanned Ae-rial Vehicles in Helping Secure a Border Charac-terized by Rough Terrain and Active Terrorists.Master’s thesis, Naval Postgraduate School,Monterey, CA.

Preston, J. 2014. Border Patrol Seeks to AddDigital Eyes to Its Ranks, New York Times(March 21); http://www.nytimes.com/2014/03/22/us/border-securitys-turn-toward-the-high-tech.html.

Soza & Company. 1996. The Theory of Search: ASimplified Explanation. Report, US CoastGuard Office of Search and Rescue; http://www.navcen.uscg.gov/pdf/theory_of_search.pdf (accessed August 20, 2016).

Sozen, V. 2014. Optimal Deployment of Un-manned Aerial Vehicles for Border Surveil-lance. Master’s thesis, Naval PostgraduateSchool, Monterey, CA.

Szechtman, R., Kress, M., Lin, K., and Cfir,D. 2007. Models of Sensor Operations forBorder Surveillance, Naval Research Logistics,Vol 55, No 1, 27–41.

Wagner, D. H., Mylander, W. C., and Sanders,T. J., eds. 1999. Naval Operations Analysis.Naval Institute Press.

Washburn, A. R. 2002. Search and detection.INFORMS.



Documents

ABSTRACT B Optimal Unmanned Vehicle Allocation for ...faculty.nps.edu/emcrapar/docs/Volkan_19.pdfVehicle Allocation for Border Surveillance Volkan S¨ozen Turkish Army [email protected]