10/2/2007 1

Optimal Placement of Suicide

Bomber Detectors

Xiaofeng Nie, Rajan Batta, Colin G. Drury, Li Lin

Department of Industrial and Systems Engineering

Research Institute for Safety and Security in Transportation

The State University of New York at Buffalo

10/2/2007 2

Framework

• Introduction

• Basic Setting and Optimization Model

• Properties

• Greedy Adding Heuristic and Branch and Bound

• Base Case and Computational Analysis

• Future Work

10/2/2007 3

Introduction

• Kaplan and Kress analyze the operational effectiveness of suicide bomber (SB) detector schemes under best-case assumptions

• Two urban environments: grid and plaza

• Two kinds of intervention: instruct to flee and hit the deck

• Under some situations, intervention will not reliably result in meaningful casualty reductions

10/2/2007 4

Introduction (cont.)• Here, we consider the optimal placement of detectors to minimize the expected casualties in a threat area where the entrances and the potential targets are known

• We divide the threat area into grids

• Some grids are blocked to model physical obstructions

• The SB detector is not perfectly reliable

• Assume that the SB will travel on the shortest path from his/her chosen entrance to the selected explosive grid

10/2/2007 5

10/2/2007 6

Basic Setting

{ }

grids unblocked ofset theis \

grids blocked all ofset theis

,2,1,,2,1:

entrances, actual ofset theis

entrances possible all ofset theis

entrances are There

grids into divided is arear rectangula The

BGU

GB

njmiijG

LlEL

E

l

nm

=•

⊂•

===•

=⊂•

•

•

ו

��

10/2/2007 7

Basic Setting (cont.)

grids blockedany crossing

without gridanother togrid one fromcan walk SB The

y probabilit with gridat bomb theexplodes and

entrance thefrom enters (SB)bomber suicideA

be grids explosive potential ofset Let the

density with ddistributePoisson

spatially are sindividual the, gridany For

•

•

⊂•

∈•

kij

ij

ij

k

US

Uij

γ

λ

10/2/2007 8

Basic Setting (cont.)

grid explosive rgetedhis/her ta

reaches SB thebefore remaining seconds 10least at

are theresuch that detection :detectionTimely

is radiusdetection effective The

unique ispath shortest that theassume WLOG,

as denoted , grid explosive to entrance

frompath shortest ugh the walk thro willSB The

•

•

•

•

τ

kijPijk

10/2/2007 9

Basic Setting (cont.)

ratedetection ousinstantane

theis and from distance within is which of

section theoflength theis where,1

fromaway

m 10least at is which ofportion thebe Let

be timely SB the

detecting ofy probabilit let the ),(every For

along SB the

detect can timely which grids all ofset the:)(

ητ

η

rsP

lep

ij

PP

p

ijNrs

P

rsijN

kij

rskij

l

rskij

kijkij

rskij

k

kij

k

rskij×−−=•

•

∈•

•

10/2/2007 10

10/2/2007 11

Basic Setting (cont.)

θy probabilitly with successful

dneutralize be could explosive thedetected, If

tlyindependen work detectors The

is casualties expected the,at explodes SB If

otherwise

in locateddetector a is thereif 0

1

define ,any For

)( Define ,

•

•

•

=

∈•

=•

ij

ij

kijk

Cij

ijx

Tij

ijNT ∪

10/2/2007 12

Events Related to the SB

10/2/2007 13

Probabilities and Casualties

)1]()1( 1[)1(

at explodes and from enters SBgiven casuality Expected

)1( 1

yprobabilit with detected, when )1( is casualties expected The

)1(

yprobabilit with detected,not when is casualties expected The

at explodes and entrance from walksSB When the

)(Nrs)(Nrs

)(Nrs

)(Nrs

kk

k

k

ij

ij

x

rskijij

ij

x

rskij

ij

x

rskij

ij

ij

x

rskij

ij

CpCp

ijk

p

C

p

C

ijk

rsrs

rs

rs

∏∏

∏

∏

∈∈

∈

∈

−−−+−

•

−−

−•

−

•

θ

θ

10/2/2007 14

Total Expected Casualties

])1()1[(

]})1()1{[(

casualties expected totalThe

y probabilit

withat explodes and from enters SB The

1 )(Nrs

1 )(Nrs

k

k

∑∑ ∏

∑∑ ∏

= ∈ ∈

= ∈ ∈

−+−=

−+−

•

•

l

k Sij

ij

ij

kij

x

rskijijkij

kij

l

k Sij

ij

ij

x

rskijij

kij

CpC

CpC

ijk

rs

rs

γθγθ

γθθ

γ

10/2/2007 15

Optimization Model

employ willedetector w ofnumber maximum theis where

},1,0{

,

..

])1()1[( 1 )(Nrs k

M

Tijx

Mx

ts

CpCMin

ij

Tij

ij

l

k Sij

ij

ij

kij

x

rskijijkijrs

∈∀∈

≤

−+−

∑

∑∑ ∏

∈

= ∈ ∈

γθγθ

10/2/2007 16

Optimization Model (cont.)

gporgrammininteger binary nonlinear a iswhich

, },1,0{

, ..

)1(

toreduces problem theconstant, is )1( Since

Let

)(1

Tijx

Mxts

pWMin

C

CW

ij

Tij

ij

ijNrs

x

rskij

l

k Sij

kij

ijkij

ijkijkij

k

rs

∈∀∈

≤

−

−•

=•

∑

∏∑∑

∈

∈= ∈

γθ

θγ

10/2/2007 17

Properties

1 be toelements

first set the and oforder decreasingin

)(in elements thesequence ,|)(| If 2. Case

)(every for 1 ,|)(| If 1. Case

grid explosive one and entrance oneonly is thereIf

solution optimal in the ,If

programnonlinear convex a is problemrelaxtion The

*

**

Mp

ijNMijN

ijNrsxMijN

ij

XMxM |T|

rskij

kk

krsk

Tij

ij

>∗

∈=≤∗

•

=≥•

•

∑∈

10/2/2007 18

Properties (cont.)

solution optimal in the grid

in that detector one locate will we,1 and grids

other all dominates which grid one exists thereIf

solution optimal in the

0 grids, least at by dominated is grid If

dominates pair, , oneleast at for

and pairs , allfor If :Dominance

*

≥

•

=•

>

≥•

M

xMuv

uvrsijkpp

ijkpp

uv

uvkijrskij

uvkijrskij

10/2/2007 19

Greedy Adding Heuristic

10/2/2007 20

A Special Case

optimal is procedure GAH by thegiven solution the

then ,)( ),,( of pairs possible allfor If

Property reSubstructu Optimal

1 heresolution w optimalan exists there

Then procedure. GAH by the 1 be set to be to variable

decisionfirst thebe Let :Property ChoiceGreedy

intersect )(set theof none wherecase specialA

*

φ=∩•

•

=

•

•

ijNijk

x

x

ijN

k

rs

rs

k

10/2/2007 21

Branch and Bound

solutions optimalany geliminatin

withoutspace feasible thedecrease will

constraint adding ,by dominated is grid If

riabledecison va ingcorrespond theeliminate

can wegrids, than moreby dominated is grid a If

properties dominance some Explore

programnonlinear convex a is problem relaxation The

rsuv xx

rsuv

M

≤

∗

∗

•

•

10/2/2007 22

Base Case

10/2/2007 23

10/2/2007 24

10/2/2007 25

Base Case Solutions

%43.0 error Relative

27.856 is

aluefunction v optimal the,1 issolution

optimal thealgorithm, Bound andBranch theusingBy

27.976 is aluefunction v

objective ingcorrespond the83, grid and 55 grid then

46, grid choosefirst will weprocedure, GAH theusingBy

Value Bound andBranch Value Bound andBranch Value GAH

*

83

*

55

*

36

==•

===

•

•

−

xxx

10/2/2007 26

Computational Analysis

Procedure GAH of Efficiency

Analysis of Robustness

Analysisy Sensitivit

•

•

•

10/2/2007 27

Sensitivity Analysis

RateDetection ousInstantane ofEffect

RadiusDetection ofEffect

Entrances ofNumber ofEffect

Placed Detectors ofNumber ofEffect

•

•

•

•

10/2/2007 28

Effect of Number of Detectors

10/2/2007 29

Effect of Number of Entrances

10/2/2007 30

Effect of Detection Radius

10/2/2007 31

Effect of Detection Rate

10/2/2007 32

Robustness Analysis

kijkij

kijkij

kijkij

εγ

εγ

εγ

+=•

+=•

+=•

055.0

050.0

045.0

onsPerturbati Three

10/2/2007 33

10/2/2007 34

10/2/2007 35

10/2/2007 36

Efficiency of GAH Procedure

10/2/2007 37

10/2/2007 38

Conclusions

• Considered how to deploy SB detectors in a threat area where the potential targets are known

• Proposed an optimization model where the objective function is the total expected casualties

• Developed two algorithms (one heuristic and one exact) and studied a base case

10/2/2007 39

Future Directions

y probabilitdetection joint heconsider t could weLater,

detectors.amony cy independen assume we work,In this

detectors of typesdifferent thesite to where

andemploy tokindeach for many how them,from

choose tohow :detectors of kinds severalConsider

•

•

10/2/2007 40

Questions?