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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
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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
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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
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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
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10/2/2007 5
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10/2/2007 6
Basic Setting
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10/2/2007 7
Basic Setting (cont.)
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10/2/2007 8
Basic Setting (cont.)
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reaches SB thebefore remaining seconds 10least at
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10/2/2007 9
Basic Setting (cont.)
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10/2/2007 10
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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
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10/2/2007 12
Events Related to the SB
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10/2/2007 13
Probabilities and Casualties
)1]()1( 1[)1(
at explodes and from enters SBgiven casuality Expected
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10/2/2007 14
Total Expected Casualties
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10/2/2007 15
Optimization Model
employ willedetector w ofnumber maximum theis where
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10/2/2007 16
Optimization Model (cont.)
gporgrammininteger binary nonlinear a iswhich
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10/2/2007 17
Properties
1 be toelements
first set the and oforder decreasingin
)(in elements thesequence ,|)(| If 2. Case
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grid explosive one and entrance oneonly is thereIf
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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
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10/2/2007 19
Greedy Adding Heuristic
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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
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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
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10/2/2007 22
Base Case
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10/2/2007 23
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10/2/2007 24
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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
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10/2/2007 26
Computational Analysis
Procedure GAH of Efficiency
Analysis of Robustness
Analysisy Sensitivit
•
•
•
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10/2/2007 27
Sensitivity Analysis
RateDetection ousInstantane ofEffect
RadiusDetection ofEffect
Entrances ofNumber ofEffect
Placed Detectors ofNumber ofEffect
•
•
•
•
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10/2/2007 28
Effect of Number of Detectors
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10/2/2007 29
Effect of Number of Entrances
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10/2/2007 30
Effect of Detection Radius
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10/2/2007 31
Effect of Detection Rate
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10/2/2007 32
Robustness Analysis
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055.0
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onsPerturbati Three
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10/2/2007 34
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10/2/2007 35
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10/2/2007 36
Efficiency of GAH Procedure
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10/2/2007 37
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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
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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
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10/2/2007 40
Questions?