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Planning Troop & Resource Deployments at Military Bases using Exponentially WeightedVoronoi DiagramsAbstract 1106 for WG-16 Presented at 2013 81.1 MORS ConferenceMay 14, 2013 @ 1400 Hrs EDT
Suchisman GangopadhyayMontgomery High School, New JerseyAdmitted as Freshman, University of Illinois, [email protected]
Suchisman Gangopadhyay
2Outline
• Premise• Troop Deployment and Voronoi• Insurgent Activity and Weighted Voronoi• Boulding’s Loss of Strength Gradient and Exponential
Weightage• Method• Observations & Results• Next steps• References
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3Premise
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Deciding on how many resources to send to a military base is a
time consuming process
Due to the unpredictable nature of war, reducing the time this process takes can win battles and save lives
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Assuming uniform land structure, resources can be deployed based on how
much land each base must administer
One of the methods of determining these areas is to
use Voronoi Diagrams
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Voronoi Diagrams tessellate a plane into sections based on how close parts of the plane are to certain points, called sites
A cell is formed around each site, consisting of the points which are closest to that site. Points on cell walls are equally close to two sites, while points
on vertices are equally close to three or more sites
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The positions of the bases can be used as sites to create a
Voronoi diagram
Resources can then be distributed in the ratio of areas
of the sites
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Unfortunately, this method does not account for the
uneven presence of insurgents
A base in a safer area may receive the same amount of resources as a base that has to protect regions that are constantly
under attack
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The use of a weighted Vornonoi can account for this
If a normal Voronoi uses the simple distance formula, then a weighted
Voronoi uses a weight to modify that distance
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One frequent type of distance formula often substituted for the regular distance
formula is “Manhattan distance”
Other types of weighting have each site holding a different weightage, which is
then added, multiplied, etc. to a distance between some point and that site
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In 1962, Kenneth Boulding devised a loss of strength
gradient
He argued that the strength of a military force was related to the distance that force had to travel
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The loss of strength gradient can be expressed as:
Due to the use of exponents in this gradient, it makes sense for the
weightage used in this Voronoi to be exponential
log[ (10 )]_ _miles e
miles per dayAdjustedPower Power
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13Method
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Step 1: Plot the coordinates of the bases on a Cartesian plane (represented by blue stars)
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Step 2: Use a Voronoi tessellation to find the land each base administers
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Step 3: Add locations of insurgent activity to the Voronoi (represented by orange dots
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17Step 4: Find the concentration of enemy activity in each cell
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Step 5: Use these concentrations as weightages to create an exponentially weighted Voronoi
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Observations & Results
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Step 6: Distribute resources using the ratios of the area of the weighted cells
Site Coordinates
Site Metrics
Area of Site Insurgent DensityWeighted
Voronoi Ratio
Recommended
Allocation of
Resources
A: (9,7) 16.866 0.474 1102 a 13,638
B: (10,7) 29.969 1.334 15875 196,469
C: (6,6) 28.459 0.808 2652 32,821
D: (3,6) 38.375 1.251 15944 197,322
E: (5,8) 29.333 0.954 4828 59,751
TOTAL 144 N.A. 40401 500,001
a. Ratio is calculated as the number of points plotted in each weighted Voronoi cell by MATLAB ®.
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This method results in more resources being given to bases in the middle of enemy activity
While still accounting for how much land each base must
administer
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When making an exponentially weighted Voronoi diagram, there are a few things one must look out for
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First, exponentially weighted Voronoi diagrams differ by scale
Therefore, it is important to be consistent with units when constructing the diagram
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Next, there are some strange structures one may see when
creating an exponential Voronoi
A cell seeming like it is subsumed within another may
actually be much bigger
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25Next Steps
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Exponential weightage is only the beginning
By using the deployment of resources in successful operations as an example, the formula for weightage can be adjusted to
replicate those results
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Another way this method can become more accurate and versatile is to account
for natural formations and enemy structures
Instead of starting with a regular Voronoi, the land each base administers could be
split based on a Voronoi weighted to account for uneven territory
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Lastly, while simple and additively weighted Voronoi diagrams can be constructed efficiently, other types of weighted Voronoi diagrams can
take much longer to create
The quicker the algorithm for creating these weighted Voronoi diagrams is, the
quicker one can respond to new information
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29Q & A
• Thanks!
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30References
I. K. Boulding, Conflict and Defense, Harper, New York, 1962, 262.
II. K. Boulding, The Meaning of the 20th Century, The Great Transition, George Allen & Unwin, London, 1965, 87
III. A.G. Boyer, N.J. Gauthier, and C.W. Johnson, The Paradoxes of Military Risk Assessment, In Proceedings of the 25th International Systems Safety Conference, Baltimore, USA, International Systems Safety Society, Unionville, VA, USA, 859-869, 0-9721385-7-9, 2007
IV. B. B. de Mesquita, The War Trap, Yale University Press, New Haven, 1981, 103-108
V. D. Lemke, Regions of War and Peace, Press Syndicate of the University of Cambridge, Cambridge, England, 2002, 71-73
VI. J.D. Vanvactor, Risk Mitigation Through a Composite Risk Management Process, The U.S. Army Risk Assessment Organization Development Journal Vol. 25 Nbr. (2007)
VII. K. Webb, The Continued Importance of Geographic Distance and Boulding's Loss of Strength Gradient, Comparative Strategy, Volume 26 Issue 4, 2007, 295-310