suchisman gangopadhyay montgomery high school, new jersey

Suchisman Gangopadhyay

1

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]

mailto:[email protected]


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


3Premise


4

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


5

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


6

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


7

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


8

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


9

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


10

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

http://en.wikipedia.org/wiki/File:2Ddim-L2norm-10site.png

http://en.wikipedia.org/wiki/File:2Ddim-L1norm-10site.png


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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


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


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


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


29Q & A

• Thanks!


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

suchisman gangopadhyay montgomery high school, new jersey

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