Planning Troop & Resource Deployments at Military Bases using Exponentially Weighted Voronoi Diagrams Abstract 1106 for WG-16 Presented at MORS Conference May 14, 1400 Hrs EDT Suchisman Gangopadhyay Montgomery High School, New Jersey Admitted as Freshman, University of Illinois, Urbana-Champaign Suchisman Gangopadhyay 1

Outline 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 Suchisman Gangopadhyay 2

Premise Suchisman Gangopadhyay 3

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 Suchisman Gangopadhyay 4

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 Suchisman Gangopadhyay 5

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 Suchisman Gangopadhyay 6

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 Suchisman Gangopadhyay 7

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 Suchisman Gangopadhyay 8

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 Suchisman Gangopadhyay 9

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 Suchisman Gangopadhyay 10

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 Suchisman Gangopadhyay 11

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 Suchisman Gangopadhyay 12

Method Suchisman Gangopadhyay 13

Step 1: Plot the coordinates of the bases on a Cartesian plane (represented by blue stars) Suchisman Gangopadhyay 14

Step 2: Use a Voronoi tessellation to find the land each base administers Suchisman Gangopadhyay 15

Step 3: Add locations of insurgent activity to the Voronoi (represented by orange dots Suchisman Gangopadhyay 16

Step 4: Find the concentration of enemy activity in each cell Suchisman Gangopadhyay 17

Step 5: Use these concentrations as weightages to create an exponentially weighted Voronoi Suchisman Gangopadhyay 18

Observations & Results Suchisman Gangopadhyay 19

Step 6: Distribute resources using the ratios of the area of the weighted cells Suchisman Gangopadhyay 20 Site Coordinates Site Metrics Area of SiteInsurgent Density Weighted Voronoi Ratio Recommended Allocation of Resources A: (9,7) a 13,638 B: (10,7) ,469 C: (6,6) ,821 D: (3,6) ,322 E: (5,8) ,751 TOTAL 144 N.A ,001 a. Ratio is calculated as the number of points plotted in each weighted Voronoi cell by MATLAB ®.

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 Suchisman Gangopadhyay 21

When making an exponentially weighted Voronoi diagram, there are a few things one must look out for Suchisman Gangopadhyay 22

First, exponentially weighted Voronoi diagrams differ by scale Therefore, it is important to be consistent with units when constructing the diagram Suchisman Gangopadhyay 23

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 Suchisman Gangopadhyay 24

Next Steps Suchisman Gangopadhyay 25

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 Suchisman Gangopadhyay 26

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 Suchisman Gangopadhyay 27

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 Suchisman Gangopadhyay 28

Q & A Thanks! Suchisman Gangopadhyay 29

References 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, , , 2007 IV.B. B. de Mesquita, The War Trap, Yale University Press, New Haven, 1981, V.D. Lemke, Regions of War and Peace, Press Syndicate of the University of Cambridge, Cambridge, England, 2002, 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, Suchisman Gangopadhyay 30