City-Data Forum US Population Geometric Median & Minimax Point: Measures of Central Tendency for Facility Planning (condo)
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10-04-2014, 03:52 AM
 Location: Virginia 1 posts, read 1,117 times Reputation: 10

EDIT: There's no "URL" button, only the check boxes at the bottom of the post, and those didn't work.
Quote:
 If you had to build a distribution center for something totally arbitrary, serving customers in the US (or North America), in what location is half of the US population within driving distance? The answer is somewhere south of Pittsburgh.
- Facilities Design Professor

I came here looking to see if I could figure out whether or not he was right, and how to generalize what he had said. I searched the boards and didn't find anything, and via relentless googling I ended up finding some stuff (mostly regarding the geometric median) both theoretical and empirical, so I figured I'd share. Disclaimer: I was a crappy engineering student and I have no ties to any companies. I've checked out a UPS distribution center before.

The typical measure of central tendency is the mean, because it's pretty simple to understand and computer: it's simply the weighted average of where everyone lives. I'm sure tons of you know about the Mean Center of United States Population, which is somewhere around Plato, MO
http://en.wikipedia.org/wiki/Mean_ce...tes_population

The problem is, the mean is not a particularly robust measurement, because it can be thrown off by outliers. There exists a Median Center of United States Population, which is "the point though which a north-south line and an east-west line each divides the total population of the country in half." This point is located somewhere in southwest Indiana. This point isn't necessarily useful but it's still a little better.

When it comes to placing things like distribution centers (and hospitals/fire etc.), a better question to ask is "What location MINIMIZES the MAXIMUM distance anyone has to travel?" OR "What location MINIMIZES the SUM/AVERAGE everyone has to travel?" The latter is the "Geometric Median" (minisum) and is a little easier to calculate than the former (minimax) - both algorithms are much harder to computer than just cranking out the mean, because they require a computer to solve an "optimization problem" which basically* works like this: "Here is an algorithm that defines the problem. Use computing power to check through all possible solutions in order to find the best one." After doing some research I found an answer to the United States Geometric Median in an (unpublished and undated but recent) paper entitled "The FedEx Problem" by Kent Morrison from the Cal Poly SLO Math Department:

Quote:
 We found the minimum to be (39◦, 87◦), a location in Greene County, Indiana, about 70 miles southwest of Indianapolis. From this point the average distance to any person in the country is 795 miles, while the average distance to Memphis (FedEx Hub) is 843 miles. ... Furthermore, the optimal location is only about 85 miles northwest of Louisville where UPS has established its main hub."
http://www.calpoly.edu/~kmorriso/Res...fedexfinal.pdf

So in short, this location in Indiana is genuinely more "central" than the simple mean location found in Missouri

Morrison goes on to point out that the solution to the geometric median for the United States population (as solved by himself) is VERY close aforementioned "Median Center of United States Population" (as solved by the US Census Bureau) which kind of surprised me. Either way, I'm still looking for an answer to the "Minimax" location if anyone has any idea of where to look!

Some resources (sorry for bad formatting):
The only relevant result on here I could find w/r/t my professor's Pittsburgh claim, with folks stating "Pittsburgh is within driving distance to 80% of the US population and Canada." and then someone stated (more accurately) "within 500 miles of 50% of the population of the United States."
Pittsburgh vs Raleigh

http://en.wikipedia.org/wiki/Mean_ce...tes_population
http://en.wikipedia.org/wiki/Median_...tes_population
http://en.wikipedia.org/wiki/Center_of_population
http://en.wikipedia.org/wiki/Geometric_median
http://en.wikipedia.org/wiki/Facility_location
http://en.wikipedia.org/wiki/Location_theory

Another point of curiosity: if you examine the images that show the location of the mean and the median over time (via the respective Wiki links), it's interesting that in 1880 they were about 100mi apart around western Ohio & Kentucky - but as of 2010 the Median had only crept westward to southwest Indiana, while the Mean has gone a few hundred miles to the southwest and is now in southern Missouri. This is a perfect example of why the (simple) median is a more robust measure of central tendency than the mean. It makes sense that the center of population, by any measure, would move to the south and to the west over time. The reason the mean location is so far west is because it's "biased" due to population growth, which brings this weighted average more towards the west. The median location remains almost the same because it's simply the spot where half of the population is east/west or north/south, and so even a major population shift wouldn't cause this to move very much.

Some interesting math-heavy stuff I came across, for further reading:

"A case study: Condorcet’s effect and medians." by BERNARD MONJARDET, published in "Journal Electronique d'Histoire des Probabilités et de la Statistique"
http://www.jehps.net/juin2008/Monjardet.pdf (2008)

"Implementing a Bayesian approach to criminal geographic profiling" by Mike O’Leary from the Department of Mathematics at Towson University (Fairly Recent)
http://pages.towson.edu/moleary/docs/2010%20ComGeo.pdf

"Optimal Location of Facilities" by Gerard Rushton from the Department of Geography at University of Iowa (1979)
http://www.rri.wvu.edu/wp-content/up...malWebBook.pdf

Other claims regarding geography that don't touch on math:

Columbus is a great city because "Even with the growth in Sunbelt states in the South and West, 147.5 million people — 48 percent of the nation’s population — still live within 500 miles of Columbus, according to the study. Pittsburgh was second on the list at 47 percent, Dayton and Cleveland were tied in third place at 46 percent, and Cincinnati was fifth at 44 percent." via "... the list produced by Upper Arlington-based Three Scale Research."
http://www.dispatch.com/content/stor...tudy-says.html

Pittsburgh wants you to know how awesome they are! "Pittsburgh is within 500 miles of more than half the U.S. population and less than a 90-minute flight from 50% of North America’s population. It's under 6 hours by car or train to 9 states, D.C. and Canada..."
http://www.planpittsburgh.com/gettin...e-travel-time/

Last edited by mikesty; 10-04-2014 at 03:54 AM.. Reason: URL help?