Location optimization: centers of gravity

Other than production facilities and distribution centers, the center of population or oth-er weighted geographic centoth-ers have been used historically even to detoth-ermine the loca-tions of entire capitals of countries. There is even a term, “forward capital” used to de-note a capital that is moved to a new location away from existing metropolises to better suit the geographic distribution of population. Possibly the most extreme example of

this is the relocation of the Brazilian capital from Rio de Janeiro to Brasília in 1956, when the new capital was built from nothing to sit in the center of Brazil. (UNESCO World Heritage Convention 2013) In a way, Russia is no stranger to this practice, as the main reason for Peter the Great to move the capital from Moscow to Saint Petersburg was to open connections to the west. Although the logic behind this was other than a geographic center – it was that of proximity to the west – the principle of moving away from current population centers is the same. Moscow has since regained its position as the capital, but Saint Petersburg is still the second largest city in the country.

The load-distance method is a means for systematic selection process to evaluate loca-tions based on proximity factors. In brief, the method finds a facility location that mini-mizes the sum of loads multiplied by the distance the load travels. Other weights, such as time, can be used to instead of or in addition to load size. (Krajewski et al. 2013, p.

119)

Center of gravity is an approximation of the optimal point to evaluate alternatives using the load-distance method. At its simplest, center of gravity can be calculated by the formulae

where x and y note the coordinates and l the load by which each coordinate is weighted (Krajewski et al. 2013, p. 119). These simple formulae are provided in textbooks such as Krajewski et al. (2013) and Stevenson (2011), which mention that it is an approxima-tion, but they do not stress the fact that in a larger geographic area, the result is nothing short of misleading, as the method ignores the spherical shape of the Earth and only gives adequate approximations for points near each other.

The shortcomings of this method can be easily demonstrated by an example: Calculat-ing the center of gravity for Saint Petersburg (59°N 30°W) and Anchorage (61°N 149°W), the aforementioned equations give the following results:

𝑥^∗ =^{59∗1+61∗1}₁₊₁ = 60 = 60°𝑁 𝑦^∗ =^{30∗1−149∗1}₁₊₁ = −59,5 ≈ 60°𝑊.

This is a point in the ocean somewhere between Greenland and Labrador – some 3300km away from the actual optimal point.

The actual optimal center of gravity for two points can be calculated by basic geomet-rics according to which the shortest distance between them on the surface of a sphere follows the great circle. Thus the point with the shortest distance to the two ends is the

center point of the great circle route. (Encyclopaedia Britannica 2013c) Saint Petersburg and Anchorage lie almost symmetrically on the same latitude on the opposite sides of the Earth, and their optimal point is close to the North Pole (90°N).

The difference between the “false” center of gravity based on formulae 1 and 2 and the

“correct” center of gravity based on geometrics is visible in figure 3.8. The “false” cen-ter of gravity indicated by the red circle is situated on the average latitude between Saint Petersburg and Anchorage, whereas the green circle showing the “correct” point follows the great circle, approximately the meridians 30° east and 150° west. Figure 3.8.

is drawn on a map centered on the North Pole, which means that great circle route for Saint Petersburg and Anchorage follows a meridian. The map projection serves the pur-pose perfectly here, as a “regular” Mercator projection distorts the northern parts of the hemisphere greatly and usually cuts out all polar regions above the 85^th parallel north (Encyclopaedia Britannica 2013d).

Figure 3.8. An example of center of gravity fallacy. "False" center of gravity in red and

"correct" in green. This extreme example was chosen deliberately, and other settings generally yield closer results even with the “false” formulae. (Map base from Wiki-media 2005)

As the figure above shows, the optimal route between Saint Petersburg and Anchorage would pass the North Pole. Crossing the arctic is possible in aviation, but when the rele-vant means of transportation are roads, railroads and waterways, as is the case in this thesis, such a route would not suite the requirements. Unrealistic routes are not a prob-lem in this thesis, however, since the locations are not as far apart as Saint Petersburg

and Anchorage. The farthest great circle distances between the westernmost and east-ernmost parts of Russia are not realistic considering the road infrastructure in northern Siberia, and they are not used as such – nor are any other great circle distances. Instead, all absolute distances are corrected with a circuity factor, which will be described later.

Also, the number of trans-Russian loads forms only a minimal part of all transportation.

In a setting with more than two points, the midpoints of great circle distances cannot be used. For a large number of points, a definitively closer approximation than formulae 1 and 2 can be achieved by converting the coordinates to three-dimensional Cartesian co-ordinates and calculating the center of gravity for those points. The result is a point in three-dimensional space, not on the surface of the Earth. A point on the surface is then extrapolated following a line from the result point and the center of the Earth. This new surface point is far closer to the optimal point than calculations based on formulae 1 and 2, and the method can easily be applied to any number of points. To find out the exact optimal point would require iterative, labor- and computer time-intensive calculations, which would yield a result only marginally more accurate than the center of gravity for Cartesian coordinates.

For this thesis, the center of gravity for three-dimensional Cartesian coordinates is used, as Russia – being the largest country in the world – is not small enough an area for for-mulae 1 and 2 to provide sufficient accuracy. Of course, road and railroad networks af-fect the actual routings of transportation, and no roads follow great circles between two points. Nevertheless, using a method that provides more accurate results for an idealized situation is also beneficial for real-world applications.

Actual road distances can be approximated by multiplying the great circle distances by a circuity factor. A circuity factor is a constant that is calculated by dividing the actual road distance between two places by their great circle distance. (Simchi-Levi et al.

2003, pp. 32-33) An example of this is given in figure 3.9.

Figure 3.9. Circuity factor for the distance between the Tatarstan plant (A) and a Com-pany X distributor in Moscow (B). "Great circle" is in parenthesis as it is represented by a straight line - not an actual great circle route, which would curve slightly in a Mercator projection. (Map base from Google Maps 2013)

In the figure above, the circuity factor for the distance between the Tatarstan plant and one of Company X’s distributors in Moscow is 1.20. Basic geometrics state that a circu-ity factor cannot be less than 1.00 (without traveling through the Earth’s interior), since the great circle distance is the shortest distance between two points.

For Russia, Ballou et al. (2002) calculate the circuity factor for Russia to be 1.37 with a standard deviation of 0.26. The average circuity factors in the study vary from 1.12 in Belarus to 2.10 in Egypt. No explanations are given, but one could speculate that Bela-rus is geographically ideal for roads with no mountains, major bodies of water or other curvatures whereas the road network in Egypt is greatly influenced by the distribution of population along the banks of the Nile. Russia, of course, has great variations be-tween different parts of the country, and the aforementioned value 1.37 is only an aver-age for entire Russia.

The reason why a circuity factor is used instead of actual road distances is the number of nodes involved. If road distances were calculated individually between each of the 83 federal subjects of Russia, thousands of separate routings on applications like Google Maps would have to be made. By using coordinates, a spreadsheet program can easily do the calculations, which are then multiplied by the circuity factor.