Zones in Macro Transport Model described with "optimal" points, part III: Second Optimal Gravity Center

This is third part of Spatial Analysis of Zones for macroscopic transport modeling. Now I focus on problem of location of "second optimal point", being transfer point for journey within zone (in this case it’s county).

Two other parts of this series can be found here, and here where You can find description of input database. The previous parts focus on: 1) Where is optimal transfer point for journeys within area 2) where is optimal transfer point for journeys from region to center of network (i.e. from region to state capital, i.e. from Kent to London, from Bretagne to Paris, from Liguria to Rome, from Silesia to Warsaw).

Problem: 

Now I assume that one point location is fixed, and it’s legal capital of area (county seat). With this constrain I search for second point that will minimize travel times within area.

 Method: 

second gravity point can be defined as follows: 

Where:

SCG          is second gravity point

P                is population

p                is zone index

i                 is index for addresses within area

d                is distance

C               center of area

Results:

Minimization problem wasn’t monotonic, and had some local minima, thus it was solved by greedy algorithm (enumeration), global optimization approach like here was not employed due to small complexity.

Following diagram show goal function values in 3d plot:

Summary:

Second Optimal Point for area can be a good hint to describe area. It may be a hint when You try to place connector, or describe travel times within Zone. It may be also a good hint when defining location of Zone Centroid.