Zones in Macro Transport Model described with "optimal" points, part II: Optimal Access Point

This post reports on results of my spatial analysis of transport zones.

I define optimal location of access point for area. Location guarantees minimal overall transport costs for journeys from area to central point of region (i.e. Capital)

I define Optimal Access Point, being point that provides minimal sum of travel times for journeys to central point of network (i.e. Capital).

Presumptions for this analysis (transport network, demand model) were described here. additionally what You can find there is analogical definition of point that minimizes travel times within area (Center of Gravity)

Method:

This analysis was done for Counties (around 50k inhabitants , 2k sq. km ). I got GIS database consisting of: 

·        Location of  place with information about their population (fig.1)

·        Transport network (roads + railway) (fig.2)

·        Administrative division (zones) (fig.1+2) (each zone was about 50k inhabitants, 2k km²)

For each zone I find an “Optimal Access Point”, being defined as follows:

Where:

P                is population

d                is distance

M               is metropolital point (central point of network)

A, B           are parameters crucial for results, namely:

if (B/A -> Inf ) -> point = center of network ,

if ( B/A -> 0) -> point ~ center of gravity for area

Solution: 

Goal function was strictly monotonic, and continuous, so optimization was straightforward, with following results

Boundary = area boundary

X = central point for area

triangle = gravity center for area

O = Optimal Location Point

Background color = goal function value

the A, and B parameters will place Optimal Access Point on line between Center of Gravity for region, and Central point of network, and line can be a curve rather than straight line

Conclusion 

Optimal Access Point location can be a good hint during decision of transit network design, i.e. one can locate multimodal stop at point that minimizes both Access Time to Central point of Network, and is close to Center of Gravity.

PS. Cartesian distance was proposed here (for simplicity), however any other distance measures can be employed, i.e. travel distance.