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I'm trying to find out how long it would take to travel by car from each of the Lower Layer Super Output Areas (LSOA) in Norfolk, England to the nearest Hospital (or just what the distance is and I can estimate drive time, if the former is too difficult/impossible!).

Any suggestions how I can do that using QGIS, OSM road data and/or Ordnance Survey Meridian 2 road data, population weighted centroids for each of the 2011 Census LSOA areas and hospital locations?

I have all of the above in QGIS (see image below).

There are 537 LSOA and four hospitals in total. Is there a way to work out the time taken and/or distance to travel without having to do it one by one?

enter image description here

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Did you look into anita graser examples of pgrouting. Take also a look into [GRASS network functions] (jcastellssala.wordpress.com/2012/05/07/…). –  Alexandre Neto May 20 '13 at 18:37
    
Are you able to do a bit of Python scripting? –  Nathan W May 21 '13 at 2:57
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You can call our RW Net 4 library from python. We have a working sample, that does something quite similar to what you describe above and it can also use Meridian 2, which we also offer in a free and improved version. The sample uses QGIS and Python.NET for the operations.

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I saw a presentation (PDF) at GISRUK'13 that made use of Routino to solve a similar problem. It makes use of the OpenStreetMap road network but I think you should be able to change that if you install it locally. But for your size of problem you could probably script the solution using the web api.

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