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I want to determine the territories in which customers to a bank has the shortest path. It can be done with desktop software packages or programming if it is needed. Does anybody has idea on how to start doing it? I want to produce a map like this and this map only the territories are important for me.

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This sounds more like Network Analyst than Spatial Analyst - do you have that extension licensed and what version of ArcGIS for Desktop are you using? – PolyGeo May 15 '13 at 10:33
yes I do. ArcGIS 10. But I need to know the algorithm as well for programming. – f.ashouri May 15 '13 at 13:49
How to do it in ArcGIS 10.0 is one question but looking for an algorithm may be best researched/posted separately. – PolyGeo May 15 '13 at 22:40

Have you looked at Service Area and/or Closest Facility in the Network Analyst Toolbox? This may be helpful. The map you posted has desire lines, which you can find in the Business Analyst Toolbox.

Another option, if you know which point goes to which facility, is Envelope to Polygon.

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If this is something that you need to do more than once, perhaps take a look at the Business Analyst Extension.

Create Territories

Create Trade Areas

enter image description here

The extension includes toolboxes which can be called from python or strung together within a model.

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