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Starting from an origin-destination dataset for a given country, where the origins are city (points) and the destination are ports (points), how can I identify the area of influence for each particular port?

Is there a tool in ArcMap which can automate the creation of the polygons for showing which region is more key to one port or another?

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With Network Analyst, using the Location Allocation tool.

Input your origins as facilities, and destinations as demand points. This should give you a result of associated points, which then can be used to create polygons using either Feature to Polygon or Construct Polygon, Help Site

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