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I have been using GIS to make travel distribution maps using UK census data using the mid super output layers and have managed to use the network analysis tool to find the driving route from the centre of the mid output area to the desired destination using closest facilities.

However I would like the map to 'combine' the routes once they meet, ie when two different origins meet on a main road towards the destination so I can see clearly how many people are using that road for this destination. Currently it just displays the individual routes as separate and doesn't add them to a single combined route, instead it just overlaps them.

I've managed to pair the number of people traveling from the origin to the destination in the routes but is there any way I can do this, where it shows exactly how many people are using each road to get to a destination according to the most used route?

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You could take the underlying road segments that your routes are built from and use their centroids to spatially join the routes and sum on number of people. So explore the spatial join tool.

But if you had looked at the ESRI Code sharing website you would have found the Count Coincident Lines tool...

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