I'm using ArcMap 10.1. When I make an attribute based buffer, the result had shown as overlapped. How can I solve this issues?
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You could use Erase (Analysis Tools), with the wider buffered roads (lines) as the erase feature (the dominant feature). This will leave the geometry (shape) of the dominant road alone and 'erase' the portion of the smaller road that overlaps it. You may have to do multiple iterations as you have more than 2 'classes' of road.
Alternatively, if you want them in the same feature class, you could just dissolve them together.