# Optimal placement of a shape on a polygon layer based on an attribute

I am using ArcGIS and I have a layer of polygons that has an attribute. I would like to overlay a circle (or other shapes in theory) of a specified size such that it would encompass the maximum amount of the attribute from the polygons. For example, placing a circle with a diameter of 5 miles over census tracts (that have a population density attribute) so that the circle will encompass the maximum population possible.

I understand that preferably the shape I'm overlaying will be large enough not to get stuck in the middle of a single polygon. I also realize that calculating a maximizing function might not be entirely straightforward as there will be partial overlaps.

I imagine that there is something already out there that does this but I'm not having any luck with searching. I don't know what this sort of analysis would be called either so that doesn't help.

If ArcGIS isn't the appropriate tool for this I'm open to other suggestions. Thanks in advance for any help you can provide!

• Would it be possible to convert the polys to centroids and use the attributes as part of a cartographic representation formula? – MLowry Aug 3 '11 at 20:14
• Maybe, how would you go about it? – mga Aug 5 '11 at 4:39

## 1 Answer

Because you contemplate partial overlaps, this is a "continuous" problem that is best solved with a raster algorithm. Therefore, first convert the polygons to a grid. Use the attribute densities for the grid values: that is, divide the attributes by the polygon areas if the attributes are given as total counts.

Compute a focal sum (or focal mean) using a circular neighborhood with 5 mile radius (or any other neighborhood shape you like). Any cell where the focal sum attains its highest value is the center of an optimal circle.

• Focal sum (and mean) results in NULL data for all points 80% of the time. When it works though, it solves the problem I was asking. This is a great start for me. Thanks for your help! – mga Aug 5 '11 at 4:44
• Focalsum (in ESRI products) works in two ways: by default, if a NoData cell lies in the neighborhood, the result is NoData. Alternatively--you can specify this--NoData cells are ignored. This is the ignore nodata option. – whuber Aug 5 '11 at 14:31