Algebraic properties of location problems with one circular barrier

The consideration of barriers to travel plays an increasingly important role in the transportation and location literature. In one of the classical papers on location problems with barriers, Katz and Cooper (1981) considered the Weber problem (often also referred to as median problem) with one circular barrier region. Considering the same problem we develop new structural results showing that the set of feasible solutions can be subdivided into a polynomial number of cells of algebraic invariance, on every convex subset of which the generally non-convex objective function is convex. These results imply improved exact and heuristic solution procedures based on convex optimization methods.