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On Insensitivities in Urban Redistricting and Facility Location

This paper considers one class of problems associated with urban service systems that dispatch vehicles from fixed facilities. Given the limited resources available, one important issue is the location of the facilities and the design of their response districts to minimize average response time in the face of spatially distributed demand patterns. Calculations with spatially homogeneous demands suggest that the mean travel time resulting from totally random distribution of facilities in the region served is reduced by only 25 per cent when the facilities are optimally distributed. This apparent insensitivity of mean travel time to facility location is pursued in detail by analyzing two classes of systems involving a pair of facilities. In the first case, a procedure is outlined for determining the optimal location of a second facility, given a position for the first facility, when no interfacility cooperation is allowed. In the second case, the same region is examined, but allowing a form of interfacility cooperation. These simple models suggest that, because of insensitivities, it may not be necessary to quantize geographical data finely and then to try laboriously to find the "optimal" solutions to redistricting and facility-location problems. Redistricting and facility location based on rather crude assumptions and an awareness of some of the heuristic properties illustrated by simple analytical models may yield mean travel times very near the minimum possible.