Yesterday I was talking with a former student, whose company provides statistical advice to clients in the service industry. He mentioned an interesting paradox about optimal location, in this case of ambulances. The UK government has various standards or targets to be achieved by public sector bodies, and one of these is about how quickly ambulances can reach an emergency. It's of the form: "In X% of calls, the ambulance must reach the emergency within Y minutes".
He said that in urban areas, it is relatively easy to deploy vehicles to achieve this, because of the short distances involved. And it is also reasonably easy in regions where there are several medium to large towns and cities and a scattered rural population, because the response to the cities dominates the statistics. But the targets are hard to achieve where there are many small to medium sized towns, none of which have sufficient demand to require a vehicle of their own. Then the solution leads to an excess of vehicles.
You can test this for yourself. Suppose that the district is an equilateral triangle, and the demand is such that one vehicle is needed. If the population is spread uniformly across it, then you place your vehicle at the centre of the triangle. If the population is concentrated and divided equally at the three corners, then the best place would still be the centre, but nobody lives there, and so the vehicle should be located at one of the corners. But that means that in two cases out of three, the vehicle must travel to one of the other settlements, and potentially violate the service standard.
I haven't seen this paradox mentioned anywhere in the literature, so you may have read it here first!
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