Coincidence conditions in multifacility location problems with positive and negative weights

In minisum multifacilty location problems one has to find locations for some new facilities, such that the weighted sum of distances between the new and a certain number of old facilities with known locations is minimized. In this kind of problem, the optimal locations of clusters of facilities frequently tend to coincide. By testing conditions for coincidence, one has the opportunity to collapse some or even all facilities coinciding at an optimal point into one. In this way, the dimension of the problem and the degree of nondifferentiability is reduced. Several conditions for coincidence have been published recently. In this paper, these conditions are extended and improved with respect to new sufficient coincidence conditions for location problems with attracting and repelling facilities. An example shows that these new conditions detect more coincidences than the conditions which are known so far, even if all facilities involved are attracting ones.