When facilities coincide: Exact optimality conditions in multifacility location

Abstract In continuous minisum multifacility location problems it has been observed that optimal solutions often involve coincidences of facilities. This paper gives a complete analysis by way of subdifferential calculus of the exact optimality conditions at such solutions. These conditions are expressed as the existence of a multidimensional conservative flow in a network, satisfying a nonlinear constraint on each edge. By way of network flow theory a description with a minimal number of variables is obtained. This procedure is illustrated by several examples, showing how it enables a sensitivity analysis of the optimal solution with respect to the weights of the problem. The optimality conditions are shown to be always directly verifiable if the network is a tree. Necessary conditions for coincidences at an optimal solution, which previously appeared in the literature, are generalized, and it is shown in which cases these are also sufficient.

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