A direct solution of allocation problems

Abstract

This paper deals with tree-structured networks that interconnect n given points A 1 ,..., A n in the plane through up to ( n – 2) auxiliary junction points S 1 ,..., S n − 2 . The problem of locating these auxiliary vertices is tackled to minimize the total cost of the connection network. The formulation of the problem is not bound by the condition that specific link costs are equal for all branches, in order to comprise a broader range of applications, such as distribution networks. Starting from a well-known mechanical analogy, the authors arrive at some mathematical expressions and conditions of their applicability that allow calculation of the optimal coordinates of the unknown S i points directly without resorting to the usual iterative procedures. The method is illustrated for the case of n = 3 points and its extension to more general cases is presented. The formulas obtained also apply to the particular case of determining Steiner minimal trees, specific link cost being constant. An example of application to electric distribution networks is also given.