Equivalent Mathematical Programming Formulations of Monotonic Tree Network Location Problems

We consider the optimization problem of locating several new facilities on a tree network, with respect to existing facilities, and to each other. The new facilities are not restricted to be at vertices of the network, but the locations are subject to constraints. Each constraint function, and the objective function, is an arbitrary, nondecreasing function of any finite collection of tree distances between new and existing facilities, and/or between distinct pairs of new facilities, and represents some sort of transport or travel cost. The new facilities are to be located so as to minimize the objective function subject to upper bounds on the constraint functions. We show that such problems are equivalent to mathematical programming problems which, when each function is expressed using only maximization and summation operations on nonnegatively weighted arguments, are linear programming problems of polynomial dimensions. The latter problems can be solved using duality theory with special purpose column generation and shortest path algorithms for column pricing.