Exact Solution of P-dispersion Problems

The p-dispersion-sum problem is the problem of locating p facilities at some of n prede-ned locations, such that the distance sum between the p facilities is maximized. The problem has applications in telecommunication (where it is desirable to disperse the transceivers in order to minimize interference problems), and in location of shops and service-stations (where the mutual competition should be minimized). Simple upper bounds for the problem are presented, and it is shown how these bounds can be tightened through a reformulation scheme which runs in O(n 3) time. A branch-and-bound algorithm is then derived, which at each branching node is able to derive the upper bounds in O(n) time. Computational experiments show that the algorithm may solve geometric problems of size up to n = 80, and weighted geometric problems of size n = 200. The related p-dispersion problem is the problem of locating p facilities such that the minimum distance between two facilities is as large as possible. Formulations and simple upper bounds are presented, and it is discussed whether a similar framework as for the p-dispersion sum problem can be used to tighten the upper bounds. A solution algorithm based on transformation of the p-dispersion problem to the p-dispersion-sum problem is nally presented, and its performance is evaluated through several computational experiments .