Relaxation method for the solution of the minimax location-allocation problem in euclidean space

Abstract

A method previously devised for the solution of the p‐center problem on a network has now been extended to solve the analogous minimax location‐allocation problem in continuous space. The essence of the method is that we choose a subset of the n points to be served and consider the circles based on one, two, or three points. Using a set‐covering algorithm we find a set of p such circles which cover the points in the relaxed problem (the one with m < n points). If this is possible, we check whether the n original points are covered by the solution; if so, we have a feasible solution to the problem. We now delete the largest circle with radius rp (which is currently an upper limit to the optimal solution) and try to find a better feasible solution. If we have a feasible solution to the relaxed problem which is not feasible to the original, we augment the relaxed problem by adding a point, preferably the one which is farthest from its nearest center. If we have a feasible solution to the original problem and we delete the largest circle and find that the relaxed problem cannot be covered by p circles, we conclude that the latest feasible solution to the original problem is optimal. An example of the solution of a problem with ten demand points and two and three service points is given in some detail. Computational data for problems of 30 demand points and 1–30 service points, and 100, 200, and 300 demand points and 1–3 service points are reported.

The conditionalp-center problem in the plane

“…The method reported in this article for conditional location is an extension of a previous work by the same authors for the unconditional problem [4]. This, in turn, is an adaptation of a relaxtion method given by Handler and Mirchandani [13] for the analogous problem in networks.…”

“…In the present work we describe an algorithm yielding optimal solutions. The complexity of this algorithm is identical to that of its predecssor [4] for the unconditional problem, and in practice, it is usually more efficient, a distinct advantage over the method suggested in [lo]. Before proceeding with the description of the algorithm, we present mathematical formulations of the unconditional and conditional p-center problems.…”

A particle swarm optimization algorithm for the continuous absolute p-center location problem with Euclidean distance

“…Using Euclidean distances in the plane/space, this problem is equivalent to finding the center of the smallest circle enclosing all points, hence the term "center" regarding this problem [1]. According to [9], usually, the utilization of minimax criterion arises when location of emergency facilities is considered. The facilities will be located in such a way that the response time to the farthest customer will be minimal.…”

“…For example, as presented in [9], suppose we need to find 3 centers for the 10 demand points represented as [♦] in blue in Figure 1. Therefore, the continuous Euclidean p-center location problem searches for the optimal location of 3 (p) points (centers) within the problem space in such a way that the maximum distance from these 3 centers to n demand points is minimum than any other 3 points in the space.…”

Aerospace technical support for DARPA Network modeling and simulation program and cognitive networks

“…It is shown in [11] and [12] that the p-center problem is NPhard (nondeterministic polynomial hard). Exact solutions are provided by [13], [14], [15], [16] all address the Euclidean distance p-center problem but involve rather inefficient schemes that do not scale to larger problems. There are also several heuristic methods for the p-center problem that are based on the assumption that the single facility location problem can be efficiently solved.…”