The conditionalp-center problem in the plane

Abstract

An algorithm is given for the conditional p-center problem, namely, the optimal location of one or more additional facilities in a region with given demand points and one or more preexisting facilities. The solution dealt with here involves the minimax criterion and Euclidean distances in two-dimensional space. The method used is a generalization to the present conditional case of a relaxation method previously developed for the unconditional p-center problems. Interestingly, its worst-case complexity is identical to that of the unconditional version, and in practice, the conditional algorithm is more efficient. Some test problems with up to 200 demand points have been solved. 0 1993 John Wiley & Sons, Inc.Location-allocation problems are those in which a number of service facilities, usually (as here) assumed identical, are to be optimally located to serve a number of given demand points. Our concern here is with the case in which the underlying universe for location and travel is the two-dimensional Euclidean space. The two major versions of the problem studied so far are the minisum problem in which the minimand is the sum of weighted distances of the demand points to their closest service facilities, and the minimax problem in which one wishes to minimize the maximum distance from any demand point to its closest center. The minisum problem was first suggested and treated by Cooper [6], and later dealt with by several investigators [3, 11,18, 211. The subject of this article, on the other hand, is related to the minimax problem, known also as the p-center problem. Common applications for this model include emergency services and communication centers. It has been discussed by a number of authors in recent years [3, 8, 9, 15, 22, 231. In a previous work by the present authors [4], a relaxation method for the minimax problem has been found to work well for fairly large problems.The location problems mentioned above deal with the simultaneous optimal location of a number of identical (uncapacitated) service centers, and the allocation of each of the demand points to its closest center. A problem which may be of no less practical significance is the conditional location problem where a

On the Continuous Fermat-Weber Problem

“…Choice of Metric. Many papers on geometric location theory have dealt with continuous sets F of feasible placements, including [2,5,13,14,16,21,37,38,39,62,63,64]. In the majority of these papers, distances are measured according to the L 1 metric.…”

Conditional location of path and tree shaped facilities on trees

“…There are also studies in the literature dealing with the point-conditional location problem in the plane. (See for example, [5,6]. )…”

The multi-period $p$-center problem with time-dependent travel times

“…The results have indicated that the algorithm can consistently generate solutions with a limited number of relocations that improve on the corresponding time-invariant optimal solutions. Future work will focus on adapting the ideas of this paper to other variants of the p-center problem, such as the capacitated p-CP (Kramer et al, 2018), the weighted p-CP (Chen and Handler, 1993), the conditional p-CP (Berman and Drezner, 2008), the α-neighbor p-CP (Chen and Chen, 2013).…”