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.

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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