Pergamon Locafim Science, Vol. 5, No. 3, 147-163 1997 pp. 0 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0966-8349/9&l $19.00 -1 0.00 PII:SO966-8349(98)00032-l THE SINGLE FACILITY LOCATION PROBLEM WITH MINIMUM DISTANCE CONSTRAINTS YAEL KONFORTY and ARIE TAMIR* Department of Statistics and Operations Research, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978. Israel (Revised version received 10 November 1997) Abstract-We consider the problem of locating a single facility (server) in the plane, where the location of the facility is restricted to be outside a specified forbidden region (neighborhood) around each demand point. Two models are discussed. In the restricted l-median model, the objective is to minimize the sum of the weighted rectilinear distances from the n customers to the facility. We present an O(n log n) algorithm for this model, improving upon the O(n”) complexity bound of the algorithm by Brimberg and Wesolowsky (1995). In the restricted l-center model the objective is to minimize the maximum of the weighted rectilinear distances between the customers and the serving facility. We present an O(n logn) algorithm for finding an optimal i-center. We also discuss some related models, involving the Euclidean norm. 0 1998 Elsevier Science Ltd. All rights reserved Key words: Planar location problems, undesirable facilities, location with forbidden regions, center and median problems. 1. INTRODUCTION One of the most active research areas within location theory in recent years has been the location of undesirable or obnoxious facilities. Facilities of this nature, such as nuclear reactors, garbage depots and chemical plants, may cause lower quality of life or even pose a severe danger to people living nearby. The location models that are mostly used in this context are those which maximize a weighted sum of the distances between the facilities and the demand points, (maxisum criterion), or the minimum of the distances between the obnoxious facilities and the customers, (maximin criterion); see Erkut and Neuman (1985) for a literature review. These models focus only on the costs associated with the obnoxious nature of the facilities, the ‘social’ costs, which decrease with the distances from the facilities, but ignore the service or transportation costs, which usually increase with these distances. For example, in the case of electrical power plants the costs of transmission (including loss of power) are not represented by the above criteria. The recent paper by Brimberg and Wesolowsky (1995) present a different approach. They describe a model for locating a single facility (server) in the plane, which considers transport- ation or service costs between the facility and a given set of n demand points, as well as social costs arising from the undesirable characteristics of the facility. Their model has a standard minisum objective to minimize transportation costs. These costs are assumed to be linearly dependent on the rectilinear distances between the facility and the demand points. The social costs are included implicitly in the form of constraints forcing the location of the *To whom correspondence should be addressed. 147