Sociedad de Estadr e lnvestigaci6n Operativa Top (1998) Vol. 6, No. 2, pp. 179-194 Polynomial Algorithms for Parametric Minquantile and Maxcovering Planar Location Problems with Locational Constraints Emilio Carrizosa Departamento de Estadlstica e 1.0. Universidad de Sevilla Tarfia s/n, 41012 Sevilla, Spain ecarriz@cica.es Frank Plastria Center ]or Industrial Location Vrije Universiteit Brussel Pieinlaan, 2, 1050 Brussels, Belgium Frank. Piastria @vub.ac.be Abstract A location is sought within some convex region of the plane for the central site of some public service to a finite number of demand points. The parametric max- covering problem consists in finding for each R > 0 the point from which the total weight of the demand points within distance R is maximal. The parametric mini- mal quantile problem asks for each percentage a the point minimising the distance necessary for covering demand points of total weight at least a. We investigate the properties of these two closely related problems and derive polynomial algorithms to solve them both in case of either (possibly inflated) Euclidean or polyhedral distances. Key Words: maximal covering, minimal quantile, single facility location, Eu- clidean distance, polyhedral distance, sensitivity analysis AMS subject classification: 90B85, 90C31. 1 Introduction Let A = {al,... , ap} C R 2 be a set of demand points, requiring for a cer- tain public service, the location of which is to be determined within a closed convex set X. Due to the public nature of the service, the minimisation of the maximum distance seems to be a plausible criterion. However, if some The research of the first author is partially supported by Grant PB96-1416-C02-02 of Ministerio de Educaci6n y Cultura, Spain Received: December 1997; Accepted: November 1998