Locating a Central Hunter on the Plane

Abstract

Protection, surveillance or other types of coverage services of mobile points call for different, asymmetric distance measures than the traditional Euclidean, rectangular or other norms used for fixed points. In this paper, the destinations are mobile points (prey) moving at fixed speeds and directions and the facility (hunter) can capture them using one of two possible strategies: either it is smart, predicting the prey's movement in order to minimize the time needed to capture it, or it is dumb, following a pursuit curve, by moving at any moment in the direction of the prey. In either case, the hunter location in a plane is sought in order to minimize the maximum time of capture of any prey. An efficient solution algorithm is developed that uses the particular geometry that both versions of this problem possess. In the case of unpre-dictable movement of prey, a worst-case type solution is proposed, which reduces to the well-known weighted Euclidean minimax location problem.

A variational inequality approach for constrained multifacility Weber problem under gauge

“…We first test CMFWP under Euclidean distances with n = 50, 100, 200, 500, 1000 and m = 2, 4, 6, 8, 10. All customers are randomly generated in [−250, 250] 2 ; the weights between customers and facilities are randomly chosen in [1,3] and the weights between facilities are chosen in [1,5]. The locational constraints are set as…”

“…In spite of the pioneering work of [33], the asymmetric distance has started to attract the researchers' interest until the recent several decades, and some progress has been made in both its theoretical and computational aspects recently, see e.g. [2,3,7,15,22,24].…”

Minmax-distance approximation and separation problems: geometrical properties

“…Interesting instances of gauges which are not norms are, for instance, the skewed gauges, used in Statistics to define a multivariate version of quantiles, [8], and also suggested in Location Theory to model transportation on an inclined plane, in presence of steady wind, [30], or to a moving target [7]. Skewed gauges have the form…”

Asymmetric distances, semidirected networks and majority in Fermat–Weber problems

“…Plastria (1992b) introduced skewed norms as a particular class of gauges, those with centrally symmetric dual unit ball (however, not necessarily with the origin as symmetry centre), or, equivalently, of the form γ (x) = N(x) + x; p , where N is a norm, p a fixed vector with dual norm less than 1, N • (p) < 1. This class of gauges encompasses the cases of movement on an inclined plane (Hodgson et al 1987), movement in presence of a fixed wind or current (Plastria 1992b), time needed for capturing a fleeing prey by a dumb or a smart hunter (Cera and Ortega 2002;Cera et al 2008) and the proposals of Drezner and Wesolowsky (1989) for asymmetric distances.…”

“…For at least two decades no further work on this topic seems to have been published, and it is only in recent years that interest in asymmetric distance problems has been revived by some location researchers (see Durier and Michelot 1985;Hodgson et al 1987;Michelot and Lefebvre 1987;Idrissi et al 1988Idrissi et al , 1989Drezner and Wesolowsky 1989;Durier 1990;Chen 1991;Plastria 1992b;Buchanan and Wesolowsky 1993;Fliege 1994Fliege , 1997Fliege , 1998Fliege , 2000Plastria 1994;Nickel 1998;Cera and Ortega 2002;Cera et al 2008). All of these contributions concern continuous problems with distances derived from gauges, the asymmetric extensions of norms (see Sect.…”