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 unpredictable movement of prey, a worst-case type solution is proposed, which reduces to the well-known weighted Euclidean minimax location problem.
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References
Plastria, F.: Continuous covering location problems. In: Hamacher, H., Drezner, Z. (eds.) Location Analysis: Theory and Applications, pp. 39–83. Springer, Berlin (2001)
Barton, J.C., Eliezer, C.J.: Pursuit curves II. Bull. Inst. Math. Appl. 31, 139–141 (1995)
Barton, J.C., Eliezer, C.J.: On pursuit curves. J. Aust. Math. Soc. 41, 358–371 (2000)
Cera, M., Ortega, F.A.: Locating the median hunter among n mobile preys on the plane. Int. J. Ind. Eng.: Theory Appl. Pract. 9, 6–15 (2002)
Plastria, F.: On destination optimality in asymmetric distance Fermat–Weber problems. Ann. Oper. Res. 40, 355–369 (1992)
Pelegrín, B., Michelot, C., Plastria, F.: On the uniqueness of optimal solutions in continuous location theory. Eur. J. Operat. Res. 20, 327–331 (1985)
Drezner, Z.: On minmax optimization problems. Math. Program. 22, 227–230 (1982)
Elzinga, D.J., Hearn, D.W.: Geometrical solutions for some minimax location problems. Transp. Sci. 6, 379–394 (1972)
Charalambous, C.: Extension of the Elzinga–Hearn algorithm to the weighted case. Operat. Res. 30, 591–594 (1982)
Hearn, D.W., Vijay, J.: Efficient algorithms for the (weighted) minimum circle problem. Operat. Res. 30, 777–795 (1982)
Welzl, E.: Smallest enclosing disk (balls and ellipsoids), new results and new trends in computer science. In: Maurer, H. (ed.) Lecture Notes in Computer Science, vol. 555, pp. 359–370. Springer, New York (1991)
Plastria, F.: Fully geometric solutions to some planar center minimax location problems. Stud. Locat. Analysis 7, 171–183 (1994)
Chrystal, G.: On the problem to construct the minimum circle enclosing n given points in the plane. Proc. Edinb. Math. Soc. 3, 30–33 (1885)
Shamos, M.I., Hoey, D.: Closest point problems. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science, pp. 151–162 (1975)
Megiddo, N.: The weighted Euclidean 1—center problem. Math. Operat. Res. 8, 498–504 (1983)
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Communicated by J.P. Crouzeix.
The work of the second and third authors was supported in part by a grant from Research Projects BFM2003-04062 and MTM2006-15054.
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Cera, M., Mesa, J.A., Ortega, F.A. et al. Locating a Central Hunter on the Plane. J Optim Theory Appl 136, 155–166 (2008). https://doi.org/10.1007/s10957-007-9293-y
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DOI: https://doi.org/10.1007/s10957-007-9293-y