This paper addresses the problem of searching for a located target in the plane by using one searcher starting its motion from the point . The searcher moves along parabolic spiral curve. The position of the target has a known distribution. We show that the distance between the target position and the searcher starting point depends on the number of revolutions, where the complete revolution is done when . Furthermore, we study this technique in the one-dimensional case (i.e., when the searcher moves with linear search technique). It is desired to get the expected value of the time for detecting the target. Illustrative examples are given to demonstrate the applicability of this technique assuming circular normal distributed estimates of the target position. 1. Introduction Search problem dates back to World War II, and the works of Koopman [1–3] and Stone [5, 6] offer a classic treatise of this area from an operational research perspective. Many variants and extensions of this problem, in a wide variety of directions, have been presented in both the statistical and operations research literature since Koopman [1] solved this problem for the unidimensional case. This problem has been discussed under some specific hypotheses by Richardson [4]. However, as pointed out by Koopman [2, 3], there is so much complexity in real search and rescue missions that any statistical model can only reflect part of the real-life situation and ours are no exception. On search theory in general, Stone [5] has given a good account of various results presently available, with some informative examples, and also has provided a rigorous mathematical treatment of the subject, for both discrete and continuous cases. On the other hand, Stone [6] provided an overview of different areas in the development of search theory, which could be designated as classical, mathematical, algorithmic, and dynamic. Also, exhaustive surveys of works realized on this topic have been given by Iida [7] and Benkoski et al. [8]. Studying of searching problem in the plane had found considerable interest among researchers due to its wide applications in our life. Feinerman et al. [9] introduced some search trajectories to find the desert Ants Nearby Treasure. They used multiple searchers without communication between them in the plane. Edelsbrunner and Maurer [10] obtained the optimal solutions for the postoffice problem. The certain point location problems in two dimensions are derived via geometric transforms from an optimal solution for the search problem in three dimensions: find the first point hit
References
[1]
B. O. Koopman, “The theory of search—III. The optimum distribution of search effort,” Operations Research, vol. 5, pp. 613–626, 1957.
[2]
B. O. Koopman, “Search and its optimization,” The American Mathematical Monthly, vol. 86, no. 7, pp. 527–540, 1979.
[3]
B. O. Koopman, “An operational critique of detection laws,” Operations Research, vol. 27, no. 1, pp. 115–133, 1979.
[4]
H. Richardson, “Search theory,” in Encyclopedia of Statistics, N. Johnson and S. Kotz, Eds., vol. 19, pp. 320–315, 1984.
[5]
L. Stone, Theory of Optimal Search, Military Applications Section, Operation Research Society of America, Arlington, Va, USA, 2nd edition, 1989.
[6]
L. D. Stone, “What's happened in search theory since the 1975 Lanchester Prize?” Operations Research, vol. 37, no. 3, pp. 501–506, 1989.
[7]
K. Iida, Studies on the Optimal Search Plan, vol. 70 of Lecture Notes in Statistics, Springer, New York, NY, USA, 1992.
[8]
S. Benkoski, M. Monticino, and J. Weisinger, “A survey of the search theory literature,” Naval Research Logistics, vol. 38, pp. 469–494, 1991.
[9]
O. Feinerman, A. Korman, Z. Lotker, and J. Sereni, “Collaborative search on the plane without communication,” in Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC '12), pp. 77–86, ACM, New York, NY, USA, 2012.
[10]
H. Edelsbrunner and H. A. Maurer, “Finding extreme points in three dimensions and solving the post-office problem in the plane,” Information Processing Letters, vol. 21, no. 1, pp. 39–47, 1985.
[11]
A. Mohamed, H. Abou Gabal, and M. El-Hadidy, “Coordinated search for a randomly located target on the plane,” European Journal of Pure and Applied Mathematics, vol. 2, no. 1, pp. 97–111, 2009.
[12]
A. Mohamed, H. Fergany, and M. El-Hadidy, “On the coordinated search problem on the plane,” Istanbul University Journal of the School of Business Administration, vol. 41, pp. 80–102, 2012.
[13]
M. El-Hadidy, “Optimal spiral search plan for a randomly located target in the plane,” International Journal of Operational Research, 2013.
[14]
A. Mohamed and M. El-Hadidy, “Coordinated search for a conditionally deterministic target motion in the plane,” European Journal of Mathematical Sciences, vol. 2, no. 3, pp. 272–295, 2013.
[15]
A. Mohamed and M. El-Hadidy, “Existence of a periodic search strategy for a parabolic spiral target motion in the plane,” Afrika Matematika, vol. 24, no. 2, pp. 145–160, 2013.
[16]
M. El-Hadidy, “Optimal searching for a helix target motion,” submitted for publication.
L. Stone, C. Keller, T. Kratzke, and J. Strumpfer, Search Analysis for the Location of the AF447 Underwater Wreckage, METRON, report to BEA, 2011.
[19]
A. Beck and M. Beck, “The linear search problem rides again,” Israel Journal of Mathematics, vol. 53, no. 3, pp. 365–372, 1986.
[20]
A. Beck and M. Beck, “The revenge of the linear search problem,” SIAM Journal on Control and Optimization, vol. 30, no. 1, pp. 112–122, 1992.
[21]
D. J. Reyniers, “Coordinated search for an object hidden on the line,” European Journal of Operational Research, vol. 95, no. 3, pp. 663–670, 1996.
[22]
D. J. Reyniers, “Co-ordinating two searchers for an object hidden on an interval,” Journal of the Operational Research Society, vol. 46, no. 11, pp. 1386–1392, 1995.
[23]
Z. T. Balkhi, “The generalized linear search problem, existence of optimal search paths,” Journal of the Operations Research Society of Japan, vol. 30, no. 4, pp. 399–421, 1987.
[24]
Z. T. Balkhi, “Generalized optimal search paths for continuous univariate random variables,” Operations Research, vol. 23, no. 1, pp. 67–96, 1987.
[25]
A. Mohamed, “The generalized search for one dimensional random Walker,” International Journal of Pure and Applied Mathematics, vol. 19, no. 3, pp. 375–387, 2005.
[26]
A. B. El-Rayes, A. M. Abd El-Moneim, and H. M. Abou Gabal, “Linear search for a Brownian target motion,” Acta Mathematica Scientia B, vol. 23, no. 3, pp. 321–327, 2003.
[27]
A. R. Washburn, Search and Detection, Postgraduate School, Monterey, Calif, USA, 2nd edition, 1989.
[28]
L. D. Stone, Theory of Optimal Search, Academic Press, New York, NY, USA, 1975.
[29]
A. Mohamed, M. Kassem, and M. El-Hadidy, “Multiplicative linear search for a Brownian target motion,” Applied Mathematical Modelling, vol. 35, no. 9, pp. 4127–4239, 2011.
[30]
A. Mohamed and M. El-Hadidy, “Optimal multiplicative generalized linear search Plan for a discrete random walker,” Journal of Optimization, vol. 2013, Article ID 706176, 13 pages, 2013.
