The Generalized Search for a Randomly Moving Target

Abdelmoneim Anwar Mohamed Teamah

Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt.

DOI: 10.4236/am.2014.54060                

Abstract

A target is assumed to move randomly on one of two disjoint lines L₁ and L₂ according to a stochastic process . We have two searchers start looking for the lost target from some points on the two lines separately. Each of the searchers moves continuously along his line in both directions of his starting point. When the target is valuable as a person lost on one of disjoint roads, or is serious as a car filled with explosives which moves randomly in one of disjoint roads, in these cases the search effort must be unrestricted and then we can use more than one searcher. In this paper we show the existence of a search plan such that the expected value of the first meeting time between the target and one of the two searchers is minimum.

Keywords

Stochastic Process; Expected Value; Linear Search; Optimal Search Plan

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Mohamed, Abd El-M.A. (2005) Generalized Search for One Dimensional Random Walker. International Journal of Pure and Applied Mathematics, 19, 375-387.
[2]	Ohsumi, A. (1991) Optimal Search for a Markovian Target. Naval Research Logistics, 38, 531-554. http://dx.doi.org/10.1002/1520-6750(199108)38:4<531::AID-NAV3220380407>3.0.CO;2-L
[3]	El-Rayes, A.B. and Abd EL-Moneim, M.A. (1989) Searching for a Randomly Moving Target. Proceeding of the Third ORMA Conference (Germany), 1, 323-329.
[4]	Washburn, A.R. (1989) Search and Detection. 2nd Edition, ORSA Books, Arlington.
[5]	El-Rayes, A.B., Abd-Elmoneim, M.A. and Abu Gabl, H.M. (2003) A Linear Search for a Brownian Target Motion. Acta Mathematica Scienta, 23B, 321-327.
[6]	Balkhi, Z.T. (1989) Generalized Optimal Search Paths for Continuous Univariate Random Variables. Recherche Operationnelle Operation Research, 23, 67-96.
[7]	El-Rayes, A.B. and Abd-Elmoneim, M.A. (1998) Oscillating Search for a Randomly Located. AMSE, 40, 29-40.
[8]	El-Rayes, A.B., Abd-ELmoneim, M.A. and Fergani, H. (1993) On the Generalized Linear Search Problem. Delta Journal, 6, 1-10.
[9]	Balkhi, Z.T. (1987) The Generalized Linear Search Problem, Existence of Optimal Search Paths. Journal of the Operations Research Society of Japan, 30, 399-420.
[10]	Rousseeuw, P. (1983) Optimal Search Paths for Random Variables. Journal of Computational and Applied Mathematics, 9, 279-286. http://dx.doi.org/10.1016/0377-0427(83)90020-1
[11]	Mohamed, Abd El-M.A. and AbuGabl, H.M. (2000) Generalized Optimal Search Paths for a Randomly Located Target. Annual Conference (Cairo) ISSR, Math. Statistics Part, 35, 17-29.
[12]	Stone, L.D. (1975) Theory of Optimal Search. Academic Press, New York.
[13]	Fristedt, B. and Heath, D. (1974) Searching for a Particle on the Real Line. Advances in Applied Probability, 6, 79-102. http://dx.doi.org/10.2307/1426208
[14]	Mohamed, Abd El-M.A., Kassem, M.A. and El-Hadidy, M.A. (2011) Multiplicative Linear Search for Brownian Target Motion. Applied Mathematical Modelling, 35, 4127-4139. http://dx.doi.org/10.1016/j.apm.2011.03.024
[15]	Mohamed, Abd El-M.A. and AbuGabl, H.M. (2008) Double Linear Search Problem for a Brownian Motion. Journal of the Egyptian Mathematical Society, 16, 99-107.