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Optimal Search for a Moving Target

  • I. M. MacPhee (a1) and B. P. Jordan (a1)

Abstract

Consider the problem of searching for a leprechaun that moves randomly between two sites. The movement is modelled with a two-state Markov chain. One of the sites is searched at each time t = 1,2,…, until the leprechaun is found. Associated with each search of site i is an overlook probability αi and a cost Ci Our aim is to determine the policy that will find the leprechaun with the minimal average cost. Let p denote the probability that the leprechaun is at site 1. Ross conjectured that an optimal policy can be defined in terms of a threshold probability P* such that site 1 is searched if and only if pP*. We show this conjecture to be correct (i) when α1 = α2 and C1 = C2, (ii) for general Ci when the overlook probabilities α, are small, and (iii) for general αi and Ci for a large range of transition laws for the movement. We also derive some properties of the optimal policy for the problem on n sites in the no-overlook case and for the case where each site has the same αi, and Ci.

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References

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1.Ahlswede, R. & Wegener, I. (1979). Search problems. Salisbury: John Wiley.
2.Assaf, D. & Sharlin-Bilitzky, A. (1994). Dynamic search for a moving target. Journal of Applied Probability 31: 438457.
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6.Ross, S.M. (1983). Introduction to stochastic dynamic programming. New York: Academic Press.
7.Weber, R.R. (1986). Optimal search for a randomly moving object. Journal of Applied Probability 23: 708717.
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Probability in the Engineering and Informational Sciences
  • ISSN: 0269-9648
  • EISSN: 1469-8951
  • URL: /core/journals/probability-in-the-engineering-and-informational-sciences
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