Continued fractions by C. D Olds

By C. D Olds

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By the same token, define the cost of the search as C = K × T. 1 of constrained search, it is required to find the strategy w* specifying the observation numbers κ* (aj,T) such that the probability p(w*,T) is maximal and the cost is constrained as © 2015 by Taylor & Francis Group, LLC C ( w*, T ) ≤ C. 48)  κ * (a j , T ) t (a j )  ,  j =2  m ∑ where n j are integer valued functions, which define the numbers of observations of the areas a j, j = 1, 2,…, m, given the available cost. 1. Finally, assume that the probability of detecting the target in the area a j, j = 1, 2,…, m, does not depend on the observation number; in other words, similarly to the sequential search, the observations do not add information about the target’s location and are defined as 1 − ε j = φ(a j, 1), where ε j is an overlook probability of nondetecting the target in the area aj given that it is somewhere in aj.

X Since q(w, t) is the probability of nondetecting the target, the probability of detecting the target during the period [0, t] using the strategy w is defined as p ( w, t ) = 1− q ( w, t ) . 4) t ∫ w ( x, τ) dτ applied in the point x up to the 0 q ( w, T ) = u ( x ) exp −κ ( x, T ) d x and p ( w, T ) = 1 − q ( w, T ) . 5) x Additional condition, which represents a “conversation” of search efforts, is specified as follows: T © 2015 by Taylor & Francis Group, LLC ∫κ ( x, T ) dx = ∫K ( t ) dt. 6) 11 Methods of Optimal Search and Screening Thus, according to the goal of the searcher, it is required to find such search strategy w that maximizes the probability p(w, t) or minimizes the probability q(w, t) of nondetecting the target.

Using the detection function φ, the probability of detecting the target during the period 0,t ) using the search strategy w is defines as follows: ∫ t ∫ p ( w, t ) = u ( x ) ϕ ( x, κ ) d x, κ ( x, t ) = w ( x, τ ) dτ. 9) 0 Let W be a set of all possible search strategies w over the domain X given the target’s probability distribution u ( x ) , x ∈ X. Then, according to the goal of the searcher, it is required to find a search strategy w* ∈ W such that the probability p(w*, T) is maximal over all strategies w ∈ W.

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