OPTIMAL SEARCH FOR AN OBJECT WITH A RANDOM LIFETIME
Teruhisa Nakai · Journal of the Operations Research Society of Japan · 1982
One stationary object is in one of n boxes with the distribution . Let F_i(t) be the distribution of the lifetime of the object in box i and suppose that F_i(t) is composed of two probability masses α_i at t = 0, β_i at t = ∞ and a probability density function f_i(t) on the interval (0, ∞) which is differentiable in t almost everywhere. Let c be the search cost per unit time. If the object is in box i and box i is searched for t hours, the object is detected with probability 1-exp(-λ_it) whether it is alive or not. We suppose that the search is continued until the object is detected whether it is alive or not. If the searcher detects the living object in box i, he obtains a reward r_i(>0). If the searcher detects the died object, no reward is obtained. The criterion is to maximize the expected return (reward minus cost) until the object is detected. We obtain necessary and sufficient conditions for a policy to be optimal. Furthermore we obtain the optimal search rate in the implicit form. Specially we obtain the optimal search policy. explicitly in the case that f_i(t) (i = 1,…, n) are differentiable in t. We consider two numerical examples and give the explicit solutions. One of them is the case of the exponential lifetime distribution and another is the case of the uniform lifetime distribution. Finally we deal with stopping problem in which the searcher is permitted to stop the. search at any time. Some results about the optimal search policy and the optimal stopping time are obtained.