A machine can manufacture any one of n Markov chains P$ (1 < y < n); the P$ are defined on the space of all sequences x = (x(m)} (1 < m < oo) and are absolutely continuous (in finite times) with respect to one another. It is assumed that chains P%> evolve in a random way, dictated by a Markov chain 9(m) with n states, so that when 0(m) =j the machine is producing Pp. One observes the o-fields of x(m) in order to determine when to inspect 0(m). With each product P$ there is associated a cost e,.

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... One inspects 8 at a sequence of times (each inspection entails a certain cost) and stops production when the state 9 = n is reached. The problem is to find an optimal sequence of inspections. This problem is reduced, in this paper, to solving a certain free boundary problem. In case n = 2 the latter problem is solved.