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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quality control for Markov chains and free boundary problems

Authors: Robert F. Anderson and Avner Friedman
Journal: Trans. Amer. Math. Soc. 246 (1978), 77-94
MSC: Primary 93E20; Secondary 49A29, 62N10
MathSciNet review: 515530
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Abstract: A machine can manufacture any one of n Markov chains $ P_x^{{\lambda _j}}\,\left( {1\, \leq \,j\, \leq \,n} \right)$; the $ P_x^{{\lambda _j}}$ are defined on the space of all sequences $ x\, = \,\left\{ {x\left( m \right)} \right\}\,\left( {1\, \leq \,m\, \leq \,\infty } \right)$ and are absolutely continuous (in finite times) with respect to one another. It is assumed that chains $ P_x^{{\lambda _j}}$ evolve in a random way, dictated by a Markov chain $ \theta \left( m \right)$ with n states, so that when $ \theta \left( m \right)\, = \,j$ the machine is producing $ P_x^{{\lambda _j}}$. One observes the $ \sigma $-fields of $ x\left( m \right)$ in order to determine when to inspect $ \theta \left( m \right)$. With each product $ P_x^{{\lambda _j}}$ there is associated a cost $ {c_j}$. One inspects $ \theta $ at a sequence of times (each inspection entails a certain cost) and stops production when the state $ \theta \, = \,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.

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Keywords: Markov chain, quality control, stopping time, inspection time, quasi variational inequality
Article copyright: © Copyright 1978 American Mathematical Society

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