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Transactions of the American Mathematical Society

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Multidimensional quality control problems and quasivariational inequalities

Authors: Robert F. Anderson and Avner Friedman
Journal: Trans. Amer. Math. Soc. 246 (1978), 31-76
MSC: Primary 93E20; Secondary 49A29, 62N10
MathSciNet review: 515529
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Abstract: A machine can manufacture any one of n m-dimensional Brownian motions with drift $ {\lambda _j}$, $ P_x^{{\lambda _j}}$, defined on the space of all paths $ x\left( t \right)\, \in \,C\left( {\left[ {0,\,\infty } \right);\,{R^m}} \right)$. It is given that the product is a random evolution dictated by a Markov process $ \theta \left( t \right)$ with n states, and that the product is $ P_x^{{\lambda _j}}$ when $ \theta \left( t \right)\, = \,j,\,1\, \leqslant \,j\, \leqslant \,n$. One observes the $ \sigma $-fields of $ x\left( t \right)$, but not of $ \theta \left( t \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 to solving a certain elliptic quasi variational inequality. The latter problem is actually solved in a rather general case.

References [Enhancements On Off] (What's this?)

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Keywords: Brownian motion, random evolution, Markov process, stopping time, optimal sequence of inspections, quality control, quasi variational inequality
Article copyright: © Copyright 1978 American Mathematical Society

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