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

DOI:
https://doi.org/10.1090/S0002-9947-1978-0515529-8

MathSciNet review:
515529

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

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