Control Charts and the Minimization of Costs

Abstract

Summary The behaviour of a machine or manufacturing process is represented by a simple model, in which two types of cost are considered: running costs and overhaul costs. Both types depend on the performance level of the machine, which changes stochastically and is tracked by means of a sequence of observations. It is assumed that the machine will run for an infinite period, interrupted by recurrent overhauls in which its performance level is modified. At any instant, it must be decided on the basis of the given observations whether or not an overhaul would be advantageous. The problem of finding an optimal decision procedure is described in terms of a control chart, and a certain functional equation is obtained, which relates the expected cost in any position and the decision procedure. Methods of approximation are considered and this leads to a reformulation of the problem in continuous time, which can be solved explicitly.

Parameter estimation in first-order autoregressive model for statistical process monitoring in the presence of data autocorrelation

“…On the other hand, the upper-boundary case of r=1 is essentially the same as the model of discrete Brownian motion for the quality characteristic employed by Bather (1963).…”

“…The paper by Bather (1963) or Kirkendall (1986) did not explicitly discuss the estimation of the parameters for the steady model, but the papers by Phadke (1982), Crowder (1986) and Crowder and Eshleman (2001) addressed this problem using variants of maximum likelihood approach. Generally, the model parameter estimation methods may be grouped into two broad categories as follows.…”

“…Use of adaptive Kalman filtering in statistical process control, in regard to the estimation of the 'unobservables', is reasonably well established. A very early application appeared in Bather (1963), who employed the steady model to arrive at a special form of Kalman Filter procedure. Its use in the area was proposed formally by Phadke (1982), who considered a random walk model which is a slightly more general version of the steady model.…”

“…Two sub-cases have been considered in detail in the literature; (i) the case when r=1, which is also known as the steady model (see e.g. Bather, 1963;Harrison, 1967), and (ii) the case when 0or o1 which is used as a model for statistical process control (SPC) in the presence of data autocorrelation (see e.g. Vasilopoulos and Stamboulis, 1978;Lu and Reynolds, 1999).…”

Empirical Bayes detection of a change in distribution

“…230 ROHANA J. KARUNAMUNI AND SHUNPU ZHANG paper is focused on Bayes sequential methods. Bayes stopping rules for the present problem were studied by Shiryayev (1963aShiryayev ( , 1963bShiryayev ( , 1978 and Bather (1963Bather ( , 1967, and later developed by Zaeks and Barzily (1981), Zacks (1983), Pollak (1985) and others.…”