Control Charts and the Minimization of Costs

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.