Nonlinear filtering and optimal quality control

Some stochastic models of optimal decision processes in quality control problems are formulated, analyzed, and solved. It is assumed that costs, positive or negative, are assigned to various events in a simple manufacturing model, such as processing an item, producing a saleable item, discarding an item for salvage, selling a lemon, etc. and models are described by giving a sequence of events, some of which are decisions to process, to abandon, to accept, to restart, … . All the models have the rather unrealistic classical information pattern of cumulative data. The object is then to find optimal procedures for minimizing the total cost incurred, first in dealing with a single item, and second, in operating until an item first passes all the tests. The policies that appear as optimal depend on such matters as whether a conditional probability given certain data exceeds a ratio of prices, and on more complex functionals of the conditional expectations in the problem. Special sufficient classes of policies are discerned, which reduce the decision problem to finding one number.