An optimal choice problem for a set of checking procedures

An item having a known initial failure probability is to be controlled by some out of a finite set of possible checks. Every check costs a certain amount of money, and budget constraints must be met. A check is characterized by the probabilities of three events: (i) letting a workable item pass, (ii) overlooking a failure if one is present, (iii) introducing a failure into a workable item. We show how any subset of checks to be employed is ordered optimally, and how the optimal subsequence of checks depends on the initial failure probability and oil the budget constraint.