Optimal inspection policies where penalty costs are proportional to the duration of the inspection interval containing the failure

Inspection problems arise in connection with systems which are subject to random failure, the nature of the system being such that the failure can only be detected by inspection, e.g. alarm systems, certain production systems, etc. An inspection policy is an increasing sequence x1,x2,...of times at which inspections are scheduled. Most commonly used inspection models assume that costs are made up of (i) an inspection cost proportional to the number of inspections, and (ii) a penalty cost due to the elapsed time between the failure and its detection. This paper considers a cost model in which the penalty cost (ii) is proportional to the duration of the inspection interval in which the failure occurs, a situation which may be more appropriate in certain production control situations. An algorithm for computing the optimal inspection policy is given for a wide class of failure distributions.