On minimum expected cost due to audits and failure

In order to study the problem of detecting an optimal sequence of random audits, this paper is devoted to determine a checking schedule minimizing the expected total cost resulting from the inspections and the possible first failure. This problem is considered in the general case in which the number of checks during an interval is described by a Poisson checking process: in the framework of the Optimal Control Theory, the analytical solution of a new problem which is related to that of detecting the optimal random audit scheme is characterized. This is achievable because in this case the control function is associated to the rate of growth of the checking intensity. In particular, the proposed representation leads to the explicit detection of the (sub-)optimal solution for exponential or uniform failure density functions and it ensures the analysis of the optimal solution dynamic in the phase-diagram framework.