A modelling procedure to optimize component safety inspection over a finite time horizon

In this paper a model of a safety inspection process is proposed for the expected consequence of inspections over a finite time horizon. A single dominant failure mode is modelled, which has considerable safety or risk consequences assumed measurable either in cost terms or in terms of the probability of failure over the time horizon. The model established extends earlier work assuming an infinite time horizon, and uses the concept of delay time and asymptotic results from the theory of renewal and renewal reward processes. The paper establishes a pragmatic procedure for formulating objective functions which may be optimized to determine the optimal inspection intervals. Merits of both the exact and asymptotic formulations of these objective functions for possible use in the inspection optimization process are considered. Although the procedure for developing an objective function over a finite time zone inspection process assumes perfect inspection, it can be generalized to the imperfect inspection case. Because of the intractability of the mathematics, it is suggested that when optimizing an inspection process over a finite time zone, an asymptotic formulation of the objective function should be optimized, and this solution then checked and if necessary refined, using simulation calculation. A numerical example illustrates the performance of the basic periodic inspection policy over different time horizons using the asymptotic solution. The results are compared with simulations performed to estimate the exact expected cost measure. It is shown that the simpler asymptotic solution is satisfactory in the case considered, especially when the time horizon is relatively long.