Probabilistic indicators of imperfect inspections used in modeling condition-based and predictive maintenance

This study proposes mathematical models for assessing the probabilistic indicators of imperfect inspections conducted when performing condition-based and predictive maintenance. The inspections used in mentioned types of maintenance differ in decision rules regarding system operability at the time of checkup. Contrary to the previous studies, we present the decision rule for each type of inspection on the time axis, which allows the formulation of the set of mutually exclusive events at discrete times. The correct and incorrect decisions correspond to true-positive, false-positive, true-negative, and false-negative events. We propose general expressions for computing the probabilities of possible decisions for both types of inspection. The paper introduces the effectiveness indicators of condition-based and predictive maintenance such as average operating costs, total error probability, and a posteriori probability of failure-free operation. We illustrate the developed approach by calculating the probabilities of correct and incorrect decisions using a specific stochastic deterioration process. The results of the calculations verify that probabilities of correct and incorrect decisions for both types of inspection are very substantially time-dependent despite the large number of published studies where these probabilities are independent of time.