# Decision Problem on Imperfect Inspections Combined under Two-Stage Inspection Policy
## 6 Conclusions  
The delay-time model of a failure process is very useful for modelling a degradation process of technical systems. It enables necessary maintenance activities to be done only when the need arises from signals of an imminent failure. Signals can only be identified by an inspection, which can be carried out using various testing methods available in a system. The methods usually differ in their costs and accuracy, and that is why choosing the best method is not a simple task. The research presented in this article considers the inspection accuracy as a decision variable of a condition-based maintenance policy. This approach is rare in the literature, and this paper represents a step towards filling this gap. For this reason, the paper presents the two-stage inspection policy model applied to a system whose failure process is in accordance with the delay-time concept. A quite new issue is the development of the model in which the second stage of an inspection is not perfect. The relaxed assumption about the unerring second inspection stage can have a significant meaning in practice, as was shown in the presented case study. The model can be used to support the selection of an inspection policy in the case when all available testing methods are imperfect or when precise methods are expensive and may be unprofitable. The situation is common in practical maintenance applications. The usefulness of the proposed model was shown in the example of a piston seal diagnosis in a hydraulic cylinder. In this case, five inspection methods were considered and only one of them, the most expensive, was perfect. It turned out that in the case in question the most fruitful inspection policy was the two-stage policy with periodic measurement of the energy consumption executed as the first inspection stage and the confirmation of possible sealing defect by the temperature measurement of the cylinder surface with a temperature sensor. The combination proved that the application of this mix can reduce the maintenance cost per unit time by up to 35% compared to the application of the one-stage, perfect inspection. In addition, the presented case study showed the relations, which were confirmed by the additional numerical example, that most two-stage inspection variants are more profitable than one-stage policy solutions if an inspection period is shorter than the expected delay time of a system. The detailed analysis of the influence of false positive error probability on expected costs in a system, together with conclusions drawn from the case study, were the basis for identifying the search rules useful for finding economic parameters of the two-stage inspection policy. The application of the rules was demonstrated in the numerical example. The example confirmed their application potential if there is a need to choose parameters of the two-stage inspection policy and all available testing methods are imperfect. Even though the rules provide a good basis for developing the simple optimization algorithm for selection of the best inspection accuracy, their results must be carefully examined with respect to the sensitivity to all the variables and parameters of the failure and maintenance process of a system. The mathematical model presented in this paper, the identified range of inspection periods for which the two-stage policy is better than the one-stage policy, and the set of defined rules when choosing effective parameters of the two-stage policy constitute the new contribution of this research to the maintenance theory. Periodic imperfect inspections are still a common practice in the industry. However, if there is more than one inspection method, they can be combined in different ways: in the form of the two-stage inspection policy, in the form of inspections with various precision executed in different inspection intervals, or in the mix of these two approaches.