# Decision Problem on Imperfect Inspections Combined under Two-Stage Inspection Policy
## 4 Applicability of Two-Stage Inspection Policy to Piston Seal Diagnosing in Hydraulic Cylinder
### Application Results of Two-Stage Inspection Policy
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Description** & **Value** \\ \hline \multirow{2}{*}{\(A_{U}\)} & The shape parameter of Weibull probability distribution & \multirow{2}{*}{1.6} \\  & modelling the period of normal functioning & \\ \hline \multirow{2}{*}{\(B_{U}\)} & The scale parameter of Weibull probability distribution & \multirow{2}{*}{2148 [mth]} \\  & modelling the period of normal functioning & \\ \hline \multirow{2}{*}{\(A_{H}\)} & The shape parameter of Weibull probability distribution & \multirow{2}{*}{10} \\  & modelling the period of the defective state (the delay time) & \\ \hline \multirow{2}{*}{\(B_{H}\)} & The scale parameter of Weibull probability distribution & \multirow{2}{*}{100 [mth]} \\  & modelling the period of the defective state (the delay time) & \\ \hline \multirow{2}{*}{\(c_{F}\)} & The unit cost of a failure & 10,000 [5] \\ \hline \multirow{2}{*}{\(c_{D}\)} & The unit cost of preventive maintenance & 2000 [5] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the failure process. effect of too rare inspections is partially eliminated by unnecessary (not resulting from a real defect) replacements. On the other hand, the low probability of a system failure (Figure 6) that might be achieved by frequent imperfect inspections may be uneconomic if the one-stage inspection with high probability of a false positive error or the two-stage inspection policy with the high probability of a false positive error of the second stage are used. In such cases (Figure 7) the expected lifetime of the system is significantly reduced compared to inspection cases with the lower probability of false positive mistake obtained by the introduction of the perfect one-stage inspection or by accurate confirmation of a defect in the second stage of the two-stage policy. This effect is clearly depicted in Figure 7. There may be two series of results observed there. The trend of increasing lifetime of the sealing for longer inspection intervals is the result of the application of inspection policies (regardless of whether it is a one- or two-stage inspection) with the higher probability of false positive mistake. However, if the first or second stage of an inspection has a low probability of false positive error, even very frequent inspections do not reduce the potential lifetime of the considered sealing. Such a tendency is visible as horizontal lines in Figure 7.  
Figure 4: The lowest expected cost per unit time found for the tested one-stage and the two-stage policies. Figure 3: The expected cost per unit time for the inspection interval \(T\) if the one-stage and various combinations of the two-stage policy are applied. Figure 5: The expected cost per unit time for all the tested combinations of the two-stage policy for the range of inspection interval \(T\) limited with respect to the most economical solutions. Figure 6: The expected number of system failures for the inspection interval \(T\) if the one-stage and various combinations of the two-stage policies are applied. Figure 7: The expected lifetime of the sealing for the inspection interval \(T\) if the one-stage and various combinations of the two-stage policy are applied. The relations resulting from the case study analysis can serve as advice when researching the economical parameters of the two-stage inspection policy. The set of rules for how to find profitable solutions of the two-stage policy will be defined in the next part of the paper.