# 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
### Failure Process and Inspection Methods
Thus, for comparison reasons, the following methods are considered: (1) the measurement of oil temperature after completing 'normal shift', (2) measurement of the periodic energy consumption by a machine where a cylinder is installed, (3) the temperature measurement of the cylinder surface by a temperature sensor mounted on the surface, executed after a standardized procedure of machine operation, (4) the temperature measurement of the cylinder surface by a thermovision camera executed after a standardized procedure of machine operation, (5) the disassembly of the cylinder and detailed visual inspection of the sealing. The probabilities of errors of both types as well as cost estimates are given in Table 2.  
Because of the lack of any objective research on the issue, the data presented in Table 2 were estimated by experts who represent an academic researcher working in the design and testing of hydraulic systems and a manager in a company producing considered hydraulic cylinders. Methods 3 and 4 are directly dependent because they use the temperature measured after a standardized procedure of machine operation of a hydraulic cylinder as a diagnostic signal. Their joint application is pointless. For this reason, a combination of the 3 rd and 4 th methods is not considered in the case. Moreover, one can notice that the temperature threshold, above which the piston seal is diagnosed as defective, can be changed by a decision maker. In such a case, the probabilities of false diagnoses would also change. The probability distributions' parameters of the time to a noticeable seal leakage were calculated based on the database on the time to a leakage of external sealings, obtained from customers' (mines') complaints reported to the manufacturer. The data from the external sealings were adopted for the estimation of the times to a failure of the internal seal. The expected time of the defective stage was approximated by the experts as well as the unit cost of replacement and failure resulting from the machine unavailability time and material costs. The necessary data are presented in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \(n\) & **The method** & \(\alpha_{n}\) & \(\beta_{n}\) & \(c_{In}\) [5] \\ \hline \multirow{2}{*}{1} & the measurement of oil temperature after completing ’normal shift’ in the oil tank with the threshold level of 70 \({}^{\circ}\)C & 0,5 & 0,3 & 5 \\ \hline \multirow{2}{*}{2} & periodic measurement of the one-week energy consumption by a machine where a cylinder is installed & 0,3 & 0,2 & 10 \\ \hline \multirow{3}{*}{3} & the temperature measurement of the cylinder surface by a temperature sensor mounted on the surface, executed after a standardized procedure of machine operation with the threshold level of 80 \({}^{\circ}\)C & 0,1 & 0,2 & 50 \\ \hline \multirow{3}{*}{4} & the temperature measurement of the cylinder surface by a thermovision camera executed after a standardized procedure of machine operation with the threshold level of 70 \({}^{\circ}\)C & 0,2 & 0,4 & 30 \\ \hline \multirow{3}{*}{5} & disassembly the cylinder and detailed visual inspection of the sealing & 0 & 0 & 300 \\ \hline \end{tabular}
\end{table} 
Table 2: Probability and cost parameters of the available inspection methods.