# 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  
The potential results of the application of the one- and two-stage inspection policies in the condition-based maintenance of the considered hydraulic system are shown in Figures 3-7. Due to the huge number of considered combinations of joined inspections, the amount of data presented in Figures 3, 6 and 7 is high. However, the aim is to illustrate the general trends in the relation between the one- and two-stage inspection policies and the accuracy of used inspection methods. To simplify, the size of symbols in the figures reflects the probability value of the false positive error of an inspection method which is applied as the first stage policy. The larger symbols represent policies in which \(\alpha_{n\_1\_1}\) and \(\alpha_{n\_1\_1}\) have higher values. The most accurate testing methods used in the first stage of an inspection are depicted with the smallest symbols. The relations depicted in Figure 3 prove explicitly that the one-stage policy is never better than the two-stage policy if the inspection period is set correctly and does not exceed the expected time of sealing's defective state, which is \(T\approx 100\) [mth] in the tested case. Even the perfect policy (\(\alpha\) 5\(1\)1 = 0, \(\beta\) 5\(1\)1 = 0) used alone gives a worse cost effect than most of the two-stage options. The comparison of the most profitable solutions found for all the tested one-stage policies and the tested variants of the two-stage policies is presented in Figure 4, while the results of all the tested two-stage variants are shown more clearly in Figure 5. Based on the results presented in the Figure 4, one may conclude that the costs savings that may be achieved by the best combination of imperfect testing methods in the considered example may be up to 35% of the total costs that are incurred if the perfect inspection is applied as the one-stage policy. In the considered case, the best combination is to use the periodic measurement of the one-week energy consumption by a machine where the cylinder is installed (2 nd inspection method) as the first stage of an inspection and 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 (3 rd inspection method) as the second stage of inspection. The costs shown in Figure 5 prove that the most profitable variants of two-stage policies consist of: a first-stage inspection with a low value of the false negative error probability \(\beta_{n\_1\_1}\) and the second stage with the low positive error probability \(\alpha_{n\_1\_2}\). Figures 6 and 7 present component effects of applying considered combinations of inspections, such as the expected number of system failures in a renewal cycle and the expected lifetime of the considered piston seal. Based on Figure 6 it may be observed that the longer inspection interval becomes, the higher the failure probability of the two-stage inspection policy is for a given inspection interval \(T\). This is mainly the effect of defects overlooked due to false negative errors in the inspection first stage and too rare inspections in comparison to the expected delay time. The disadvantage of the low accuracy of the inspection's first stage may be compensated by more frequent controls. If the one-stage imperfect inspection policy is applied, the negative