# Decision Problem on Imperfect Inspections Combined under Two-Stage Inspection Policy
## 5 Discussion
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Description** & **Value** \\ \hline \(A_{U}\) & The shape parameter of Weibull probability distribution modelling the period of & 3,5 \\  & normal functioning & 70 [units of time] \\ \hline \(B_{U}\) & The scale parameter of Weibull probability distribution modelling the period of & 70 [units of time] \\  & normal functioning & \\ \hline \(A_{H}\) & The shape parameter of Weibull probability distribution modelling the delay time length & 3,5 \\ \hline \(B_{H}\) & The scale parameter of Weibull probability distribution modelling the delay time length & 30 [units of time] \\ \hline \(c_{F}\) & The unit cost of a failure & 10,000 [money units] \\ \hline \(c_{D}\) & The unit cost of a preventive replacement & 1000 [money units] \\ \hline \(\alpha_{1}\) & The probability of false positive error of the 1 st inspection method & 0,4 \\ \hline \(\beta_{1}\) & The probability of false positive error of the 1 st inspection method & 0,1 \\ \hline \(c_{ln\_1}\) & The unit cost of the 1 st inspection method & 16 \\ \hline \(\alpha_{2}\) & The probability of false positive error of the 2 nd inspection method & 0,2 \\ \hline \(\beta_{2}\) & The probability of false positive error of the 2 nd inspection method & 0,3 \\ \hline \(c_{ln\_2}\) & The unit cost of the 2 nd inspection method & 24 \\ \hline \(\alpha_{3}\) & The probability of false positive error of the 3 rd inspection method & 0,1 \\ \hline \(\beta_{3}\) & The probability of false positive error of the 3 rd inspection method & 0 \\ \hline \(c_{ln\_3}\) & The unit cost of the 3 rd inspection method & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of system reliability, costs, and accuracies of inspection methods. If one considers the application of the 1 st method in the first stage of the inspection, the initial inspection interval should be reduced at least twice (\(w=2\)) to decrease the probability of defect overlooking: \(\beta_{1}\)\({}^{\text{w}}=\)\(\beta_{1}\)\({}^{2}\) = 0,01 \(\approx\) 0, \(T_{1}\approx\) 15 [units of time], \(c_{ln1}\) = 16 [units of money] per this period and \(c_{ln1}\)/\(T_{1}\approx\) 1.05 [units of money/units of time]. If the inspection period is reduced three times (\(w=3\)): \(\beta_{1}\)\({}^{\text{w}}=\)\(\beta_{1}\)\({}^{3}\) = 0,001 \(\approx\) 0, \(T_{1}\approx\) 10 [units of time], \(c_{ln1}\) = 16 [units of money] per this period and \(c_{ln1}\)/\(T_{1}\approx\) 1,6 [units of money/units of time]. According to the necessary condition regarding the profitability of the application of a given method in the first inspection stage, it seems to be more reasonable to use the 1 st method as the first stage with the frequency \(T_{1}\approx\) 15 or \(T_{1}\approx\) 10 [units of time], because the following condition is met: \(c_{ln1}\)/\(T_{1}<c_{ln3}\)/\(T_{3}\) for both variants. After choosing the method applied in the first stage, the methods with the lowest \(\alpha_{n}\), should be considered for the need of the second stage of the inspection: \(n=3\), \(\alpha_{3}\) = 0,1 or \(n=2\), \(\alpha_{2}\) = 0,2. The false positive mistake from the combination of the 1 st method applied in the first stage and 2 nd method in the second stage is: \(a_{1}\)\(\cdot\)\(a_{2}\) = 0,4.0,2 = 0,08 with the additional unit cost \(c_{ln2}\) = 24 [units of money] per every \(T_{1}\approx\) 15 [units of time], \(c_{ln2}\)/\(T_{1}\) = 1,6 [units of money/units of time] or per \(T_{1}\approx\) 10 [units of time], \(c_{ln2}\)/\(T_{1}\) = 2,4 [units of money/units of time]. The length of \(T_{1}\) depends on the chosen inspection interval for the first stage of an inspection;  
The false positive mistake of the combination of 1 st and 3 rd methods is: \(a_{1}\)\(\cdot\)\(a_{3}\) = 0,4.0,1 = 0,04 with the additional unit cost \(c_{n2}\) = 80 [units of money] per every \(T_{1}\approx\) 15 [units of time], \(c_{ln3}\)/\(T_{1}\approx\) 5,33 [units of money/units of time] or per \(T_{1}\approx\) 10 [units of time], \(c_{ln3}\)/\(T_{1}\approx\) 8 [units of money/units of time].