# Decision Problem on Imperfect Inspections Combined under Two-Stage Inspection Policy
## 3 Two-Stage Inspection Policy with Imperfect Second Stage-Methodology
### Mathematical Model of Two-Stage Inspection Policy  
To develop the model, at first, all the necessary probabilities of events in the system must be considered: of a system failure, its preventive replacement, and inspection. To estimate the probability of a system failure in time, one needs to consider possible scenarios that may lead to a system failure in the \(i\)-th inspection interval:  
1. the defect appeared in the system during a previous inspection interval; however, it became undiscovered due to false negative inspections until the failure moment; \begin{table}
\begin{tabular}{c c c} \hline \multirow{2}{*}{**Real state of the System**} & \multicolumn{2}{c}{**Probability that the System Is Diagnosed as**} \\ \cline{2-3}  & **Good** & **Defected** \\ \hline Good & 1-\(\alpha_{n\_II\_1}\) & \(\alpha_{n\_II\_1}\) \\ Defected & \(\beta_{n\_II\_1}\) & 1-\(\beta_{n\_II\_1}\) \\ \hline \end{tabular}
\end{table} 
Table 1: The probability of 1 st stage inspection results dependent on a real state of the system. 2. the defect appeared after the (\(i\)-1)-th inspection, and the system failed during the same inspection interval. Figure 2 presents an example of the possible scenarios of a failure process development that results in a failure during the third inspection interval. The first possible scenario in Figure 2a assumes that a defect appears in the first interval period and becomes unrevealed by the two following inspections in \(t_{1}\) and \(t_{2}\) due to false negative mistakes, but the system works with the defect until a failure, which happens during the third inspection interval. The scenario in Figure 2b assumes that a defect appears in the second inspection interval, becomes undiscovered in \(t_{2}\), and system fails in the third interval. Figure 2c presents the third possible scenario leading to a failure in the third inspection interval. The defect arises in the third inspection interval and a failure happens in the same interval. In Figure 2b,c there are also indicated the inspections in \(t_{1}\) (Figure 2b) and in \(t_{1}\) and \(t_{2}\) (Figure 2c) which could have diagnosed an unreal defect during normal operation due to a false positive error. In such scenarios the inspected system would be replaced without a real need and a part of its potential lifetime would be lost. However, it could happen in other possible scenarios of a failure process. Based on the analysis of underlying assumptions and scenarios, the probability of a system failure in the \(i\)-th inspection interval: \(t\in\spherical_{i-1}\), \(t_{i}\)), for the two-stage imperfect inspection policy, if the \(n\_\_\_1\) inspection level is used for the first stage of an inspection and the \(n\_\_\_1\_2\) level for the second stage, may be expressed:  
\[ \begin{split} F_{II}(t_{i},T,n\_II\_1,n\_II\_2)=\sum\limits_{z=1}^ {z=i}P\{t_{z-1}<U<t_{z}\ \wedge\ t_{i-1}<U+H<t_{i}\}\cdot\beta_{n\_\_\_1}{i-z}\cdot(1-\alpha_{n\_ \_1}_{\_1}\cdot\alpha_{n\_\_1}_{\_2})^{z-1}=\\ =\sum\limits_{z=1}^{z=i}\int_{t_{z-1}}^{t_{z}}f_{U}(x)\cdot[F_{H} (t_{i}-x)-F_{H}(max\{t_{i-1}-x,0\})]dx\cdot\beta_{n\_\_\_1}{i-z}\cdot(1- \alpha_{n\_\_1}_{\_1}\cdot\alpha_{n\_\_1}_{\_2})^{z-1}\end{split}  \tag{1}\]  
The above formula incorporates the facts that a false negative mistake may occur during every inspection after defect coming with the probability \(\beta_{n\_\_\_1}{\_1}\) (the false mistake probability of the first stage of an inspection), whilst a false positive error is limited by the joint accuracy of the first-stage inspection and the second-stage confirmation. Formula 1 is  
Figure 2: Alternative scenarios of a failure process development resulting in a failure during the third inspection interval. valid only for the cases where \(\beta_{n\_II\_1}>0\). If the probability of a false negative mistake is equal to 0, to avoid the expression \(0^{0}\), Equation (1) changes into:  
\[