# Decision Problem on Imperfect Inspections Combined under Two-Stage Inspection Policy
## 3 Two-Stage Inspection Policy with Imperfect Second Stage-Methodology
### Assumptions and Notation  
At first, basic assumptions and nomenclature taken while modelling are presented.  
1. The system in consideration is a one-unit system that may be in one of the three states: normal functioning, functioning with a defect that shows approaching failure and failed;
2. To reveal a defect in the system, an inspection must be carried out, whilst the transition of the system to its failure state is observed instantaneously without any testing;
3. There is more than one inspection method available in the system. They are called inspection methods/levels. These methods differ in cost and accuracy. The application of a lower-level inspection is cheaper and more prone to an incorrect positive diagnosis than a higher-level inspection. The methods are independent of each other;
4. The system is inspected every \(T\) period with an imperfect testing method chosen from the available ones, and that constitutes the first stage of inspection. If the first-stage inspection diagnoses a defect in the system, additional confirmation (the second stage of inspection) of this result with a higher-level method is carried out. If the defect  
Figure 1: The methodology of the research study.  
really exists, the second-stage inspection cannot neglect it, but it can reveal a false positive diagnosis;
5. The accuracy of a level/testing method is defined by the probability of a false positive and a false negative mistake. The probabilities are constant in time;
6. If a defect is confirmed in the system, the system is perfectly renewed (replaced), and its new renewal cycle begins;
7. Inspection and replacement times are negligible compared to the inspection interval and are not considered in the model;
8. The inspection and maintenance process generates the cost of a system replacement, the cost of a system failure, and the inspection cost (dependent on an applied inspection level). The main notations used when modelling are presented below:
* \(U\beta\)-the random variable representing the duration of the normal functioning period;
* \(H\)-the random variable representing the delay time length;
* \(f_{u}(x)\), \(F_{u}(x)\)-respectively, the probability density (pdf) and the cumulative distribution (cdf) function of the variable \(U\);
* \(f_{H}(y)\), \(F_{H}(y)\)-respectively, pdf and cdf of the variable \(H\);
* \(T\)-the length of an inspection interval;
* \(i\)-the number of an inspection interval: \(i=1\), \(2\), \(3\), \(\ldots\), \(\infty\);
* \(t_{i}\)-the moment of \(i\)-th inspection ending the \(i\)-th inspection interval, \(t_{i}=T*i\);
* \(n\_II\_k\)-the number of \(n\)-th inspection method (level) applied in the \(k\)-th stage of an inspection, \(n=1\), \(2\), \(3\), \(\ldots\), \(N\), \(k=1\), \(2\) if the two-stage policy is applied in the system;
* \(n\_I\_k\)-the number of \(n\)-th inspection method (level) applied in the \(k\)-th stage of an inspection, \(n=1\), \(2\), \(3\), \(\ldots\), \(N\), \(k=1\) if the one-stage policy is applied in the system;
* \(\alpha_{n\_I\_k}\), \(\alpha_{n\_II\_k}\)-respectively, the probability of a false positive diagnosis if the \(n\)-th level of inspection is applied in the \(k\)-th inspection stage of the one-stage or the two-stage inspection policy, \(\alpha_{n\_II\_1}\geq\alpha_{n\_II\_2}\), \(\alpha_{N\_II\_2}=0\);
* \(\beta_{n\_I\_k}\), \(\beta_{n\_II\_k}\)-respectively, the probability of a false negative diagnosis if the \(n\)-th level of inspection is applied in the \(k\)-th inspection stage of the one-stage or the two-stage inspection policy. The dependency between the probability of errors and the real state of the system, for the example when the \(n\)-th inspection level is applied in the first inspection stage, is presented in Table 1;
* \(c_{\text{F}}\)-the unit cost of a system failure;  
* \(c_{\text{H}\_k}\)-the unit cost of the \(n\)-th level inspection level used in the \(k\)-th stage of an inspection;
* \(c_{\text{D}}\)-the unit cost of a system replacement;
* \(\max\{x,y\}\)-maximum value of \(x\) and \(y\).