# Decision Problem on Imperfect Inspections Combined under Two-Stage Inspection Policy
## 5 Discussion
The dependencies may be summarized in the following way:  
* the inspection interval of a system cannot be longer than the expected length of its delay-time, and this expected delay-time may be treated as the initial length of the inspection interval for the two-stage policy;
* the first stage of an inspection should use an imperfect testing method with low probability of false negative mistake only if its unit cost is lower than the cost of the available perfect method (if a perfect method exists);
* the inspection interval should be reduced 'w' times (for 'w' satisfying the condition: \((\beta_{n\_II,1})^{w}\approx 0\)) if \(n\)-th inspection method is applied as the first stage of the inspection;
* the necessary (but insufficient) condition for \(n\)-th method applied as the first inspection stage of the two-stage inspection policy to be more economic than \(m\)-th method is: \(c_{In\_n}/T_{n}<c_{In\_m}/T_{m}\), where: \(c_{In\_n}\), \(c_{In\_m}\) are unit costs of application of respectively \(n\)-th and \(m\)-th method, and \(T_{n}\), \(T_{m}\) are the reduced inspection intervals respectively for \(n\)-th and \(m\)-th method;
* the second stage of an inspection should utilize the testing method with the low probability of false positive mistake;the necessary (but insufficient) condition for \(m\)-th method applied as the second inspection stage of the two-stage inspection policy to be more economic than \(p\)-th method is similar to the first stage condition: \(c_{ln\_m}/T_{n}<c_{ln\_p}/T_{n}\). However, this time, the denominator of the expression must represent the inspection interval chosen for the needs of the first stage of the inspection. The rules presented above are very simplified and they define necessary, but not sufficient conditions, for a given combination of imperfect inspection methods to be most profitable out of all possible combinations. Nevertheless, they may be used to support economical solutions searching in a very easy way, which is their main advantage. To confirm the validity of the rules, an additional numerical case is presented below. The aim of this new, numerical example is to check whether the rules are also valid for a case with quite different system reliability characteristics than in the case study as well as different inspection costs and probability of errors. The assumptions for the example are as follows:  
1. there are three inspection methods that can be used separately as the one-stage inspection policy or jointly as the two-stage policy;
2. the data on system reliability, costs, and accuracies of inspection methods are presented in Table 4. The unit cost of inspection was established as a function: \(c_{ln}=100\cdot(1-2\alpha_{n})\cdot(1-2\beta_{n})\) to relate the inspection costs and its precision. To satisfy the rules defined in this paragraph, the initial inspection period should be established first: \(T\approx 30\) [units of time], which is the value close to the expected delay time. Then, the possible methods for the first stage of inspection should be examined. The methods with the lowest values of \(\beta_{n}\), preferred in this example, are: \(n=3\), \(\beta_{3}=0\) or \(n=1\), \(\beta_{1}=0\), \(1\). * If one decides to use the 3 rd method, there is no need to reduce the initial inspection period because \(\beta_{3}=0\) and the near-the-best inspection period is \(T_{3}\approx 30\) [units of time], \(c_{ln\_3}=80\) [units of money] per this period, \(c_{ln\_3}/T_{3}\approx 2.7\) [units of money/units of time];