# Decision Problem on Imperfect Inspections Combined under Two-Stage Inspection Policy
## 2 Literature Review
Only authors in [7; 15] present the choice of inspection method explicitly as a decision problem and make the effort to develop the optimization algorithm for the inspection level picking. The second method of inspection accuracy management is the combining of two inspection methods with different possibilities of revealing defects in a common inspection policy. They are carried out in a system in different inspection intervals [16; 17; 18; 40]. The number of publications modelling this issue is not sufficient, but higher than in the first group. In [16] [PERSON] considers a system in which two kinds of inspection are applied. The minor inspection, carried out every \(T\) period, may reveal a minor defect with a limited probability, but it is perfect when there is a severe defect in the system. The major inspection, applied every \(iT\), perfectly detects minor and severe defects. [PERSON] compares the policy of minor and major inspections with the policy that uses only one of these types. He concludes that the policy with two kinds of inspections is better for the tested case; however, its profitability strongly depends on system characteristics. The decision variables considered in the model are the periods between minor and major inspections, and the time of the delay since the severe defect was discovered until the following replacement. In [17; 40] [PERSON] uses the delay-time concept to consider multiple nested inspections in a multicomponent production system. The inspections are perfect; however, every higher level of inspection tests a wider part of the system. This is, in a way, an imperfect inspection with different accuracy when looking from the point of view of the whole system. Inspections of different levels are executed in different intervals, and the cost-optimal intervals for them are searched in the paper. In this paragraph the group of papers which deal with imperfect inspections applied in a system with different frequency [21; 22; 26; 41] should also be mentioned. In fact, they consider the application of the same inspection method during the lifetime of a system but optimize inspection intervals dividing the lifetime into phases resulting, for example, from 'weak' and'strong' elements working in a system [21] or system aging [26]. Even though neither of these models treats inspection accuracy as a decision variable like the papers [16; 17; 18; 40], they all confirm the fact that the right management of imperfect inspection frequency is 'generally better than the one-phase inspection' [41] (p. 1) policy. The third way of inspection accuracy management, which is the point of interest in this paper, is the two-stage inspection policy. In this concept an imperfect or initial inspection is executed and after discovering defect a perfect inspection is applied to confirm the system state [19; 20; 21; 22; 23; 24; 25]. There are few works on the matter and none of them considers application of two imperfect inspections in a common inspection policy. [PERSON] et al. [21] model a protection system composed of weak and strong elements, which is inspected periodically with methods that can be false negative and false positive. They consider one-stage and two-stage policies and search the best inspection intervals to reveal defects of weak and strong elements, and the moment of a planned, preventive maintenance. The two-stage policy appears to be about 25% more expensive than the one-stage policy in their case study. In contrast, in [22] [PERSON] et al. consider the same system operating in a finite time horizon. This time, the authors confirm the cost-predominance of the two-stage inspection policy over the one-stage. The two-stage inspection policy for a two-state system is also examined in [25]. [PERSON] develops the analytical model of operating and maintenance costs in a system with a one-stage failure process, and the conclusion from his research is that the two-stage inspection policy can be better than the one-stage policy, but the decision should be made after a careful comparison of both policies.