# Decision Problem on Imperfect Inspections Combined under Two-Stage Inspection Policy
## 1 Introduction
Inspection methods that can be used in such cases are, e.g., visual inspections made physically by specially trained staff or remotely by less accurate but cheaper unmanned aerial vehicles [10; 11; 12], dimension, shape, or other non-destructive measurement of all (highly expensive) or only critical components (cheaper solution) of an object [13; 14]. The lower scale examples of elements subjected to imperfect inspections are blades of a pump whose perfect inspection usually requires pump disassembly, steel rope suspending an elevator which can be perfectly diagnosed only by expensive non-destructive methods, or machine internal sealings whose perfect diagnosis may be impossible without cost- and time-consuming disassembly of the machine. In practise such defects, if not monitored continuously, may be periodically diagnosed by visual, sound, thermic, vibration, or other signals with different methods. The methods are not always precise and may indicate a system condition with some limited accuracy but often with much lower cost and inspection time than available perfect methods. These practical examples show that sometimes it is impossible or unprofitable to use perfect inspection methods when cheaper, but less accurate ones are available. Their advantages may be used jointly to obtain satisfying cost and reliability results. It leads to the conclusion that the inspection accuracy can be treated as a decision variable in the condition-based maintenance optimization problem. The purposeful management of inspection accuracy applied in a technical system which is periodically inspected is not very extensively explored in the literature. However, [PERSON] in his review on the delay-time modelling [1] points out that modelling the inspection level is one of the topics that deserves further research. The management of periodic inspection accuracy is met in the literature in the form of three basic approaches. The first is just a simple comparison of the results of the available inspection methods and deciding which is the best for the sake of a chosen optimization criterion [7; 15]. The second is the combination of two inspection methods with different accuracies and different frequencies in a common inspection policy [16; 17; 18]. The third approach, which is the point of preoccupation of this paper, is the application of the two-stage inspection policy. In such a policy, 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]. The detailed analysis of literature on this issue is presented in the next section of the paper. It has shown that there is an insufficient number of models and methods that may support decision makers in their optimization dilemmas if available inspection methods differ in costs and accuracies. Most literature models consider the usage of a single imperfect method or two methods jointly, an imperfect and a perfect, under the same inspection policy. They never assume that all inspection methods applied under one policy are imperfect, nor evaluate results that a such solution can have on inspected system results. This shortages in the inspection accuracy planning and managing have been the premise for defining the goal of this paper. The objective is to develop the mathematical model of the two-stage inspection policy in which the second stage can be imperfect and present the results of the research study and its economic potential. The results enable the definition of relations which can support decision process concerning the selection of periodic inspection policy if perfect methods are unavailable or economically unjustified.