# Decision Problem on Imperfect Inspections Combined under Two-Stage Inspection Policy
## 2 Literature Review  
Although a great development in diagnosing methods of system conditions can presently be observed, imperfect inspections are still common in industry [1]. The accuracy of an inspection may be influenced by, among others, kind of a measured signal, measurement errors, measurer experience, limited knowledge about the relations between technical parameters measured in a failure process, and the reliability or functional state of an inspected system [19]. The accuracy is quantified by the probability of false positive or false negative inspection error [21; 26; 27; 28; 29]. Most literature models deal with the impact of false negative and false positive mistakes on the maintenance process results by the optimization of various process parameters: time between inspections [15; 30; 31; 32; 33], time to the first inspection [15], the number of inspections in various phases of a system operation [34], the number of inspections before a planned replacement [16; 24; 28], or the length of replacement delay after defect recognition [32]. Very few papers consider the fact that the accuracy of periodic inspection can be intentionally shaped or chosen, even though in the literature it is pointed out that the cost of inspection is usually an increasing function of its accuracy [35]. Such a dependency between inspection accuracy and its cost causes that a decision maker has a choice and can optimize inspection policy by choosing the best inspection method or methods in systems which may be diagnosed with different methods. Such a decision should be taken after identifying optimization constraints important in the maintained system and after definition of the optimization criteria, which is usually expressed by system cost or availability. In this way, the inspection accuracy becomes one of the decision variablesof a maintenance policy. However, there is still a lack of optimization algorithms that might be used to support the optimization process. Models that are typically developed in the literature allow one to estimate inspection and maintenance policy results for a given inspection method. They, however, do not define optimization (or rationalization) algorithms that could support the decision process. The approaches that are used to manage and optimize methods of periodic inspections may be divided into three groups, which were briefly presented in the previous section of the paper. The first approach assumes that if there is more than one inspection method available in a system, the results of each method for various values of decision variables (mainly inspection interval length) are evaluated, and then the best solution is chosen. Such an approach is presented in [15]. [PERSON] et al. consider a one-element system in which three levels of inspections may be applied for detection of crack initiation. They differ with costs and the probability of false negative mistake. The probability of the error decreases over time as the crack propagates. The authors develop the mathematical cost model of the maintenance process and propose a procedure to optimize its cost with respect to the period between inspections, their level, and the time to the first inspection. In [7] [PERSON] et al. consider the problem of various inspections methods applied in maintenance of warehouse infrastructure. They model the degradation in time of multiple components such as belts and pulleys and use the genetic algorithms method to find the best combination of three available inspection methods in the system. The literature models found which present this approach are limited to the positions [7; 15]. Other models available in the literature focus on the results modelling of imperfect inspection application in a system with given assumptions and treat inspection accuracy as a given parameter of the maintenance policy (e.g., [28; 31; 36; 37; 38; 39]). Such models may be easily applied to evaluate results of available inspection methods varying due to their accuracy.