# Classification of Systems and Maintenance Models
## 3 Classification of Systems
When the consequences of a failure can be evaluated economically, it is necessary to use technical-economic indicators of the MRS in the form of the average operating profit per unit of time (AOPUT) or average maintenance cost per unit of time (AMCUT). The possibility of an economic evaluation of the consequences of failure exists only for systems that do not affect safety, that is, for subclasses \(I_{13}\), \(I_{14}\), and \(I_{25}\). If, for some reason, material damage from a system failure that belongs to one of these subclasses cannot be quantified, then complex reliability indicators, namely inherent availability and achieved availability, should be used as MRS effectiveness indicators. If a system failure leads to the non-fulfillment of the assigned safety-critical functions, then the consequences of failure cannot be evaluated. Indeed, the failure of safety-critical systems in the nuclear, chemical, aviation, military, etc. industries may result in many casualties. Moreover, it is not known how many victims there will be. For example, two accidents at the nuclear power plants of Chornobyl and Fukushima killed tens of thousands of people [79; 80]. The cost of the loss of human lives cannot be estimated in such cases. Therefore, when planning and optimizing the maintenance of safety-critical systems, criteria are used that do not include the cost of the failure consequences. In such cases, the dominant factor in evaluating the consequences of a failure is the inability to perform a task. As higher demands are placed on the reliability of safety-critical systems during their intended use, it is logical that meeting these demands comes at a price. Therefore, when evaluating the MRS effectiveness for subclasses \(I_{11}\) and \(I_{12}\), it is recommended to utilize the \"reliability-costs\" criterion, which demands the presence of two effectiveness indicators. The first identifies the operational reliability of the system, whereas the second characterizes maintenance costs. As indicators of reliability, we used the a posteriori probability of failure-free operation (APFFO) for non-repairable systems and the operational probability of failure-free operation (OPFFO) for repairable systems. The APFFO and OPFFO can be determined both for periodic and sequential inspections. The APFFO is the conditional probability of failure-free operation of the system on the interval \((t_{k},t)\), provided that the system can be used for its intended purpose based on the inspection results at moments \(t_{1},\ldots,t_{k}\). From the definition of APFFO, it follows that for non-repairable systems, the moments of inspections are assigned until the system is rejected. Once the system is replaced, the inspection schedule is restarted, beginning with the first inspection moment. Under the OPFFO, we refer to the probability of failure-free operation of the system over the operating time interval \((t_{k},t)\), considering that at moments \(t_{1},\ldots,t_{k}\), maintenance was carried out, including inspection and restoration of systems judged as inoperable. From the definition of OPFFO, it follows that for repairable systems, inspection times are assigned considering the repair of rejected systems. The definitions of the APFFO and OPFFO for systems with continuous condition monitoring are as follows:  
The APFFO is the conditional probability of the system's failure-free operation during the interval \((0,\,\tau_{p})\), assuming that the system has not been rejected by the condition-monitoring results, where \(\tau_{p}\) is the periodicity of maintenance. The OPFFO is the probability of failure-free operation of the system in the interval \((0,\,\tau_{p})\), considering that, at any moment, the system can be rejected as inoperable and then restored. Maintenance costs are characterized by unit costs (UC), that is, the average costs per unit time of system use. Moreover, for non-repairable systems, UC includes inspection and replacement costs, whereas, for repairable systems, UC includes inspection and repair costs.