Optimal preventive 'maintenance-first or -last' policies with generalized imperfect maintenance models

This paper presents modified preventive maintenance policies for an operating system that works at random processing times and is imperfectly maintained. The system may suffer from one of the two types of failures based on a time-dependent imperfect maintenance mechanism: type-Ⅰ (minor) failure, which can be rectified by minimal repair, and type-Ⅱ (catastrophic) failure, which can be removed by corrective maintenance. When the system needs to be maintained, two policies "preventive maintenance-first (PMF) and preventive maintenance-last (PML)" may be applied. In each maintenance interval, before any type-Ⅱ failure occurs, the system is maintained at a planned time \begin{document}$ T $\end{document} or at the completion of a working time, whichever occurs first and last, which are called PMF and PML, respectively. After any maintenance activity, the system improves but its failure characteristic is also altered. At the \begin{document}$ N $\end{document}-th maintenance, the system is replaced rather than maintained. For each policy, the optimal preventive maintenance schedule (\begin{document}$ T $\end{document}, \begin{document}$ N $\end{document})\begin{document}$ ^{*} $\end{document} that minimizes the mean cost rate function is derived analytically and determined numerically in terms of its existence and uniqueness. The proposed models provide a general framework for analyzing the maintenance policies in reliability theory.