Given the deteriorative nature of industrial systems, implementation of advanced Preventive Maintenance (PM) strategies becomes of paramount importance to cope with the maintenance needs of ever-changing complex industrial and safety-critical systems. Conventionally, PM approaches are developed based on the perfect maintenance assumption, i.e., the underlying system is renewed to the as-good-as-new state after each preventive repair, or corrective maintenance action. Such an assumption, however, is not realistic in applications such as military machinery, power generation networks, and Cyber-Physical Systems (CPS), rendering conventional PM strategies impractical. In such application domains, it is not feasible to perform all the required maintenance actions during the available time leading to imperfect maintenance. Overlooking imperfect maintenance is critically problematic as it further deteriorates the reliability of the underlying system shortening its life span. Therefore, it is critical to perform optimal maintenance decisions under imperfect maintenance assumptions. While Geometric Process (GP) is broadly used for imperfect maintenance modeling and analysis of repairable systems, its utilization to describe the failure mechanism of production systems/processes is still in its infancy. Existing works, typically, consider restrictive assumptions to simplify the maintenance models, which limits their applicability. This paper addresses this gap and proposes a rigorous mathematical model without the incorporation of restrictive assumptions. More specifically, we consider a system for which the operational states are observable through inspections performed at specified time points and only the failure state is immediately observable. Upon the inspection, if the system is found to be in a partially-failed state, a Minor Repair (MIR) action is conducted. The effect of MIR is imperfect and MIR can only be conducted a maximum number of N times during a production cy
