# Decision Problem on Imperfect Inspections Combined under Two-Stage Inspection Policy
## 1 Introduction  
Deterioration in time is an inevitable characteristic of many technical systems. A suitable preventive or predictive maintenance policy, when applied in a system, increases system safety due to the elimination of unexpected failures, facilitates scheduling of maintenance operations and resource usage, as well as increases the system's availability. Most technical systems do not fail suddenly, and symptoms of a future failure may be observed some time before the failure will occur [1]. If the symptom is known and discovered when it exists, maintenance decisions and actions can be taken to protect a system from a future failure. This assumption is the basis for the condition-based maintenance strategy. The delay-time concept introduced by [PERSON] [2] can be considered as a simplified condition-based maintenance model with only binary states space [1]. It reflects the two-stage character of a failure process and is extensively used in the literature to model and optimize condition-based maintenance policies. The delay-time concept considers that there are two subperiods in the course of system operation: from the new to an initial point when a defect can be first observed, and then from that point to a failure. The time from the initial point of an identifiable defect to a failure is called the delay-time [1, 2, 3]. Reviews of the delay-time models may be found, e.g., in [1, 4, 5]. The condition-based maintenance (CBM) strategy has gained a lot of attention from practitioners and scientists over the last few decades. Potential savings resulting from the predictive maintenance application in comparison to the preventive maintenance applied alone in industrial systems are significant. They are estimated to be about 25-35% ofmaintenance costs, 70-75% of the number of unexpected failures in a system, and 35-45% of the maintenance operations time [6]. The application of CBM in a system requires correct system state identification to learn whether a failure is imminent or if the system can still operate. However, such identification requires additional expenditures on inspection (i.e., device, labour, logistic, and system unavailability costs), apart from typically considered costs of failure (i.e., system unavailability and restoring costs, safety consequences cost) and maintenance (i.e., system restoration and unavailability cost as well as maintenance equipment and labour costs) [7]. This trade-off between CBM costs makes the maintenance policy optimization a complex task and requires the right identification of optimization criteria (e.g., a cost ratio or system availability) and problem constraints (e.g., limited budget, limited time or labour resources, availability of inspection and rehabilitation methods, etc.) [8]. The most advanced form of system diagnosis is continuous monitoring, but it often happens that it is more profitable to use periodic inspections to reveal an actual condition of a system [9]. Despite the existence of advanced testing methods, there are also cases of technical systems where unerring inspection methods are unavailable or application of a perfect method is very expensive and thus, it is unprofitable. The term of a perfect inspection means the inspection method which always allows one to correctly diagnose a real system condition. Examples of complex technical systems where only imperfect inspections are often applied are critical facilities [8; 10; 11; 12] (e.g., buildings, bridges, power lines, transportation infrastructure, etc.) where perfect methods cannot be used due to scale of the facility, a budget-limiting (e.g., a large bridge) or defect phenomenon (e.g., internal structural abnormalities of a building). Nevertheless, the potential failure cost of such an object can be huge and for this reason the system condition must be diagnosed.