# Time-Dependent Probabilistic Model for Hierarchical Structure in Failure Mode and Effect Analysis
## 1 Introduction
[PERSON] [33] developed a new tool for reliability and FMEA by integrating the conventional aspects of the popular failure mode and criticality analysis (FMECA) procedure with economic considerations. Here, FMECA was approached as a multi-criteria decision-making technique that integrates four different factors, as follows: The chance of failure, the chance of non-detection, the severity, and the expected cost. Based on [PERSON]'s [33] study, [PERSON] [34] suggested a new method, priority-cost FMECA, that allows for the calculation of a new RPN and the introduction of the concept of profitability, taking the corrective action cost into consideration. [PERSON] and [PERSON] [11] introduced a new methodology, life cost-based FMEA, that measures risk in terms of cost. Life cost-based FMEA is useful to compare and select design alternatives that can reduce the overall life-cycle cost of a particular system. [PERSON] [13] developed an approach to improve FMEA from an economic perspective by considering the estimated cost of external faults in the evaluation of the severity of failures. [PERSON] [12] proposed the use of a FMEA tool based on fuzzy utility cost estimation to overcome the limitations of conventional FMEA techniques. The utility theory accounted for the nonlinear relationship between the cost incurred in the event of a failure and used ordinal ranking. [PERSON] et al. [22] suggested the expected loss model, in which failure was dependent on time and the system was periodically monitored to prevent failures during its mission period. The loss due to each failure mode was assumed to depend on the remaining mission period of the system. Some researchers have accounted for time or a hierarchical structure. For instance, [PERSON] et al. [35] assumed an exponential probability distribution for the evaluation of occurrence and detection times for failures and their corresponding causes, and the expected severity was derived for each FC. A new risk metric (REM) was defined as the product of the FC occurrence rate and the expected severity of the corresponding failure. In another study by [PERSON] et al. [23], both time and a two-level hierarchical structure were employed to evaluate risks of the failure mode and its corresponding cause. Assuming the severity of failure depends on the length of the delayed time for corrective action, a severity model was developed. Using the expected severity, a risk priority metric was defined. Regarding the hierarchical structure, [PERSON] [36] introduced the failure mode factors and effects analysis (FMFEA) technique that allowed hierarchical representations of causes and ripple effects for the failure mode. [PERSON] et al. [37] tried to systematically identify the root causes of failure modes based on a their proposed FMEA sheet and scientifically evaluated the relevant risk priorities for hierarchical FCs. [PERSON] et al. [32] presented the failures hierarchically through FTA and conductedrisk assessment through FMEA. All critical risk assessments were performed from the system level to the component level. To the best of our knowledge, there is no study that simultaneously considers the hierarchical failure structure and the elapsed time after a failure in the FMEA. This paper is intended to develop a hierarchical time-dependent FMEA (HTD-FMEA). If the occurrence of a specific failure is represented in a hierarchical structure, the system (top-level) failure will be caused by at least one of its FCs (bottom level). It will take time for the system failure to occur after the FCs, and the elapsed time from each FC to the system failure will be different from the others. Moreover, the system failure can lead to a serious accident, such as product failure or leakage of a hazardous substance, and the loss due to such an accident will vary depending on the detection time. If the FC is detected before the system failure occurs, only the constant cost of repairing the FC will be required.