# Time-Dependent Probabilistic Model for Hierarchical Structure in Failure Mode and Effect Analysis
## 1 Introduction  
For assuring the safety and reliability of a system, engineers identify all possible failures and their corresponding failure-causes (FCs) and take actions to reduce the occurrence of FCs or mitigate the loss due to their consequences. Failure mode and effect analysis (FMEA) is a structured method used for defining, identifying, and mitigating known and/or potential causes of failures, problems, errors, etc. in systems, designs, processes, and services prior to them being made accessible to customers [1]. Since its introduction as an analysis tool for failure mitigation, FMEA has been extensively used in a wide range of industrial applications, including, but not limited to, automotive, semiconductor design, aerospace and aviation, healthcare, and steel manufacturing applications [2; 3; 4; 5]. In traditional FMEA approaches, the risk associated with an FC is usually evaluated using the risk priority number (RPN), which corresponds to the mathematical product of the occurrence (O), severity (S), and detection (D) of an FC. However, this traditional FMEA risk-evaluation approach has often been extensively criticized in extant literature for a variety of reasons [6]. The major shortcomings of the traditional RPN-based approach are as follows: (i) The relative importance among O, S, and D is not considered [7; 8; 9; 10]; (ii) the three risk factors are difficult to precisely evaluate [4; 11; 12; 13; 14; 15; 16; 17]; (iii) interdependencies among various failure modes and effects are not considered [1; 5; 18; 19]; (iv) the method depends on experts' intuition and experience rather than the scientific method to estimate the three risk components [20; 21]; and (v) there is no consideration of possible hierarchical relationships among failures [6; 22; 23]. Many researchers have attempted to improve the RPN-based risk-evaluation method of FMEA. Some researchers have considered facilitating the assessment of the three risk-evaluation factors--O, S, and D--by adopting fuzzy logic-based approaches [4; 5; 7; 10; 12; 19; 24; 25; 26] or fuzzy rule-based Bayesianreasoning approaches [27]. Other researchers [9; 28] employed both grey and fuzzy theories in FMEA. [PERSON] et al. [2] proposed a systematic approach to evaluate potential failures using a service-specific FMEA and grey relational analysis. [PERSON] [29] evaluated the structure of hierarchy and interdependence of corrective action, then calculated the weight of corrective action through the analytic network process (ANP). [PERSON] et al. [30] employed both FMEA and fuzzy analytic hierarchy process (FAHP) techniques to recognize risk-evaluation criteria. [PERSON] and [PERSON] [31] tried to identify possible interactions among the principal causes of failure by integrating ANP and a multi-criteria decision-making technique. Some researchers have proposed an integrated FMEA technique combining the various methods, i.e., the decision-making method [18; 20]; ordered weighted geometric averaging (OWGA) [8]; the linguistic distribution assessment method [16]; Dempster-Shafer theory (DST) [15]; fault-tree analysis (FTA) [32]; robust data-envelopment analysis (RDEA) [21]; and failure modes, mechanisms, and effect analysis (FMMEA) [17]. [PERSON] [14] suggested a method based on treating the numerical initial measurements as estimates of the location parameters of probability distributions, which enables objective consideration of the uncertainty inherent in such measurements and computation of the probabilities of each potential failure being the most important according to each criterion. [PERSON] & [PERSON] [1] presented a general model to explain the functional relationship among the three following factors: Severity, probability of occurrence, and detection. Based on the functional relationship, they calculated the risk of failure mode according to the conditional probability of three factors. Some researchers have proposed cost-based FMEA technology.