# Time-Dependent Probabilistic Model for Hierarchical Structure in Failure Mode and Effect Analysis
## 2 Background  
To ensure the safety and reliability of a system, engineers identify all independent failures and take action to reduce the occurrences of failure or mitigate the severity of their consequences [1; 20; 38]. To prevent failure, it is necessary to systematically analyze the FC and take measures to eliminate the root causes. The FTA as a deductive methodology is a widely used technique to describe the causes in a hierarchical way, but it does not possess a feature to easily assess the risk of failure [32; 39]. However, the FMEA does not have a feature to analyze failures hierarchically, but it possesses a feature to focus the analysis on the most critical FC. Some researchers have tried to supplement the shortcoming of FMEA based FTA [23; 32; 36; 37; 39]. [PERSON] et al. [32] performed a case study of failure analysis of an additive manufacturing system for metal printing, the Metal FAB1 of Additive Industries (AI), by combining FTA and FMEA. In their study, failure analysis was performed at the system, function, and component levels. Figure 1 shows a key part of the resulting fault tree at the functional level; to simplify the diagram, some branches are not depicted. The functional failure mode is the \"Powder layer deposit failure\" at the 0 th level (system failure, SF). There are 14 intermediate events (subsystem failures, SSFs), marked with a white box, and 9 component failure modes (FCs), marked with a blue circle. Subsequently, FMEA is performed to risk prioritize the identified component failure modes by means of the RPN (\(=\mathrm{O}\times\mathrm{S}\times\mathrm{D}\)). It seems logical to analyze the failures hierarchically and determine the risk priorities for the component failure modes (i.e., FCs). This is because, to prevent the occurrence of the functional failure mode, the critical component failure mode must be removed first.  
In Figure 1, the occurrence of \"Pulley failure\" or \"Recoater encoder failure\" eventually leads to \"Powder layer deposit failure\" along each failure path. These two failure paths can be described with the random occurrence and detection time of each failure, as shown in Figure 2. For Case A, the FC (i.e., \"Recoater encoder failure\") was detected long after the SF (i.e., \"Power layer deposit\") occurred. In this case, the loss from the failure may be expected to include the cost \(\theta\) of the failure effect, along with the constant cost \(\alpha\) and \(\beta\) for repairing the FC and SF, respectively. On the contrary, in Case B, the FC (i.e., \"Pulley failure\") was detected before the SF occurred; in this case, only the constant repair cost \(\alpha\) is expected.  
Figure 1: Functional level fault tree analysis (FTA) (partial), adopted from [32].  
Figure 2: Loss depending on the detectable time in hierarchical failure occurrence.  
If these failures occur as a result of cause and effect in a hierarchical structure, all failures will occur sequentially from the FC. Furthermore, in general, the FC can take an extremely short time or a fairly long time before it is detected. If any FC is found before the SF, only the foreseeable cost for repair will be expended. However, after the SF, the repair cost, as well as the opportunity cost of handling the failure effects, will be borne.