# Time-Dependent Probabilistic Approach of Failure Mode and Effect Analysis
## 1 Introduction  
Failure mode and effect analysis (FMEA) is an analysis technique for defining, identifying, and eliminating known and/or potential failures, problems, errors, etc. from systems, designs, processes, and/or services before they reach the customer [1]. The results of the FMEA can help analysts to identify and correct the failure modes that have a detrimental effect on the system and improve its performance during the stages of design and production [2]. Since its introduction as an analysis tool for reducing failures, FMEA has been extensively used in a wide range of industries, including automotive, semiconductor, aircraft medical, and steel industries [3; 4; 5; 6]. In traditional FMEA approaches, the risk associated with a failure-cause 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 a failure-cause (i.e., \(RPN=O\times S\times D\), where \(O\) is related to failure occurrence, \(D\) to failure detection, and \(S\) to failure severity). _RPN_ evaluation of a potential failure-cause requires evaluation of three risk factors via the use of the 1-to-10 scale. The higher the _RPN_ of a failure-cause, the greater is the associated risk with regard to system/product reliability. However, this traditional FMEA risk-evaluation approach has often been extensively criticized in extant literature for a variety of reasons [2]. Noteworthy drawbacks of the traditional _RPN_-based approach include (i) no consideration of relative importance among \(O\), \(S\), and \(D\) parameters [7; 8; 9; 10]; (ii) difficulties involved in precise evaluation of the three risk factors [5; 11; 12; 13; 14; 15; 16; 17]; (iii) no consideration of interdependencies among different failure-causes and the corresponding effects [6; 18; 19; 20]; and (iv) over-dependence on expert intuition and experience instead of scientific methods for evaluation of the three risk components [21; 22]. Many researchers have attempted to overcome the above-mentioned drawbacks, thereby improving the FMEA risk-evaluation method in the process. [PERSON] et al. [2] reported that the most popular FMEA approach corresponds to the fuzzy rule-based system [10; 23; 24; 25], followed by the grey theory [9; 10], cost-based model [11; 12; 13], AHP/ANP [7; 19; 26; 27], and linear programming. Others included an integration-based approach [8; 15; 16; 21; 22; 28] and probability-based methodology [14; 29]. Apart from these approaches, the use of a time-dependent probabilistic approach for FMEA risk-evaluation has been proposed in this study. In the proposed model, two additional factors--time (failure occurrence and detection) and loss (repair and opportunity cost)--have been considered owing to them being directly associated with the criticality of the risks. Typically, failures occur on account of the occurrence of even one of their causes. Additionally, occurrence of a failure may not be detected in real time, and there may exist some intermediate time delay. The loss incurred owing to the effect of a failure depends on this time delay. If a potential failure-cause can be detected prior to occurrence of actual failure, only a certain repair cost would be incurred. However, in the event of an actual failure, substantial monetary loss, including the expense due to failure and the corresponding repair cost, may be incurred. Figure 1 depicts typical scenarios of failure occurrence along with crude estimates of the expected loss depending on the time elapsed between failure occurrences and detection of their possible causes. The scenarios are based on the assumption that (i) each cause can independently lead to failure; (ii) the time elapsed between occurrence of a probable cause and that of the consequent failure is different in each case; (iii) the time required for detection of the failure-cause is different in each case; and (iv) failure effects are realized after failure occurrence.