# Reliability Analysis of Vehicle Braking System Based on Hyperellipsoidal Dynamic Bayesian Network
## 2 Basic Theory of Reliability Analysis
### Fuzzy Theory  
In engineering practice, the presence of numerous uncertainties presents a significant challenge to the safety and reliability of systems. The traditional Bayesian network employs conditional probability tables to express the likelihood of event occurrence in a binary manner. However, this simplified treatment often proves inadequate for fully reflecting the true state of complex systems. To address this limitation, this paper introduces the T-S polymorphic fault tree method, which achieves a more detailed and practical representation of system behavior by employing flexible T-S gate rules to construct conditional probability tables in Bayesian networks. Nevertheless, in practical engineering applications, the precise values of the T-S gate rules (i.e., conditional probability tables) are frequently challenging to ascertain directly from observational data, particularly in the absence of sufficient historical data support [12]. In response to this challenge, this paper puts forth a solution informed by the expertise of professionals in the field. In instances where direct numericalization of the T-S gate rules is not a viable option, a group decision-making approach is employed to gather the knowledge and judgment of multiple experts, with the aim of deriving these rules. In order to transform the subjective evaluations of the experts into quantifiable probability values, this paper introduces the concept of fuzzy set theory. This method permits the utilization of fuzzy affiliation functions to transform experts' fuzzy evaluations into concrete probability estimates, thereby providing a reliable foundation for the construction of T-S gate rules. (1) Determine the fuzzy linguistic evaluation level and the corresponding fuzzy probability. In order to obtain the T-S gate rule for each non-bottom event, a 7-level expert language variable is employed as a means of relating the expert's judgment to the fuzzy number. The probability of occurrence and the degree of affiliation are based on the different evaluation languages [13]. The natural language levels are as follows: \"Very High\", \"High\", \"Fairly High\", \"Medium\", \"Fairly Low\", \"Low\", and \"Very Low\". The corresponding fuzzy number form is shown in Table 1.  
2 The fuzzy probabilities assigned by each expert are then combined. The evaluations given by the experts are used to obtain the corresponding fuzzy numbers in each form, and then the evaluations are combined to obtain a unified fuzzy number. This process often uses the arithmetic average method and weighted average method, taking into account the experts' familiarity with the field, engineering application capabilities, and other aspects that may have many differences, which lead to different degrees of fit of their respective evaluation information to the problem, give each expert different evaluation weights, and come to the evaluation results comprehensively. There are two principal methods for determining the weights: subjective and objective. This paper employs the objective method for synthesizing the evaluation of each expert. The weight of each expert is calculated by determining the degree of similarity between the evaluation fuzzy number of each expert and the evaluation fuzzy number resulting from the averaging process. A higher degree of similarity indicates a higher quality of the expert evaluation, which is reflected in a higher corresponding weight. Conversely, a lower weight is assigned to the expert. The method is weighed by the differences in evaluation among experts, effectively avoiding the potential issue of an expert's own weight being disproportionately large, which could lead to significant discrepancies in the evaluation results and introduce uncertainty into the overall assessment.