A safety instrumented system (SIS) is extensively used to prevent or reduce risk. The Probability of Failure to perform its intended functions on Demand (PFD) of a practical SIS may vary within multiple safety integrity levels (SIL) due to uncertainties relevant to input parameters. An SIS will be considered to be unsafe when its PFD is greater than a prescribed value, and unsafety probability (UP) is employed to measure the unsafety degree of the investigated SIS in this work. Redundancy architecture is commonly employed to improve the reliability of an SIS. This paper investigates the effects of multiple uncertain input parameters on the UP of an SIS with the k‐out‐of‐n redundancy arrangement. We derive the detailed formulations of the sensitivity for such effects and we also discuss the physical meaning of the proposed sensitivity. An example is employed to demonstrate the proposed sensitivity and the results show that the estimated sensitivity values will be kept to be unchanged with respect to the redundancy when the redundancy is high, whereas the results will vary with the redundancy when the redundancy is low. Meanwhile, we provide a comparison of the results for the truncated uncertain parameters and the non‐truncated uncertain parameters. The results for truncated uncertain parameters and non‐truncated uncertain parameters will approach the same values as the truncated region of the uncertain parameter reduces to zero. The results show that we can approximate the sensitivities of the truncated parameters by the ones of the non‐truncated parameters with less computation time when the truncated region is less than 10−3.
Various uncertain factors exist in the practical systems. Random variables, uncertain-but-bounded variables and fuzzy variables are commonly employed to measure these uncertain factors. Random variables are usually employed to define uncertain factors with sufficient samples to accurately estimate probability density functions (PDFs). Uncertain-but-bounded variables are usually employed to define uncertain factors with limited samples that cannot accurately estimate PDFs but can precisely decide variation ranges of uncertain factors. Fuzzy variables can commonly be employed to define uncertain factors with epistemic uncertainty relevant to human knowledge and expert experience. This paper focuses on the practical systems subjected to epistemic uncertainty measured by fuzzy variables and uncertainty with limited samples measured by uncertain-but-bounded variables. The uncertainty propagation of the systems with fuzzy variables described by a membership function and uncertain-but-bounded variables defined by a multi-ellipsoid convex set is investigated. The combination of the membership levels method for fuzzy variables and the non-probabilistic reliability index for uncertain-but-bounded variables is employed to solve the uncertainty propagation. Uncertainty propagation is sued to calculate the membership function of the non-probabilistic reliability index, which is defined by a nested optimization problem at each membership level when all fuzzy variables degenerate into intervals. Finally, three methods are employed to seek the membership function of the non-probabilistic reliability index. Various examples are utilized to demonstrate the applicability of the model and the efficiency of the proposed method.
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