Logical differential calculus for calculation of Birnbaum importance of non-coherent system. (November 2021)

• Record Type: Journal Article
• Authors: Zaitseva, Elena; Levashenko, Vitaly; Sedlacek, Peter; Kvassay, Miroslav; Rabcan, Jan
• Is Part Of: Reliability engineering & system safety. Volume 215(2021)
• Publication Date: 2021-11
• Subjects: Reliability analysis -- Importance measures -- Birnbaum importance measure -- Non-coherent system -- Structure function -- Logic differential calculus -- Direct partial Boolean derivative -- Boolean derivative; Reliability (Engineering) -- Periodicals; System safety -- Periodicals; Industrial safety -- Periodicals; Fiabilité -- Périodiques; Sécurité des systèmes -- Périodiques; Sécurité du travail -- Périodiques; 620.00452
• DOI: 10.1016/j.ress.2021.107829
• Languages: English
• ISSNs: 0951-8320

Abstract

Abstract: The Birnbaum importance is one of the measures often used in importance analysis. Many algorithms for the calculation of this measure have been proposed for coherent systems. Several interpretations of the Birnbaum importance for non-coherent systems also exist. These interpretations are usually based on the prime implicants of the structure function of the system. However, identification of the prime implicants is a complex problem. In this paper, we develop a method for computation of Birnbaum importance using Boolean expression (algebraic representation) and truth table (matrix representation). Both of these representations are calculated based on Logical Differential Calculus, in particular, Boolean derivatives. The Boolean expression derived based on Boolean derivatives is equal to known definitions of the Birnbaum importance. The matrix approach for the calculation of the Birnbaum importance based on Boolean derivatives is novel. This approach can be used for calculation of the Birnbaum importance for the structure function represented by truth table and does not need transformation of the structure function into Boolean expression. An important advantage of the proposed approach (in algebraic or matrix form) is a possibility to define the Birnbaum importance for the evaluation of the non-coherent system failure (restore) with respect to simultaneous changes of states of several system components. Highlights: The Boolean expression of Birnbaum importance for non-coherent system is proposed. The Logical Differential Calculus is developed for importance analysis of non-coherent system. The Birnbaum importance is defined based on Boolean Derivatives. The Birnbaum importance of several components is defined based on Boolean Derivatives. The matrix approach for Birnbaum importance is proposed