Analysis of minimal cut and path sets based on direct partial Boolean derivatives

Abstract

Principal steps in reliability engineering include estimation of system reliability and identification and quantification of situations which cause a system failure. There are several techniques that can be used to solve these tasks. Some of them are based on minimal cut (path) sets, which represent minimal sets of basic events, whose simultaneous occurrence leads to a failure (repair) of the system. In this article, applications of minimal cut (path) sets in reliability analysis are summarized, and their connection with direct partial Boolean derivatives is studied. In reliability analysis, direct partial Boolean derivatives identify situations in which a failure (repair) of one system component results in a system failure (repair). Therefore, they reveal the influence of one system component on the whole system. However, minimal cut (path) sets define the influence of a simultaneous failure (repair) of a group of system components on the system activity. Therefore, there should be some correlation between direct partial Boolean derivatives and minimal cut (path) sets. This correlation is studied in this article, and as a result, new algorithms for identification of minimal cut (path) sets are proposed based on this correlation.

New Matrix Based Algorithm for Calculation of Importance Measures

“…Authors of studies [2,5,6] shown that the reliability analysis of system can be implemented by application of Logical Differential Calculus. The algorithms for calculation of frequency indices have been studied in [6].…”

“…The algorithms for calculation of frequency indices have been studied in [6]. The definition and computation of IMs based on Logical Differential Calculus, in particular Direct Partial Boolean Derivatives (DPBD), have been proposed and investigated in [2,5]. These derivatives allow investigating of the function value change depending on the change of the value of the function variable.…”

“…The computational complexity of the calculation of IMs based on the system structure function depends on the system dimension (number of system components). Authors of papers [5,7] propose to use the Binary Decision Diagram (BDD) for the structure function representation to decrease the computational complexity of algorithms for reliability analysis. The application of BDD in importance analysis of system and IMs calculation have been considered in [5,8].…”

“…Authors of papers [5,7] propose to use the Binary Decision Diagram (BDD) for the structure function representation to decrease the computational complexity of algorithms for reliability analysis. The application of BDD in importance analysis of system and IMs calculation have been considered in [5,8]. Other approach of this computational complexity decreasing is the use of parallel procedure [9,10].…”

A Hierarchical Approach for Fast Calculating Minimal Cut Sets of a Microgrid

“…Minimal path sets are obtained by incidence-matrix method and then inverted to minimal cut sets in [9,10]. Literature [11] summarizes the applications of minimal cut sets and proposes a new algorithm for minimal cut sets based on correlation with direct partial Boolean. Literature [12] proposes a new algorithm based on Boolean satisfiability problem to solve the minimal cut sets in fault tree analysis.…”