Fast Bdd Truncation Method for Efficient Top Event Probability Calculation

A Binary Decision Diagram (BDD) [1-3] provides an efficient representation and manipulation of Boolean formulae, and it was shown that a BDD is effective in the diverse fields of computer science and reliability [4]. Bryant [3] popularized the use of a BDD by developing a set of algorithms for an efficient construction and manipulation of BDDs. The BDD algorithm was applied to a reliability analysis [5,6] and has been investigated to solve large fault trees and importance measures [7-10]. The BDD algorithm has become a very popular method to calculate an exact top event probability (TEP) of a small or intermediate size reliability problem [5-10]. Figure 1 displays the relations among fault tree solving methods, such as a BDD algorithm, a ZBDD (Zero-suppressed BDD) algorithm, and an algorithm based on traditional Boolean algebra. The BDD algorithm generates a BDD structure by solving a fault tree. On the other hand, an algorithm based on traditional Boolean algebra generates minimal cut sets (MCSs). An MCS is a minimal combination of basic events that causes a top event. The MCSs could be expressed by a ZBDD structure, which is a factorized form of MCSs [11]. As shown in Fig. 1, a BDD structure could be reduced to a ZBDD structure by truncating and subsuming the BDD structure. However, there had been no method to generate a ZBDD structure directly from a fault tree before the development of the ZBDD algorithm [12]. The BDD algorithm calculates an exact TEP since it does not employ any approximations such as a truncation, a rare-event approximation, and a delete-term approximation. All details regarding a BDD and a ZBBD are explained in Sections 1.2 and 1.3, and the delete-term approximation is illustrated in Appendix A. Since the BDD algorithm is highly time and memory consuming, especially for large problems, it has been difficult to solve large reliability problems such as fault trees for accident sequences in a Probabilistic Safety Assessment (PSA). In order to solve a large complex problem with limited computational resources, numerous attempts have been made to minimize BDD size, which is measured by the number of nodes in the BDD structure (see Sections 1.2 to 1.4). The BDD size is heavily dependent on the choice of the variable ordering for a BDD construction (see Section 1.3 and Appendix B). As shown in Fig.1, a BDD truncation cannot be applied to the BDD algorithm (see Section 2.2).