1. Fault Tree Analysis Using Binary Decision Diagrams
Lab Seminar
May 4th, 2006
Seung Ki, Shin

2. Contents
Introduction
Classification of Fault Trees
Shortcoming of Conventional Fault Tree Analysis
Binary Decision Diagrams Approach
Example of Non-Coherent Fault Tree
Summary & Further Study
References

3. Introduction
The fault tree diagram itself is an excellent way of deriving the failure logic for a system.
Conventional fault tree analysis techniques have several disadvantages when analyzing some kinds of fault trees.
It is possible to overcome these disadvantages using Binary Decision Diagram (BDD) method.

4. Classification of Fault Trees
Static
Coherent
Non-coherent
Dynamic
Static fault tree : Composed of Boolean gates
Dynamic fault tree : Adding sequential notion to static fault tree

5. Classification of Fault Trees
Coherent fault tree
Logical gates are restricted to AND and OR gates.
Top event is described in terms of Minimal Cut Sets.
Minimal Cut Set
: Combination of component failure events which are necessary and sufficient to cause the top event.
ex)
Non-coherent fault tree
Inverse gates besides AND and OR gates (NOT, NAND, NOR, and XOR gates)
Top event is described with Prime Implicants from Boolean algebra.
Prime Implicant
: Combination of basic events (success or failure) which is both necessary and sufficient to cause the top event.

6. Shortcoming of Conventional Fault Tree Analysis
Inclusion-Exclusion Expansion (IEE)
( are minimal cut sets/prime implicants )
For complex systems an analysis may produce hundreds of thousands of minimal cut sets. Then it is impossible to calculate the exact probability using IEE.
Truncation of the expansion is used to simplify the calculation.
It is justified for coherent fault trees.
For non-coherent fault trees, this approximation is not valid and creates considerable inaccuracies in evaluating top event probability.

7. Shortcoming of Conventional Fault Tree Analysis
The prime implicants are frequently reduced to their coherent approximations by assuming any working states for the components in the expression are set to TRUE.
This approximation may induce considerable inaccuracies.

8. Binary Decision Diagrams Approach
The binary decision diagram (BDD) method was utilized by Bryant and later developed by Rauzy.
BDD provides an alternative logic form to the fault tree structure to express the system failure causes.
Exact system failure probability can be deduced without the need to resort to any approximations.
The BDD structure has the additional advantage that its quantification does not require the minimal cut sets/prime implicants.

9. Binary Decision Diagrams Approach1
Terminal Node
<Fault Tree>
<Binary Decision Diagram>

10. Binary Decision Diagrams Approach
Conventional Method
Minimal Cut Sets :
Probability of Top Event (Inclusion-Exclusion Expansion)
Binary Decision Diagrams Approach
Disjoint Path :
Probability of Top Event
* Due to the binary branching each path in the BDD is mutually exclusive and so the probability of system failure is obtained by simply summing the probability of each disjoint path leading to a terminal one node.

11. Example of Non-Coherent Fault Tree
- Each component failure probability : 0.1

12. Example of Non-Coherent Fault Tree
Exact Calculation Using IEE Method
Min cut set :
Probability of top event :
Two Conventional Approximations
Truncation (after one term)
Coherent approximation
Min cut set becomes
Very inaccurate !

13. Example of Non-Coherent Fault Tree
Binary Decision Diagram Approach 1
Disjoint Path
Probability
0.081
Total
Simple & Exact !

14. Summary & Further Study
When analyzing non-coherent systems, it is shown that analysis methods based on traditional fault tree analysis are both inaccurate and inefficient.
It has been shown that analysis procedures based on binary decision diagrams to represent the system failure logic can produce all minimal cut sets for problems which defeat conventional approaches.
The size of the resulting BDD is determined by the ordering that has to be given to the basic events in the fault tree before the BDD is constructed.
To improve the efficiency of the BDD analysis, it is important to seek a BDD of minimal size when a certain fault tree is given.

15. References
J.D. Andrews, S.J. Dunnett, “Event Tree Analysis Using Binary Decision Diagrams”, IEEE, 2000.
A. Rauzy, “New algorithms for fault trees analysis”, Reliability Engineering and System Safety, 1993.
R.M. Sinnamon, J.D. Andrews, “Improved Efficiency in Qualitative Fault Tree Analysis”, Quality and Reliability Engineering International, 1997.