Criticality Importance
The Birnbaum importance measure, IB(A), is useful, but it does not directly consider how likely the event A is to occur. For instance, in the previous example, IBA=(1.0–p{B})does not even involve the probability of the event A. This could lead to assigning high importance values to events that are very unlikely to occur and may be very difficult to improve. Remember, an event with a low probability of occurring in a fault tree is an event that has already been improved, so further improvement may be difficult to obtain. Therefore, in an attempt to focus only on those events that truly are important (which not only lead to the top event but also are more likely to occur and may reasonably be improved), a modified Birnbaum importance measure known as a Criticality importance measure is used.
The Criticality importance measure is defined as:
The Criticality importance measure modifies the Birnbaum importance measure by:
• Adjusting for the relative probability of the basic event to reflect how likely the event is to occur and how feasible it is to improve the event (which makes it easier to focus on the truly important basic events).
• Conditioning on the occurrence of the top event X to restrict the measure to evaluating the effect of the basic event , not the probability of the top event X (which makes it possible to compare basic events between fault trees).
Now, the Criticality importance measure, , for the earlier OR gate example, where and , is to be calculated. The probability of the top event, the event = {A or B} is first calculated:
is the probability of the top event occurring. is the probability of event occurring. P{ } is the probability of event not occurring.
P{A and B} is the probability of both events A and B occurring. P{A or B} is the probability of either event A or event B or both events occurring.
If events A and B are independent, then P{A and B} = . Therefore:
Based on earlier calculations:
IB{A}=0.8
P{A}=0.1 and
P{X}=0.28
Therefore, the Criticality importance measure is given by:
Similar calculations for event B yields:
IB(B)=0.9
And
Now, consider the Criticality importance measure for the AND gate, where:
Here,
Thus:
Similarly,
So that:
Given independence of the basic events, all of the basic events under an AND gate will have the same Criticality importance measure. Thus, the Criticality importance measure is uninformative for AND gates.