Unequal Event Probabilities
Recall that a fault tree is a negative outcome analysis, which means that improvement in a system reduces the probability of the top event. Suppose now that event A has a probability of 0.1, and event B has a probability of 0.2. Event B is now more likely to happen than event A.
It would follow that these events would have different values of a given importance measure. However, which event will have the higher importance measure value? That is, to lower the probability of occurrence of the top event, which event should receive more effort for improvement? Also, does the system structure determine the values of the importance measures? That is, will it matter whether the events are connected by an OR gate or an AND gate?
OR Gate
In considering the answer to the last question, assume that events A and B are connected by an OR gate. Then, the top event, event X, occurs if either event A or event B occurs. Also assume that the development efforts cost the same for a given improvement of either event, that is, for a given reduction in the probability of occurrence of either event A or event B. The more probable event, event B in this example, is the event that more often leads to the occurrence of event X. In this case, placing the development efforts into reducing the occurrence of event B (the more probable event) can reduce the occurrence of event X more than an equal effort spent reducing the occurrence of event A.
The Birnbaum importance measure for event A is 0.8, and the Birnbaum importance measure for event B is 0.9. This indicates that development effort should be directed toward reducing the occurrence of event B because it yields the maximum reduction in the occurrence of event X, the top event. Similarly, the Criticality and the Fussell-Vesely importance measures also both give event B the greater importance.
To reinforce this, take the probabilities of events A and B to their respective extremes. Let A have a probability of 0.01 (it hardly ever occurs), and let B have a probability of 0.99 (it nearly always occurs). Then, development efforts directed at event A are wasted (because it hardly ever causes the occurrence of the top event). All of the development effort should obviously be directed towards improving (or reducing the probability of) event B.
AND Gate
Now, assume that events A and B are now connected by an AND gate instead of an OR gate. The top event, event X, occurs only if both events A and B occur. Assume once again that the development efforts cost the same for a given improvement of either event. The least probable event, event A in this example, is the event that more often leads to the non-occurrence of event X. That is, X can only occur if A occurs. (Of course, B must also occur in order for X to occur.) Here, a greater reduction in the occurrence of event X is gained by placing development efforts into further reducing the occurrence of event A (the least probable event) than by placing development efforts into reducing the occurrence of event B.
If the Birnbaum importance measure for event A is calculated, it is now 0.2 while this value for event B is 0.1. This indicates that to maximise the reduction in the occurrence of event X, the top event, development effort should be directed at reducing the occurrence of event A. In the case of the AND gate, for reasons discussed later in this section, the Criticality and the Fussell-Vesely importance measures both give event A and event B the same importance.
Again, to reinforce this, take the probabilities of events A and B to their respective extremes. Assume that A has the probability of 0.01 (it hardly ever occurs), and B has the probability of 1.0 (it always occurs). Then, development efforts directed at eliminating that last 0.01 probability of event A is the best course of action because it ensures that the top event never occurs. No development effort should be directed towards improving event B.
To this point, it will be seen that with an OR gate, development effort is devoted to reducing the probability of occurrence of the most likely to occur event. This is the event with the highest importance measure and is most likely to cause the occurrence of the top event.
Also, it has been shown that with an AND gate, development effort is directed towards reducing the probability of occurrence of the event that is least likely to occur. This is the event with the highest importance measure and is most likely to single-handedly prevent the occurrence of the top event.
Because the importance measures say a different thing in each case, these conclusions may, at first, seem counter intuitive. However, consider an argument by analogy with the RBD. In a series system (OR gate in fault tree), the importance measures indicate that the least reliable component should be improved to reduce the differences between components in the system. The natural thought about engineering is to focus on improving the worst component, until every component works perfectly.
However, in a parallel system (AND gate in fault tree), the importance measures invite one to improve the most reliable component, increasing the differences between components in the system. Assuming equal development costs, this makes sense. In a parallel system (an AND gate in fault tree), the most reliable component (that event which is least likely to occur) is the component that is most likely to be the last component to fail (event to occur) before the system fails (the top event occurs).