importance. Top importance measures can also be calculated that give the sensitivity of the top event probability to an increase or decrease in the probability of any event in the fault tree. Both absolute and relative importance measures can be calculated.
What is often useful about the top event importances is that they generally show that relatively few events contribute to the top event probability. In many past FTAs, less than 20% of the basic events in the fault tree were important contributors, contributing more than 90% of the top event probability. Moreover, the importances of events in the fault tree generally cluster in groups that differ by orders of magnitude from one another. In these cases, the importances are so dramatically different that they are generally not dependent on the preciseness of the data used in the FTA.
In addition to providing the significance of the contributors, the top importances can be used to allocate resources. These resources might include testing and maintenance resources, inspection resources, upgrade resources, quality control requirements and a wide variety of other resource expenditures. By using the top importances, resources can be optimally adjusted to minimize total resource expenditures while maintaining the top event probability, thus providing a win-win situation. Alternatively, for a given resource expenditure such as for upgrades or for maintenance, the top importances can be used to allocate resources to minimize the top event probability. This aids decision makers in obtaining the “biggest bang for the buck” by providing an objective assessment using systematic methodologies, with associated software if needed, to supplement and complement their subjective information.
These optimizations have been used in various industries to reduce resources by as much as 40% while at the same time maintaining or decreasing the top event probability. An advantage of these optimal allocation approaches is that relative risk importances can be used, which have generally smaller uncertainties than absolute values. Moreover, uncertainties in the importances can also be handled.
In addition to allocating resources, the importances can be used to assign allowed downtimes and repair times (actually a type of resource), to focus diagnostic activities in identifying the causes of a top event, and to focus design activities and requirements in design applications.
Four basic types of top importances can be calculated for the different types of applications that are described above [1]. The basic importance measures that can be calculated for each event in the fault tree are:
Fussell-Vesely (F-V) Importance—the contribution of the event to the top event probability.
This importance measure is sometimes call the Top Contribution Importance. Both the absolute and the relative F-V importance are determinable for every event modeled in the fault tree, not only for the basic events, but for every higher-level event and contributor as well. This provides a numerical significance of all the fault tree elements and allows them to be prioritized. The F-V importance is calculated by summing all the causes (minimal cut sets) of the top event involving the particular event.
Risk Reduction Worth (RRW)—the decrease in the probability of the top event if a given
Sensitivity. This measure is related to the previous F-V importance. The risk reduction worth
for a basic event shows the decrease in the probability of the top event that would be obtained if the lower level event, e.g., the failure, did not occur. It thus gives the maximum reduction in the top probability for the upgrade of an item. Both the absolute value and relative value of the risk reduction worth are determinable for every event and contributor modeled in the fault tree. The risk reduction worth is normally calculated by re-quantifying the fault tree or the minimum cut sets with the probability of the given event set to 0.0. This calculation and that for Risk Achievement Worth and Birnbaum’s importance measure (below) are similar to a partial derivative, in that all other event probabilities are held constant.
Risk Achievement Worth (RAW)—the increase in the top event probability if a given event
occurs. This importance measure can also be called the Top Increase Sensitivity. The risk achievement worth shows where prevention activities should be focused to assure failures don’t occur. Since the failures with largest risk achievement worth have the largest system impacts, these are the failures that should be prevented. The risk achievement worth also shows the important events for contingency planning: those events having the largest risk achievement worth have the biggest impacts and should be the priority events considered in contingency plans and reaction plans. Again, both the absolute and relative risk achievement worth are obtainable for every event and contributor modeled in the fault tree. The risk achievement worth is normally calculated by re-quantifying the fault tree or the minimum cut sets with the probability of the given event set 1.0.
Birnbaum’s Importance Measure (BM)—the rate of change in the top event probability as a
result of the change in the probability of a given event. BM is equivalent to a sensitivity analysis and can be calculated by first calculating the top event probability with the probability of the given event set to 1.0 and then subtracting the top event probability with the probability of the given event set to 0.0. Because of the way BM is formulated, it does not account for the probability of an event. BM is related to RAW and RRW. When these are expressed on an interval scale (absolute value), BM = RAW + RRW.
The above importance and sensitivity measures can be calculated, not only for the fault tree, but also for its equivalent success tree. When applied to the success tree, the measures give the importance of an event not occurring. The top event is now the nonoccurrence of the undesired event and each event is the event nonoccurrence. Therefore, when applied to an event in the success tree, the F-V importance gives the contribution of the nonoccurrence of the event to the nonoccurrence of the top event. The risk reduction worth gives the decrease in the nonoccurrence probability of the top event if the event nonoccurrence probability were zero, i.e., if the event did occur. The risk achievement worth gives the increase in the nonoccurrence probability of the top if the nonoccurrence probability of the event were 1.0, i.e., if the event were assured not to occur. Thus, the importance measures for the success tree give equivalent information as for the fault tree, but from a nonoccurrence, or success, standpoint.