Minimal Cut Sets

The result of the ETA is a Hst of combiaations of equipment and human failures that ate sufficient to result ia the accident (71). These combiaations of failures are known as minimal cut sets. Each minimal cut set is the smallest set of equipment and human failures that are sufficient to cause the accident if all the failures ia that minimal set exist simultaneously. Thus a minimal cut set is logically equivalent to the undesired accident stated ia terms of equipment failures and human errors. 

Fault Tree Solution. Solving the fault tree means obtaining the minimal cut sets. The minimal cut sets are all the combinations of equipment failures that can result in the fault tree TOP event. Computer programs are requked to determine the minimal cut sets for large fault trees (72). The solution method has four steps ... 

Minimal cut sets are then ranked. Two factors are considered in the ranking procedure. The first factor considers stmcture, ie, a one-event minimal cut set is more important than a two-event minimal cut set. The implication is that one event is more likely to occur than two events, two events are more likely than three events, and so on. The second factor considers ranking within equal-size minimal cut sets. The general ranking rules consider the probabihty of human error, active equipment failure, and passive equipment failure (73). 

J. B. EusseU, E. B. Henry, and H. N. MarshaU, MOCUS A Computer Program to Obtain Minimal Cut Sets from Fault Trees, Report 1156, Aerojet Nuclear Co., Idaho EaUs, Idaho, 1974. 

In connection witli fault trees, cut sets, and minimal cut sets are defined as follows. A cut set is a basic event or intersection of basic events that will cause the top event to occur. A minimal cut set is a cut set tliat is not a subset of any other cut set it may also be defined as the smallest combination of... 

The only cut set tliat is a subset is CG, which is a subset of G. Eliminating CG yields the following minimal cut sets ... 

The probability of the top set T is the probability of the union of the events represented by the minimal cut sets. This probability usually can be approximated by the sum of the probabilities of these events. Suppose that all tlie basic events are independent and tliat tlieir probabilities are... 

A fault tree is, itself, a Boolean equation relating basic events to the top event. The equation can be analyzed quantitatively or qualitatively by hand or by using computer code(s). If it is analyzed quantitatively, the probabilities or frequencies of the intermediate events and the top event are calculated. If it is analyzed qualitatively, a list of the failure combinations that can cause the top event is generated. These combinations are known as cut sets. A minimal cut set (MCS) is the smallest combination of basic events that, if they occur or exist simultaneously, cause the top event. These combinations are termed "minimal" because all of the basic events in a MCS must occur if the top event is to occur. Thus, a list of MCSs represents the known ways the top event can occur, stated in terms of equipment failures, human errors, and associated circumstances. 

Once the fault tree has been fully drawn, a number of computations can be performed. The first computation determines the minimal cut sets (or min cut sets). The minimal cut sets are... 

The minimal cut sets are useful for determining the various ways in which a top event could occur. Some of the mimimal cut sets have a higher probability than others. For instance, a set involving just two events is more likely than a set involving three. Similarly, a set involving human interaction is more likely to fail than one involving hardware alone. Based on these simple rules, the minimal cut sets are ordered with respect to failure probability. The higher probability sets are examined carefully to determine whether additional safety systems are required. 

The minimal cut sets are determined using a procedure developed by Fussell and Vesely.4 The procedure is best described using an example. 

Determine the minimal cut sets for the fault tree of Example 11-5. 

The procedure does not always deliver the minimal cut sets. Sometimes a set might be of the following form ... 

The second and third sets are supersets of the first basic set because events 1 and 2 are in common. The supersets are eliminated to produce the minimal cut sets. 

The other procedure is to use tl e minimal cut sets. This procedure approaches the exact result only if the probabilities of all the events are small. In general, this result provides a number that is larger than the actual probability. This approach assumes that the probability cross-product terms shown in Equation 11-10 are negligible. 

The minimal cut sets represent the various failure modes. For Example 11-6 events 1,3 or 2,3 or 1,4 or 2,4 could cause the top event. To estimate the overall failure probability, the probabilities from the cut sets are added together. For this case... 

Fault trees are also used to determine the minimal cut sets. The minimal cut sets provide enormous insight into the various ways for top events to occur. Some companies adopt a control strategy to have all their minimal cut sets be a product of four or more independent failures. This, of course, increases the reliability of the system significantly. 

Finally, the entire fault tree procedure enables the application of computers. Software is available for graphically constructing fault trees, determining the minimal cut sets, and calculating failure probabilities. Reference libraries containing failure probabilities for various types of process equipment can also be included. 

The problem of Example 11-5 is somewhat unrealistic in that it is highly likely that the operator will notice the high pressure even if the alarm and shutdown systems are not functioning. Draw a fault tree using an inhibit gate to include this situation. Determine the minimal cut sets. If the operator fails to notice the high pressure in 1 out of 4 occasions, what is the new probability of the top event ... 

Compute the MTBF, failure rate, reliability, and probability of failure of the top event of the system shown in Figure 11-19. Also show the minimal cut sets. 

After repair, the component is returned to the working state. Minimal cut set analysis is a mathematical technique for developing and providing probability estimates for the combinations of basic component failures and/or human error probabilities, which are necessary and sufficient to result in the occurrence of the top event. 

S. Klamt, Generalized concept of minimal cut sets in biochemical networks. Biosystems 83(2 3), 233 247 (2006). 

---