Minimum cut set

A runaway chemical reaction can occur if coolers fail (A) or if there is a bad chemical batch (B). Coolers fail only if both cooler 1 (C) and cooler 2 (D) fail. A bad chemical batch occurs if tliere is a wrong mi. (E) or a process upset (F). A wrong mi. occurs only if tliere is an operator error (G) and instrmiient failure (H). Construct a fault tree and obtain tlie minimum cut sets. 

Where p is the complete rate of the plans, % m is the number of minimum cut sets in the standard fault tree n is the number of lost basic events of the plans to evaluate p e is the weight of basic event e,. 

A Fault Tree may have a number of cut sets, each of which being a set of basic events that together cause the top (undesired) event to occur. The minimum cut set is the one with the least amount of events that can cause the top event to occur. The critical path is the highest probabihty cut set (i.e. most probable cause of the top event) [Ericson (2005), Chapter 11]. 

The final result allows the easy calculation of the probability of A1 and shows, moreover, the minimal paths ( minimum cut sets , that is the minimum number of components involved) which may lead to the final event (the top event) Al. They are A, B, F, G and (C D). The calculation of the final probability... 

Cut sets and minimum cut sets should then be determined. A cut set is a group of base events, which, if they all occur, will cause the top event to occur. Using the rules of Boolean algebra, redundant base events can be eliminated and combined so that groups of base events containing the minimum number of events that could cause the top event to occur can be determined. They are called minimum cut sets. An examination of the minimum cut sets aids in identifying the base events that contribute most to the undesired top event and in determining the most effective ways to reduce the likelihood of top event occurrence. 

For complex projects, determining cut sets and/or minimum cut sets is usually helpful in the analysis. 

As a practical matter, the modern-day analyst uses appropriate computer software to propagate quantified data and to determine cut sets and minimum cut sets. Computer assistance is also highly recommended for drawing and/or constructing fault trees. 

For academic purposes, brief explanations of how to calculate probabilities and how to determine minimum cut sets follow. For the purist (or masochist) who chooses to attempt to perform fault tree analysis manually, a good statistics course emphasizing Boolean algebra is recommended. 

Cut sets and minimum cut sets can be determined manually by assigning letters to all gates in the tree and assigning numbers to all base events (circles). In both cases, start at the top of the tree and work down the tree from top to bottom.Then systematically build a matrix by replacing the letter for each gate with the combination of numbers and/or letters that are inputs for that gate. 

Rgpire 15-9 Cut Set Example No. 1. In this example, each base event appears only once in the tree, therefore the cut sets derived are the minimum cut sets. 

CUT SETS ARE CONTAINED IN MATRIX (5). THESE CAN BE REDUCED AS SHOWN IN MATRIX (6) TO PRODUCE A MINIMUM CUT SET OF EVENTS 1 2. 

Figure 15-10 Cut Set Example No. 2. In this example, base events 1 and 2 both appear more than once, therefore it is possible to reduce the total cut sets by first eliminating redundant numbers in each row (matrix 5) and then eliminating rows which contain all of the events found in a shorter row (matrix 6) to produce the minimum cut set of events 1 and 2. This example uses the same fault tree configuration as one used by the Certified Safety Professional (CSP) examination in the Examination Information booklet, fifth edition, 1989 by the Board of Certified Safety Professionals (pp. 27-29).

Explain the general procedure for reducing a list of cut sets to a list of minimum cut sets. 

A cut set is a listing taken directly from the fault tree of the events, all of which must occur to cause the top event to happen. Cut sets are the unique combinations of failures that can cause a top event to happen. A minimum cut set (MCS) is the smallest combination of primary or basic events that can result in the top event or undesired incident. This means that when any basic event is removed from an MCS, then the remaining events collectively are no longer a cut set. Thus ... 

The above rules can be used to obtain the minimum cut sets leading to a top event in a fault tree. The occurrence probability of a top event can then be obtained from the associated minimum cut sets. The following two mini-trees are used to demonstrate how the occurrence probability of a top event can be obtained ... 

---