14) The
minimal
cut sets for
two
campo nents
semes
System
Answers
Answer:
The minimal cut sets for the top event are a group of sets consisting of the smallest combinations of basic events that result in the occurrence of the top event. They represent all the ways in which the basic events cause the top event [52]. The equivalent Boolean algebra function of Fig. 6.1 can be expressed as:
(6.1)
By applying the equivalent Boolean algebra operation, the final Boolean expression of the top event can be obtained as:
(6.2)
It can be seen from Eq. (6.2) that the top event is composed of two second-order minimal cut sets: , ; two third-order minimal cut sets: , ; and four fourth-order minimal cut sets: , , , . All the eight minimal cut
Step-by-step explanation:
Minimal Cut-Set Method
The minimal cut-set method is another alternative for the analytical calculation of the network's reliability. A minimal cut-set () of a graph is a cut whose cut-set has the smallest number of elements. In a water network a cut-set is a set of edges (pipes) that when removed the water flow between node i and j is not possible. The failure probability of a cut-set is equal to the probability that all pipes of the cut-set have failed, thus it is a parallel system and its probability is equal to:
(5.5)
where nc is the number of pipes of the mth cut-set. There are various algorithms for finding the minimum cut-set of a graph [65]. According to Eq. (5.5), cut-sets with many pipes have very small probabilities and therefore can be omitted.
If the minimum cut-set method is followed, the network reliability is equal to the probability of occurrence of at least one cut. Moreover, the terms with many unions (e.g., greater or equal to three) of cut-sets can be neglected, since their probabilities are very small. Therefore, if M is the number of minimal cuts between i and j and if we use F to denote the minimum cuts, the exact calculation of the failure probability can be obtained using Eq. (5.3), taking into account that
(5.6)
Moving back to the network of Fig. 5.3, the network has four possible cuts: ; ; ; and , while the failure probability of every pipe is . The cuts are the same with the paths of the path enumeration method, but they have been selected following a different rationale. The failure probability of each cut is , and . Cuts and can be neglected since they have very small probabilities and thus the failure probability will be . Assuming or , the path enumeration method produces an exact value equal to , while the minimal cut method yields .
METHODS FOR ANALYSIS OF COMPLEX RELIABILITY NETWORKS
M.T. Todinov, in Risk-Based Reliability Analysis and Generic Principles for Risk Reduction, 2007
3.5 DRAWBACKS OF THE METHODS FOR SYSTEM RELIABILITY ANALYSIS BASED ON MINIMUM PATH SETS AND MINIMUM CUT SETS
The main drawback of methods based on minimal paths and minimal cut sets is that the number of minimal paths or cut sets increases quickly with increasing the size of the system. For large systems, the increase of the number of paths and cut sets leads to a combinatorial explosion, which can be demonstrated by using the simple network in Fig. 3.6.
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Figure 3.6. A simple network with MNminimal cut sets.
For the parallel–series network in Fig. 3.6, a minimal cut set is present if a single component fails in each of the N branches composed of M components arranged in series. Since each branch can fail in M different ways, the number of ways in which all parallel branches can fail is MN. Even for a system containing only two components in a branch (M = 2), for N = 50 branches in parallel we already have 250 different cut sets, which is a very large number. Not only the manipulation of such a large number of cut sets is impossible, but even their storage is a problem.
Preliminary System Safety Assessment
Peng Wang, in Civil Aircraft Electrical Power System Safety Assessment, 2017
5.7.3.2.3 The Accurate Calculation When Hidden Failures Exist in the Minimal Cut Set
A hidden failure may occur in any flight during its maintenance interval. However, the failure condition would not occur until the failures of the rest events in the minimal cut set occurred.
The accurate calculation process is as follows: Firstly, think out each flight case that the hidde