## Determination of Minimal Cut Sets by Analytical Methods

The analytical method uses Boolean algebra operations in order to transform a fault tree into minimal cut sets. In contrast to the simulation methods, it does not require information on component failure behaviour [242]. This is only needed for calculating the failure probability of the system. It finds all minimal cut sets of a system. In order to avoid difficulties with computer capacity, a cut-off criterion must be applied [242].

Basically two approaches may be used in the method, i.e. the "Top-down" approach, in which the algorithm starts with the undesired event represented by the Top gate working its way down to the basic events, and the "Bottom-up", where the calculation is initiated at the level of basic events, and ends with the undesired event. In order to understand the method the "Top-down" approach to be discussed in this section, and is described in [86]. In this method the tree is presented by a matrix in which the entry of a "1" indicates a connection and a "0" means that there is none. For example, matrix A0(in Fig. F.2) is the representation of a fault tree (Fig. F.1). Rows of matrix represent the "OR" gates (upper part) and "AND" gates (lower part). The columns are divided into three blocks, i.e.: basic events, OR-gates, and AND-gates.

Figure F.1 The fault tree is presented by a matrix

Gate no. 10 11 12

16 17

Basic events 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 00000000 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 00000000 00000000 00000000

0000000 0000000 1 1 0 0 0 0 0 0000000 0000000 0 0 0 1 1 0 0 0000000 0 0 1 0 0 1 0

### Figure F.2 The fault tree is presented by a matrix

The minimal cut sets of the fault tree can be determined by transformation of the matrix A0 into a form which only contains "0" in the two blocks on the right side, which represent the gates. This is achieved by replacing the gates systematically by their entries. In the case of an "AND" gate, all entries figure in original row, where the replacement is made. If a gate is of the "OR" type, for each entries a new row opened [86]. The algorithm starts with the top gate. In this example, replacing gate "16" in the matrix A0 by entries, which lead to:

Basic events Gates

0 00000001|0000000 0" 000000000|00000001

The first row of the matrix A1 already contains a representation in terms of components (only component "9"). Therefore, it is retained unchanged in further step. Replacement of gate "17" in the second row, which represent an "AND" gate, lead to:

Basic events 0 00000001 000000000

Gates

In the next step gate "12" in the second row of A2is replaced. It is "OR" gate, which implies that for each its entries a new row is opened. This lead to:

Basic events Gates

0 00000001|0000000 0" 000000000|10000100 000000000|01000100

These procedures are repeated until all the gates side contains only "0". The final result is shown in the matrix A..

123456789

Basic events

10 11 12 13 14 15 1617

Gates

0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

0 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |

It can be seen that A4 only has entries different from "0" in the block which corresponds to the components of the tree. Each row represents a minimal cut set. The procedure above gives all the cut sets of the tree. These do not necessarily have to be minimal [86]. Therefore nonminimal cut sets have to be eliminated after the cut sets have been obtained. This applies also if a minimal cut set appears several times, then it is retained only once. Both elimination steps are realized using Boolean operations [86].

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