CUT SET TRANSFORMATION TOP TOP OR AND AND AND AND 1 Component Cut Sets 2 Component Cut Sets 3 Component Cut Sets 4 Component Cut Sets 5 Component Cut Sets

DEFINITIONS: Cut set Minimal Cut Set Path Set Minimal Path Set A Cut set is a collection of basic events; if all theses basic events occur, the top event is guaranteed to occur. Minimal Cut Set A minimal cut set is such that if any basic event is removed from the set, the remaining events collectively are no longer a cut set. Path Set A path set is a collection of basic events; if none of the events in the sets occur, the top event is guaranteed not to occur. Minimal Path Set A minimal path set is a path set such that if any basic event is removed from the set, the remaining events collectively are no longer a path set.

1 2 4 3 5 6 [ Example ] Minimum Cut Sets OR 1 AND OR OR 2 OR 4 OR 3 5 6 Minimum Cut Sets {1},{2,4},{2,5},{2,6},{3,4},{3,5},{3,6} i.e. {1,2,3} and {1,4,5,6} are the minimum Path Sets

PROCEDURE FOR FINDING CUT SETS 1) Uniquely identify all gates and basic events. 2) Resolve all gates into basic events. 3) Remove duplicate events within sets. 4) Delete all supersets (sets that contain another set as a subset).

STEP 1: TOP Event Fault Event 1 Fault Event 2 BASIC Event 1 Fault D BASIC Event 1 Fault Event 1 BASIC Event 2 BASIC Event 4 C 1 2 4 BASIC Event 2 BASIC Event 3 2 3 SAMPLE FAULT TREE

STEP 2: The second step is to resolve all the gates into BASIC events. This is done in a matrix format, beginning with the TOP event and proceeding through the matrix until all gates are resolved. Gates are resolved by replacing them in the matrix with their inputs. The TOP event is always the first entry in the matrix and is entered in the first column of the first row . There are two rules for entering the remaining information in the matrix: the OR-gate rule, and the AND-gate rule.

The OR-gate rule: When resolving an OR gate in the matrix, the first input of the OR gate replaces the gate identifier in the matrix, and all other inputs to the OR gate are inserted in the next available row, one input per row. The next available row means the next empty row where the OR gate appeared, these entries must be entered (repeated) in all the rows that contain the gate’s inputs. The AND-gate rule: When resolving an AND gate in the matrix, the first input to the AND gate replaces the gate identifier in the matrix, and the other inputs to the AND gate are inserted in the next available column, one input per column, on the same row that the AND gate appeared on.

MATRIX FOR RESOLVING GATES IN SAMPLE FAULT TREE STEP 2: (b) (a) MATRIX FOR RESOLVING GATES IN SAMPLE FAULT TREE

MATRIX FOR RESOLVING GATES IN SAMPLE FAULT TREE STEP 2:(續1) (d) (c) MATRIX FOR RESOLVING GATES IN SAMPLE FAULT TREE

MATRIX FOR RESOLVING GATES IN SAMPLE FAULT TREE STEP 2:(續2) (f) (e) MATRIX FOR RESOLVING GATES IN SAMPLE FAULT TREE

STEP 3: set 1: 1, 2, 2 1, 2 set 2: 1, 2, 4 1, 2, 4 set 3: 1, 2, 3 1, 2, 3 set 4: 1, 3, 4 1, 3, 4 STEP 4: minimum cut sets: {1, 2}, {1, 3, 4}

Path Sets: A B B 1 1 D D D 2 4 C C 1 1 2 4 2 4 C 2 3 2 3 {1} {2,4} {2,3}