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A path set P is a set of components, which by functioning ensures that the system is functioning. A path set is said to be minimal if it cannot be reduced without losing its status as a path set. A cut set K is a set of components, which by failing causes the system to fail. A cut set is said to be minimal if it cannot be reduced without losing its status as a cut set. We refer to these minimal sets as minimal path and cut sets or vectors (MPSs, MPVs and MCSs, MCVs).

Consider the minimal path sets of the system, P1, P2,…, Pp, and the minimal cut sets of the system, K1, K2,…, Kk. The reliability of the system is given by the union of all minimal path sets. The unreliability is given by the probability that at least one minimal cut set occurs.

Example 1.4

Consider a bridge structure with five edges, E1,…, E5, as shown in Figure 1.11:

1 Find the minimal path sets and the minimal cut sets of the system.

2 Calculate the reliability of the system if the reliability of each component is p.

Figure 1.11 Bridge system.

Solution

1 The minimal path sets areThe minimal cut sets are

2 The reliability of the system is calculated by the union of the path sets: