A simple algorithm to search for all d-MPs with unreliable nodes

Many real-world systems are multistate systems composed of multistate components in which the reliability can be computed in terms of the lower bound points of level d, by formulating in terms of either the d-minimal paths (d-MPs) or d-minimal cutsets (d-MCs). Such systems (electric power, transportation, etc.) may be regarded as flow networks whose arcs have independent, discrete, limited and multivalued random capacities. A simple method is proposed to search for all d-MPs for network reliability in a system subject to both arc and node failures. The proposed method does not require re-enumeration for all of the d-MPs or MPs for the additional the node failure consideration. Because only the sum of the flow through into (or from) each unreliable node of the d-MPs which found in the network with perfect nodes is calculated, the proposed algorithm is easier to understand and implement. With considering unreliable nodes, this method is also more realistic and valuable for performing the reliability analysis in an existing network. The computational complexity of the proposed algorithm is analyzed and compared with the existing methods. One example is illustrated to show how all d-MPs are generated in a network with arc and node failures solved by the proposed algorithm.