Technical Notes on "A new approach to the d-MC problem"

Abstract: System reliability is the probability of the maximum flow in a stochastic-flow network from the source node to the sink node being more than a demand level d. There are several approaches to compute system reliability using upper boundary points, called d-MinCuts (d-MCs). Search for all the d-MCs in a stochastic-flow network is an NP-hard problem. Here, a work proposed by Yeh [Yeh WC. A new approach to the d-MC problem. Reliab Eng and Syst Saf 2002; 77(2): 201-206.] for determining all the d-MCs is investigated. Two results (Lemma 3 and Theorem 5) are shown to be incorrect and their correct versions are established. Also, the complexity result (Theorem 6) is shown to be incorrect and the correct count is provided.