A reliability index to measure multi-state flow network considering capacity restoration level and maintenance cost

In this study, every edge of a multi -state flow network (MFN), in addition to multiple capacities, are endowed with a maintenance cost. The existing studies measure the maintenance cost integrated reliability index of an MFN with an assumption that all edges are maintained to restore the network from a degraded state to the largest state, and in contrast, this study takes one step further to consider a more general case that the network is restored from a degraded state to an expected state (including but not limited to the largest state). This study proposes a method to simultaneously evaluate the index MR(d,D,T), defined as the probability that a network can be restored from a degraded state of capacity level below d to an expected state of capacity level not less than D and the maintenance cost is within a limit T, and the index URd defined as the probability that network is in a degraded state. The maintenance cost integrated reliability R(d,D,T) is then calculated, which is the ratio of MR(d,D,T) to URd denoting the ability of an MFN to restore from a failure state to an expected state under the limited maintenance cost. Numerical examples are presented to demonstrate the reliability index R(d,D,T).