A Fast Algorithm for Quickest Path Reliability Evaluations in Multi-State Flow Networks

Many real-world multi-state systems can be modeled as multistate flow networks (MFN) such that the net flow into and out of a node (excluding the source and target nodes) is equal to zero, e.g., distribution systems and supply chains. The quickest path (QP) reliability problem is to evaluate the probability, i.e.,
$R_{(d, t)\hbox{-}{\rm QP}}$
, that at least
$d$
units of data can be sent from the source node to the sink node through a single special minimal path (MP) within
$t$
units of time in an MFN. Such a special MP is called a
$(d, t)\hbox{-}{\rm QP}$
here. In this study, a novel algorithm based on depth-first-search (DFS) is proposed to search for all
$(d, t)\hbox{-}{\rm QPs}$
without solving two NP-hard problems: finding all minimal paths (MPs) in advance, and removing all infeasible
$(d, t)\hbox{-}{\rm QPs}$
candidates. The correctness of the proposed Depth-First-Search (DFS)-based algorithm is proven, and an example is provided to illustrate the generation of all
$(d, t)\hbox{-}{\rm QPs}$
. Furthermore, the analysis of the algorithm's computational complexity and computer experiments indicate that it is more efficient than known algorithms.