A novel multi-distribution multi-state flow network and its reliability optimization problem

The traditional multi-state flow network (MFN) reliability problem is a well-known NP-hard problem, and has been an active area of research for the past four decades. In the MFN reliability problem, the state distribution, i.e. the states and the occurrence probabilities of each arc, is known and fixed. However, the notations used in the MFN problems never considered the state distribution. In addition, each arc has only one state distribution, and this limits the application of the MFN. Thus, the notations relating to the state distribution are added or redefined in this study, and a novel budget-allocation multi-distribution MFN reliability problem is defined and proposed by considering networks with more than one state distribution under different budget allocations. Furthermore, a new algorithm is proposed to solve the proposed novel NP-Hard problem. The correctness and time complexity of the proposed algorithm are analyzed, and one benchmark example is given to demonstrate how to optimize the budget-allocation multi-distribution MFN reliability under different budget constraints.