An Efficient Method to Find All d-MPs in Multistate Two-Terminal Networks

Abstract

Many systems in the real-world are multi-state systems composed of multi-state components (nodes and arcs) in which the reliability of the system can be calculated in terms of the lower bound points of level d, called d-Minimal Paths (d-MPs). Such systems (oil/gas production, power transmission and distribution networks, etc.) may be considered as network flows whose arcs have independent and discrete random capacities. In this paper, we present a new algorithm to generate all d-MPs candidates for all d levels. Our algorithm is based on the incremental enumeration of the different d-MPs. In the beginning, we determine the list of all minimal paths (also called 1-MPs). The latter will then be used to generate the 2-MPs. Then from 1-MPS and 2-MPs we generate 3-MPs . . . and so on. The greatest contribution of our method is the use of a technique which makes it possible to know, at each level i, which minimal paths can lead to valid (i+1)-MPs without resorting to an additional validation process. This has reduced the execution time compared to the existing algorithms to find all d-MPs for all possible d values.

Evaluating multi-state systems reliability with a new improved method

“…In [21], Lin offered a mathematical formulation based on three restrictions for obtaining the candidates for d-MPs from all previously enumerated MPs. Lamalem [25] presented a novel method for listing all d-MPs for all d levels. This method utilizes the network's routes matrix to generate new d-MP candidates from (d-1)-MP d-MP candidates.…”

“…The enumeration of all d-minimal pathways (d-MPs) for a certain level d is one of the available approaches [10], [19]- [25] for computing the system dependability of a multi-state system. There are two ways provided in the literature for locating all d-MPs for a certain d. The first one generates d-MPs using direct methods, and the second approach generates d-MPs using pre-enumerated minimal paths (MPs).…”

An Improved Vectorization Algorithm to Solve the d-MP Problem

“…One can model many real-world systems such as transportation networks, computer and communication networks, distribution systems, and power transmission systems as SFNs [4,13,22,38]. Therefore, the reliability evaluation of SFNs has been a very attractive problem in the last decades, and several direct and indirect algorithms have been proposed in the literature to address it [1,4,5,7,8,9,10,11,12,13,14,16,17,18,22,23,25,26,27,28,29,30,31,34,35,36,37,39].…”

“…The network reliability, R d , can be assessed indirectly in terms of minimal paths (MPs) [4,9,11,12,17,23,25,30,31,34,36,39] or minimal cuts (MCs) [1,5,7,10,13,14,16,22,27,28,29,35] through a three-stage process; (1) calculating all the MPs or MCs, (2) determining all the d-MPs in the case of MPs or d-minimal cuts (d-MC) in the case of MCs, and (3) calculating a union probability as the network reliability.…”

A simple vectorization algorithm to address the d-MP problem without generating duplicate candidates

“…This is why the problem of reliability evaluation of MFNs has turned to be of great importance [2]. The reliability of two-terminal MFN can be evaluated indirectly in terms of minimal paths (MPs) [2,4,6,7,13,14,16,18] or minimal cuts (MCs) [3,5,[8][9][10][11]15] through a three-stage process; (1) determining all the MPs or MCs, (2) finding all the d-MP s or d-MC s, which are called the d-MP or d-MC problem, and (3) calculating a union probability as the network reliability. The focus of this work is on the second stage and the d-MP problem, and hence we assume that all the MPs are available in advance.…”