Minimal Cut Vectors Research Articles

Reliability estimation and optimisation of multistate flow networks using a conditional Monte Carlo method

The Monte Carlo (MC) method is a practical approach to estimating the reliability of large multistate flow networks (MSFNs) in reality, e.g. transportation systems and computer networks. However, deriving an accurate reliability estimate using the crude MC method is computational expensive. This research proposes a conditional MC method to estimate the reliability of a MSFN using the minimal path vectors to level d (d-MPs) and minimal cut vectors to level d (d-MCs). A recursive method is developed to select d-MPs and d-MCs that incur a narrow gap between upper and lower reliability bounds. Then, state vectors are conditionally sampled in a recursive manner using matrix operations. The conditional MC method is embedded in the genetic algorithm (GA) to optimise system reliability. A ranking and selection procedure is used in GA to allocate simulation efforts to different solutions. Numerical studies validate that the proposed conditional MC method can obtain a more accurate reliability estimate than the crude MC method within the same computation time. The improved GA that includes the conditional MC method also outperforms the original GA in reliability optimisation.

An improved algorithm for reliability bounds of multistate networks

Indirect approaches based on minimal path vectors (d-MPs) and/or minimal cut vectors (d-MCs) are reported to be efficient for the reliability evaluation of multistate networks. Given the need to find more efficient evaluation methods for exact reliability, such techniques may still be cumbersome when the size of the network and the states of component are relatively large. Alternatively, computing reliability bounds can provide approximated reliability with less computational effort. Based on Bai’s exact and indirect reliability evaluation algorithm, an improved algorithm is proposed in this study, which provides sequences of upper and lower reliability bounds of multistate networks. Novel heuristic rules with a pre-specified value to filter less important sets of unspecified states are then developed and incorporated into the algorithm. Computational experiments comparing the proposed methods with an existing direct bounding algorithm show that the new algorithms can provide tight reliability bounds with less computational effort, especially for the proposed algorithm with heuristic L1.

A Novel Multistate Minimal Cut Vectors Problem and Its Algorithm

Network reliability is always a powerful index for measuring the performance of the multistate flow network (MFN) that is commonly used to model real-world applications. The multistate minimal cut (MC) vector is a special vector, and it is one of the most popular tools in evaluating the reliability of MFNs. Due to its importance, a novel multistate MC vectors problem is proposed by extending the traditional multistate MC vectors problem from a specific fixed flow to all different flows for more applications. To solve this novel problem, a new algorithm is proposed. The time complexity and demonstration of the proposed algorithm are analyzed, and examples are provided. An experiment is also conducted to compare the proposed multistate MC vectors algorithm and the multistate minimal path vectors algorithm (another important method of calculating MFN reliability).

Reliability evaluation of multistate networks: An improved algorithm using state-space decomposition and experimental comparison

This article introduces an improved algorithm using State-Space Decomposition for exact reliability evaluation of multistate networks given all minimal path vectors (d-MPs for short). We make two main contributions to the area. First, during each recursive call for the decomposition process, we find that the set of d-MPs can be decomposed recursively, and only those qualified d-MPs from a previous set of unspecified states are needed. Second, an improved heuristic rule is proposed choose an appropriate d-MP to decompose each set of unspecified states. Then, efficiency investigations of the proposed algorithm are conducted using hypothetical networks by changing one of the following network parameters while fixing the others, namely, the number of components, the number of d-MPs, and the number of states for each component. Efficiency investigations on networks with known structures are also conducted. Based on the computational experiments, it is found that (i) the proposed algorithm is more efficient than existing algorithms using the state-space decomposition method; (ii) the proposed algorithm is more efficient than existing algorithms using the Recursive Sum of Disjoint Products method when the number of d-MPs is not too small; and (iii) the indirect approach incorporating the proposed algorithm is more efficient than existing direct approaches. Guidelines for choosing the appropriate algorithm are provided. In addition, an algorithm is developed for network reliability evaluation given all minimal cut vectors (d-MCs for short).

Approximate reliability of multi‐state two‐terminal networks by stochastic analysis

Reliability is an important feature in the design and maintenance of a large-scale system. Usually, for a two-terminal network reliability is a measure for the connectivity between the source and sink nodes. Various approaches have been presented to evaluate system reliability; however, they become cumbersome or prohibitive due to the large computational complexity, especially when multiple states are considered for nodes. In this study, a stochastic approach is presented for estimating corresponding reliability. Randomly permuted sequences with fixed numbers of multiple values are generalized from non-Bernoulli binary sequences to model the multi-state property. State probabilities are represented by randomly permuted sequences to improve both computational efficiency and accuracy. Stochastic models are then constructed for arcs and nodes with different capacities. The proposed stochastic analysis is capable of predicting reliability at high accuracy and without a need for constructing the commonly-used but complex multi-state minimal cut vectors. Non-exponential distributions and correlated signals are readily handled by the stochastic approach for a general network. Results obtained through the stochastic analysis are compared with exact values and those found by Monte Carlo simulation. The accuracy, efficiency and scalability of the stochastic approach are assessed by evaluating several case studies.

Evaluation of Algorithms for Identification of Minimal Cut Vectors and Minimal Path Vectors in Multi-State Systems

Minimal Cut Vectors (MCVs) and Minimal Path Vectors (MPVs) are one of the key concepts of reliability analysis. They allow us to estimate system availability or to analyze influence of individual system components on the entire system. However, the main problem of their use, especially in reliability analysis of complex systems, lies in their identification. Several algorithms have been proposed to solve this task. Some of the most universal ones are based on logical differential calculus. These algorithms use integrated direct partial logic derivatives to find situations that can correspond to the MCVs (MPVs) and a special type of logic conjunction to select only those situations that really agree with the MCVs (MPVs). In this paper, we summarize the ideas behind these algorithms in more formal way and present results of some experiments performed to study their time complexity.

Ordering Heuristics for Reliability Evaluation of Multistate Networks

This paper develops ordering heuristics to improve the efficiency of reliability evaluation for multistate two-terminal networks given all minimal path vectors ( $ d $ -MPs for short). In the existing methods, all $d$ -MPs are treated equally. However, we find that the importance of each $d$ -MP is different, and different orderings affect the efficiency of reliability evaluation. Based on the above observations, we introduce the length definitions for $d$ -MPs in a multistate two-terminal network, and develop four ordering heuristics, called O1, O2, O3, and O4, to improve the efficiency of the Recursive Sum of Disjoint Products (RSDP) method for evaluating network reliability. The results show that the proposed ordering heuristics can significantly improve the reliability evaluation efficiency, and O1 performs the best among the four methods. In addition, an ordering heuristic is developed for the reliability evaluation of multistate two-terminal networks given all minimal cut vectors ( $d$ -MCs).

Performance Evaluation of Generalized Multi-State<tex>$k$</tex>-Out-of-<tex>$n$</tex>Systems

The generalized multi-state k-out-of-n:G system model defined by Huang provides more flexibilities for modeling of multi-state systems. However, the performance evaluation algorithm they proposed for such systems is not efficient, and it is applicable only when the k/sub i/ values follow a monotonic pattern. In this paper, we defined the concept of generalized multi-state k-out-of-n:F systems. There is an equivalent generalized multi-state k-out-of-n:G system with respect to each generalized multi-state k-out-of-n:F system, and vice versa. The form of minimal cut vector for generalized multi-state k-out-of-n:F systems is presented. An efficient recursive algorithm based on minimal cut vectors is developed to evaluate the state distributions of a generalized multi-state k-out-of-n:F system. Thus, a generalized multi-state k-out-of-n:G system can first be transformed to the equivalent generalized multi-state k-out-of-n:F system, and then be evaluated using the proposed recursive algorithm. Numerical examples are given to illustrate the effectiveness and efficiencies of the proposed recursive algorithms.

A Monte-Carlo simulation approach for approximating multi-state two-terminal reliability

This paper describes a Monte-Carlo (MC) simulation methodology for estimating the reliability of a multi-state network. The problem under consideration involves multi-state two-terminal reliability (M2TR) computation. Previous approaches have relied on enumeration or on the computation of multi-state minimal cut vectors (MMCV) and the application of inclusion/exclusion formulae. This paper discusses issues related to the reliability calculation process based on MMCV. For large systems with even a relatively small number of component states, reliability computation can become prohibitive or inaccurate using current methods. The major focus of this paper is to present and compare a new MC simulation approach that obtains accurate approximations to the actual M2TR. The methodology uses MC to generate system state vectors. Once a vector is obtained, it is compared to the set of MMCV to determine whether the capacity of the vector satisfies the required demand. Examples are used to illustrate and validate the methodology. The estimates of the simulation approach are compared to exact and approximation procedures from solution quality and computational effort perspectives. Results obtained from the simulation approach show that for relatively large networks, the maximum absolute relative error between the simulation and the actual M2TR is less than 0.9%, yet when considering approximation formulae, this error can be as large as 18.97%. Finally, the paper discusses that the MC approach consistently yields accurate results while the accuracy of the bounding methodologies can be dependant on components that have considerable impact on the system design.