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The reliability evaluation of multistate networks is primarily using minimal path (cut) vectors, namely d-MPs (d-MCs), which are the lower (upper) boundary vectors that satisfy the system demand d. The generation of all d-MPs (d-MCs) is an NP-hard problem. Thus, it is important to develop an efficient algorithm to search for d-MPs (d-MCs). Existing algorithms searching for d-MPs based on MPs all generate duplicate d-MP candidates, and extra steps are required to detect and remove those duplicates. In this article, we propose an improved algorithm to search for d-MPs for all d levels without generating duplicate d-MP candidates for two-terminal multistate networks. First, we discover and prove the mechanism of generating duplicate d-MP candidates based on MPs. Second, a novel method is proposed to prevent generating duplicate d-MP candidates. Third, an improved algorithm is developed by combining the proposed method to search for d-MPs without generating duplicate d-MP candidates for all d levels. Through computational experiments, it is found that the proposed algorithm is more efficient than existing algorithms for finding all d-MPs for all possible d values. [ABSTRACT FROM AUTHOR]

The multi-state k-out-of-n system model finds wide applications in industry, and has been extensively studied in recent years. This model has also been generalized to the multi-state weighted k-out-of-n system model. Recursive methods, and universal generating functions (UGF) are two primary algorithms for exact performance evaluation of multi-state k-out-of-n systems. However the computational burden becomes the crucial factor when there is a "dimension damnation" problem caused by the increase in the number of components in the system, and the number of possible states a component may be in. In situations wherein exact values of system reliability are not necessary, we may use more efficient algorithms to approximate system reliability. In this paper, we develop a comprehensive framework for reliability approximation of multi-state weighted k-out-of-n systems. Two fuzzy based multi-state weighted k-out-of-n system models are defined. Procedures for building these two models from the conventional models are also introduced. The fuzzy recursive methods, and fuzzy UGF techniques are developed to evaluate such systems. The clustering technique, and curve fitting method are used to determine the fuzzy weights, and probabilities of states in the models. [ABSTRACT FROM AUTHOR]

Algorithms have been available for exact performance evaluation of multi-state k-out-of-n systems. However, especially for complex systems with a large number of components, and a large number of possible states, obtaining "reliability bounds" would be an interesting, significant issue. Reliability bounds will give us a range of the system reliability in a much shorter computation time, which allow us to make decisions more efficiently. The systems under consideration are multi-state k-out-of-n systems with i.i.d. components. We will focus on the probability of the system in states below a certain state d, denoted by Qsd. Based on the recursive algorithm proposed by Zuo & Tian [14] for performance evaluation of multi-state k-out-of-n systems with i.i.d. components, a reliability bounding approach is developed in this paper. The upper, and lower bounds of Qsd are calculated by reducing the length of the k vector when using the recursive algorithm. Using the bounding approach, we can obtain a good estimate of the exact Qsd value while significantly reducing the computation time. This approach is attractive, especially to complex systems with a large number of components, and a large number of possible states. A numerical example is used to illustrate the significance of the proposed bounding approach. [ABSTRACT FROM AUTHOR]

The generalized multi-state k-out-of-n:G system model defined by Huang et al. [5] 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 ki 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. [ABSTRACT FROM AUTHOR]