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Reliability evaluation is a useful tool for network performance demonstration, and therefore has become an increasingly relevant task in network design and operation. The multistate minimal cut (called d -MC for short) method is a popular algorithm for reliability evaluation of multistate flow networks. This article focuses on d -MC problem, and presents some new results to improve its solution. Grounded on the newly obtained results, an algorithm is proposed to find d -MCs, together with discussions on its time complexity that compares favorably with the existing methods. An illustration of the presented algorithm is provided by solving a simple example. Furthermore, numerical experiments and a case study are conducted to demonstrate the advantage and the utility of the proposed algorithm, respectively.

In this paper, each edge in a multi-state flow network is characterized by both multi-valued capacities and a maintenance cost. Accordingly, the performance of a multi-state flow network is measured by capacity reliability as well as maintenance budget limit. An integrated performance indicator MR(d,b) is defined as the probability that a multi-state flow network can provide the required capacity level d while satisfying the maintenance budget limit b. An algorithm is developed to calculate MR(d,b) exactly. Unlike the exhaustive algorithm that enumerates each capacity vector one by one, the developed algorithm iteratively separates capacity vectors satisfying the required capacity level d from the universal space, and then checks whether they meet the maintenance budget limit b. Furthermore, the efficiency of the developed algorithm is validated via numerical experiments.

Reliability evaluation of multi-state systems gives a reasonable demonstration of system performance, and thus is of great importance to their planning, designing and operation. One of the common methods for reliability evaluation is using d -minimal cuts ( d -MCs). This paper proposes a new method to solve the d -MC problem. Specifically, several efforts have been devoted to searching for all d -MCs from two aspects: (i) A new technique is developed to calculate lower capacity bounds of edges which are appropriately used to determine some real d -MCs without any verification, and further to reduce the number of d -MC candidates; (ii) A new approach is put forward to correctly and effectively detect duplicate d -MCs, and the approach brings important insights into the underlying reason why a d -MC derived from one MC can be generated from another MC once again. A simple example and a real case study of the LCD monitor delivery are provided to illustrate the solution procedure, and the utility of the proposed algorithm, respectively. In addition, numerical experiments conducted on four benchmark networks show that the proposed algorithm outperforms a newly developed method in the literature.

This paper presents a d -minimal cut based algorithm to evaluate the performance index R d +1 of a distribution network, defined as the probability that a specified demand d + 1 can be successfully distributed through stochastic arc capacities from the source to the destination. To improve the efficiency of solving d -minimal cuts, a novel technique is developed to determine the minimal capacities of arcs. Also, two new judging criteria are proposed to detect duplicate d -minimal cuts. Both theoretical and computational results indicate that our algorithm outperforms the existing methods. Furthermore, a real case study is provided to illustrate the application of the algorithm.

The cooperative output regulation problem for a class of nonlinear uncertain multi-agent systems is considered. Based on the distributed internal model, the problem is firstly transformed into a global stabilization problem of the augmented system. Then, using the backstepping design method, a distributed control law with its stability analysis is proposed to solve the global stabilization problem of the augmented system. Finally, a numerical simulation is made to show the efficacy of the analytic results.

This paper presents a (d, c)-minimal paths based algorithm to evaluate the reliability index R(d,c), defined as the probability that the source distributes a demand d successfully to the destination with the total distribution cost not exceeding budget constraint c. The proposed algorithm employs two schemes to reduce the search space of (d, c)-minimal paths: (1) by proposing some conditions, an improved method for solving (d, c)-minimal paths is developed; (2) an existing decomposition technique is applied to limit the search space. Computational results show a clear advantage of the proposed algorithm in seeking (d, c)-minimal paths.

A more practical and desirable performance index of multi-state systems is the two-terminal reliability for level (d, c) (2TRd, c), defined as the probability that d units of flow can be transmitted from the source node to the sink node with the total cost less than or equal to c. In this article, a simple algorithm is developed to calculate 2TRd, c in terms of (d, c)-MPs. Two major advantages of the proposed algorithm include: (1) as of now, it is the only algorithm that searches for (d, c)-MPs without requiring all minimal paths (MPs) and the procedure of transforming feasible solutions; (2) it is more practical and efficient in solving (d, c)-MP problem in contrast to the best-known method. An example is provided to illustrate the generation of (d, c)-MPs by using the presented algorithm, and 2TRd, c is thus evaluated. Furthermore, the computational experiments are conducted to verify the performance of the presented algorithm.