A simple algorithm for evaluating the k-out-of-n network reliability Wei-Chang Yeh * e-Integration & Collaboration, Department of Industrial Engineering, Feng Chia University, P.O. Box 67-100, Taichung 408, Taiwan, ROC Received 13 August 2003; accepted 9 September 2003 Abstract Evaluating the network reliability is an important topic in the planning, designing, and control of systems. The minimal cut set (MC, an edge set) is one of the major and fundamental tools for evaluating network reliability. A k-out-of-n MC is a special MC in a k-out-of-n network in which some nodes must receive at least k flows from their n input edges, where k is an integer number between 1 and n: In this study, an alternative method is given first to define a MC using a node set (called MCN) in k-out-of-n networks. A very simple algorithm based on some intuitive theorems that characterize the structure of the MCN and the relationship between MC and MCN is developed to solve the k-out-of-n network reliability by finding the k-out-of-n MCs between two special nodes. The proposed algorithm is not only easier to understand and implement, but is also better than the existing algorithm. The correctness of the proposed algorithm will be analyzed and proven. Two examples are illustrated to show how all k-out-of-n MCs are generated and verified in a k-out-of-n network using the proposed algorithm. The reliability of one example is then computing using one example. q 2003 Elsevier Ltd. All rights reserved. Keywords: Network reliability; k-out-of-n; Minimal cuts; Algorithm 1. Introduction In recent years, network reliability theory has been applied extensively in many real-world systems such as computer and communication systems, power transmission and distribution systems, transportation systems, oil/gas production systems [1–3], etc. System reliability plays important roles in our modern society [4–10]. The evaluation of the network reliability is a NP-hard problem [33]. Reliability evaluation approaches exploit a variety of tools for system modeling and reliability index calculation. Among the most popular tools are network- based algorithms founded in terms of either MCs or MPs [11–35]. Unfortunately, to search for all MCs or MPs is also a NP-hard problem [33]. Between MCs and minimal paths (MPs), MCs readily provide the list of events that cause network failures [34,35]. Hence, MC applications are more effective. The traditional network without k-out-of-n nodes always assumes that a node can generate output flows to its of all output edges if and only if any flow from any one of its input edges exits. In real applications, it is necessary to extend traditional networks without k-out-of-n nodes to k-out-of-n networks. If k ¼ 1 for all nodes, then the k-out-of-n network becomes a traditional network. For example, the voter output the result in an n-version programming system only when there are at least k modules that have the same results, where k is a defined value [36]. Similar applications are observed in other redundant systems [36]. To the author’s best knowledge, Tan first investigated the search for enumerating the k-out-of-n network reliability [36]. He used an iterative method and Boolean algebra to generate the set of all k-out-of-n MCs (i.e. with k-out-of-n nodes), starting with the source node and ending with the sink node. However, Tan’s algorithm failed to solve the k- out-of-n MCs between a pair of nodes in an efficient way. Tan’s algorithm is applicable only to acyclic directed k-out- of-n networks [36]. In real-life cases, many networks such as computer and telecommunications include cycles for redundancy. The need for a new efficient algorithm to enumerate the k-out-of-n network reliability thus arises. The purpose of this paper is to develop a more efficient and intuitive algorithm than the existing algorithm proposed by Tan [36] to enumerate the k-out-of-n general network reliability in terms of all k-out-of-n MCs. The proposed algorithm is based on some simple concepts that were found 0951-8320/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2003.09.018 Reliability Engineering and System Safety 83 (2004) 93–101 www.elsevier.com/locate/ress * Corresponding author. Tel.: þ886-4-2321-6168; fax: þ 886-4-2208- 4168. E-mail address: yeh@ieee.org (W.-C. Yeh).