This research studies four special types of systems: k-out-of-n:F systems, k-out-of-n:G systems, consecutive-k-out-of-n:F systems, and consecutive-k-out-of-n:G systems. A k-out-of-n:F system fails if and only if at least k of its n components fail. A k-out-of-n:G system is good if and only if at least k of its n components are good. A consecutive-k-out-of-n:F system is a sequence of n ordered components such that the system works if and only if less than k consecutive components fail. A consecutive-k-out-of-n:G system consists of an ordered sequence of n components such that the system works if and only if at least k consecutive components work. The consecutive-k-out-of-n systems are further divided into linear systems and circular systems corresponding to the cases where the components are ordered along a line and a circle, respectively;After the reliability evaluation of the k-out-of-n systems and the reliability evaluation and optimal design of the consecutive-k-out-of-n systems are reviewed. The properties of these systems are further investigated. Next, this research concentrates on the optimal design of the consecutive-k-out-of-n systems. An arrangement of components is optimal if it maximizes the system\u27s reliability. An optimal arrangement is invariant if it depends only upon the ordering of component reliabilities but not their actual values. Theorems are developed to identify invariant optimal designs of some consecutive systems. Other theorems are provided proving that there are no invariant optimal configurations for some consecutive systems. For those systems where invariant optimal designs do not exist, a heuristic method is provided to find at least suboptimal solutions. Two case studies are presented to show the applications of the theoretical results developed in this study.
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