A consecutive-k-out of-n:F system consists of an ordered sequence of n components such that the system fails if and only if at least k consecutive components fail. A consecutive-k-out-n:G system consists of a sequence of n ordered components such that the system works if and only if at least k consecutive components work. For a consecutive-k-out-of-n system, the optimal design problem deals with permutations of components with different reliabilities. A design (permutation) is optimal if it maximizes the system reliability. An optimal design is invariant if it depends solely on the ranking of component reliabilities. This paper summarizes results for the invariant optimal designs of consecutive-k-out-of-n systems, identifies invariant optimal design of some consecutive-k-out-of-n systems, proves that there do not exist invariant optimal designs for other consecutive k-out-of-n systems, and thus completes the theories on invariant optimal designs of these systems. For those systems without invariant optimal designs, a heuristic method and a randomization method are presented to provide at least suboptimal designs.
