A depth first search algorithm for optimal arrangements in a circular consecutive-k-out-of-n:F system

A circular consecutive-k-out-of-n:F system consists of n components arranged along a circular path. This system fails if k or more consecutive components fail. As the number of components n increases, the amount of calculations to obtain a solution within a reasonable computing time would be too much, except for the special cases of k = 1, 2, n - 2, n - 1, n. Therefore, most of the recent studies have focused on proposing algorithms to obtain quasi optimal arrangements. It would be meaningful for such studies, however, to obtain the exact optimal arrangement of a moderate-sized system and investigate its characteristics. In this paper, we propose an efficient algorithm for solving this problem. We developed this algorithm to reduce the search space by eliminating arrangements with equal system reliability produced by rotation and/or reversal of certain arrangements. In addition, this algorithm also prevents redundant calculations by memorizing the calculated values, when depth first search is conducted, based on Hwang's formula calculating the reliability of a circular consecutive-k-out-of-n:F system. We conducted some numerical experiments in order to evaluate this algorithm. The results of the experiments show that our proposed algorithm can reduce computing time approximately by a factor of 4n as compared to an exhaustive search.