Analysis for a trend on the optimal arrangements in a multi-state consecutive-k-out-of-n:F system

In traditional reliability theory, both the system and its components are allowed to take two possible states: either working or failed. In the binary context, a system with n components in sequence is called a consecutive-k-out-of-n:F system if the system fails whenever at least k consecutive components in the system fail. Many studies have appeared over the years dealing with the reliability evaluation of binary consecutive-k-out-of-n:F system. For consecutive-k-out-of-n:F system, one of the most important problem in this system is the optimal arrangement problem wherein the solution is obtained by the arrangement of components with maximum system reliability. On the other hand, in the multi-state consecutive-k-out-of-n:F system, both the components and the system are allowed to be in M+1 possible state. One of the important problem in the multi-state consecutive-k-out-of-n:F system is the optimal arrangement problem wherein the solution is obtained by the arrangement of components with maximum expectation of system states. (Note: k means the number of consecutive components, whose failed components lead to the system failure. Though the number is not a scalar and should be expressed as a vector (k
₁
,k
₂
,...,k
_M
).) In general, the optimal arrangements depend on the values of component state probabilities. It is known, however, that for some case, optimal arrangements do not depend on the values of component state probabilities but on the ranks of component state probabilities. In this study, we consider optimal component arrangement for a multi-state consecutive-k-out-of-n:F system which maximize the expectation of the system state when these components are arranged to the positions in the system. First, we investigated characteristics of optimal arrangements in multi-state consecutive-k-out-of-n:F systems when M=3 and maximum k
_l
(l=1,2,3) is 2, by using analytical approach. We proposed characteristics of optimal arrangement in the systems when n=3, 4 and 5. From these results, we obtained characteristics and trends of conditional optimal arrangements with M=3 and maximum k
_l
s are 2. Next, we investigated characteristics of optimal arrangements in multi-state consecutive-k-out-of-n:F systems by using heuristic approach. From these results, we obtained a trend of optimal arrangements with other parameters.