Reliability Improvement of Circular k-Out-of-N: G Balanced Systems Through Center of Gravity

This paper considers a circular
$k$
-out-of-
$n$
: G balanced system equipped with homogeneous and stationary units. Such a system consists of
$n$
identical and circularly arranged units where the system is considered as reliable if at least
$k$
units are operational while maintaining a specific balance condition. Building on the previous two research studies which proposed two such balance conditions based on symmetry and proportionality concepts, this paper introduces a new balance definition that incorporates the concept of the center of gravity. According to the proposed balance definition, a circular
$k$
-out-of-
$n$
: G balanced system is considered balanced if its center of gravity formed by the operating units is located at the geometric origin of the system. This new balance condition is not only simple but also advantageous as it covers the previously introduced two balance conditions. We investigate the inclusion relationship between the three balance conditions through mathematical proofs and several examples. To evaluate the system reliability, we apply the minimum tie-set method in which the system is interpreted as a parallel system consists of minimum tie-sets. A descriptive numerical example is introduced to explain the reliability evaluation procedure and extensive numerical studies verified the consistent system reliability improvement resulting from the proposed balance definition.