Reliability importance and invariant optimal allocation

The reliability importance of a component is a partial derivative of the system reliability with respect to this component reliability. When all components are i.i.d., the reliability importance is called the B-importance. Relationships between reliability allocation and the reliability importance for general coherent systems are explored. The invariant optimal allocation is an allocation related only to the relative ordering rather than the magnitude of the component reliabilities. A strong heuristic method (LK heuristic) is developed to search for an ideal allocation through the application of the reliability importance. The following conclusions are drawn: if there exists an invariant optimal allocation for a system, the optimal allocation is to assign component reliabilities according to the B-importance ordering. Furthermore, the allocation generated by the LK heuristic is the optimal allocation.