Complementation of Multi-State System Success to Obtain System Failure and Utilization of Both Boolean Functions in Checking Reliability Expressions

This chapter addresses two important useful contributions that extend binary reliability techniques to multi-state reliability techniques, namely: (a) the problem of complementation or inversion of the function of system success to that of system failure (or equivalently, of deriving the logical minimal cutsets in terms of the logical minimal paths), and (b) the associated problem of hand-checking of a symbolic reliability expression, which relies on complete knowledge of system success and system failure. The paper deals specifically with the reliability of a multi-state delivery network with multiple suppliers and deteriorating production capacity. It presents two complementation procedures, one via the application of multi-state De Morgan’s rules, and the other via the multi-state Boole-Shannon expansion. The chapter also illustrates one prominent case in which this complementation is needed, as it outlines a method for checking the reliability of the multi-state system in terms of its logical minimal paths and logical minimal cutsets.